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

:up:

@Metaphysician Undercover I think you would approve of Wittgenstein's view. He was a finitist, and a math anti-realist. He didn't believe in set theory. He thought it was bullshit.

When we say, e.g., that “there are an infinite number of even numbers” we mean that we have a mathematical technique or rule for generating even numbers which is limitless, which is markedly different from a limited technique or rule for generating a finite number of numbers, such as 1–100,000,000. “We learn an endless technique,” says Wittgenstein (RFM V, §19), “but what is in question here is not some gigantic extension.” — SEP

A second strong indication that the later Wittgenstein maintains his finitism is his continued and consistent treatment of ‘propositions’ of the type “There are three consecutive 7s in the decimal expansion of π” (hereafter ‘PIC’).[4] In the middle period, PIC (and its putative negation, ¬PIC, namely, “It is not the case that there are three consecutive 7s in the decimal expansion of π”) is not a meaningful mathematical “statement at all” (WVC 81–82: Footnote #1). On Wittgenstein's intermediate view, PIC—like FLT, GC, and the Fundamental Theorem of Algebra—is not a mathematical proposition because we do not have in hand an applicable decision procedure by which we can decide it in a particular calculus. For this reason, we can only meaningfully state finitistic propositions regarding the expansion of π, such as “There exist three consecutive 7s in the first 10,000 places of the expansion of π” (WVC 71; 81–82, Footnote #1). — SEP

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. — frank

Can you set this out clearly, so we can see what you are claiming?

Can you set this out clearly, so we can see what you are claiming? — Banno

Sure. Prior to the 19th Century, a convergent series would have been treated as if it reaches the limit, though it would have been ok to say it's actually just approaching it. In the 19th Century, they decided that it doesn't just approach it, it actually gets there because the function is continuous. This doesn't really make a lot of sense to me, but I haven't finished reading about it.

I'll tell you a story to illustrate how it used to be. A student was studying electronics and was confused to find that on a test, the resistance across two points was specified as infinite. The student was asked to state what the voltage across this span would be.

The student tried to apply Ohm's law, voltage = current x resistance. So the voltage would be zero (the current) times infinity (the resistance). Except, looking again, that would mean that the voltage divided by zero = infinity. Which makes no sense.

The student went to the professor after the test and asked what had gone wrong with Ohm's Law, and he was told: "Oh, the resistance isn't really infinite. It's infinite for all practical purposes. It's just really big. We multiply the really big resistance by the really small current, and we get 12. We know it's 12 because the power source is a 12V battery."

The student walked away re-committed to paying attention to practical purposes. If you get too entranced by the philosophy, you'll realize there's no way anything is actually the way we say it is.

OK. I see that "choice sequence" arises in mathematical constructivism. All these years and never came across it. Flipping a coin over and over.

I think you would approve of Wittgenstein's view. He was a finitist, and a math anti-realist. He didn't believe in set theory. He thought it was bullshit. — frank

I agree. Wittgenstein understood set theory is platonism, and rejected it as an inadequate representation of thinking. Thinking is the private property of subjects and is therefore inherently subjective. Platonism presents us with the products of thinking as something independent from the act of thinking, these are what we call "thoughts".

But this neglects a very important feature of thinking which is communication. We present our thoughts to each other through communication. When we allow that communication must be represented as a necessary aspect of "thoughts", then the true "objects" produced by thinking are the spoken and written symbols (leading toward nominalism), rather than some ideas which are called "thoughts".

The issue is that when we accept the reality, that communication is a necessary aspect of what is commonly called "thought", then it becomes very clear that the other representation of "thought" as some sort of ideas which are produced by thinking, has no grounding accept in platonism. The notion that thinking produces some sort of objects, ideas, is a misrepresentation, because what is actually produced is a system of symbols. And the existence of those supposed objects (ideas) have no grounding accept in platonism. So platonism is a false representation because it does not account for the role of symbols in the act of thinking.

Banno clearly takes the platonist perspective, which ignores the role of symbols, and we can see this from the following.

