Banno

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 i...

Well, again, that needs some finesse: This is well worth working through, as well as was he right? My contention - and I haven't put it together into ...

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

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?...

"1, 2, 3..." isn't rigorous, of course - there are many different ways to continue the sequence. ? is rigorous - well, at least more rigourous. So the...

: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 ...

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 t...

What nonsense. Platonism treats mathematical propositions as descriptions of independently existing objects; psychologism treats them as reports of me...

Your potted history is inaccurate; but any so brief account will be. Leibniz, Euler, and even Newton routinely identified infinite sums with finite va...

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... l...

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

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 ...

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 t...

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

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

You're mostly just playing sillybuggers as it stands.

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 ...

Failure to commit. Again.

Still not seeing much here. Chat says 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?

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 ...

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...

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...

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 limi...

Being obvious to Meta is not a proof. Always keep in mind that Meta argues that there are no numbers between 1 and 3. This is exactly arse about. The ...

...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 ...

: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 ...

That stipulation is what ? 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 lim...

The meaning of of this was just given. L is the limit of the sequence (a_n) iff for every \varepsilon > 0 there exists N such that for all n \ge N, \l...

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 \vare...

Were it says The limit will be called the sum of the series. :meh:

, I just gave a proof involving a sequence that gives the exact value of the limit: zero. \lim_{n \to \infty} \frac{1}{n} = 0 This is a counter instan...

Because he was looking at Numerical Analysis not Real Analysis.

Ok. Details? Simple example of a limit with an exact value Consider the sequence a_n = \frac{1}{n} Claim \lim_{n \to \infty} \frac{1}{n} = 0 Proof (?–...

Well, you can play with all that if you like - some of what you say here looks muddled. The salient bit today is that a limit is not a rounding off. T...

well I haven’t had an exam on it in 50 years… Not just Cauchy. Tell me where I’m wrong if you can.

There's a need to be clear here that representation is Michael's word. Neural nets of course do not function by representing one thing as another. the...

:smile: Back a few pages I began a bit on the definition of a limit. I got as far as completeness and the least upper bound. Every nonempty set of rea...

Interesting. A worthy topic - a more intricate form of "rounding off"? :wink: I'll defer to your experience. My understanding is that what I said hold...

But the argument is not that I directly see X, because that is little more than a rhetorical ploy on the part of the indirect realist. At issue is whe...

So "I see X" is true if we directly see X or if we indirectly see X and yet they do not collapse into one? Not following that at all. So you say "I se...

Good. then the two collapse into one. And you have now agreed that "I see the apple" is true, and "I see a mental image of the apple" misleading. "fir...

So the strawberry is actually grey? Notice how you here work with the merely philosophical construct "the-strawberry-as-it-is-in-itself"? We never get...

Hokum. You conflate "I see an apple" and "I indirectly see an apple". Again, that "naive realist" is no more than a foil against which to draw the sup...