## Thursday, 6 October 2011

### Ultra-finitism, types and tokens

Ultra-finitism is not discussed much in standard philosophy of mathematics literature. In the philosophy of mathematics literature, what seems to amount to the same view is called "strict finitism". Here are four pieces on the topic:
-- a 1975 Synthese article, "Wang's Paradox", by Michael Dummett;
-- a 1982 Synthese article, "Strict Finitism", by Crispin Wright;
-- a 2005 PhD dissertation, "Strict Finitism as a Foundation for Mathematics" (University of Glasgow) by Jim Mawby; ;
-- and a 2007 Aristotelian Society article, "Strict Finitism Refuted?", by Ofra Magidor.
Discussions of strict finitism have often attempted to refute the view by invoking Wang's paradox, noting that "being feasible" is a vague notion susceptible to a sorites paradox.

My own view is that strict finitism tends to be ignored for a much simpler reason: it seems confused - it conflates types and tokens (see premise (U) below). Tokens are, for example, physical inscriptions of linguistic expressions. A token $t$ of the canonical numeral "$SS......S0$" is a physical inscription of it. But numerals themselves are not tokens. Linguistic expressions are finite sequences of symbol types. So, the numeral $\underline{n}$ is a finite sequence of length $n+1$. This is an abstract object, not a physical one.

Philosophers sceptical of abstract entities reject mathematical reality in its entirety; so the appropriate position for someone with such views is nominalism (or, as it has come to be know more recently, fictionalism): i.e., the view that there are no numbers, or sets, or functions, etc. (W.V. Quine & Nelson Goodman 1947; Hartry Field 1980). Of course, all will agree that there are tokens. But nominalists do not think numbers are tokens. There is a very large literature on nominalism/fictionalism.

So, if I understand ultra-finitism correctly, the argument for ultra-finitism is something like this:
(P) There is an upper limit on how many tokens there are, or even feasibly could be.
(U) All numbers are tokens
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(C) One must be dubious about large numbers.
The crucial assumption in ultra-finitism is the reductionist assumption (U): a reduction of numbers to (physical) tokens. Considerations about the non-feasibility of exponentiation justify physical premise (P), not reductionist premise (U); and everyone agrees with (P), unless they "go modal" (as Quine once put it, "the cure is worse than the disease").
Three questions:
1. Is this reconstruction right? If not, is there a better one?
2. If it is right, what is the justification for the ultra-finitist assumption (U)?
3. Perhaps ultra-finitists are nominalists/fictionalists, but unfamiliar with the philosophical literature on the topic?

1. In "The Number of Numbers" (http://tinyurl.com/3wrkfsq) Greg Landini discusses finitism in the context of the an interpretation of Russellian logicism that rejects the thesis that numbers are objects. Accordingly, the token/type point that you raise here may not affect the view Landini describes.

2. Nelson in Elements (p. 11) gives the label "finitary credo" to (U), which is spelled out as "for every number x there exists a numeral d such that x is d." However, he explicitly *rejects* the credo and, indeed, uses this rejection as the basis for his rejection of induction. (His argument is a bit tenuous because he seems to assume that the only possible justification for induction is the finitary credo. Therefore, by rejecting the credo, he has provided a satisfactory justification for a rejection of induction.)

Maybe there are ultra-finitists who reason as you have said. My guess is there are as many reasons as there are ultra-finitists, so a discussion about what ultra-finitists think, as if there is some uniformity, is likely not to go very far.

For instance I hold a certain dubiousness towards large numbers, but I certainly don't reason as you propose. On the contrary, I simply see a lack of good reasons to accept their existence. In short I think the burden of proof is clearly on the other side. If you think there are very large numbers, then it is you, by positing the existence of something, who must come up with a good reason why we should believe in them. (What is your argument?)

3. Great - thanks for the pointer.
(In the post, I'm musing about how to figure out what mathematicians who are ultra-finitists (Nelson, Zeilberger and some others) mean when they sketch their philosophical views. Actually, I think (3) is closest to what they mean: the view is really nominalism. But I should probably write out some detailed quotes though.)

