Mathematics - 2018-03-21 (page 1 of 5)

0celo7

00:00

You just need to know that a bounded sequence has a convergent subsequence, aka Bolzano-Weierstrass.

If you don't have that basic result, you're kind of screwed.

user525966

Does this work if we use an actual example so I can see better how you're using this notation?

Cows

@0celo7 so can you share some wisdom on how to deal with this?

0celo7

@user525966 Consider $1-x^2$ on $[-1,1]$

user525966

okay

0celo7

@Cows No idea, I don't know how to integrate

Zee

00:01

Hey @TedShifrin

user525966

(max value will end up being at x=0)

0celo7

Yes

user525966

just for my own reference

Cows

@0celo7 is there a clever alternative to integration that you can share?

0celo7

Mathematica

user525966

00:02

so we want to show that the following must be false: $\forall d \in [-1,1], \exists x \in [-1,1]: f(x) > f(d)$

correct?

0celo7

Ugh, what?

Is there some reason why you're insisting on the set theory

Cows

@0celo7 don't laugh but I don't know how to do this integral in Mathematica. I tried and it hates either it or me lol. Can you try it, please? and if it works tell me the code you used

user525966

I defined maximum value as $\exists d \in [a,b] : \forall x \in [a,b], ~f(x) \leq f(d)$

Zee

I mean that’s a reasonable statement, I don’t see the set theory

Akiva Weinberger

@Cows What did you start with?

Cows

00:05

@AkivaWeinberger you mean in mathematica or for the project?

Joe Shmo

sup yall

0celo7

@user525966 Ok that's not wrong, but writing this out in terms of logical quantifiers isn't helping.

Cows

@AkivaWeinberger integrate[Sqrt[-3 - 12 t + 3 t^2 - 6 t^3 + 2 t^4], t]

Joe Shmo

does anyone have a good reference for an introductory topological dynamics text?

0celo7

Do you know Bolzano-Weierstrass/Heine-Borel or not?

Cows

00:06

@0celo7 integrate[Sqrt[-3 - 12 t + 3 t^2 - 6 t^3 + 2 t^4], t]

user525966

My concern here is that I don't want to prove something obvious with something else "obvious", I want it to be a natural outcome

i want to know why we need to do x, y, z, etc to get the proof

I do not

0celo7

Because it's a nontrivial result.

Cows

@AkivaWeinberger mathematica just gives me back what i typed in

0celo7

Just like the intermediate value theorem. There's no reason for it to be easy to prove.

Cows

@AkivaWeinberger mathematica just give me back what i gave it

0celo7

00:07

@Cows Mathematica commands start with capital letters, so what you wrote cannot be correct.

user525966

for example the concept of supremum almost feels like cheating because it has maximum practically built into it, so I don't see why this lets us prove something when it's basically the same as our original statement

anakhronizein

Anyone good with highest weight stuff with reps?

In particular, Lie algebras.

Joe Shmo

@user525966 what?

0celo7

@user525966 What?

Zee

All of math is about cheating , if you can get away with doing it , then do it , whatever it takes to make mathematical progress

Akiva Weinberger

00:09

@Cows For the project… What $\int\sqrt{2t^4-6t^3+3t^2-12t-3}dt$ the simplification of something? What led you to that integral?

Cows

@0celo7 what is this (-(1/4) + t/3) Sqrt[-3 - 12 t + 3 t^2 - 6 t^3 + 2 t^4] +
1/4 (-((42 (t -
Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 + 2 #1^4 &,
2])^2 (-EllipticF[
ArcSin[\[Sqrt](((t -
Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 + 2 #1^4 &,
1]) (Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 + 2 #1^4 &,
2] - Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 +
2 #1^4 &, 4]))/((t -
Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 + 2 #1^4 &,
2]) (Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 + 2 #1^4 &,
1] - Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 +
2 #1^4 &,
4])))], -(((Root[-3 - 12 #1 + 3 #1^2 - 6 #1^3 +

0celo7

Look, @Cows, I don't care. Sorry.

Cows

@0celo7 wow hehe

0celo7

You never gave a good reason for doing this integral.

Akiva Weinberger

Oh God @Cows

Well, the "Root" stuff refers to roots of the polynomial, I guess

Various "Elliptic" functions are special functions

user525966

00:10

Like when we start talking about "bounds" or "maxes" these notions practically get around the proof by asserting the max exists or something, to me that doesn't feel like a proof

Cows

@0celo7 I am trying to calculate the error from a cubic polynomial approximation of part of a circle. Please don't tell anyone lol

user525966

feels more like kicking the can down the road to some other concept that just takes something a true as an axiom somewhere

Akiva Weinberger

You're gonna want to start with a construction of the reals.

