Mathematics - 2017-06-29 (page 1 of 9)

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Akiva Weinberger

12:07 AM

OK, I found a counterexample to my question, but it's huge.

$(39,52,0)$, $(20,-15,60)$, and $(48,-36,-25)$

Hi, DogAteMy, Semiclassic.

Akiva Weinberger

Hello!

After a 24-hour day of travel, I'm back. I guess you're used to that with your Israeli pilgrimages.

hi

heya MikeM

12:09 AM

I locked my friend's keys inside his house.

Akiva Weinberger

I haven't mentioned this on chat yet, but I've actually been vacationing in Argentina since Sunday!

From the house to the hotel was 22 hours...

oooh, exciting, DogAteMy.

That was a talented move, MikeM.

You have an issue with keys.

Akiva Weinberger

We went to the Iguazu Waterfalls in the north of the country, which meant an extra flight from Buenos Aires to there.

Now we're going back down to Buenos Aires to visit it for real

Well, say hi to mr @Pedro whilst you're there.

I have issues, broadly.

12:11 AM

I would not presume to go that far, Mike.

Waiting for a locksmith.

Well, perhaps Godot will come by in the meantime.

I'm still punchy after my 24-hour travel day and little sleep. I'm not your age any more :P

Isn't punchy, like, overwhelmingly chipper?

I would interpret that as being an aggressive morning person.

punchy generally means not making a whole lot of sense.

Akiva Weinberger

"Don't" punch "criticize" punch "my" punch "word choices!" punch

12:20 AM

sends DogAteMy to bed without any supper

12:53 AM

-1

Q: Is there a multivariate integer function f(x,y) that returns the number of factors of y in x with a closed form?

I'm looking for a simple function in terms of the elementary and exponential functions that when passed an integer x and integer y, it returns the number of times that y can be divided from x such that the result is an integer. I believe that it is something along the lines of $\lfloor\log_y(x)\r...

number-theory elementary-number-theory functions logarithms

@TedShifrin any clue on how to answer this?

Akiva Weinberger

@ForeverMozart So the digits of the numeral "60"

Hey there everyone!

@Daminark yoo

How's it going?

Not bad

Had a semi productive day

You?

1:05 AM

It was pretty solid

I had a semi classical day. Get it? There's a pun.

@Zee I had a massively productive day. Wrote 4000 words of documentation.

1:30 AM

a tree is a connected graph with no cycles. How many non-isomorphic tree with 5 vertices exist

@Typhon lol

@Simple Google says: graphclasses.org/smallgraphs.html#nodes5

Not all of those are trees, but any of the trees will be in there.

There's also a question about "how many non-isomorphic trees on five vertices exist" on the main site.

2:10 AM

It's always better to tell the truth since lying makes you stupid

@Semiclassical thanks, this question is from the math subject GRE practice book. I have hard time to solve since I know none graph theory

mmkay

@Zee Current politics being an object lesson in that...

2:34 AM

mathb.in/147773

^ Anyone wanna check I think it's wrong so far

2:47 AM

still waiting for my day. quantum mechanical

@Semiclassical haha

@Avantgarde you see errors in mathb.in/147773

I think in that case it's the converse

you may have a point. hard to sort out cause vs. effect there.

@Zophikel what's that? I can't read it atm, I'm reading something else

2:53 AM

@Avantgarde just justifying some assumptions made on the following statement: $\triangle_{h}f(z) \rightarrow f'(z), \triangle_{h}g(z) \rightarrow g'(z).$

                   ^ I think i'm on the right track and I found and fixed some errors but I doubt myself :(

What does downvote mean

It means to vote down

But how does it affect you

your account at least

Get enough of those and your gonna have to make a new account

what how

my question got downvoted four times why

3:07 AM

I suppose couse you didn't show you put any effort into understanding the question

Idk man

I did

thats a weird system

I know but you probably wanna add some math, like maybe , I tried blah blah blah and it didn't work

true

I didn't know about that

Or maybe people here hate competitions... idk

I like them

3:09 AM

Ya, I like those people too

Are most here in college

i would say most people here know enough math at least at the college level

Does calculus bc count as college level math

It's HS/first year college

Or as we used to call community college grade 13

I think the problem people have with competition problems is that they share some common features with homework problems

for one, there's usually little in the way of context or motivation for the problems; for another, the tendency of the OP tends to be "I dunno how to do this, you tell me" which doesn't play well.

