Well, yes - conceptual clarification. It's the elimination of the muddle of "first-person phenomenal experience", a philosophical fiction, bringing with it metaphysical baggage that isn’t doing real explanatory work; it presumes a private, ineffable inner object that is unnecessary to account for perception. What remains, after discarding the fiction, is simple: we see apples (or tennis balls), not mental phantasms. The causal/physiological machinery can be acknowledged without turning it into a metaphysical object.

No, I'll leave the baiting to you.

It kind of looks like your alternative involves people walking into a fictional world and pretending it's all real, drawing conclusions based on it's reality, when they know good and well it's all a lie. — frank

That's were you live.

Well, finitism doesn't automatically reject set theory. Arguing in terms of 'isms' will not get us as far as setting out the detail. some might see ZFC or other set theories from a finitist perspective, treating infinite sets as symbolic devices or potentialities, without committing to their actual existence. Finitism rejects the Platonist reading of infinite sets, but I think I've shown that there is at least one alternative.

I'm happy to call him a finitist, for what that's worth - the interesting thing is how that plays out.

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 rule 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.

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.

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.

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.

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.

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...?

No, the claim is that we do not directly see the tennis. We still indirectly see the tennis, much like when watching it on TV. — Michael

If you like. then it is the indirect realist who introduces "direct" and "indirect", and who is going to haver to explain their use.

The point about the "direct realist" being a straw man.

And back to Austin's account of seeing things directly and indirectly.

And again, when you sit in the front row of the Rod Laver Arena, you are not seeing some mental phantasm of the tennis.

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.

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.

An interesting read, bringing us back to the Stoics in the context of disability.

The Post Paralysis Peace Paradox

I don't think it's fire-walled... let me know.

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?

This argument that "we see tennis, even if on TV; therefore direct realism is true" is ridiculous. — Michael

Quite so. And it's not what was argued. The indirect realist makes the ridiculous claim that even when you are at the Rod Laver Arena, you do not see the tennis, but an image of the tennis. That we do not have "direct" perception of the tennis when watching it on the screen, or in the front row.

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.

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

...you should cut me some slack... — frank

Meh. You seem more interested in the drama than the maths.

You're mostly just playing sillybuggers as it stands.

I enjoy these chances to exercise my math muscles a bit more directly than usual, — Srap Tasmaner

Yes! What I'm finding interesting here are the links to set theory and first order logic, but it's a strain to recall the little undergrad calculus I did study.

I've tried to present my working as explicitly as possible - and ChatGPT is invaluable here, for both checking arguments and formatting Mathjax. I'd have hoped that if there were real objections, the objector would at least take the trouble to set them out formally.

Ok. We can drop it. — frank

Failure to commit. Again.

Still not seeing much here. Chat says

Pointwise convergence tells you that each point eventually settles down; uniform convergence tells you that the process itself settles down everywhere at once, which is why only the latter supports treating the limit as the genuine sum.

and here we are dealing with real analysis and uniform convergence, so this is stuff is peripheral..?

:wink: Your longest thread so far... are you happy with it?

I was trying to understand it myself. — frank

Why? As in, where does it fit?

Yep. Indeed, it's not mathematics that is the topic here - one of the resources I was using described nonstandard analysis as saving mathematics from the philosophers.

Notice that @jgill, our resident mathematician, shows only passing interest here. Maths doesn't much care, and part of getting the conceptual work right might well be explaining why it doesn't matter. Nothing essential to his mathematical work turns on the choice.

At the core the difference might be seen as between an approach the closes of mathematical possibilities by saying "you can't do that" and an approach the encourages trying stuff out. One stance says: only methods that fit a preferred ontology count as legitimate; the other says try it and see whether it can be made rigorous...

Which in turn comes back to two different ways of doing philosophy.

If you - who avoids commitment at every turn - can set out why it's relevant, I might have a go.

As it stands, you're just being a bit of an arse hole, not wanting to address the content here but to play with personalities instead.

:yawn:

Fuck off.

Understanding builds in the defence.

Would it be better to attack it without understanding it?

