Keynes On Probability Oberbauer

On the Origins of Certain Arguments in J.M. Keynes Theory of Probability Mag. Dr. Maximilian Oberbauer October 19, 2008 Abstract The object of this paper is to raise awareness concerning the importance of philosophical concepts to the process of formulating mathematical theories. As an example the question ’What is a probability’ is discussed. Serveral philosophical ideas of what the word ’is’ might or might not mean in this context are presented. With their help J.M. Keynes juxtaposition of his an J. Venns opinion on what a probabilities is, is discussed.

Contents 1 Introduction 1.1 A note on style . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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2 Common notions

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3 Other Peoples Opinions 3.1 According to Thomas Aquinas . . . . . 3.2 According to Francisco Suarez . . . . . . 3.3 According to Thomas Hobbes . . . . . . 3.4 Circles and Thunderstorms reconsidered

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4 What is a Probability?

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5 Conclusion

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1

Introduction

This paper is concerned with a seemingly simple question: What do we mean, when we say: “There is a probability”? The question has a far greater double: What do we mean, when we say: “There is a mathematical object”? There are to concepts involved here; the existence of something and the thing that exists. This paper is basically concerned with the meaning of the word is in these sentences; it will be argued that this presents a (formidable) problem. To make obvious how this problem appears and affects mathematical theory one of its appearances will be examined in greater detail: John Maynard Keynes attempt to define what a probability is, and his attempt to distinguish his idea from the ’frequency theory’ of probability. This is a helpful example for various reasons, first of all, because Keynes does not just present one but two different approaches, and secondly, because (as will become obvious) Keynes has at best a clouded idea, what the problem of his discussion is. Accordingly, this paper is organized in three steps:

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Step one: In order to get a quick and readily understandable idea of the problem, a collection of intuitive ideas will be presented. They are supposed to convey a feeling of what we are talking about. Step two: Three (pertinent) concepts of being will be discussed. These concepts are ”snapshots’ taken from the works of Thomas Aquinas, Francisco Suarez and Thomas Hobbes. No discussion of the respective theories of this Philosophers are intended; the purpose is to isolate some tools. Step three: After briefly discussing some fundamental ideas of Keynes theory of probability, we will follow his attempt to distinguish his ideas from those of John Venn, one of the early proponents of the frequency theory of probabilty. It will be shown, that both theories are best understood as answers to the question: What is a probability? Employing the tools gained in step two, it will become clear that neither Keynes nor Venn have seen this clearly; furthermore it will become clear that problems and advantages of both theories can be examined (fruitfully) using the tools developed here. In conclusion: the object of this paper is to raise awareness, not to argue for the final settlement of a debate in one or the other way; no argument, for example, is made, that the tools here employed are sufficient or appropriate; more tools, better tools would be welcome. The argument is that ideas concerning the meaning of the word ’is’ — that is: arguments concerned with some concept of existence, be they expressed by using the word ’is’ or not — are important in the construction of mathematical theories.

1.1

A note on style

Large sections of this paper are written in a rather colloquial style. There are several reasons for this: First of all, the topic is not something (in my experience) usually discussed by mathematicians. On the contrary; part of this paper is intended to show, that even if such a disscussion takes place, people, quite litterally, don’t know what they are talking about. Once again, this paper is about raising awareness, not about producing results, in particular the section common notions is meant to be a ’soft introduction’; it’s main purpose is the accumulation of arguments and questions; the need for the succeding philosophical arguments should become clear, or, at any rate, be obvious once these questions are put forward in earnest. An analogie might help: in a paper concerned with geometry the author often has the considerable advantage to illustrate the point by showing figures, graphs and charts. There is not figure to be drawn here, but there are metaphors that help (I hope) the understanding along. We will, for example, discuss John Venn and the frequency theory of probability, and a dolls house — the doll’s house being the central object of meditation. This is so obviousley silly that (I hope) nobody is going to take it seriously. That has the — much underrated — advantage that there will be no confusion between our modell and ’the world’, that is to say, between what is used as a crutch and what we want to explain to ourselves. Precision and clarity are by no means the prerogative of an ability to express something in symbolic form. [Carnap, 1958, p. 1 ff]

