1
Introduction

1.1 Why a New Book on Probability?

There exist numerous treatments of this topic, many of which are very good, and others continue to appear. To add one more would certainly be a presumptuous undertaking if I thought in terms of doing something better, and a useless undertaking if I were to content myself with producing something similar to the 'standard' type. Instead, the purpose is a different one: it is that already essentially contained in the dedication to Beniamino Segre

[who about twenty years ago pressed me to write it as a necessary document for clarifying one point of view in its entirety.]

Segre was with me at the International Congress of the Philosophy of Science (Paris 1949), and it was on the occasion of the discussions developed there on the theme of probability that he expressed to me, in persuasive and peremptory terms, a truth, perhaps obvious, but which only since appeared to me as an obligation, difficult but unavoidable.

'Only a complete treatment, inspired by a well-defined point of view and collecting together the different objections and innovations, showing how the whole theory results in coherence in all of its parts, can turn out to be convincing. Only in this way is it possible to avoid the criticisms to which fragmentary expositions easily give rise since, to a person who in looking for a completed theory interprets them within the framework of a different point of view, they can seem to lead unavoidably to contradictions.'

These are Segre's words, or, at least, the gist of them.

It follows that the requirements of the present treatment are twofold: first of all to clarify, exhaustively, the conceptual premises, and then to give an essentially complete exposition of the calculus of probability and its applications in order to establish the adequacy of the interpretations deriving from those premises. In saying 'essentially' complete, I mean that what matters is to develop each topic just as far as is necessary to avoid conceptual misunderstandings. From then on, the reader could follow any other book without finding great difficulty in making those modifications that are needed in order to translate it, if such be desired, according to the point of view that will be taken here. Apart from these conceptual exigencies, each topic will also be developed, in terms of the content, to an extent sufficient for the treatment to turn out to be adequate for the needs of the average reader.

1.2 What are the Mathematical Differences?

1.2.1. If I thought I were writing for readers absolutely innocent of probabilistic-statistical concepts, I could present, with no difficulty, the theory of probability in the way I judge to be meaningful. In such a case, it would not even have been necessary to say that the treatment contains something new and, except possibly under the heading of information, that different points of view exist. The actual situation is very different, however, and we cannot expect any sudden change.

My estimation is that another fifty years will be needed to overcome the present situation, but perhaps even this is too optimistic. It is based on the consideration that about thirty years were required for ideas born in Europe (Ramsey, 1926; de Finetti, 1931) to begin to take root in America (even though B.O. Koopman (1940) had come to them in a similar form). Supposing that the same amount of time might be required for them to establish themselves there, and then the same amount of time to return, we arrive at the year 2020.

It would obviously be impossible and absurd to discuss in advance concepts and, even worse, differences between concepts to whose clarification we will be devoting all of what follows; however, much less might be useful (and, anyway, will have to suffice for the time being). It will be sufficient to make certain summary remarks that are intended to exemplify, explain and anticipate for the reader certain differences in attitude that could disorientate him, and leave him undecided between continuing without understanding or, on the other hand, stopping reading altogether. It will be necessary to show that the 'wherefore' exists and to give at least an idea of the 'wherefore', and of the 'wherefores', even without anticipating the 'wherefore' of every single case (which can only be seen and gone into in depth at the appropriate time and place).

1.2.2. From a mathematical point of view, it will certainly seem to the reader that either by desire or through ineptitude I complicate simple things; introducing captious objections concerning aspects that modern developments in mathematical analysis have definitively dealt with. Why do I myself not also conform to the introduction of such developments into the calculus of probability? Is it a question of incomprehension? Of misoneism? Of affectation in preferring to use the tools of the craftsman in an era of automation which allows mass production even of brains - both electronic and human?

The 'wherefore', as I see it, is a different one. To me, mathematics is an instrument that should conform itself strictly to the exigencies of the field in which it is to be applied. One cannot impose, for their own convenience, axioms not required for essential reasons, or actually in conflict with them.

I do not think that it is appropriate to speak of 'incomprehension'. I have followed through, and appreciated, the reasons pro (which are the ones usually put forward), but I found the reasons contra (which are usually neglected) more valid, and even preclusive.

I do not think that one can talk of misoneism. I am, in fact, very much in favour of innovation and against any form of conservatism (but only after due consideration, and not by submission to the tyrannical caprice of fashion). Fashion has its use in that it continuously throws up novelties, guarding against fossilization; in view of such a function, it is wise to tolerate with goodwill even those things we do not like. It is not wise, however, to submit to passively adapting our own taste, or accepting its validity beyond the limits that correspond to our own dutiful, critical examination.

I do not think that one can talk of 'affectation' either. If anything, the type of 'affectation' that is congenial to my taste would consist of making everything simple, intuitive and informal. Thus, when I raise 'subtle' questions, it means that, in my opinion, one simply cannot avoid doing so.

1.2.3. The 'wherefore' of the choice of mathematical apparatus, which the reader might find irksome, resides, therefore, in the 'wherefores' related to the specific meaning of probability, and of the theory that makes it an object of study. Such 'wherefores' depend, in part, on the adoption of this or that particular point of view with regard to the concept and meaning of probability, and to the basis from which derives the possibility of reasoning about it, and of translating such reasoning into calculations. Many of the 'wherefores' seem to me, however, also to be valid for all, or many, of the different concepts (perhaps with different force and different explanations). In any case, the critical analysis is more specifically hinged on the conception that we follow here, and which will appear more and more clear (and, hopefully, natural) as the reader proceeds to the end - provided he or she has the patience to do so.

1.3 What are the Conceptual Differences?

1.3.1. Meanwhile, for those who are not aware of it, it is necessary to mention that in the conception we follow and sustain here only subjective probabilities exist - that is, the degree of belief in the occurrence of an event attributed by a given person at a given instant and with a given set of information. This is in contrast to other conceptions that limit themselves to special types of cases in which they attribute meaning to 'objective probabilities' (for instance, cases of symmetry as for dice etc., 'statistical' cases of 'repeatable' events, etc.). This said, it is necessary to add at once that we have no interest, at least for now, either in a discussion, or in taking up a position, about the 'philosophical' aspects of the dispute; in fact, it would be premature and prejudicial because it would entangle the examination of each concrete point in a web of metaphysical misunderstandings.

Instead, we are interested, on the contrary, in clearly understanding what one means according to one's own conception and in one's own language, and learning to enter into this conception and language in its motivations and implications (even if provisionally, in order to be able to make pertinent criticism later on). This is, it seems to me, an inviolable methodological need.

1.3.2. There is nothing more disappointing than to hear repeated, presented as 'criticisms', clichés so superficial that it is not possible to infer whether the speaker has even read the arguments developed to confute them and clear them up, or has read them without understanding anything, or else has understood them back to front. The fault could be that of obscure presentation, but a somewhat more meaningful reaction would be required in order to be able to specify accurately, and to correct, those points which lend themselves to misunderstanding.

The fault may be the incompleteness of the preceding, more or...