You're aware that the issues of the century before last were solved using an axiomatisation of the continuum - along the lines started earlier in this thread - and then nonstandard analysis showed they weren't such a problem, anyway...?

So...?

Apt.

And you suppose that to be an end to it?

We set out the sequence $a_n = (1/2)^n$, or the sum $\sum_{n=1}^{\infty}\left(\frac{1}{2}\right)^n$, then find that the limit is 1.

One might set the limit to one then look for a sequence, but of course there is more than one such sequence... quite a few more.

Your posts make less and less sense as we proceed.

Obviously, there is always "a little but more" in terms of how close we can get to the limit. that is implied by your definition of "limit". — Metaphysician Undercover

Being obvious to Meta is not a proof.

Always keep in mind that Meta argues that there are no numbers between 1 and 3.

The sequence is designed, and produced from the limit. — Metaphysician Undercover

This is exactly arse about. The limit is a result of the sequence. Those who care to look can see exactly that in the proofs offered earlier.

The relevant issue is that when I see the tennis match on television I do not have direct perception of the tennis match. — Michael

...and yet you saw the tennis. Thank you for such an apt example. The indirect realist is the one insisting that you never saw the tennis, only every pixels on a screen. For the rest of us, those pixels are part of watching the tennis. The causal chain is not the epistemic chain.

That the apple causes the experience isn't that it's a constituent of the experience. — Michael

This and your quote appear to be a constipated way of saying that one only sees the apple if there is an apple. Sure. At issue is whether one sees the apple or a "representation" of the apple. In your now well-beaten dead horse, one sees the apple as it was ten seconds ago. But somehow you conclude that one is therefore not seeing the apple. How that works escapes me.

:meh:

The intuition goes: Given that there are real numbers, and given that our sequence can get as close as we like to some number, let's call that number the limit of the sequence.

Added: the pedagogic problem - it's not a mathematical problem - is how to dissipate the notion that the limit is "a little bit more" than the sequence? Notice that the limit is set out in terms of the sequence - the limit is provided by the sequence alone! so the limit results form the sequence. But it need not be one of the elements of the sequence. It's not something the sequence reaches toward — it is a property of the sequence itself. The limit isn't something the sequence is trying to get to; it's a concise description of how the sequence behaves. The sequence doesn't "know about" or "aim for" its limit - the limit is simply our label for a pattern in the sequence's terms.

The conclusion "x=0" is not valid without a further stipulation that there can be nothing between the least ε and zero. — Metaphysician Undercover

That stipulation is what ℝ is. It is not an extra, and it does not make the argument that there is a limit circular.

It is not a stipulation about limits.

The limit will be called the sum of the series. — Banno

The meaning of of this was just given.
$L$ is the limit of the sequence $(a_n)$ iff
for every $\varepsilon \gt 0$
there exists $N$ such that for all $n \ge N$,
$\lvert a_n - L \rvert \lt \varepsilon$

it says: The terms of the sequence can be made as close as you like to L by going far enough out in the sequence.

Importantly, there is no little bit left over because in the real numbers there is no positive number smaller than every positive number. So if the difference between $a_n$ and L can be made smaller than any positive number you choose, the difference must actually be zero.

But yes, I am getting a bit sick of working on the tags... especially since folk seem to ignore them.

Stipulate that the limit is the value, then use that as a premise in proving an instance of this. — Metaphysician Undercover

You misread.
What is stipulated is what is meant by a limit:

Definition (limit of a sequence)
$L$ is the limit of the sequence $(a_n)$ iff
for every $\varepsilon \gt 0$
there exists $N$ such that for all $n \ge N$,
$\lvert a_n - L \rvert \lt \varepsilon$

If ∣x∣<ε for every ε>0, then x=0 is not a stipulation about limits; it is a theorem about the real numbers, derived from the order structure of ℝ.

The structure of the argument is:

• Define the limit (ε–N).
• Assume: the ε–N condition holds for some L.
• Introduce the independent fact about ℝ: no nonzero real can be smaller than every positive real.
• Conclude: therefore the difference is zero → exact equality

We are nto doing numerical analysis.