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Common notions

We start with a (regrettably not funny) variation of a common way to tell a joke: What is the difference between a thunderstorm, a circle, a stone and E 0 = c∇ × B − 4πJ ? It is possible to distinguish two groups here: a thunderstorm and a stone are natural objects, a circle and E 0 = c∇ × B − 4πJ are mathematical objects. One difference between a natural and a mathematical object is their connection to the science that studies them. Maybe this is understood best by using (the idea of) time: Stones and Thunderstorms have been around for a lot longer than the sciences studying them; one of the questions that, therefore, does not arise, is as to the nature of the relationship between a thunderstorm and meteorology. A thunderstorm is something in and by itself. Being by itself is meant to signify being — existing — without the help or aid of humans. At this point the notion of time can be discarded; Thunderstorms are — they

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exist — without the help of humans, that is to say there presence — there beeing now — does not depend on them. But what about the breeze brought forth by a ventilator? Clearly, one would say, that breeze would not not exist without the help of humane beings, and stops being there once it the ventilator is swiched of. A verbal distinction will help: the difference between the breeze and the thunderstorm is the way in which they have been occasioned, that is the cause of their existence is different, not their way of existing. Another example: A computer has clearly been put together by humane beings (and, moreover, is the result of a lot of intellectual endeauvour by humande beings), however, once it is assebled, it is — it exists — without the help of a humane being. (It is helpful to) What about circles and E 0 = c∇ × B − 4πJ ? E 0 = c∇ × B − 4πJ is one of Maxwell’s equations; it is used as an example, because, as opposed to thunderstorms, we can say that it is — occasioned — by a human being (Maxwell). (Not any humane being — Maxwell ; somebody who has a name, a date of birth, and so on.) As a matter of simplicity we shall use x2 + y 2 = r2 instead of E 0 = c∇ × B − 4πJ ; both equations can be substituted for one another, because in our context they do have the same meaning. They are algebraic formulas. In particular x2 + y 2 = r2 is used to descirbe a circle. The word circle, however, is not just used to describe an algebraic expression, but also brings to mind the image of a circle, like one that is drawn by the help of a compass. A circle so drawn is, obviousley, comparable to a thunderstorm; it is by itself, and without the help of a humane being, whereas x2 + y 2 = r2 is a circle, only if one understands the meaning of the expression. We shall ad to this an additional complication: what about statements of properties of such objects: The tenth proposition of the third book of Euclid’s Elements: [Eucild, 2005, p. 7] states that a circle does not cut another circle at more than two points My Edition refers to 3.5, 3.1 and 1.11; 3.1 refers to 1.11, 1.9, 1.8 and definition 10; 1.11 refers to 1.8, 1.3, 1.1, and again definition 10. The point of this exercise is obvious: All of Euclid’s propositions are constructed by use of 23 Definitions, 5 Postulates and 5 Common notions. There are of course subsequent definitions and there is considerable doubt whether Euclid’s Definitions and are sufficient, well formulated and so on. Notice, that the point of interest is not something that can possibly exist; or even be imagined: One may imagine to circles cutting each other, and one may infer from this imagination that they cannot have more than two points of intersection, but this is, as stated, an inference, or a conclusion, or an observation, or a theorem, but it is definitively not something that can exist by itself. (To understand this point, you may ask yourself, whether you have ever met, drunken or touched a conclusion.) The example, however, gives a hint, as to the way it came into existence: it refers us to previous propositions. That is normally understood as providing certainty. For example, Thomas Hobbes discovered Euclid and his continuing love for geometry in a gentlemen’s library in Paris: The Elements lay there, opened on the page containing 1.47 — the proposition concerned with Pythagoras law, nothing less — and Hobbes could trace it back right to the Postulates.[Aubrey, 1999] That is to say, he read it backwards, in which case the propositions can be understood as being conclusively demonstrated. Reading Euclid forwards, however, they appear to be constructed from or by use of the first sentences. Constructed with care and diligence, and a lot of effort. They did not appear by themselves, as opposed to storms, which do appear without any humane help or effort. This gives rise to questions about their existence. To better understand what questions may be asked at this point, it seems advisable to get some help concerning the meaning of the word existence. What, exactly, does it mean to be there?