I quickly read the Landini paper, but it's very dense in Russell-Frege notation! It describes a Russellian type-theoretic logicism, with an axiom of infinity, and then one can see what happens without that axiom (or with modifications). We think of our finitely many tokens as the individuals, and numbers are second-level properties rather than objects. But then we lose some arithmetic.

4. a, thanks for the comment; your hint that I need to get some exact arguments from ultrafinitists written down is right!
Ultra-finitism just isn't there in the usual Phil Maths literature (though there is lots on nominalism), so I'd want to get a bit clearer what the view is. But I'm fairly sure that the epistemologically bottom level that they're happy with consists in tokens and token computations, rather than numbers; so, I think (3) is roughly the right interpretation - they're nominalists, rather than strict finitists.

I take the point entirely about the burden of proof. (Though there are Russellian arguments that over-egged scepticism is as unreasonable as over-egged confidence.)
People will have difference grounds for accepting arithmetic (and more) as true; e.g., direct intuition, logicism, structuralism, modal constructiblity. For me, however, the reasons are Quinian: we postulate things for scientific reasons to account for our flux of experience. So, epistemologically, I see no difference of kind between my hands, the cells they are composed of, neutrinos and $\mathbb{R}^4$. (And, the homeric gods and characters from Dickens, but they do me no use in science.) I do have some sympathy with Gödel's view that the human mind can cognize abstract mathematical reality, but I'll stick with the Quinian epistemology for the moment!

Here's another argument, this time directed against some sort of ultra-finitist position. I written down a (token of a) formula $B$ with 61 symbols. There is a mathematical proof that it is inconsistent, but also a proofone that the shortest derivation of its inconsistency will never be tokened. The mathematical proof isn't ultra-finitistically acceptable. Is it inconsistent? If there is no token witness that $B$ is inconsistent, is a merely existential guarantee, based on mathematical considerations, good enough?

Yes, I noticed what Nelson calls the finitary credo (which he goes on to argue against): I think he meant, "for any x, there is a numeral d such that Val(d) = x". This is a theorem of PA, of course, but it depends on a bit of induction (my guess is $I \Delta_0$).

The usual justification of induction is: $\mathbb{N}$ is the closure of $\{0\}$ under successor. I guess that Nelson counts that as circular.

But there's an epistemological stand-off here.
He rejects hypothetico-deductivism, and wants to build up from very meagre beginnings. I accept hypothetico-deductivism. I open a physics textbook and I accept the mathematical theory given as true; and I accept the mixed mathematical laws as approximately true.
I say, "How do you justify the predictions of quantum electrodynamics?". The reply is, "Our best theories are only useful instruments, and their utility doesn't imply their truth: also, like all past theories, they will be overthrown". And on it goes ...

In the end, I think this becomes a debate about scientific instrumentalism vs. scientific realism (and connected to the theory of how mathematics is applied).

5. "For me, however, the reasons are Quinian: we postulate things for scientific reasons to account for our flux of experience. "

I think you want to be careful here, because if you mean that *all* of arithmetic is postulated, then (x)(y)(z)((x+y)=z=>(y+x)=z) may turn out to be false if our experiences change. This does not seem right to me. Rather, there is a core of arithmetic which is analytically true and so does not depend on Quinean justifications. Otherwise, I have a certain sympathy with your Quinean view that there is a non-core part which is postulated and is accepted because it has worked. If you accept this view as correct, though, then there are some natural questions: which part of mathematics is analytic and which postulates can only be justified because they work? Also, you basically committed yourself to the view that if someone can come up with simpler axiomatizations that also work, then mathematics will need to change.

6. (As I just yesterday remembered, Shaughan Lavine's book on set theory has some sort of strictly finite set theory, based on Mycielski.)

a, "Rather, there is a core of arithmetic which is analytically true and so does not depend on Quinean justifications."

This is something I can drop on my epistemological picture, and replace "core" with a matter of epistemic degree, rather than a sharp semantic boundary. Some things will be more core than others, of course, yes.

