Yep. Yet the limit is not something the sequence is chasing, but a property of the sequence as a whole...? — Banno

It depends on whether you are referring to a recursive sequence or to a choice-sequence.

A recursive-sequence is an algorithm for generating a sequence prefix of any finite length, where a limit refers to a convergence property of the algorithm, as opposed to referring to a property of any prefix that is generated by using the algorithm.

On the other hand, a choice-sequence S is an unfinishable sequence of choices that is both

- Dedekind finite - meaning we don't have an injection N --> S.
- Of unbounded length - meaning we don't have an injection S --> {0,1,2,..n} for any finite n.

Such potentially infinite sequences do not possess a limit unless the choices are made in accordance with an epsilon-delta strategy that obeys the definition of "limit". So in this case, we can speak of approaching a limit, because Eloise and Abelard are endlessly cooperating to produce a strategy for continuing a live sequence that literally approaches their desired limit, as opposed to the previous case of Eloise having a one-move winning-strategy when competing against Abelard for proving a convergence property of a dead algorithm.

Unfortunately, ZFC grounded classical mathematics cannot formally recognize potentially infinite (live) sequences due to the axiom of Choice that "finishes" them. Hence there is a clash between common-sense mathematical intuition (i.e. intuitionism) on the one hand, that correctly thinks of infinite sequences as referring to either unfinishable processes or algorithms, versus the formal straight-jacket imposed by the timeless world of ZFC, that cannot express the notion of a live process approaching a limit.

By default, classical mathematics is implied when talking about calculus, and even though ZFC isn't explictly assumed in textbook discussions of calculus, the logic they appeal to when discussing logical concepts such as limits, is classical in which calculus proofs are inductive proofs, which aren't applicable when reasoning about choice sequences, whose proof-theory is coinductive.

(I've never read a textbook definition of a limit as a two-player game - but they nevertheless informally appeal to such games when encouraging students to rote learn - a short term pedegogical payoff leading to long-term confusion after the students forget the game-theoretic reasoning behind the proofs)

So I enjoy these chances to exercise my math muscles a bit more directly than usual, and I take deep offense at Metaphysician Undercover's repeated dismissal of mathematics as a tissue of lies, half-truths, and obfuscations. — Srap Tasmaner

I don't understand this feeling of offense. This is philosophy, and what we do is critical thinking, and therefore criticize. What I don't get, is that many people think it's acceptable, even warranted and expected, that we criticize metaphysical principles, yet some of the same people believe it's for some reason unacceptable, and offensive to criticize mathematical principles. Where is the consistency in this type of attitude?

What I apprehend here is that some people take mathematics as a sort of religion. So in the same way that some people get seriously offended when their "God" is criticized, some others get seriously offended when their "mathematics" is criticized.

Such potentially infinite sequences do not possess a limit unless the choices are made in accordance with an epsilon-delta strategy that obeys the definition of "limit". So in this case, we can speak of approaching a limit, because Eloise and Abelard are endlessly cooperating to produce a strategy for continuing a live sequence that literally approaches their desired limit, as opposed to the previous case of Eloise having a one-move winning-strategy when competing against Abelard for proving a convergence property of a dead algorithm. — sime

This is what @Banno seems to be in denial of. The intent behind creating the infinite sequence, is to create an infinite sequence. This implies that the so-called "limit", as defined by Banno, is prior to the sequence, as a requirement for the creation of the sequence.

On the other hand, we could look at the infinite extension of pi, as an unintentional infinite sequence. Notice, that now there is no "limit". This exposes the nature of "the limit", it is a concept which serves the purpose of creating an infinite sequence. When an infinite sequence is created unintentionally, there is no "limit".

This leads to a question about the intentionality of the infinity which is the natural numbers. If this is an unintentional infinite sequence, we ought to assume that there is no limit. But if it is intentional, then there ought to be some sort of limit, as the source of its creation.

Grammar Psychology tricking so many here. :lol:

Infinity is a word, that presents the concept of "indefinite continuation" in terms of a beings and unity. Just like we think of the numbers 1 and 2 in terms of being and unity, these things in themselves... so obviously that there is infinitely many things in themselves between these two things in themselves... as grammatical objects these things in themselves are seen as limits. People will see "infinity" as the thing in itself (only 1 infinity), or they will see that infinite meta regress between two things in themselves (Zeno's paradox [infinite infinities]).

Infinity isn't a known truth in terms of indefinite continuation in reality.

It's only possible in meta.

