Mathematics - 2023-08-02

冥王 Hades

00:39

@leslietownes gonna tell this to my dad

@TedShifrin it’s unfortunately “growing” really quickly. My sore throat has turned into a runny nose and headache

@robjohn I actually found that using vectors here was extremely effective especially in order to prove that second statement

haven’t really found a purely trig solution yet

Jakobian

01:14

@冥王Hades that's bacteria eating away at your throat. Just wait till they get to your brain.

one potato two potato

07:50

Now I finally spare some time to study GGT

The Empty String Photographer

08:38

Quiz: which mathematics operator/s have the lowest precedence?

user223626865

Additions and subtractions in order from left to right.

The far right being the lowest.

The Empty String Photographer

@user223626865 wrong!

user223626865

>8(

The Empty String Photographer

@user223626865 Do you want to know the correct answer?

user223626865

Ok

The Empty String Photographer

08:48

@user223626865 Equality operators! $=<>\not=$ etc.

user223626865

Define: operator

The Empty String Photographer

@user223626865 something that takes at least 1 number and outputs another number (0 or 1 in the case of equality operators).

Jakobian

09:22

@TheEmptyStringPhotographer that's a relation

冥王 Hades

10:19

@Jakobian am I about to get dumber?

Jakobian

11:00

Of course. That's a cold and common flu symptome

Peter

11:41

Does anyone know where one can download the Carmichael-numbers upto $10^{21}$ ? There are "only" about $20$ million ones , so this does not need much space.

Xander Henderson

@冥王Hades Honestly, when I am coming down with a cold, the very first symptom that I notice is that I get dumber. I get really bad at basic arithmetic, for example. My teaching gets quite loopy.

sunny

11:56

When someone uses a derivative rule, e.g. the product rule, the quotient rule, etc., they are assuming the point of differentiability is an interior point of the domain, right? In other words, these rules don't hold at boundary points of the domain, right?

I know there is the concept of left-hand and right-hand derivatives, but are there corresponding rules for these derivatives?

Xander Henderson

@sunny Do you know how to prove any of those differentiation rules?

Have you tried running the proofs with one sided limits?

Does anything change?

sunny

12:11

Hmm, yeah, here's a proof of the product rule: $$\begin{align} (fg)'(x) &= \lim_{h \rightarrow 0}\frac{f(x+h)g(x+h) - f(x)g(x)}{h} \\&= \lim_{h \rightarrow 0}\frac{f(x+h)g(x+h) - f(x+h)g(x) + f(x+h)g(x) - f(x)g(x)}{h} \\ &= \lim_{h \rightarrow 0}\left(\frac{f(x+h)(g(x+h)-g(x))}{h} + \frac{g(x)(f(x+h)-f(x))}{h}\right) \\ &= \lim_{h \rightarrow 0}f(x+h)\frac{g(x+h)-g(x)}{h} + \lim_{h \rightarrow 0}g(x)\frac{f(x+h)-f(x)}{h} \tag{*} \\ &= f(x)g'(x) + g(x)f'(x). \end{align}$$

The only thing that would change is the evaluation of the limits in $(*)$.

Xander Henderson

@sunny Does it change?

sunny

@XanderHenderson Well, I think it does. First of all, $g'(x)$ and $f'(x)$ would change to $g'(x^+)$ and $f'(x^+)$ respectively, where $h(x^+)$ is the right-hand derivative of $h$ at $x$. I'm unsure how $f(x)$ and $g(x)$ would change. In the proof, we are using the continuity of $f$ at $x$ to evaluate $\lim_{h \rightarrow 0}f(x+h)$, however, now $x$ is a boundary point, so I'm unsure if "right-hand continuity" implies "right-hand differentiability".

I guess $g(x)$ would stay the same, since it doesn't depend on $h$.

Xander Henderson

@sunny Continuity does not imply differentiability (though the converse holds). However, left- (right-)differentiability should imply left- (right-)continuity, no?

sunny

Oh yeah, I mixed it up.

I guess we would get $f(x^+)$ instead of $f(x)$, where $f(x^+)$ is the right-hand limit of $f$ at $x$. So to summarize, instead of $f(x)g'(x) + g(x)f'(x)$ we would have $f(x^+)g'(x^+) + g(x)f'(x^+)$, which looks considerably different.

Seems like this result is of little use. For instance, if we'd be interested in finding extreme values, then we'd just check the derivative at the interior points, and then evaluate the function at the boundary points to compare with the critical points.

In that context, computing the derivative at a boundary point seems like it gives little information.

