Pluralist Mathematics, Minimalist Philosophy: Hans Reichenbach. Drunken Risibility.

Hans Reichenbach relativized the notion of the constitutive a priori. The key observation concerns the fundamental difference between definitions in pure geometry and definitions in physical geometry. In pure geometry there are two kinds of definition: first, there are the familiar explicit definitions; second, there are implicit definitions, that is the kind of definition whereby such fundamental terms as ‘point’, ‘line’, and ‘surface’ are to derive their meaning from the fundamental axioms governing them. But in physical geometry a new kind of definition emerges – that of a physical (or coordinative) definition:

The physical definition takes the meaning of the concept for granted and coordinates to it a physical thing; it is a coordinative definition. Physical definitions, therefore, consist in the coordination of a mathematical definition to a “piece of reality”; one might call them real definitions. (Reichenbach, 8)

Now there are two important points about physical definitions. First, some such correlation between a piece of mathematics and “a piece of physical reality” is necessary if one is to articulate the laws of physics (e.g. consider “force-free moving bodies travel in straight lines”). Second, given a piece of pure mathematics there is a great deal of freedom in choosing the coordinative definitions linking it to “a piece of physical reality”, since… coordinative definitions are arbitrary, and “truth” and “falsehood” are not applicable to them. So we have here a conception of the a priori which (by the first point) is constitutive (of the empirical significance of the laws of physics) and (by the second point) is relative. Moreover, on Reichenbach’s view, in choosing between two empirically equivalent theories that involve different coordinative definitions, there is no issue of “truth” – there is only the issue of simplicity. In his discussion of Einstein’s particular definition of simultaneity, after noting its simplicity, Reichenbach writes: “This simplicity has nothing to do with the truth of the theory. The truth of the axioms decides the empirical truth, and every theory compatible with them which does not add new empirical assumptions is equally true.” (p 11)

Now, Reichenbach went beyond this and he held a more radical thesis – in addition to advocating pluralism with respect to physical geometry (something made possible by the free element in coordinative definitions), he advocated pluralism with respect to pure mathematics (such as arithmetic and set theory). According to Reichenbach, this view is made possible by the axiomatic conception of Hilbert, wherein axioms are treated as “implicit definitions” of the fundamental terms:

The problem of the axioms of mathematics was solved by the discovery that they are definitions, that is, arbitrary stipulations which are neither true nor false, and that only the logical properties of a system – its consistency, independence, uniqueness, and completeness – can be subjects of critical investigation. (p 3)

It needs to be stressed here that Reichenbach is extending the Hilbertian thesis concerning implicit definitions since although Hilbert held this thesis with regard to formal geometry he did not hold it with regard to arithmetic.

On this view there is a plurality of consistent formal systems and the notions of “truth” and “falsehood” do not apply to these systems; the only issue in choosing one system over another is one of convenience for the purpose at hand and this is brought out by investigating their metamathematical properties, something that falls within the provenance of “critical investigation”, where there is a question of truth and falsehood. This radical form of pluralism came to be challenged by Gödel’s discovery of the incompleteness theorems. To begin with, through the arithmetization of syntax, the metamathematical notions that Reichenbach takes to fall within the provenance of “critical investigation” were themselves seen to be a part of arithmetic. Thus, one cannot, on pain of inconsistency, say that there is a question of truth and falsehood with regard to the former but not the latter. More importantly, the incompleteness theorems buttressed the view that truth outstrips consistency. This is most clearly seen using Rosser’s strengthening of the first incompleteness theorem as follows: Let T be an axiom system of arithmetic that (a) falls within the provenance of “critical investigation” and (b) is sufficiently strong to prove the incompleteness theorem. A natural choice for such an axiom system is Primitive Recursive Arithmetic (PRA) but much weaker systems suffice, for example, IΔ₀ + exp. Either of these systems can be taken as T. Assuming that T is consistent (something which falls within the provenance of “critical investigation”), by Rosser’s strengthening of the first incompleteness theorem, there is a Π₀¹-sentence φ such that (provably within T + Con(T )) both T + φ and T + ¬φ are consistent. However, not both systems are equally legitimate. For it is easily seen that if a Π₀¹-sentence φ is independent from such a theory, then it must be true. The point being that T is ∑₁⁰-complete (provably so in T). So, although T + ¬φ is consistent, it proves a false arithmetical statement.

