Re: ​Mathematics with the potential infinite - some inexhaustible?

[James Moody]

Response Part I
---------------------------

I don't want to get too far into philosophy, but I would like to connect
this to Descartes and Chomsky.

Descartes argued from his capacity to understand "the infinite" to the
existence of God. The basic idea which we can extract from his argument is
this:

1)  Our minds are obviously finite, and finiteness can only beget more
finiteness, so the notion of "the infinite" cannot possibly originate from
our own minds.

2) Thus it must originate externally, from some source that must be (truly)
infinite.

3) ?????

4) God (a benevolent, omnipotent being who cares about me and doesn't want
to deceive me) thus exists and ensures that I am capable of finding the
truth.

5) I can directly trust "clear and distinct" impressions of mathematical
truths, such as those in planar geometry, and because God exists I can also
trust my senses to lead me to the truth about the external world.

Descartes imagines hypotheticals where a demon convinces you of false
things like 5+7=11, or the existence of a planar triangle whose internal
angles add up to 270°, perhaps by short-circuiting your brain when counting
or similar. But certain truths could not be totally synthesized by a finite
but powerful malfeasant demon, and one such truth would be the
understanding of the infinite (e.g., that there are infinitely many natural
numbers).

Descartes' argument here sounds at least vaguely similar to naive
infinitist arguments that might say things like, "I can picture in my head
the natural numbers N, extending to infinity, so that proves there exists a
model of PA!". It also seems related to Kronecker's statement: "God made
the integers, all else is the work of man".

We might have reason to doubt the soundness of Descartes' reasoning here,
of course. For example, not to make an ad hominem, but Descartes also (in
the same line of work) concluded quite confidently that the soul attached
to the body via the pineal gland, and that all other animals (to whom we
are in fact genetically related) are nothing more than automatons, neither
feeling nor thinking, like the clockwork mechanical toys popular in his
day, unlike humans who, via a soul, had the capacity to actually think and
make intentional decisions. Darwin's theory should have been the final nail
in the coffin for this worldview, but even the recent Cambridge Declaration
on Consciousness has not yet dispelled in the general public some of these
unsubstantiated notions of a hard category distinction between human and
non-human animal cognition.

Chomsky breaks from Descartes in a few ways, but also rescues part of his
worldview in the process. According to Chomsky, all humans and only humans
(except those with significant brain damage / abnormalities) are capable of
understanding the infinite, because they have a "language organ" which
allows then to understand "generative grammar". This is a hard line
distinguishing human cognition from any other kind of animal cognition,
because the line between "finite" and "infinite" is a hard line, according
to Chomsky. The difference with Descartes is that rather than lying in the
existence of a soul, this hard difference lies in the existence of a
biological structure in our brains that supposedly does not exist at all in
other animals (at least not in a functional form).

But it is interesting here: Chomsky is actually claiming that a *finite*
brain structure is capable of understanding the infinite, because
understanding the infinite is nothing more than being able to understand a
language with a generative grammar which can generate infinitely many
distinct sentences. If you can understand a finite set of rules describing
the axiom schema of PA, then I guess you understand "the infinite", whether
or not you think those axioms are "True with a capital T". That's very
different from Descartes' idea of us starting with an idea of the infinite,
which would claim that our rules we write down governing the infinite are
secondary to a more fundamental idea that *actually exists* external to us
and that did not originate within us.

If we follow Chomsky, then "understanding the infinite" is something that
only requires a finite generative structure. A finitist could thus accept
that we can "understand the infinite" while denying the existence of any
actual infinities, because "understanding the infinite" does not entail the
*existence* of the infinite. We could even argue a finitist / potentialist
could accept the consistency of PA, or even of ZFC, if those consistency
statements are expressed as along the lines of "it is impossible to find a
contradiction from these axioms following these inference rules" rather
than "there exists in the Platonic realm a completed infinite object, so
that if a supernatural being  completes the supertask of checking each
axiom on that infinite object, they will find that all axioms do in indeed
hold". To put it another way: the finitist is perfectly happy to say it is
*impossible* for the dog running in circles to catch their tail, and this
does not require them to believe that infinite ordinals actually exist.

Response Part II
---------------------------

Going back to the original points by Dennis and Martin, the "limit of 2^-n"
has a kind of self-similarity, similar to the self-similarity that exists
in some of the well-known examples of fractals. This self-similarity makes
it exceptionally easy for a finitist / potentialist to make claims about
them, because once you imagine one step of the limiting procedure, you've
basically imagined *all possible* steps of the limiting procedure, since
they are similar to each other. This allows you to easily prove things
about that procedure, e.g. by recursion. The potentialist could claim that
the procedure "(a) start with 1, (b) divide it by two, and (c) go to step
b" never terminates, and even claim that it is impossible for it to
terminate (no supertasks). At the same time, they could accept that a
"limit" of this procedure exists, because there is a unique number, in this
case 0, so that for every possible epsilon, there possibly exists a number
N so that for any possible n > N, the value at n-th step of this procedure
is no more than epsilon away from 0. That is a completely potentialist
definition, and a potentialist I think would have no problem proving this
using the self-similarity of the procedure without ever assuming infinities
exist. They could also happily prove that this limit is equal to the limit
of the sequence 3^-n.

