catfish

You always are fishing. It's what you do. What I so rudely call "failure to commit". — Banno

I'm actually Socrates. I forgot to tell you.

Baloney.

No, I'll leave the baiting to you. — Banno

Ok. I'm going fishing.

That's were you live. — Banno

Don't bait me into giving my sociological report on this thread. :cool:

I agree with Hanover that if the question is supposed to be metaphysics, it isn't resolvable.

I agree with Banno, that indirect realists do claim to know the truth about the world by way of their senses.

I agree with Michael that indirect realism is undeniable if we accept contemporary science.

Who else?

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. — Banno

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.

Is that how you see math?

How would you characterize the view of your opponent?

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

It's a rejection of set theory. We wouldn't even talk about the extension of the real numbers. There is no extension.

Well, again, that needs some finesse: — Banno

If you keep reading, the SEP explains that the arguments that he wasn't a finitist are weak.

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

He would say there's no fact of the matter regarding who is right. As you mentioned before, there is no change in practice if we accept or reject finitism.

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:

This thread is a war of definitions.

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.

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

That is correct.

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:

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.

@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

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:

I feel your pain, but it is the result of applying a philosophical approach to mathematics. Philosophers love to dwell in the past and compare one master with another, one ancient idea with another. Mathematics is a social agreement and looks forward, not backward. I've taught engineering calculus, although its been quite a while ago, and the notion of the convergence of, say a power series, is fundamental. An analytic function is defined by convergent power series. — jgill

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.

I understand. In my approach to any topic, I need a skeleton, and then I put flesh on it. As it grows, my comprehension grows. So with any philosophical topic, I need to know what the skeleton is: what is the bone of contention? What are the different arguments? What are their strengths and weaknesses? That way I can take something a person says and see where it goes on the skeleton. I'm aware that other people start with the flesh and sometimes never really coordinate it all with a skeleton. I just can't do that. My brain doesn't work that way.

You're mostly just playing sillybuggers as it stands. — Banno

I'm really not. I learned calculus in an engineering setting. It never really occurred to me that anyone thought the sum of a convergent series actually equals the limit. At face value, that really makes no sense. It turns out Newton would agree with me. Leibniz would not. So this conflict is at the beginning of this kind of math. Since people have been struggling with it for like 300 years, you should cut me some slack for trying to get it. :razz:

Failure to commit. Again. — Banno

What? I thought you were telling me to drop it. :lol:

I wasn't being critical. Defending an idea without understanding it is a sign of a conservative spirit. Nothing wrong with that.


Ok. We can drop it.

Why? As in, where does it fit? — Banno

Point-wise convergence is considered to be a weak explanation for how a series converges in a way that allows us to say the limit is the sum.

Uniform convergence is considered to be the stronger explanation.

It's become the custom to express the two ideas in mathematical terminology, which doesn't do much for me. I need a "verbal" explanation. It appears you can't get that without whole biographies of everyone involved.

Actually, I was trying to understand it myself. It's tough finding an explanation in plain english.

Fuck off. — Banno

Oh dear. It seems you can't.

Would it be better to attack it without understanding it? — Banno

No. Can you explain the difference between point-wise and uniform convergence?

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...? — Banno

I read about it, yes. Cauchy's original solution was eventually rejected in favor of Weierstrass's solution. As has been mentioned, this is history that usually isn't taught. I know this irritates you, but what's most interesting to me is the way people defend it when they don't actually understand it.

If representation is just the function of neural states, your philosophical view stops being indirect and becomes more of scientific direct realism. — Richard B

No.

And you suppose that to be an end to it? — Banno

Absolutely. Never to come up again. :grin:

My first calculus teacher was awesome. He told us this story about when he was young and he dropped some mercury on the floor. They tried to sweep it up with a broom and it turned into a "blue bloody million" little balls of mercury.

The history of mathematics is a worthy subject in itself, but that is not the topic here. — SophistiCat

Cauchy, Weierstrass, and Riemann saved calculus from mounting criticism that it doesn't make sense. So, that did become the topic. We've finished talking about it now, though. :grin:

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

Yes. I don't think either of us is interested in arguing about what that actually means. Let's leave it.

But the point I made is that "point particle" is a conception of convenience, designed for the purpose of representing interactions. It does not represent how the electron is actually conceived as existing. The electron is modeled as a "point particle", but it does not exist that way, the probability cloud is a better representation (though still very inadequate) of how electrons exist. — Metaphysician Undercover

I actually took a deep dive on this at one point. The electron is, in fact, conceived by scientists as a point. It's startling, but true.

You're right that per Cauchy, the sum of the series is the limit. However, the devil is in the details. The sum and the limit are never equal. see here.

I'm not concerned about credibility or showing that I'm working. :grin:

But an electron is conceived as a point.
— frank

I don't think so — Metaphysician Undercover

It is. An electron is a point particle.

Ok. Details? — Banno

Yea, you're wrong.

It's also an answer to this, I think. — Srap Tasmaner

How so?

The limit is not “almost” the value. — Banno

That's incorrect.

Tell me where I’m wrong if you can. — Banno

According to Zvi Rosen, the sum and the limit are not equal (according to Cauchy). They're just as close as we "want" them to be. This is Cauchy's definition of a limit:

When the values successively attributed to the same variable approach indefinitely a fixed value, eventually differing from it by as little as one could wish, that fixed value is called the limit of all the others. — Cauchy

So it's true that the idea of an infinitesimal was removed, but the idea of infinitely small remained, and we added the idea of "as small as we want."

We'll see what the other's said.

Wait, what? — AmadeusD

You don't have confidence that you can tell what's true and real?

That is to say, none of this discussion is responsive to the metaphysical question of what the fundamental constitution of reality is. As in, what is the apple in the noumena? — Hanover

That is correct. Both sides of this argument start with irrational confidence in our ability to discern what is true and real. Neither side proposes to build a bridge to that confidence. As you noted, there is no bridge to it. You just have it.

Starting with that confidence, we observe by way of anatomy and physiology that perception of the world appears to be constructed by the brain out of discreet electrical impulses. As you note, this is not a metaphysical argument, it's a scientific fact.

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. — Banno

This is all from proofs by Cauchy that I don't understand. Do you understand it?