International Conference Ars Conjectandi 1713 – 2013

October 15 -16, 2013. Basel, Switzerland

Edith Dudley Sylla

Jacob Bernoulli, changing views

Ars Conjectandi

Tercentennial� This is our opportunity to focus on Jacob

Bernoulli and Ars Conjectandi rather

than treating it as one important link in a

chain of influence from the

correspondence of Pascal and Fermat in

1654 to the axiomatization of probability

by Kolmogorov in 1933.

� This focus enables us to see Bernoulli’s

work much more clearly and accurately.

Why choose the composition?

� By looking at the development of the

ideas that went into Ars Conjectandi

from the earliest evidence about 1685

up through Bernoulli’s death in 1705,

along with Bernoulli’s other writing in the

same years, we have a better chance of

understanding Bernoulli’s perspective on

the application of mathematics to the art

of conjecturing.

Distortions of the picture

� The received picture of the work on

mathematical probability of Jacob Bernoulli

and of what is taken to be his main

contribution to mathematical probability –

his “weak law of large numbers” – has

many larger and smaller distortions.

� E.g. the statement that Jacob’s nephew

Nicolaus Bernoulli edited Ars Conjectandi,

which is repeated ad infinitum.

Compounding the error: The Unfinished

Game (New York: Basic Books, 2008):

� “Undeterred by Leibniz’s reply, Bernoulli

continued his investigation, and in the

two years that remained to him, he

made considerable progress. (After he

died, in 1705, his nephew Nikolaus

Bernoulli began organizing his uncle’s

results into publishable form, a task so

challenging that it took him eight years

before Ars Conjectandi was published.)”

The evidence: Nicolaus Bernoulli

in his prefatory letter to AC� “The publishers might indeed have hoped

that the brother of the deceased…would supply what was missing…. They also thought of giving the job to me… but I could not undertake it because I was away traveling. When I came back into the country and was asked again, I declined the job because I felt myself unequal to it…I also advised that the treatise, already in large part printed, should be communicated to the public in the form in which the Author left it.”

The evidence: Johann Bernoulli

� Letter of Johann Bernoulli to Pierre Varignon, 29 April 1713: “I can tell you that neither the publisher, nor the printer, nor the copyeditor understands the slightest thing about the subject matter that the book treats. Perhaps the neatness with which my late brother normally copied his writings will supply the defect of the ignorance of these people.”

What really happened

� When Jacob Bernoulli knew he would soon die, he left instructions for what should be done with his unpublished work. He did not want his brother, Johann Bernoulli, or his nephew, Nicolaus Bernoulli to handle his papers. Instead, he asked that his former student Jacob Hermann organize his Nachlass and and send information for éloges and that his son take his papers to Pierre Varignon in Paris, so he could recommend what was worth publishing.

Nicolaus Bernoulli “the younger”

� Jacob’s son, who was also named Nicolaus Bernoulli and who, like his cousin, was born in 1687, went to Paris to study painting and interacted with Pierre Varignon, but he did not take his father’s papers.

� The manuscript of Ars conjectandiremained with Jacob’s immediate family until it was taken to the Thurneysen brothers to be published in 1713.

The Bernoulli mathematicians

� The Bernoullis are “the most renowned

family in the history of mathematical

sciences” (Stephen Stigler)

� This leads to a confusion between

Bernoulli mathematicians – those who

have the same first name are given

numbers to keep them straight – and

also between the mathematicians and

those who had other careers.

Nicolaus Bernoullis alive in 1705.

� Jacob’s father Nicolaus Bernoulli (spice merchant), 1623 – 1708.

� Jacob’s brother Nicolaus Bernoulli “the elder” (painter), 1662 – 1716.

� Jacob’s son Nicolaus Bernoulli “the younger” (painter), 1687 – 1769.

� Jacob’s nephew Nicolaus I Bernoulli, son of Nicolaus “the elder,” 1687 – 1759.

� Jacob’s nephew Nicolaus II Bernoulli, son of Johann I Bernoulli, 1695 – 1726.

