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The Emergence of Probability: A Philosophical Study of Early Ideas About Probability Induction and Statistical Inference
Unknown Binding
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Ian Hacking
54 books154 followersIan Hacking is Professor Emeritus of Philosophy at the University of Toronto, specialised in the History of Science.
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May 4, 2026
A quite interesting breakdown of how probability, as we currently understand it, emerged as a concept—or, really, as two dueling concepts with the same name—in Europe in the 17th and 18th centuries. What I found fascinating was the way in which probability could not emerge except through the evolution of how Europe understood the nature of truth/fact/knowledge/reality. “Whether motivated by God, or by gaming, or by commerce, or by the law, the same kind of idea emerged simultaneously in many minds.”
At the time of the Renaissance, there was “knowledge,” which only related to things that were proved by (logical) (philosophical) demonstration, and “opinion,” which related to testimony, and to wise things written in books by wise people, essentially. “Probable,” at the time, meant something like “approved by the wise.” If you wanted truth, you could look to pure logic, or to what authorities/the wise had written down in the past. The “high sciences” dealt in knowledge alone, while the “low sciences” dealt in opinion. There were two kinds of evidence - authority, and testimony. The “evidence of things,” and the concept of proof by induction, did not yet exist.
So, Hacking writes, “we are concerned with the transformation from the study of books to the study of nature.” How did it happen? Well, “[a] new kind of testimony was accepted: the testimony of nature which, like any authority, was to be read.” So came “the doctrine of signatures. Each thing has a signature and the physician must master the signatures. Signatures are ultimately derived from the sentences in the stars, but a bountiful God has made them legible on earth. Everything is written.” There was an effort to show that these signatures, these names, this language that exists to describe them must be natural, inevitable, fundamentally true.
“The discovery that all names are conventional,” i.e., rather than natural, inevitable true, correct, “thunders us into modern philosophy.” Only then could Europe distinguish between “natural” signs—i.e., “the evidence of things”—and “arbitrary” signs—i.e., the names of things, in any particular language. Thus “natural” signs became “internal” evidence - the evidence gained from comparing the fact of a specific thing to our understanding of what that thing is typically like, or what tends to happen with that thing in nature. This evidence is gathered not through logical analysis from first principles, or from referring to the wisdom of the past, but from observation and experimentation of what happens, or can possibly happen, in reality.
There were two main areas of interested observation at the time. The interwoven development of these ideas means that our current understanding of probability is really two concepts in one: “aleatory probability,” having to do with mathematical ratios and ideas of “fair” prices, which arose out of gambling—thanks, Pascal!—and “epistemic probability,” or the idea of something being qualitatively more likely to be true, which seems to have been heavily influenced by legal thought—thanks, Leibniz! (“Aleatory probabilities have to do with the physical state of coins or mortal humans. Epistemic probabilities concern our knowledge.”)
On the gambling side, Hacking points out that there are thousands of years of human history involving wagers and games of change. “Yet theories of frequency, betting, randomness, and probability appear” in Europe “only recently. No one knows why.”
(Hacking is very clear that he’s only dealing with the European origins of probability, and that there is no reason to expect that anywhere else in the world developed in the same way, or at the same pace. He points, for example, to a passage in the Mahabharata which implies a nuanced understanding of probability that Europe wouldn’t attain for another millennium and change. He says—in 1975, when this book was first published)—“[i]t is reasonable to guess, then, that a good deal of Indian probability lore is at present unknown to us.” And another possible factor he points out: “Perhaps a symbolism that makes addition and multiplication easy,” i.e., Arabic numerals, “is a prerequisite for any rich concept of probability.” As evidence for this, he points out that “the old word for chance, namely ‘hazard’, is as arabic as ‘algebra’. The first European probabilists were Italian, solving North African problems.” Elsewhere, he writes, “Naturally I here make no claim about Sanskrit or Greek concepts of evidence. I am concerned with a specific lack at a particular time […] We cannot even speculate about how another concept of probability might have emerged elsewhere at another time, from the transformations in another culture.”)
Probability theory as pertains to gambling is usually attributed to Pascal, his correspondence with Fermat, his involvement with the Duke of Roannez’s salon in Paris, and his associates at Port-Royal, although Hacking makes sure to point out that this is not a scenario involving a lone genius; these ideas were arising with many people across Europe, all at once and often through correspondence between them all, and their conversations in Paris, and their study of each others’ publications and unpublished manuscripts. Interestingly, although “[m]ost seventeenth century writers thought that Latin was the most suitable language for expanding mathematics,” and “[t]he vernaculars were not deemed rich enough in the burgeoning vocabulary of pure mathematics,” “the doctrine of chances is applied mathematics arising from vulgar practices and has plenty of terminology” available in the vernacular. The difficulty with probability wasn’t talking about it in Dutch, or English, or French, or German, but with figuring out which Latin words might fit.
