Calculation Thinking Computational Thinking
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Calculation - Thinking - Computational
Thinking
Seventeenth-Century Perspectives on
Computational ScienceMichael S. Mahoney
Princeton University
Published in Folkerts, Menso; Seising, Rudolf (Hg.):Form, Zahl, Ordnung. Studien zur Wissenschafts- und
Technikgeschichte. Ivo Schneider zum 65. Geburtstag(Boethius: Texte und Abhandlungen zur Geschichte der Mathematik
und der Naturwissenschaften), Stuttgart: Franz Steiner Verlag, 2003.
Festschriften allow one to begin on a personal note. Over thirty years ago Ihad the pleasure of daily conversations with Ivo Schneider during a year's
sabbatical at the Institut fr Geschichte der Naturwissenschaften in Munich.
We resumed those conversations two years later when Ivo spent a year as
visiting professor at Princeton. At the time, we shared a focus on
mathematics and the mathematical sciences in the 16th and 17th centuries,
and our long talks, often on the way to and from the coffee shop, gave me
an opportunity to sound out my ideas with a knowledgeable, imaginative,
and articulate colleague. Each of us has since moved off into other areas, in
my case to the history of computing since 1945. Yet those conversationsremain pertinent. For, the two subjects are not as far apart as they might
seem at first glance. They both involve the emergence of new disciplines, or
of new ways of thinking about old disciplines: in the 17th century, symbolic
algebra and the new mode of analytical reasoning that it fostered; more
recently, theoretical computer science as a mathematical discipline.
On the one hand, the two subjects display considerable continuity of theme.
Theoretical computer science has drawn its mathematical structure from
developments in abstract algebra that in turn exemplify many of the themes
that informed the first efforts in symbolic algebra in the 17th century. In that
sense, this essay completes a circle that I opened up during that year in
Munich with a lecture on "Die Anfnge der algebraischen Denkweise im 17.
Jahrhundert."(1) It may complete it in another sense as well, namely by
marking the end of algebraic thought as it was conceived in the 17 th century,
at least as far as it was thought to capture the relation of mathematics to the
world.
1. Published under that t itle
in the short-lived but
important journal founded
by Schneider and Eberhard
Schmauderer,Rete:
Strukturgeschichte der
Naturwissenschaften
1,1(1971), 15-31; English
trans., "The Beginnings of
Algebraic Thought in the
17th Century", in S.
Gaukroger (ed.),Descartes:Philosophy, Mathematics
and Physics (Sussex: The
Harvester Press/Totowa,
NJ: Barnes and Noble
Books, 1980), Chap.5.
Among the topics on our
daily walks was the
question of "s tructural
history" and Ivo's plans for
a journal devoted to it.
Let me start, then, with a brief account of the emergence of a new
mathematical world in the 17th century and then jump to the new world of
mathematics being created by means of the computer, of which we have
only limited mathematical understanding.
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I. Cutting the World Up and Putting it Back
Together with Mathematics
Toward the close of the 16th century, the French lawyer and mathematician
Franois Vite created a new symbolic algebra, or "logistic of species",
designed to extend the heuristic power of algebra from arithmetic to
mathematics in general and thereby to provide a new tool for carrying outthe form of mathematical reasoning which the Greeks had called "analysis".
He called his new algebra the "analytic art" (or "art of analysis").(2)
Essentially, it went beyond the old algebra by using symbols to denote both
knowns and unknowns (his convention was to use vowels for unknowns and
consonants for knowns; Descartes chose to work from opposite ends of the
alphabet) and thus to separate the form of the relationship among knowns
and unknowns from any particular values, indeed from any particular kind of
quantity, that they might represent. That is, an equation expressed a
relationship among things that could be added to one another, subtractedfrom one another, and so on. What interested Vite and his successors was
not so much the solution of an equation as the structure of the relation that it
expressed.
2. Franois Vite,In artem
analyticen isagoge (Tours,1591; republ. in Opera, ed.