Confusion between brothers

Johann and Jacob Bernoulli

� Jacob Bernoulli, 1654 – 1705

� Johann I Bernoulli, 1667 – 1748

� Then there is the next generation:

� Nicolaus I Bernoulli, 1687 – 1759

� Daniel Bernoulli, 1700 – 1782

� Nicolaus II Bernoulli, 1695 – 1726

� Johann II Bernoulli, 1710 – 1790

Concentrating on Jacob Bernoulli

� Makes it easier to see how slowly the

epistemic concept of probability was

coupled to the mathematics of games of

chance.

� Bernoulli’s point was that a responsible

person in making decisions should pay

attention to frequencies, costs, and

benefits.

Jacob Bernoulli. Professor at

Basel University and Fellow of

the Royal Societies of France

and Prussia. Most celebrated

mathematician,

The Art of Conjecturing

Posthumous work

To which is added:

Treatise on Infinite Series

And Letter written in French

On the Game of Tennis

Emblem with the words:

Make haste slowly

Basel:

Thurneysen Brothers expense. 1713.

Ars Conjectandi: the complete book

1).Thurneysen Brothers, Basel, 1713.

2). Reprint: Culture et Civilisation,

Bruxelles, 1968.

3. Translated to English: Jacob Bernoulli,

The Art of Conjecturing together with

Letter to a Friend on Sets in Court

Tennis, trans., introd. and notes by Edith

Dudley Sylla, Johns Hopkins University

Press, 2006.

Content of 1713 volume:

� 1. The Art of Conjecturing (239 pages)

� 2. On infinite series (66 pages)

� 3. Letter to a Friend on Parts (Parties)

in the Game of Tennis (in French, 35

pages)

On infinite series

Originally appeared in M.A. oral exams of

Jacob Bernoulli’s students, Jacob Bernoulli

presiding:

Part I. defended by Jacob Fritz, 1689

Part II. defended by Jerome Beck 1692

Part III. defended by Jacob Hermann 1696

Part IV. def. by Nicolaus Harscher 1698

Part V. def. by Nicolaus I Bernoulli 1704

Letter to a friend complements

the book Ars Conjectandi (=AC)

� This is the best example we have of how Bernoulli expected that his proposal to learn ratios of possibilities or “cases” a posteriori or by experience would work in developing the art of conjecturing.

� Before proving his fundamental theorem, Bernoulli expected that the mathematics of Huygens’ On reckoning, and the math of combinations and permutations would be his main tools.

Letter to a Friend cont.

� For predicting who will win in tennis,

Bernoulli proposes observing the results

of a couple hundred strokes, not

thousands. He is not aiming for very

high probability of precise accuracy.

This means he was likely not

discouraged by the need for thousands

of observations to obtain a probability of

a thousand to one as he calculates in

Ars Conjectandi.

Make haste slowly

� Emblem on title

page of AC includes

the words “Festina

lente”, meaning

“Make Haste

Slowly.” Whoever

chose this emblem

or why, it was

appropriate to Jacob

Bernoulli – slow but

sure progress.

Bernoulli working on AC in 1680s

� As the 11th corollary to the triple

problems – arithmetical, geometrical,

and astronomical – that Bernoulli offered

in competition for the vacant chair of

mathematics at Basel in 1687 was a

problem “From the art of conjecturing”

concerning a lottery with 16,000 tickets

and how the expectation would increase

with the removal of 1000 blank tickets.

Contents of Ars Conjectandi

� Part I. Republication of Christiaan

Huygens, On reckoning in games of

chance, with comments by Jacob Bernoulli.

� Part II. On combinations and permutations.

� Part III. More complicated games.

� Part IV. Application of the foregoing

mathematics to civil, moral, and economic

matters. Ends with fundamental proposition

(breaks off incomplete).

Bernoulli and Huygens.

� Interpreting the mathematics of games of chance.