冥王 Hades

12:33

@XanderHenderson Huh, serves as a perfect handicap for an upcoming math contest

Xander Henderson

@sunny Depends on what you are interested in.

冥王 Hades

There’s a professor here who holds a record for cracking a large number of walnuts with his elbows.

The guy has elbows made of titanium

Joe

13:15

@sunny: While differentiability of a function $f$ at $x$ can be defined whenever $x$ is in the domain of $f$ and $x$ is a limit point of the domain of $f$, I don't think this is a particularly useful notion. The most important theorems of calculus, such as the mean value theorem, require $f$ to be differentiable on an open interval $(a,b)$. (Note that each point of this interval is an interior point.)

@sunny: If you define the derivative for functions $f:\mathbb Q\to\mathbb R$, for instance, then a function could have derivative zero everywhere without being constant!

I don't want to claim that differentiability in the looser sense is a useless notion, but its properties are much more pathological.

sunny

@Joe thank you for sharing some insight. Why would this be the case?

Joe

@sunny: Let $f(x)$ equal $0$ whenever $x^2<2$ and equal $1$ whenever $x^2>2$. Then, you can check that $f$ has derivative zero on its domain.

@sunny: For functions $f:\mathbb R\to\mathbb R$, you can use the mean value theorem to show that if the derivative is zero everywhere then $f$ is constant. However, there is no corresponding "mean value theorem" for functions $f:\mathbb Q\to\mathbb R$.

@sunny: Going even deeper, this is because the central theorems of calculus require the completness of the real numbers. Without completeness, things go completely wild...

See this thread for more details. I have posted an answer before, but the other answers are actually probably more comprehensive

sunny

13:31

Interesting, thanks.

Joe

@sunny: Sorry, the domain of $f$ in my above comment is intended to be $\mathbb Q$, if that was not clear.

@sunny: Also of interest is this thread about "gaps" or "holes" in the rational number systems. Intuitively speaking, $\mathbb Q$ has "holes" at points like $\sqrt2$, and $\mathbb R$ "fills" those holes

sunny

cool

Joe

@sunny: My favourite mathematics book is Michael Spivak's Calculus. I have never read a book with better exposition. It begins by stating the axioms of the real numbers, but initially it omits the "completeness" axiom. Then, when he is about to prove his first "big" theorem about $\mathbb R$, he shows why completeness is the missing piece in the jigsaw, and explains how the theorem cannot be proven without first postulating the completeness of the real numbers.

sunny

@Joe cool, do you have the 4th version? :)

Joe

At the end of the book, he constructs the real numbers from $\mathbb Q$, thus showing that there is a number system (more precisely, a field) that satisfies the axioms that $\mathbb R$ is postulated to have.

I do have the fourth version, yes.

@sunny

user 85795

13:47

How about Hardy's textbook :^)

sunny

@Joe I also have a copy of the book. I can not pass any judgement on the book yet, since I have read too little of it. However, one comment I have is that in the definition of continuity, he defines it as $\lim_{x\to c}f(x)=f(c)$. However, this only holds for limit points of the domain of $f$. This is just one example where I wish he had been more careful or worded it differently. For example, in Rudin's PMA this is a separate theorem.

Joe

@sunny: This is one drawback of Spivak's book, but I don't think it is major. Spivak only defines continuity and differentiability at interior points, but he is not explicit about it. Your objection is correct. However, I will note that again, the most important theorems of calculus assume that $f$ is continuous on an interval (open or closed). So we are only considering continuity at points when $x$ is an interior point of the domain.

(Ok, to say that $f$ is continuous on $[a,b]$ means that $f$ is continuous on $(a,b)$, and $f$ is right continuous at $a$, and left continuous at $b$. So technically, it is not just interior points we care about.)

But we are certainly not usually interested in continuity for functions $\mathbb Q\to\mathbb Q$, except when we want to demonstrate just how pathological notion it is!

@sunny: I think it is sensible to consider "right-hand" continuity at $x$, provided that $f$ is defined on an interval $[x,x+\delta)$ for some $\delta>0$. But to define it any more generally is not so useful.

@user85795: I have never read Hardy's textbook, but I have heard good things about it. Personally, I would delay the construction of the real numbers to the end of an analysis introduction, since we are much more interested in their axiomatic properties rather than any specific set-theoretic construction of them. But it is a matter of taste, of course.

user 85795

Good point.