F. van Schooten, Leiden,
1646)
The main task of his ars analytica, which distinguished it from all previous
algebras, was the investigation of the constitutio aequationum, i.e. the
structure of equations: how they are constituted and how they are related to
one another. In a series of treatises Vite set forth techniques for the analysis
and transformation of equations. He thus established the prototype, inessence if not in historical fact, for Book III of Descartes' Gomtrie, that
misnamed treatise on the theory of equations, where Descartes showed just
how an nth-degree polynomial is the product ofn linear binomials, how the
coefficients of the polynomial are the result of combinations of the roots, and
thus not only how terms can be removed by reducing or augmenting the
roots but also how one might imagine roots that do not correspond to real
numbers, although structurally they must be there.
xn
+a
1x
n-1
+... + a
n = (x
-r1)(
x - r2)...(
x - rn)
a1 = -(r1 + r2 + ... + rn), etc.
x3 - 1 = (x- 1)(x - a)(x - b); a = ?, b = ?
In short, the new symbolism and its associated techniques enabled one to
talk about equations and the number and nature of their solutions, even
without solving them. In retrospect and put anachronistically, Vite's analytic
art was a language of metamathematics as well as mathematics.
As such, it gave rise to three lines of development of interest to the matter at
hand. First, as already noted, it made symbolic algebra the study of the
abstract structures of mathematics. Second, in providing a language for
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talking about mathematical reasoning, it became a model for talking about
reasoning in general, that is, it suggested a form of symbolic logic by which
reasoning could be viewed as a kind of calculation. Third, it stimulated a
program of mathematical research that extended the heuristic power of
algebra into new realms and made the analytic art the language of
mathematical science. Let me begin with the last point and then return later
to the first two.
In his work, Vite set out an agenda which I have termed the "analytic
program" and which called for the application of his art to the works cited in
Book VII of Pappus of Alexandria'sMathematical Collection as
constituting the field of analysis, or as Newton happily phrased it, the
"Treasury of Analysis". The algebraic geometries devised by Descartes and
Fermat addressed this agenda, as did Fermat's new methods of maxima and
minima and of tangents. In the latter case, the application of the art carried
symbolic algebra into the realm of the indefinitely small, or infinitesimal, and
linked it, in ways not pertinent to the current discussion, to independent
efforts at recapturing the techniques of quadrature, or the determination of
the area of curved figures, associated with the name of Archimedes.
Methods of tangents and techniques of quadrature developed alongside one
another through much of the 17th century, in many cases without reference
to algebra.(3) As is well known, it was the signal achievement of Newton and
Leibniz to establish the inverse relationship between them. Both did so in the
language of symbolic algebra, and Leibniz in particular signaled the
importance of the symbolism. The 'd' was to be construed as an operation
on the quantity to which it was prefixed. Differentiation constituted a "certain
modification" (quaedam modificatio) of a quantity, and the rules governing
that modification gave rise to a new realm of structures to be analyzed.(4)
3. For the state of the art
just prior to the work of
Newton and Leibniz, see
my "Barrow's Mathematics:
Between Ancients and
Moderns", in M. Feingold
(ed.),Before Newton: The
Life and Times of Isaac
Barrow (Cambridge:
Cambridge UniversityPress, 1990), Chap. 3
4. For more extended
discussions on this and
what follows, see my
"Infinitesimals and
Transcendent Relations:
The Mathematics of
Motion in the Late
Seventeenth Century", in
D.C. Lindberg and R.S.
Wes tman (eds.),
Reappraisals of the
Scientific Revolution
(Cambridge: Cambridge
University Press , 1990),
Chap. 12, and "The
Mathematical Realm of
Nature", in D.E. Garber et
al. (eds.), Cambridge
History of Seventeenth-
Century Philosophy
(Cambridge: CambridgeUniversity Press , 1998),
Vol. I, pp. 702-55.
Although the calculus was not created for the sake of doing mechanics, it
was set to assume that role at the time Newton'sPrincipia appeared. It is,
of course, ironic that analytic mechanics was couched in the terms of
Leibniz's calculus rather than Newton's own fluxions, but the translation into
algebraic terms involved more than symbolism. In keeping with the heuristic
goals that had motivated Vite in the first place, Euler pointed to the difficulty
posed by Newton's geometrical original:
Newton'sMathematical Principles of Natural Philosophy,
by which the science of motion has gained its greatest
increases, is written in a style not much unlike [the synthetic
geometrical style of the Ancients]. But what obtains for all
writings that are composed without analysis holds most of all
for mechanics: even if the reader be convinced of the truth of
the things set forth, nevertheless he cannot attain a sufficiently
clear and distinct knowledge of them; so that, if the samequestions be the slightest bit changed, he may hardly be able to
resolve them on his own, unless he himself look to analysis and
evolve the same propositions by the analytical method.(5)
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By bringing out the essential structure of problems, algebraic analysis (Euler
would consider the phrase redundant) made clear how they and their
solutions were related to one another. One could not only do mathematics
but could see how the mathematics was done.5.Mechanica sive motus
scientia analytice exposita
(St. Petersburg, 1736),
Preface, [iv].