� Part I of Ars Conjectandi. Bernoulli’s comments on Huygens’ De ratiociniis in ludo aleae.

� Huygens’ book was not about mathematical probability.

Huygens’ work is key for Bernoulli

� The initial mathematical tools for Bernoulli’s Part I came from Huygens’ On reckoning in games of chance.

� Huygens’ work was based on earlier practical or commercial arithmetic books.

� These in turn had inherited a great deal from Arabic books on algebra, which had been applied in Islamic inheritance mathematics, where a frequent issue was the dissolution, because of death, of a business partnership before the end planned in the initial contract.

Huygens’ fundamental principle

� “A person’s lot or expectation to obtain

something in a game of chance should

be judged to be worth as much as an

amount such that, if he had it, he could

arrive again at a like lot or expectation

contending under fair conditions.”

Bernoulli comment on Huygens

� “I will try to demonstrate [Huygens’

fundamental principle] by reasoning that

is more popular…and more adapted to

common comprehension. I posit only

this as an axiom or definition: Anyone

may expect, or should be said to expect,

just as much as he will acquire without

fail.”

Acquire without fail?

� This only follows by what Bernoulli

called “institutional necessity,” namely

only if there are other people willing to

play games and rules for the games or,

in economics, other people willing to

take part in business partnerships and

legal structures governing business

activity.

Books II and III

� Adding a more thorough study of

combinations and permutations to the

mathematics already found in Huygens’

book and in Bernoulli’s notes (e.g.

algebra, analysis, infinite series,

logarithms, intersection of curves on

graphs, etc.)

� Bernoulli’s lecture De Arte Combinandi

when he became dean 1692

Book IV. Applications to civil,

moral and economic problems

� Definitions of certainty, probability,

necessity, and contingency.

� Physical, hypothetical, and contractual

or institutional necessity.

� Contingency, fortune, and luck.

� Knowledge and conjecture. The art of

conjecture.

� Pertinent general axioms.

Definition: art of conjecture

� “To conjecture about something is to measure its probability. Therefore we define the art of conjecture, or stochastics, as the art of measuring the probabilities as exactly as possible, to the end that, in our judgments and actions, we may always choose or follow that which has been found to be better, more satisfactory, safer, or more carefully considered. On this alone turns all the wisdom of the philosopher and all the practical judgment of the statesman.”

General axioms or ethical rules for the

art of conjecturing. Rule 1.

� There is no place for conjectures in

matters in which one may reach

complete certainty.

Rule 2.

� It is not sufficient to weigh one or

another argument. Instead we must

bring together all arguments that we can

come to know and that seem in any way

to work toward a proof of the thing.

Rule 3.

� We should pay attention not only to

those arguments that serve to prove a

thing, but also to all those that can be

adduced for the contrary, so that, when

both groups have been properly

weighed, it may be established which

arguments preponderate.

Rule 4.

� Remote and universal arguments are

sufficient for making judgments about

universals, but when we make

conjectures about individuals, we also

need, if they are at all availaable,

arguments that are closer and more

particular to those individuals.

Rule 5.

� In matters that are uncertain and open

to doubt, we should suspend our actions

until we learn more. But if the occasion

for action brooks no delay, then between

two actions we should always choose

the one that seems more appropriate,

safer, more carefully considered, or

more probable, even if neither action is

such in a positive sense.

Rules 6 and 7.

� What may help in some case and can

harm in none is to be preferred to that

which in no case either helps or harms.

� We should not judge the value of human

actions by their results, since sometimes

the most foolish actions enjoy the best

success, while the most prudent actions

have the worst result.

Rule 8.

� In our judgments we should be careful

not to attribute more weight to things

than they have. Nor should we consider

something that is more probable than its

alternatives to be absolutely certain or

force it on others. For it is necessary

that the confidence we ascribe to any

particular thing be proportioned to the

degree of certainty the thing has….

Rule 9.

� Because…it is rarely possible to obtain certainty … complete in every respect, necessity and use ordain that what is only morally certain be taken as absolutely certain.

