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The modern axiomatic approach to probability was established by Kolmogorov nearly 70 years ago. According to Wikipedia, the first ideas connected to a mathematical theory of probability arose with Cardano, Fermat, Pascal during the 16th century.

I find quite surprising that probability as a mathematical theory has emerged so late in time. In particular, as far as I know, ancient Greeks, that were skilled geometers (and mathematicians in general), did not even try to address this problem. One could argue that probability was not necessary at that time from a practical viewpoint while, for instance, geometry was more useful in everyday life. I think that this is not completely true. Indeed, it is well-known that ancient Greeks were used to bet or to play games that involved dices and, therefore, probability.

Hence, my question: is it true that the first attempts to connect probability and mathematics date back to the 16th century? If the answer is affirmative, what are the most probable reasons of this late development? And in particular, why didn't the ancient Greeks give any contribution to this branch of mathematics despite its practical usefulness?

Thanks.

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This is a good point, I mused about it too. First, Pythagoreans and Plato had a very high minded idea of mathematics, gambling would have been seen as a lowly pursuit. This in itself does not explain it however. Plato's successors at the Academy considered mechanics lowly too, for it "uses bodies needing much vulgar manual labour, mechanics was thus distinguished as falling outside of geometry, and, since it was for a long time disregarded by philosophy, it has become one of the military arts", as Plutarch relates. That did not stop Ctesibius (the inventor of the water organ, first feedback controlled water clock, and an early head of the Alexandrian Museum), Archimedes and Apollonius. But gambling is not even part of the military art, and we should not underestimate the scarcity of ancient sources, none of Ctesibius's or Apollonius's works on mechanics survive, and of Archimedes's only one does, On Floating Bodies, and even that one was preserved for centuries in a single copy, according to Russo. One can only imagine how medieval monks might have felt about copying manuscripts on gambling.

There was also a strong streak of fatalism, both in Greek mythology and philosophy, recall Homer's Iliad, the three Fates, the myth of Oedipus, the oracle of Delphi, and the influential Stoic philosophy, which validated divination by asserting unbreakable causal chains through which the present predetermines the future (what we call hard determinism). Chrysippus, the leader of Stoics, was not a superstitious hack, he invented propositional logic two millenia before Leibniz and Frege, and was held on a par with Aristotle. His works did not survive either, by the way. Nor did Hipparchus's rather sophisticated combinatiorial computation related to Chrysippus's logic (the number of compound propositions that can be formed from ten simple ones). If people thought that "random" outcomes could be divined or influenced, the oracle of Delphi and sibyls were a better option than binomial coefficients. Epicureans, who opposed fatalism and recognized true chance ("atoms swerve"), were in turn hostile to mathematics, because... Plato was for it.

Gambling was not the only place, where ancients could have used probability and statistics, according to Gingerich "the real world of observed data is rarely as clean as the pure mathematics of Euclid. As Ptolemy wrestled with errors of measurement without any error theory, he was repeatedly forced into compromises to reconcile discordant observations... Clearly Ptolemy’s notion of an observation involved a certain level of processing and not raw data. Interpolation between other observations to produce an “observation” on a critical day required for symmetry is by no means excluded". Traces of this "processing" are multiple in the Almagest, but Ptolemy's "methods" appear to be sporadic, and he is silent on the subject.

This "blind spot" is not confined to Greeks, we find no probabilistic ideas in Babylonian sources, whose commercial prowess seems to invite them, nor in ancient China where the Book of Change, I Ching, with its elaborate permutations of sticks, does too. Ironically, I Ching was used for divination. In the end, I am not sure that we can be more definitive than "the stars just weren't aligned for it" before Cardano.

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  • $\begingroup$ (+1) Especially for "the oracle of Delphi and sibyls were (thought to be) a better option than binomial coefficients". By the way, the sayings of these professional oracles can be seen as carefully constructed "fair coins", so that the ex post interpretation of them could be contradictory in a perfectly balanced way. $\endgroup$ – Alecos Papadopoulos Mar 11 '16 at 13:26
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Imho the Greeks would have considered probabilities as a sophism - the attempt to produce knowledge out of ignorance. And even today they are still not far from truth: except for the frequentist interpretation which is objective, the rest is subjective (mostly the Bayesian kind). Kolmogoroff axiomatic evades the interpretation problem but so it is also known as theory of measure and thus would not be recognized as a separate endeavor from geometry. And one should not forget Bertrand's paradoxes that continue to poison probabilistic arguings.