Euler's point concerned mathematics rather than mechanics, but the twowere so wrapped up in one another that in the 18th century analytic
mechanics was considered a branch of mathematics rather than of physics.
A few decades later, Lagrange took pride in the absence of diagrams from
hisMcanique analitique (1788):
No drawings are to be found in this work. The methods I set
out there require neither constructions nor geometric or
mechanical arguments, but only algebraic operations subject to
a regular and uniform process. Those who love analysis will
take pleasure in seeing mechanics become a new branch of it
and will be grateful to me for having thus extended its
domain.(6)
6. "On ne trouvera point de
Figures dans cet Ouvrage.
Les mthodes que j'y
expose ne demandent ni
constructions, ni
raisonnemens
gomtriques ou
mcaniques , mais
seulement des oprations
algbriques, assujetties
une marche rgulier et
uniforme. Ceux qui aiment
l'Analyse, verront avecplaisir la Mcanique en
devenir une nouvelle
branche, et me sauront gr
d'avoir tendu ainsi le
domaine." Avertissement.
The equations of the infinitesimal calculus had become the sole vehicle of
mechanics, the unchallenged means of mechanical thought. With the
Principia, and especially with its translation into the calculus, the
effectiveness of mechanics rested on that of mathematics. Proposition 41 of
Book I shows what that means. It is important to grasp the profound
implication of the condition in the statement of the problem (NB it is a
problem, not a theorem):
Assuming any
sort of centripetal
force, and
granting the
quadrature of
curvilinear
figures, required
are both the
trajectories in
which the bodies
move and the
times of motions
in the trajectories
found.(7)
7. Isaac Newton,
Philosophiae naturalis
principia mathematica
(London, 1687), 127.
[emphasis added]
In this proposition Newton maps the motion of an orbiting body on the left
onto a graph of motion at the "atomic" level on the right. The orbit and the
position of the body on it at any given time are thus captured mathematically,
provided that one can determine the area under the curves abzv and dcxw,
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i.e. that one can integrate the equations of motion. As Pierre Varignon put it,
after translating Newton's scheme into the two basic "rules", velocity v =
ds/dtand forcey = (ds/dx)(dds/dt2), wherex is measured along the axis
AC from A ands is measured along the curve VIK from V,
As to how these two rules are to be used, I say for now that,
being given any two of the seven curves noted above [curves
relating distance, time, force, and velocity in variouscombinations], that is to say, the equations of two taken at will,
one will always be able to find the five others,supposing the
required integrations and the solution of the equations that
may be encountered[emphasis added].(8)
8. Pierre Varignon, "Du
mouvement en gnral par
toutes sortes de courbes;
& des forces centrales, tantcentrifuges que
centreptes, ncessaires
aux corps qui les
dcrivent",Mmoires de
l'Acadmie Royale des
Sciences (1700), 86.
The condition linked the success of mathematical physics to that of the
calculus. It was the job of the calculus to secure those integrations and
solutions, and that is where its practitioners directed their efforts over thenext centuries. The intellectual satisfaction derived from reductionist
explanations depended on the capacity of the mathematics to carry out the
integration that provided the reduction, in the sense of showing that the
behavior at the reduced level did produce or correspond to the behavior at
the observable level.
The situation did not change with the shift from central-force physics to other
models of physical action. Once couched in the terms of the calculus, the
effectiveness of the physical model and its capacity to convey understanding
depended on the capacity of the calculus to provide a solution to thedifferential equations that resulted from analysis. In some cases, it was a
matter of calculating, as in expansion into series and term-by-term
integration. In other cases, it was a matter of exploiting the power of
algebraic analysis to explore structural relationships among problems and
thus to determine conditions of solvability or, in some cases, to prove
unsolvability, as in the case of the general quintic. Over the course of the
eighteenth century, what began as a search for the algorithms that made
integration as straightforward and mechanical as differentiation ended in a
theory that settled for analysis into families of curves reducible to canonicalforms.