I think that can go too far, of course, but I can see why competition problems get short shrift on the main site.

3:31 AM

Plus I also think math in some ways is too noble to boil down to a competition, it's one of the few fields where the top don't treat the average as dirt

Unlike philosophy for example...

I don't like maths

Then why are you pursuing math?

jk

Does \lim_{n\to\infty}{1\over n}\sum_{k=1-n^2}^0e^{k/n}=\int_{-\infty}^0 e^xdx

Does $\lim_{n\to\infty}{1\over n}\sum_{k=1-n^2}^0e^{k/n}=\int_{-\infty}^0 e^xdx$

Guys any help?

4:18 AM

Sometimes when things get too quiet, I suspect am the only one in the world

Akiva Weinberger

Well it's $\lim_{N\to\infty}\int_{-N}^0e^x\operatorname d\!x$

and the finite integral can be made into a Riemann sum like normal

That might give you what you have, I don't know

@Zee Everyone online is a bot except you.

the joy of solipsism

(personally, solipsism creeps me the f* out. I'm already responsible for enough of my own bad decisions; being responsible for the entire world is a horrific thought.)

@AkivaWeinberger don't mess with my head, am a recovering solipsist

Akiva Weinberger

"I didn't know there were so many solipsists, I thought I was the only one"

4

lol

reminds me of an old but good joke.

"Did you hear about the dyslexic insomniac agnostic? He stayed up every night, wondering if there really was a dog."

4:23 AM

Kek

hahah

I'm the philosophy and infinity-categories bot btw

@Zee

No, you aren't infinite categories yet

We are both wanna be

Lol I mean, soon though, I'm sitting on algebraic topology lectures right now and the professor is p categorical

Ya but it takes time to get brain washed, I know you are there when I ask you about a real analysis question and you start talking about an n topos

4:26 AM

My understanding of category theory stuff is strictly at the level of slogans.

Am not sure it goes beyond that...

e.g. a category is a collection of objects and morphisms between them

a 2-category is (somehow) a category of categories?

and then up from there.

So, given two categories, you can talk about a functor going from one to the other

user84215

What would happen if the "infinity" did not exist in mathematics ?

I thought it was a catagory with a mapping between the mappings

4:27 AM

Which takes the objects of one to those of another, as well as the morphisms

Hmm, sounds plausible.

@aminliverpool don't bring philosophy here, how dare you

And functors have to respect composition (either in the same order, covariant, or in reverse, contravariant)

As well as identity

so a 2-category has the morphisms between morphisms?

I think so

4:28 AM

Basically, a 2-category is the class of categories and functors between them being the morphisms

Hmm.

I think the problem I have with this is that I always want to have some concrete example.

I suppose homotopy?

With set theory, a set of sets is easy enough.

So, an example of a functor is a forgetful functor

What kind of functor? /s

4:29 AM

O goes to 0 by a continoues mapping of mappings between the two

If you have a category of sets and some specialized morphisms, a forgetful functor loses structure of both

@zee hmm, fair.

user84215

It is not philosophy. My question is: which theories and theorems depend on the "infinity" concept ?

The easy example is THE forgetful functor, takes some category and drops everything to sets

so homotopy would be kinda like a 2-category?

4:30 AM

I hate easy examples...

And forgets about morphisms as, say homomorphisms or homeomorphisms or linear mappings or whatever, and just thinks of them as functions

Well, this example ends up being rather convenient since the forgetful functor is adjoint to the free functor

Say, you take a set and create the free abelian group generated by it

Oh here we go... the brain washing is on going

Brainwashing would require I understand it.