A corollary to the math on wagers was the math on births and deaths - the beginning of statistics. The first tables of birth and death information were published in London because of “a desire know about the current state of the plague.” This data could clearly be put to much more use beyond that, for developing public policy: “Graunt could plot the astonishing growth of the city, and also prove that much of the increase was due to immigration, not procreation. He could also show that despite the horrors of plague, the decrease in population of the worst epidemic was always made good within two years.” These tables, and the math associated with them, also began to become more important because of their use in calculating annuity prices. At the time, annuities were a major source of public funding in Europe due to Christian ideas about usury which discouraged the use of loans, and people would often buy annuities for themselves or jointly with spouses, friends, and even shipmates. (I’m sure if you dig, Ishmael will have had something to say about this.)
While gambling and annuities provided a field for development of aleatory probability, the law was the source of much epistemic probability, i.e., qualitative probability, where evidence might be mixed, where interests might be complex and contingent, where probability is always conditional on what we know, now. Of course, this is where things get confusing. “We now return to our starting point. Leibniz had learned from the law that probability is a relation between hypotheses and evidence. But he learned from the doctrine of chances that probabilities are a matter of physical propensities. Even now no philosopher has satisfactorily combined these two discoveries.”
In any case, the modern idea(s) of probability came into being. Hacking glosses past various efforts to show that consistent statistical results are prove that God exists (God the watchmaker, etc.), discusses Bernoulli’s “masterpiece of permanent value” (“Once a research program is under way the occasional masterpiece of permanent value often has three distinct characteristics. It does something almost completely new, which, although much in the air at the time, has never before crystallized, but, once written down, sets the direction for all future enquiry. Secondly, it epitomizes what everyone has known for a long time but has been unable to state succinctly. Thirdly, and much less often noticed, it ends certain possible lines of development, which, until that node in history, were perfectly open but now become closed.”), and ends with a short description of Hume’s “sceptical problem of induction,” as prompted by the acceptance of this new probabilistic conception of truth/fact/reality.
The end of this book is not the end of the story - Hacking has apparently written a sequel, The Taming of Chance, which apparently takes this history… if not up to the present, at least further along. (Here, he writes, “Only the advent of the quantum theory has made it possible to conceive of statistical regularity as a brute and irreducible fact of nature. And there is still a small but valiant program of ‘hidden variable’ theorists who contend that the gross statistical laws must be the manifestation of some as yet unspecified deterministic laws.” Presumably there’s more to be said about this!) I’ll read it - The Emergence of Probability was quite readable for an academic book. The short chapters, frankly, were a game changer.
I’m including below a few other quotes which I liked but couldn’t find a place for above:
—On escape from the Enlightenment paradigm: “I’m inviting the reader to imagine, first of all, that there is a space of possible theories about probability that has been rather constant from 1660 to the present. Secondly, this space resulted from a transformation upon some quite different conceptual structure. Thirdly, some characteristics of that prior structure, themselves quite forgotten, have impressed themselves on our present scheme of thought. Fourth: perhaps an understanding of our space and its preconditions can liberate us from the cycle of probability theories that has trapped us for so long.” For what it’s worth, comparing the European experience to other regions of the world might also help move us toward this “liberation.”
—On “high” versus “low” sciences: “It would be amazing if a Paracelsus were an ‘influence’ on a Pascal or Leibniz. The mathematicians despised what they knew of the occult. Yet their contempt for those earlier hermetical figures does not preclude the possibility that whatever these geometers thought about opinion, they thought in a conceptual space that was the legacy of the very empirics whom they scorned. The intellectual objects about which, and in which, the new mathematicians thought had been formed in the crucibles of the alchemists and the vials of the physicians.”
—On the significance of probability theory for theology: “In the Thomistic theology, God, insofar as he could be conceived, was an object of knowledge, not opinion. Pascal had dared to make God an object of decision, not belief. Wilkins, ushering in a more complacent age, dared to make God an object of probable opinion.”
—The case for a universal basic income, from 1662: “Graunt recommends a guaranteed annual wage. He reasons as follows: (i) London is teaming with beggars. (ii) Hardly anyone dies of starvation. (iii) Therefore the national wealth already feeds them. (iv) They shouldn’t be put to work, for their produce will be shoddy and the Dutch (who at Ypres already subsidize idlers) will gain British trade. (v) So at no extra cost to the Nation we should feed them and keep them from defiling our thoroughfares by begging.”
—And a funny P.S. about that same guy: “later his business was burnt down, he fell into Catholicism, and he became somewhat withdrawn.” What a way to go out in history!