Despite the successes of analysis, it became increasingly clear that in many
cases, for example the n-body problem, the move from differential equation
to finite form could be accomplished only by numerical calculation, that is by
reducing the analytical expressions to explicit summations iterated over small
intervals. One could do that by hand, but it was clearly a job suited more for
a machine. The story of the development of mechanical computing devices,
both analog and digital, during the nineteenth and early twentieth centuries
has been recounted many times, and I do not want to retrace that story
here.(9) What is important is that, as far as a mathematical understanding of
the world is concerned, the turn to mechanical calculation has from the
outset been a matter offaute de mieux. A numerical solution may produce
9. See, for example, William
Aspray (ed.), Computing
Before Computers (Ames:
Iowa State University
Press, 1990).
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from the basic relations specific values to be matched against measurements,
but it generally brings very little insight into how those values reflect the
working of the underlying relationships. One may, of course, experiment with
various initial values and try to discern how the outcome changes, but doing
so does not bring insights of the sort provided by
relating work to energy by way of force
and momentum. Numerical solutions do not reveal how the system works
because they hide precisely the intermediate (mediating) relationships that
lead from the behavior of the parts to that of the whole.
In the seventeenth century, the interactive development of algebra and
mechanics led to an analytic view of the world that characterized scientific
thought for the next three centuries. The invention of symbolic algebra and its
extension into the realm of the infinitesimal ultimately provided a powerful
mathematical tool for the study of the world as matter in motion. What made
the tool so powerful was that the algebra that lay at its foundation could beused not only to do mathematics but to talk about it as well. Not only could
one solve problems using algebra, but one could use the same algebra to
analyze questions of solvability. Algebra and the calculus not only captured
the world in mathematical structures but also provided the tools for analyzing
those structures mathematically.
More than simply a means of thinking about mathematics, symbolic algebra
was considered a means of thinking about thought itself. Descartes was only
the first of a line of thinkers down to the present who have pictured the
workings of the mind as a form of calculation, or as Hobbes put it, ofratiocination. Looking toward a universal characteristic, Leibniz foresaw a
time when matters of controversy could be resolved by sitting down and
calculating. From Boole'sAlgebra of Thought, through Frege's
Begriffsschriftand Russell's and Whitehead'sPrincipia mathematica (the
title no coincidence), algebra formed the link between mathematics and logic
and thus provided a means of thinking about thought itself.
From the outset, algebra was also associated with the notion of a mechanical
procedure. Algebra proceeded by straightforward rules, by what Leibniztermed "algorithms", thus giving new and fateful meaning to a word that had
been synonymous with reckoning since the 12th century.(10) That is what
made it appealing to him as a vehicle of logic: one could move from
premisses to conclusions by calculation.(11) Or rather, one could carry out
logical analysis in the way one did algebraic analysis: by following the rules of
algebra. That is what lay behind the notion of mechanizing logic; it is what lay
behind Turing's idea of capturing the notion of computability in an abstract
machine.(12)
10. "Algorithm" derived
from "algorismus", which
in turn was the Latin form
of al-Khwarizmi, the author
of the first Arabic treatise
on calculation with
"Indian" numbers, the
decimal place-valuesystem. Translated by
Robert of Chester in the
12th century, theLiber
algoritmi de numero
indorum became the basis
of a series of textbooks on
arithmetic, the most widely
read of which was John of
Holywood'sAlgorismus
vulgaris (ca. 1220). The
word acquired a 'th' in the
17th century by a back-
formation evidently based
on the assumption that the
word was originally Greek.
11. "... quando orientur
controvers iae, non magis
disputatione opus erit inter
duos philosophos, quam
inter duos Computistas.
Sufficiet enim calamos in
manus sumere sederequead abacos, et s ibi mutuo
(accito si placet amico)
dicere: calculemus."Die
philosophischen Schriften
von Gottfried Wilhelm
Leibniz, ed. C. J. Gerhardt
(Berlin, 1890, VII, 200. (I
thank Siegfried Probst for
locating this source and
pos ting it to the Historia
Mathematica list.)
