FOM Digest, Vol 242, Issue 10
Vaughan Pratt
pratt at cs.stanford.edu
Mon Feb 13 02:40:33 EST 2023
Martin Dowd wrote, "A line segment in 3-dimensional Euclidean space
has uncountably many points."
Since no one else has challenged Martin's statement, let me do so here.
Just as Russell had no need of the axiom "God exists", Euclid had no
need of the axiom (or theorem) that his eponymous space has
uncountably many points.
When your only tools are a straightedge and compass, you don't need
anything more than the constructible points, aka real algebraic
geometry with only linear and quadratic polynomials. And if you
insist on Euclidean space being topologically complete, then
furthermore you need the limits of sequences of contructible points
when those limits exist. The example I gave of 2/pi as the limit of
the the midpoint of the base of the prototypical pie slice in
(essentially) Archimedes' method of approximating pi shows that this
completion of the constructible points introduces new points including
transcendentals such as pi. Yet even that completion is only
countable, assuming the sequences themselves are constructible (true
for Archimedes' method) and not merely the individual elements.
When your tools include a smartphone or other computer, you need at
most the computable points, and its completion via computable
sequences (more constrained than general Cauchy sequences) is likewise
countable.
Vaughan Pratt
Vaughan Pratt
On Mon, Feb 6, 2023 at 6:17 AM <fom-request at cs.nyu.edu> wrote:
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> Today's Topics:
>
> 1. Actual and Potential infinite (JOSEPH SHIPMAN)
> 2. Re: ?Mathematics with the potential infinite - some
> inexhaustible? (martdowd at aol.com)
> 3. Re: A question about finitism (James Moody)
>
>
> ----------------------------------------------------------------------
>
> Message: 1
> Date: Sun, 5 Feb 2023 10:32:39 -0500
> From: JOSEPH SHIPMAN <joeshipman at aol.com>
> To: Foundations of Mathematics <fom at cs.nyu.edu>
> Subject: Actual and Potential infinite
> Message-ID: <B1E51F28-5182-4EBD-8F05-84BEBF3FBEC3 at aol.com>
> Content-Type: text/plain; charset=utf-8
>
> ?Most of the discussion that is going on about ?actual? and ?potential? infinite does not seem to involve any theorems, so let me ask a couple of questions that focus on mathematical practice.
>
> 1) Is there any proposition statable in the language of first order arithmetic, which is a theorem of ZF but not of PA, which can be said not to require ?actual infinity? to prove?
> 2) Can you name two theorems T1 and T2 statable in the language of analysis (second order arithmetic), whose *statements* are of the same logical type, such that T1 requires ?potential infinity? but not ?actual infinity?, but T2 requires ?actual infinity?, for some reasonable meaning of ?requires??
>
> I am struggling to understand how this discussion, when reduced to questions of what can be proven, is about anything other than the cut point in the hierarchy of logical strength that appears between PA and PA+Con(PA).
>
> ? JS
>
> Sent from my iPhone
>
>
> ------------------------------
>
> Message: 2
> Date: Sun, 5 Feb 2023 16:37:50 +0000 (UTC)
> From: martdowd at aol.com
> To: "fom at cs.nyu.edu" <fom at cs.nyu.edu>
> Subject: Re: ?Mathematics with the potential infinite - some
> inexhaustible?
> Message-ID: <805410823.485428.1675615070087 at mail.yahoo.com>
> Content-Type: text/plain; charset="utf-8"
>
> Dennis Hamilton writes:
>
> Now it is certainly a fair point that obtaining zero as the limit as n goes to infinity of 2^-n is the limit of an infinite sequence.? What is bothering me is the idea that because the sequence is infinite, it is therefore not something we can experience.
> This is true of mathematical objects in general.? An integer n can be "experienced" in everyday life in various ways.? But what about Z_n, the ring of integers mod n?? Also, infinity can be experienced in some ways.? A line segment in 3-dimensional Euclidean space has uncountably many points.? Everyday life example of countably infinite sets seem more involved.
>
> Martin Dowd
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> Message: 3
> Date: Sun, 5 Feb 2023 12:30:46 -0500
> From: James Moody <jmsmdy at gmail.com>
> To: Arnon Avron <aa at tauex.tau.ac.il>
> Cc: fom at cs.nyu.edu, Vaughan Pratt <pratt at cs.stanford.edu>, Haim
> Gaifman <hg17 at columbia.edu>
> Subject: Re: A question about finitism
> Message-ID:
> <CADrhAnjwJjJo7jEEDUPDkQHo-bFOa0u5ozHPOaPfr67qesdfmQ at mail.gmail.com>
> Content-Type: text/plain; charset="utf-8"
>
> 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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