Almagest

Undergraduate Journal

in the History and Philosophy of Science

2015

Copyright © by the Authors

All rights reserved

First printed September 2015

Cover Image: William Cuningham,"Coelifer Atlas" from The Cosmographical

Glasse. London: John Day, 1559.

Editorial Board

Co-Editors in Chief

Robert Fraser

Rory Harder

Editors

Zachery Brown

Joel Burkholder

Gavin Lee

Mirka Loiselle

Emma Pask

Zoe Sebastien

Contents

Foreword

Galenic Medicine

Felix Walpole ....................................................................................... 9

Wundt in Translation: Wundtian Introspection and/as

Psychological Phenomenology

Charles Dalrymple-Fraser ................................................................ 31

Culture and the Nature of Mathematics

Caitlin MacLeod ................................................................................ 45

Holism Cannot Save Structuralism

Kyle Da Silva ..................................................................................... 56

Einstein, Bohr and the Vienna Circle

Robert Wesley .................................................................................... 66

The Similarities of Thought Experiments and Computer

Simulations: A First Approach

Natalie Morcos .................................................................................. 80

7

Foreword

“Philosophy of science without history of science is empty; his-

tory of science without philosophy of science is blind” (Lakatos,

1970)

We are pleased to present the seminal publication of Almagest: Undergraduate

Journal in the History and Philosophy of Science. The creation of this journal

is a part of the recent efforts of many dedicated students to strengthen the com-

munity of undergraduates at the University of Toronto who study the History

and Philosophy of Science and Technology (HPS). Notably, at the beginning

of last academic year, Deb Mazer and Felix Walpole founded the HPS Under-

graduate Society; a university recognized course union that runs academic and

social events for HPS students. Part of their vision was the establishment of an

undergraduate journal to showcase excellent work done by students studying

in the field. At the beginning of this academic year, the current editorial board

was assembled. The finished product you see before you is the result of the

hard work of this editorial board and of course the authors whose papers we

have chosen to publish.

HPS is a diverse discipline, and we have tried to reflect this in the papers

we have chosen to publish. The papers “Culture and the Nature of Mathemat-

ics” and “Holism Cannot Save Structuralism” reflect two distinct ways of ap-

proaching the science of mathematics within HPS: historical investigation and

philosophical analysis. It is interesting to note that, as “Culture and the Nature

of Mathematics” suggests, these two approaches are not necessarily in conflict,

and can certainly complement each other. The three papers “Galenic Medi-

cine”, “Einstein, Bohr, and the Vienna Circle”, and “Wundt in Translation”

cover specific, important episodes in the history of science, and do so by look-

ing at individual actors, social institutions, and intellectual movements in var-

ious ways. We are pleased to note that “Culture and the Nature of Mathemat-

ics”, “Holism Cannot Save Structuralism”, “The Similarities of Thought Ex-

periments and Computer Simulations”, and “Galenic Medicine”, all draw on

amazing work done by University of Toronto faculty associated with the Insti-

tute for the History and Philosophy of Science and Technology. This testifies

to the overall strength of the HPS community at the University of Toronto.

We would to thank Caitlin MacLeod and Charles Dalrymple-Fraser who

refereed throughout the reviewing process, despite not being on the editorial

8

board. We would also like to thank our sponsors – the IHPST, ASSU, among

others – whose financial support made this publication possible. A very special

thank you is due to Professor Hakob Barseghyan for his constant support and

for his generous help with the journal’s layout. Overall, we hope that you enjoy

reading the papers in this journal, and that the project we have begun continues

well into the future.

Sincerely,

The 2014-15 Editorial Board

9

Galenic Medicine Felix Walpole

In recent years, the attempt to provide an overarching theory of scientific

change has been abandoned in favor of particularist approaches that claim the

history of science can only be studied in a piecemeal fashion. Inherent to the

particularist method is a belief that no general theory of scientific change is

possible, let alone practical—an attitude that follows directly from the failure

of twentieth century philosophers of science to provide a working model.

However, Hakob Barseghyan’s Theory of Scientific Change (TSC) appears to

demonstrate that a general theory of scientific change is not only possible, but

in fact desirable.1 Armed with four laws of scientific change, the TSC attempts

to explain transitions between scientific worldviews, and between the accepted

theories and employed methods that define them. This research paper will ap-

ply Barseghyan’s TSC to historically account for the subject of Galenic medi-

cine, a topic that has yet to be sufficiently addressed.2 Before beginning my

analysis, I will first provide a brief description of the framework endorsed by

the TSC.

At any moment of the history of science, there are certain theories that

the scientific community of the time accepts as the best available descriptions

of their respective domains. In addition to accepted theories, there are also

methods of assessment employed by the scientific community of the time. This

set of all accepted theories and employed methods constitutes what can be re-

ferred to as the scientific mosaic of the time. The TSC endorses four basic laws

that describe the dynamic interaction that occurs between theories and meth-

ods, including how they change over time. Here are the laws:

1 Hakob Barseghyan, The Laws of Scientific Change (Springer, 2015). Although attempts

to create a working TSC have suffered from a long history of failure, Barseghyan

successfully argues that a TSC is both possible and desirable in the metatheoretical part

of his theory, Scope. Perhaps most importantly, Barseghyan illustrates how it is not

possible to make sense of “facts” unless they are interpreted in light of some theory, See

Introduction, viii.

2 There have been surprisingly few philosophical texts devoted specifically to the rise and

decline of Galenism. Owsei Temkin, Galenism: Rise and Decline of a Medical

Philosophy (Ithaca, N.Y.: Cornell University Press, 1973) appears to be first and only of

its genre.

10

The First Law: An element of the mosaic remains in the mosaic

unless replaced by some other elements.

The Second Law: In order to become accepted into the mosaic, a

theory is assessed by the methods actually employed at the time.

The Third Law: A method becomes employed only when it is de-

ducible from other employed methods and accepted theories of the

time.

The Zeroth Law: At any moment of time, the elements of the sci-

entific mosaic are compatible with each other.

Furthermore, there are numerous significant theorems derivable from these

laws; however, a basic formulation of the laws will suffice for the purposes of

this essay.

My analysis begins by surveying the development of the Galenic tradi-

tion within the context of the Aristotelian scientific mosaic—wherein the the-

ories of Plato, Aristotle, as well as Hippocratic humoral theory were consoli-

dated into an inclusive medical tradition. The essay will then shift focus to

consider the relationship between anatomical knowledge, medicinal theory,

and medical practice within the context of the ‘Scientific Revolution’ in which

an odd paradox appears: despite the acceptance of new mechanistic philoso-

phies and anatomical theories that in many ways appeared to contradict Ga-

lenic medical theory, it appears that physicians continued to use characteristi-

cally Galenic medical practices (such as bloodletting) well into the eighteenth

century. Although such findings could indicate a need to endorse a particular-

ist approach to explain such incongruities, I argue that this would be pre-emp-

tive move that arises from a miscomprehension of the intricate relationship

between medical theory and medical practice. This confusion could be effec-

tively resolved by employing Barseghyan’s distinction between acceptance,

use, and pursuit. By endorsing this distinction it is possible to account for the

scientific community’s acceptance of increasingly sophisticated anatomical

theories while physicians continued to use seemingly ancient treatments that

were justified by Galenic humoral theory. I conclude that it was only after the

dissolution of the long-held distinction between anatomical knowledge and

medicinal theory in the late eighteenth century that Galenic theory can truly be

said to have vacated the scientific mosaic.

11

I. Aristotelian Galenism

At its most fundamental level, Galenic medicine can be characterized as a long

standing medical tradition in which university trained physicians utilized a set

of medical theories to remedy the ‘humoral imbalances’ of their patients and

return them to their ‘natural temperaments’.3 This humoral medical tradition

thrived in various forms for over a span of nearly two millennia; from the time

of Hippocrates and the ancient Greeks, through Galen and the pragmatic Ro-

mans, Avicenna and the Golden Age of Islamic science, and throughout the

European Renaissance. Although the tradition underwent various modifica-

tions, the fundamental elements of the tradition remained accepted until the

seventeenth century wherein occurred what is typically known as the ‘Scien-

tific Revolution’. In the context of the sixteenth and seventeenth centuries, new

theories and observations challenged both Galen’s medical authority but also

the fundamental pillars of the Aristotelian scientific mosaic in which it had

developed. To make sense of success and eventual replacement of Galenic

medicine, it is necessary to outline in detail the theories and methods that were

inherent to this tradition. Such an account must therefore consider humoral

Galenic medicine in relation to the ‘Aristotelian world view’ in which it devel-

oped and flourished.

Like most components of ‘pre-modern science’, the humoral under-

standing of the body that characterized Galenic medicine relied heavily on the

Aristotelian notion of the four elements. To a Galenic physician, ill-health was

defined by an imbalance of the body’s natural fluids – its ‘four humors’ –

which altered a person’s ‘natural temperament’, its ‘natural’ constitution.4 The

humoral theory involves a number of complex relations which must be out-

lined to understand both the Galenic understanding of health as well as the

Aristotelian Worldview more generally. Each of the four humors corresponded

with one of the four natural temperaments, the four Aristotelian elements, the

four seasons, and four vital organs.5 The humor blood corresponded with a

sanguine temperament, Air, spring, and the heart—Yellow bile corresponded

with a choleric temperament, Fire, summer, and the liver—Black bile with a

melancholic temperament, Earth, autumn, and the spleen—and phlegm with a

3 Vivian Nutton, Ancient medicine (London: Routledge, 2004), 22. 4 Nancy G. Siraisi, Medieval & Early Renaissance Medicine: An Introduction to

Knowledge and Practice (Chicago: University of Chicago Press, 1990), 2. 5 Nutton, Ancient medicine, 38. See chart below.

12

phlegmatic temperament, Water, winter, and the brain. The humor assigned to

each temperament resulted from four possible combinations of qualities.6

Choleric was characterized by hot and dry qualities—sanguine wet and hot—

melancholic dry and cold—and phlegmatic cold and wet. All disease was un-

derstood to result from imbalance of the humors which deterred the body from

its natural composition.7 To remedy such imbalances, Galenic physicians ap-

plied ‘contrary’ measures to balance the humors:8 sanguine opposed melan-

cholic, and choleric opposed phlegmatic.

Therefore if a patient had a fever, an ailment characterized by an imbal-

ance of the humors resulting in their temperament being unnaturally wet and

hot, contrary measures would be taken to balance the humors by prescribing

dry and cold remedies. Each individual patient had a ‘natural temperament’,

i.e., a person naturally melancholic would have a temperament characterized

by dry and cold qualities in contrast to the hot and wet qualities of a sanguine

temperament.

6 Ibid. 7 Roy Porter, Blood and Guts: A Short History of Medicine (New York: W.W. Norton,

2004), 27. 8 Ibid., 30.

13

The seasons too, affected the body’s regulation of the humors, and

therefore the physician’s prescription.9 Depending on the patient’s tempera-

ment, they would be more susceptible to certain diseases at certain times of

year.10 Astrology therefore played a key role in medicinal theory throughout

the Aristotelian mosaic. Each bodily organ corresponded to a zodiac sign, and

was particularly vulnerable to a planet’s direct influence on the body’s tem-

perament. The moon, for example, had a strong impact on the brain, as the

moon and brain both shared phlegmatic qualities (cold and wet).11 Although

the physician was the arbiter of the healing process, health and healing was

highly individual and dependent on the patient. The role of the physician was

to artificially ‘assist’ nature in restoring the patient’s body to its natural tem-

perament.12 Only rarely were characteristically intrusive interventions such as

surgery employed, and only in extreme circumstances.13 An ideal physician

therefore had an intricate understanding of Aristotelian natural philosophy but

also knew the patient well, (preferably since birth), thus making them more

likely to know their patient’s natural temperament.14 Most commonly, the phy-

sician aided the balancing of the patient’s natural temperament and thus guided

the body back to health via a prescription of regimen, diet and exercise – a

process most commonly referred to as the six-non naturals.15 It is clear that the

justifications held by physicians for prescribing treatment, relied entirely on

humoral theory that was engrained with other elements of the Aristotelian sci-

entific mosaic.

Within the context of the Aristotelian scientific mosaic, Galenic medi-

cine absorbed the theories of Plato and Aristotle, as well as Hippocratic hu-

moral theory into one grand, all-inclusive medical tradition. Medicinal

9 Hippocrates, W. H. S. Jones, Paul Potter, On Airs, Waters and Places (London:

Heinemann, 1923), 2. 10 Ibid. 11 Porter, Blood and Guts: A Short History of Medicine, 26. 12 Siraisi, Medieval & Early Renaissance Medicine: An Introduction to Knowledge and

Practice, 2. 13 Nutton, Ancient medicine, 38. 14 Roy Porter, “The patient in England, c. 1660-1800,” in Medicine in Society: Historical

Essays, ed. Andrew Wear (Cambridge: Cambridge University Press, 1992), 92-93. In

both the ancient and medieval world, such a luxury was only ever realized by the

incredibly wealthy. 15 Nutton, Ancient medicine, 38.

14

knowledge was intricately interwoven with other elements of the scientific mo-

saic such as physics, horology, and astrology, which were all integrated under

the umbrella of the Aristotelian mosaic, and remained so well into the renais-

sance.16 After the decline of the Roman Empire, Europe fell into disarray and

the Galenic medical tradition would have been lost if it were not for its preser-

vation in the Eastern Empires where it thrived throughout the Islamic Golden

Age (circa. 900 AD).17 Islamic physicians and medical theorists relied on the

ancient knowledge of the Ancient Greek’s and Romans, yet made many mod-

ifications of their own. Most important were contributions from Rhazes and

Ibn Sina (Avicenna) who implemented a systematization of empirical, experi-

mental and quantitative approaches to the practice of science and medicine that

later became absorbed in the western Renaissance.18 By the twelfth century,

Galenic theory had been absorbed back into the Aristotelian medieval world

view, and would continue to flourish throughout the Renaissance as one of the

key authoritative bodies of knowledge attributed to the ancients.19

Margarita Philosophica (Feinburg 1503)

Zodiac Man.

A chart featured in an early modern ency-

clopedia,compiled by Gregor Reisch. The

chart showcases the vital organs and their

corresponding zodiacal sign.

16 Roger French, William Harvey's Natural Philosophy (Cambridge University Press:

Cambridge, 1994), 35. 17 Jon Mcginnis. Avicenna (England: Oxford University Press, 2010), 20. 18 Sami K. Harmarneh, Health Sciences in Early Islam: Volume 1, ed. Munawar A. Anees

(Zahra Publication: Washington, 1983), 155. 19 French, William Harvey's Natural Philosophy, 18.

15

Despite the various modifications that the Galenic tradition endured

throughout the medieval era, the humoral theory, as has been discussed, re-

mained fundamentally intact. Similarly, despite constant debate over the spe-

cifics of how the Aristotelian universe operated, the medieval scientific mo-

saic, as well as its medical community, remained committed to a number of

fundamental Aristotelian beliefs. For instance, the general belief that the mac-

rocosm had a direct influence on the microcosm remained unquestioned, as

was the belief that the superlunary world had a direct influence on the human

body. Indeed, it is the interconnectedness and reliance on a number of funda-

mental beliefs in the Aristotelian mosaic that made it so cohesive and immune

to change. While retaining its fundamental reliance on humoral theory, the Ga-

lenic medicinal theory that flourished throughout the Renaissance had become

characteristically more ‘anatomical’.20 Increasingly, theorists came to focus on

the ‘natural’ function of organs that had of course been created by a higher

agency.21 This allowed for Galenic doctrine to be modified and accepted in

Christian Western Europe throughout the Middle Ages and the Renaissance,

therefore satisfying the requirements of the method of the time. A Galenic phy-

sician not only knew by experience the kinds of diseases a patient may be suf-

fering from, but also understood the causes and hence, also, by rational prog-

nosis, the outcome.

20 Porter, Blood and Guts, 55. 21 French, William Harvey's Natural Philosophy, 6.

16

Petrus Apianus, Cosmographia (1539)

The Aristotelian Universe: celestial and terrestrial spheres

The Galenic practice of bloodletting, for example, was theoretically

supported and practiced because it could “relate therapeutic techniques to the

structure of the body, to its functioning, and from here by a microcosmic-mac-

rocosmic parallel to the fundamentals of the world picture.”22 Galenic medical

theory therefore used new anatomical findings to make sense of the ancient

teachings of Galen, providing physiological explanations that started from

structure, to function, to malfunction, to treatment.

II. Galenism and the ‘Scientific Revolution’

In 1628, William Harvey published De Motu Cordis, in which he presented his

observation of blood circulation. This observation directly conflicted with the

Galenic explanations of arterial and venous blood, the role of the liver in pro-

ducing venous ‘nutritional’ blood, and hence the very tripartite system integral

to the Galenic system.23 For this reason, Harvey was apprehensive to publish

his observations, knowing full well the many ways in which they could be used

22 Ibid., 5. 23 Andrew Wear, “Early Modern Europe 1500-1700,” in The Western Medical Tradition

(Cambridge: Cambridge University Press, 1998), 326. For details of Galen’s tripartite

physiology see Nutton, Ancient medicine, 230

17

to support the emerging mechanistic theories characteristic of seventeenth cen-

tury English experimentalism, as well as René Descartes’ mechanism in

France. As an ardent Galenist, Harvey was clear that he did not want his ob-

servations to be used to ‘overthrow’ Galenic medicine, but that he was contrib-

uting to it.24 And yet, Harvey is often referred to by contemporary historians

as having most ardently contributed to the decline of Galenic theory.25 Regard-

less of Harvey’s Aristotelian education under Fabricus or his individual adher-

ence to Galenism, by the time that Harvey had died in 1657 the battle for cir-

culation had been won.26 A useful indicator can be found in the treatises of

physician John Floyer, who notes that he was taught Harvey’s anatomy while

at Queens College in Oxford in the mid 1650’s.27 This provides evidence that

at the level of the scientific community, Harvey’s theory of blood circulation

was accepted as the best available description of human anatomy at Oxford,

the heart of the English enlightenment, since the mid sixteenth century.