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. — Banno

We need to consider the role of symbols in representation, to understand the thinking which is being represented in these situations. Consider the following two ways to represent the natural numbers, "1, 2, 3, ...", and "N". Would you agree that these two symbolizations each signify something different? The latter represents a complete object which we know as 'the natural numbers". The former represents an endless sequence, which by that understanding, could never be complete.

With respect to those two distinct ways of representing "the natural numbers", would you agree that it is possible, even acceptable, and conventional, to represent what we know as "the natural numbers", in two contradictory ways? One symbolization means something, the other means something else, and the two contradict each other. This implies that there is two contradictory ways to understand what "the natural numbers" means, depending on the symbolization employed in usage.

@Srap Tasmaner.
This proposition, that what is meant by "the natural numbers" has contradictory meaning depending on the application, ought not be taken as an offence. You ought to accept it as a proposed description of the reality of mathematics, and judge honestly whether it is a true description or not.

And, the idea that an "object" within a highly specialized field of study like mathematics has contradictory definitions ought not be surprising to you. Take a look for example at the difference between rest mass, and relativistic mass in physics for example. The concept of "mass" has contradictory meaning depending on the application. This is just a description of the reality of human knowledge.

The student tried to apply Ohm's law, voltage = current x resistance. So the voltage would be zero (the current) times infinity (the resistance). Except, looking again, that would mean that the voltage divided by zero = infinity. Which makes no sense. — frank

Yeah, because the student doesn't understand basic math. If resistance is infinite then you can't tell what voltage is being applied - unless, of course, you have another piece of information available, such as a place on the diagram where it clearly tells you what it is!

Anyway, I don't know what point this wooly analogy is supposed to illustrate, other than wooly thinking. Which, I suppose, is appropriate when it follows this:

Prior to the 19th Century, a convergent series would have been treated as if it reaches the limit, though it would have been ok to say it's actually just approaching it. In the 19th Century, they decided that it doesn't just approach it, it actually gets there because the function is continuous. — frank

Yeah, because the student doesn't understand basic math. If resistance is infinite then you can't tell what voltage is being applied - unless, of course, you have another piece of information available, such as a place on the diagram where it clearly tells you what it is! — SophistiCat

It sounds like you're saying that zero times infinity equals 12. :lol:

I agree. Wittgenstein understood set theory is platonism — Metaphysician Undercover

That is correct.

Your potted history is inaccurate; but any so brief account will be. Leibniz, Euler, and even Newton routinely identified infinite sums with finite values. They did not treat limits as approximations, indeed they did not systematically distinguish approximation from equality in the modern sense. For them, infinitesimals existed as actual quantities—smaller than any finite number but not zero, and hence dy/dx was literally a fraction. Limits reconceptualised these by showing that for every ε > 0 (no matter how small), there exists some δ > 0 such that whenever 0 < |x - a| < δ, we have |f(x) - L| < ε.

Now some students see the ε-δ definition and think: "For any ε > 0, we can get within ε of L. So we can get arbitrarily close to L. But we never actually reach L. Therefore the limit only approximates L"

Here the student misses the quantification: For All ε > 0, the value is within ε of L.
This isn't saying we can get close. It's saying we can get closer than any specified distance. The universal quantifier ∀ε is doing the heavy lifting. And bear in mind that ∀x F(x) is ~∃x ~f(x). This is an existential statement: there is a limit; it has a specific value, and is not an approximation.

This is a pedagogic point, not a mathematical one - and one that I learn is brushed over in the schools of engineering, perhaps because their use of rounding is so routine.

And to that history we can add the return of infinitesimals, this time with a firm foundation, in the development of non-standard analysis and hyperreal numbers, *ℝ, whcih includes positive numbers smaller than any standard real, but not zero.

For two centuries, students were told: "Leibniz and Euler were being sloppy. Infinitesimals don't exist. Here's the rigorous way (ε-δ)." Then Robinson showed "Actually, infinitesimals exist just fine. The old guys were on to something."