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Other Peoples Opinions

Having amassed problems and questions, we shall try to present ’solutions’ in an orderly manner. In particular three authors and their ideas will be examined. It should be noted, that this is an attempt to gather some ideas, not to comperhensively explain their positions on these issues. 3

3.1

According to Thomas Aquinas

Thomas Aquinas [Aquinas, 2008, c.1] discusses the difference between things that are potentially and things that are actually. Something that is ’in potentia’, is ’in possibility’, whereas something that is ’in actu’, in, literally, ’in action’ The latin verb ’ago’ means to do, to perform, to cause something; used as a noun ’actio’ means action, a deed, or a performance. The relationship between a possibility and an actuality is one of completeness: something that is merely in potentia is incomplete, whereas being actually means being complete. (His example is sperm and menstrual blood, which are potentially a human being.) . . . non habet esse ex eo quod advenit, sed per se habet esse completum. Things actually existing ’do not have being because of what comes to them, but by themselves they have complete existence.’1 To understand this, it is necessary to consider the context: Thomas tries to explain what the difference between subject and matter is, and he says that matter by itself is incomplete. There never is just matter, a thing is always something. Pure matter would be pure being there; if, however, we talk about being there, we already imply a place (and maybe also time), so one can easily see, that something always has a place and a time; in addition it also has properties. It is interesting to note, that we are speaking of an ’it’. What is this ’it’ ? It is a single object; Thomas example is a humane being, that is to say a single entity.

3.2

According to Francisco Suarez

Franciso Suarez tries to clarify what being actually (as opposed to potentially) means. It always means being this or that. One can always ask: what is that thing? [Suarez, 2008, 2, IV, 6] Quid est id per quod respondemus ad quaestionem quid sit res. Note that Suarez frames this as a question, as a mental exercise If we ask: ’What is that, by which we respond to the question what is the thing?’, we are making an observation from the point of view of a human being. Note, that this is in contrast to Thomas approach; no observer is necessary for Thomas argument. Consequently, Suarez has to make an additional qualification: Things actually existing ’. . . are they just made up by the intellect.’ (in latin: ’. . . neque est mere conficta per intellectum’ [Suarez, 2008, 2, IV, 7]). That is to say, ’. . . they are neither fictional nor chimeras, . . . ’ (In latin: ’. . . id est non fictam nec chymericam. . . ’ [Suarez, 2008, 2, IV, 6)]) It is worth the while to pause for a moment and to consider what this involves; Thomas says, things that are potentially are lacking; Suarez says they are not at all (omnio nihil) [Suarez, 2008, 31, II, 1], mere figments of our imagination. Notice that non being and being fictional are taken to mean the same. But since being imaginary does not quite mean being nothing, Suarz has to add a qualification, which is why he states, that he is not talking about imaginary things. He has to add that qualification, because he cannot conceivably mean that chimeras (or fictional object in gerneral) are not — he has already pointed out not only that they are, but also how they are (they are fictions). The difference between existing fictionally and existing actually is not a property; The very same thing (having the same properties) can be either actually or fictionally.

3.3

According to Thomas Hobbes

Thomas Hobbes [Hobbes, 1985, ch. 46; p. 689] states (quite categorically):2 The World . . . I mean . . . the Universe . . . is corporeal, that is to say, Body; and hath the dimensions of Magnitude, namely, Length, Bredth, and Depth: . . . every part of the Universe is Body, and that which is not Body is no part of the Universe: and because the Universe is All, that which is no part of it is nothing, and consequently nowhere. 1 The 2 The

translations in this paper are (exclusively) mine. spelling is that of the 1651 edition.

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That is clear enough; it should just be noted, that Hobbes version of existing actually, that is, of not being imaginary, is being bodily.

