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A Treatise on Probability: The Connection Between Philosophy and the History of Science
John Maynard Keynes, 1st Baron Keynes, CB (1883-1946) was a British economist whose ideas have profoundly affected the theory and practice of modern macroeconomics, as well as the economic policies of governments. He greatly refined earlier work on the causes of business cycles, and advocated the use of fiscal and monetary measures to mitigate the adverse effects of economic recessions and depressions. His ideas are the basis for the school of thought known as Keynesian economics, and its various offshoots. "A Treatise on Probability," originally published in 1921, launched the "logical-relationist" theory.
550 pages, Paperback
First published January 1, 1921
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About the author
John Maynard Keynes
414 books713 followersJohn Maynard Keynes, 1st Baron Keynes (CB, FBA), was an English economist particularly known for his influence in the theory and practice of modern macroeconomics.
Keynes married Russian ballerina Lydia Lopokova in 1925.
NB: Not to be confused with his father who also was an economist. See John Neville Keynes.
Keynes married Russian ballerina Lydia Lopokova in 1925.
NB: Not to be confused with his father who also was an economist. See John Neville Keynes.
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Displaying 1 - 15 of 15 reviews
May 14, 2023
The conventional view among probabilists of today regarding the philosophical foundations of their discipline would surely be an undiluted Kolmogorovianism (q.v. our reviews of the outstanding Soviet mathematician A.N. Kolmogorov’s pioneering Grundbegriffe der Wahrscheinlichkeitsrechnung of 1933 here and here), dented perhaps by a sizable contingent of frequentists, Fisherians or von Misesians. But there is another, if largely neglected entry in the lists: John Maynard Keynes’ A Treatise on Probability (originally published by the Cambridge University Press in 1920 and here reprinted unabridged in lithographic reproduction by the Wildside Press LLC). Long before he entered government service as a controversial macroeconomist during the downturn of the 1930’s, Keynes had achieved distinction as a professor of applied mathematics as is evident from the authority with which he writes in the present volume. While he may not have stood out so much back in his day, among other authors on the subject of probability such as Émil Borel [Éléments de la théorie des probabilités, 1909] and Henri Poincaré [Calcul des probabilités, 1912], in our time he figures rather as a lonely relic for sticking to a non-quantitative perspective on its philosophical foundation – precisely why it promises to be such a rewarding enterprise to read him once again today!
For we are habituated to a predominance of the statistical-mechanical paradigm of the theoretical physicist, for which a strictly numerical methodology is indeed appropriate and which lends itself so nicely to a mathematical formulation. This accomplished, all the wonderful techniques of modern analysis become applicable to problems of probability and very deep theorems can be proved. Not only this, we have every right to suppose them relevant to the natural world. Why, then, bother to resurrect Keynes’ dusty old treatise?
The real world is not the same as the natural world, but a superset of the latter. The complications of the theory of probability issue from two sources: first, technical difficulties in formulating the problem and in solving the implied calculations, such as any undergraduate physics major encounters in a course on statistical thermodynamics. A great deal of ingenuity has in fact been poured into the development of an art by which to facilitate the practical overcoming of such challenges – but pace Kolmogorov this does not by a long stretch constitute the whole of the theory! The second source, then, lies in the process of assessing whether the statistical assumptions underlying a given probabilistic model do in fact hold up when it is brought to bear on actual instances in the real world, as anyone who has had occasion more than to dabble in so-called data science and mining knows very well. But here seasoned judgment counts for a lot more than calculational facility, which is all the mere numbers man has. So, once confronted with real-world applications, it is by no means evident that all probabilities must be numerical and Keynes’ perspective becomes pertinent. At any rate, it behooves everyone in our time to run through Keynes’ arguments once more merely to have his complacency perturbed – we who may be superlative calculators but who have all but lost any capacity for prudence in decisions bearing upon the meanings of things.