� It would be useful, accordingly, if definite limits for moral certainty were established by the authority of the magistracy. For instance, it might be determined whether 99/100 of certainty suffices or whether 999/1000 is required.

Bernoulli’s fundamental proposition.

� The last completed section of Ars

Conjectandi contains Bernoulli’s proof of

a fundamental proposition, now called

the “weak law of large numbers.”

� It does not contain further applications of

the proposition to sample legal, moral,

and economic problems as planned.

Bernoulli’s introduction to the proof of

his principal proposition:

� “It was shown in the preceding chapter

how from the numbers of cases in which

arguments for things can exist or not

exist, indicate or not indicate, or also

indicate the contrary, and from the

forces of proving proportionate to them,

the probabilities of things can be

reduced to calculation and evaluated.”

Bernoulli’s introduction, cont.

� “From this it resulted that the only thing

needed for correctly forming conjectures

on any matter is to determine the

numbers of these cases accurately and

then to determine how much more easily

some can happen than others.”

Bernoulli’s introduction, cont.

� “But here we come to a halt, for this can hardly ever be done. Indeed, it can hardly be done anywhere except in games of chance. The originators of these games took pains to make them equitable by arranging that the numbers of cases resulting in profit or loss be definite and known and that all the cases happen equally easily. But this by no means takes place with most other effects that depend on the operation of nature or on human will.”

Bernoulli’s introduction, cont.

� “Nevertheless, another way is open to us

by which we may obtain what is sought.

What cannot be ascertained a priori, may

at least be found out a posteriori from the

results many times observed in similar

situations, since it should be presumed

that something can happen or not happen

in the future in as many cases as it was

observed to happen or not to happen in

similar circumstances in the past.”

Bernoulli’s introduction, cont.

� “This empirical way of determining the

number of cases by experiment is neither

new nor uncommon…. Neither should it

escape anyone that to judge in this way

concerning some future event it would not

suffice to take one or another experiment,

but a great abundance of experiments

would be required.”

Bernoulli’s introduction, cont.

� “Even the most foolish person, by some

instinct of nature, alone and with no previous

instruction (which is truly astonishing) has

discovered that the more observations of this

sort are made the less danger there will be of

error. But although this is naturally known to

everyone, the demonstration by which it can

be inferred from the principles of the art [of

conjecturing] is hardly known at all, and,

accordingly it is incumbent on us to expound it

here.”

Bernoulli’s introduction, cont.

� “But I would consider that I had not achieved enough if I limited myself to demonstrating this one thing, of which no one is ignorant. Something else remains to think about, which perhaps no one has considered up to this point. It remains, namely, to ask whether, as the number of observations increases, so the probability increases of obtaining the true ratio between the numbers of cases in which some event can happen and not happen, such that this probabilitiy may eventually exceed any given degree of certainty.”

Bernoulli’s introduction, cont.

� “Or whether, instead, the problem has

an asymptote, so to speak: whether, that

is, there is some degree of certainty that

may never be exceeded no matter how

far the number of observations is

multiplied, so that, for example, we may

never be certain that we have

discovered the true ratio of cases with

more than a half or two-thirds or three-

fourth parts of certainty.”

Urn model for the question

� “Suppose that there are hidden in an urn…three thousand white tokens and two thousand black, and that , in order to investigate their number by experiments, you take out one token after another (but each time putting back the one that you have taken out before you choose the following one, lest the number of tokens in the urn be diminished) and you observe how many times a white token comes out and how many times a black one.”

Urn model for the question, cont.

� “It is asked whether you can do this so many times that it becomes ten, a hundred, a thousand, etc. times more probable (that is, that in the end it becomes morally certain) that the numbers of times in which you have chosen a white and in which you have chosen a black will have to each other the same ratio of three to two that the numbers of tokens or of cases secretly enjoy than some other different ratio.”