3.4

Circles and Thunderstorms reconsidered

Having gathered some concepts, we will no apply them. The first question to arise is, whether matematical objects have being because of what comes to them? (Aquinas) That is to say, does Proposition 3.10 exist because of it’s properties? The answer, clearly is, yes; that is the point of the exercise. (The point of having axioms, postulates and a way of combining them.) The next point is whether mathematical ojects are fictional; one can argue that a circle is not fictional, physical bodies do have the shape of circles, clearly they are not fictional, a line drawn with a compass is not fictional. Is that line a circle, or is it just a representation of a circle? The equation x2 + y 2 = r2 is not an equation of a (single) circle but the equation of any circle (in an Euclidian plan, with the center at the origion, etc.). Following Aquinas and Suarez we have to focus on single entities. A statement like: ’a circle does not cut another circle at more than two points’ moreover, does refer to two geometrical figures, however it’s substance is not concerned with anything like figure, but with what can and what cannot be. Mathematical object are, of course, not just figures. We may think about criterias for the convergence of infinite series, and apply the Thomas Hobbes test. While one can at least imagine a figure of geometry (with no than three dimensions), to be somewhere, a criteria for convergence is no somewhere, however, that does not mean it is nothing. Or does it? Hobbes opposition to arithemetic is often subject either to ridicule or simply bewilderment [?, p. 70 ff]. However, given his philosophical priciples this is not inconsitent; The figures of geometrie have ’Length, Bredth, and Depth’, whereas arthimetical expressions have not. Some of the might signify a figure, that is describe in some way, and then, the meight not. Going forward, the importand thing to realize is, that although mathematical objects clearly do exist — if not, we would not be able to talk about them — but that their way of existence is somewhat problematic. One important distinction is between being fictional and being actual, being actual may mean being corporal, whereas being fictional definitvely means not being corporeal.

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What is a Probability?

Having accumulated some opinions, we will now be able to follow a discussion that is something like a mirror image of the question raised before. J.M. Keynes maintains that probabilities are logical relations between two or more arguments, that is to say, they are fundamentally thoughts. (They happen, so to speak, between your ears and behind your eyes, and definitively no place else.) To explain why this should be the way we think of them, he contrasts his opinions with those of John Venn, who, Keynes says, brought forward the opinion that probabilities are observable statistical frequencies. Venn maintained that, if there is a series of events that have something in common, and something different, the occurrence of what is different can have an observable frequency, he then identifies the observed frequency with the probability that any event will occur.[Keynes, 2004, p. 92] Another adherent of the frequency theory of probability M.G. Bulmer gives an example: There is a coin (that is, what the events have in common), it is tossed repeatedly (the series), and the occurrence of head or tail (the difference) has a frequency.[Bulmer, 1965, p. 2] I shall link the opinion that the objects of mathematics are thoughts to the opinion that probabilities are logical relations, and conversely the opinion that they are statistical frequencies to the opinion that they have a being distinguished from thoughts alone. Keynes summarizes [Keynes, 2004, pp. 93] Venns opinion: Venns maintains an empirical view of logic; the distinctive characteristics of probability prevail, in Venns opinion, principally in properties of natural kinds,

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as distinguished from things artificial. The existence of a . . . probability series can be established only by experience. Keynes summs up his summary of Venn opinion: If we want to discover what is in reality a series of things, not a series of our own conceptions, we have to appeal to the things themselves. If we try to imagine how Venn must have understood reality, we may think of reality being like a dolls house: 1. There is a reality, like there is a doll’s house. There are things that are in reality, like dolls in a doll’s house. It is possible to be in a doll’s house, as well as it is possible to be someplace else. Likewise it is possible to be in reality, as well as elsewhere, for example something can be in our own conceptions. (If one is to extend this thought, it would be possible to say, that things can also — that is: in addition – be in a virtual reality.) 2. Interestingly enough, adding dolls of our own design just doesn’t do: Things artificial are not allowed. 3. Notice, there are not just dolls in the house: there are also series, which can be found there no less than dolls. Like Thomas Aquinas Venn thinks it to be important that what he observes is independent, has it’s being because of itself. If one were to pursue Aquinas idea further, one would find that humane action is, just like Venn, what motivates the thinking about independence. Like Suarez Venn thinks about the difference between what actually and in fact is, and what is just fiction. This, however, requires a more elaborate explanation: Keynes, as said, summarizes Venn’s opinion by juxtaposing a series of things 3 with a series of our own conceptions. Suarez writings are highly technical and abstract, and, fittingly, he uses the word esse in all its grammatical alterations to talk about a being – that is to say, about what is. He also talks at some length about the different meanings these alterations have.[Suarez, 2008, 2, IV] The word esse is a highly abstract term, encompassing literally anything, one can possibly talk about. The English language uses the words like ‘anything’, ‘something’, ‘everything’ and ‘nothing’ in about the same way. In the passages quoted here, however, Suarez uses the word ‘res’; easily translated as ‘thing’. Notice that words like anything, something, etc. contain the word ‘thing’, but do not necessarily refer to something tangible, like a vase, a coin, a doll or a human being, whereas the Latin word ‘res’ always does. That is the word ’res’ always refers to a Hobbes style entity, one that his dimensions. To give an example: One can say: ‘I’ve got something like a thought, an idea or a plan’ whereas it would be meaningless to speak of ‘res aliquam quasi idea mentis’. Why does Suarez use the word ‘res’ ? Because he is not speaking about some being, any being, but about something that is a Thomas Hobbes style body. He describes that kind of being in various ways, on way is to say it has such properties as enable a sensual perception, another way is to say it is not imagined, it is not fictional, which is exactly Venn’s juxtaposition. Venn, however, ‘identifies’ – Keynes choice of words not mine – series of probabilities with series of things. That is problematic, because Venn cannot possibly believe that a series is a corporeal object. Keynes says, Venn has ’an empirical view’ of logic. [Keynes, 2004, p. 93] To understand what this meas, we shall turn to Bulmer, whose opinions on the subject I take to be very similar to Venns. Bulmers discussion of what happens, when a coin is tossed is not a thought experiment. He makes that perfectly clear by telling the story of a Danish mathematician, who, being imprisoned, had nothing better to do but to toss a coin ten thousand times. (He got 5067 times head, and 4933 times tails.) [Bulmer, 1965, p. 2 f] Bulmer adds, that it would be reasonable to suppose, if the experiment were to be continued indefinitely, by – one can only assume – the Danish mathematician who in the mean time has been transferred to (a mathematicians version of) Dantes Inferno, the Results would be split evenly. The point of this little stories is to avoid the inconvenience of explaining the difference between an observation, a series of observations, and a mathematical formula. 3 The