To make the notion of "computable" as clear and simple as possible, Alan
Turing proposed in 1936 a mechanical model of what a human does when
12. On the mechanization of
logic, see Sybille Krmer,
Symbolische Maschinen:
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computing:
We may compare a man in the process of computing a real
number to a machine which is only capable of a finite number
of conditions q1, q2, ..., qRwhich will be called "m-
configurations". The machine is supplied with a "tape" (the
analogue of paper) running through it, and divided into sections
(called "squares") each capable of bearing a "symbol".(13)
Turing imagined, then, a tape divided into cells, each containing one of a
finite number of symbols. The tape passes through a machine that can read
the contents of a cell, write to it, and move the tape one cell in either
direction. What the machine does depends on its current state, which
includes a signal to read or write, a signal to move the tape right or left, and
a shift to the next state. The number of states is finite, and the set of states
corresponds to the computation. Since a state may be described in terms of
three symbols (read/write, shift right/left, next state), a computation may itselfbe expressed as a sequence of symbols, which can also be placed on the
tape, thus making possible a universal machine that can read a computation
and then carry it out by emulating the machine described by it.
Die Idee der
Formalisierung in
geschichtlichem Abri
(Darmstadt:
Wissenschaftliche
Buchgesellschaft, 1988)
and Martin Davis, The
Universal Machine: The
Road from Leibniz to
Turing(NY/London: W.W.Norton, 2000).
13. "On Computable
Numbers, with an
Application to the
Entscheidungsproblem",
Proceedings of the London
Mathematical Society,
ser.2, vol. 42(1936), 230-
265; at 231.
Turing's machine, or rather his monograph, belonged to the then current
agenda of mathematical logic. TheEntscheidungsproblem stemmed from
David Hilbert's program of formalizing mathematics; as stated in the
textbook he wrote with W. Ackermann,
TheEntscheidungsproblem is solved when one knows a
procedure by which one can decide in a finite number of
operations whether a given logical expression is generally valid
or is satisfiable. The solution of theEntscheidungsproblem is
of fundamental importance for the theory of all fields, the
theorems of which are at all capable of logical development
from finitely many axioms.(14)
14. D. Hilbert and W.
Ackermann, Grundzge
der theoretischen Logik
(Berlin: Springer, 1928), 73-
4: Das
Entscheidungsproblem ist
gelst, wenn man ein
Verfahren kennt, das bei
einem vorgelegten
logischen Ausdruck durch
endlich viele Operationen
die Entscheidung ber die
Allgemeingltigkeit bzw.
Erfllbarkeit erlaubt. Die
Lsung des
Entscheidungsproblems ist
fr die Theorie allerGebiete, deren Stze
berhaupt einer logischen
Entwickelbarkeit aus
endlich vielen Axiomen
fhig sind, von
grundstzlicher
Wichtigkeit.
Turing designed his machine to compute theEntscheidungsproblem, or
rather to show that it was uncomputable. Just as he was submitting his paper
to the London Mathematical Society, Alonzo Church published an articlewhich anticipated Turing's results by means of a different sort of logical
calculus, namely the lambda calculus, and an equivalent notion of
"computability", which Church called "effective calculability". With the help
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of W.H.A. Newman, for whose course Turing originally wrote his paper,
Turing went to study with Church at Princeton, where he subsequently
showed that his machine had the same power as Church's lambda calculus
or Stephen Kleene's recursive function theory for determining the range and
limitations of axiom systems for mathematics.(15) For the next several years,
all these schemes remained abstract devices of metamathematics. For quite
independent reasons, the war changed that.
15. Stephen C. Kleene,
"Origins of Recursive
Function Theory",Annals
of the History of
Computing3,1(1981), 52-67
II. Thinking Numerically - Computational
Thinking: The Computer
The Harvard Mark I and ENIAC marked the culmination of the
development of a mechanical calculator, and the latter marked the turning
point to electronic digital computation. John von Neumann first encountered
ENIAC in his role as mathematical physicist looking for means of rapid
numerical solution of non-linear partial differential equations.(16) But as he
talked with ENIAC's creators about the next version of their device, he
shifted his focus to a different sets of concerns. In the concept of the stored
program he not only saw a means of achieving a working device with the
capacity in principle to behave as a Turing machine, but he had a vision also
of what that capacity might mean for doing science. By analogy with
organisms viewed as natural automata, computers as artificial automata had
the potential to grow with the problems they were meant to solve. In
particular, he contemplated the conditions under which an automaton couldreplicate itself. While von Neumann imagined an actual machine floating in a
primeval sea of components, his colleague Stanislaw Ulam suggested instead
the model of a cellular automaton, that is a two-dimensional array of cells
each containing a finite automaton which changes its state as a function of the
states of the cells surrounding it. One could then ask about the possible
configurations of cells that would be capable of reproducing themselves in
the space of the cellular automaton.(17)
16. On von Neumann, see
William Aspray,John von
Neumann and the Origins
of Modern Computing
(Cambridge: MIT Press ,
1990).
