- The numbers "run out", at some
*finite*point. - Numbers
*are*physical objects. - There are
*no*numbers (i.e., nominalism). - There is a number such that it is
*not**concretely realized*.

*are*, or how they're related to other things, etc. But I think the first simply rests on changing the

*meaning*of "number" from "element of $\mathbb{N}$" to "element of finite $A \subseteq \mathbb{N}$", and it is easy to define modifications of the axioms of arithmetic (allowing the successor, addition and multiplication functions to be "partial") and models of such systems with domain $A \subseteq \mathbb{N}$, with $A$ finite; the second seems to me to be a confusion (you can't purchase the number 57 at Tesco's or put $(\omega, <)$ on an optical bench); the third is fine, and for all I know, true -- but it is simply

*nominalism*(and has a huge research literature devoted to it).

The last is the most plausible ontological view, and merely says that beyond a certain level, no larger numbers are

*concretely realized*. But note that this view is perfectly compatible with the existence of $\mathbb{N}$, and in fact with the existence of much, much more. It may well be true, a physical fact, that few (out of the transfinitely many) abstracta are

*concretely realized*, because of the finiteness of the physical world. But this is not a particularly sceptical view. In fact, it is rather like

*Plato's view*.

Still, these doctrines (1)-(4) make no mention of epistemic matters: proof, evidence, justification, etc. One can think of ultra-finitism as an

*epistemological*view, rather similar to positivism's view of how the world is known (i.e., by direct contact with sense experience). We can introduce an epistemic element to this by the following epistemological doctrine, which is a form of constructivism:

Ordinary finitists (who accept the numbers as a potential infinity and the computational operations on them), and constructivists more liberally, accept theToken Cognizability (TC)

Ajustificatonfor asserting the existence of a number $n$ is aof $n$.tokenconstruction

__notion of a__

*modal*__construction. For example, a formula $\phi$ is provable if it__

*possible**be proven (even if, for practical reasons, it cannot in practice). So, one can change the modality used to define finitism to a much much stronger one, meaning roughly, "*

__could____can in practice__", and thereby get ultra-finitism.

So that's the idea. In more detail, frequently large numbers are defined by

*function term*s (usually arithmetic), $t$. Examples of function terms might be:

$sssss\underline{0}$But a function term $t$ is a (syntactically) complex expression, and has not "directly informed" you what number it

$5 \times (29 + 2)$

$2^{1000}$

$2^{2^{2^{2^{2}}}}$

*denotes*, what its

*value*, $val(t)$ is. In formalized systems of arithmetic, such as $Q$, $I \Sigma_1$, $PA$, etc., there is a

*canonical*means of referring to numbers. These are the canonical numerals:

$\underline{0}$(Notations vary. Sometimes people, including me, use a prime notation for successors, e.g., $\underline{0}^{\prime \prime \prime}$, instead.)

$s\underline{0}$

$ss\underline{0}$

$sss\underline{0}$

and so on

These can be defined by a primitive recursion $n \mapsto \underline{n}$ by:

$\underline{0} := \underline{0}$If $F$ is a system of formalized arithmetic, then usually, for any function term $t$, the system proves an equation $t = \underline{n}$. where $n = val(t)$. So, one may then say:

$\underline{n+1} : = s\underline{n}$

A construction of $n$ is a canonical numeral for $n$Finally, numerals, constructions, reductions and proofs, thus defined, are

A reduction of $t$ is a proof of an equation $t = \underline{n}$, for some $n$.

*abstract objects*. A numeral is a

*finite sequence*. A construction is a

*finite sequence*. And a finite sequence on $A$ is a function $\sigma: I \to A$, where $I < \omega$ is a finite initial segment of the ordinals ($I$ is the index set), we can then write:

$\sigma = (a_0, a_1, \dots) = (a_i \mid i \in I)$.Now the canonical numeral for the number $n$ has size, or length, roughly equal to $n$ itself. And a reduction for a term $t$ will have size at least as large as its value $val(t)$.

But despite being abstract entities, these numerals, constructions and reduction sometimes have physical, or concrete,

*tokens*. The tokens are entities produced by

*cognition*, and

*intended*to be tokens of the relevant numerals or proofs. So, we can define:

A token construction of $n$ is atokenof a canonical numeral for $n$

A token reduction of $t$ is atokenof a proof of an equation $t = \underline{n}$, for some $n$.

Then, we can note that the actual world satisfies the following condition:

(FIN) There(To be more exact, this is so unless we allow rather strange entities to count as "tokens", where these tokens areareterms for which there is no (actual)tokenreduction.

*unintended*. E.g., random patterns in the sand, or peculiar tiny regions of spacetime, shaped like very long sequences of "S"s, that no one has ever seen.)

It then follows from this, along with

*Token Cognizablity*, that:

There are terms for which there is no justification for asserting the existence of their value.This

*explains*the view of many ultra-finitists that we should be

*sceptical*of very large (finite) numbers. So, ultra-finitism is now an epistemological view, based primarily on the two assumptions:

A response to this is to request aToken Cognizability (TC): Ajustificatonfor asserting the existence of a number $n$ is aof $n$.tokenconstruction

Finiteness(FIN): Thereareterms for which there is no (actual)tokenreduction.

*justification*for the assumptions required here for this argument to go through: in particular, the epistemological doctrine of

*Token Cognizability*. Why should it be true that a reason for asserting the existence of a number must involve a

*token construction*of it?

Why can one not have

*indirect means*for asserting the existence of the numbers? We have

__indirect__means for asserting the existence of electrons and quasars and long-dead dinosaurs. Why can we not have

__indirect__means for asserting the existence of (all) the numbers, and the completed set $\mathbb{N}$ of them all?

There's another finitism via a theorem of Mycielski:

ReplyDelete"If φ is a sentence in the language of T and φ' is a regular relativization of φ, then φ is a theorem of T if and only if φ' is a theorem of Fin(T)."

books.google.com/books?id=GvGqRYifGpMC&pg=PA273

There is no new math, just a finitistic math isomorphic (the proofs are one-to-one) to "standard" math.

The philosophical name is Intentionalism:

'Platonism is unsatisfactory because it violates our instinctive drive to obey Ockham's principle of parsimony.

Intentionalism says that pure mathematics is a description of finite structures consisting of finitely many imagined objects.

The term intentionalism is chosen for its contrast with extensionalism which accepts actually infinite sets and leads naturally to Platonism."

poesophicalbits.blogspot.com/2013/04/intentionalism.html