Next, on to an analysis of the contents of Keynes’ intriguing text. In five parts, this copious volume visits every imaginable aspect of the philosophical problem, ranging from the most fundamental ideas on probability and its relation to knowledge (part i) to the fundamental theorems about inference in the necessary or probable cases (part ii), more deeply probing reflections on induction and analogy (part iii), the meaning of objective chance and randomness (part iv) and lastly, the foundations of statistical inference (part v) as exhibited in the law of large numbers and in the passage from à priori to à posteriori probabilities.
For someone sharing this reviewer’s temperament, the first part will be by far the most arresting. Part ii, while indispensable to any fully worked out account of the theory of probability, contains mainly theorems stating the simplest properties of absolute and conditional probabilities whose derivations do not seem to call upon any very deep insights. In this regard, one might suppose it comparable to the painstaking investigations into the foundations of arithmetic carried out by Frege, Husserl, Russell, Whitehead et al. – not in any event this reviewer’s cup of tea! Keynes’ comments in part iii have considerably more intrinsic interest to one philosophically disposed, bearing as they do upon essential epistemology. Aside: the few formulae strewn across the pages of part iii are schematic; since most of the argument in this part proceeds by way of plain text, it is very readable and Keynes’ style, to boot, pleasant to take in. All the same, what one gets from it is largely background knowledge or worthwhile distinctions to keep in mind, nothing all that constructive in itself.
A similar comment would apply to part iv on the notions of objective chance and of randomness. Criticism of the views of Hume, Condorcet, Cournot, Poincaré, Bertrand and de Morgan on selected questions. Part iv is rounded out with a somewhat idiosyncratic treatment of probability as it applies to conduct of human affairs – not too taxing, an entertaining diversion from statistical physics. Part v concerns statistical inference, or as one might prefer to say these days, hypothesis testing. Keynes’ exposition tends to devolve into minute criticisms of shortcomings in contemporary authors – apart from impressing upon the reader the always to be respected fact that arguments in the theory of probability can turn out to be quite subtle, this part would appear to have comparatively little relevance to mathematicians in our time.
So to this reviewer at any rate the meat of Keynes’ treatise is to be found almost entirely in the 125 pages of part i, to which we presently return. For Keynes, probability is associated with degrees of rational belief: what about the possibility of measuring it? Nobody else seems ever so much as to raise a question such as this. Keynes considers comparisons of relatively probability possible even in the absence of any knowledge as to supposed absolute probabilities. He illustrates his ideas in a diagram on pp. 40-43, intended to depict how among all possible pairs of probabilities, some may be comparable while others remain incomparable. The following chapter concerns the principle of indifference asserting that in the absence of relevant knowledge, one is to suppose every possibility equally likely, as in the throw of an unloaded die with six sides: the probability to land on any given face should be 1/6. In Keynes’ hands, with reference to current debates, the principle of indifference begins to seem questionable, or at least tricky to put into practice. In our terms: how could one have a canonical measure on the sample space by which to demarcate equally likely elementary events when it is some such very measure we are trying to define in the first place? Is the principle of indifference ultimately circular? The following section on the weight of arguments will always be topical: for instance in the philosophy of science, how much weight to assign to simplicity, naturality etc. in a theoretical model? Keynes delivers a somewhat cryptic passage on this score: ‘The weight, to speak metaphorically, measures the sum of the favourable and unfavourable evidence, the probability measures the difference’ (p. 85).
Two isolated remarks are in order here. On p. 16, Keynes introduces a distinction between probable knowledge versus vague knowledge – the latter a very interesting concept: when we suppose we know x, what we actually know might only be some y obtained from x via the application of a forgetful functor, but not nevertheless nothing; i.e., we may be mistaken about x but right about y. On pp. 40-44, rules governing partial ordering of probabilities – could a version of these be wrought into some instantiation of what would deserve to be named a quantum probability, i.e., to explain whence the amplitude of the wavefunction originates?
