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The Provenance of Pure Reason
by William Tait
Contents
This is a collection of essays dating from 1881-2002 together with such supplementary material (introduction and endnotes) as was thought necessary to reflect the development of Tait's ideas over the period. Many of these papers are also available from Tait's website (Chs 2,4,8,10,11,12), and my reading of the book, which I have on loan, will therefore first address those chapters which I don't have in electronic copy.
The introduction is quite short and is intended strictly as a supplement to rather than an introduction to the papers, almost exclusively concerned with the non-historical material.
Tait observes that "the crux" of understanding Hilbert's conception of finitist mathematics is in the answer to the question how general claims about natural numbers can be proven without assuming the infinitude of numbers or some other infinite totality. This chapter attempts to answer that question, and also argues that finitist reasoning is primitive recursive reasoning.
2. Remarks on Finitism
Appendix to Chapters 1 and 2
Tait identifies the Truth/Proof problem as the problem of establishing that there is a connection between a mathematical proposition being provable and its being true (of the platonic entities which are its subject matter). His aim in this chapter is to defend Platonism in mathematics against this attack.
An explication and defence of what Tait calls "realism" in contrast with "super-realism" (which is what many others call "realism"). Tait's position is very close to that of Hilbert.
5. The Law of Excluded middle and the Axiom of Choice
6. Constructing Cardinals from Below
7. Plato's Second-Best Method
8. Noesis: Plato on Exact Science
My comments on Tait's paper on Kripke's analysis of some ideas of Wittgenstein. To do with following rules.
10. Frege versus Cantor and Dedekind: On the Concept of Number
A summary of the content of Cantor's 1983 paper "Foundations of a General Theory of Manifolds" and a discussion of how this relates to the paradoxes of set theory.
12. Goedel's Unpublished Papers on Foundations of Mathematics
Introduction
The introduction is quite short and is intended strictly as a supplement to rather than an introduction to the papers, almost exclusively concerned with the non-historical material.

The introduction is in six sections preceded by a bit. These include discussions of some of the issues and references to the chapters which provide further discussion. My notes replicate some of these references (Ch. nn). Apart from saying something about what is said later, he makes clear that, notwithstanding the title of the book, it is exclusively concerned with Mathematics and its Philosophy. He takes the view that the province of pure reason is mathematics, and that reason in other domains is in some way dependent upon mathematical reasoning.

I am sympathetic to this message, subject only to some doubt about whether mathematics construes itself quite broadly enough. I am wiling to believe that all sound deductive reasoning is reducible to set theory, and would be happy for mathematics to claim the whole as its domain, but feel that many of the applications of logic (for example in information technology) would be rejected as mathematics proper by some mathematicians.

1. The Axiomatic Method

Here Tait affirms that "the only conception of mathematics itself that I believe is the axiomatic conception" which he attributes (with some qualifications) to Hilbert, Plato (Chs. 7 & 8), Cantor (Ch. 11 sec. 6) and Dedekind. He discusses briefly some of the dissent from this position.

2. Existence and the Specter of Formalism

I think the main thrust here is to affirm a notion of (mathematical) existence which is taken from Hilbert and also finds in Cantor (Ch. 11), and to distance the axiomatic method from the more extreme varieties of formalism in which meaning is denied (which is not Hilbert's).

Makes distinction between "abstract" and "concrete" maths (Ch. 4), by which he means, having "lots of models" and "providing the tools for constructing such models" respectively. In neither case are these theories "formal" in the sense of "just defining games with symbols".

Argues against "univocal" notion of existence, possessed by material but not by mathematical

He defends his position against those of Frege, Dummett, Dedekind, Quine, Kant and Gödel.

Often talks of the meaning of existence "as given by the existential quantifier", which I didn't understand, though I don't have a problem with the position he identifies in Hilbert and Cantor.

3. The Theory of Types

This is about constructive type theories, about propositions as types rather than classical type theories.

Begins with historical sketch: Euclid, Brouwer, Heyting, Curry, Howard. Points out that propositions as types works for classical as well as constructive mathematics,

Tait aknowledges here that he has made some mistakes in his material relating to propositions as types for classical mathematics, and is evidently retrenching in his introduction relative to the positions he has take in some of his papers (Chs. 3 & 5). He sounds much more convinced by what he has taken from Hilbert than by propositions as types.

