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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Derivability from Relational Logic of Charles Sanders Peirce to Essential Laws of Quantum Mechanics

Charles Sanders Peirce made important contributions in logic, where he invented and elaborated novel system of logical syntax and fundamental logical concepts. The starting point is the binary relation S_iRS_j between the two ‘individual terms’ (subjects) S_j and S_i. In a short hand notation we represent this relation by R_ij. Relations may be composed: whenever we have relations of the form R_ij, R_jl, a third transitive relation R_il emerges following the rule

R_ijR_kl = δ_jkR_il —– (1)

In ordinary logic the individual subject is the starting point and it is defined as a member of a set. Peirce considered the individual as the aggregate of all its relations

S_i = ∑_j R_ij —– (2)

The individual S_i thus defined is an eigenstate of the R_ii relation

R_iiS_i = S_i —– (3)

The relations R_ii are idempotent

R²_ii = R_ii —– (4)

and they span the identity

∑_i R_ii = 1 —– (5)

The Peircean logical structure bears resemblance to category theory. In categories the concept of transformation (transition, map, morphism or arrow) enjoys an autonomous, primary and irreducible role. A category consists of objects A, B, C,… and arrows (morphisms) f, g, h,… . Each arrow f is assigned an object A as domain and an object B as codomain, indicated by writing f : A → B. If g is an arrow g : B → C with domain B, the codomain of f, then f and g can be “composed” to give an arrow gof : A → C. The composition obeys the associative law ho(gof) = (hog)of. For each object A there is an arrow 1_A : A → A called the identity arrow of A. The analogy with the relational logic of Peirce is evident, R_ij stands as an arrow, the composition rule is manifested in equation (1) and the identity arrow for A ≡ S_i is R_ii.

R_ij may receive multiple interpretations: as a transition from the j state to the i state, as a measurement process that rejects all impinging systems except those in the state j and permits only systems in the state i to emerge from the apparatus, as a transformation replacing the j state by the i state. We proceed to a representation of R_ij

R_ij = |r_i⟩⟨r_j| —– (6)

where state ⟨r_i | is the dual of the state |r_i⟩ and they obey the orthonormal condition

⟨r_i |r_j⟩ = δ_ij —– (7)

It is immediately seen that our representation satisfies the composition rule equation (1). The completeness, equation (5), takes the form

∑_n|r_i⟩⟨r_i|=1 —– (8)

All relations remain satisfied if we replace the state |r_i⟩ by |ξ_i⟩ where

|ξ_i⟩ = 1/√N ∑_n |r_i⟩⟨r_n| —– (9)

with N the number of states. Thus we verify Peirce’s suggestion, equation (2), and the state |r_i⟩ is derived as the sum of all its interactions with the other states. R_ij acts as a projection, transferring from one r state to another r state

R_ij |r_k⟩ = δ_jk |r_i⟩ —– (10)

We may think also of another property characterizing our states and define a corresponding operator

Q_ij = |q_i⟩⟨q_j | —– (11)

with

Q_ij |q_k⟩ = δ_jk |q_i⟩ —– (12)

and

∑_n |q_i⟩⟨q_i| = 1 —– (13)

Successive measurements of the q-ness and r-ness of the states is provided by the operator

R_ijQ_kl = |r_i⟩⟨r_j |q_k⟩⟨q_l | = ⟨r_j |q_k⟩ S_il —– (14)

with

S_il = |r_i⟩⟨q_l | —– (15)

Considering the matrix elements of an operator A as Anm = ⟨rn |A |rm⟩ we find for the trace

Tr(S_il) = ∑_n ⟨r_n |S_il |r_n⟩ = ⟨q_l |r_i⟩ —– (16)

From the above relation we deduce

Tr(R_ij) = δ_ij —– (17)

Any operator can be expressed as a linear superposition of the R_ij

A = ∑_i,j A_ijR_ij —– (18)

with

A_ij =Tr(AR_ji) —– (19)

The individual states could be redefined

|r_i⟩ → e^iφ_i |r_i⟩ —– (20)

|q_i⟩ → e^iθ_i |q_i⟩ —– (21)

without affecting the corresponding composition laws. However the overlap number ⟨r_i |q_j⟩ changes and therefore we need an invariant formulation for the transition |r_i⟩ → |q_j⟩. This is provided by the trace of the closed operation R_iiQ_jjR_ii

Tr(R_iiQ_jjR_ii) ≡ p(q_j, r_i) = |⟨r_i |q_j⟩|² —– (22)

The completeness relation, equation (13), guarantees that p(q_j, r_i) may assume the role of a probability since

∑_j p(q_j, r_i) = 1 —– (23)

We discover that starting from the relational logic of Peirce we obtain all the essential laws of Quantum Mechanics. Our derivation underlines the outmost relational nature of Quantum Mechanics and goes in parallel with the analysis of the quantum algebra of microscopic measurement.