24 Temkin, Galenism; Rise and Decline of a Medical Philosophy, 152. In the preface to De

Motu Cordis, Harvey devotes the work to the King, who he compares to the heart, the

moral and central figure of the nation. Above all, Harvey understood the circulation of

blood as an anatomical exercise. 25 Historians of science have often treated his observation as a “cornerstone of modern

physiology”, one that “isolated and solved a limited problem and thus cleared the way

for progress in modern science”. For discussion, see Temkin, Galenism; Rise and

Decline of a Medical Philosophy, 156-157 26 Ibid., 153; French, William Harvey's Natural Philosophy, 2. 27 Mark Jenner, “Bathing and Baptism: Sir John Floyer and the Politics of Cold Bathing,”

in Refiguring Revolutions: Aesthetics and Politics from the English Revolution to the

Romantic Revolution, eds. Kevin Sharpe and Steven N. Zwicker (Berkeley: University

of California Press, 1998), 199. Interestingly, Floyer’s practice provides the first known

account of a physician using Harvey’s observations of blood circulation to measure the

pulse of a patient by using a stopwatch to measure heartbeats per minute, and therefore

the first to apply Harvey’s theory of blood circulation in a therapeutic sense. Curiously,

Floyer was like Harvey, an ardent Galenist who opposed the mechanistic philosophy of

the ‘moderns’ in favour of the ancients. For discussion, see Floyer’s two books The

ancient Psychrolousia revived: or, an essay to prove cold bathing both safe and useful

[…] Containing an Account of many Eminent Cures done by the Cold Baths in England.

Together with a Short Discourse of the wonderful Virtues of the Bath-Waters on decayed

Stomachs, drank Hot from the Pump (London: printed for Sam. Smith and Benj.

Walford, at the Prince's Arms in St. Paul's Church-Yard, 1702) and The physician's

pulse-watch; […] which is grounded on the Observation of the Pulse, is recommended.

To which is added, An Extract out of Andrew Cleyer, concerning the Chinese Art of

Feeling the Pulse (London: printed for Sam. Smith and Benj. Walford, at the Prince's-

Arms in St. Paul's Church-Yard, 1707).

18

Harvey’s observations can be understood as a direct extension of the

work of Vesaluis’s De Humani Corporis Fabrica (1543) that attempted to

identify all of Galen’s anatomical errors, as well as Realdo Columbo’s that

immediately preceded his own.28 These observational anatomists challenged

the authority of the Galenic anatomical tradition, and can be understood to

endorse a ‘new anatomy’. Until Vesalius, the authoritative Galenic text on

anatomy had been Mondino’s Anathomia Corporis Humani, completed in

1316 (except in France where they preferred Gui de Chauliac).29 Anatomical

teaching had relied primarily on a commentary of Mondino’s work, using a

method characteristic of the Aristotelian discours.30 Throughout the fifteenth

century, anatomists such as Berengario encouraged a trend towards a more ob-

servational anatomy in which he described ‘things’ rather than words. Beren-

gario’s satisfied the requirements of the method of the time by writing in a

fashion that addressed an academic audience who had expectations about the

form argument could take. Readers expected proof of novel anatomical dis-

coveries, such as Berengario’s finding, contra Galen, that the rete mirabile did

not exist in the human brain – a discovery later corroborated by Vesalius.

28 Wear, “Early Modern Europe 1500-1700,” 273. 29 French, William Harvey's Natural Philosophy, 18. 30 Ibid., 20.

19

Mondino, Anathomica Corporis Humani

(1316).

A lecturer (learned physician) reads

from Mondino’s text, while a demon-

strator points to the appropriate body

parts excavated by the barber-surgeon.

Vesalius was the first to break with this

age old tradition, and perform dissec-

tions personally.

Novel observations in the fifteenth century were introduced as proposi-

tions, as though they could be handled dialectically in the characteristically

Aristotelian fashion.31 In addition, Berengario relied on demonstration via the

use of public anatomies and illustrations – what could generally be referred to

as a physical logic – that were new ways of demonstrating anatomical structure

to the reader. The new anatomy remained acceptable by sectioning itself off

from medicine by giving it firstly a religious and secondly a philosophical pur-

pose.32Anatomy was therefore accepted as an autonomous and practical busi-

ness – an experimental one. This notion is indicative of the division between

medical theory and anatomical knowledge that continued throughout the re-

naissance, and arguably until the nineteenth century.

By the time that Vesalius published De Fabrica, the scientific commu-

nity of the time had for two decades relied solely on the authority of Galen’s

On the Use of the Parts and Anatomical Procedures.33 This reliance can be

understood from within the context of the Renaissance that revered the classi-

cal knowledge of the ancients. To be accepted within the community, Vesalius

31 Ibid., 20. 32 Mondino exemplifies this tendency in his dedication the pope. Ibid., 25. 33 Ibid., 21; Wear, “Early Modern Europe 1500-1700.”

20

presented De Fabrica in a humanist framework and claimed to be restoring the

older and purer human anatomy practiced in areas such as ancient Alexandria.

Specifically, Vesalius praised the work of Herophilus and Erasistratus, who

were known to approve of human dissection.34 Vesalius distrusted Galen’s ob-

servations and rightly suspected that he had never dissected a human body.

Above all else, he emphasized the value of direct observation, as would Harvey

a century later, and he was the first to personally perform dissections.35 Vesa-

lius successfully became the new anatomical authority; De Fabrica was con-

temporarily regarded as the most sophisticated anatomical work ever written,

which won him the position of personal physician to Emperor Charles I of the

Holy Roman Empire.36 More importantly, Vesalius contributed to an important

transition towards the experimental culture characteristic of the Enlighten-

ment. As noted by Roger French, “Vesalius was more experimental than his

predecessors and less analytical than his successors; but above all it was clear

that anatomical research now had to be experimental on the Galenic model.”37

Therefore Vesalius indeed endorsed an ‘experimental Galenism’, yet main-

tained a distinction between anatomical theory and medical theory. Notably,

Vesalius’s treatments differed little to Galen’s; his main contention was to im-

prove the accuracy of Galen’s anatomical work.38 Similarly, the first half of

De Motu Cordis presented the already known inconsistencies in Galen’s theo-

ries in an unprecedentedly clear and organized manner without making any

comment on the implications of his findings for accepted medicinal theory.

Although Harvey’s research programme and the reasons for pursuing

anatomy were deeply rooted in Aristotelian natural philosophy, his theory be-

came accepted at Oxford, at the heart of the Enlightenment and Baconian ex-

perimental philosophy that later actualized into the infamous institution, the

Royal Society of London for the Improvement of Natural Knowledge. In

France, René Descartes had incorporated Harvey’s theory of blood circulation

to substantiate his mechanistic philosophy – to show the body to be analogous

34 It is worth noting that it was not until the sixteenth century that anatomists could legally

dissect human cadavers. Only in 1555 after reforms made by Queen Elizabeth I could

anatomists legally dissect cadavers in England. Charles Joseph Singer, A Short History

of Anatomy from the Greeks to Harvey (New York: Dover Publications, 1957), 173;

French, William Harvey's Natural Philosophy, 32. 35 Ibid. 36 Wear, “Early Modern Europe 1500-1700,” 273 37 French, William Harvey's Natural Philosophy, 29. 38 Ibid.

21

to a machine and the heart analogous to a pump in Discours Sur La Méthod

(1637).39 Therefore despite Harvey’s intent, his observations were used to sub-

stantiate the new mechanical and experimental philosophies of Descartes, and

the Royal Society. These mechanistic philosophies actively challenged the the-

oretical elements of the Aristotelian mosaic, of which Galenic humoral theory

had at its roots. Yet as noted by Barseghyan, despite popular opinion, the shift

in worldviews characterized as the so-called Scientific Revolution, did not oc-

cur until the end of the seventeenth century.40 Therefore Harvey’s anatomical

observations were accepted within the English Aristotelian mosaic by the mid-

seventeenth century at Oxford for they satisfied the Aristotelian method: Har-

vey’s observations were accepted as superior to Galen’s due to the Aristotelian

manner in which he presented them; he made clear that he arrived at his con-

clusions intuitively, confirmed by experience.41 Further, although his observa-

tions appear from a modern perspective to be incompatible with Galenic med-

ical humoral theory (recall that blood circulation contradicts Galen’s theory

that blood was produced in the liver and absorbed by the body) Harvey’s ob-

servations were acceptable because of the distinction between anatomical

knowledge and medical theory, and can therefore be seen to accord with

Barseghyan’s second and zeroth law.42 The confusion that results from the fact

that Descartes, and institutions such as the Royal Society, endorsed Harvey’s

observations for their potential to substantiate their experimental and mecha-

nistic philosophies, can be resolved by placing the acceptance of Harvey’s the-

ory in its proper context. As argued by Andrew Wear, it is due to this confusion

that Harvey has mistakenly been characterized by particuarlist historians as

the “last ancient”, the “first modern”, or paradoxically, as a conflation of the

two.43 These confusions can be remedied by considering Harvey’s anatomy

within the confines of Barseghyan’s TSC, which allows for a logical explana-

tion of the acceptance of his work at the community level.

39 Domenico Bertoloni Meli, Mechanism, Experiment, Disease: Marcello Malpighi and

Seventeenth-Century Anatomy (Baltimore: Johns Hopkins University Press, 2011), 4 40 Barseghyan, A Theory of Scientific Change, 37. 41 Harvey is specifically referred to as having practiced as an “experiential

Aristotelianism.” French, William Harvey's Natural Philosophy, 310. 42 Barseghyan, A Theory of Scientific Change, 117. 43 Andrew Wear, “William Harvey and the ‘Way of the Anatomists’,” History of Science,

XXI (1983), 223

22

III. Galenic Theory—A New Perspective

It was within the eighteenth century that the Aristotelian-scholastic natural phi-

losophy, with its theory of elements and four causes, its geocentric cosmology,

and its Aristotelian laws of motion, were replaced by elements of a Newtonian

worldview in which new mechanical theories and experimental methods were

adopted.44 As argued by Roy Porter, “By 1700, advances in gross anatomy and

physiology were thus firing hopes of a truly philosophical understanding of

the body’s structures and functions, cast in the language of the prestigious sci-

ences of mechanics, mathematics and chemistry.”45 On the one hand, medici-

nal theory was clearly influenced by the new mechanical philosophy and ex-

perimental culture of the eighteenth century scientific community, and yet

there can still be seen a deep adherence to Galenic humoral theory. This ap-

pears to challenge the rather simplistic analysis offered by historians such as

Lester S. King who have portrayed seventeenth and eighteenth century medi-

cal thinkers as having taken part in a mechanical medical revolution which

overthrew the dogmatism of medieval Galenism and paved the way for modern

medicine.46 It is here that the essay will consider to what extent Barseghyan’s

notions of acceptance, use and pursuit, can be applied to sift through such

apparent confusion and provide a logical explanation of the continuation of

Galenic medicine throughout the eighteenth century.

Barseghyan’s terms of acceptance, use, and pursuit have allowed for

his TSC to resolve a number of issues inherent to the work of previous philos-

ophers of science, yet it is not clear that they necessarily apply to medicine.

The emblematic example provided by Barseghyan is that of physics and bridge

building: although a contemporary physicist may accept general relativity as

the best available description of reality, she would use classical Newtonian

mechanics when building a bridge.47 Further, she can pursue other theories

such as String Theory, while accepting General Relativity as the best available

description of its domain – theoretical physics.48 To treat medicine in such a

way would require that accepted medical theories be characterized as those

44 Barseghyan, A Theory of Scientific Change, 37. 45 Porter, Blood and Guts, 66-67. 46 Lester S. King, The Road to Medical Enlightenment (Elsevier Publishing Company:

New York, 1970), 172. King refers to this process as a “quarrel between the Ancients

and the Moderns.” 47 Barseghyan, A Theory of Scientific Change, 28. 48 Ibid., 28-30.

23

taught in universities, and perhaps endorsed by institutions such as the Royal

Society. Correspondingly, a physician uses medical theory to treat a patient.

By endorsing this distinction, a physician could accept the sophisticated ana-

tomical depictions of the body while using Galenic humoral theory to treat

patients. Indeed, it appears that this was often the case, for as noted by Andrew

Wear, although anatomical investigations increased in prevalence and sophis-

tication throughout late seventeenth and eighteenth centuries, they rarely pro-

vided a means to new or ‘better’ working medical remedies.49 Yet if a physician

used Galenic medical practices because they proved better in healing their pa-

tients, are they not also to be considered the best available description of their

object – i.e., healing? This paradox is complicated further by the fact that most,

if not all, notable medical theorists were physicians themselves. Further, uni-

versities taught both anatomical theory, and also pathology, which continued

to rely on Galenic humoral theory, as did powerful institutions such as the Col-

lege of Physicians London.50 As I have suggested, making sense of this odd

phenomenon can be achieved by considering the distinction between anatom-

ical knowledge and medicine that is nicely drawn out via an analysis of Mar-

cello Malpighi and the Bologna Controversy.

In 1701, Malpighi defended himself against the various criticisms posed

by the university faculty at Bologna, specifically by Giovani Sbaraglia and

Paolo Mini, for his adherence to microscopy, and to the value of anatomical

investigation.51 Malpighi’s attackers at the university denounced Malpighi

principally due to his belief that both anatomy and microscopy had potential

value to medicine, claiming that “knowledge of the marvelous conformation

of these entities (anatomical structures of the body) will not advance the art of

curing the sick”.52 Throughout his career, Malpighi actively endorsed the value

of microscopy to medicine, and utilized it to create new medical theories, as

well as to his practice as a physician.53

49 Andrew Wear, “Making Sense of Health and the Environment in Early Modern

England,” in Medicine in society: Historical essays, ed. Andrew Wear (Cambridge:

Cambridge University Press, 1992), 120. 50 Jenner, “Bathing and Baptism: Sir John Floyer and the Politics of Cold Bathing,” 199. 51 Meli, Mechanism, Experiment, Disease, 312 52 Catherine Wilson, The Invisible World: Early Modern Philosophy and the Invention of

the Microscope (Princeton University Press: New Jersey, 1995), 232 53 Ibid.

24

De Motu Cordis (1628)

Harvey supplied an “ocular

demonstration” of the function

of the valves in the veins,

valves which were confirmed

by Marcello Malpighi’s ana-

tomical work with the micro-

scope.

He famously used the microscope to corroborate Harvey’s observation

of blood circulation, as he located the capillaries in the heart, the ‘tiny doors’

that had been invisible to Harvey for whom microscopes were not yet availa-

ble.54 Yet Malpighi also used the microscope to isolate particles in urine that

were invisible to the naked eye, which factored into his treatment of the pa-

tient.55

The controversy between Malpighi and his peers clearly indicates that

his experiments were criticized because they did not satisfy the employed

method of the time and thus violated Barseghyan’s third law of scientific

change. A similar observation is made by Barseghyan in his analysis of the

conditions in which Galileo’s theories were condemned by the scientific com-

munity of the time.56 Despite the increasing prevalence of anatomical investi-

gation in the seventeenth and eighteenth century, figures such as Malpighi who

endorsed the value of anatomy to medicine were rare until the end of the eight-

eenth century. Indeed, the rarity of such figures may explain modern scholars’

fascination with figures such as Malpighi who appear to have held high the

torch of ‘modern’ medicine against the restrictive and ancient attitudes of their

54 Meli, Mechanism, Experiment and Disease, 312 55 Wilson, The Invisible World: Early Modern Philosophy and the Invention of the

Microscope, 235 quoting Malpighi’s Riposta. As also noted by Meli, Malpighi

continued to prescribe remedies that were characteristically Galenic. Further, he

opposed the mechanism of his contemporaries such as Bellini and Basille, and historians

should therefore refrain from labelling him as distinctly ‘modern’. For discussion see

Meli, Mechanism Experiment and Disease, 313. 56 Barseghyan, A Theory of Scientific Change, 44. Interestingly, Malpighi was referred to

by his contemporaries as the Galileo of medicine. For discussion, see Meli, Mechanism,

Experiment and Disease, 312.

25

contemporaries.57 It is therefore possible to assume that the scientific commu-

nity of the eighteenth century could, and did continue to accept increasingly

sophisticated anatomical representations of the body, while maintaining hu-

moral theory to be the best way to treat a patient. In this manner physicians

used Galenic treatments while accepting ‘modern’ anatomical descriptions of

the body in conjunction with the new mechanical philosophy as the best de-

scription of reality.58 Therefore, as can similarly been seen to have occurred in

physics, medicine can be seen to have theoretical and practical components.

Just as a physicist would refrain from using general relativity to build a bridge,

in most cases seventeenth and eighteenth century physicians found no use for

anatomy in practical medicine.59

Over the course of the eighteenth century, the distinction between ana-

tomical knowledge and medical theory began to erode as it became increas-

ingly evident that anatomical knowledge was indeed valuable to medicine.60

An influential figure in this process was the Dutch anatomist Herman Boer-

haave (1668-1738), who speculated from his anatomical investigations that the

body acted as a ‘plumbing network’, and that medicine should concern itself

with maintaining a healthy flow of the body’s interior substances.61 Although

Boerhraave integrated his anatomical investigations into his medical theory,

they did not displace an adherence to humoral theory, but modified it:

Health was maintained by the free and vigorous movement of

fluids in the vascular system, sickness explained in terms of

blockages, constrictions or stagnation […] the old humoral em-

phasis upon balance had thus been preserved but translated into

mechanical and hydrostatic idioms.62

57 See introduction to International Malpighi Symposium. Pietro M. Motta, Microscopy of

reproduction and development: a dynamic approach: a celebrative symposium on

Marcello Malpighi (1694-1994): proceedings of the 2nd International Malpighi

Symposium, held in Rome, Italy, September 14-16, 1995 (Rome: Antonio Delfino

editore, 1997). 58 This line of reasoning appears to be consistent with the analysis of Owsei Temkin who

notes that “as a guide to medical practice, Galenism did survive in spite of the new

philosophy”. For discussion see the chapter “Fall and Afterlife” within his Rise and

Decline of a Medical Philosophy. 59 Wear, “Early Modern Europe 1500-1700,” 287. 60 Ibid., 290-291. 61 Porter, Blood and Guts, 67 62 Ibid.

26

Therefore, Boerhaave, arguably one of the most prestigious medical theorists

of the eighteenth century, still perceived the cause of disease, and thus the

treatment of disease, in humoral terms refashioned in the language of the new

mechanical philosophy. However, this appears to be the last time that humoral

theory can be seen to have been involved in the acceptance of a medical theory.