4. Constructive Mathematics
5. Set Theory
This section describes how Chapter 6 (constructing cardinals from below) differs from previous versions of that paper. This consists (in my words) in avoiding the presumption that the extension of the concept "pure well-founded set" is well-defined.
6. Plato
This section considers some of the criticism of Chapters 7 and 8 since their first publication, without making any concessions.
1. Finitism
Tait observes that "the crux" of understanding Hilbert's conception of finitist mathematics is in the answer to the question how general claims about natural numbers can be proven without assuming the infinitude of numbers or some other infinite totality. This chapter attempts to answer that question, and also argues that finitist reasoning is primitive recursive reasoning.
What is Finitism?

One of the main aims of this chapter is to argue that finitistic reasoning is essentially that formalised in the formal system which is known as "PRA" (Primitive Recursive Arithmetic). I was hoping that this would begin with an account of what Tait takes finitism to be, and then proceed to an argument to the effect that under this conception finitistic reasoning corresponds to proof in PRA.

I thought at first that this was indeed what he was doing, and that the little story about not assuming the infinitude of numbers or the existence of any infinite totalities which I reproduced above was the definition of finitism from which Tait would proceed. Perhaps Tait thinks it is, but there no serious attempt to connect the discussion with this simple picture. To understand what he is writing about, you really need already to have a fairly good knowledge of what Hilbert's conception of finitism is, and to know that there is a great deal more to it than this talk about the infinite.

The issue is presented as ontological, about how mathematics can be done with a limited ontology in which there are no infinite totalities. What I really need first is to be told why this represents a problem either for Peano arithmetic as a whole (in which only natural numbers exist), or any extension to PA obtained by adding as axioms true statements of arithmetic.

When Tait proceeds with his presentation of finitist reasoning, he starts immediately with a notion of proof which seemed to me of much later origin, and radically different from that known to Hilbert, i.e. that which accompanies the "propositions as types" idiom. How I asked myself, was this connected with the question of infinity? What is it about avoiding infinite totalities which prevents us from using a plain first order logic for reasoning about numbers (which as it happens had not actually been invented when Hilbert starting talking about finitism).

A "proof" of a general arithmetic proposition is a function which yields for each argument a proof of that instance of the general proposition. But of course, simply presenting such a function (even if that weren't in the present context itself problematic) does not constitue a proof (for if it were, as he observes, a proof of the consistency of ZF would be readily forthcoming). You have of course to "prove" in some other sense, that this function does what it is supposed to do. The "propositions as types" enthusiasts appear at first sight to be offering a radical new notion of proof to replace the one we had, but we find that these proofs are not checkable, they are not decidable. You have to have an old fashioned proof to establish that you have one of the new proofs.

So it immediately seems to me that Tait is immersed in problems which have not been connected with finitism and which seem to be entirely a product of his introduction of constructive proof methods.

So I left the chapter in frustration and looked around at what else was in the book, eventually returning to read the second of his two chapters on finitism. Here I noticed in the next paper a reference to Zach's PhD thesis, which seemed a good place to look for a clearer picture of what Hilbert's finitism was.

I downloaded Zach's thesis, which was very helpful, and quickly got a slightly better understanding of finitism, including that it was a part of Hilbert's conception of finitist proof that the proof of a generalisation would involve exhibiting a function which from any object in the domain of discourse (the numbers) would contruct a proof of the instantiation of the generalisation to that object. OK, so we don't actually have "propositions as types", but we do have a very constructive notion of proof, and perhaps the connection between this and infinity is accepted from Brouwer.

Generally, Tait has launched into his account of finitism in a way which is very bound up with more modern constructivist ideas, and the result is that it is completely unclear to me which of the problems which he is addressing are really inherent in finitism and which are merely problems in the constructive machinery with which he is attempting to give an account of finitism.

3. Truth and Proof: The Platonism of Mathematics
Tait identifies the Truth/Proof problem as the problem of establishing that there is a connection between a mathematical proposition being provable and its being true (of the platonic entities which are its subject matter). His aim in this chapter is to defend Platonism in mathematics against this attack.
1

First Tait described the Truth/Proof problem. I think the essential point is that abstract or Platonic mathematical objects are by hypothesis causally isolated from the material universe, and for this reason we can have no knowledge of them. Whatever evidence we may have for some proposition about them is causally disconnected from them and therefore would remain the same even if the abstract realm were entirely different, or even non-existent.