To close out the present review, let us advert to scattered membra disjecta from parts ii-v which strike our fancy. It is quite interesting that there can exist laws of error which lead to convergence to the median rather than to the mean, or to the geometric or harmonic mean etc. [pp. 224-248]. To see the reason why, consider that in general mean of f does not equal f of mean, so it depends on what the relevant measure is for the class of phenomena in question [pp. 241-242]. Rejection of discordant observations can be a subtle issue, based on experience [pp. 247-248]. On pp. 340-341, Keynes serves up several good points on Paley’s argument from design. On inferring causes behind the phenomena when not self-evident [p. 343]; telepathy or existence of spirits – is it possible for any evidence to be convincing?
On p. 346, remarkable occurrences may be very improbable before they happen but permit no large number of alternatives: something further must be required if what actually occurs is to derive any peculiar significance apart from its being remarkable: say, clarity, correlation with something else not intrinsically germane?
The peculiar virtue of prediction or predesignation is altogether imaginary. The number of instances examined and the analogy between them are the essential points, and the question as to whether a particular hypothesis happens to be propounded before or after their examination is quite irrelevant. (p. 349) – apply this to the history of science?
Another quotation from Keynes, at the head of the chapter on conduct:
Given as our basis what knowledge we actually have, the probable, as I have said, is that which it is rational for us to believe. This is not a definition. For it is not rational for us to believe that the probable is true; it is only rational to have a probable belief in it or to believe it in preference to alternative beliefs. To believe one thing in preference to another, as distinct from believing the first true or more probable and the second false or less probable, must have reference to action and must be a loose way of expressing the propriety of acting on one hypothesis rather than on another. We might put it, therefore, that the probable is the hypothesis on which it is rational for us to act. It is, however, not so simple as this, for the obvious reason that of two hypotheses it may be rational to act on the less probable if it leads to the greater good. We cannot say more at present than that the probability of a hypothesis is one of the things to be determined and taken account of before acting on it. (p. 351) – apply this idea to the climate crisis?
Critique of the utilitarian calculus [pp. 355ff].
Bernoulli’s second axiom: when computing a probability, always condition on all the available information (p. 368, see also p. 84).
There is no direct relation between the truth of a proposition and its probability. (p. 369)
The importance of probability can only be derived from the judgment that it is rational to be guided by it in action. (p. 369)
Let us investigate the generalisation that the proportion of male to female births is m. The fact that the aggregate statistics for England during the nineteenth century yield the proportion m would go no way towards justifying the statement that the proportion of male births in Cambridge next year is likely to approximate to m. Our argument would be no better if our statistics, instead of relating to England during the nineteenth century, covered all the descendants of Adam. But if we were able to break up our aggregate series of instances into a series of sub-series, classified according to a great variety of principles, as for example by date, by season, by locality, by the class of the parents, by the sex of previous children, and so forth, and if the proportion of male births throughout these sub-series showed a significant stability in the neighborhood of m, then indeed we have an argument worth something. Otherwise we must either abandon our generalisation, amplify its conclusions, or modify its conclusion. (pp. 466-467)
The problem of hyperparameters (p. 471); how to estimate them in the absence of any priors? Maximum likelihood of a generative model? Cf. p. 478, permanent versus accidental causes. What about when an estimate of the parameter is part of the experiment, thus reducing the number of degrees of freedom by one?
In summary, Keynes’ peculiar treatise on probability can be warmly recommended to those few who retain the intellectual integrity to research the foundations of a subject in a never-ending quest to refine one’s own understanding of things. This reviewer at all events is convinced that a qualitative approach to the foundations of probability has its virtues, in much the same way as does the field of metamathematics and proof theory, inspired by Hilbert and Gödel. Nobody supposes that metamathematics ought to supplant ordinary mathematics per se, recourse to which will always be demanded by real-world problems, merely that one should occupy a portion of one’s attention with the former so as to sharpen his insight into the latter.
For we are habituated to a predominance of the statistical-mechanical paradigm of the theoretical physicist, for which a strictly numerical methodology is indeed appropriate and which lends itself so nicely to a mathematical formulation. This accomplished, all the wonderful techniques of modern analysis become applicable to problems of probability and very deep theorems can be proved. Not only this, we have every right to suppose them relevant to the natural world. Why, then, bother to resurrect Keynes’ dusty old treatise?