After the death of Boerhaave in the mid-eighteenth century, many medical the-

orists understood the workings of the body by appealing to vitalist forces

which became increasingly consistent with emerging chemical theories. Georg

Ernst Stahl, the founder of the distinguished medical school at the University

of Halle, presumed the presence of an immaterial soul (anima), which presided

and actively sustained vitalist power in organisms (as opposed to Descartes’

notion of a ‘ghost in the machine’).63 To contemporary medical theorists, it

was not clear whether bodily processes such as digestion were the result of an

internal vitalist force, chemical action of gastric acids or of the stomach’s me-

chanical activities of churning and pulverizing.64 However, it is clear that all

of these competing theories were trying to satisfy the requirements of the em-

ployed method of the time – the hypothetico-deductive method. In London,

anatomist John Hunter expanded upon Stahl’s theory to propose a ‘life princi-

ple’ to account for properties that elevated living organisms above gross inan-

imate matter – a life-force that lay in the blood.65 His physiological theory was

substantiated by the chemical theorists such as Joseph Black who introduced

the concept of ‘fixed air’ that was crucial to further theoretical understandings

of respiration.66 In France, Lavoisier explained the passage of gases into the

lungs: the air inhaled was converted into and exhaled as black’s ‘fixed air’

which Lavoisier termed Carbon Dioxide, through which it became clear that

oxygen was indispensable to life.67

It is unquestionable that these mechanistic, chemical, and experimental

investigations transformed how disease was to be understood over the course

of the eighteenth century. Such transitions culminated at the end of the eight-

eenth century to finally dissolve the distinction between medical theory and

anatomical investigation. It is arguable that the definitive point in which anat-

63 Ibid., 68 64 Ibid. 65 Ibid., 70. 66 Ibid. 67 Ibid.

27

omy and medicine reconciled, and thus displaced humoral theory from the sci-

entific mosaic, was the acceptance of Giovanni Battista Morgagni’s theory of

pathological anatomy. Morgagni’s De Sedibus et Causis Morborum (On the

Sites and Causes of Disease, 1761) drew on some 700 autopsies to demonstrate

how the cause of disease was found in the internal organs.68 The acceptance of

his theory shows that the causal nature of disease had been reconsidered;

Morgagni’s investigations shifted emphasis from symptoms, which had up un-

til this point revealed the internal disorder of the humors, to the site of disease

which could be determined via anatomical investigation. Disease was thus no

longer understood as an abnormal condition of the whole organism (due to the

disruption of the humors), but as a result of a specific ‘diseased’ part of the

body.69 By showing how symptoms of disease could be traced anatomically

via experiment, Morgagni’s investigations satisfied the requirements of the

scientific mosaic and united pathology and anatomy for the first time. As noted

by Temkin, following Morgagni’s investigations, pathological anatomy be-

came the cornerstone of nineteenth century medicine, and was accepted in

Paris, Vienna, Dublin and London.70 Further, pathological anatomy paved the

way for the rise of clinical medicine in the nineteenth century in which Galenic

pathology eventually ceased to be practiced by the majority of physicians.71

Conclusion

This essay has attempted to incorporate the history of Galenic medicine into

the framework of Barseghyan’s TSC. After surveying the history of its devel-

opment in the ancient world, and its consolidation throughout the Aristotelian

mosaic, Galenic medicine was considered in relation to the ‘Scientific Revo-

lution’, in which time the theories inherent to the Aristotelian mosaic, includ-

ing Galenic medicine, were forcibly challenged. Many histories of medicine

within the context of the ‘Scientific Revolution’ provide either a simplistic ac-

count of a wholesale transition between ‘ancient’ and ‘modern’ medicine, or

become lost in a sea of particularist narratives. This essay has employed

Barseghyan’s distinction between acceptance, use, and pursuit to resolve the

68 Temkin, 137-138. 69 Ibid., 138. 70 Ibid. See also, Fiona Hutton, The Study of Anatomy in Britain, 1700-1900 (London :

Pickering & Chatto, 2013), 1. 71 Porter, Blood and Guts, 73.

28

confusion, and to tease out the intricate connection between medical theory

and medical practice. My analysis has revealed an essential distinction be-

tween anatomical knowledge and medicine to be crucially important: despite

the adoption of a new mechanistic philosophy, experimental method and a

more sophisticated anatomy in the seventeenth century, Galenic medicine re-

mained central to both medical practice and theory until the adoption of ‘patho-

logical anatomy’ in the late in the eighteenth century. Despite a number of

competing vitalist and mechanical medical theories, Morgagni’s De Sedibus et

Causis Morborum (1761), which was accepted throughout the European uni-

versities towards the end of the eighteenth centuries, was instrumental in unit-

ing anatomy and medicine and paving the way for nineteenth century clinical

medicine. The essay has therefore provided a conceptual historical framework

through which further analysis of the late eighteenth and early nineteenth cen-

tury could provide further substantiation.

29

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Floyer, John, Sir. The ancient Psychrolousia revived: or, an essay to prove

cold bathing both safe and useful […] Containing an Account of

many Eminent Cures done by the Cold Baths in England. Together

with a Short Discourse of the wonderful Virtues of the Bath-Waters

on decayed Stomachs, drank Hot from the Pump. London: printed

for Sam. Smith and Benj. Walford, at the Prince's Arms in St. Paul's

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—.The physician's pulse-watch; […] which is grounded on the Observation

of the Pulse, is recommended. To which is added, An Extract out of

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Arms in St. Paul's Church-Yard, 1707.

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Places. Hippocrates. London: Heinemann, 1923.

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ing & Chatto, 2013.

Jenner, Mark. “Bathing and Baptism: Sir John Floyer and the Politics of Cold

Bathing.” Refiguring revolutions: aesthetics and politics from the

English revolution to the Romantic revolution, edited by Kevin

Sharpe and Steven N. Zwicker, 197-217. Berkeley: University of

California Press, 1998.

King, Lester S. The Road to Medical Enlightenment. Elsevier Publishing

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Meli, Domenico Bertoloni. Mechanism, Experiment, Disease: Marcello Mal-

pighi and Seventeenth-Century Anatomy. Baltimore: Johns Hopkins

University Press, 2011.

30

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Mcginnis, Jon. Avicenna. England: Oxford University Press, 2010.

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approach: a celebrative symposium on Marcello Malpighi (1694-

1994): proceedings of the 2nd International Malpighi Symposium

held in Rome, Italy, September 14-16, 1995. Rome: Antonio Delfino

editore, 1997.

Porter, Roy. “The patient in England, c. 1660-1800.” Medicine in society:

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Temkin, Owsei. Galenism: Rise and Decline of a Medical Philosophy. Ithaca,

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—.”Making Sense of Health and the Environment in Early Modern Eng-

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31

Wundt in Translation: Wundtian Introspection

and/as Psychological Phenomenology Charles Dalrymple-Fraser

“For a topic of rather central importance in the emergence of

modern psychology, introspection has not been accorded the

historical attention it deserves. From the global statements and

glib generalizations that abound one might easily get the im-

pression that introspection always meant the same thing, irre-

spective of time and place. That, of course, is far from true, and

if we are to avoid historically unjustified generalizations about

“paradigms” and so forth, we will have to develop a far more

differentiated view of the topic than presently prevails.” (Dan-

ziger, 1980)

In recent years, historians of psychology have suggested that the accounts of

Wundt common to the American history of psychology are misattributed, and

that Wundt was not an advocate of introspection as we know it. Whereas the

common understanding of Wundt’s introspection matches closely with Wil-

liam James’s account—“Introspective Observation […] needs hardly to be de-

fined. It means, of course, looking into our own minds and reporting what we

there discover” (James, 1890, p. 135)—Wundt’s account features a few further

qualifications. The purpose of this paper is to defend the claim that Wundt’s

account of innere Wahrnehmung, while commonly translated as “introspec-

tion”, is better understood as a phenomenological practice. In particular, it will

be argued that (a) there is reason to believe that Titchener mistranslated Wundt

into English and that Wundt’s original project does not fit the standard intro-

spective model, and (b) the distinctions between introspection and phenome-

nology recently set out by Gallagher and Sørensen (2006) finds Wundt’s work

more phenomenological than introspective. In this way, we will find that

Wundt’s experimental method is more relevant today than is typically thought.

This paper has four sections. First, I suggest that Titchener’s translations

of Wundt resulted in a misrepresentation of Wundt’s work; support for which

is located in other literary reviews. In section two, I further support the claim

that Wundt’s innere Wahrnehmung is not equal to “introspection”, by demon-

strating how a careful reading of some of Wundt’s more available texts can

overcome objections levied against introspection at large. In section three, I

situate Wundt’s innere Wahrnehmung in the introspection–phenomenology

32

framework set out by Gallagher and Sørensen (2006), and demonstrate that a

closer translation of innere Wahrnehmung in the present day might be “phe-

nomenological observation”. I provide a brief summary and conclusion in sec-

tion four.

I Titchener and Problems in the Translation of Wundt

There have been many historical difficulties in translating Wundt, resulting in

very few of his works being made available in English. Still, there is a small

corpus of Wundt’s works available in English, most of which were penned by

one of his students, Edward B. Titchener. Recently, the accuracy of Titchener’s

translations has come into question, and it has become suspect whether Titch-

ener’s Wundt is the same psychologist as Wundt himself. In what follows, I

will briefly outline the evidence against Titchener’s translation, though keep-

ing in mind the important role Titchener’s translations played in making

known Wundt’s work.

I.i Translating Wundt

It will not serve us merely to analyze the accuracy of Titchener’s trans-

lations, unless we consider also the context in which the translations were

done. Titchener has often been credited as the student who most brought

Wundt to America, through his dedication to translating Wundt, his devotion

to bringing introspectionism to the United States, and in his relationship to

Edwin G. Boring—considered by many to be the most influential historian of

psychology (Blumenthal, 1979; Blumenthal, 1980; Costall, 2006 Dnziger,

1980). Indeed, Titchener began working on translations of Wundt’s Grundzüge

de Physiologischen Psychologie72 in 1887: three years before Titchener would

even begin his studies with Wundt in Leipzig (Tweney & Yachanin, 1980).

This eagerness evidences Titchener’s belief in the import of translation.

Further evidence of Titchener’s commitment to translating Wundt is

found in his steadfastness. When Titchener began translating the Grundzüge in

1887, it was in its third edition. Unfortunately, Wundt’s own frequent revisions

of the work led to Titchener’s abandonment of a translation of the third edition,

turning instead to translate Wundt’s revised fourth edition in 1893 (Tweney &

Yachanin, 1980). Yet, again, Titchener was forced to redirect his translation

72 The Principles of Physiological Psychology.

33

efforts to the fifth edition, after further revisions by Wundt, leading to his sub-

sequent claim that “One does not undertake the task of translating a large work

for the third time and in mature life with the enthusiasm that one brings to it

as a young student” (Titchener, in Wundt, 1904, p. x). Given waning enthusi-

asm for the task, Titchener only translated the first volume of the fifth edition

Grundzüge before turning his attention instead to the translation of other works

(Tweney & Yachanin, 1980).

A final dynamic of import is Titchener’s relation to Edwin G. Boring.

Boring, as above noted, is often considered the most influential historian of

psychology, and is most renowned for his A History of Experimental Psychol-

ogy (1929, revised in 1950). In his History, Boring expounds Wundt as “the

father of experimental psychology” (1950, p. 317), and credits the Grundzüge

as “the most important book in the history of modern psychology” (1950, p.

322). The History was well read, and it is to this text that many historians of

psychology cast the blame for the apparent misconceptions about Wundt’s

works (Blumenthal, 1977; Blumenthal, 1979; Danziger, 1979; Danziger, 1980;

Leahey, 1981; Tweney & Yachanin, 1980). Yet, as Danziger (1980) points out,

and as Blumenthal (1977, 1979) also suggests, “it appears that Boring fre-

quently based his account on E. B. Titchener’s opinions and special interpre-

tations rather than on a careful direct examination of the foreign sources them-

selves” (Danziger, 1980, p. 241).

Accordingly, given Titchener’s substantial role in bringing Wundt to

America, it seems that Titchener is to blame—if there is blame to be cast—for

the American misconceptions of Wundt which historians have uncovered. Yet,

before turning to a brief characterization of the errors in translation, it is again

worth noting that without Titchener’s contributions—erroneous as they may

be found—it is doubtful that Wundt’s name would have remained with us to-

day (see Tweney & Yachanin, 1980, for a further analysis of Titchener’s role

in Americanizating Wundt).

I.ii Problems with the Translation of Wundt

Historians of psychology have recently found Titchener’s translations

of Wundt to be misinformed, even though Titchener studied directly under

Wundt (Blumenthal, 1977; Costall, 2006; Danziger, 1980; Leahey, 1981;

Tweney & Yachanin, 1980). It is beyond the scope of this paper to detail the

34

errors in Titchener’s translations of Wundt.73 Rather, in what remains of this

section, I will briefly indicate a few problems with Titchener’s use of “intro-

spection”, and a few possible reasons for his errors. In the next section, I will

further explicate Wundt’s account by demonstrating his responses to those ob-

jections lobbied against the account of “introspection” Titchener brought to

America.

Danziger (1980) provides the most comprehensive account of error in

Titchener’s translations. Here we will focus on the translation of “introspec-

tion”. In particular, Danziger notes that Titchener fails to distinguish between

Wundt’s accounts of “self-observation” (Selbstbeobachtung) and “internal

perception” (innere Wahrehmung). Wundt insists the two concepts are distinct,

but Titchener has translated both as “introspection” (Danziger, 1980).74 Failing

to account for these differences, according to Danziger, critically undermines

the force of Wundt’s argument. However, Wundt’s account makes use of this

bifurcation to specifically defend against objections which are frequently

raised to common notions of introspection, objections which Wundt consid-

ered valid (Danziger, 1980). Titchener’s conflation, then, critically undermines

Wundt’s account by conflating it with theories Wundt found objectionable (cf.

Danziger, 1980). Hence we must wonder, why would a direct student of

Wundt’s be prone to such a mistake as Danziger and others suggest?

There are three main suggestions in circulation. The first is that Wundt

was largely untranslatable: aside from his frequent tendency to revise his ac-

counts, Wundt’s writing has been considered “too parochially German for

expostation […] One can, in English, not even properly translate the thing!”

(van Hoorn & Verhave, 1980, p. 108; see also Tweney & Yachanin, 1980). Yet,

Titchener tried, and his difficulties with translation are well recorded:

Now that there is growing up an American school of psychol-

ogy, which promises to be only second in importance to the Ger-

man, […] it becomes imperatively necessary for us to have at

our disposal a working outfit of technical terms in our own lan-

guage. (Titchener, 1895, p. 78)

[To his own translator] Your difficulty with the vocabulary is

precisely the same difficulty that I had myself nearly thirty years

73 For a rather comprehensive account of what follows, see Danziger, 1980 74 Unfortunately, the text which presumably makes this distinction most clear,

Selbstbeobachtung und innere Wahrnehmung, has not been translated into English.

Here, we must side with Danziger’s translation.

35

ago when I was translating Wundt and Kulpe. At that time there

was no technical English vocabulary either for quantitative ex-

perimental psychology or for psycho-physics. (Titchener, 1922,

p. 1, as cited in Tweney & Yachanin, 1980).

Second, and relatedly, Titchener seems to have believed that students

ought to read texts in their original French or German, and that translations

were necessary only for novices and those newly learning a language (Titche-

ner, 1922, p. 3, as cited in Tweney & Yachanin, 1980). In this way, it is con-

ceivable that Titchener translated Wundt only approximately, with the expec-

tation that anyone in serious study or interest would turn to the original Ger-

man; it is unlikely that Titchener anticipated the scholarship that Boring’s read-

ership would later bring upon Wundt. Finally, it has been noted that Titchener

identified his own account of introspection as stemming more from a British

tradition than from a Wundtian one, and several other differences in the phi-

losophy of mind and psychological approaches have been drawn between

Titchener and Wundt (see Tweney & Yachin, 1980; Blumenthal, 1975; Dan-

ziger, 1980; for further accounts of the differences). Hence, it is conceivable

also that Titchener’s translations were simply influenced by his own philoso-

phies.

Regardless of the actual reason for the errors in translation, and despite

any well-meaning intentions that may underlay them, Titchener’s Wundt ad-

vocates for an account of introspection different from Wundt’s innere

Wahrnehmung, and this mistranslation remains to be corrected. In what fol-

lows, I will briefly survey some critical objections to introspection, and

demonstrate how a careful reading of Wundt is capable of responding to them,

thereby indicating further differences between Wundt’s work and our common

notions of introspection.

II. Wundt and Objections to Introspection

In this section, I will consider two main objections to introspection: (a) that

introspection is unreliable and unscientific, and (b) that the very act of intro-

specting alters the objects of introspection. It is not my intent to demonstrate

that introspection is capable of dispelling these concerns; rather, I intend to

36

demonstrate that apparent instances of introspection in translations of Wundt75

seem at least to suggest a difference between Wundt’s account and the standard

account against which these objections are raised (for a detailed discussion of

the problems of introspection, see commentary by Frith & Lau, 2006, and

Howe, 1991; see also, Lyons, 1986).

II.i The Unreliability of Introspection

Perhaps the most common objection to introspection is that it is unreli-

able: the individual variances make any generalizations impossible, and it is

not clear whether subjects are capable of distinguishing the intended objects

of introspections from hallucinations or fictions.76 Yet, Wundt seems to ac-

count for just these concerns. Indeed, in his Outlines of Psychology (1902), he

presents at length a characterization of psychological reflection as necessarily

experimental:

If we apply these considerations to psychology, it is obvious at

once, from the very nature of its subject-matter, that exact ob-

servation is here possible only in the form of experimental ob-

servation […] In order to investigate with exactness the rise and

progress of these processes, their composition out of various

components, and the interrelations of these components, we

must be able first of all to bring about their beginning at will.

This is possible here, as in all cases, only through experiment,

not through observation. (Wundt, 1902, pp. 24-5; emphasis

original)

Here, and in near passages, Wundt advocates for an exclusively experi-

mental psychology, where the objects of analysis are studied only insofar as

we are able to control for their appearances.77 Indeed, Wundt’s methods were

explicitly experimental, and his laboratory in Leipzig was filled only with the

most up-to-date tools (see, for an overview, Popplestone & McPherson,

75 While it is difficult to confirm, based on the availability of translations, it should seem

that the instances which follow seem to map onto Danzigers’s (1980) above account of

innere Wahrnehmung. 76 This is the main objection raised by Watson (1913) against introspection and toward

behaviorism. It is interesting to note, however, that Watson does not make any appeal to

Wundt, but rather calls Titchener out by name. 77 To go further, from a more internalist perspective, Wundt believed that introspection

qua innere Wahrehmung itself was a more immediate and objective datum than those of

the other sciences, for scientific data were but abstractions from those experiences

presented.