3
Tait's first response to the Truth/Proof problem is simply to observe that sceptical arguments are eqally available to refute the inference to material entities in the observable world. The analogy holds good because, though sceptical arguments can be brought against this kind of mathematical knowledge, they can equally be brought against empirical science.

For my part I find this unconvincing. In the case of material entities there is no claim of causal independence, quite the opposite. The problem arises from the causal isolation of abstract entities, material entities are not causally independent, that is how we have knowledge of them.
5
In this section Tait denies that a Tarskian model of truth is relevant to his conception of Platonism. He denies that the relationship between mathematics and the objects of mathematics is that of an interpretation.
2

With reference to Dummett and Benacerraf a brief discussion of the taking of talk about sensible objects as an unproblematic paradigm case of the application of the correspondence theory of truth. It is alleged that the correspondence theory of truth which when applied to mathematics leads to Platonism, is thought applicable to mathematics because of an analogy with empirical language and facts which does not hold because of the supposed causal isolation of the mathematical objects.

4
The critics point to the use of senses to obtain evidence on which our knowledge of sensible objects is based and ask what comparable source of information is available on which our knowledge of mathematical objects is based. Tait observes that many philosophers have sought to place mathematical intuition in this role, but he (quite rightly in my view) argues that mathematical intuition only becomes useful later down the track, when a mathematical theory is already established rather than in establishing the axioms.

I thought he was on the point of saying something worthwhile about the basis of our knowledge of mathematical objects, but instead he decides to argue that perception does not play the role suggested in our knowledge of sensible objects.
4. Beyond the Axioms: The Question of Objectivity in Mathematics
An explication and defence of what Tait calls "realism" in contrast with "super-realism" (which is what many others call "realism"). Tait's position is very close to that of Hilbert.
Discussion of Main Points
Some Points of Interest

These are the points of greatest interest to me.

  1. Existence means existence follows from axioms
  2. concurrence of Cantor in (1)
  3. Tait calls above "realism" and uses "super-realism" for additional requirement of really exising
  4. For Hilbert truth (in other matters as well) is being provable from axioms, and Tait concurs
  5. Tait distinguished between concrete and abstract mathematics, all this applies only to concrete.
  6. Need consistency, which is hard to establish.
  7. Incompleteness is larger problem, Tait doesn't have a very good position on this, bearing in mind that it undermines his criterion of truth.
  8. The challenge posed by cultural relativism
Some Counterpositions
Here are some of my contrary views.
  1. Because of formal incompleteness I admit informal semantics, a sentence being true if its truth is entailed by the semantics as a whole
  2. nice
  3. The notion of realism is too well known for it to be worth trying to redefine it. Tait's super-realism is realism and another name is needed for his realism. Objectivism would have been nice if it weren't associated with Ayn Rand, so for me semanticism would be OK. This is because my view is that truth is objective to the extent that it is entailed by the semantics.
  4. I have to admit informal semantics.
  5. For me the distinction is of importance, but truth in both conrete and abstract mathematics is objective in the same way. Where the semantics admits more than one model then a sentence is true iff it is true in all models, which is the consequence of taking entailment from the semantics as the criterion.
  6. sure
  7. Admitting informal semantics solves this problem.
  8. truth is relative to (language + semantics) and only to culture because different cultures have difference languages.
Some More Detailed Notes
1

Tait regards "the axiomatic conception of mathematics" in its modern version, due to Hilbert. He attributes to Hilbert the view that "the demands of mathematical existence and truth are entirely satisfied by a suitable axiomatisation". The idea here is, that however the axioms are arrived at, once the axioms are stipulated questions of truth and existence are then well-defined.

He also cites Cantor as of like mind, quoting him thus:

"First we may regard the whole numbers as real in so far as, on the basis of definitions, they occupy and entirely determinate place in our understanding ..."

though Cantor does not go as far as Hilbert in expecting these definitions to be supplied by an axiomatisation.

Tait seems to think this way only in relation to "concrete mathematics" (e.g. number theory and analysis) rather than abstract mathematics (group theory or geometry). "For, obviously, the questions of objective existence and truth concern only this part.".

2

Discussion of the "superrealist" objection, that even if the axioms are consistent there may not in fact exist a model of them.

3

Discussion of problem posed by incompleteness, in relation to whether an undecidable proposition A have a truth value and if not how this can be reconciled with the provability of "A ∨ ¬ A".