What is to be shown� “It is to be shown that so many experiments

can be taken that it becomes any given

number of times (say c times) more likely

that the number of fertile observations will fall

between these bounds than outside them,

that is that the ratio of the number of fertile to

the number of all the observations will have a

ratio that is neither more than

(r + 1)/ t nor less than (r – 1) t.”

Proof of the proposition

� The proposition is proved on the basis of

five lemmas.

� The proof is mathematically rigorous.

� The proof is pure not applied or

concrete mathematics. It does not

represent a physical law.

The proof of the theorem & its interpretation.

� Assuming that you have a binomial distribution, the proof shows that if higher and higher powers of the binomial are taken, more and more of the distribution will be concentrated around the center of the distribution, and this is without limit.

� Afterwards, interpreting the terms of the binomial distribution as representing combinations of observed results, the theorem can be used to show how many observations are needed for what degree of probability that the observed ratios will be within a certain interval around the unknown underlying ratio.

Interpretations of binomial expansions.

� (r + s)2 = r2 + 2rs + s2

� Interpretation: in tossing a coin (one coin twice or two coins together once) there are twice as many ways to get one heads and one tails (i.e. rs) as there are to get two heads (r2 ) or two tails (s2).

� (r + s)4 = r4 + 4 r3s + 6 r2s2 + 4 rs3 + s4

� Interpretation: Between the bounds of 1:4 and 3:4 surrounding 2:4 on each side there are fourteen possible outcomes compared to only two outcomes (4:4 and 0:4) outside the bounds. Now take much higher powers.

Bernoulli’s view of his result

� “Nota bene: I esteem this discovery

more than if I had given the quadrature

of the circle itself, which, even if it were

found very great, would be of little use.”

� In Meditationes (his research notebook),

art. 173 (Bernoulli 1975: 88)

Why does the book stop here?

� As Bernoulli told Leibniz in 1703, the lack of suitable legal, moral, or economic applications held up the completion his book after he had finished the proof of his fundamental theorem.

� Sylla: The theorem made it imperative to find applications that used ratios of outcomes found by experiment or repeated observation, as exemplified in Bernoulli’s Letter to a Friend. These Bernoulli lacked.

Looking for suitable problems

� On 3 October 1703 Bernoulli wrote to Leibniz:

� “I would very much like to know, dear sir, from whom you have it that a theory of estimating probabilities has been cultivated by me. It is true that several years ago I took great pleasure in this sort of speculations, so that I could hardly have thought any more about them. I had the desire to write a treatise on this matter. But I often put it aside for whole years because my natural laziness compounded by my illnesses made me most reluctant to get to writing. I often wished for a secretary who could easily understand my ideas and put them down on paper.”

Bernoulli to Leibniz October 1703, cont.

� “Nevertheless, I have completed most of the book, but there is lacking the most important part, in which I teach how the principles of the art of conjecturing are applied to civil, moral, and economic matters. [This I would do after] having solved finally a very singular problem, of very great difficulty and utility. I sent this solution already twelve years ago to my brother, even if he, having been asked about the same subject by the Marquis de l’Hôpital, may have hid the truth, playing down my work in his own interest.”mmSylla (2006), p. 36.

Examples mentioned in Part IV.

� Weighing of evidence in court cases

� When can someone who has

disappeared be assumed dead?

� Dice games

� Validity of contracts

� Relative life expectancies (this last

would have been the most promising)

Urgent requests to Leibniz

� In the last years of his life, in letters to

G. W. Leibniz, Bernoulli asked several

times to borrow a copy of Jan de Witt’s

work in Dutch on annuities (Leibniz

could not find the copy he had

misplaced somewhere among his

papers).

Jan de Witt on annunities

� De Witt’s book actually used Huygen’s

formulas in calculating the different

expectations involved in lifetime

annuities and permanent annunities, so

it might not have been as helpful as

Bernoulli imagined.

� What was needed was data to calculate

life expectanies, expectancies for two

lives, etc.

Leibniz also had questions about

Bernoulli’s fundamental theorem

� In his replies, Leibniz (not knowing the

details) also questioned the

assumptions of the proof of Bernoulli’s

principal proposition.