37

1980).78 Indeed, the quality of experimental apparatus was such that Titchener

predicted in 1989 that “A large percentage of students will doubtless continue

to spend a year in Germany, for the sake of acquiring the language and seeing

the German equipments” (Titchener, 1898, p. 330).

But regardless of technical availability, it is clear that Wundt had no

intent to engage in a form of introspection which was unkempt nor which could

not be contrasted between experiments or brought about by will. Indeed, if one

is today inclined to say of Wundt’s program that it was overly subjective, this

inclination would likely be rooted only in ignorance of the technological limi-

tations of the time. Certainly, Wundt professed his best to follow an objective

measure, and to arrange his Leipzig laboratory to this effect; and if or where

he failed at a sufficient degree of objectivity, it is likely a mark of the techno-

logical availability of the time.79

II.ii The Interference of Introspection

Another common objection to introspection is that the act of introspect-

ing alters those contents upon which one introspects: that the act of introspect-

ing involves an atypical attitude to one’s thoughts, and thus results in atypical

data which are not conducive to a study of mental objects at all. Yet, Wundt

seems to recognize and anticipate this. His solution comes in two parts.

First, Wundt makes a qualitative distinction between different aspects

of mental contents, in a way reminiscent of Hume’s content–feeling distinc-

tion:

[…] There are regularly other properties which are immediately

recognized as accompanying processes. These are partly feel-

ings which are characteristic of particular forms of apprehen-

sion and apperception, partly sensations of a somewhat variable

character. [These feelings] vary according to the different con-

ditions under which the entrance takes place. (Wundt, 1902,

236-237)

78 A detailed account of the required minimum capacities of a chronometer for reaction

time studies, for example, can be found in Wundt (1902), pp. 220-222. 79 Indeed, in one anecdote, James Cattell recounts his apprehension to bring to Wundt’s

attention an error in the precision and calibration of a device, which was supposed to be

used to measure reaction times. When notified, Wundt was very cordial and thanked and

cited Cattell for the notice. Shortly after, the lab received a more precise instrument.

(Sokal, 1980; see also Benjamin, 2006, pp. 59-61).

38

First, the new content may force itself on the attention suddenly

and without preparatory affective influence; this we call passive

apperception. [….] Secondly, the new content may be preceded

by the preparatory affective influences mentioned above, and as

a result the attention may be concentrated upon this content

even before it arrives; this we call active apperception. (Ibid,

238; emphasis original)

Then, he maps this notion to how introspection qua innere Wahrehmung

is under experimental control, and can be generated externally:

Passive apperception may be studied by the use of unexpected

impressions, and active, by the use of expected impressions.

(Ibid., 239)

Consequently, Wundt finds that changes in mental contents can be ex-

perimentally controlled for. Hence, concerns of interference can be met and

controlled for in Wundt’s system.

Whether or not Wundt was correct, it is clear that his system was capa-

ble of responding to objections which were raised against introspection. Ac-

cordingly, there seems further reason to believe that Wundt’s psychology was

not intended to be understood as introspection, at least as we consider it today;

nor should it be considered equal to the introspection of Wundt’s time (cf.

James, 1890, p. 135), else those stern objections raised against introspection-

ists by the onset of behaviorism would surely find stake in his claims. At this

point, then, it might be useful to consider alternative translations of Wundt’s

innere Wahrehmung.

III. Wundt and Psychological Phenomenology

A recent paper by Gallagher and Sørensen (2006) attempts to detail the differ-

ences between introspection and phenomenological psychology, in order to

support the methodological validity of phenomenological psychology over in-

trospection. In particular, they distinguish two different forms or procedures

of introspection: (a) weak introspection, in which one gives a verbal report of

one’s experience as a subject, and where the focus is on the experience of a

stimulus; and (b) strong introspection, in which the focus in on one’s own

mental experience, involving a direct examination of one’s own mental pro-

cesses. Furthermore, this strong introspection is distinguished from phenome-

nology: “The phenomenological field of research does not concern private

39

thoughts, but intersubjectively accessible modes of appearance” (Zahavi,

2003, p. 54). That is, phenomenologists are not interested in subjects’ experi-

ences qua those private and subjective experiences; rather, they are interested

in how a structure can be drawn from those experiences which is representative

of common experience. From this, Gallagher and Sørensen conclude that phe-

nomenological psychology is a more justifiable methodological approach to

psychology than introspection, and begin to detail the roles which phenome-

nological psychology may play in psychology as a whole. So where does

Wundt’s work fit in this paradigm?

Certainly, we have seen that Wundt’s work strives for objectivity be-

yond weak introspection, but does it extend beyond strong introspection? Un-

fortunately, those of Wundt’s texts which would be of most use to this exami-

nation are without available translation,80 so those more explicit points will

have to be contained. And, though we may point to the conclusions of his stud-

ies to demonstrate how Wundt attempts to abstract information from individ-

uals into a general psychological science, these will not satisfy our purposes

here: that the ends appear similar says nothing of the means. However, there

seems to be supportive evidence in Wundt’s accounts of folk psychology,

which indicate that his program is indeed interested in the intersubjective

structures which can be derived from—and structured toward—common ex-

perience:

Matured consciousness stands continually in relation to the

mental community in which it has a receptive and an active part.

(Ibid, p. 331)

If we abstract from the knowing subject in our treatment of the

world of experience, that world appears as a manifold of inter-

acting substances; if, on the contrary, we regard the world of

experience as the total content of the experience of the subject

including the subject itself, then the world appears as a manifold

of interrelated occurrences. […] All these terms serve to desig-

nate, not different spheres of experience, but different supple-

mentary points of view in the consideration of an experience

which is presented to us as an absolute unity. (Ibid, pp. 357-8)

[…] History has been conceived as a course of events, which

not only exhibits an orderly sequence from an objective point of

view, but which is also subjectively experienced as a nexus by

80 In particular, Selbstbeobachtung und innere Wahrnehmung.

40

the individuals concerned. In the one case, history is a recon-

struction, on the basis of external observation, of the inner con-

nection of phenomena. (Wundt, 1916, p. 512)

It is difficult to illustrate the specific significance of these passages

without a wealth of context, but the relevance may be made clear with regard

to the language Wundt here employs. Here he talks about connections of phe-

nomena, both within an individual and within a community. In particular, he

talks about the congruence between an objective and subjective position, both

where they act as different perceptions of the same content and how they stand

in relation to a broader mental community. Indeed, we might draw a phenom-

enological psychology from Wundt’s experiments and his attempts to objectify

the internal experience. But we also here find the fragments of his account of

intersubjectivity: the subject as experiencing a world of interrelated occur-

rences. And, it is not a far step to relate this back to Wundt’s experimental

methodology, nor to reflect that he indeed draws out a structure for the com-

mon experience from the subjective experience. Accordingly, it seems that

Wundt’s account is better translated as a phenomenological psychology than

introspection, according to this account.

Finally, this attribution of a phenomenological psychology to Wundt is

not an ahistorical appropriation; the account Gallagher and Sørensen forward

is congruent with the phenomenology of Wundt’s own lifetime. Indeed, Ed-

mund Husserl (1859-1938) is considered the founder of the phenomenology,

on whose works Gallagher and Sørensen (2006) build their account. Moreover,

it is widely known that Husserl and Wundt read and commented upon each

other’s works; Husserl even attended some of Wundt’s lectures when he was

a student at Leipzig in 1876 (Husserl & Moran, 2002, p. xvii; Rollinger, 1999,

p. 15). Furthermore, Wundt was known to be influenced by other philosophers,

such as Kant and Hegel, who have been considered forerunners of phenome-

nology (for example, see Wundt, 1916, p. 520). It is thus not anachronistic to

consider Wundt a phenomenological psychologist (and given the time, perhaps

the first phenomenological psychologist), granted both that his account fits

Gallagher and Sørensen’s (2006) definition, and that he read and communi-

cated with the father of phenomenology, whose account of phenomenology

provided the basis for Gallagher and Sørensen’s own position.

41

Conclusion

It has been shown that the Americanization of Wundt was a mistranslation of

Wundt’s real work; a mistranslation which was caused by Titchener and prop-

agated by Boring, even if neither had the malicious intent to do so. In doing

so, Titchener and Boring presented Wundt as an introspectionist, in a way

which strawmanned Wundt’s original bifurcation of introspection into

Selbstbeobachtung and innere Wahrnehmung. It was shown that Wundt more

accurately seems to fit the contemporary account of phenomenological psy-

chology forwarded by Gallagher and Sørensen (2006), and that such a catego-

rization does not appear ahistorical. This reclassification of Wundt’s work not

only paints him in a better light, against a landscape of concern about the ve-

racity of introspection, but also paves the way for further exploration into his

work, particularly as the fields of neurophenomenology and phenomenological

psychology advance. Accordingly, it seems apt to consider translating Wundt

as a phenomenological psychologist, rather than an introspectionist, and to

take this consideration as a jumping-off point for further studies of his work.

42

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45

Culture and the Nature of Mathematics Caitlin MacLeod

Tracing the history of mathematics is a notably different process than learning

concepts in a mathematics classroom. These ideas are not accessed as one pre-

formed body of knowledge, but rather emerge gradually, developing through

the works of individuals building on each other’s knowledge. And yet, the ex-

tent to which the social context of mathematicians affects mathematical con-

tent is a matter of debate. Ethnomathematics is an area of study quite relevant

to this debate. As a field, ethnomathematics has several areas of focus, and

through the pursuit of these topics aims primarily to improve mathematical

education and combat eurocentrism. In general, however, ethnomathematics

examines the relationship between culture and mathematical knowledge.81 As

a result, ethnomathematical findings may be discomfiting as they challenge

assumptions regarding the nature of mathematics and mathematical

knowledge.

This paper is separated into three main sections. In the first, I will offer

an overview of what ethnomathematics is and what it aims to accomplish. In

the second portion of the paper, I will discuss the implications of ethnomathe-

matical research on our understanding of the nature of mathematics. Finally,

in the third and final section, I will examine how concepts introduced in the

earlier parts apply to particular instances in the history of mathematics. In do-

ing so, I intend to demonstrate how, in addition to its primary goals, ethno-

mathematics seriously challenges our common conceptions of the nature of

mathematics.

To begin, I will offer a brief overview of the central aims of ethnomath-

ematics, in order to demonstrate how these goals are distinct from, and yet lend

themselves to furthering our understanding of mathematical knowledge. In

general, ethnomathematics studies the relationship between culture and math-

ematics in order to discover how mathematical knowledge is acquired and

transmitted, thus contributing to a more nuanced understanding of the

knowledge itself. A significant factor in understanding the relationship be-

tween mathematics and culture is knowing how the body of knowledge of a

81 M. Ascher, Ethnnomathematics: A Multicultural View of Mathematical Ideas (Belmont,

California: Wadsworth, 1991), 186.

46

given culture is perceived by outsiders. This, in turn, is shaped by the ways

individuals from one culture may conceive of or characterize the culture of

another.

Given the colonial and imperial histories of many European countries,

it is hardly surprising that the European gaze is the lens through which many

of the world’s cultures are perceived. Within this tradition of colonization is

an undercurrent of undeniably racist assumptions.82 These assumptions, which

view all other groups as inherently inferior in terms of intelligence, cultural

structures and more, work to position Western culture, practices and beliefs as

the default. This has supported Western knowledge as universal truth, with dis-

agreement from other groups being taken as signs of inferiority.83

A significant role in this conception of other cultures as inferior has

been played by perceptions of how cultures develop. In tracing their own his-

tory, European thinkers have constructed an archetype of the linear progression

of society, with contemporary Western society as the pinnacle of civilization.

This linear progression from era to era is taken as the inevitable process re-

gardless of environmental factors.84 As Europeans came into contact with other

cultures, a group’s place in this evolutionary motif became an identifier of how

developed they were.85 Alongside this model of development, societies were

characterized as stages of human life, with ‘primitive’ groups assigned the at-

tributes of children.86 In this way, it was argued that individuals from these

other cultures had only the faculties and abilities of children, and so the devel-

opment of their knowledge was constrained by their limited ability to reason.87

These Eurocentric assumptions influence the way the history of mathe-

matics is presented. The path to contemporary mathematical knowledge is per-

ceived as a progression of ideas within the European tradition, from the An-

cient Mediterranean through the universities of Europe.88 As a response to this,

82 James Stigler and Ronald Barnes, “Culture and Mathematics Learning,” Review of

Research in Education, Vol. 15 (1989), 253. 83 Ascher, Ethnomathematics, 191. 84 Ascher, Ethnomathematics, 189. 85 Stigler and Barnes, “Culture and Mathematics Learning,” 255. 86 Ascher, Ethnomathematics, 191. 87 Stigler and Barnes, “Culture and Mathematics Learning,” 256.

88 Paulus Gerdes, “Survey of Current Work in Ethnomathematics,” in Ethnomathematics:

Challenging Eurocentrism in Mathematics Education, eds. Arthur B. Powell and

Marilyn Frankenstein (Albany, New York: State University of New York Press, 1997),

337

47

ethnomathematics aims to present the mathematics of other, non-Western so-

cieties in the cultural contexts in which they were developed.89 This serves to

break a feedback cycle, where the mathematical knowledge of other cultures

continues to be treated as trivial because it has been dismissed as trivial in the

past. By better understanding the cultural context from which mathematical

understanding emerges, it becomes possible to do justice to the knowledge of

these other cultures as valuable pieces of knowledge in their own right.90

The other focus of ethnomathematics is mathematical education. This,

too, bears eurocentrism in mind as it seeks how best to introduce learners from

different cultural backgrounds to mathematics.91 The issue to be confronted is

that the mathematics taught in formal education is westernised, regardless of

the social and institutional context in which it is being learned.92 In the case of

Navajo Indians, for example, who view space and time in terms of dynamic

and ongoing processes, Pixten and Francois suggest approaching geometry

with consideration for the Navajo worldview, rather than by insisting on a Eu-

clidian view of the world.93 In this way, ethnomathematics can guide the use

of cultural context to enable students to grasp ‘universal’ mathematical con-

cepts by presenting them through the use of the cultural elements to which the

concepts are connected.

These two aims of ethnomathematics both pertain to cultural biases

within the study of mathematics. What is more, they investigate the ways in

which mathematical knowledge is developed, perceived and transmitted.94 As

a result, the insights gained in their discussion of Eurocentrism, cultural con-

text, and mathematical knowledge also reflect back on Western mathematics.

As ethnomathematics dislodges the mathematics of the European tradition as

both the default and the objective universal, it becomes not only reasonable

89 Ascher, Ethnomathematics, 191.

90 Stigler and Barnes, “Culture and Mathematics Learning,” 256.

91 Rik Pixton and Karen François, “Ethnomathematics in Practice,” in Philosophical

Dimensions in Mathematics Education, eds. Karen François and Jean Paul Van

Bendegem (New York: Springer, 2007), 214.

92 Gerdes, “Survey of Current Work in Ethnomathematics,” 337.

93 “Ethnomathematics in Practice.”

94 Carmen Batanero and Carmen Díaz, “The Meaning and Understanding of

Mathematics,” in Philosophical Dimensions in Mathematics Education, eds. Karen

François and Jean Paul Van Bendegem (New York: Springer, 2007), 107.

48

but possible to consider the effects of culture on the contents of Western math-

ematics.95 Furthermore, by positioning Western mathematics as one of several

possible alternatives, ethnomathematics begins to challenge status of West-

ern,mathematics as a collection of objective and necessary facts, concerning

entities which exist independently of the humans who discover them.

This question of how we obtain mathematical knowledge is central to

understanding what this knowledge is about. In this next section, I will discuss

how the suggestions made by ethnomathematics relate to several perspectives

on the nature of mathematics. These views originate in philosophical discus-

sion, but are essentially connected to historical considerations. Indeed, without

an understanding of what mathematics is, it would be impossible to discern

which developments in human thought can be characterized as stages in the

progress of mathematics.96 Here, the fundamental question to be addressed is

whether mathematical facts are discovered or created.97 Rather than offering a

definitive answer, ethnomathematics puts forward a more nuanced understand-

ing of the issues at hand by suggesting arguments for each side of this debate.

When one claims that mathematical knowledge is something to be dis-

covered, one is suggesting that mathematical entities exist independently of

humans, and that mathematical statements have objective truth.98 This view

can be tied to realism, which posits that there is a distinct reality which would

exist whether or not there were humans to conceive of it.99 Platonism can also

be tied to this perspective, as it alleges that mathematical objects are distinct

entities which exist outside of any particular instance of them.100

The views of Platonism and realism are consistent with the ethnomath-

ematical finding that certain elements of mathematics are universal to all cul-

tures. It has been found that counting is intrinsic to being human, and so num-

bers are universal to human culture, despite the many forms by which they

95 Thomas E. Gilsdorf, Introduction to Cultural Mathematics: With Case Studies in the

Otomies and Incas (Hoboken, New Jersey: John Wiley and Sons, 2012), 12. 96 Joong Fang and Kaoru Takayama, Sociology of Mathematics and Mathematicians

(Hauppage, New York: Paida Press), 191-198. 97 Ascher, Ethnomathematics, 7. 98 Michele Friend, Introducing Philosophy of Mathematics (Stocksfield: Acumen, 2007),

23. 99 Ibid., 26. 100 James R. Brown, Philosophy of Mathematics: Introduction to the World of proofs and

Pictures (Oxford: Routledge, 1999), 12.

49

may be represented.101 The fact that numbers are, in this way, equally accessi-

ble regardless of environment or social structure supports the view that these,

and other similarly accessible mathematical concepts, are independent entities

which humans can discover, rather than being the product of a particular set of

circumstances.

The fact remains, however, that culture and circumstances do affect the

contents of a society’s mathematical knowledge.102 For example, when we

consider the cases of European and Sioux representations of space, we see the

effect of external beliefs in shaping mathematical concepts. The Sioux con-

ceive of space – and by extension, geometry – in terms of circles because in

the Sioux worldview, the ‘Power of the World’ works in circles, reflected in

the roundness of the sky, the whirls of the wind and the cycle of the seasons.