If we agree that Plato has had some influence on people with scientific-philosophic interests from the next few generations, it is rather obvious why they could not conceive the idea of probability except as ignorance. For Plato it was the maths and ideas (forms) that were really real while the changing physical world consisted of mere shadows. Time being excluded from this reality it is difficult to conceive probabilities. And of course there is nothing negative, - unknown or uncertain, in the domain of true knowledge.

Aristotle had no interest in mathematics so we cans skip him and his early followers. The popular Stoics held a doctrine of eternal recurrence which is obviously incompatible with probabilities.

So except for the atomists it is difficult to see who would have entertained seriously the idea of probability theory but apparently they had an other agenda.

Re:refs. Aristotle's Rhetoric (Book III ch5) gives advice for good style:

"avoid ambiguities, unless, indeed, you definitely desire to be ambiguous, as those do who have nothing to say but are pretending to mean something. Diviners use these vague generalities about the matter in hand because their predictions are thus, as a rule, less likely to be falsified. We are more likely to be right, in the game of "odd and even," if we simply guess "even" or "odd" than if we guess at the actual number; and the oracle-monger is more likely to be right if he simply says that a thing will happen than if he says when it will happen, and therefore he refuses to add a definite date. All these ambiguities have the same sort of effect, and are to be avoided unless we have some such object as that mentioned".

Ian Hacking has asserted that the emergence of (modern) probabilities is an 'epsitemic shift', that is a conceptual revolution and his book The Emergence of probabilities (1975) remains a (much discussed) classic. A more recent book is The Science of Conjecture: Evidence and Probability Before Pascal by James Franklin (2001).

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  • $\begingroup$ Ahh thank you for explicitly mentioning the "interpretation problem." I always had this thought in the back of my mind, but I never knew it was "a thing." Would you know any good books or other sources to read up on this? $\endgroup$ – Ovi Jul 24 at 17:35
  • $\begingroup$ Perhaps the SEP article Interpretations of Probability (2011) plato.stanford.edu/entries/probability-interpret $\endgroup$ – sand1 Jul 26 at 22:40
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I hope another answer is okay. Absolutely nothing of what follows is mine. The source[1] is given at the end. That main concern of that article, however, is de Finetti's theory of subjective probability and Hume's problem of induction.

It has been suggested that an important step for the development of ideas of probability is the notion of what Jerzy Neyman termed Fundamental Probability Sets (FPS): a set of outcomes that are all equally likely. Such sets naturally allow us to assign numbers to every outcome. In fact, this is how the subject is still introduced in school.

Suppose we suppress what we understand about probability, and how we currently often use terms such as equally possible, equally probable and equally likely interchangeably; and consider the following heuristic, it still seems very suggestive:

If all alternative outcomes are equally likely then none of them should obtain.

The idea being that if any one of the alternatives were to obtain it would break the symmetry that led them to be included in the FPS in the first place. (I think this exposes the circularity at the heart of Bernoulli's defining equally probable on the basis of equipossibility, that Reichenbach talks about)

According to G.E.L. Owen (wiki) the above heuristic is a 'very Greek pattern of argument'. It has a name: Ou Mallon. It is employed, for example, by Anaximander to argue that the Earth is fixed at the centre of the universe, by Parmenides to argue that the universe is uncreated.

Thus, even if the Greeks were mathematically proficient, their existing pattern of reasoning blocked them from identifying FPSs and using it to develop notions of probability, since for them 'equally likely' means none of the outcomes should be possible.

A counter view of sorts is voiced in the first chapter of Hacking's brilliant The Emergence of Probability, called an Absent Family of Ideas (this whole chapter is worth reading in connection to your question). While not addressing this particular point, Hacking dismisses the general argument that people in ancient times failed to notice FPSes.

[1] Zabell, S. L. Symmetry and Its Discontents; Skyrms, B., & Harper, W. L. (eds). Causation, Chance, and Credence, Vol. I, 155-190. (1988). Kluwer Academic.

P.S.: An interesting flip in perspective seems to happen by the 1700s. The symmetry conditions, that for the Greeks would block an outcome from obtaining, becomes the very grounds for chance to operate.

Consider this passage from Hume, A Treatise of Human Nature, 1739:

Since therefore an entire indifference is essential to chance, no one chance can possibly be superior to another, otherwise than as it is compos’d of a superior number of equal chances. For if we affirm that one chance can, after any other manner, be superior to another, we must at the same time affirm, that there is something, which gives it the superiority, and determines the event rather to that side than the other: That is, in other words, we must allow of a cause, and destroy the supposition of chance; which we had before establish'd. A perfect and total indifference is essential to chance, and one total indifference can never in itself be either superior or inferior to another. This truth is not peculiar to my system, but is acknowledg'd by every one, that forms calculations concerning chances.

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