Brian Tung

general recent conversations

$Mathematics$ Discussion on question by some kid tr…

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Brian Tung

Jul 18 11:22

There's a way to intuit the general form of the Taylor series in the same way Euler solved the Basel problem; if I get the chance, I might write that up here, since it's kind of fun.

$Mathematics$ Discussion on question by Hola: Famou…

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Brian Tung

Jul 11 14:19

There's probably something about chess played on (or chess problems posed on) a non-standard board, potentially with non-standard topology (like a torus).

$Mathematics$ Discussion on question by Darmani V: …

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Brian Tung

Jan 22 10:03

@DanielV: That's a fair point. Not to mention that I think the question is somewhat predicated on a misconception: that people who think that $0.999\ldots \not= 1$ have some logical argument behind it. Frankly, I'm surprised this question has gotten as much attention as it has. But I'm not sure that closing it as a duplicate is quite right, either.

Brian Tung

Jan 22 10:02

@ThomasAndrews: One of my favorites. I should hunt down my copy and re-read it...

Brian Tung

Jan 22 10:02

In general, I'm with @ThomasAndrews here: They don't have even a false proof of $0.999\ldots \not= 1$; they consider the idea that $0.999\ldots = 1$ to be the thing that is unusual and needs proof. To them, the default is that if it looks different, it is different until proven otherwise.

Brian Tung

Jan 22 10:02

A single data point (yet one that I think is not uncommon): Back in college some large-ish number of years ago, one of my friends who had trouble with this insisted that if $0.999\ldots$ had an infinite number of $9$s in it, then surely by adding $0.000\text{[an infinite number of $0$s]}1$, one would obtain $1$, and since it wasn't $0$ one was adding, they had to be different. Obviously, this line of reasoning isn't sound, but I'd guess it's not rare, either.

$Mathematics$ Discussion on question by Hudjefa: A …

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Brian Tung

Jan 20 10:26

Where did the question come from, though?

Brian Tung

Jan 20 10:26

It would help if you could provide us some context for this question, such as: (a) Is this homework? (b) If so, what course are you taking? (c) What specific topic are you covering at the moment? (d) What do you know that you think might be connected? (e) If you're stuck, what are you stuck on? For example, do you know what to apply, but don't know how to apply it, or do you not know what to apply? Please edit these facts into your original post, not as responses to this comment, as comments may be deleted without warning.

Brian Tung

Jan 20 10:26

Is there anything in the problem context that justifies independence?

$Mathematics$ Discussion on question by Bao Yu Bo: …

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Brian Tung

Nov 5, 2024 11:19

Here too. One of these could probably be chosen as a duplicate. (I don't fault the OP for not finding these if English is not their native language.)

$Mathematics$ Discussion on question by Sam: Is ord…

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Brian Tung

Oct 22, 2024 16:51

For what it's worth, I often have trouble figuring out what comments are the referents to which other comments, and this thread (such as it is) is one example. I have no dog in this fight, but the progression is somewhat unclear to me.

$Mathematics$ Discussion on question by Display nam…

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Brian Tung

Oct 8, 2024 09:06

Huh OK. Sorry, I didn't mean to offend, but this line of reasoning—somewhat reminiscent of the ontological argument—does not work (logically, or otherwise). As I said, I'm generally thankful for that, because it means that many evils are not guaranteed to exist, and even the good that is not guaranteed to exist, at least provides a motivation to do good.

Brian Tung

Oct 8, 2024 09:06

Sorry, do you actually think this line of reasoning works? I assumed you thought it was specious and were wondering how it failed.

Brian Tung

Oct 8, 2024 09:06

Well, the same reasoning would apply to war, famine, disease, pestilence, and a whole host of other evils, so I at least am thankful that this line of reasoning is full of crap. :-)

Brian Tung

Oct 8, 2024 09:06

What's the motivation behind this question, if I might ask?

$Mathematics$ Mathematics

Associated with Math.SE; for both general discussion & math qu...

Brian Tung

Oct 6, 2024 22:01

I've always pronounced it as HOH-muh-toh-pee. But also I say huh-MAH-luh-jee, so who knows.

Brian Tung

Oct 6, 2024 21:53

@Joe: No worries! I don't expect you to know the consensus (or even if there is one), though if you did know, that'd have been great! However, I'm still curious how you pronounce it (if you have occasion to).

Brian Tung

Oct 6, 2024 21:38

Is there any consensus on how to pronounce "homotopy" in English? Do people say huh-MAH-tu-pee, or HOH-muh-toh-pee, or something else entirely?