—I also enjoyed Hacking’s thoughts on Leibniz: “this is another instance of Leibniz’s infuriating ability to get the right answer by an unjustifiable inference from the wrong data,” and on an early publication of his: “I cannot tell how good a piece of work it is. It seems to be a curious mixture of logical insights combined with a jejune and oversimplified view of legal process.” There’s also a whole section on what Leibniz really meant by his “we’re living in the best of all possible worlds” line, which apparently has to do with Leibniz having his own special definitions of “best” and “possible.”
At the time of the Renaissance, there was “knowledge,” which only related to things that were proved by (logical) (philosophical) demonstration, and “opinion,” which related to testimony, and to wise things written in books by wise people, essentially. “Probable,” at the time, meant something like “approved by the wise.” If you wanted truth, you could look to pure logic, or to what authorities/the wise had written down in the past. The “high sciences” dealt in knowledge alone, while the “low sciences” dealt in opinion. There were two kinds of evidence - authority, and testimony. The “evidence of things,” and the concept of proof by induction, did not yet exist.
So, Hacking writes, “we are concerned with the transformation from the study of books to the study of nature.” How did it happen? Well, “[a] new kind of testimony was accepted: the testimony of nature which, like any authority, was to be read.” So came “the doctrine of signatures. Each thing has a signature and the physician must master the signatures. Signatures are ultimately derived from the sentences in the stars, but a bountiful God has made them legible on earth. Everything is written.” There was an effort to show that these signatures, these names, this language that exists to describe them must be natural, inevitable, fundamentally true.
“The discovery that all names are conventional,” i.e., rather than natural, inevitable true, correct, “thunders us into modern philosophy.” Only then could Europe distinguish between “natural” signs—i.e., “the evidence of things”—and “arbitrary” signs—i.e., the names of things, in any particular language. Thus “natural” signs became “internal” evidence - the evidence gained from comparing the fact of a specific thing to our understanding of what that thing is typically like, or what tends to happen with that thing in nature. This evidence is gathered not through logical analysis from first principles, or from referring to the wisdom of the past, but from observation and experimentation of what happens, or can possibly happen, in reality.
There were two main areas of interested observation at the time. The interwoven development of these ideas means that our current understanding of probability is really two concepts in one: “aleatory probability,” having to do with mathematical ratios and ideas of “fair” prices, which arose out of gambling—thanks, Pascal!—and “epistemic probability,” or the idea of something being qualitatively more likely to be true, which seems to have been heavily influenced by legal thought—thanks, Leibniz! (“Aleatory probabilities have to do with the physical state of coins or mortal humans. Epistemic probabilities concern our knowledge.”)
On the gambling side, Hacking points out that there are thousands of years of human history involving wagers and games of change. “Yet theories of frequency, betting, randomness, and probability appear” in Europe “only recently. No one knows why.”
(Hacking is very clear that he’s only dealing with the European origins of probability, and that there is no reason to expect that anywhere else in the world developed in the same way, or at the same pace. He points, for example, to a passage in the Mahabharata which implies a nuanced understanding of probability that Europe wouldn’t attain for another millennium and change. He says—in 1975, when this book was first published)—“[i]t is reasonable to guess, then, that a good deal of Indian probability lore is at present unknown to us.” And another possible factor he points out: “Perhaps a symbolism that makes addition and multiplication easy,” i.e., Arabic numerals, “is a prerequisite for any rich concept of probability.” As evidence for this, he points out that “the old word for chance, namely ‘hazard’, is as arabic as ‘algebra’. The first European probabilists were Italian, solving North African problems.” Elsewhere, he writes, “Naturally I here make no claim about Sanskrit or Greek concepts of evidence. I am concerned with a specific lack at a particular time […] We cannot even speculate about how another concept of probability might have emerged elsewhere at another time, from the transformations in another culture.”)
Probability theory as pertains to gambling is usually attributed to Pascal, his correspondence with Fermat, his involvement with the Duke of Roannez’s salon in Paris, and his associates at Port-Royal, although Hacking makes sure to point out that this is not a scenario involving a lone genius; these ideas were arising with many people across Europe, all at once and often through correspondence between them all, and their conversations in Paris, and their study of each others’ publications and unpublished manuscripts. Interestingly, although “[m]ost seventeenth century writers thought that Latin was the most suitable language for expanding mathematics,” and “[t]he vernaculars were not deemed rich enough in the burgeoning vocabulary of pure mathematics,” “the doctrine of chances is applied mathematics arising from vulgar practices and has plenty of terminology” available in the vernacular. The difficulty with probability wasn’t talking about it in Dutch, or English, or French, or German, but with figuring out which Latin words might fit.