By contrast, the intuitiveness of Euclidian space for Westerners reflects the

European perspective that nature and human activity alike work along straight

lines which optimize efficiency.103 Here, we see evidence for the view that

mathematical knowledge is constructed. In these cases, the mathematics is de-

veloped in response to a particular set of values, and so different mathematics

develop in different contexts.

With respect to Western mathematics, the conclusion that at least some

of mathematics is merely contingent on circumstance suggests that some of

our mathematical knowledge could have been different from how it is. In this

way, cases can be made for two different views: constructivism and conven-

tionalism. Constructivism says that knowledge is created piece by piece as a

product of our minds. Going even further, conventionalism says knowledge is

merely a set of agreed upon behaviours with no deeper meaning. The relation-

ship with culture could suggest that mathematical knowledge has arisen out of

the existing contents of people’s minds—contents which have been deter-

mined by cultural context.104 This would support the conventionalist view. On

the other hand, the fact that mathematical concepts can be shown to have de-

veloped out of social needs also lends credence to the constructivist conception

101 Ascher, Ethnomathematics, 6. 102 Brian Martin, “Mathematics and Social Interests,” in Ethnomathematics: Challenging

Eurocentrism in Mathematics Education, eds. Arthur B. Powell and Marilyn

Frankenstein (Albany: State University of New York Press, 1997), 155. 103 Ascher, Ethnomathematics, 124-125. 104 Friend, Introducing Philosophy of Mathematics, 101.

50

of mathematics as a set of rules agreed upon because they are useful tools built

on a useful logical foundation.105

These philosophical considerations give a conceptual overview of the

perspectives in the on-going debate regarding the nature of mathematics. In

the third section of this paper, I will attempt to illustrate the conflict with the

use of examples which demonstrate tension between Platonist intuitions and

the conventionalist and constructivist implications of the cultural study of the

issue.

There are numerous cases from around the world and throughout history

which illustrate this relationship between mathematics and culture. In an effort

to restrict the scope of this paper and to retain its focus, the examples I will

discuss in this final section will be drawn from a limited range of time and

place: Babylonian and Greek mathematics in Antiquity. This focus on the de-

velopment of Western mathematics throughout Antiquity is a deliberate at-

tempt to concentrate the questions raised by ethnomathematical considerations

on the roots of current mathematical thought. In this way, I hope to make clear

how ethnomathematics may be troubling for contemporary mathematical stud-

ies.

The Babylonian example which I will discuss is that of Plimpton 322,

a cuneiform tablet inscribed with a table of Pythagorean triplets (sets of three

values which satisfy the relation a2+b2=c2). Robson describes the process of

discerning possible interpretations of the tablet as using, “language, history

and archaeology, social context, as well as the network of mathematical con-

cepts within which the article was created.”106 As one of several thousand

known documents from ancient Iraq, Plimpton 322 relates to other artefacts in

three important ways. First, it can be situated amongst similar mathematical

tables. Second, connections can be drawn to other instances where the words

of Plimpton 322’s column headings were used. Third, Plimpton 322 is one of

many tablets found in the same area.107 These similarities to other artefacts

indicate, among other things, the influence of scribal training, and the use of

certain techniques developed by palace and temple administrators in the

105 Brown, Philosophy of Mathematics, 13. 106 Eleanor Robson, “Words and Pictures: New Light on Plimpton 322,” American

Mathematical Monthly, Vol. 109, No. 2 (2001), 106. 107 Ibid., 109.

51

area.108 This knowledge of the tablet’s cultural background informs Robson’s

analysis of its mathematical significance.

The cultural connections seen in the study of Plimpton 322 demonstrate

that prosaic applications may have impact on both the content and the form of

mathematical practices in a given context. Practical concerns may guide the

progress of mathematics in both of these ways. In terms of method, traditions

developed for social uses may become the dominant ways of performing math-

ematical operations which may in turn impact which forms and directions of

inquiry are possible. The contents may be directly affected as social factors

determine where mathematical work is concentrated. As those areas which are

relevant to immediate needs are emphasised, they will be the points from

which new pathways to mathematical knowledge stem: consider, for example,

the advances in statistics and probability theory brought about by actuarial

problems.

As a final example, take the rise of abstraction in Ancient Greek math-

ematics, a characteristic that is still prized today. Farrington describes this as

part of an educational reform belonging to a political and social program. Dur-

ing the rise of democracy in Greek society, the reason of the Citizens – a rigidly

delineated group of men – was prized, while the contrasting slave class was

viewed with contempt.109 The value placed on logic and reason led the emer-

gence of proofs as an important tool, while the disdain for the slave class

caused practical or useful knowledge to be viewed as ‘vulgar’.110 In this way,

the emergence of ‘acceptable’ mathematical methods took parallel changes in

the Greek social structure.

The reflection of cultural values in the practice of mathematics is clear

in this case. Social influence is not limited to determining which elements of

an independent mathematical reality are revealed. Instead, cultural values

shape the very contents of mathematical knowledge as they dictate what

should and should not count as ‘mathematics’.111 Indeed, since Ancient Greek

mathematics is a direct ancestor of the mathematical knowledge of today, even

more so than that of Ancient Babylon, these values are built into the core of

the Western mathematical tradition. The presence of subjective values in the

108 Ibid., 118. 109 B. Farrington, “The Rise of Abstract Science Among the Greeks,” Centaurus, Vol. 3

(1953), 36-37. 110 Ibid., 37; Fang and Takayama, Sociology of Mathematics and Mathematicians, 198. 111 Fang and Takayama, Sociology of Mathematics and Mathematicians, 191.

52

content of mathematics is very much a challenge to the assumption that math-

ematical developments are objective and rational.112

Understanding the ways that environmental concerns shape the trajec-

tory of mathematics give rise to the possibility that current mathematical

knowledge could have come to exist with fundamental differences.113 If, over

the course of the history of Western mathematics, different areas had been em-

phasized or other techniques developed, the result could have been a very dif-

ferent set of mathematical truths. This suggests a view far more in keeping

with the conventionalist or constructivist perspectives that knowledge is built

up and contingent, rather than to the seemingly intuitive view that mathemati-

cal knowledge exists independently and is uncovered logically and ration-

ally.114

With this in mind, it is important to make note of Wittgenstein’s distinc-

tion between mathematics as a system and the practice of doing mathemat-

ics.115 With respect to the question of discovery versus creation, or objectivity

versus subjectivity, it is both conceivable and consistent with ethnomathemat-

ical findings to suppose that it may only be the practice of mathematics that is

significantly influenced by cultural factors. Meanwhile, although these cultur-

ally-determined customs and methods may affect which parts of an objective,

Platonist reality are discovered, the mathematical contents themselves could

be entities whose existence is independent from culture.116

Ethnomathematics is a relatively recent development in the study of

mathematics. Despite, and perhaps because of, its sometimes startling com-

mentary on central tenets of Western mathematical thought, ethnomathemati-

cal considerations continue to be mainly restricted to promoting education and

combating eurocentrism. Nevertheless, there are many ways in which ethno-

mathematics adds another dimension to our understanding of mathematics in

general. By considering the effects of culture on mathematical knowledge, we

are forced to question where that knowledge comes from, how it is acquired,

and whether mathematical entities exist independently and universally. What

112 Martin, “Ethnomathematics and Social Interests,” 155. 113 Ibid., 158. 114 Van Kerkhove, Bart, “A Place for Education in the Contemporary Philosophy of

Mathematics,” in Philosophical Dimensions in Mathematics Education, eds. Karen

François and Jean Paul Van Bendegem (New York: Springer, 2007), 184. 115 Stigler and Barnes, “Culture and Mathematics Learning,” 257. 116 Brown, Philosophy of Mathematics, 12.

53

is more, such questions are brought out of the realm of philosophy – philoso-

phy of mathematics, epistemology, and metaphysics – to become matters of

history and social science. As a result, the development of ethnomathematics

can serve to present well-established issues in the discussion of mathematics

in a new light, and so offer new insights into the nature of mathematics. Even

without providing concrete or definitive answers to this question, these new

perspectives can prove enriching for our understanding of mathematics and its

history.

54

References

Asher, M. Ethnomathematics: A Multicultural View of Mathematical Ideas.

Belmont, California: Wadsworth, 1991.

Batanero, Carmen and Díaz, Carmen. “The Meaning and Understanding of

Mathematics.” Philosophical Dimensions in Mathematics Educa-

tion, edited by Karen François and Jean Paul Van Bendegem. New

York: Springer, 2007.

Brown, James R. Philosophy of Mathematics: An Introduction to the World of

Proofs and Pictures. Oxford: Routledge, 1999.

Dunham, William. Journey Through Genius. New York: John Wiley and

Sons, 1990.

Fang, Joon and Takayama, Kaoru. Sociology of Mathematics and Mathemati-

cians. Hauppage, New York: Paida Press, 1975.

Farrington, B. “The Rise of Abstract Science Among the Greeks.” Centaurus,

Vol. 3 (1953): 32-39.

Friend, Michele. Introducing Philosophy of Mathematics. Stocksfield: Acu-

men, 2007.

Gerdes, Paulus. “Survey of Current Work in Ethnomathematics” Ethnomath-

ematics: Challenging Eurocentrism in Mathematics Education, ed-

ited by Arthur B. Powell and Marilyn Frankenstein. Albany, New

York: State University of New York, 1997.

Gilsdorf, Thomas E. Introduction to Cultural Mathematics: With Case Stud-

ies in the Otomies and Incas. Hoboken, New Jersey: John Wiley and

Sons, 2012.

Martin, Brian. “Mathematics and Social Interests.” Ethnomathematics: Chal-

lenging Eurocentrism in Mathematics Education, edited by Arthur

B. Powell and Marilyn Frankenstein. Albany, New York: State Uni-

versity of New York Press, 1997.

Pixton, Rik and François, Karen. “Ethnomathematics in Practice.” Philo-

sophical Dimensions in Mathematics Education, edited by Karen

François and Jean Paul Van Bendegem. New York: Springer, 2007.

55

Robson, Eleanor. “Words and Pictures: New Light on Plimpton 322.” Ameri-

can Mathematical Monthly, Vol. 109, No. 2 (2001): 105-120.

Stigler, James and Barnes, Ronald. Culture and Mathematics Learning.” Re-

view of Research in Education, Vol. 15 (1989): 253-306.

Van Kerkhove, Bart. “A Place for Education in the Contemporary Philosophy

of Mathematics,” Philosophical Dimensions in Mathematics Educa-

tion, edited by Karen François and Jean Paul Van Bendegem. New

York: Springer, 2007.

56

Holism Cannot Save Structuralism Kyle Da Silva

Structuralism is a theory of mathematics that views mathematical entities as

essentially structures or patterns. Mathematical entities are to be viewed as

structures both epistemologically and ontologically. To get a clearer idea of

what this means we can use some common mathematical objects; the natural

number sequence (1,2,3,4,5 . . .). Take a natural number; for example 4. Ac-

cording to structuralism the number 4 will not be an independent object. 4 will

only be a place in a structure. In this case it will be the natural number structure

although it could also be others. There is nothing to the number 4 other than

the relations which it bears to the other spots in the structure. That is the on-

tology of the number 4. Epistemologically 4 is not to be viewed as an object

either. If we have 4 eggs arranged together on a table in front of us, we do not

see an instantiation of 4 but of the 4-structure. The 4-structure is a sub-structure

of the natural numbers. These may seem like small differences at first but they

have important consequences. A system is a concrete pattern between objects;

as with the four eggs above. A structure is the abstract object corresponding to

a system.

To understand the main epistemological problem we first need to artic-

ulate Benacerraf’s dilemma. Benacerraf’s dilemma as is currently understood

was articulated in Paul Benacerraf’s paper “Mathematical Truth”.117 The di-

lemma is about the theories of philosophy of mathematics. On one hand if the

theory is anti-realist (mathematical sentences do not refer to anything) then we

have to come up with a semantic theory different from the regular semantic

theory of science and common language. But anti-realist theories have an easy

time explaining how we come to know about mathematics. On the other hand

are realist theories, which have a straight forward semantics. However since

mathematical entities are abstract on realist accounts, they have an epistemo-

logical problem. The problem is that since mathematical entities are abstract

we have to be able explain how flesh and blood mathematicians can have ac-

cess to mathematical entities that are causally inert. Moreover since the prob-

lems are tied to features of realism and anti-realism it seems as if no matter

what theory you choose, there will be a crucial flaw.

117 Paul Benacerraf “Mathematical Truth”, Journal of Philosophy, 70(1973) 661-679.

57

For structuralism to be an independent epistemological theory of phi-

losophy of mathematics it needs to have a story for the epistemology of math-

ematics separate from other theories. In “Can Ante Rem Structuralism Solve

the Access Problem?” Fraser Macbride argues that it does not have such a

story. Stewart Shapiro responds in a paper “Epistemology of Mathematics:

What Are The Questions? What Count As Answers?”. Part of Shapiro’s re-

sponse uses “entitlement”, specifically Crispin Wright’s variant. “Entitlement”

is a concept used in defense against sceptics, we have beliefs that we are enti-

tled to even if we cannot justify them. Shapiro argues that we can use the con-

cept of entitlement on mathematical structures, and then find justification post

hoc when we find that it works. In this paper, I will explicate Shapiro’s enti-

tlement argument. Then I will argue that Shapiro cannot utilize entitlement

while remaining a holist, because conditions for entitlement are at odds with

holistic justification. Furthermore, I will argue that once entitlement is taken

out of the picture, Shapiro’s argument amounts to an indispensability argu-

ment. If this is the case, then Shapiro’s structuralism is merely an add-on, ra-

ther than an independent theory.

I will very briefly explain Shapiro’s structuralism. Shapiro is a structur-

alist with regards to mathematics. This is the view that mathematical entities

are essentially structures. Mathematical objects are identified with places in

structures. For example, the number one is identified with a place in the natural

number structure. Shapiro’s particular view of structuralism is called ante rem

structuralism. Ante rem structuralism says that structures are abstract, and in-

dependent of the systems in the world which instantiate them.118 This brand of

structuralism is subject to the access problem like any other philosophical the-

ory of mathematics, but in particular has trouble because the entities which it

identifies mathematical objects with are abstract and independent. Shapiro

must explain not only how we come to beliefs about these mathematical enti-

ties, but also how we have warrant for these beliefs.119

Shapiro offers an explanation, consisting of three stages: abstraction,

projection, and description. Abstraction is the movement from token to type.

From an instantiated system we move from the objects in the system, to the

structure which they are instantiating. For example, seeing three objects then

118 Fraser Macbride. “Can Ante Rem Structuralism Solve The Access Problem?”

Philosophical Quarterly Vol. 58, No. 230 (January 2008). 155-156. 119 Ibid. 157.

58

abstracting the 3-structure, which is instantiated of any 3 objects. Projection is

the move from this simple abstraction to recognizing that multiple of these

smaller structures (1-structure, 2-structure, 3-structure…) are part of an over-

arching structure. Description is the final part, where we move from projec-

tions, which do not give us knowledge of all the entities required for mathe-

matics, to descriptions which do. If we can describe a structure, and the de-

scription is coherent, then we are guaranteed the existence of the structures.120

While this explanation is short and inadequate as a full account of Shapiro’s

proposed epistemology, it is sufficient for the present discussion. Fraser Mac-

Bride’s critique, to which Shapiro is responding, grants abstraction but takes

issue with projection. MacBride’s issues come with the transition between:

1) Particular knowledge that a given structure has a successor

distinct from it

2) General knowledge that all structures, given or otherwise,

have their own distinct successors.121

To be able to justify this move, MacBride argues that we have to be able

to deduce 2) from 1).122 The argument is that to be able to deduce 2) from 1)

we would need to grasp truths which are more general than 2). The point of

the epistemological project is to explain how we go from particular to general,

so such a deduction would not be satisfactory.123 While MacBride offers more

arguments against Shapiro’s epistemology, it is this argument which Shapiro

is primarily responding to in the section of his paper in which I am interested.

I will now move on to an explication of Shapiro’s response to Mac-

Bride’s argument. In section II Shapiro begins by urging the use of holism to

solve the problems that MacBride has put forward.124 Roughly, holism is the

view that our beliefs are not to be justified on the basis of other more basic

beliefs. Instead holists argue that our beliefs are justified based off of our entire

belief system for each belief. He goes on to use the example of Zermelo and

120 Ibid. 157-158. 121 Ibid. 160. 122 Ibid. 160. 123 Ibid. 160. 124 Stewart Shapiro. “Epistemology of Mathematics: What are the Questions? What Count

as Answers?” The Philosophical Quarterly. Vol. 61, No. 242 (January 2011). 140.

59

the axiom of choice to explain how such a justification would work.125 The

idea being that the axiom of choice was justified not by proof, but by being

essential to other proofs.126 Similarly, the justification of 2) comes not from a

deduction from 1) but from 2) being important in “our intellectual enter-

prise”.127 Shapiro does not rest here however; he goes on in section III to pro-

vide a more detailed picture of this justification using the notion of entitlement.

Specifically, Shapiro utilizes Crispin Wright’s notion of entitlement. Shapiro

asserts that while Wright’s notion is from within a foundationalist framework,

he can adopt it for holism.128 Entitlement is employed by Wright to combat

material world scepticism. The main idea being that there are certain presup-

positions which we are entitled to; we cannot give them warrant but we are

nevertheless entitled to them. Thus when we are faced with a sceptical argu-

ment, we do not need to give up our beliefs about the material world.129 Wright

gives two conditions for a presupposition, P, to be warranted:

(i) We have no sufficient reason to believe that P is untrue.

(ii) The attempt to justify P would involve further presuppo-

sitions in turn of no more secure a prior standing . . . and

so on without limit; so that someone pursuing the relevant

enquiry who accepted that there is nevertheless an onus to

justify P would implicitly undertake a commitment to an

infinite regress of justificatory projects, each concerned to

vindicate the presuppositions of its predecessor.130

Shapiro argues that MacBride’s argument for why 2) cannot be deduced

from 1) satisfies (ii) of the entitlement conditions.131 As stated above, Mac-

Bride argues that 2) cannot be deduced from 1) without recourse to principles

125 Ibid. 141. 126 While a footnote is insufficient in many respects to explain the axiom of choice I can

give its salient features. The axiom of choice is an axiom in set theory which has some

counterintuitive consequences which had led to people doubting its truth. However the

axiom of choice allows for a lot of mathematical work to be done and some important

proofs were done using the axiom of choice which justified place as an axiom. 127 Ibid. 142. 128 Ibid. 143. 129 Crispin Wright. “Warrant for Nothing (and Foundations for Free)?” Proceedings of the

Aristotelian Society, Supp. Vol. 78 (2004). 191. 130 Ibid. 191-192. 131 Shapiro, 145.