The real world is not the same as the natural world, but a superset of the latter. The complications of the theory of probability issue from two sources: first, technical difficulties in formulating the problem and in solving the implied calculations, such as any undergraduate physics major encounters in a course on statistical thermodynamics. A great deal of ingenuity has in fact been poured into the development of an art by which to facilitate the practical overcoming of such challenges – but pace Kolmogorov this does not by a long stretch constitute the whole of the theory! The second source, then, lies in the process of assessing whether the statistical assumptions underlying a given probabilistic model do in fact hold up when it is brought to bear on actual instances in the real world, as anyone who has had occasion more than to dabble in so-called data science and mining knows very well. But here seasoned judgment counts for a lot more than calculational facility, which is all the mere numbers man has. So, once confronted with real-world applications, it is by no means evident that all probabilities must be numerical and Keynes’ perspective becomes pertinent. At any rate, it behooves everyone in our time to run through Keynes’ arguments once more merely to have his complacency perturbed – we who may be superlative calculators but who have all but lost any capacity for prudence in decisions bearing upon the meanings of things.
Next, on to an analysis of the contents of Keynes’ intriguing text. In five parts, this copious volume visits every imaginable aspect of the philosophical problem, ranging from the most fundamental ideas on probability and its relation to knowledge (part i) to the fundamental theorems about inference in the necessary or probable cases (part ii), more deeply probing reflections on induction and analogy (part iii), the meaning of objective chance and randomness (part iv) and lastly, the foundations of statistical inference (part v) as exhibited in the law of large numbers and in the passage from à priori to à posteriori probabilities.
For someone sharing this reviewer’s temperament, the first part will be by far the most arresting. Part ii, while indispensable to any fully worked out account of the theory of probability, contains mainly theorems stating the simplest properties of absolute and conditional probabilities whose derivations do not seem to call upon any very deep insights. In this regard, one might suppose it comparable to the painstaking investigations into the foundations of arithmetic carried out by Frege, Husserl, Russell, Whitehead et al. – not in any event this reviewer’s cup of tea! Keynes’ comments in part iii have considerably more intrinsic interest to one philosophically disposed, bearing as they do upon essential epistemology. Aside: the few formulae strewn across the pages of part iii are schematic; since most of the argument in this part proceeds by way of plain text, it is very readable and Keynes’ style, to boot, pleasant to take in. All the same, what one gets from it is largely background knowledge or worthwhile distinctions to keep in mind, nothing all that constructive in itself.
A similar comment would apply to part iv on the notions of objective chance and of randomness. Criticism of the views of Hume, Condorcet, Cournot, Poincaré, Bertrand and de Morgan on selected questions. Part iv is rounded out with a somewhat idiosyncratic treatment of probability as it applies to conduct of human affairs – not too taxing, an entertaining diversion from statistical physics. Part v concerns statistical inference, or as one might prefer to say these days, hypothesis testing. Keynes’ exposition tends to devolve into minute criticisms of shortcomings in contemporary authors – apart from impressing upon the reader the always to be respected fact that arguments in the theory of probability can turn out to be quite subtle, this part would appear to have comparatively little relevance to mathematicians in our time.
