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

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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.

Unique Derivative Operator: Reparametrization. Metric Part 2.

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Moving on from first part.

Suppose ∇ is a derivative operator, and gab is a metric, on the manifold M. Then ∇ is compatible with gab iff ∇a gbc = 0.

Suppose γ is an arbitrary smooth curve with tangent field ξa and λa is an arbitrary smooth field on γ satisfying ξnnλa = 0. Then

ξnn(gabλaλb) = gabλaξnnλb + gabλbξnnλa + λaλbξnngab

= λaλbξnngab

Suppose first that ∇ngab = 0. Then it follows immediately that ξnngabλaλb = 0. So ∇ is compatible with gab. Suppose next that ∇ is compatible with gab. Then ∀ choices of γ and λa (satisfying ξnnλa =0), we have λaλbξnngab = 0. Since the choice of λa (at any particular point) is arbitrary and gab is symmetric, it follows that ξnngab = 0. But this must be true for arbitrary ξa (at any particular point), and so we have ∇ngab = 0.

Note that the condition of compatibility is also equivalent to ∇agbc = 0. Hence,

0 = gbnaδcn = gbna(gnrgrc) = gbngnragrc + gbngrcagnr

= δbragrc + gbngrcagnr = ∇agbc + gbngrcagnr.

So if ∇agbc = 0,it follows immediately that ∇agbc = 0. Conversely, if ∇agbc =0, then gbngrcagnr = 0. And therefore,

0 = gpbgscgbngrcagnr = δnpδrsagnr = ∇agps

The basic fact about compatible derivative operators is the following.

Suppose gab is a metric on the manifold M. Then there is a unique derivative operator on M that is compatible with gab.

It turns out that if a manifold admits a metric, then it necessarily satisfies the countable cover condition. And then it guarantees the existence of a derivative operator.) We do prove that if M admits a derivative operator ∇, then it admits exactly one ∇′ that is compatible with gab.

Every derivative operator ∇′ on M can be realized as ∇′ = (∇, Cabc), where Cabc is a smooth, symmetric field on M. Now

∇′agbc = ∇agbc + gnc Cnab + gbn Cnac = ∇agbc + Ccab + Cbac. So ∇′ will be compatible with gab (i.e., ∇′agbc = 0) iff

agbc = −Ccab − Cbac —– (1)

Thus it suffices for us to prove that there exists a unique smooth, symmetric field Cabc on M satisfying equation (1). To do so, we write equation (1) twice more after permuting the indices:

cgab = −Cbca − Cacb,

bgac = −Ccba − Cabc

If we subtract these two from the first equation, and use the fact that Cabc is symmetric in (b, c), we get

Cabc = 1/2 (∇agbc − ∇bgac − ∇cgab) —– (2)

and, therefore,

Cabc = 1/2 gan (∇ngbc − ∇bgnc − ∇cgnb) —– (3)

This establishes uniqueness. But clearly the field Cabc defined by equation (3) is smooth, symmetric, and satisfies equation (1). So we have existence as well.

In the case of positive definite metrics, there is another way to capture the significance of compatibility of derivative operators with metrics. Suppose the metric gab on M is positive definite and γ : [s1, s2] → M is a smooth curve on M. We associate with γ a length

|γ| = ∫s1s2 gabξaξb ds,

where ξa is the tangent field to γ. This assigned length is invariant under reparametrization. For suppose σ : [t1, t2] → [s1, s2] is a diffeomorphism we shall write s = σ(t) and ξ′a is the tangent field of γ′ = γ ◦ σ : [t1, t2] → M. Then

ξ′a = ξads/dt

We may as well require that the reparametrization preserve the orientation of the original curve – i.e., require that σ (t1) = s1 and σ (t2) = s2. In this case, ds/dt > 0 everywhere. (Only small changes are needed if we allow the reparametrization to reverse the orientation of the curve. In that case, ds/dt < 0 everywhere.) It

follows that

|γ’| = ∫t1t2 (gabξ′aξ′b)1/2 dt = ∫t1t2 (gabξaξb)1/2 ds/dt

= ∫s1s2 (gabξaξb)1/2 ds = |γ|

Let us say that γ : I → M is a curve from p to q if I is of the form [s1, s2], p = γ(s1), and q = γ(s2). In this (positive definite) case, we take the distance from p to q to be

d(p,q)=g.l.b. |γ|:γ is a smooth curve from p to q.

Further, we say that a curve γ : I → M is minimal if, for all s ∈ I, ∃ an ε > 0 such that, for all s1, s2 ∈ I with s1 ≤ s ≤ s2, if s2 − s1 < ε and if γ′ = γ|[s1, s2] (the restriction of γ to [s1, s2]), then |γ′| = d(γ(s1), γ(s2)) . Intuitively, minimal curves are “locally shortest curves.” Certainly they need not be “shortest curves” outright. (Consider, for example, two points on the “equator” of a two-sphere that are not antipodal to one another. An equatorial curve running from one to the other the “long way” qualifies as a minimal curve.)

One can characterize the unique derivative operator compatible with a positive definite metric gab in terms of the latter’s associated minimal curves. But in doing so, one has to pay attention to parametrization.

Let us say that a smooth curve γ : I → M with tangent field ξa is parametrized by arc length if ∀ ξa, gabξaξb = 1. In this case, if I = [s1, s2], then

|γ| = ∫s1s2 (gabξaξb)1/2 ds = ∫s1s2 1.ds = s2 – s1

Any non-trivial smooth curve can always be reparametrized by arc length.

Coding Information While Operators Fail to Commute. Drunken Risibility.

Suppose ∇ is a derivative operator on the manifold M. Then there is a (unique) smooth tensor field Rabcd on M such that for all smooth fields ξb,

Rabcd ξb = −2∇[cd] ξa —– (1)

Uniqueness is immediate since any two fields that satisfied this condition would agree in their action on all vectors ξb at all points. For existence, we introduce a field Rabcd and do so in such a way that it is clear that it satisfies the required condition. Let p be any point in M and let ξ’b be any vector at p. We define Rabcd ξ’b by considering any smooth field ξb on M that assumes the value ξ’b at p and setting Rabcdξ’b = −2∇[cda. It suffices to verify that the choice of the field ξb plays no role. For this it suffices to show that if ηb is a smooth field on M that vanishes at p, then necessarily ∇[cd] ηb vanishes at p as well. (For then we can apply this result, taking ηb to be the difference between any two candidates for ξb.)

The usual argument works. Let λa be any smooth field on M. Then we have,

0 = ∇[cd] (ηaλa) = ∇[c ηad] λa + ηa[cd] λa  + (∇[c λ|a|) (∇d] ηa) + λa ∇[cd] ηa —– (2)

It is to be noted that in the third term of the final sum the vertical lines around the index indicate that it is not to be included in the anti-symmetrization. Now the first and third terms in that sum cancel each other. And the second vanishes at p. So we have 0= λa∇[cda at p. But the field λa can be chosen so that it assumes any particular value at p. So ∇[cd] ηa = 0 at p.

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Rabcd is called the Riemann curvature tensor field (associated with ∇). It codes information about the degree to which the operators ∇c and ∇d fail to commute.