Re[2]: Mathematics with the potential infinite - some inexhaustible?
Matthias
matthias.eberl at mail.de
Tue Feb 14 12:19:56 EST 2023
Thanks for the interesting post of James Moody. I agree with has beed
said there, and want to add small comments.
<--
The trick is that we just think of "limit" as a different kind of
mathematical object declared by fiat...
-->
I would say that a sequence is of a higher type, e.g. nat -> rat
(rational numbers), and the limit is of base type (e.g. abstract real
numbers).
<--
I don't see anything here requiring the potentialist to accept anything
other than procedures with finite descriptions, do you?
-->
A potentialist should/could also accept procedures that are not
definable by a description in some specific language. A lawless choice
sequence, or random sequence should also be permitted.
<--
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.
-->
I think that from a potentialist's perspective it makes even sense to
see infinitesimals as indefinitely small, but finite distances. This is
possible since all mathematical investigations are finite and use only a
finite initial part of every sequence or process. These indefinitely
small distances depend on the context of the investigation, but one can
consistently assume their existence.
Basically, the idea of an actual infinite set of all natural numbers,
infinitely many instances of a schema, a gapless number line,
infinitesimals, all of them are fictions or extrapolations of an
arbitrary large set of numbers, arbitrary many applications of a schema,
arbitrary dense sets of points / divisions of the line and arbitrarily
small distances between two points. These fictions are the result of
applying the idea that these big (or small) entities should exist
independent of how often the procedures were advanced.
Matthias
------ Originalnachricht ------
Von "James Moody" <jmsmdy at gmail.com>
An martdowd at aol.com
Cc fom at cs.nyu.edu
Datum 12.02.2023 17:34:17
Betreff Re: Mathematics with the potential infinite - some
inexhaustible?
>
>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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