Brian Tung

Oct 3, 2024 07:08

@PM2Ring Not actually British, but I can't help hearing this: youtube.com/watch?v=f9XWSeKdOuc

Brian Tung

Oct 3, 2024 07:02

Why on earth is this question (math.stackexchange.com/questions/4979570/…) up-voted +3 when there are sincere albeit mundane questions being blitzed by down-votes? Not to say this question is insincere, but is there something substantial to this question that I'm missing?

$Mathematics$ Discussion on question by Antonio Mar…

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Brian Tung

Sep 9, 2024 11:17

What is your question? You say "Could someone help me?" Help you do what, exactly? If it's to verify your proof, then the wall of text is a problem, obviously. If it's something else, please clarify.

$Mathematics$ Discussion on question by Jonathan Xu…

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Brian Tung

Aug 13, 2024 18:14

It really wouldn't, but I went ahead and did it anyway. Please check whether the result is as desired. (Global search and replace makes it fairly straightforward.)

$Mathematics$ Discussion between Krishiv Mohan and …

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Brian Tung

Jun 5, 2024 05:06

Regarding the question itself, a solution does not seem immediately obvious to me. It is not surprising that in the week or so that has gone by, no one has come up with one. It is a somewhat arcane art.

Brian Tung

Jun 5, 2024 05:04

I think it's five users.

As indicated by the votes on the question itself, people find it interesting. On that basis, I think there will be interest in re-opening it. But you have thus far come across as reluctant to improve the question (and your responses may come across to some as a bit brusque). Again, I appreciate that it does not seem that way to you, but please trust my judgment on the community here as one who's been here rather longer than you have. :-)

Brian Tung

Jun 5, 2024 05:01

It is relevant that your math coach has not worked on dissection problems with you. If your math coach specifically told you that there is a solution, then even that is relevant. I can appreciate that you find this trivial information, but please do not be reluctant to add it. You would be acting (or rather not acting) against your own interests.

Brian Tung

Jun 5, 2024 04:59

I think you misunderstand. Multiple users must vote for the question to be re-opened. What you are doing here is putting in additional information to encourage readers to vote to re-open the question. I cannot do that unilaterally. Perhaps a moderator could, but I am not a moderator.

Brian Tung

Jun 5, 2024 04:55

Have you done other dissection problems with your math coach? Which ones have you done that might be helpful here? Your math coach presumably believes that you are capable of finding the answer to this one, so knowing what they know you've done would be relevant information.

Brian Tung

Jun 5, 2024 04:49

I would recommend that you add to your post that your math coach is your source for this question and that they are the ones who know there is a solution. I am trying to make the question more likely to be reopened, so it is in your own interest to add as much information as you can to the post, even if it seems tangential to you. (It cannot be answered until it is reopened.)

Brian Tung

Jun 5, 2024 04:49

OK, that's good information! You can put that in the original post. I assume your math coach informed you that there was indeed a solution.

Brian Tung

Jun 5, 2024 04:49

It may be irrelevant to you, but it is not irrelevant to all readers. We get plenty of questions for which there is no solution. It is useful for many of them to know where the question comes from (note that two people have asked, fairly respectfully, and you have declined to answer). Please do not treat the question disrespectfully. Perhaps you did not mean it to be disrespectful, but it comes across that way.

$Mathematics$ Discussion on question by Princess Mi…

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Brian Tung

May 25, 2024 20:39

I'll probably have to stop considering this question for now, since even after all your clarifications, I'm still having trouble making out what situation you're setting up, and what it is exactly that you're asking. I don't mean to suggest that it's your fault or anything (whatever that would mean), but I suspect it's not in my interests for now to continue thinking about this. Sorry!

Brian Tung

May 25, 2024 20:39

"there are no assumptions about this structure beyond what we know; we just know that this is the set of numbers, which element thereof is the distinguished 0, how succession is defined, and how addition is defined": The second part is rather a lot of what we know about the natural numbers! And if succession and addition are already defined, then what is left to be circular?

Brian Tung

May 25, 2024 20:39

I would not say that circularity is inherently inconsistent*, but it is more prone to inconsistency, and it's not a good way to do exposition even if it turns out to be consistent in a given instance. Perhaps the successor function and the addition operation could be defined circularly, in a way that is consistent, but there's not really a good reason to do so, and it would still give people an uneasy feeling just because of the circularity. ¶ *Now someone will mention So-and-So's Theorem that says that from any circular definition can be derived both a statement and its negation. <shrug>!

Brian Tung

May 25, 2024 20:39

"I am assuming that the number system is defined as the usual $\mathbb{N}$..." I don't have the book in question; what properties are assumed to be satisfied by "the usual $\mathbb{N}$" here?