So to this reviewer at any rate the meat of Keynes’ treatise is to be found almost entirely in the 125 pages of part i, to which we presently return. For Keynes, probability is associated with degrees of rational belief: what about the possibility of measuring it? Nobody else seems ever so much as to raise a question such as this. Keynes considers comparisons of relatively probability possible even in the absence of any knowledge as to supposed absolute probabilities. He illustrates his ideas in a diagram on pp. 40-43, intended to depict how among all possible pairs of probabilities, some may be comparable while others remain incomparable. The following chapter concerns the principle of indifference asserting that in the absence of relevant knowledge, one is to suppose every possibility equally likely, as in the throw of an unloaded die with six sides: the probability to land on any given face should be 1/6. In Keynes’ hands, with reference to current debates, the principle of indifference begins to seem questionable, or at least tricky to put into practice. In our terms: how could one have a canonical measure on the sample space by which to demarcate equally likely elementary events when it is some such very measure we are trying to define in the first place? Is the principle of indifference ultimately circular? The following section on the weight of arguments will always be topical: for instance in the philosophy of science, how much weight to assign to simplicity, naturality etc. in a theoretical model? Keynes delivers a somewhat cryptic passage on this score: ‘The weight, to speak metaphorically, measures the sum of the favourable and unfavourable evidence, the probability measures the difference’ (p. 85).
Two isolated remarks are in order here. On p. 16, Keynes introduces a distinction between probable knowledge versus vague knowledge – the latter a very interesting concept: when we suppose we know x, what we actually know might only be some y obtained from x via the application of a forgetful functor, but not nevertheless nothing; i.e., we may be mistaken about x but right about y. On pp. 40-44, rules governing partial ordering of probabilities – could a version of these be wrought into some instantiation of what would deserve to be named a quantum probability, i.e., to explain whence the amplitude of the wavefunction originates?
To close out the present review, let us advert to scattered membra disjecta from parts ii-v which strike our fancy. It is quite interesting that there can exist laws of error which lead to convergence to the median rather than to the mean, or to the geometric or harmonic mean etc. [pp. 224-248]. To see the reason why, consider that in general mean of f does not equal f of mean, so it depends on what the relevant measure is for the class of phenomena in question [pp. 241-242]. Rejection of discordant observations can be a subtle issue, based on experience [pp. 247-248]. On pp. 340-341, Keynes serves up several good points on Paley’s argument from design. On inferring causes behind the phenomena when not self-evident [p. 343]; telepathy or existence of spirits – is it possible for any evidence to be convincing?
On p. 346, remarkable occurrences may be very improbable before they happen but permit no large number of alternatives: something further must be required if what actually occurs is to derive any peculiar significance apart from its being remarkable: say, clarity, correlation with something else not intrinsically germane?
The peculiar virtue of prediction or predesignation is altogether imaginary. The number of instances examined and the analogy between them are the essential points, and the question as to whether a particular hypothesis happens to be propounded before or after their examination is quite irrelevant. (p. 349) – apply this to the history of science?
Another quotation from Keynes, at the head of the chapter on conduct:
Given as our basis what knowledge we actually have, the probable, as I have said, is that which it is rational for us to believe. This is not a definition. For it is not rational for us to believe that the probable is true; it is only rational to have a probable belief in it or to believe it in preference to alternative beliefs. To believe one thing in preference to another, as distinct from believing the first true or more probable and the second false or less probable, must have reference to action and must be a loose way of expressing the propriety of acting on one hypothesis rather than on another. We might put it, therefore, that the probable is the hypothesis on which it is rational for us to act. It is, however, not so simple as this, for the obvious reason that of two hypotheses it may be rational to act on the less probable if it leads to the greater good. We cannot say more at present than that the probability of a hypothesis is one of the things to be determined and taken account of before acting on it. (p. 351) – apply this idea to the climate crisis?
Critique of the utilitarian calculus [pp. 355ff].
Bernoulli’s second axiom: when computing a probability, always condition on all the available information (p. 368, see also p. 84).