60

more general than 2). Thus attempts at justification only lead us to “presuppo-

sitions in turn of no more secure a prior standing”. Shapiro goes on to say that

MacBride did not argue for condition (i) not being met.132 Therefore we can

conclude that 2) is something to which we are entitled.

However Shapiro does not rest with 2) as merely entitled. Shapiro states

that in the anti-foundationalist (holistic) framework, P can evolve from an en-

titlement to an outright belief.133 Shapiro does not give detailed account of how

this works, but gestures to the discussion of Zermelo and the Axiom of Choice

earlier in his paper.134 The idea seems to be that we can accept an entitlement

into our holistic web, and when we see the importance it has within the web,

it evolves from entitlement to outright belief. The overall argument is not

wholly different from the one given when discussing Zermelo and the Axiom

of Choice; it has been given a layer. What entitlement does for Shapiro is ex-

plain why we can accept something like 2) in any form in the first place. Once

we have, we can provide the justification post-hoc.

I will now argue that the notion of entitlement Crispin Wright expounds

and the holism which Shapiro utilizes in his argument against MacBride are at

odds. Shapiro must give up either entitlement or holism, in either case his ar-

gument against MacBride does not work. The argument will focus on Wright’s

second condition of entitlement and it’s applicability to holism. The second

condition is a test for entitlement. How it works is that we try to justify the

potential entitlement, if the attempt to justify only leads us to less certainty and

condition (i) obtains then we have an entitlement. In the foundationalist set-

ting, this works. Justification for the foundationalist proceeds by inference

from more basic beliefs, terminating in the basic belief. If we were to attempt

to justify but the inference could only be made from a less certain or secure

belief, and the following inference from an even less certain one, we would

not be justifying at all.

In adapting entitlement to his holistic framework, Shapiro tweaks some

aspects of Crispin Wright’s notion of entitlement. He argues against a sharp

distinction between beliefs and entitlements on the grounds that holism does

not recognize such sharp divisions; the contents of the holistic web are in

132 Ibid. 145. 133 Ibid. 146. 134 Ibid. 146.

61

flux.135 He also argues that entitlement can extend to matters of ontology,

where Wright claims it cannot.136 Whether or not Shapiro is correct to make

these tweaks is not at issue here, this is merely to point out that Shapiro has

made tweaks to accommodate a different theory of justification. However,

Shapiro does not tweak Wright’s conditions for entitlement. He takes them at

face value. Condition (i) seems to be neutral to either a holistic or foundation-

alist theory of justification so it is not at issue. Condition (ii) is not, it is based

on an attempt to justify. If entitlement is to accommodate holism, then Shapiro

must be able to utilize condition (ii) using holistic justification.

While Shapiro does not give us a full account of holism, there is enough

of an account of what justification is in this holistic setting to see if condition

(ii) works within it. In discussing Zermelo and the Axiom of Choice, Shapiro

states that any proposed axiom “must pay its dues by playing a role in the

systematization of an established and successful practice.”137 Commenting on

the argument at hand shortly after Shapiro claims “We do not prove the prop-

osition, from self-evident premises; instead, we recognize its role in our intel-

lectual enterprise”.138 And in a brief description of holism Shapiro states that

“a working hypothesis, or even a blind guess … can become an established

belief or known fact if it proves fruitful, serving a central essential role in a

successful system”.139 From these remarks on holism, we can extrapolate quite

a bit about the type of holistic justification with which Shapiro is working.

Justification is conferred by playing a role in a system which is established and

successful. It is not conferred via inference from premises as in foundational-

ism.

Returning to condition (ii), I wish to examine the part which stipulates

that justification of P must involve presuppositions of a less secure standing

than P. If a holist is to attempt to justify something then what happens, as per

Shapiro’s remarks, is the belief/presupposition/blind guess in question is put

into a role in the holistic systematization. If it can fill the role and is fruitful in

the system, it is justified. If it does not, then it is rejected. So for a proposition

to be entitled, an attempt to justify would need to lead to using propositions

which are of a less secure standing than the proposition in question. However,

135 Ibid. 145-146. 136 Ibid. 146. 137 Ibid. 141. 138 Ibid. 142. 139 Ibid. 140.

62

in the holistic perspective the proposition is placed within a systematization

which is “established and successful”.140 Therefore any holistic justification is

going to involve propositions which are of a more secure standing than the

proposition in question. This is because to justify just is to put the proposition

to be justified in a system with more secure beliefs/knowledge/facts/etc. Any

attempt to justify a proposition will lead either to its justification or to its re-

jection. It either plays the role fruitfully or it does not. Shapiro uses entitlement

to get 2) into the position to be justified, but entitlement for the holist requires

that 2) be put into that position anyway. If we have gone far enough to ask if

2) is entitled or not, we have already gone beyond the point where it is either

justified or it is not. Therefore the holist cannot use this notion of entitlement.

Shapiro claimed that entitlement, albeit reworked to fit holism, plays a

key role in his picture.141 However, given the argument we might think that in

fact entitlement is not that key, since in the process of entitling 2), we have

justified it. What entitlement did was give Shapiro a non-arbitrary reason for

accepting 2) into his holistic system. Given the holistic framework, one might

argue that it doesn’t matter. As long as we get to the justification, the route

does not matter. I will now argue that if we rest with the argument as it stood

in section II (the holism without entitlement) then the argument just is the in-

dispensability argument.

The indispensability argument is an argument made for mathematical

realism based on the relationship between mathematics and science. The rela-

tionship seems to be such that we could not account for our scientific

knowledge without mathematics. Thus we can argue that mathematics is indis-

pensable to science, and in so far as we are committed to science we are com-

mitted to mathematics also being true.142 The most famous example of this

argument comes from W.V.O. Quine and Hilary Putnam, which is recreated on

the Stanford Encyclopedia of Philosophy as:

(P1) We ought to have ontological commitment to all and only the enti-

ties that are indispensable to our best scientific theories.

(P2) Mathematical entities are indispensable to our best scientific theo-

ries.

140 See note 21. 141 Ibid. 146. 142 Mark Colyvan. “Indispensability Arguments in the Philosophy of Mathematics” The

Stanford Encyclopedia of Philosophy (Fall 2014 Edition), Edward N. Zalta (ed.).

Accessed Dec. 1st 2014.

63

(C) We ought to have ontological commitment to mathematical enti-

ties.143

The argument’s conclusion is ontological in nature but the argument

itself is a justification for mathematical realism. Shapiro’s argument for justi-

fying 2) is not exactly the argument stated above. He does not explicitly state

(P1) and his conclusion does not directly deal with ontological commitment.

He does however make a claim very similar to (P2). A proposition gains justi-

fication “if it proves fruitful, serving a central essential role in a successful

system”144 If a proposition plays an essential and central role in a holistic sys-

tematization then it would be indispensable to that systematization based

merely off it being essential. While an analogue of (P1) is not stated by

Shapiro, we can find one in section II, particularly from the line just quoted. If

we construe (P1) as about justification, then we get:

(P1*) We ought to adopt all and only the propositions which are

indispensible to our holistic systematization.

This seems too strong and is not actually reflective of the claims Shapiro

has made regarding holism. However we can change it slightly to get:

(P1**) We ought to adopt all the propositions which are indis-

pensible to our holistic systematization.

We cannot say that only these propositions are justified, but Shapiro’s

remarks imply at least that indispensability is a sufficient condition for a prop-

osition being justified. However, (P1**) is enough to make Shapiro’s argument

an indispensability argument since what is justifying 2) is its indispensability

from the holistic systematization.

Indispensability is a problem for Shapiro for two main reasons. The

most simple is that Shapiro’s stated view is not indispensability. The second

and more damaging reason is that indispensability does not require structural-

ism. Indispensability does not say anything about mathematical entities other

than that they exist (or at the very least that we should be committed to their

existence). If indispensability is to be an argument for structuralism, then we

need specific arguments as to why structures are indispensable but traditional

mathematical objects are not. Furthermore, if we take Shapiro’s argument as

143 Ibid. 144 Shapiro, 140.

64

an indispensability argument then his stratified epistemology is doing no work.

It might be a description of how mathematical novices come to beliefs about

mathematics, but it plays no role in the justification of our knowledge of math-

ematics. At the very least, it seems if we take the above arguments seriously

then Shapiro has to do a lot of reworking with his theory.

Entitlement is important to Shapiro’s theory because while it still uses

a holistic argument as justification, it at least uses the stratified epistemology

to get to entitlement. Insofar as we take structuralism to be necessary to the

stratified epistemology, then Shapiro’s theory is essentially structuralist. If we

take away entitlement then we are left with an argument that is merely the

indispensability argument, which does not require structuralism. We could still

add structuralism on to the indispensability argument, but it would no longer

be a genuine response to the Benacerraf dilemma.

65

References

Colyvan, Mark. “Indispensability Arguments in the Philosophy of Mathemat-

ics” The Stanford Encyclopedia of Philosophy (Fall 2014 Edition),

Edward N. Zalta (ed.). Accessed Dec. 1, 2014.

MacBride, Fraser. “Can Ante Rem Structuralism Solve The Access Prob-

lem?” Philosophical Quarterly Vol. 58, No. 230 (January 2008).

155-164.

Shapiro, Stewart. “Epistemology of Mathematics: What are the Questions?

What Count as Answers?” The Philosophical Quarterly. Vol. 61,

No. 242 (January 2011). 130-150.

Wright, Crispin. “Warrant for Nothing (and Foundations for Free)?” Proceed-

ings of the Aristotelian Society, Supp. Vol. 78 (2004). 167-212.

66

Einstein, Bohr and the Vienna Circle Robert Wesley

It is well accepted that the philosophy of the Vienna Circle had a significant

influence on physicists and the development of physical theories in the early

20th century. This is no less true for the two most prolific physicists of the era:

Neils Bohr and Albert Einstein. While many other physicists were simply in-

fluenced by logical positivism as a movement, both Einstein and Bohr inter-

acted frequently and intimately not only with core members of the Vienna Cir-

cle itself, but also many of its periphery allies. The influence of logical posi-

tivism on these two physicists is then best explained in terms of intimate inter-

actions between particular people, that is, actual dialogues between individuals

rather than as reactions to ideological movements.

Einstein and Bohr had different reactions to their positivist colleagues.

This is nowhere better illustrated than in their famous 1935 EPR (Short for

Einstein-Podolsky-Rosen) debate. This essay will be comprised of four main

sections. The first will sketch out some preliminaries concerning what consti-

tuted membership in the Vienna Circle. The second and third will present in-

teractions between the members of the Vienna Circle and Einstein and Bohr,

respectively. The fourth will illustrate the differences in their reactions to log-

ical positivism through Einstein and Bohr’s EPR dialogue.145

I. Membership in the Vienna Circle

The Vienna Circle was a group of philosophers, physicists and mathemati-

cians who met weekly from 1924 to 1936 at the University of Vienna to discuss

central issues in the philosophy of science. The core group members included

its unofficial leader, philosopher Moritz Schlick, along with physicist Philipp

Frank, social scientist Otto Neurath, his wife, mathematician Olga Hahn-Neu-

rath, philosopher Viktor Kraft, and mathematicians Hans Hahn, Theodor

Radacovic and Gustav Bergman. Philosopher Rudolf Carnap was the most sig-

nificant later addition to the group in 1926, but other students of Schlick and

Hahn (such as Kurt Gödel) later joined the group as well.146

145 A timeline of significant events has been added at the end of this paper for reference. 146 Thomas Uebel, “Vienna Circle,” in The Stanford Encyclopedia of Philosophy, ed.

Edward N. Zalta, §2.1.

67

But to limit membership of the circle to just the aforementioned indi-

viduals would be to misunderstand the significance of referring to them as a

group. Indeed, most historians of science no longer recognize the Circle as

having a single unified agenda. Instead, they cast the Circle as featuring many

distinct philosophies which differ considerably despite agreeing on a few cen-

tral points.147 Broadly construed, these philosophies had the following in com-

mon: firstly, their attempts to renew empiricism, secondly, their desire for a

unity of all science with philosophy, and thirdly, their re- quirement of a crite-

rion of meaning. According to this criterion, the only meaningful statements

were synthetic statements of the empirical sciences; and of these only those

that were empirically testable.148 Inheriting Ludwig Wittgenstein’s view from

the Tractatus, analytic statements did not carry factual meaning and, therefore,

only showed ways of representing the world.149 This lead to a general rejection

not only of metaphysics, but of most philosophy outside of empirical science

(e.g. ethics, aesthetics, etc.) on account of its being apparently meaningless.150

It is then best to expand membership in the Vienna Circle to include

philosophical allies to this central project. Prominent allies and sympathizers

include the Berlin Society for Empirical Society (especially Hans Reichen-

bach), Gustav Hempel from Berlin, Jorgen Jorgensen from Denmark and A.J.

Ayer from the UK.151 The Circle’s 1929 manifesto also lists Einstein, Bertand

Russell and Wittgenstein as its “leading representatives,” although Wittgen-

stein vocally resented this claim.152

It is also important to mention notable influences on the Circle, the most

significant being the physicists Ernst Mach, Hermann von Helmholtz and Lud-

wig Boltzmann (especially Mach, whose work on positivism was essential to

the positivists’ meaning criterion). Other influences included mathematicians

David Hilbert and Felix Klein, the logicists Gottlob Frege, Russell and Witt-

genstein, the British empiricists (especially David Hume) and, to some extent,

the French conventionalists Henri Poincare and Pierre Duhem.153 It will be

147 Ibid. 148 Ibid. 149 Ludwig Wittgenstein, Tractatus Logico-Philosophicus, trans. David Pears and Brian

McGuiness (London: Routledge, 2001), 4.121. 150 Uebel, “Vienna Circle,” §3.1; Sarkar Sahotra, The Scientific Conception of the World:

The Vienna Circle (Dordrecht: Reidel, 1973), 329-339. 151 Uebel, “Vienna Circle,” §2.1. 152 Sahotra, The Scientific Conception of the World. 153 Uebel, “Vienna Circle,” §2.1.

68

shown shortly that Einstein was not only one of the most the important con-

temporaries of the Circle, but one of its most important influences as well.

II. Einstein and the Vienna Circle

Einstein shared many of his philosophical influences with the Vienna Circle.

From a young age Einstein exposed himself to philosophy, first reading Kant’s

Critique of Pure Reason by the age of 13.154 Throughout his physics education

Einstein continued to read the work of prolific philosophers of physics such as

Mach, Hume, Duhem, Poincare, John Stuart Mill and others. But of these phi-

losophers, the three who most affected Einstein’s thinking were Mach, Hume

and Duhem.155

Einstein clearly believed that much of his success in physics was due to

his philosophical influences. After Mach’s death in 1916 Einstein writes:

How does it happen that a properly endowed natural scientist comes to

concern himself with epistemology? Is there no more valuable work in his spe-

cialty? I hear many of my colleagues saying, and I sense it from many more,

that they feel this way. I cannot share this sentiment. When I think of the ablest

students whom I have encountered in my teaching […] I can affirm that they

had a vigorous interest in epistemology.156

What Einstein owed most from Mach’s teachings was the following

view concerning the historical nature of physical theories: that one must look

at the historical context of a physical concept to understand its argumentative

force.157 Einstein also cited Mach’s positivism as an influence on his theory of

Special Relativity; he shared this influence with the logical positivists of the

Vienna Circle.

However, Einstein’s early acceptance of Mach’s positivism can confuse

the subtle nature his influence had on the Vienna Circle. The founders of the

154 Don Howard, “Einstein and the Development of Twentieth-Century Philosophy of

Science,” in The Cambridge Companion to Einstein, eds. Michel Janssen and Christoph

Lehner (New York: Cambridge University Press, 2014), 356. 155 Ibid., 356-357. 156 From a memorial notice quoted in Don Howard, “Albert Einstein as a Philosopher of

Science,” Physics Today (Dec. 2005), 34-35. 157 Howard, “Einstein and the Development of Twentieth-Century Philosophy of Science,”

358; Don Howard, “Einstein’s Philosophy of Science,” in The Stanford Encyclopedia of

Philosophy, §1; Paul Pojman, “Ernst Mach,” in The Stanford Encyclopedia of

Philosophy, §4.1.

69

Circle saw Einstein’s theories of relativity as examples of positivist epistemol-

ogy put into action, and (correctly) identified its Machian roots. For this rea-

son, many members of the Circle, such as Neurath and Schlick, considered

Einstein among their greatest influences.158 As mentioned earlier, this led them

to list Einstein as one of their “leading representatives” in their 1929 Mani-

festo. However, as soon as Einstein began receiving praise from the Circle, he

realized that his own views concerning his theories were much more accurately

reflected in Duhem’s conventionalism than in Mach’s positivism. Frank’s

reading of Einstein is probably closer to the truth than any of his contemporar-

ies, as it correctly recognizes Einstein’s appeal to conventionalism.159

Duhem’s conventionalism falls out of his thesis of theoretical holism.

His view was that hypotheses should be tested as part of an entire theory, rather

than in isolation. If a conflict arose between theory and evidence, specific parts

of the theory could be altered until the evidence fit the theory as a whole.

Which parts are ultimately chosen were then purely conventional.160 As Du-

hem’s views were in direct conflict with those of the Vienna Circle, much of

Einstein’s conversation with the Circle concerned this disagreement.

Einstein was in personal correspondence with several members of the

Vi- enna Circle before their meetings even began. In 1907, Einstein sent some

suggestions to Frank concerning a book Frank had recently written on causal-

ity. The two physicists became lifelong friends, with Frank eventually becom-

ing Einstein’s biographer in 1947.161 Schlick began his correspondence with

Einstein in 1915 after he sent Einstein a paper he had written concerning the

philosophical significance of the theory of relativity. Throughout the 1920s the

two had a particularly close relationship, as Einstein worked to both promote

Schlick’s career and to arrange an English translation of some of Schlick’s

work.162 Reichenbach first met Einstein as a student when he audited Ein-

stein’s lectures at the University of Berlin. Einstein was at first unimpressed

158 Uebel, “Vienna Circle,” §3.7; Rudolf Carnap, “Intellectual Autobiography,” in The

Philosophy of Rudolf Carnap, ed. Paul Arthur Schilpp (Illinois; Open Court, 1963), 20. 159 Howard, “Einstein and the Development of Twentieth-Century Philosophy of Science,”

354-355. 160 Roger Ariew, “Pierre Duhem,” in The Stanford Encyclopedia of Philosophy,§ 2.1. 161 Howard, “Einstein and the Development of Twentieth-Century Philosophy of Science,”

361. 162 Ibid.