I meant him getting brain washed, I don't have much concern about you getting brain washed

I mean they're a useful way of looking at things

Sure they can seem like a lot of blah

4:33 AM

It's a bit like being a character in a Lovecraft story; you can't be driven mad by the Necronomicon if you can't actually read it.

But the fact that you are able to think about linear isomorphisms both in terms of how they operate on elements and in terms of, they're closed under composition and can thus be made into morphisms on the category of vector spaces (in contrast to something like, say, lower rank ones) is helpful

user84215

Please speak about infinity.

Peter May actually said you basically can't do anything real without some categorical language (or hiding it). And you can't do algebraic topology/geometry without legit category theory

@Daminark lol, prof may got his own agenda

No thanks.

4:35 AM

@aminliverpool the existence of infinite sets are pretty important

Like, you'd lose the real numbers and all of analysis without it

I presume there's an argument to be made for categorical thinking, especially in a modern context.

Also even as much as $\mathbb{Z}$

But I don't really care. Not my problem.

I don't think catagory theory is useless

Am glad it was invented and is being studied

But it is overrated...

My attitude: "I'm glad it was invented and is being studied...by people other than me."

4:37 AM

Lol

I mean there is some respect in which you can think about things categorically, and that can clean up a lot of exposition. So if you are inclined to think like that, it's nice for sure. If not, whatever

user84215

you can do mathematical analysis without assuming the existence of the set of whole real numbers.

@Zee Okay the whole "____ is overrated" shtick is getting pretty bs at this point

@Daminark alright alright I'll back off

out of curiousity, what's something you consider _under_rated?

4:39 AM

Like, that's presuming a value system that you can just drape over everyone, so either rigorously prove that you can do that and then all other value systems will be declared bs

it's all a matter of taste not rigor

Well you're trying to make your taste apply to people other than yourself

@Semiclassical I suppose something like PDE is underrated

Hey everyone

So like, that statement has just as much value as the polar opposite, either do infinity-topoi or you're just wasting your life

4:40 AM

I suspect the more involved question is: Underrated by whom?

Within the realm of applied math, for instance, PDE's are huuuge

@Daminark pure math

user84215

the answer to my question ?

To be fair I also know people who tend to just be like "Psh, analysis, who cares?"

Akiva Weinberger

All proofs have finite length.

@Daminark something like microlocal analysis is not as popular as Algebraic geomtry even though I think the former is better

Akiva Weinberger

4:41 AM

So, in that sense, math has no infinity.

And they're also talking nonsense, the viable claim is "I prefer ___ to ___"

Microlocal analysis is a phrase I feel like I should know.

"Better" ¯_(ツ)_/¯

¯_(ツ)_/¯

Shit

¯\_(ツ)_/¯

That's better

user84215

We do not need the axiom of infinity

My own personal standard: If it's related somehow to QFT, it's probably important but I probably won't understand it well.

4:43 AM

@Semiclassical Maybe to some Diff Geometer's I would think (shrugs)

@Daminark your views about math seems to me rather nihilistic

All value is subjective

I disagree

Fine, I'm gonna state the correct value system then

The point of math is to gather tools to do number theory

The point of any subject that uses math is to inspire that math

I actually agree to some extent

The point of algebra*

I would put in your statement

Akiva Weinberger

@aminliverpool If you replace the axiom of infinity by "no infinite set exists," you get something equivalent to PA.

Nah @Zee, the only reason analysis exists is to develop the zeta function

4:46 AM

@Daminark lolololol

user84215

Why ?

The point of "pure math", perhaps. The point of applied math is to do engineering :P

Akiva Weinberger

The equivalence goes through weird binary stuffs.