There is no direct relation between the truth of a proposition and its probability. (p. 369)
The importance of probability can only be derived from the judgment that it is rational to be guided by it in action. (p. 369)
Let us investigate the generalisation that the proportion of male to female births is m. The fact that the aggregate statistics for England during the nineteenth century yield the proportion m would go no way towards justifying the statement that the proportion of male births in Cambridge next year is likely to approximate to m. Our argument would be no better if our statistics, instead of relating to England during the nineteenth century, covered all the descendants of Adam. But if we were able to break up our aggregate series of instances into a series of sub-series, classified according to a great variety of principles, as for example by date, by season, by locality, by the class of the parents, by the sex of previous children, and so forth, and if the proportion of male births throughout these sub-series showed a significant stability in the neighborhood of m, then indeed we have an argument worth something. Otherwise we must either abandon our generalisation, amplify its conclusions, or modify its conclusion. (pp. 466-467)
The problem of hyperparameters (p. 471); how to estimate them in the absence of any priors? Maximum likelihood of a generative model? Cf. p. 478, permanent versus accidental causes. What about when an estimate of the parameter is part of the experiment, thus reducing the number of degrees of freedom by one?
In summary, Keynes’ peculiar treatise on probability can be warmly recommended to those few who retain the intellectual integrity to research the foundations of a subject in a never-ending quest to refine one’s own understanding of things. This reviewer at all events is convinced that a qualitative approach to the foundations of probability has its virtues, in much the same way as does the field of metamathematics and proof theory, inspired by Hilbert and Gödel. Nobody supposes that metamathematics ought to supplant ordinary mathematics per se, recourse to which will always be demanded by real-world problems, merely that one should occupy a portion of one’s attention with the former so as to sharpen his insight into the latter.
October 29, 2007
despite being wrong in almost all of his economic theories, keynes knows how to rock out the philosophy of probability. i'm a big fan of this book.
Want to read
May 14, 2011Available to read free in public domain on Project Gutenberg here.
September 28, 2019
It is really a fantastic work discussing (mostly) the philosophy of probability. It was written nearly 100 years ago so sometimes the writing tends to drag a bit or is not concise. However, it is well worth the time. I would recommend specifically a close read of parts I, IV, and V. Also chapter XXX is excellent. I would skim parts II and III unless you really want to get into the weeds on derivations and theorems.
All in all, it is still an extremely relevant contribution to the philosophical underpinnings of probability and statistics.
All in all, it is still an extremely relevant contribution to the philosophical underpinnings of probability and statistics.
November 2, 2020
[found in the following list]
Books with the best decision theoretic and philosophical foundation
by Michael Emmett Brady
The following books will provide an optimal understanding of how one should study and organize the data and observations that comprise the social sciences. These books provide a broad foundation in logical, epistemological, and philosophical techniques that are sound and valid. A reader who masters these books will quickly grasp the complex, dynamic, nonlinear aspects of social science systems as they evolve through time.
1. The General Theory of Employment, Interest, and Money - John Maynard Keynes
2. A Treatise on Probability - John Maynard Keynes
3. Risk, Uncertainty and Profit - Frank H. Knight
4. The Theory of Economic Development - Joseph A. Schumpter
5. The Wealth of Nations - Adam Smith
6. Risk, Ambiguity and Decision - Daniel Ellsberg
7. The (Mis)behavior of Markets - Beniot Mandelbrot and Richard L. Hudson
8. Probability, Econometrics and Truth - Hugo A. Keuzenkamp
9. The Unbround Prometheus: Technological Change and the Industrial Development in Western Europe from 1750 to the Present - Second Edition - David S. Landes
10. The Laws of Thought - George Boole
11. The Black Swan - Nassim Nicholas Taleb
12. Fooled by Randomness - Nassim Nicholas Taleb
13. J.M. Keynes Theory of Decision Making, Induction and Analogy - Michael Emmett Brady
Books with the best decision theoretic and philosophical foundation
by Michael Emmett Brady
The following books will provide an optimal understanding of how one should study and organize the data and observations that comprise the social sciences. These books provide a broad foundation in logical, epistemological, and philosophical techniques that are sound and valid. A reader who masters these books will quickly grasp the complex, dynamic, nonlinear aspects of social science systems as they evolve through time.
