Monday, 24 June 2013

Concrete Realization and Ultra-Finitism

Suppose $w$ is a possible world at which there are exactly $15$ concreta instantiating some causal connectedness relation. Let $I = \{1, \dots, 15\}$, Let $\mathsf{Conc}$ be the property of being concrete and let $\mathsf{Causconn}$ be the relation of being causally connected.

Then, the categorical diagram description of the world $w$ looks like this:
$\exists x_1, \dots, x_{15}$
$[\bigwedge \{x_i \neq x_j \mid i \neq j \in I\} \wedge $
$\forall z(\mathsf{Conc}(z) \to \bigvee \{z = x_i \mid i \in I\}) \wedge $
$\bigwedge \{\mathsf{Conc}(x_i) \mid i \in I\} \wedge$
$\neg \mathsf{Causconn}(x_1,x_1) \wedge \mathsf{Causconn}(x_1,x_2) \wedge
(where the final clause says how the causal connectedness relation $\mathsf{Causconn}$ is instantiated in $w$. The details don't matter.)

Let $\kappa$ be a cardinal. Define:
$\mathsf{Real}_C(\kappa)$ := $\exists X(\forall x(x \in X \to \mathsf{Conc}(x)) \wedge \kappa = |X|)$
This means:
"$\kappa$ is concretely realized by some set of concreta".
$w \models \mathsf{Real}_C(\kappa)$ if and only if $0 \leq \kappa \leq 15$.
That is, every number up to $15$ is concretely realized (in $w$).

The reason for setting things up like this (modally) is to try and define Ultra-Finitism as charitably as possible, so that it doesn't sound bonkers.

One interpretation of ultra-finitism is that, somehow, the natural numbers "run out", at some point. I can't make much sense of it. If successor, $s : \mathbb{N} \to \mathbb{N}$, is a total injective function with $0$ outside its range, there must be infinitely many successors of $0$. Perhaps the idea is that successor is "really" non-total? One can easily define such models. (E.g,  "arithmetic with a top", as discussed in work on Bounded Arithmetic; e.g., Hájek & Pudlák, 1993, Metamathematics of First-Order Arithmetic, Ch IV, Sc 2; and in this 2002 paper by Neil Thapen) But really, that is not what we mean by the natural numbers.

A second interpretation is that ultra-finitism is the view that numbers are physical objects. And some ultra-finitists talk as if they do hold these strange beliefs. But surely everyone knows that numbers aren't things that move around, have mass, etc.! Numeral tokens yes. But numbers?

A third interpretation of ultra-finitism is simply that it is the view that there are no numbers (i.e., nominalism), but that there are numeral tokens (which are physical things). But this is not really a finitist position (for mathematics) at all, except in a trivial sense (setting the number of numbers as 0). It's nominalism and nominalism has a very large research literature devoted to it. If ultra-finitists are nominalists, then the solution to their troubles is easy: go to a library and study the literature!

A fourth, and what seems to me probably the only conceptually stable view, is that ultra-finitism is the view that only strictly finitely many numbers are realized concretely. This is by no means crazy. If that is the correct view, then ultra-finitism (at size $n \in \omega$)---denoted $UF_n$---can now be defined as follows:
$UF_n := $ for all $w$, $w \not\models \mathsf{Real}_C(n)$
which means: no world $w$ concretely realizes the number $n$.


  1. Hi Jeff,

    It seems that you'd have to restrict the universal quantifier in UF_n to some kind of "physically" possible worlds or at least specify further, what is meant by "possible world" here. Given what we know in physics today, is it inconsistent to assume an infinite number of physical objects? If yes, this makes perfect sense then.

    When we talk about logically (or mathematically) possible worlds, UF_n seems to be false though (e.g. by talking N as domain). But it might as well be, that I'm just another brainwashed Platonist.

  2. AJ JA,

    If we take the physics of the standard model, say, physical fields are functions (scalars, tensor fields, n-forms, etc.) on the manifold M of spacetime events. The events must be taken to be concreta (otherwise the laws of physics would be necessities, not contingencies); but there are infinitely many of these events, since a manifold is "locally like" R^4. Also, there seems to be no upper bounds on the cardinalities of concreta in logically or metaphysically possible worlds. It makes sense to consider a world w at which there are k many concreta, with k any infinite cardinal.

    Having said that, there are arguments that physics of the actual world should restrict itself to finiteness in various ways; and one can try and discretize and finitize spacetime in various ways. (E.g., "Casual Set Theory".)



  3. This may be a little off the mark ... but it's possible that after a certain point, the numbers wrap around. That is, the natural numbers are really the integers mod p for some huge prime p. Why a prime? Because the integers mod a prime are a *field* -- you can add, subtract, multiply, and divide.

    In Penrose's The Road to Reality he even mentions that at least one physicist does in fact hold this view: that the natural numbers in the physical universe are actually the integers mod p. After some huge number, they wrap around to zero.

    Not saying this is true ... or false ... just possible. Certainly it's a position an ultra-finitist would take comfort from.

  4. Thanks, Anon

    Yes, right, Penrose mentions the idea that perhaps physical spacetime is based on the finite field $\mathbb{Z}_p$ for some prime $p$. Still, I don't think that Penrose means to suggest that we're confused about what $\mathbb{N}$ is, and that $\mathbb{N} = \mathbb{Z}_p$ after all. Rather he means that perhaps physical spacetime is like that, for all we know (an empirical question), not that the usual well-known numbers are like that (a purely mathematical one).

    On the other hand, the mathematician and ultra-finitist Doron Zeilberger does suggest something very like this: that the numbers, or the real line, are, in fact, $\mathbb{Z}_p$. E.g., he explains here,