17. Arthur Burks, "VonNeumann's Self-
Reproducing Automata", in
Papers of John von
Neumann on Computing
and Computer Theory, ed.
William Aspray and Arthur
Burks (Cambridge,
MA/London: MIT Press;
Los Angeles/San
Francisco: Tomash
Publishers, 1987), 491-552.
Von Neumann also pointed to a fundamental problem posed by the use of
the computer as a means of thinking about the world, and indeed about
thinking itself. To the extent that science seeks mathematical understanding,
that is understanding that has the certainty and analytical transparency of
mathematics, then one needed a mathematical understanding of the
computer. As of the early 1950s, no such mathematical theory of the
computer existed, and von Neumann could only vaguely discern its likely
shape:
There exists today a very elaborate system of formal logic, and,
specifically, of logic as applied to mathematics. This is a
discipline with many good sides, but also with certain serious
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weaknesses. This is not the occasion to enlarge upon the good
sides, which I certainly have no intention to belittle. About the
inadequacies, however, this may be said: Everybody who has
worked in formal logic will confirm that it is one of the
technically most refractory parts of mathematics. The reason
for this is that it deals with rigid, all-or-none concepts, and has
very little contact with the continuous concept of the real or of
the complex number, that is, with mathematical analysis. Yetanalysis is the technically most successful and best-elaborated
part of mathematics. Thus formal logic is, by the nature of its
approach, cut off from the best cultivated portions of
mathematics, and forced onto the most difficult part of the
mathematical terrain, into combinatorics.
The theory of automata, of the digital, all-or-none type, as
discussed up to now, is certainly a chapter in formal logic. It
will have to be, from the mathematical point of view,
combinatory rather than analytical.(18)
Neither here nor in later lectures did von Neumann elaborate on the nature
of that combinatory mathematics, nor suggest from what areas of current
mathematical research it might be drawn.
Over the two decades following von Neumann's work on automata,
researchers from a variety of disciplines converged on a mathematical theory
of computation, composed of three main branches: the theory of automata
and formal languages, the theory of algorithms and computational
complexity, and formal semantics.(19) The core of the first field came to lie in
the correlation between four classes of finite automata ranging from the
sequential circuit to the Turing machine and the four classes of phrase
structure grammars set forth by Noam Chomsky in his classic paper of
1959.(20) With each class goes a particular body of mathematical structures
and techniques.
18. John von Neumann,
"On a logical and general
theory of automata" in
Cerebral Mechanisms inBehavior--The Hixon
Symposium, ed. L.A.
Jeffries (New York: Wiley,
1951), 1-31; repr. in Papers,
391-431; at 406.
19. For more detail see my
"Computers and
Mathematics: The Search
for a Discipline of
Computer Science", in J.
Echeverra, A. Ibarra and T.Mormann (eds.), The Space
of Mathematics
(Berlin/New York: De
Gruyter, 1992), 347-61, and
"Computer Science: The
Search for a Mathematical
Theory", in John Krige and
Dominique Pestre (eds.),
Science in the 20th
Century (Amsterdam:
Harwood Academic
Publishers, 1997), Chap. 31.
20. Noam Chomsky, "On
certain formal properties of
grammars",Information
and Control2,2(1959), 137-
167.
Two features of the mathematics warrant particular attention. First, as the
study of sequences of symbols and of the transformations carried out on
them, theoretical computer science became a field of application for the most
abstract structures of modern algebra: semigroups, lattices, finite Boolean
algebras, -algebras, categories. Indeed, it soon gave rise to what otherwise
might have seemed the faintly contradictory notion of "applied abstract
algebra".(21) Second, as the computer became a point of convergence for a
variety of scientific interests, among them mathematics and logic, electrical
engineering, artificial intelligence, neurophysiology, linguistics, and computer
programming, algebra served to reveal the abstract structures common to
these enterprises. Once established, the mathematics of computation then
became a means of thinking about the sciences, in particular about questionsthat have resisted traditional reductionist approaches. Two examples of
particular importance to biology are Aristide Lindenmayer's L-systems, an
application of formal language theory to patterns of growth, and, more
21. See, for example, Garrett
Birkhoff and Thomas C.