$0.\dot{9}$ really does equal 1.

Wittgenstein understood set theory is platonism — Metaphysician Undercover

What nonsense. Platonism treats mathematical propositions as descriptions of independently existing objects; psychologism treats them as reports of mental acts. Both misunderstand mathematics, which consists in public techniques governed by rules.

0.9̇ really does equal 1 — Banno

Really? 0.999... = 1 ?

Ask ChatGPT about the popularity of NSA. It is on target.

Consider the following two ways to represent the natural numbers, "1, 2, 3, ...", and "N". Would you agree that these two symbolizations each signify something different? — Metaphysician Undercover

Depends on whether the first symbolism is time dependent. Does counting actually require temporal steps. Can you think of 1,2,3 as instantaneous? Just speculating.

Really? 0.999... = 1 ?

Ask ChatGPT about the popularity of NSA. It is on target. — jgill

:smile:

And so maths is a game that never ends...?

So far as I can see, $0.\dot{9}$ is not an infinitesimal, but a real.

And again we must avoid mixing up a completed infinite definition with a process imagined as still going on.

I'm tempted to use a constitutive definition here, that 0.999... counts as 1.

I don't think that leads to any contradictions, and cleans things up nicely. But...?

I'm not familiar with

.9̇ — Banno

Depends on whether the first symbolism is time dependent. Does counting actually require temporal steps. Can you think of 1,2,3 as instantaneous? Just speculating. — jgill

"1, 2, 3..." isn't rigorous, of course - there are many different ways to continue the sequence. ℕ is rigorous - well, at least more rigourous. So there is a sense in which they are the same only if the sequence is continued in a certain way... the rule is shown in the doing, as Wittgenstein put it.

Yet it is also stated in the definition of ℕ

However, this is not time-dependent. We understand what it would be to continue the sequence correctly or incorrectly, without doing so.

Might be the encoding. a nine with a dot over it, marking repetition.

$\dot{9}$ = 999999...

I'll go back and edit.


Edit: Ah - it's an English/USA thing? - do you use $\overline{9}$ ? Interesting. Down another rabbit hole. This is the sort of of thing I was taught, and taught to others: Recurring Decimals.

I can think of 1,2,3 as instantaneous, so also 1,2,3,... which makes the latter the same as N.

What babble, but entertaining. :cool:

Both misunderstand mathematics, which consists in public techniques governed by rules. — Banno

Wittgenstein would agree with this view, and it's why he rejected set theory. I posted a couple of quotes above that show that.

Wittgenstein would agree with this view, and it's why he rejected set theory. — frank

Platonism treats numbers as independently existing; psychologism treats them as things in the mind; Wittgenstein showed how they are a public, social practice.

W. didn't reject set theory What he rejected was both the platonic and psychological interpretations of set theory, together with the false antipathy that thinks we must choose one or the other.

W. didn't reject set theory - indeed, he uses it in his writing. What he rejected was both the platonic and psychological interpretations of set theory, — Banno

According to the SEP he was a finitist. You're imagining that there is some finitist approach to set theory. There isn't. Look it up. :brow:

According to the SEP he was a finitist. — frank

Well, again, that needs some finesse:

Though commentators and critics do not agree as to whether the later Wittgenstein is still a finitist and whether, if he is, his finitism is as radical as his intermediate rejection of unbounded mathematical quantification (Maddy 1986: 300–301, 310), the overwhelming evidence indicates that the later Wittgenstein still rejects the actual infinite (RFM V, §21; Zettel §274, 1947) and infinite mathematical extensions. — Stanford

This is well worth working through, as well as was he right?

My contention - and I haven't put it together into a PhD yet, so it is incomplete - is that he lacked, or missed, the mechanism that allows us to move from a definition to a quantification, the "counts as" of the constitutive definition.

That's the direction taken by Austin, and then Searle, and a large part of why their work is worth considering alongside that finitism. We bring things into existence by with we do with words, in a way that Wittgenstein might not have recognised.