Wednesday, 26 June 2013

Epistemic Interpretation of Ultra-Finitism

In the previous post on ultra-finitism, I tried out four different interpretations of ultra-finitism (by "numbers" I mean elements of $\mathbb{N}$; or of $\omega$, if you prefer):
  1. The numbers "run out", at some finite point.
  2. Numbers are physical objects.
  3. There are no numbers (i.e., nominalism).
  4. There is a number such that it is not concretely realized.
All of these are ontological views, saying how the numbers 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:
Token Cognizability (TC)
A justificaton for asserting the existence of a number $n$ is a token construction of $n$.
Ordinary finitists (who accept the numbers as a potential infinity and the computational operations on them), and constructivists more liberally, accept the modal notion of a possible construction. For example, a formula $\phi$ is provable if it could 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, "can in practice", and thereby get ultra-finitism.

So that's the idea. In more detail, frequently large numbers are defined by function terms (usually arithmetic), $t$. Examples of function terms might be:
$5 \times (29 + 2)$
But a function term $t$ is a (syntactically) complex expression, and has not "directly informed" you what number it 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:
and so on
(Notations vary. Sometimes people, including me, use a prime notation for successors, e.g., $\underline{0}^{\prime \prime \prime}$, instead.)

These can be defined by a primitive recursion $n \mapsto \underline{n}$ by:
$\underline{0} := \underline{0}$
$\underline{n+1} : = s\underline{n}$
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:
A construction of $n$ is a canonical numeral for $n$
A reduction of $t$ is a proof of an equation $t = \underline{n}$, for some $n$. 
Finally, numerals, constructions, reductions and proofs, thus defined, are 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 a token of a canonical numeral for $n$
A token reduction of $t$ is a token of 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 are terms for which there is no (actual) token reduction.
(To be more exact, this is so unless we allow rather strange entities to count as "tokens", where these tokens are 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:
Token Cognizability (TC): A justificaton for asserting the existence of a number $n$ is a token construction of $n$.
Finiteness (FIN): There are terms for which there is no (actual) token reduction.
A response to this is to request a 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?

1 comment:

  1. There's another finitism via a theorem of Mycielski:

    "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)."

    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."