There are two classic examples which some (e.g., St Andrews neo-logicism) have been argued to be analytic:

Hume's Principle: $\#(X) = \#(Y) \leftrightarrow X \sim Y$

Extensionality: $\forall x \forall y(\forall z(z \in x \leftrightarrow z \in y) \rightarrow x = y)$

Once pointed out, and given their role and use, which has been extensively explored, it seems reasonable to say that these are inside the Quinian core - but the core is epistemic, not semantic.

"Also, you basically committed yourself to the view that if someone can come up with simpler axiomatizations that also work, then mathematics will need to change."

Indeed. But such arguments are about "what cannot be logically ruled out with certainty". *That* standard applies only to the philosophy seminar room! Rather, rational thinking works with degrees of uncertainty, presumably constrained by a probabilistic credence function of some sort. So, we rationally assign negligible probability to the prediction that mathematicians will (for whatever reasons - internal consistency, coherence, sensory experience, physics, etc.) eventually reject $PA$.

Perhaps a strictly finite set theory might in fact be the optimal approach for someone interested in strict finitism.

7. So I think an ultrafinitist position can be made compatible with yours. The ultrafinitist would, I think, be willing to predict that the non-core can be replaceable with core when scientific justifications are needed. After all, the world as we experience it is finite.

8. "So I think an ultrafinitist position can be made compatible with yours."

If we are to analyse a philosophical view, we must write it down as precisely as possible, isolate its basic assumptions and subject it to criticism.
UF consists in the certain philosophical claims:

(U) Each number has a physical token.
(Fin-Tok) There is a largest, "finite", physical token.

The crucial claim is (U), which is false, as it contradicts science: for science postulates a countable supply of numbers, a transfinite supply of sets, functions, sequences, along with proofs, derivations, languages, propositions, models, vector spaces, manifolds, Lie groups, fibre bundles, etc, etc.

"After all, the world as we experience it is finite."

I take science as the epistemically best account of what the world is like: science states (indeed, presupposes) that there are numbers, functions, sets, propositions, vector spaces, fibre bundles, manifolds, etc.. etc.
This is not finite.

The role of experience is epistemic, not ontological. So, that experience is finite has no logical relevance to ontology, unless one insists on an idealist reduction of the world to subjective experiences:

(Idealism) Each thing is subjectively experienced.
(cf., Berkeley's "esse est percipi")

But such philosophical opinions have little rational basis and contradict science, for science itself tells me that there are physical objects which do not affect my sense organs.

All in all, we simply have a philosophical opinion which contradicts science and has no greater rational basis than the belief that the world was created 14 seconds ago.

9. "UF consists in the certain philosophical claims:

(U) Each number has a physical token."

Well, and Nelson explicitly rejects (U), as I pointed out. So which ultrafinitists are you talking about?

"for science postulates a countable supply of numbers, a transfinite supply of sets, functions, sequences, along with proofs, derivations, languages, propositions, models, vector spaces, manifolds, Lie groups, fibre bundles, etc, etc."

Perhaps. (Science postulates a transfinite number of sets? Please...) But as you said you have a Quinean perspective, you are willing to jettison all of that should mathematics suitable for science be developed which doesn't need all of that baggage. And as I said the ultrafinitist is willing to make the prediction that this is possible.

"So, that experience is finite has no logical relevance to ontology."

Sure it does. Think about it.

"...unless one insists on an idealist reduction of the world to subjective experiences."

That, IMHO, is a strawman.

Anyway, it's pretty clear that this is the end of the road. Bye.

10. a, "Well, and Nelson explicitly rejects (U), as I pointed out"

But I think Nelson means "physical token" when he says "numeral". If so, he thinks, "each number has an actual physical token".

"Science postulates a transfinite number of sets? Please..."

Would you agree that if $\phi : M \rightarrow \mathbb{R}$ is a scalar field on a spacetime model $(M, g_{ab}, T_{ab})$, then the cardinality of $\phi$ is transfinite?

"Anyway, it's pretty clear that this is the end of the road. Bye."

Ah, that's just your strict finitism talking. There is no end of the road, so do return!

11. I think three-realm finitism is a version of the idea
first realm is where most useful finite numbers are
second realm is where gaps of describability start to appear
third realm the only aspect of the numbers is their potential describability