Leibniz argued, for instance:

� Unlike tokens in an urn, we don’t know

how many different sorts of germs of

disease may be in the human body, and,

in any case, new diseases might arise.

� Bernoulli answered that if there were

new diseases, the given data and

calculations based on them would have

to be changed accordingly.

What the theorem potentially allows

� The theorem shows that learning ratios of possible outcomes (the unknown “cases” lying behind effects) by experience is feasible and that there is no intrinsic limit to its reliability if enough observations are made.

� Implication: Natural and human underlying cases, while far more multifarious and complex, will behave like tokens hidden in an urn, dice, or playing cards. But is it true?

How did Bernoulli justify his assumption of a

binomial distribution? By reference to God.

� “Objectively, certainty means nothing

else than the truth of the present or

future existence of the thing….”

� “In themselves and objectively, all things

under the sun, which are, were, or will

be, always have the highest certainty.

This is evident concerning past and

present things, since by the very fact

that they are or were, these things

cannot not exist or not have existed.”

Bernoulli’s justification, cont.

� “Nor should there be any doubt about

future things, which in like manner, even if

not by the necessity of some inevitable

fate, nevertheless by divine foreknowledge

and predetermination, cannot not be in the

future. Unless, indeed, whatever will be will

occur with certainty, it is not apparent how

the praise of the highest Creator’s

omniscience and omnipotence can prevail.”

Ars Conjectandi and the origins of

mathematical probability

� Jerzy Neyman (1976):

� “With an unavoidable degree of

oversimplification, one might say that

the theory of probability started in 1713,

with the publication of the book Ars

Conjectandi, by Jacob Bernoulli.”

Reasons to begin mathematical

probability with Ars Conjectandi

� Bernoulli is the first to mathematize

probability so named and not only

expectations in games of chance.

� Bernoulli proposes taking mathematics

beyond games of chance to civil, moral,

and economic decisions.

� His fundamental proposition implies that

one may learn relative likelihoods from

experience or a posteriori.

First Objection

� Mathematical probability is usually

thought to have started with the 1654

correspondence of Pascal and Fermat,

which led to Huygens’ On reckoning in

games of chance.

Reply: Pascal and Fermat did not

influence Jacob Bernoulli

� The important input to Ars Conjectandi

was Huygens’ De ratiociniis in ludo

aleae. Huygens had not seen Fermat’s

and Pascal’s methods and neither had

Bernoulli. Bernoulli had not seen

Pascal’s work on the arithmetic triangle,

but in any case he already knew most of

what it contained.

Huygens is important but his concern is

not mathematical probability.

� The mathematics of games of chance

was based on lot or expectation. It was

not understood to involve something

called “probability.”

The central role of lot or expectation

� Huygens’ and Bernoulli’s conceptions of

calculations in games of chance treat games

like business contracts or partnerships.

� The assumption is that business partnerships

should be equitable or fair. What you stand to

gain (your expectation) should depend on

your investment of capital (money or

equipment such as ships) and labor.

Second Objection

� Bernoulli’s fundamental theorem has

narrow application, since it applies only

to binomial distributions.

� True, but Bernoulli’s work quickly led

(even before it was published) to the

work of others, especially Abraham De

Moivre, who enlarged the compass of

mathematical probability.

Siméon Denis Poisson, 1836:

� In the preamble that I read some months ago at the Academy … I considered the law of large numbers as a fact that we observe in things of all kinds.... One should not confuse this general law with the beautiful theorem of Jacob Bernoulli, on the demonstration of which he meditated, as is known, for twenty years. According to this theorem, outcomes occur very nearly, in a long series of trials, in proportion to their respective probabilities. But one should not lose sight of the fact that he supposes that the chances remain constant, while, to the contrary, the chances of physical phenomena and of moral matters almost always very continually, without any regularity and often to a great extent.

