A question about finitism

[James Moody]

Arnon,

You object that Haim is making claims about "every m", which you suggest
violates the principles of finitism. And more generally, you reason
finitist mathematics is incoherent because it cannot be expressed in only
finitist language.

 Your argument against finitism is similar to the argument against Popper's
claim that falsifiability is essential for a (scientific) claim to be
meaningful. Popper's falsifiability, like finitism / potentialism, imposes
a limit on concepts we are allowed to use in a particular discipline. The
objection there is that the essentialness of "falsifiability" is itself not
a falsifiable claim, so by Popper's own logic it is meaningless.

But it is possible for Popper's supporters to get around this by admitting
that the essentialness of falsifiability is not a meaningful *scientific*
claim, but that it could still be a meaningful meta-scientific claim in the
philosophy of science. Likewise, a supporter of potentialism could admit
that some uses of quantification over "all" potential objects could be
useful in making meta-claims about mathematics as a whole, while denying
that those objects "actually" exist. They could also claim that certain
conditions hold *necessarily* of any mathematical object that might ever
potentially exist, even if only finitely many mathematical objects actually
exist right now.

To make a non-mathematical analogy using a classic example from analytic
philosophy, consider the claim "all "bachelors are unmarried". One
interpretation of this statement might be: of all 700m bachelors alive
right now, each of them happens to not be married. The classical infinitist
mathematical interpretation of this might be something closer to this: the
(infinite) set of all abstract objects that are bachelors is contained in
the (infinite) set of all abstract objects that are unmarried. But there is
a middle ground which denies the existence of an "abstract set of all
bachelors" in some Platonic realm containing all possible bachelors, but
still claims something stronger than the contingent claim that "all 700
people alive today who are bachelors happen to be unmarried". That would be
to claim that the (analytic) concept of "bachelor" contains the (analytic)
concept of "unmarried". Some might theorize that this implies *infinitely
many* distinct claims about the infinitely many bachelors that could ever
exist. But you could also say that this analytic claim is something
entirely different from synthetic (scientific) claims that quantify over
(actual) objects, and that you can accept it even if you believe that it is
*impossible* for there to exist infinite sets (of bachelors, or more
generally).

James


On Sun, Feb 5, 2023, 10:05 AM Arnon Avron <aa at tauex.tau.ac.il> wrote:

> Dear Haim,
>
> I was never able to understand the coherence of `finitism'.
>
> For example, in your first reply to Vaughan you wrote:
>
> "Only structures based on proper initial segments of the natural numbers:
> {0, 1, 2,…, *m*} are accepted as legitimate, but for every *m, *if *n* >
> *m*, one accepts also the extension based on {0, "we v1, …., *m*, *m*
> +1,…., *n*}."
>
> First question: at least to me it seems that if one understands that X is
> an initial segment of the natural numbers, it means that somehow
>    he understands that there is something that X s an initial segment of,
> so he understands that there is the  collection of the natural numbers.
>    So why pretending not to understand that collection?
>
> Second question: you explicitly wrote that "*for every m*​, if n>m ...".
> But if I understood you correctly (almost certainly I did not) a finitist
> is not allowed
>    to make claims about *every m*​!
>
> And the final question: is there any way for a finitist to explain his
> principles (even to himself!)
>    without violating these principles? I doubt it...
>
> Best regards,
>
> Arnon
>
>
>
>
>
>
>
> ------------------------------
> *From:* FOM <fom-bounces at cs.nyu.edu> on behalf of Haim Gaifman <
> hg17 at columbia.edu>
> *Sent:* Sunday, January 29, 2023 5:03 AM
> *To:* Vaughan Pratt <pratt at cs.stanford.edu>
> *Cc:* fom at cs.nyu.edu <fom at cs.nyu.edu>
> *Subject:* Re: ​Re: Mathematics with the potential infinite
>
> Dear Vaughan,
> Long time no hear no see, and it is very nice to hear from you.
> The restriction of subscribing only to potential infinities (which can be
> traced back to Aristoteles) is Hilbert’s so called *finitist* position;
> Abraham Robinson agrees with him. Only structures based on proper initial
> segments of the natural numbers: {0, 1, 2,…, *m*} are accepted as
> legitimate, but for every *m, *if *n* > *m*, one accepts also the
> extension based on {0, 1, …., *m*, *m*+1,…., *n*}. The functions and/or
> relations that come with these structures are the usual functions and/or
> relations of PA (Peano Arithmetic). Of course, the functions are partial
> functions  , given the restrictions on the domain.
>
> PA, which is based on the standard model *N *of natural numbers, is much
> much… stronger than the theories
> that arise within the framework of potential infinity.
> One such interesting theory has been proposed by Skolem
> and is known as PRA for Primitive Recursive Arithmetic.
>
> Now your question, if I understand you correctly, asks for a way of
> describing an uncountable structure using only potential infinities.This
> would be impossible, unless you allow countable non-standard model for the
> theory linearly ordered groups.
>
> Best, Haim Gaifman
>
> On Jan 28, 2023, at 3:13 AM, Vaughan Pratt <pratt at cs.stanford.edu> wrote:
>
> My apologies for not having previously followed threads on this topic.
> However after seeing Stephen Simpson's message just now (Friday) it
> occurred to me to ask whether an uncountable set could be described using
> only potential infinities, for example the real numbers (R, *, 0, <=) as a
> linearly ordered group under addition, compatibly ordered in the sense that
> each of the group multiplication's two arguments is monotone: if x <= y
> then x*z <= y*z, and likewise for the right argument.  (* = +.)
>
> Define a *geodesic* to be a nondegenerate linearly ordered group (G, *, 0,
> <=).  (Although G is not assumed abelian, the linear order makes it
> abelian.)  Examples include the integers, the dyadic rationals, every field
> between the rationals and the reals, and many non-Archimedean extensions
> thereof.
>
> Call a geodesic G *gapless* when (i) it is dense, and (ii) for every
> nonempty suborder (U, <=) of (G, <=) having no least element, and every
> nonempty suborder (L, <=) of (G, <=) with L < U and having no greatest
> element, such that there is at most one element of G between L and U; then
> there exists an element of G between L and U.
>
> I claim that every gapless geodesic is isomorphic to R with the above
> structure.
>
> (Proof outline: Take any element x of G with 0 < x and pair 0 and x with 0
> and 1 in R.  Pair the integers in R with the subgroup of G generated by x,
> cyclic and therefore abelian.  Repeatedly divide the intervals in (n, n+1)
> in G into two equal parts and pair the results with the dyadic rationals in
> (0,1), a dense set.  Pair each dyadic irrational q in R with the unique x
> given by the gaplessness condition for any L and U in G whose counterpart
> in R converges to q from each side.  Lastly, G must be Archimedean or there
> would be an empty gap between the finite and infinite elements of G.)
>
> 1.  Do these definitions, claims, and constructions meet the criteria for
> only potential infinities?
>
> 2.  Can R be shown to be uncountable using only potential infinities?
>
> (Those familiar with Otto Hoelder's 1901 paper showing that every
> Archimedean linearly ordered group is isomorphic to some subgroup of R
> under addition, which may be anywhere between Z and R, may see some
> similarity of ideas in the above.)
>
> Vaughan Pratt
>
>
>
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