1. The General Theory of Employment, Interest, and Money - John Maynard Keynes
2. A Treatise on Probability - John Maynard Keynes
3. Risk, Uncertainty and Profit - Frank H. Knight
4. The Theory of Economic Development - Joseph A. Schumpter
5. The Wealth of Nations - Adam Smith
6. Risk, Ambiguity and Decision - Daniel Ellsberg
7. The (Mis)behavior of Markets - Beniot Mandelbrot and Richard L. Hudson
8. Probability, Econometrics and Truth - Hugo A. Keuzenkamp
9. The Unbround Prometheus: Technological Change and the Industrial Development in Western Europe from 1750 to the Present - Second Edition - David S. Landes
10. The Laws of Thought - George Boole
11. The Black Swan - Nassim Nicholas Taleb
12. Fooled by Randomness - Nassim Nicholas Taleb
13. J.M. Keynes Theory of Decision Making, Induction and Analogy - Michael Emmett Brady
October 19, 2020
Timeless book on epistemology and philosophy of probability. Predicts the importance of overfitting in modern statistics - the reliance on data and computing power rather than sound inductive process.
It gives a complete review of probability up to the early 1900s. From Bernouilli and Hume to Laplace to Tchebycheff and many in between.
Some concepts are dated and the 1900’s writing style slows down the reading, but to paraphrase Dalio I suggest you favor the great (ie. this) over the new.
It gives a complete review of probability up to the early 1900s. From Bernouilli and Hume to Laplace to Tchebycheff and many in between.
Some concepts are dated and the 1900’s writing style slows down the reading, but to paraphrase Dalio I suggest you favor the great (ie. this) over the new.
May 23, 2020
Keynes spends a good deal of time focused on the real-world implications of stylized assumptions embedded in statistical theorems. While it sometimes feels like a bit of a straw man, the broader lens is absolutely essential for practitioners attempting to rely statistical inference; so the book is a good reminder just how little we can know with confidence from apparently-robust statistics. The book also lends itself reasonably well to picking out the interesting sections and skipping the drier math.
April 7, 2020
Deep, thoughtful approach to the logicist foundations of probability. Unconventional by current standards. Stimulating. Not a good intro for a novice. Excellent.
April 18, 2023
An elegant way of thinking about uncertainty in the world - though Keynes' mathematical proofs completely went over my head. Probability is less about objective truth as it is the understanding of what we can and can't know for certain between discrete and correlated events. Only then can we draw reasoned conclusions.
"The natural man is disposed to the opinion that probability is essentially connected with the inductions of experience and, if he is a little more sophisticated, with the Laws of Causation and of the Uniformity of Nature. As Aristotle says, 'the probable is that which usually happens.' Events do not always occur in accordance with the expectations of experience; but the laws of experience afford us a good ground for supposing that they usually will."
"Probability is likeliness to be true ... the grounds of it are, in short, these two following. Firsts, the conformity of anything with our own knowledge, observation, and experience. Secondly, the testimony of others, vouching their observation and experience."
"When we are accustomed to see two impressions conjoined together, the appearance or idea of the one immediately carries us to the idea of the other ... Thus all probable reasoning is nothing but a species of sensation. 'Tis not solely in poetry and music, we must follow our taste and sentiment, but likewise in philosophy."
"The fundamental tenet of a frequency theory of probability is, then, that the probability of a proposition always depends upon referring it to some class whose truth-frequency is known within wide or narrow limits."
"The natural man is disposed to the opinion that probability is essentially connected with the inductions of experience and, if he is a little more sophisticated, with the Laws of Causation and of the Uniformity of Nature. As Aristotle says, 'the probable is that which usually happens.' Events do not always occur in accordance with the expectations of experience; but the laws of experience afford us a good ground for supposing that they usually will."
"Probability is likeliness to be true ... the grounds of it are, in short, these two following. Firsts, the conformity of anything with our own knowledge, observation, and experience. Secondly, the testimony of others, vouching their observation and experience."
"When we are accustomed to see two impressions conjoined together, the appearance or idea of the one immediately carries us to the idea of the other ... Thus all probable reasoning is nothing but a species of sensation. 'Tis not solely in poetry and music, we must follow our taste and sentiment, but likewise in philosophy."