Mathematics vs. physics

� Jacob Bernoulli has in mind a clear

distinction between pure or abstract

mathematics and concrete mathematics.

� What Bernoulli offered in Ars

Conjectandi was the beginnings of a

concrete mathematical discipline, but

without its empirical foundation.

Jacob Bernoulli, Theses Logicae de conversione et

oppositione enunciationum (12 February 1686)

� Miscellaneous Thesis XI. The concrete

mathematical disciplines such as physics,

medicine, astronomy, optics, statics, ballistics

(and if you wish astrology), etc., add to abstract

mathematics only certain principles, as

foundations. [These principles] are partly proved

elsewhere and partly drawn from experience

alone. On these principles one may reason

further with no less geometrical rigor than one

reasons in abstract mathematics on the basis of

common notions or innate axioms.

Miscellaneous Thesis XI, continued.

� Thus physics presupposes the laws of motion; medicine supposes the fabric of the human body; astronomy the fabric or system of the world; astrology the influx of the stars on sublunar things and that the fate of men, of cities, and regions, depends on the configuration of the heaven that obtains when they are brought forth into the light or take on their original parts; catoptrics assumes that the angles of incidence and reflection are equal; dioptrics that the sines of the angles of incidence and refraction are proportional; statics that the moments increase with distance from the fulcrum; ballistics that the spaces traversed by a falling weight are as the squares of the times.

Miscellaneous Thesis XII

� Thesis XII. Whence it is clear that the certitude of these sciences depends uniquely on the certitude of these principles and not on the mode of forming conclusions, all of which should be deduced from the principles by the most evident reasoning. This is the reason why abstract mathematics is of unconquered certainty, why astrology is vain and futile, while the others are of a middle certainty between these two. This is because such are the principles on which they are erected (Bernoulli 1744: 233 – 234).

Where does ‘probability’ enter

Bernoulli’s fundamental theorem?

� Only in the probability that the observed ratio of outcomes will fall within the chosen interval rather than outside:

� “It is to be shown that so many experiments can be taken that it becomes any given number of times (say c times) more likely [verisimilior] that the number of fertile observations will fall between these bounds than outside them, that is, that the number of fertile to the number of all the observations will have a ratio that is neither more than (r + 1)/t nor less than (r - 1)/t .

Bernoulli’s probability is epistemic

� Although the sorts of mathematics that

appear in Ars Conjectandi would later

become part of mathematical probability,

Bernoulli defined ‘probability’ as degree

of certainty in what he called the

subjective sense. On the other hand, he

thought that judgments made in civil,

moral, and economic questions should

take account of relative frequencies.

Probability of knowledge

� “Subjectively, certainty is the measure of

our knowledge concerning the truth.”

� “Seen in relation to us, the certainty of

things is not the same for all things, but

varies in many ways, increasing and

decreasing…. Other things receive a less

perfect measure of certainty in our minds,

greater or less in proportion as there are

more or fewer probabilities that persuade

us that the thing is, will be, or was.”

For Bernoulli, probability is not a

property of things, but of opinions.

� “It is most certain given the position, velocity, and the distance of a die from the gaming table at the moment when it leaves the hand of the thrower, that the die cannot fall other than the way it actually does fall.”

� “these effects follow from their own proximate causes no less necessarily than the phenomena of eclipses follow from the motion of the heavenly bodies.”

Bernoulli’s definition of probability

� “Probability, indeed, is degree of certainty, and differs from the latter as a part differs from the whole. Truly, if complete and absolute certainty, which we represent by the letter a or by 1, is supposed, for the sake of argument, to be composed of five parts or probabilities, of which three argue for the existence or future existence of some outcome and the others argue against it, then that outcome will be said to have 3a/5 of 3/5 of certainty.”

Is this mathematical probability?

� In Ars Conjectandi the mathematics of

games of chance as found in Huygens’

On reckoning in games of chance, and

the mathematics of combinations and

permutations were to be applied by

analogy to decision making in legal,

moral, and economic problems. But this

does not mean that probability was in

things.