A corollary to the math on wagers was the math on births and deaths - the beginning of statistics. The first tables of birth and death information were published in London because of “a desire know about the current state of the plague.” This data could clearly be put to much more use beyond that, for developing public policy: “Graunt could plot the astonishing growth of the city, and also prove that much of the increase was due to immigration, not procreation. He could also show that despite the horrors of plague, the decrease in population of the worst epidemic was always made good within two years.” These tables, and the math associated with them, also began to become more important because of their use in calculating annuity prices. At the time, annuities were a major source of public funding in Europe due to Christian ideas about usury which discouraged the use of loans, and people would often buy annuities for themselves or jointly with spouses, friends, and even shipmates. (I’m sure if you dig, Ishmael will have had something to say about this.)
While gambling and annuities provided a field for development of aleatory probability, the law was the source of much epistemic probability, i.e., qualitative probability, where evidence might be mixed, where interests might be complex and contingent, where probability is always conditional on what we know, now. Of course, this is where things get confusing. “We now return to our starting point. Leibniz had learned from the law that probability is a relation between hypotheses and evidence. But he learned from the doctrine of chances that probabilities are a matter of physical propensities. Even now no philosopher has satisfactorily combined these two discoveries.”
In any case, the modern idea(s) of probability came into being. Hacking glosses past various efforts to show that consistent statistical results are prove that God exists (God the watchmaker, etc.), discusses Bernoulli’s “masterpiece of permanent value” (“Once a research program is under way the occasional masterpiece of permanent value often has three distinct characteristics. It does something almost completely new, which, although much in the air at the time, has never before crystallized, but, once written down, sets the direction for all future enquiry. Secondly, it epitomizes what everyone has known for a long time but has been unable to state succinctly. Thirdly, and much less often noticed, it ends certain possible lines of development, which, until that node in history, were perfectly open but now become closed.”), and ends with a short description of Hume’s “sceptical problem of induction,” as prompted by the acceptance of this new probabilistic conception of truth/fact/reality.
The end of this book is not the end of the story - Hacking has apparently written a sequel, The Taming of Chance, which apparently takes this history… if not up to the present, at least further along. (Here, he writes, “Only the advent of the quantum theory has made it possible to conceive of statistical regularity as a brute and irreducible fact of nature. And there is still a small but valiant program of ‘hidden variable’ theorists who contend that the gross statistical laws must be the manifestation of some as yet unspecified deterministic laws.” Presumably there’s more to be said about this!) I’ll read it - The Emergence of Probability was quite readable for an academic book. The short chapters, frankly, were a game changer.
I’m including below a few other quotes which I liked but couldn’t find a place for above:
—On escape from the Enlightenment paradigm: “I’m inviting the reader to imagine, first of all, that there is a space of possible theories about probability that has been rather constant from 1660 to the present. Secondly, this space resulted from a transformation upon some quite different conceptual structure. Thirdly, some characteristics of that prior structure, themselves quite forgotten, have impressed themselves on our present scheme of thought. Fourth: perhaps an understanding of our space and its preconditions can liberate us from the cycle of probability theories that has trapped us for so long.” For what it’s worth, comparing the European experience to other regions of the world might also help move us toward this “liberation.”
—On “high” versus “low” sciences: “It would be amazing if a Paracelsus were an ‘influence’ on a Pascal or Leibniz. The mathematicians despised what they knew of the occult. Yet their contempt for those earlier hermetical figures does not preclude the possibility that whatever these geometers thought about opinion, they thought in a conceptual space that was the legacy of the very empirics whom they scorned. The intellectual objects about which, and in which, the new mathematicians thought had been formed in the crucibles of the alchemists and the vials of the physicians.”
—On the significance of probability theory for theology: “In the Thomistic theology, God, insofar as he could be conceived, was an object of knowledge, not opinion. Pascal had dared to make God an object of decision, not belief. Wilkins, ushering in a more complacent age, dared to make God an object of probable opinion.”
—The case for a universal basic income, from 1662: “Graunt recommends a guaranteed annual wage. He reasons as follows: (i) London is teaming with beggars. (ii) Hardly anyone dies of starvation. (iii) Therefore the national wealth already feeds them. (iv) They shouldn’t be put to work, for their produce will be shoddy and the Dutch (who at Ypres already subsidize idlers) will gain British trade. (v) So at no extra cost to the Nation we should feed them and keep them from defiling our thoroughfares by begging.”
—And a funny P.S. about that same guy: “later his business was burnt down, he fell into Catholicism, and he became somewhat withdrawn.” What a way to go out in history!
—I also enjoyed Hacking’s thoughts on Leibniz: “this is another instance of Leibniz’s infuriating ability to get the right answer by an unjustifiable inference from the wrong data,” and on an early publication of his: “I cannot tell how good a piece of work it is. It seems to be a curious mixture of logical insights combined with a jejune and oversimplified view of legal process.” There’s also a whole section on what Leibniz really meant by his “we’re living in the best of all possible worlds” line, which apparently has to do with Leibniz having his own special definitions of “best” and “possible.”
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