Bartee,Modern AppliedAlgebra (New York:
McGraw-Hill, 1970) and
Rudolf Lidl and Gunter Pilz,
Applied Abstract Algebra
(NY: Springer, 1984).
22. Aristide Lindenmayer,
"Mathematical models for
cellular interactions in
development",Journal of
Theoretical Biology
18(1968), 280-99, 300-15. W.Fontana and Leo W. Buss,
"The barrier of objects:
From dynamical systems to
bounded organizations ", in
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recently, Walter Fontana's and Leo Buss's theory of biological organization
based on the model of the lambda calculus.(22)
J. Casti and A. Karlqvist
(eds.),Boundaries and
Barriers (Reading, MA:
Addison-Wesley, 1996),
56-116.
III. Generating the World
The computer is essential to those new approaches to biology, as it is to the
application of cellular automata to a range of physical, biological, ecological,
and economic investigations.(23) It is not a matter of calculating numbers
where analytic solutions are not possible, but rather of defining the local
interactions of a large number of elements of a system and then letting the
system evolve computationally, because we have neither the time nor the
mental capacity to derive that system. For example, rather than seeking a
numerical approximation to the non-linear partial differential equations of
fluid flow, one models the interaction of neighboring particles and displays
the result graphically. Instead of a mathematical function, what emerges is a
picture of the evolving system; an analytical solution is replaced by the stages
of a time series.
23. For a general view, see
Gary William Flake, The
Computational Beauty of
Nature: Computer
Explorations of Fractals,
Chaos, Complex Systems,
and Adaptation
(Cambridge, Mass: MIT
Press, 1998).
In other applications, the results may include new elements or new forms of
interaction among them. In particular, the system as a whole may acquire
new properties, which emerge when the interactions among the elements
reach a certain level of complexity. Precisely because the properties are a
product of complexity, that is, of the system itself, they cannot be reduced
analytically to the properties of the constituent elements. The current state of
mathematics does not suffice to gain analytical insight into the structures of
such systems, and hence, although the computer by its nature is
mathematical, we do not have means of understanding its mathematics, or
rather the computation does not afford mathematical understanding, certainly
not in the sense of Newton'sPrincipia.
In a certain sense, the notion of complexity as an emergent property of
systems governed locally by simple relationships may have lain inherent in themechanistic world view set forth in the seventeenth century. What was real
was matter in motion. Matter had no essential properties other than mass or
bulk, by which pieces of it occupied space to the exclusion of other pieces.
Motion was a matter of change of place with respect to time, brought about
by pieces of matter pushing one another around. None of this was directly
observable; rather, it underlay observation itself. What the senses perceived,
each in its own way, was changing patterns of matter impinging on matter.
The complexity of the world lay in the complexity of those patterns of
interaction. For Descartes, the behavior of heavy bodies referred to as"gravity" emerged from the interaction of the particles of the vortex
surrounding the earth.
At one level, this remains the case. The ultimate particles may embody a
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different catalog of essential properties, the laws of interaction may take a
different form, but nothing of the "new" science challenges the role of those
particles as the ultimate building blocks of the physical world. Similarly,
nobody doubts that life as we know it is a chemical phenomenon, resting in
principle on the interaction of fundamental particles. People now speak of
"carbon-based" life, using the qualifier to suggest that there could be some
other form but in so doing also accepting and reinforcing the premiss that life
is a form of chemistry reflecting the potential inherent in the physicalproperties of carbon and hydrogen, which properties themselves emerge
from the different numbers and configurations of the electrons, protons, and
neutrons that constitute the atoms
What has changed is the attitude toward the means of expressing the
relationships among the fundamental particles and of transforming
expressions at one level into expressions at another level. In thePrincipia,
Newton could capture the basic relationship of bodies attracting one another
by the expression ma = mm'/r
2
, where a by definition is d
2
S/dt
2
. Movingfrom small particles to large bodies was facilitated by being able to show that
forces among the constituents of a body conjoined to act as a single force
concentrated at its center of mass. The equation relating the forces of two
bodies acting on one another over a distance proved mathematically
tractable, in the sense that one could solve it in closed analytic form.