Mathematics and philosophy

� Like Huygens and van Schooten, Bernoulli wanted to demonstrate the capability of mathematics to serve in all the activities of daily and civic life. Both Jacob Bernoulli and Johann I Bernoulli in their public lectures extolled the power of mathematics.

� Nevertheless he was also astute in epistemology or the philosophy of science.

Basing decisions on evidence

� In the Art of Conjecture, the decision

maker was to use relative frequencies

as part of the evidence to increase the

probability of making a good decision.

So probabilities and frequencies were

closely related, but they were not the

same thing

Bernoulli’s character and goals

� More evidence than I have time to

review indicates that Bernoulli hoped to

contribute to human betterment by the

writing of his book. He thought that

mathematics was the best possible tool

for solving problems.

� Because of the so-called Streitschriften

between Jacob and Johann Bernoulli,

he has been misrepresented.

Evidence of Bernoulli’s character

� In the Acta Eruditorum of December 1695, after discussing some of his controversies with Johann, Jacob wrote:

� “… so that those who enjoy the history of invention may know who deserves what credit… It is better here to take the attitude of Columbus, who, to his friends who envied him for his discovery of the new world, was led to respond, not by boasting or exaggerating the magnitude

AE 1695 continued

� of his discovery, but by proposing to

them a question about something

simple. In this modest way he wanted to

imply to them that they could have found

the same thing if they had looked…..

Indeed I am far from thinking that we

can give ourselves much glory fo rthe

difficulty or subtlety of what we have

discovered. I am more persuaded that

AE 1695 continued

� we [prob. Jacob and Johann] cannot brag

of anything that could not have come into

the mind of anyone of modest talents

instructed in our principles. Just as in

nature, so in science, there are no leaps.

All our thinking, like a quantity, grows by

steps and gradually increases, so that form

one degree to the next following, it only

requires, so to speak, an infinitely small

step…. The reason why all of us do not

AE 1695 cont.

� discover everything, is that there is so

great a multitude of things that no one

has sufficient leisure or occasion to pay

attention to everything…. Two similar

investigators with equal ingenuity may

observe the same unknown lands and

the two may return home loaded with

new spoils, which neither one, who

plundered the land alone, could carry.”

Battier’s memorial speech, 1705

� “Certainly our Bernoulli aspired in his

papers to do that which would reveal the

traces of his Creator in His works. Nor is

the contemplation of mathematics so sterile

that it does not supply assistance in

improving the quality of our civil life….

Even those singlemindedly devoted to

mathematical contemplations, who do not

wish to advance mechanical operations by

the application of mathematics -- thinking

Battier, continued

� like the ancient Plato and Archimedes

that it is beneath the dignity of the free

man – even those, by their efforts and

vigils, discover new principles that later

are applied by others in the invention of

the most useful things which

marvelously serve the human race.

Thus no one has justly said that the

vigils of mathematicians are not fruitful.”

Jacob Bernoulli’s emblem and motto

� Jacob Bernoulli had chosen as his

emblem Phaeton, who stole his father’s

chariot, drove too close to the Sun, and

fell. Phaeton’s horse has wings.

� Jacob chose as his motto:

� Ad astra invito patre or Invito patre

sidera verso (to the stars despite my

father; vs. the usual Invita Minerva)

Phaeton: Ad astra invito patre

(to the stars despite my father)

Phaeton, Seeking the stars (i.e. choosing mathematics) crashing?

Trouble with his father?

� The fact that Phaeton crashes brings in

a bit of irony or self-effacement to

Bernoulli’s choices.

� When Bernoulli was suspended from the

faculty senate for his part in the 1691

Wesen or uprising in Basel, his father

helped to bring about his reinstatement.

Mathematics and philosophy

� When we are able to focus on Jacob

Bernoulli, we can understand what he

meant by his fundamental proposition

more clearly.

� We can also see that the philosophical

context into which he put his Art of

Conjecturing is of interest and relevance

as well as his mathematics.