@Daminark yes who cares about all of science

Or geometry

Or engineering

The science is only there because it helps us to think of analysis, which then helps us learn number theory

4:48 AM

Or social science

Akiva Weinberger

Like, you define $f$ that maps the empty set to $0$, and maps the set $\{x_1,x_2,\dots,x_n\}$ to $2^{f(x_1)}+2^{f(x_2)}+\dotsb+2^{f(x_n)}$, IIRC

@Dami, I can't tell if you're being sarcastic or not

Akiva Weinberger

which is doable without infinite sets

Psh, the point of geometry is to eventually get up to algebraic geometry, which then helps us learn number theory.

user84215

Ignoring mathematical logic, Which theories and theorems in mathematics depend on the "infinity" concept ?

4:49 AM

What's the point of number theory?

@aminliverpool almost all

And the point of engineering is to get good enough infrastructure to learn more about number theory.

Akiva Weinberger

@aminliverpool Real analysis, for one

And the point of social science is....umm...hang on a bit.

Akiva Weinberger

Only a subset of topology is doable without infinitary objects

(I don't actually know enough social science at the academic level to make a value claim, but I couldn't resist.)

4:51 AM

@Zee Number theory is an end in itself

@Semiclassical just think of game theory

@Daminark so just a game?

user84215

We can deal with limit, differentiation, integration and so on without considering the "infinity" concept.

Sounds pretty nihilistic to me

@Zee No, it's the goal of human existence

Akiva Weinberger

@aminliverpool What about the real numbers in the first place?

4:52 AM

@Daminark yes, the goal of human exsistence is to play this game?

@Perturbative I'm developing with a value system which a priori is equally valid as Zee's analysis-centric one

@Zee Number theory is not a game

Akiva Weinberger

You can't even describe most of them in a finitary fashion (unlike the integers or rationals)

@Daminark how is it different than a game?

@Dami Ahhh I see

user84215

We do not need the existence of the set of whole real numbers.

4:53 AM

@Zee Channeling your inner Wittgenstein?

user84215

to do limit and ...

@Semiclassical hahaha

In the same sense as analysis is different from a game

Or let's say...

Akiva Weinberger

@aminliverpool Nearly every (discontinuous) real function is essentially an infinitary object

I mean your counter is probably that analysis is a means to an end of whatever

Akiva Weinberger

4:54 AM

You cannot do real analysis in PA.

@Daminark well now you want me to answer my own question which I will do but I wanna hear your response first

You know the triangle of needs, right?

Ya

So it turns out that the lowest one is actually number theory

You put number theory as self transcended?

4:55 AM

And the highest one

Omg

Brilliant.

Actually the whole triangle is secretly number theory

Akiva Weinberger

@aminliverpool In a sense, the "infinity" in ZFC isn't really an infinity anyway. The symbol $\Bbb N$ is just a symbol such that $n\in\Bbb N$ is true for any whole number $n$

Geometry maybe as well, since the object itself is a triangle

4:55 AM

I can't even process this...

Akiva Weinberger

There, I just described $\Bbb N$ without an infinity. Note that any single proof will only use finitely many $n$, since proofs have finite length.

@aminliverpool

user84215

You can define them in finitary way

The problem nowadays with analysis is that it's going in a speculative direction such that it's increasingly more difficult to make the results number theoretic

Akiva Weinberger

All of first order logic is kind of finitary. You're messing with finitely many symbols according to finitely many rules.

Arguments about infinity are done by finite humans and therefore can only be finitely interesting. They can, however, be arbitrarily close to being infinitely boring.

Akiva Weinberger

4:57 AM

ZFC is a first order logic system.

@Daminark you lose a point for this whole argument and you gain a point for the triangle thing

Infinity categories might also potentially be ends in themselves, we have to see

Akiva Weinberger

You can program a computer to check a proof written in the language of ZFC.

@Zee What I've been saying is clearly nonsense. As clearly as everything you've been saying about how PDEs are underrated and everything else is overrated

Akiva Weinberger

And any given proof will take up a finite amount of memory and processing time.

4:58 AM

@Daminark not everything else, am not even a PDEer

Differential geomtry is underrated too

Okay fine whatever, that's not my point

Geometric measure theory too

Functional analysis is rated fairly

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