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.

The sum and the limit are never equal. see here. — frank

Were it says

If, for increasing values of n, the sum Sn approaches a certain limit S, the series will be called convergent and the limit in question will be called the sum of the series.

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 instance to your insistence. The "=" is not an approximation.

So if you would keep your credibility, show your working.

Damn keyboard keeps sticking.

Then why did you say to@jgill, "a more intricate form of 'rounding off'"? — Metaphysician Undercover

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 (ε–N)
Let $\varepsilon \gt 0$ be arbitrary.

Choose $N \gt \frac{1}{\varepsilon}$.

Then for all $n \ge N$,
$\left| \frac{1}{n} - 0 \right| = \frac{1}{n} \lt \varepsilon$

Since this holds for every positive $\varepsilon$, the difference between $\frac{1}{n}$ and $0$ can be made smaller than any positive real number.

Therefore,
$\lim_{n \to \infty} \frac{1}{n} = 0$

Conclusion
There is no “infinitely small but non-zero” remainder. In ℝ, being smaller than every positive real number forces equality with zero.

With the help of ChatGPT. Let me know if it's wrong. Looks OK to me.

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.

To which we might add, as a corollary, The limit is not “almost” the value.

well I haven’t had an exam on it in 50 years…

Not just Cauchy.


Tell me where I’m wrong if you can.

And the answer is that one sees the apple by constructing a representation of the apple. — Banno

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. they function by modifying weightings. It’s just a pattern of activations and weights, with no intrinsic “aboutness” or semantic content.

Better to say they model, in a statistical, functional sense.

: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 real numbers that is bounded above has a least upper bound in ℝ, the smallest real number that is greater than or equal to every element of the set. It's the existence of this number that guarantees the existence of a limit when one uses the sequence in a calculation... if it's a monotone increasing sequence that is bounded above...

But as you found, the interesting stuff is the variations on these themes. The thread is focused on a small, very specific region of maths, and mostly failing to get a good handle on even that.

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 holds for classical convergence in Real Analysis.

Given that "I see X" is true if "I indirectly see X" is true, it is a non sequitur to argue that if "I see X" is true then "I directly see X" is true. — Michael

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 whether one sees the apple or a representation of the apple.

And the answer is that one sees the apple by constructing a representation of the apple.

I could say "I saw Alcaraz defeat Djokovic in tennis" or I could say "I saw images on my computer screen". — Michael

Yep. Different placements of the Markov Blanket.

What we should not say is that we never saw Alcaraz defeat Djokovic, only ever images of Alcaraz defeating Djokovic.

It's an example of seeing an apple without an apple being a constituent of the experience. — Michael

:meh: This gaslights itself.

In your example, the apple causes the pattern of light that is seen ten seconds later. Hence the apple is a constituent of the experience.

No they don't. — Michael

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 see the apple" is true, and so is "I see the mental representation of the apple", and you want to claim these are the same? But it is clear that an apple is different to a mental representation of an apple. You can't make a pie with a mental representation.

Going over the already dispelled though experiment doesn't help you here.

No I don't. "I see X" is true if we directly see X or if we indirectly see X. — Michael

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. "first-person phenomenal experience" is philosophical fluff.

Naive realists say that apples are "constituents" of first-person phenomenal experience... — Michael

So indirect realists say that apples are not "constituents" of our seeing apples? How's that?

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 to taste or see "the-strawberry-as-it-is-in-itself" not becasue of any limitation on our senses, but becasue it's not a thing. "the-strawberry-as-it-is-in-itself" is already interpreted.

As being grey, apparently.

You always conflate "I see an apple" and "I directly see an apple". — Michael

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 supposed "indirect" account. That indirect account is misleading. What one sees is the apple, not a mental image or whatever.

No it isn't. Indirect realism says that what we see is not the apple.

It suffices as a refutation of indirect realism. What we see are apples, not mental images of apples. Seeing an apple is constructing a model of that apple. That model is of the apple, and is not what is seen.

The representation is the machinery, not the seen object.

At 10:00:25 there is no apple, only first-person phenomenal experience with subjective character — described as "seeing a red apple" — and this first-person phenomenal experience with subjective character is a mental representation of an apple that no longer exists. — Michael

Notice that the conclusion, that we see "only first-person phenomenal experience with subjective character", is not argued for but merely asserted? You are repeatedly presuming that what we see is a "first-person phenomenal experience with subjective character", and not an apple.

That's not how language works. We can talk about the first-person phenomenal experience, but that does not mean that we cannot talk about the apple, including seeing the apple.

Your example continues to confuse the causal chain with the epistemic outcome.

You are misrepresenting the grammar of "seeing a mental representation". — Michael

Seeing an apple is constructing a mental representation, if you like - it depends where one places the Markov Blanket. But one does not see a mental representation. One sees an apple.

light doesn't move — DifferentiatingEgg

:meh:

What does it mean to see the apple as it was? — Michael

Just that.

Again, what we see is the apple, and not a memory, a sense datum, a representation, an image,
or anything “in the mind”. Sure, the causal chain that is seeing the apple includes a delay, but so what.

Your argument is merely rhetorical, a play on the word "direct". What one sees is the apple with a ten-second delay. What one does not see is some mental representation of the apple as it was ten seconds ago.

The difference between the limit and the sum is an infinitely small number. — frank

For a convergent series the sum is defined as the limit. There is no residual “infinitely small difference” between the sum and the limit. The sum is the limit. Partial sums are less than the limit, but their difference goes to zero in the standard real number system.