word things is italicized by Keynes.

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Keynes criticism of Venns theory is revealing not for what it says, but for what is missing. Keyes criticizes Venns theory as being much to narrow, because “it excludes a great number of judgments which ... deal with probability.” [Keynes, 2004, p. 95 f] Venn (and Bulmer) exclude subjective or perceived probabilities, because they think that subjective probabilities cannot be measured. Not, because they think they do not exist. Keynes main criticism is that it is not at all clear what occasions the selection of this and that as a series. What, for example, occasions us to perceive the repeated tossing of a coin as a series? Keynes point is that Venn has to resolve himself, as to object of his inquiry. Decisions like that cannot be said to emanate for anything or object, for they do not tell us what is to be considered relevant or not. That is a clever point, but if one is to consider the basic ideas of Keynes approach, at best a sideshow. Keynes does not even attempt to make the case, that Venn must (somehow) believe, that series of probabilities exist, just like corporeal objects exist. (Even so, my guess would be, that confronted with the argument, Venn would vigorously deny that. But, clearly, this is what his theory implies.)

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Conclusion

Reading a textbook about statistics, like Bulmers, questions like the ones raised here appear if at all only as short notes at the beginning, before things Bulmer doubtlessly considers the obvious object of mathematics, like: long equations, graphs, charts and related things are discussed. The principal aim of this discussion was to say, that this is nonsensical. If one reads Keynes Treatise one encounters not just one but a whole series of very diverse ontological theories and commitments. It is to Keynes great credit, one should add, that he –– as opposed to so many other theoreticians —- at least notices that he has (more than one) philosophical problem. The choices and commitments underlying the different theories are not just of philosophical interest, but vital to the deyelopment of mathematical theories themselves.

References [Aquinas, 2008] Aquinas, T. (2008). http://www.corpusthomisticum/open.html.

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principiis

naturae.

[Aubrey, 1999] Aubrey, J. (1999). Humane Nature and de corpore politico, chapter The Brief Life, pages 231 – 244. Oxford University Press. [Bulmer, 1965] Bulmer, M. (1979 (1965)). Principles of Statistics. Dover Publications. [Carnap, 1958] Carnap, R. (1958). Introduction to Symboloic logic and its Applications. Dover Publications. Inc., New York. [Eucild, 2005] Eucild (2005). Euclids Elements in Translation, chapter English Translation. o.A. [Hobbes, 1985] Hobbes, T. (1985). Leviathan. Penguin Books, London. [Keynes, 2004] Keynes, J. M. (2004). A Treatise on Probability. Dover Publications. [Suarez, 2008] Suarez, F. (2008). Disputationes metaphysicae. bochum.de/Michael.Renemann/Suarez/index.html.

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