And you suppose that to be an end to it?

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.

Apt.

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...?

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.

Understanding builds in the defence.

Would it be better to attack it without understanding it?

Your denial never ceases to amaze me.

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

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

Fuck off.

Fuck off. — Banno

Oh dear. It seems you can't.

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:

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

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.

I was trying to understand it myself. — frank

Why? As in, where does it fit?

Living the life

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

but what's most interesting to me is the way people defend it when they don't actually understand it — frank

Gee, I don't know, frank. Isn't that mostly what people do here, no matter what the topic? Or: isn't that the claim of their opponents, should there be an actual debate? @Banno claims not to be a platonist, and @Metaphysician Undercover claims he is anyway—that Banno either doesn't understand his own position or that he doesn't actually hold the position he thinks he does.

And so far as that goes, this is par for the course among real philosophers, not just amateurs like us.

Much like @SophistiCat, there was a time in my life when I could have demonstrated Dedekind cuts for you and proved the Mean Value Theorem on demand. Nowadays, no. Much of the little knowledge of mathematics I once had is gone, along with my undergraduate expertise, but my appreciation of mathematics, the love of mathematics I've had since I was a kid, that remains. Sometimes I like these math threads because it's a chance for me to brush up, blow away some of the cobwebs, and it's a chance to look at math.

I was probably never all that good at math, much as I loved it, but even though I no longer have at my fingertips even the fingertips of the body of mathematical knowledge, I have never stopped looking at the world mathematically. 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.

Yes, we don't always understand everything we're talking about. What else is new? But it's a challenge. I like trying to understand things, and the best way I know of determining whether I do is trying to explain it myself. If I can't, I have some work to do. What else is new? I always have work to do.

Too many participants in too many discussions here evince no such desire to understand. I can take it on faith that they're participating in good faith, but I could not prove from their posts that they are not simply trolling. Maybe some people think the same of me, but I hope not, and if I thought so it would bother me, and I'd rethink how I write. (This is not hypothetical. I have had an analogous experience on the forum.)

By the way, if there's something mathematical you want to know and wikipedia doesn't work for you—some of its mathematics articles are not exactly for the general reader, in my experience—and you can't find another website with a nice explanation, you don't want TPF, you want Stack Exchange. There will be material there that's over your head, sure, but there are also people that know what they're talking about and put a surprising amount of effort into explaining it.

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..?

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.

Ok. We can drop it. — frank

Failure to commit. Again.

Failure to commit. Again. — Banno

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

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.

Defending an idea without understanding it is a sign of a conservative spirit. — frank

You're talking about dogma, I get that, but I think you're missing another possibility.

The other possibility is the sort of thing suggested by Mercier and Sperber in The Enigma of Reason. If you think of reason not primarily as a system a solitary individual would use to deduce one truth from another, that sort of thing, but instead as a tool for critiquing the views of others and supporting your own view against objections raised by others—if, in short, you see it primarily in its social function, then the sort of thing we do around here makes a little more sense.

It's very late in the day, of course, and some people, the sort of people who have devoted some time to systematic thought (logic, mathematics, law, and so on), have been able to internalize the process, and we think of the usage we see there as the norm.

But in its origin, the important thing is the process of communal decision-making and communal understanding. Seen in that light, it's no surprise that we are pretty good at spotting the flaws in the ideas of others and not so good at spotting the flaws in our own ideas. And it also makes sense that logic and argument tend toward dichotomy, black and white, true and false, right and wrong.

Why? Because in the group discussion, each individual is not responsible for figuring it all out on their own; they are responsible for bringing a view to the group and advocating for it, and everyone else does the same. You give reasons to support your view not as an explanation for how you came to hold that view—you probably don't really know that—but to build support among others.

If you start with a view that doesn't hold up, you'll discover that as others critique it, and you begin to see its weakness. But you won't have that experience if you don't bring your idea forward. In hindsight, it might very well look like you were advocating a position you didn't fully understand, but so what? The whole point was to put it to the test. If it failed, so be it, and you're the better for it.

So, no, I don't think it's always just a matter of defending that old time religion, or a conservative mindset. In some cases, it's just playing your part.

You're mostly just playing sillybuggers as it stands.

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:

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

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

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 — frank

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.

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.

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