Joe

14:05

@user85795: Also, I think it is absurd to posit that a real number really is a Dedekind cut, but I suppose this is a philosophical view rather than a mathematical view. I think it is better to think of real numbers as just a primitive notion (they are just numbers!), similar to how we think of sets as being primitive. The set-theoretic construction of the real numbers, which demonstrates more or less that the axioms of $\mathbb R$ do not lead to a contradiction...

give me the luxury of being able to hold this view.

user 85795

I will grant thee this luxury on the pedagogical grounds of noncontradiction :-)

Joe

@user85795: Much appreciated! Anyway, there is lot of literature on philosophical psoition – known as Structuralism – which essentially holds that it is misguided to ask what something like $\mathbb R$ is. Instead, we should ask what properties characterise it up to isomorphism.

Shaun

14:43

Hi :) Maybe someone here might be interested in helping with this:

4

Q: How to install the mathematical GAP-software on an Android phone?

This was posted on MSE originally. Please note that my only training in programming of any kind is in the context of mathematics, specifically group theory. Thus, please use minimal technical language. The Problem: I'm trying to get GAP on my Android phone. (I'm using a ZTE Blade V8. I doubt th...

installation

I've never been able to get GAP running on my phone.

I asked the question back in 2018, when I was using GAP a lot. I use GAP a lot again, so it would be nice to have a portable version.

I know there's this:

sagecell.sagemath.org

But that requires an internet connection and only does small calculations.

user 858770

15:59

@GratefulDisciple the quote has many implications for mindfulness meditation

in English Language & Usage: Multi-Layered Discourse Room, 7 hours ago, by user223626865

“All of us want to be completely alive, to live one hundred percent in the present moment. What prevents us? More urgently, how can we bring about such a state of mind?
The great American psychologist [William James] gives us a clue in a quotation I found in a most unexpected place, Vogue magazine. This is a direct quotation: ‘The faculty of voluntarily bringing back a wandering attention, over and over again, is the very root of judgment, character, and will. An education which should include this faculty would be the education par excellence.’

Btw, thanks for your link :-)

Jakobian

16:37

@Joe strongly disagree

that's something that a category theory junkie would say

user 726941

Please don't call people "junkies."

Jakobian

> humorous: A junkie is also a person who enjoys or is interested in a particular activity to an extreme

why would I?

user 726941

A junkie is also a person addicted to a particular drug to an extreme.

Do you have any real experience with such a person?

Jakobian

Anyway what I think it's wrong to ask what properties characterize $\mathbb{R}$ up to isomorphism. Instead we should ask what we can also think of as $\mathbb{R}$. But this is deeper than just isomorphism. Category theory restricts us to think in only one category at a time, but we can think of $\mathbb{R}$ as existig in various multiple categories at the same time

The general idea of structuralism, I think I agree with though

I wouldn't say it's misguided to say what $\mathbb{R}$ is though, instead, those properties define in part what it is for us

The reason why we call something a table is because it's something resembling a table, can be used as a table.
What it is, is defined by the properties that we attach to it

I don't think I said anything deep however. It seems though as the story writes itself with this one

Jakobian

17:09

I think sometimes students ask questions about what something is that I feel like are devoid of any meaning, though

@user726941 I think that's irrelevant because in the context it was clear I wasn't talking about drug addicts

if I made you relive some bad memories, well I'm sad about that, but it's not my fault if that happened

I wasn't using language in offensive way

Joe

@Jakobian: Okay, but what is your answer to "what is $\mathbb R$" then?

Jakobian

real numbers

Ted Shifrin

Hard to imagine what those are.

Joe

@Jakobian: Doesn't that just invite the question – what are real numbers?

Jakobian

Unique (up to field isomorphism) archimedean real-closed field

Ted Shifrin

17:20

Abstract balderdash.

Joe

@Jakobian: If you only specify what they are up to isomorphism, then aren't you agreeing with me that we should be care about their properties rather than any specific model?

Jakobian

@Jakobian as explained here

s.harp

wait

Jakobian

I'm specifying what $\mathbb{R}$ is, but that doesn't mean I can't think of something else as $\mathbb{R}$ in other context as well

s.harp

how did you ping your own message? there is no reply option for me on my own messages

Jakobian

17:24

go to permalink

then write : and then the number of message

what I want to say, that I don't like the word isomorphism here

Ted Shifrin

Important skill for narcissists.