4
Discussion of the challenge posed by incompleteness, e.g. on the objective truth value of CH.
5
Discussion of the challenge posed by post-modern cultural relativity.
9. Wittgenstein and the "Skeptical Paradoxes"
My comments on Tait's paper on Kripke's analysis of some ideas of Wittgenstein. To do with following rules.

1

The paradox as presented in [Kripke82] is described as consisting in two skeptical paradoxes from Wittgenstein's Investigations [Wittgenstein53] and their skeptial "resolution".

2

3

4

5 Ascription of Mental States

6

11. Cantor's 'Grundlagen' and the Paradoxes of Set Theory
A summary of the content of Cantor's 1983 paper "Foundations of a General Theory of Manifolds" and a discussion of how this relates to the paradoxes of set theory.
1. Cantor's Pre-Grundlagen Achievements in Set Theory

I note of this the following points:

  1. That Cantor's notion of set is relative to some well defined domain.
  2. That explicity makes weak the conditions for something being a set, it is necessary only for it to be "determined" whether an object of the domain does or does not belong to the set, it doesn't matter how it is determined.
  3. Only two sizes of infinite number had been established.

Tait here makes much of (1) which for Tait makes this notion of set, which Tait calls the logical notion of set and associates with second order logic, and which does not suffer from the paradoxes of "naive" set theory. I can't say myself that I think this a different notion of set in any way, but the supposition that a set is drawn from some already defined domain does suffice to keep things in order. The problems only arise when we seek, or worse, assume or postulate, a domain which is closed under convenient set forming operations (notably comprehension).

3. The Grundlagen and the Paradoxes of Set Theory

This paper marks the transition from what Tait calls the "purely logical" notion of set (discussed by Tait in section 1) and the one we find in modern set theory, where the domain of discourse is created by the iteration of set formation starting from some (usually empty) collection of urelements.

This transition might have lead Cantor to what has become known as "naive" set theory, in which various contradictions are derivable, but Cantor's procedure is more cautious, and involves no presumption that every collection of sets is itself a set.

Tait says that there is no conclusive evidence that at the time of writing this paper Cantor was aware specifically of the Burali-Forti paradox. He shows that Cantor's collection of numbers, Ω cannot be a set, and abserves that "Cantor clearly knew this simple theorem" without indicating the text on which he bases this claim. He therefore takes it that Cantor knows that a paradox would arise if he admitted Ω as a set. However, this is not, Tait says, the Burali-Forte paradox, which is specifically about the conception of ordinals as order types of well-ordered sets.

Personally I can't see what there is in the Burali-Forte paradox which depends upon any particular (reasonable) conception of the ordinals, and so I doubt the significance of the distinction between the paradox which Cantor would have anticipated from admitting Ω as a set and Burali-Forte.

2. Summary of the Content of the Grundlagen
  1. The paper concerns the theory of transfinite numbers, which he defines as those obtained by iterating two principles of generation, taking successors and taking limits.
  2. The number classes thus obtained now give an infinite sequence of cardinal numbers.
  3. He analyses the notion of "counting number" Anzahl extending it to the transfinite and introducing the notion of a well-ordering. The initial segments of the number sequence are what we think of as ordinals, but this term is later introduced by Cantor for the order type of a well-ordered set.
  4. The well-ordering principle, that every set is well-orderable, is proposed "as a fundamental law of thought"
  5. The distincion between sets and "what later came to be known as proper classes" is introduced. Cantor talks of his transfinite numbers as determinate infinities and of the absolute infinite, represented by the totality of infinite numbers.
  6. Cantor defends the autonomy of "free mathematics", this include the ontological position described above under "objectivity", which effectively detaches mathematics from metaphysics. Mathematics need take account only of what he calls the "immanent reality" of its concepts. It is noted here that Hallett Hallett has different reading of Cantor, taking him to be a Platonist. Purket is also cited as having a different reading, taking consistency to be insufficient.
  7. Cantor defends "classical mathematics", against Kronecker's constructivism.
4. What Numbers/Sets of Numbers are there?

Mainly given over to disagreeing with Lavine who took Cantor to have defined a set as a collection in one-one correspondence with a proper initial segment of the numbers.

Cantor does say that any collection which is smaller or equipollent with a set is itself a set, which is essentially the axiom of replacement, and hence, "suitably understood" Cantor's hierarchy yields all numbers less than the smallest weakly inaccessible number.


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