You might think I've talked myself into a corner, because haven't I thus
also committed the potentialist to accept the existence of \pi, whose
decimal expansion has a non-repeating infinite series of digits? But in
fact, I think you can consistently call yourself a potentialist and accept
the existence of a number \pi defined only through a limit (assuming we
reject the "bendy line" construction). The trick is that we just think of
"limit" as a different kind of mathematical object declared by fiat, like
"i" in the Gaussian integers. This new type of object embeds the usual
rational numbers via limits of procedures like "(a) start with p/q, (b) go
to step a", and the synthetic order relation (defined using an epsilon-N
style definition) matches the order relation on rational numbers. This is
not so surprising: a finitist would presumably have no problem accepting
similar "synthetic" objects like rational numbers, which are usually
defined as equivalence classes of pairs of integers. And indeed, the same
"issue" appears in ZFC foundations: we claim "the limit of 2^-n is 0", but
that's not literally true in ZFC! It is *not* equal to 0, the ordinal!
Recall, the real number 0 is not the same set in ZFC as the rational number
0 which is not the same set in ZFC as the integer 0, which is not the same
set in ZFC as the ordinal 0 (a.k.a. {}). These all embed into each other
preserving their structure, however, and I don't see why a potentialist
couldn't make a similar move accepting the existence of an abstract entity,
"pi" defined as "the limit of this procedure Euler came up", and allowing
them to sensibly claim things like "pi > 3", all while denying that the
completed decimal expansion of pi exists as an object.

I don't see anything here requiring the potentialist to accept anything
other than procedures with finite descriptions, do you? And if you object
to the apparent semantic shift going on, that seems like the potentialist
is redefining already established terms, presumably you would also have to
object to the semantic shift that occurs when mathematicians using ZFC
foundations implicitly identify distinct sets with each other, thus
creating a semantic shift in the notion of "="?

Response Part III
----------------------------

Others have mentioned "PA versus PA+Con(PA)" or "ZFC minus Infinity verus
ZFC" as cutoff points for a potentialist. I'm not sure those are actually
cutoff points, however, because I think a potentialist can entertain those
axiom (schemas) if they believe it is impossible to derive a contradiction
from them. They could just deny that everything they describe is "real". I
can believe that the axiom of infinity or AC or WKL_0 cannot actually lead
to contradiction, even if I don't believe the hypothetical objects whose
existence they assert actually exist. As long as I am confident these do
not contradict the truth of any sentence I can establish through finitistic
means, I can entertain them as a potentialist and maybe even use them as
"safe shortcuts" in reasoning (like conservative extensions). In the case
of "procedures defining fast Cauchy sequences", it is probably safe for a
potentialist to use the infinitist picture of a completed real number line
as a shortcut to reason about such procedures.

But I think the potentialist actually has a more accurate picture than the
infinitist. After all, if you say you believe in "the" "completed" real
number line, someone else might say "well, there is also a completed real
number line in a countable transitive model of ZFC, are you referring to
that one, or the real number line in some other model of ZFC?" The
potentialist can view the picture of a well-defined completed infinity as a
useful fiction, and in that case the usefulness does not depend on actual
existence, and so it should pose no problems for the potentialist to
entertain ZFC axioms while denying both the Platonist view that they
describe a unique actually existing set theoretic universe V, and also
comfortably remain agnostic about issues such as CH, since any consistent
fiction is as good as any other in terms of finitistic consequences.
Although maybe if I understood Harvey Friedman's recent work, I would have
a different view.

Another connection to make here would be about "infinitessimal analysis".
Originally thought to be inconsistent, Abraham Robinson showed that it is
consistent, altough it makes essential use of non-standard models. Many
infinitists claimed that infinitessimals do not exist, and now maybe they
have to claim that they must exist, as Robinson does not use any
controversial axioms. A potentialist on the other hand could claim that
infinitessimals have always been fictions, never having to admit to any
mistake about the existence/non-existence of any mathematical object. The
only thing at stake for the potentialist was whether that fiction was
useful, which still hinges on its consistency, but does not have the same
philosophical weight for the potentialist because it has no impact on their
ontology.

James


On Mon, Feb 6, 2023, 9:16 AM <martdowd at aol.com> wrote:

> 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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