"The fundamental tenet of a frequency theory of probability is, then, that the probability of a proposition always depends upon referring it to some class whose truth-frequency is known within wide or narrow limits."
December 8, 2022
Came across this via Nassim Taleb's recommendation.
Read Part I in detail, skipped Part II (per suggestion of Keynes), skimmed Part III, and read Part IV and Part V in detail.
I think I understood the whole of Part I, almost none of Part III, and about two thirds of Part IV and V. The mathematics involved are definitely comprehensible to anyone who has an undergraduate mathematics degree. This is perhaps not surprising since this book was published in 1921, while a lot of the techniques learned in undergraduate statistics courses are from the 1960s or later.
Philosophically, the most interesting parts are Chapters XXVII to XXXII. In particular, Chapter XXVII Paragraph 4 sets up the discussion in subsequent chapters, which reveal important implicit assumptions under probabilistic statements that may be taken as granted. Nassim Taleb has taken some of the text in these chapters almost directly, and I think Taleb has good reasons to do so.
There is also a quick note on ethics in the context of probability in Chapter XXVI. The material there could lead to discussions around the idea of rationality.
I am sure a re-reading will prove valuable, not the least because it took me four months of on-and-off reading to finish this book, and by the time I reached Part V I hardly remembered anything from Part I...
Read Part I in detail, skipped Part II (per suggestion of Keynes), skimmed Part III, and read Part IV and Part V in detail.
I think I understood the whole of Part I, almost none of Part III, and about two thirds of Part IV and V. The mathematics involved are definitely comprehensible to anyone who has an undergraduate mathematics degree. This is perhaps not surprising since this book was published in 1921, while a lot of the techniques learned in undergraduate statistics courses are from the 1960s or later.
Philosophically, the most interesting parts are Chapters XXVII to XXXII. In particular, Chapter XXVII Paragraph 4 sets up the discussion in subsequent chapters, which reveal important implicit assumptions under probabilistic statements that may be taken as granted. Nassim Taleb has taken some of the text in these chapters almost directly, and I think Taleb has good reasons to do so.
There is also a quick note on ethics in the context of probability in Chapter XXVI. The material there could lead to discussions around the idea of rationality.
I am sure a re-reading will prove valuable, not the least because it took me four months of on-and-off reading to finish this book, and by the time I reached Part V I hardly remembered anything from Part I...
December 8, 2020
This may quite possibly be the most personally influential book I have read in my life.
Not only did I find myself with an entirely fresh perspective on the reality of probability & statistics, but also the experience of thought itself.
In particular I found Keynes argument as to the true bounds of human rationality & what is within & without it in terms of probability & statistical analysis, I found truly monumental.
This thinking is introduce so eloquently in terms of viewing probability a matter of epistemology & only a portion presenting as valid within mathematical science.
Findings are ultimately reached in terms of constructing all thought in terms of proposition chains & the perceived persistence of such chains determine the level of personal investment & the reality of such chains determine the robustness or consequentially the fragility in ones thinking.
Not only did I find myself with an entirely fresh perspective on the reality of probability & statistics, but also the experience of thought itself.
In particular I found Keynes argument as to the true bounds of human rationality & what is within & without it in terms of probability & statistical analysis, I found truly monumental.
This thinking is introduce so eloquently in terms of viewing probability a matter of epistemology & only a portion presenting as valid within mathematical science.
Findings are ultimately reached in terms of constructing all thought in terms of proposition chains & the perceived persistence of such chains determine the level of personal investment & the reality of such chains determine the robustness or consequentially the fragility in ones thinking.
January 6, 2022
Very Very dense read. Thorough and brilliant.
December 15, 2022
After page 150, this book became too complex and so over my head that I could not read it with my normal full intellectual diligence.
November 5, 2021
An excellent book which I highly recommend. Very useful in preparation for working on complexity areas
January 19, 2020
Some books need to be read more than once to be fully appreciated. This is one of them. It combines the probability theory with its philosophical and historical aspects.
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