Unfortunately, the equation for three or more bodies, needed for any precise
mathematical account of the motion of the planets and in particular for the
motion of the moon about the earth, did not yield so easily to the techniques
of the calculus.
The subsequent articulation of the mechanical model of the physical world
increasingly challenged the capacity of mathematics to transform descriptions
at the level of the fundamental elements into descriptions at the level of direct
experience. The main root of the modern computer leads directly from the
need to substitute numerical approximations for differential equations that
could not be solved in closed form. That was especially the case for systems
of non-linear partial differential equations. The complexity of the systems
they described did not lie in the equations but in their solutions, and it was
admittedly a complexity that could not be captured in closed form. Now themodels are not expressed in a general differential equation characterizing the
whole system, which equation is then solved analytically or calculated with
the aid of a computer. Rather, they are described at the local level by means
of interactions with the immediate neighborhood, and the result is then
generated. Thus the sciences seem to have given up on mathematical
explanation.(24)
24. For the most recent and
perhaps most extreme
argument against thecapacity of traditional
mathematics and
mathematical phys ics to
encompass the complexity
of the world, see Stephen
Wolfram,A New Kind of
Science (Champaign, IL:
Wolfram Media, Inc. 2002).
Not entirely, however, and not without a struggle. John Holland, a pioneer in
the application of cellular automata to biology and the creator of geneticalgorithms, shows that some in the new field are not yet ready to surrender
the insights of mathematical analysis. In the concluding chapter ofHidden
Order: How Adaptation Builds Complexity, Holland looks "Toward
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Theory" and "the general principles that will deepen our understanding ofall
complex adaptive systems [cas]". As a point of departure he insists that:
Mathematics is our sine qua non on this part of the journey.
Fortunately, we need not delve into the details to describe the
form of the mathematics and what it can contribute; the details
will probably change anyhow, as we close in on our
destination. Mathematics has a critical role because it alongenables us to formulate rigorous generalizations, or principles.
Neither physical experiments nor computer-based experiments,
on their own, can provide such generalizations. Physical
experiments usually are limited to supplying input and
constraints for rigorous models, because the experiments
themselves are rarely described in a language that permits
deductive exploration. Computer-based experiments have
rigorous descriptions, but they deal only in specifics. A well-
designed mathematical model, on the other hand, generalizes
the particulars revealed by physical experiments, computer-
based models, and interdisciplinary comparisons. Furthermore,
the tools of mathematics provide rigorous derivations and
predictions applicable to all cas. Only mathematics can take us
the full distance.(25)
25. John H. Holland,
Hidden Order: How
Adaptation Builds
Complexity (Reading, MA:
Addison-Wesley,
1995)161-2.
Details aside, Holland's goal, with which he associates his colleagues at the
Santa Fe Institute, reflects a vision of mathematics that he and they share
with mathematicians from Descartes to von Neumann.
As von Neumann insisted in 1948, the mathematics will be different. To
meet Holland's needs it "[will have to] depart from traditional approaches to
emphasize persistent features of the far-from-equilibrium evolutionary
trajectories generated by recombination."(26) Nonetheless, his sketch of the
specific form the mathematics might take suggests that it will depart from
traditional approaches along branches rather than across chasms, and that it
will be algebraic. As the most recent work of Fontana on the lambda
calculus applied to chemistry suggests, it will be a mathematics of adecidedly modern sort. That is to be expected. The Turing machine is a
modern concept, conceived by a thinker who was nothing if not a
reductionist. His 1936 paper sets on computation, and thus on computing
machines, limits that are no less firm and no less universally accepted than
the constraints of the laws of thermodynamics or of the constant speed of
light.
26.Ibid., 171-2.
Today we confront the question of whether the computer, the newest and
leading medium of scientific thought can be comprehended mathematically,
i.e. in some way algebraically or analytically. If so, then it will be viewed as
the newest chapter of a history that began in the 17th century with the
beginning of algebraic thought. If not, then perhaps fifty years from now
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someone will be giving a lecture on the topic of "The End of Algebraic
Thought in the 20th Century."