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

H_Reichenbach

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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