Jakobian

because it implies that I already am in some sort of setting

but as a mathematician I want to think about objects fluidly

I want to be able to think of the topological space $\mathbb{R}$ and the totally ordered field $\mathbb{R}$ and not distinguish between them

still thinking of them as the same object

Joe

@Jakobian: If you want to think of $\mathbb R$ fluidly, where it denotes any complete ordered field, then surely you can't make a definite claim about what $\mathbb R$ is, beyond: it is a complete ordered field. And if that is the case, then you have specified its strucutral properties, and what it is up to isomorphism, but not what it is in the set-theoretic sense

Jakobian

I disagree with you. The reason is that you're still trying to be formal about it and put it in a setting

well, of course I am talking about one object it just isn't really relevant what it is

Joe

@Jakobian: Well, I'd say the informal view is: the real numbers are just numbers, and they have the properties of a complete ordered field. And that informal description definitely doesn't specify what $\mathbb R$ is beyond saying what it is up to isomorphism. So I think even the informal view doesn't say anything about what $\mathbb R$ is as a set beyond saying it is a collection of numbers. If you want to think of numbers as being unique, then I suppose that does specify $\mathbb R$ exactly

Jakobian

17:37

But you're thinking of $\mathbb{R}$ as a complete ordered field. What if I wanted to think of it as a topological space? I wouldn't think of it as a distinct object

$\mathbb{R}$ is not just a complete ordered field, and if we were to define it this way, it wouldn't exhaust what it is at all

Joe

@Jakobian: In my view, depending on what operations $\mathbb R$ is equipped with, it definitely has different properties, and so it should be thought of as different. For instance, as a group, $\mathbb R$ is isomorphic to $\mathbb C$, but not as a field. So I think even in everyday practice it is best to distinguish the different cases

Jakobian

I just think that $\mathbb{R}$ as an object is something that isomorphism cannot capture, and only us thinking about it can capture it

because it cannot be fully formalized, only formalized when in a particular setting

i.e. particular category

so it's not quite like a mathematical definition, but more of a definition akin to linguistics

I hope you understand what I mean, since it's hard to capture this thought precisely for me

Joe

@Jakobian: Yes, I do. I'm sorry but I need to go now, but I can talk in about half an hour

Jakobian

Alright. See you

冥王 Hades

19:20

Man this cold really has put me completely out of commission

I got too cocky and thought a cold can’t do anything to me, and decided to register for the math contest anyway

Ted Shifrin

19:50

I hate to say it, Hades, but that sounds more Covid-ish. Take care of yourself.

冥王 Hades

20:13

@TedShifrin you’re not wrong, my doctor told me it could potentially be COVID and to get tested immediately. I’m gonna get tested in a few hours

PM 2Ring

67

Q: Moderation strike: Results of negotiations

We have reached the following conclusions during negotiations between community-selected strike representatives and representatives of Stack Exchange, Inc. This aims to address most of the concerns detailed in the strike letter, the initial strike announcement, and the conditions outlined in the ...

discussion community company moderation-strike

Ted Shifrin

20:33

@PM2Ring That’s encouraging!

冥王 Hades

20:44

This does put a smile on my face

PM 2Ring

I'm pleasantly surprised. IMHO, it's one of the most positive signs that we've seen from Stack Exchange Inc, in years.

OTOH, it's a shame that we had to suffer through almost 2 months of strike to get here...

Ted Shifrin

Yes, and I only accused one live purported mathematician of writing a ChatGPT answer. His feelings were hurt and he reported my comment as rude and …

PM 2Ring

21:13

Oops. ;) It's going to get trickier, as ChatGPT improves, and as people start sounding more like ChatGPT due to reading too much of its output...

leslie townes

4

love this new feature

PM 2Ring

21:26

Leslie is being childish. Again. ;)

leslie townes

PM: goose honk

PM 2Ring

I wonder if Ted is distantly related to Lalo Schifrin...

monoidaltransform

21:36

If $X$ is compact metric space, and $U$ is an open set; and $x\in U$, is $\sup{ r>0 : B(x,r)\subseteq U\}$ actually maximum?

Ted Shifrin

@PM2Ring Nope. But composer Seymour was my dad.

@PM2Ring Still, you mean?

@monoidaltransform Huh?

monoidaltransform

not maximum I mean finite

Ted Shifrin

Of course it’s finite.

What’s an a priori upper bound?

monoidaltransform

diameter of X

Ted Shifrin

Right.

Jakobian

21:44

@monoidaltransform you forgot a slash in the left bracket

冥王 Hades

23:30

It looks like I have COVID after all @TedShifrin

How can I, the God of Underworld, succumb to such an illness

Ted Shifrin

23:59

Ted is not surprised.

Transcript for

$Mathematics$ Mathematics