70

with Reichenbach’s work, but allied with him later in Reichenbach’s philo-

sophical career, even arranging a Chair position for him with Planck at the

University of Berlin.163

In the early 1920s, Einstein’s Theory of Relativity found itself under

fire from the Neo-Kantians such as Ernst Cassirer. Clearly, Neo-Kantian met-

aphysics disagreed with the non-Euclidean geometry of General Relativity, so

they sought to attack these central notions of the theory. They argued that a

priori status should not be granted of the geometry of spacetime, but, instead,

to its topographical structure.164

Much of the correspondence between Einstein, Frank, Schlick and

Reichenbach at this time concerned their joint goal of defending relativity from

the Neo-Kantians. Einstein provides constant commentary on Schlick’s works

toward this goal between 1918 and 1924.165 In his early career, Reichenbach

actually published works defending Kant from relativity, but he later allied

with Schlick in 1924. Together they attempted to formulate an ac- count of

convention in physics that undermined the Neo-Kantians. However, while

these two believed they were simply further arguing Einstein’s own views, it

is clear their positivistic interpretations differ considerably from Einstein’s

Duhemian holism.166 While Schlick and Reichenbach argued that empirical

content attaches to theories one proposition at a time, Einstein, again, follow-

ing Duhem, held that it attaches only to whole theories.167 Still, this issue rep-

resents a significant part of Einstein’s interactions with the positivists.

Much of Einstein’s interactions with Carnap occurred years later, after

both men had moved to the United States. Carnap writes about his conversa-

tions with Einstein in the period of 1952-1954 at Princeton University:

On one occasion Einstein said that he wished to raise an objection

against positivism concerning the question of the reality of the physical

world... I explained that we had abandoned these earlier positivistic views, that

we did no longer believe in a ”rock- bottom basis of knowledge”; and I men-

tioned Neurath’s simile that our task is to reconstruct the ship while it is float-

ing on the ocean.168

163 Ibid. 164 Ibid., 362. 165 Ibid., 363. 166 Ibid., 364. 167 Ibid., 368. 168 Carnap, “Intellectual Autobiography,” 37.

71

This quote best illustrates the continued influence of the positivist’s

work on Einstein’s thought, even in Einstein’s later years. Clearly these inter-

actions left a lasting on impression on Einstein, even after the EPR debate with

Bohr.

III. Bohr and the Vienna Circle

The majority of Bohr’s interactions with the positivists came much later than

Einstein’s, with most occurring in the 1930s after Bohr’s views on comple-

mentarity in quantum mechanics were well-established. The general reaction

of the Circle was positive; three of the core members, Neurath, Frank, and

Schlick, argued that the view’s motivations were positivistic.

Neurath first came to Denmark in 1934 when invited by Jorgensen to

give a series of lectures at The Society for Philosophy and Psychology. He

writes to Carnap about his first experience with Bohr who had attended two of

these lectures:

Bohr. Idiosyncratic. An intense man. Came to two lectures and joined

the discussion enthusiastically […] Basic line: he does not want to be consid-

ered a metaphysician. And he is able to express himself relatively non-meta-

physically, when he is careful […] But he possesses certain basic attitudes

which agree with mine, e.g. that in science one cannot clear up everything at

once […] Obviously tries to come into agreement with us.169

Neurath detects a similarity between Bohr’s complementarity and his

own positivistic views, but is unsatisfied with Bohr’s presentation of them. Put

simply, complementarity was Bohr’s view that the results of quantum experi-

ments are inherently dependent on the measuring devices used. A full descrip-

tion of a quantum object could only be achieved through multiple complimen-

tary experiments, each described using terms in classical mechanics, such as

energy, momentum and spacetime coordinates.170 In 1934, Bohr’s position

could be seen as treating quantum objects as Kantian things-in-themselves,

existing independently of our experience of them (a nonsensical notion to Neu-

rath). But following his EPR debate with Einstein, Bohr expressed that such

169 From a letter Neurath wrote to Carnap quoted in Jan Faye, “Niels Bohr and the Vienna

Circle,” in The Vienna Circle in the Nordic Countries, eds. Juha Manninen and Friedrich

Stadler (New York: Springer, 2010), 34. 170 Jan Faye, “Copenhagen Interpretation of Quantum Mechanics,” in The Stanford

Encyclopedia of Philosophy, §4.

72

objects exist only in so far as they can be observed empirically, thus aligning

his view with those held by Neurath and Carnap.171

After their meeting, Bohr and Neurath remained in letter correspond-

ence for several years.172 During this time, Bohr was also in correspondence

with physicist Frank. In 1936, Frank writes to Bohr saying that he believes

Bohr’s position to be fundamentally positivistic, while Einstein’s is metaphys-

ical; Bohr agrees.173

But by far the biggest positivist influence on Bohr came from his inter-

actions with his fellow Dane, and colleague at the University of Copenhagen,

the philosopher Jørgensen. It has been argued that Jorgensen was not a logical

positivist. But in his own analysis of logical positivism in his 1951 work The

Development of Logical Positivism, Jørgensen argues the movement cannot

be seen as a single view, but as many competing views (the current author

shares this opinion).174 Given his consistent correspondence with the members

of the Vienna Circle in the 1930s, his frequent endorsement of their philoso-

phy, and his contributions to the task of uniting psychology with the natural

sciences, if Jørgensen was not a logical positivist himself, he was at least their

greatest Danish spokesperson.175

Throughout the 1930s, Jorgensen was a vocal supporter of Bohr’s the-

ories, endorsing them as positivistic over Einstein’s metaphysical criticisms.

In 1934 he writes:

But as far as I know it appears that none of these objections can stand a

closer criticism, and therefore one must think that Bohr’s and his fellow parti-

sans’ view suits the present experiences best, yes, that we up to now do not

know any other view which accords with the experience.176

Jorgensen believed that the Copenhagen interpretation of quantum me-

chanics was not a consequence of the epistemology of logical positivism, but

an example of a justification for it. In 1937 he writes:

171 Faye, “Niels Bohr and the Vienna Circle,” 36-37. 172 Ibid., 35. 173 Ibid., 37. 174 Jørgen Jørgensen, “The Development of Logical Empiricism,” International

Encyclopedia of Unified Science Vol. 2, No. 9 (Chicago: The University of Chicago

Press, 1951). 1. 175 Faye, “Niels Bohr and the Vienna Circle,” 40-41; Carl Henrik Koch, “Jørgen Jørgensen

and Logical Positivism,” in The Vienna Circle in the Nordic Countries, eds. Juha

Manninen and Friedrich Stadler (New York: Springer, 2010), 153-155. 176 Quoted in Faye, “Niels Bohr and the Vienna Circle,” 42.

73

Quite a different matter is that the results of atomic physics may serve

to support a positivistic epistemology, since not only does quantum mechanics

show that even a fundamental notion like the concept of causation is not abso-

lutely necessary to physics but it also points out the danger of operating with

assumptions (for instance of “causal determinateness”) which cannot in prin-

ciple be verified.177

It seems likely that such a position was reached by Jorgensen through

frequent discussions with Bohr. One can conclude that he acted as the most

important intermediary figure between Bohr and the Vienna Circle.

The extent to which Bohr interacted with the logical positivists is prob-

ably best demonstrated by his participation in the 2nd International Congress

for the Unity of Science, which took place from June 21-26, 1936. The confer-

ence was held at Carlsberg’s honorary residence, where Bohr was living at the

time. In attendance were Jorgensen, Frank, Neureth, Hempel, Karl Popper and

many other associated philosophers and physicists.178 Carnap, Reichenbach

and Schlick had expressed wishes to attend, but were unable to for various

reasons. Tragically, Schlick had been denied a travel permit from Austria, al-

lowing him to be shot and killed by a deranged student in front of the Univer-

sity of Vienna on June 22. It was at the conference that the rest of the Circle

received this grim news.179

This conference was, perhaps, the last real meeting of the Vienna Circle,

given that its members generally disbanded after Schlick’s death. Quantum

mechanics was one of the central focuses, with Jorgensen and Frank delivering

lectures dedicated to the subject. Schlick’s final paper was also read to the

audience; all three of these presentations praised Bohr for his positivistic solu-

tion to Einstein’s EPR criticism.180

IV. Bohr and Einstein: The EPR Debate

The difference between the influence that logical positivism had on Einstein

and Bohr is perhaps best illustrated by their famous EPR debate in 1935. In

the years leading up to the debate Einstein had tried several times to refute

177 Jørgen Jørgensen, “Causality and Quantum Mechanics,” Theoria, Vol. 1 (1937), 116-

117. 178 Faye, “Niels Bohr and the Vienna Circle,” 33. 179 Ibid. 180 Ibid., 39.

74

Heisenberg and Bohr’s Copenhagen interpretation by posing thought-experi-

ments which sought to provide violations of the uncertainty principle. But in

each case Bohr had found a flaw.181 These defeats convinced Einstein that the

theory could not be defeated in such a way. However, he held onto the reser-

vation that such a theory was still incomplete in that it did not represent reality

fully.182 In his 1935 paper with co-authors Podolsky and Rosen, Einstein ar-

gues that any complete theory must have a Criterion of Reality:

If, without in any way disturbing a system, we can predict with certainty

(i.e., with probability equal to unity) the value of a physical quantity, then there

exists an element of physical reality corresponding to this physical quantity.183

Any description of a physical system by its wave function cannot pro-

vide a corresponding element to both a quantum object’s linear momentum

and position simultaneously. Hence, if both of these elements had real concur-

rently- existing counterparts, such a description would be deemed incomplete

by this criterion. Einstein then tries to illustrate that it can be shown that both

linear momentum and position exist simultaneously for quantum objects.184 To

be brief, Einstein proposes a thought experiment involving two previously in-

teracting systems whose post-interaction joint state can be written as a super-

position of the eigenstates of the two systems (or more simply, two entangled

systems). He shows that if one assumes locality (from relativity), it is possible

to know both the linear momentum and position of one system since one of

these values can be inferred from the measurement made on the other with a

probability of 1.185 Therefore, Einstein seems to have shown that Bohr’s theory

cannot be complete.186

181 Howard, “Revisiting the Einstein-Bohr Dialogue,” 1. 182 Arthur Fine, “The Einstein-Podolsky-Rosen Argument in Quantum Theory,” in The

Stanford Encyclopedia of Philosophy, §1.1. 183 A. Einstein, B. Podolsky, and N. Rosen, “Can Quantum-Mechanical Description of

Physical Reality be Considered Complete?,” Physical Review, Vol. 47 (May, 1935), 777. 184 Ibid., 777-778; Fine, “The Einstein-Podolsky-Rosen Argument in Quantum Theory,”

§1.2. 185 Ibid., 779-780; Howard, “Revisiting the Einstein-Bohr Dialogue,” 24.

186 Einstein himself provided an alternate version of this argument, fo-

cusing instead on the separability of real existing states of affairs, and how

they cannot effect each other superluminally. Since Bohr’s response also re-

futes this position, I exclude this alternate argument for the sake of brevity. See

75

Of course, Bohr once again had a reply to Einstein’s argument. In a

response published later that year he argued that although there is no classical

disturbance to one system by measuring the other, measuring one system none-

theless influences the nature of what we can measure of the other system. This

is to say, by choosing to measure an observable in one system, we change the

experimental procedures needed to classically describe the other system.187

Einstein’s argument then fails because his Criterion of Existence cannot be

properly applied to his thought experiment.

The influence of their interactions with members of the Vienna Circle

is evident on both sides of this debate. Einstein’s position is profoundly anti-

positivistic. His Criterion of Existence makes blatant reference to the real ex-

istence of objects outside of what is empirically testable. Surely this sort of

talk would be considered meaningless by the positivists. Moreover, his use of

a priori assumptions—one being locality—gives his argument a metaphysical

character. The EPR argument can then be seen as Einstein’s final rejection of

positivism, starkly contrasting his admitted influence from Mach in his earlier

years. It is no doubt that this metaphysical position was arrived at by Einstein

at least partly by exploring the consequences of positivism with Schlick,

Reichenbach and Frank.

Bohr’s response seems to be extremely positivistic in nature. Bohr’s

central point is that Einstein cannot use the ambiguous talk of his Criterion of

Meaning. He says that to use language this way makes assumptions outside of

what can be empirically tested. But this is almost identical to the positivist’s

charge that such talk is meaningless! Indeed, as before mentioned, Reichen-

bach and Schlick congratulated Bohr on the positivistic argument he employs

against Einstein.188 His employment of postivistic views in this argument di-

verges from his first interpretations of quantum mechanics. Clearly, in earlier

arguments, Bohr makes use of realism-like talk when talking about the exist-

ence of the atom.189 It is, therefore, reason- able to believe that the shift in

Fine, “The Einstein-Podolski-Rosen Argument in Quantum Theory,” §1.3 and

Howard, “Revisiting the Einstein-Bohr Dialogue,” 25-27. 187 N. Bohr, “Can Quantum-Mechanical Description of Physical Reality be Considered

Complete?,” Physical Review, Vol. 48 (Oct., 1935), 696-700. This is explained in Fine,

“The Einstein-Podolski-Rosen Argument in Quantum Theory,” §1.2 and Howard,

“Revisiting the Einstein-Bohr Dialogue,” 27-31. 188 Faye, “Niels Bohr and the Vienna Circle,” 39. 189 Ibid., 7.

76

thinking employed in Bohr’s response to the EPR argument was formed

through his interactions with Neurath and other members of the Vienna Circle.

Conclusion

In the 20th century, logical positivism had a clear influence on the development

of physical theories. Part of this influence can surely be traced back to the

frequent interactions between Einstein, Bohr and the members of the Vienna

Circle. This essay has tried to capture the nature of this influence, but it is not

a complete picture. A deeper analysis might go further to show exactly how

the influenced views of Einstein and Bohr then affected the physics commu-

nity as a whole. Such a project may be worthwhile: much of Einstein and

Bohr’s work is uncontroversially considered foundational to our modern phys-

ics.

So, to return to the teachings of Mach—who influenced all three of Ein-

stein, Bohr and the Vienna Circle—we must continue to look at the historical

context of our physical theories to properly understand their argumentative

force.

77

Timeline of Significant Events (1905 - 1939)

1905 Einstein’s Annus Mirabilis (Special Relativity, PE Effect, etc.)

-

-

1908 Frank, Hahn & Neureth Begin Meeting at University of Vienna

-

1910 First Volume of Principia Mathematica Published

1911 Rutherford Discovers Atomic Nucleus

-

1913 Bohr Model of the Atom

1914 World War I Begins

1915 General Relativity

1916 Ernst Mach Dies

-

1918 World War I Ends

-

1920 Bohr and Einstein Meet for the First Time in Berlin

1921 Tractatus Logico Philosophicus

-

-

1924 Vienna Circle Begins its Meetings

1925 Spin Discovered; Matter’s Wave-Particle Duality Discovered

1926 Carnap Joins the Vienna Circle

1927 Uncertainty, Complimentarity & The Copenhagen Interpretation

-

1929 Vienna Circle Publishes Manifesto

-

1931 Gödel’s Incompleteness Theorems

-

-

1935 EPR Argument

1936 2nd ICUS; Schlick Killed; Vienna Circle Disbands

-

-

1939 World War II Begins

78

References

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sidered Complete?.” Physical Review, Vol. 48 (Oct., 1935): 696-702.

Carnap, Rudolf. “Intellectual Autobiography.” The Philosophy of Rudolph

Carnap, edited by Paul Arthur Schilpp. Illinois: Open Court, 1963.

Einstein, A,. Podolsky, B., and Rosen, N. “Can Quantum-Mechanical De-

scription of Physical Reality Be Considered Complete?” Physical

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ford Encyclopedia of Philosophy, edited by Edward N. Zalta. Sub-

stantive revision July 24, 2014. http://plato.stanford.edu/entries/qm-

copenhagen/.

—.“Niels Bohr and the Vienna Circle.” The Vienna Circle in the Nordic

Countries, edited by Juha Manninen and Friedrich Stadler. New

York: Springer, 2010.

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In The Stanford Encyclopedia of Philosophy, edited by Edward N.

Zalta. Substantive revision November 5, 2013. http://plato.stan-

ford.edu/entries/qt-epr/.

Howard, Don. “Albert Einstein as a Philosopher of Science.” Physics Today

(Dec. 2005): 34-40.

—.“Einstein and the Development of Twentieth-Century Philosophy of Sci-

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Janssen and Christoph Lehner. New York: Cambridge University

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—.“Einstein’s Philosophy of Science.” In The Stanford Encyclopedia of Phi-

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2004. http://plato.stanford.edu/entries/einstein-philscience/.

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—.“Revisiting the Einstein-Bohr Dialogue.” Accessed online at

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Bohr%20Dialogue.pdf.

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(1937): 115-117.

—.“The Development of Logical Empiricism.” International Encyclopedia

of Unified Science. Vol. 2, No. 9. Chicago: The University of Chi-

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Koch, Carl Henrik. “Jorgen Jorgensen and Logical Positivism.” In The Vi-

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Friedrich Stadler. New York: Springer, 2010

Pojman, Paul. “Ernst Mach.” In The Stanford Enclyopedia of Philosophy, ed-

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Sarkar, Sahotra. The Scientific Conception of the World: The Vienna Circle.

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80

The Similarities of Thought Experiments and

Computer Simulations: A First Approach Natalie Morcos

Thought experiments and computer simulations are both investigatory tools

that allow us to reason about real world scenarios without requiring that those

scenarios actually obtain. In the following paper, I will compare these two tools

and argue that, while there are notable similarities between computer simula-

tions and thought experiments, a perfect parallel is not possible. This is be-

cause computer simulations lack the full the explanatory power of thought ex-

periments. I will begin by giving Winsberg’s definition of computer simula-

tions and by highlighting a few of the ways in which this definition of com-

puter simulations seems to fit with our initial, intuitive understanding of

thought-experiments. Next, I will pin down more precisely what is meant by

the term “though experiment”. I will present three characterizations of thought

experiments, Brown’s (2011), Norton’s (1991), and Nersessian’s (1993) and I

will show that although each of these accounts affords notable similarities be-

tween thought experiments and computer simulations, each also attributes an

explanatory power to the former that cannot be adequately captured by the

latter, thus preventing a perfect parallel. I will conclude by suggesting possible

courses for further investigation.