Grammar Psychology tricking so many here. :lol: — DifferentiatingEgg

Care to remedy the confusion, per the unspoken goal and purpose of most philosophies, or merely take pot-shots from a place whose elevation and understanding I'd frankly question. :wink:

Was editing as you were typing.

But there ya go.

Grammar Psychology trick fuckin yall...

it's for some reason unacceptable, and offensive to criticize mathematical principles — Metaphysician Undercover

What I apprehend here is that some people take mathematics as a sort of religion. — Metaphysician Undercover

Yes, I attach value to mathematics, but that's like saying I attach value to logic or to language or, you know, to thinking. The basis of mathematics is woven into the way we think, and mathematics itself is primarily a matter of doing that more systematically, more self-consciously, more carefully, more reflectively. The way many on this forum say you can't escape philosophy or metaphysics, I believe you can't escape mathematics, or at least that primordial mathematics of apprehending structure and relation.

When you say you are critiquing mathematical principles, here's what I imagine: you open your math book to page 1; there's a definition there, maybe it strikes you as questionable in some way; you announce that mathematics is built on a faulty foundation and close the book. "It's all rubbish!" You never make it past what you describe as the "principles" which you reject.

So, on the one hand, I think you're simply making a mistake to think that the definition you read on page 1 is the foundation of anything. We are the foundation of mathematics. The definitions and all that, they come later. And, on the other hand, even if mathematics did have the structure you think it does, so that attacking some definition did amount to attacking the entire edifice of mathematics in one blow, I would still disapprove of your failure to engage in the material past page 1. It's childish. Maybe what the adults are doing is foolish, but the evidence for that is not a child, who doesn't understand what they're doing, announcing that it's "dumb."

Recently, one of my supervisors was explaining something to a bunch of us, and she insisted that what she was talking about was true "not theoretically, but mathematically." Put that in your pipe and smoke it.

Yes, I attach value to mathematics, but that's like saying I attach value to logic or to language or, you know, to thinking. The basis of mathematics is woven into the way we think, and mathematics itself is primarily a matter of doing that more systematically, more self-consciously, more carefully, more reflectively. The way many on this forum say you can't escape philosophy or metaphysics, I believe you can't escape mathematics, or at least that primordial mathematics of apprehending structure and relation. — Srap Tasmaner

OK, so you believe that mathematics is very much comparable to metaphysics, as I suggested. Do you also believe that to maintain consistency, if a philosopher believes that there is a need to be critical of metaphysical principles, that same philosopher ought to also believe that there is a need to be critical of mathematical principles?

When you say you are critiquing mathematical principles, here's what I imagine: you open your math book to page 1; there's a definition there, maybe it strikes you as questionable in some way; you announce that mathematics is built on a faulty foundation and close the book. "It's all rubbish!" You never make it past what you describe as the "principles" which you reject. — Srap Tasmaner

Your imagination misleads you then.

First a small point. If mathematics is a practice, as I have argued here, then it's not a surprise that one might changing from a recursive approach that is able to treat infinite sequences as a whole, to sequences of choice that do not.

But this does not invalidate ZFC nor the axiom of choice, nor need we conclude that a limit is something the sequence approaches dynamically rather than a property of the sequence as a completed object.

And the larger point: At issue is whether there is one basic ontology for mathematics. Sime is seeking to replace one ontology with another, to insist that we should think of infinite sequences as processes or algorithms, not completed totalities.

This in contrast to the Wittgensteinian approach, ontological questions dissolve into grammar and use.

What has not been shown is that something goes wrong, concretely, in classical practice if sequences are treated as completed totalities.

I notice numerous posters have the same attitude: that math is somehow immune from philosophical inquiry, and that if it's all built on nonsense, that's ok. I think it's really unfortunate that people got that impression. It's arrogant ignorance — frank

Not at all. But philosophically examining mathematics requires knowing something of the subject. Otherwise it becomes a babble of word definitions. Philosophy of mathematics as an academic subject is certainly alive and well, practiced by those familiar with foundations and at least something of the branches of math.

That is not to say philosophical discussions of math here on TPF is inappropriate, but merely speculative and more concerned with how words are interpreted. That's fine. Actually, I am curious about "choice sequences" - an example perhaps?

What has not been shown is that something goes wrong, concretely, in classical practice if sequences are treated as completed totalities. — Banno

Indeed. Neither has it been shown that something goes wrong in practice if we treat a convergent series as unequal to the limit by an infinitesimal amount.

Philosophy of mathematics as an academic subject is certainly alive and well, practiced by those familiar with foundations and at least something of the branches of math. — jgill

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