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Task of the Philosopher. Thought of the Day 75.0

Poincaré in Science and Method discusses how “reasonable” axioms (theories) are chosen. In a section which is intended to cool down the expectations put in the “logistic” project, he points out the problem as follows:

Even admitting that it has been established that all theorems can be deduced by purely analytical processes, by simple logical combinations of a finite number of axioms, and that these axioms are nothing but conventions, the philosopher would still retain the right to seek the origin of these conventions, and to ask why they were judged preferable to the contrary conventions.

[ …] A selection must be made out of all the constructions that can be combined with the materials furnished by logic. the true geometrician makes this decision judiciously, because he is guided by a sure instinct, or by some vague consciousness of I know not what profounder and more hidden geometry, which alone gives a value to the constructed edifice.

Hence, Poincaré sees the task of the philosophers to be the explanation of how conventions came to be. At the end of the quotation, Poincaré tries to give such an explanation, namely in referring to an “instinct” (in the sequel, he mentions briefly that one can obviously ask where such an instinct comes from, but he gives no answer to this question). The pragmatist position to be developed will lead to an essentially similar, but more complete and clear point of view.

According to Poincaré’s definition, the task of the philosopher starts where that of the mathematician ends: for a mathematician, a result is right if he or she has a proof, that means, if the result can be logically deduced from the axioms; that one has to adopt some axioms is seen as a necessary evil, and one perhaps puts some energy in the project to minimize the number of axioms (this might have been how set theory become thought of as a foundation of mathematics). A philosopher, however, wants to understand why exactly these axioms and no other were chosen. In particular, the philosopher is concerned with the question whether the chosen axioms actually grasp the intended model. This question is justified since formal definitions are not automatically sufficient to grasp the intention of a concept; at the same time, the question is methodologically very hard, since ultimately a concept is available in mathematical proof only by a formal explication. At any rate, it becomes clear that the task of the philosopher is related to a criterion problem.

Georg Kreisel thinks that we do indeed have the capacity to decide whether a given model was intended or not:

many formal independence proofs consist in the construction of models which we recognize to be different from the intended notion. It is a fact of experience that one can be honest about such matters! When we are shown a ‘non-standard’ model we can honestly say that it was not intended. [ . . . ] If it so happens that the intended notion is not formally definable this may be a useful thing to know about the notion, but it does not cast doubt on its objectivity.

Poincaré could not yet know (but he was experienced enough a mathematician to “feel”) that axiom systems quite often fail to grasp the intended model. It was seldom the work of professional philosophers and often the byproduct of the actual mathematical work to point out such discrepancies.

Following Kant, one defines the task of epistemology thus: to determine the conditions of the possibility of the cognition of objects. Now, what is meant by “cognition of objects”? It is meant that we have an insight into (the truth of) propositions about the objects (we can then speak about the propositions as facts); and epistemology asks what are the conditions for the possibility of such an insight. Hence, epistemology is not concerned with what objects are (ontology), but with what (and how) we can know about them (ways of access). This notwithstanding, both things are intimately related, especially, in the Peircean stream of pragmatist philosophy. The 19th century (in particular Helmholtz) stressed against Kant the importance of physiological conditions for this access to objects. Nevertheless, epistemology is concerned with logic and not with the brain. Pragmatism puts the accent on the means of cognition – to which also the brain belongs.

Kant in his epistemology stressed that the object depends on the subject, or, more precisely, that the cognition of an object depends on the means of cognition used by the subject. For him, the decisive means of cognition was reason; thus, his epistemology was to a large degree critique of reason. Other philosophers disagreed about this special role of reason but shared the view that the task of philosophy is to criticise the means of cognition. For all of them, philosophy has to point out about what we can speak “legitimately”. Such a critical approach is implicitly contained in Poincaré’s description of the task of the philosopher.

Reichenbach decomposes the task of epistemology into different parts: guiding, justification and limitation of cognition. While justification is usually considered as the most important of the three aspects, the “task of the philosopher” as specified above following Poincaré is not limited to it. Indeed, the question why just certain axioms and no others were chosen is obviously a question concerning the guiding principles of cognition: which criteria are at work? Mathematics presents itself at its various historical stages as the result of a series of decisions on questions of the kind “Which objects should we consider? Which definitions should we make? Which theorems should we try to prove?” etc. – for short: instances of the “criterion problem”. Epistemology, has all the task to evoke these criteria – used but not evoked by the researchers themselves. For after all, these criteria cannot be without effect on the conditions for the possibility of cognition of the objects which one has decided to consider. (In turn, the conditions for this possibility in general determine the range of objects from which one has to choose.) However, such an epistemology has not the task to resolve the criterion problem normatively (that means to prescribe for the scientist which choices he has to make).