$Mathematics$ Discussion on question by Elvis: Is $…

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Brian Tung

May 24, 2024 00:33

@RobArthan: Again, I'm under no confusion as to why the intuition fails. But it does indeed fail. :-)

Brian Tung

May 24, 2024 00:33

@RobArthan: I would say that I know that, but I would not say that it helps my intuition per se. ¶ I understand, intellectually, what's going on (at least to some extent). But my intuition tells me that if I have a string of beads, it can be cut in as many places as there are beads. So my intuition is simply not very good when it comes to infinity; it has to be tweaked by things I know to be true.

Brian Tung

May 24, 2024 00:33

(cont'd) The sense that these constructions mean more than they do is heightened, I think, by their coolness. I think Dedekind cuts are very cool and counter-intuitive. (How is it that you take a countable and totally ordered field like the rationals, and somehow you can make an uncountable field simply by making a single cut in them?) But they are not really (!) what the reals are. Just imagine having to schlep that formalism around all the way through real analysis.

Brian Tung

May 24, 2024 00:33

I think the observation that @Arthur makes—that everyone [who encounters this issue] has to wrestle with it at some point—is exactly on point. Not only that they have to contend with what it means, which is relatively straightforward, but also with how significant it is. Because it's easy to get the impression that the constructions of the various domains is what they "really" are. In my opinion, those domains are not their constructions. The constructions are how we keep from having to have special theories for each of them, instead of just having set theory (and logic).

$Mathematics$ Discussion on question by Benjamim Jú…

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Brian Tung

Dec 1, 2023 16:25

After that, a year is roughly $86400 \times 365 \approx 100000 \times 13/15 \times 365 \approx 3.2 \times 10^7$ seconds, so $3.2 \times 10^{13}$ password attempts. So yes, $14$ years.

Brian Tung

Dec 1, 2023 16:25

It's hard to tell what you're asking. Are you asking how to do mental calculation of $68^8$? The way I would do that personally is $64^8\times (17/16)^8 \approx 2^{48}\sqrt{e} \approx 10^{14.45} \times 1.65 \approx 2.8 \times 10^{14} \times 1.65 \approx 4.6 \times 10^{14}$. But that's highly idiosyncratic. So what are you actually asking?

$Mathematics$ Discussion on question by user122424:…

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Brian Tung

Dec 1, 2023 16:23

Dunno, I'd have to work on it for a bit. Maybe someone recognizes it? Or maybe it might be easier to work from the middle expression to the top expression?

Brian Tung

Dec 1, 2023 16:23

Apparently, you know that the middle line is the correct result; how do you know this?

$Mathematics$ Discussion on question by Mike_bb: Ko…

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Brian Tung

Oct 22, 2023 16:02

I'm afraid I probably can't. In the first place, I don't have the time to do that, but more so, I don't know the prerequisites for this field in anything like the necessary comprehensiveness to explain the situation more broadly. You might try reaching out in one of the chats here to see if you can find someone who'd be willing to help you, but this thread is beginning to venture outside the Q&A realm that is the mainstream of this site. Sorry!

Brian Tung

Oct 22, 2023 16:02

I don't mean just related to synthetic differential geometry. What mathematics courses have you taken in general? Linear algebra, single and multivariable calculus, real and complex analysis, geometry, number theory, differential equations, integration theory, probability theory, topology, category theory, you name it? ¶ I sense gaps in your understanding that suggest an incomplete background. Only you can say for sure what you know, but you must be scrupulously honest with yourself. I don't wish to quell your enthusiasm, but are you sure you understand everything as well as you think you do?

Brian Tung

Oct 22, 2023 16:02

That's an illustration, not a verification. May I ask what your mathematics background is—what courses you've taken, etc?

Brian Tung

Oct 22, 2023 16:02

P.S. I would put that link in your question above, by the way, and mention it as part of your context. Your question is still somewhat confusing with that link, but without it, it's pretty unmotivated.

Brian Tung

Oct 22, 2023 16:02

The proof is non-constructive; that is, it doesn't identify what other $\varepsilon \not= 0$ there must be in $\Delta$. So how can you check whether $\varepsilon^2 = 0$? (Other than it must be by virtue of being in $\Delta$.) ¶ I second @user10354138's comment. Are you sure you have a firm command of category theory and topos theory?

Brian Tung

Oct 22, 2023 16:02

Yes, that's right. What are you asking then? Are you having trouble understanding the proof?

Brian Tung

Oct 22, 2023 16:02

Yes, and...so? That proof just points out that $\Delta$ can't only contain $0$, because then there isn't a unique $b$. What does this have to do with finding the infinitesimals? I guess I don't understand exactly what you're asking.

Brian Tung

Oct 22, 2023 16:02

Wha-ah? It basically says all such mappings $\Delta \to R$ are linear. It doesn't prove that $\Delta$ is non-empty.