Defining computer simulations is no easy task. Winsberg (2014) states

that “[n]o single definition of computer simulation is appropriate.” However,

he does go on to offer three possible definitions: one narrow, one broad, and

one compositional. Narrowly defined, a computer simulation is a program that

explores the approximate behaviour of a mathematical model of some real

world system using step by step methods and running on a computer. It con-

tinuously takes in the state of the system at some time t and outputs the result-

ant state at time t+1. The resultant state is then fed back into the program in

its next iteration, and might also be displayed as a visualization, often designed

to mimic the output of some scientific instrument. A broader characterization

of computer simulations encompasses not only the program, but the develop-

mental process as well, including the choice of model, the implementation, the

calculation of output, and the visualization and study of the results. The third

approach Winsberg offers is to define computer simulations compositionally:

81

first clarifying the idea of a simulation, then restricting that idea to the set sim-

ulations executed via program by a digital computer. A simulation here can be

understood as a “system that is believed, or hoped, to have dynamical behavior

[sic.] that is similar enough to some other system such that the former can be

studied to learn about the latter” (Winsberg, ibid.). So, compositionally, a com-

puter simulation can be defined as a system that meets these requirements, but

which, in addition, is carried out programmatically by a computer.

With this rough understanding of computer simulations and an sem-

naticintuitive, cursory understanding of thought experiments as experiments

that we perform in our minds, it seems that some similarities between the two

are immediately visible. We might say that thought experiments and computer

simulations are similar in that neither is able to procure new information about

the world — new derivations, perhaps, or new perspectives, even, but not truly

new information. The isolated thought processes we carry out in our minds do

not have access to the external world and they can, therefore, only make use

of information already present in our minds, that is, information already

known. Similarly, a computer simulation can only make use of its initial con-

ditions and of the behavioural rules coded into its program. Like thoughts,

which are isolated in the human mind, computer simulations do not have ac-

cess to the outside world; they only have access to the information already

present in their limited environments.

Another way in which computer simulations and thought experiments

initially seem quite similar is in their reliance on correctness of representation

in order to yield factually accurate conclusions. Humphreys (2014, 110) argues

that a system S provides a core simulation of an object or process B only if S

presents a model that correctly represents B, either dynamically or statically.

Without that correctness of representation, the link between S and B is signif-

icantly weakened and the accuracy of any conclusions drawn from the simu-

lation must be called into question. Similarly, Parker asserts that a central ques-

tion in the epistemology of simulation is: “[w]hat warrants our taking a com-

puter simulation to be a severe test of some hypothesis about the natural world?

That is, what warrants our concluding that the simulation would be unlikely to

give the results that it in fact gave, if the hypothesis of interest were false?

(2008, 380)” Without representational correctness, which relies, in a sense, on

the integrity of the isomorphism between the model and reality, we must ques-

tion the conclusions drawn from the simulation, and this seems to be the case

with thought experiments, too.

82

For example, consider the following pair of thought experiments con-

cerning the possibility of a moving earth. Long ago, Aristotle presented a case

against the earth being in motion. He said the earth could not be moving, for

if it were, a cannon ball dropped from a tower would not land at the tower’s

foot, but rather at a distance from it (i.e. at the point where the foot of tower

had been when the ball was dropped, prior to the earth and the tower moving

forward). Moreover, a ball thrown from East to West (without loss of general-

ity, assuming motion along this axis) would not travel the same distance as one

thrown West to East— an outcome which, to Aristotle, seemed obviously ab-

surd. Centuries later, Galileo, arguing for the opposite conclusion, asked the

thought experimenter to imagine tossing a ball back and forth in the cabin of a

ship uniformly traversing the ocean or alternatively to imagine a drop of water

falling in this same cabin, and to notice that the imagined event unfolds indis-

tinguishably from how it would have if the ship had been at rest in a harbour.

The Galilean experiment completely undermines Aristotle’s representation of

the world, showing that the model of physics Aristotle had used in order to

draw his conclusions was not actually true to reality. As Brown says, these

examples show that “thought experiments, even the most ingenious, are quite

fallible” (2011, 33). Thus, much like the correctness of a computer simula-

tion’s results, the correctness of a thought experiment’s conclusions ultimately

relies on how correctly the modeled phenomena represent nature.

Despite the cursory appeal of these similarities, the comparison between

computer simulations and thought experiments is not as harmonious as it ini-

tially appears. The above comparisons proceed from a largely empiricist stand-

point, the assertion that thought experiments cannot yield new knowledge es-

pecially so. However, Brown (2011, 2014) insists that there are certain thought

experiments that cannot be accounted for empirically, declaring that some

thought experiments are indeed capable of producing new knowledge. In order

to support his claims, Brown proposes a platonist view of thought experiments

in which a priori access to nature is possible. He also proposes a taxonomy

according to which it is possible to classify thought experiments. This taxon-

omy will be employed throughout the remainder of the paper, even divorced

from Brown’s platonism, in order to help illustrate the overlaps and differences

between the competing accounts of thought experiments that will be discussed.

Brown’s taxonomy is two-tiered. At the first level, thought experiments

are classified as either constructive (positive) or destructive (negative). Next,

each classification is further sub-divided according the way in which the

83

thought experiment arrives at its positive or negative conclusion. Constructive

thought experiments may be either mediative, conjectural, or direct. Destruc-

tive thought experiments, on the other hand, work by a) highlighting internal

contractions in a theory, b) demonstrating how the theory conflicts with other

well-established facts or theories, c) showing via conceptual distinction that

some previously postulated result does not follow, or d) rejecting a previously

described phenomenon. Galileo’s thought experiment above, which under-

mined Aristotle’s previously postulated thought experiment on the topic of a

stationary vs. a moving earth, is an instance of this last class.

Mediative thought experiments, the first subcategory of positive

thought experiments, help us to see the consequences of a theory, often serving

a pedagogical role. Importantly, they do not aid in the development of a theory

but rather proceed from one that is already well articulated (2011, 35, 38). A

historical example is Newton’s cannon, an imaginary cannon placed atop a

very tall mountain and used to illustrate how the moon could effectively “fall”

into orbit around the earth. The second positive subcategory is conjectural

thought experiments. Like mediative thought experiments, these do not di-

rectly produce new theories. However, unlike mediative thought experiments,

they do not start from a given theory either. Instead, they establish or present

some problematic phenomenon. A clarifying theory may then be conjectured

afterwards in order to resolve the presented problem, but the new theory is not

properly contained in the thought experiment itself. For example, the ship of

Theseus sets up the problem of identity over time, but does not resolve it. Sim-

ilarly, Newton’s bucket sets the stage for postulating absolute space but does

not necessarily imply its existence.

While these first two classes employ strategies that are relatively un-

problematic for computer simulations, the last subcategory of positive thought

experiments, namely direct thought experiments, seems to be exclusive of

computer simulations. Direct thought experiments are thought experiments

that actually do establish a new theory. The legitimacy of this class requires a

platonist attitude since, in this class, a direct thought experiment starts with a

well known phenomenon and then proceeds to arrive at a novel conclusion

without any further empirical observation; these thought experiments contain,

in a sense, a moment of revelation, the apprehension of some natural truth with

the mind’s eye. Examples of direct experiments are the epitaph of Stevinus

(occasionally also called Stevin’s statics) in which a chain of balls is imagined

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hanging over a triangle in order to establish the impossibility of perpetual mo-

tion, and Einstein’s elevator, in which Einstein establishes that an object’s be-

haviour within a gravitational field is equivalent to its behaviour in an accel-

erated inertial frame. Though platonists argue that thought experiments are ca-

pable of arriving at new knowledge in this way, it is extremely unclear how

this would work, exactly, and what a moment of revelation would even look

like.

In the destructive or negative branch of the taxonomy, the subcategories

are less rigidly defined and lack precise names. However, one subcategory is

the kind of thought experiment that shows something by means of a reductio

ad absurdum, that is, by showing how some theory or set of assumptions en-

tails a contradiction, either internally or in tandem with another well-estab-

lished position. For example, though its conclusion has now come to be widely

accepted, Schrödinger originally proposed his famous cat scenario in order to

emphasize the absurdity of the then fledgeling theory of quantum mechanics.

The result of his thought experiment, which requires an imagined cat to be

thought of as both alive and dead simultaneously, violates an everyday under-

standing of living things. Another kind of destructive thought experiments is

the kind that attacks other, preexisting thought experiments. They attempt to

show that some important conclusion drawn from the attacked thought exper-

iment does not follow. The destructive thought experiment does so by either

constructing a scenario that satisfies all of the previous thought experiment’s

original requirements but denies its conclusion, or by attacking the previous

thought experiment at the phenomenological level, rejecting the reality of the

claimed phenomena. An example of the first strategy would be Poincare’s disc,

a hypothetical universe constructed to invalidate the conclusion of Lucretius’

argument for infinite space; an example of the second would be Galileo’s

aforementioned thought experiment that was put forth to attack Aristotle’s

“proof" against the possibility of a moving earth.

John Norton, another philosopher interested in the epistemology of

thought experiments, rejects Brown’s platonic characterization of thought ex-

periments. In particular, he rejects the legitimacy of direct thought experiments

which, according to Brown, are somehow able to generate new knowledge.

Instead, Norton presents a fiercely empiricist account, insisting that to hold

any alternative view, which would be to “suppose that thought experiments

provide[d] some new and even mysterious route to knowledge of the physical

world,” would be foolish and absurd (1991, 129). Norton characterizes thought

85

experiments as premise-based arguments that can be evaluated propositionally.

He asserts that, when it comes to thought experimental results, “there is only

one non-controversial source from which this information can come: it is elic-

ited from information we already have by an identifiable argument, although

that argument might not be laid out in detail in the statement of the thought

experiment” (Norton 1991, 129).

At first glance, computer simulations seem compatible with Norton’s

empiricist characterization. By denying the ability of thought experiments to

produce any new knowledge, Norton has effectively eliminated the incompa-

rability between thought experiments and computer simulations which

Brown’s belief in direct thought experiments had introduced. It is tempting, at

this point, to accept Norton’s empiricism over Brown’s platonism so as to be

able to conclude that yes, one can draw a perfect parallel between computer

simulations and thought experiments. To do so, however, would be a mistake.

Referring back to the types of thought experiments identified by Brown’s tax-

onomy, we see that Norton’s view also presents problems for a seamless com-

parison between thought experiments and computer simulations. This is be-

cause Norton’s view is able to account for all of the destructive thought exper-

imental subcategories, including the reductio ad absurdum strategy, which

lends itself easily to propositional reconstruction. This is not a strategy avail-

able to computer simulations. A computer simulation, at the bare bones level,

is a computer program which must be able to run and must be able to produce

some intelligible output. In order to do so, however, it cannot contain an inter-

nal contradiction. The system modelled by the computer simulation must be

consistent with itself. This limitation in computer programs, a limitation Nor-

ton’s account shows is not possessed by thought experiments, prevents com-

puter simulations from being entirely compatible with thought experiments as

they are characterized by Norton’s empiricist account.

Though their inability to capture the reductio ad absurdum strategy puts

computer simulations at odds with Norton’s thought experimental framework,

this limitation in computer simulations shines favourably on the possibility of

drawing a parallel between computer simulations and thought experiments as

they are defined by Nancy Nersessian’s mental modelling account. Nersessian

believes that “thought experimenting is a form of ‘simulative model-based rea-

soning’. That is, thought experimenters reason by manipulating mental models

of the situation depicted in the thought experimental narrative” (1993, 292).

86

The parallel with computer simulations is immediately obvious: computer sim-

ulation aided inquiry is indeed “simulative model-based reasoning”(ibid).

Nersessian continues, saying that thought experiments are “the construction of

a dynamical model in the mind by the scientist who imagines a sequence of

events and processes and infers outcomes” (ibid). Again, a parallel presents

itself. This is stunningly similar to the way in which Winsberg (2014) describes

computer simulations as starting with a system in a given state and applying

pre-described behavioural rules in order to output a resultant state for exami-

nation. Nersessian goes on to say that for each mentally modelled thought ex-

periment, a narrative is constructed “in order to communicate the experiment

to others, i.e., to get them to construct and run the corresponding simulation

and presumably obtain the same outcomes”(ibid). This too finds an analogue

in computer simulations: the constructed narrative can be thought of as the

underlying computer program, which allows the experiment to be transferred

to a different agent and run again in the same way.

So far, Nersessian’s account seems conducive to the drawing of a seam-

less parallel between computer simulations and thought experiments. There

are strong similarities between her characterization of mental models and

Winsberg’s characterization of computer simulations. Indeed, Nersessian’s

theory shows thought experiments to struggle representationally just like com-

puter simulations . Neither thought experiments nor computer simulations are

able to account for the subcategory of destructive thought experiments that

explore contradictions. To illustrate, consider the example of Gallileo’s falling

balls. Galileo imagines a light and a heavy ball dropped from a tower falling

to the ground. In line with his preconceived idea that heavier objects fall faster,

he imagines the heavy ball making contact with the ground first. Next, he ima-

gines repeating the experiment, but with the two balls together attached by a

string and dropped as a composite object. The composite object is heavier than

the heavy ball, so, by his initial theory, should fall faster. However, since it is

composed of both the heavy and the light ball, it also seems reasonable for him

to assert that the speed with which the composite object would fall would be

somewhere between the speeds of the initial balls, faster than one, but slower

than the other. Galileo realizes the contradiction he’s stumbled upon and con-

cludes instead that all bodies must fall at the same rate.

While it is possible to construct a mental model of the first phase of this

experiment in which the two balls fall separately, and while it is possible to

construct a mental model of the final conclusion, it does not seem possible to

87

construct a singular model rich enough to account for the experiment as a

whole. Brown presses Nersessian on this issue. With regards to the problematic

composite object, he asks “[h]ow will it move? […] The mental model could

show us one outcome or the other, but it cannot show us the contradiction”

(2011, 116). Brown asserts that what actually happens is that we employ “rea-

soning that is independent of the mental model,” in order to see that “the par-

adox can be resolved by having [all] bodies fall at the same rate,” a solution

for which we are then once again able to construct a model (ibid). However,

“[t]he second mental model comes after some crucial independent work is

done” (ibid). We cannot construct the model of the falling object until we know

how it is supposed to fall, and this exact limitation is found in computer simu-

lations as well. It is not possible to program a computer simulation of a falling

object unless the way in which the object is supposed to fall is already known.

The rules governing the object’s behaviour must be codified in the simulation

at its inception; they cannot be read off of the simulation after the fact. Thus,

computer simulations, like thought experiments on Nersessian’s mental mod-

elling account, are unable to represent situations in which an internal contra-

diction is explored. By denying the possibility of destructive reductio ad ab-

surdum thought experiments, Nersessian’s account of thought experiments

seems to have an advantage over Norton’s propositional one (in which recon-

structing contradictions is possible) when it comes to drawing a perfect parallel

between computer simulations and thought experiments.

However, though Nersessian’s account parallels computer simulations

in many ways, it still allows for the acquisition of new knowledge through a

priori access to the world, something that computer simulations are unable to

facilitate. Nersessian allows for the possibility of “discovering” new things

about the mentally modelled situations, supporting Brown’s direct construc-

tive subcategory of thought experiments. Nersessian’s model’s support for this

class of thought experiment renders her thought experimental account incom-

patible with computer simulations. Thus, much like Brown’s and Norton’s ac-

counts, Nersessian’s account does not allow us to draw a seamless parallel

between computer simulations and thought experiments either.

To summarize, though at first it seems there are some obvious and at-

tractive similarities between computer simulations and thought experiments,

upon further examination, none of the three thought experimental accounts

presented here allow us to draw a perfect parallel between the former and the

latter. Brown’s platonist views grant the thought experimenter a priori access

88

to nature, something computer simulations are unable to facilitate, which pre-

vents a seamless comparison. However, even eschewing Brown’s platonist ac-

count and opting instead for Norton’s empirically driven propositional model,

our efforts to draw a perfect parallel are no more successful. Norton’s theory

grants thought experiments greater explanatory power than that of computer

simulations. His propositional account is able to reconstruct thought experi-

ments in which internal contradictions are examined, whereas computer sim-

ulations are unable model such scenarios. Finally, though Nancy Nersessian’s

mental modelling account of thought experiments says they cannot model in-

ternal contradictions, the same way computer simulations cannot, her account

also does not allow for a perfect parallel to be drawn between the two. On her

account, the thought experimenter is again able to discover natural facts a pri-

ori, and so, much like Brown’s, Nersessian’s theory allows for direct thought

experiments, while computer simulations are unable to account for this partic-

ular subcategory. In conclusion, computer simulations seem only to succeed as

thought experiments for mediative and conjectural constructive thought exper-

iments, and for destructive thought experiments in which no internal contra-

diction is explored. Not one of the three characterizations explored in this pa-

per allows us to draw a perfect parallel between computer simulations and

thought experiments.

The above exploration should be considered no more than a first ap-

proach on this matter. Next steps could include examining further characteri-

zations of thought experiments, beyond the three mentioned here, to see if

computer simulations land more squarely in their explanatory nets. Alternately,

another course of further investigation could be to bring Brown’s subcategory

of direct constructive thought experiments under severe scrutiny. This subcat-

egory is the most problematic for the task at hand, and it is a category which

Brown himself admits may be called into question: “The contrast between di-

rect and mediative [thought experiments] may just be a matter of degree”

(2011, 41). Following this admission, Brown goes on to reify his belief in the

distinction, but the seed of potential criticism is there. If one were able to un-

dermine the category of direct constructive thought experiments, instead re-

classifying them as mediative and foregoing the possibility of accessing new

information, one would likely be able to seamlessly compare computer simu-

lations to thought experiments as they are understood on Nersessian’s mental

modelling account. This is undoubtedly an attractive possibility, however, as

89

it stands, our attempts to show that computer simulations are analogous to

thought experiments have been unsuccessful.

90

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