You can't prove anything about the reals without rigorously defining what the reals ARE.

Zee

Complete rigor is nonsense anyway , see Godel

Akiva Weinberger

There are two popular ways to do with: Dedekind cuts and equivalence classes of Cauchy sequences. (One can prove that the two approaches are equivalent.)

user525966

00:12

i know the peano axioms for natural numbers, I never got to reals yet

Zee

So you only need rigor that’s enough to make your bosses happy

Akiva Weinberger

@Zee Gödel just means we can't prove everything, but we still can prove a whole lot of stuff

Cows

@0celo7 I mentioned how it came up. So is there something better I could do to get this?

@0celo7 to get a much nicer error?

Zee

@AkivaWeinberger you can prove whatever you want with the right set of axioms so my statement holds

Akiva Weinberger

I can prove anything if you give me contradictory axioms

I don't see the relevance

Cows

00:14

@AkivaWeinberger yes i see. .

@AkivaWeinberger can I make Mathematica give me this result in latex form?

anakhronizein

@Zee I don't see how Godel means you cannot have complete rigour.

Zee

Ok Let’s try a little exercise

Cows

@0celo7 Can I make Mathematica output this as latex?

Akiva Weinberger

@Cows I don't know enough about Mathematica

Zee

Give me a trivial but rigoures math statement

Akiva Weinberger

00:16

Do you really need the indefinite integral? If you want a definite integral you could just get it to give you a decimal expansion

anakhronizein

See any statement in Principia Mathematica, @Zee

Akiva Weinberger

@Zee $\forall X,\exists Y:X\in Y$

anakhronizein

Or anything written in Coq.

Akiva Weinberger

$\forall x\in\Bbb R,\exists y:x<y$

Zee

Can you write that in English , I still can’t get latex to work

Akiva Weinberger

00:17

First one: for every (set) X, there exists a Y such that X is in Y

Second one: for every number x, there exists a y such that x<y

Zee

Well that’s a bad example since those statements are already not well defined

Joe Shmo

@user525966 the reals are defined by the very axiom you are alluding to.

Akiva Weinberger

@JoeShmo Well, not really

anakhronizein

Why are they not well-defined?

Zee

What the heck is Y ?

Joe Shmo

00:19

@AkivaWeinberger well, what axiom, and how so? :-)

Zee

Dummy symbol ?

Akiva Weinberger

One way to define the reals is to say they are the unique ordered field with the least upper bound property (I think that's all the conditions), but we don't know that such a thing exists until we construct one

@Zee A set!

And then we construct one using Dedekind cuts or Cauchy sequences or what have you

Zee

what does it mean for x to be in Y? As a subset ? As an element ?

Akiva Weinberger

Element

Eric Silva

@Zee It has much more to do with social effects and the strong history and tradition of academic work on leftism

Zee

00:21

Ok , I still don’t know whats meant by a set but go ahead and prove this to me

Akiva Weinberger

∀X,∃Y:X∈Y

Joe Shmo

@AkivaWeinberger so we need the least upper bound property. What's the issue?

Cows

@0celo7 you know. . . you were somewhat rude dude. . . next time you act that way I might have to respond in kind. I was just asking what I was looking at. you could say I don't know rather than you don't care. Not that I care if you care, but . . .like I said kinda rude dude

Akiva Weinberger

@JoeShmo I don't think you'd define the reals by the least upper bound property, I think you'd define them by whatever construction you use. I guess the point doesn't really matter though

@Zee It follows from the axiom of pairing in ZFC

The axiom of pairing says that $\forall x\forall y\exists z(x\in z\land y\in z)$; take $x=y=X$ and $z=Y$

Obviously you're gonna have to reduce stuff to axioms, and you can't reduce the axioms to anything further. I just don't see the connection to Gödel.

Zee

Well what godel tells me , is that we should not lay axioms and fit math to them but the other way around

Leaky Nun

00:25

the long boi in relative homology is basically a snake?

@AkivaWeinberger

Zee

And complete rigor is complete nonsense as well , that is true by common sense not even logic

“Mathematical logic is the least rigorous of all the sciences”

That’s what Gromov said in a talk once

Joe Shmo

@AkivaWeinberger if i remember correctly from model theory, any real closed field is elementarily equivalent to the reals. I.e., the l.u.b. property defines the reals in a very real (no pun intended) sense.

Leaky Nun

@JoeShmo l.u.b. is not a first order property

Semiclassical

For my example of a rigorous statement I’d probably do something in graph theory

Cows

I am attempting to use TexForm so I can see what Mathematica is outputting in a way I can understand

Joe Shmo

00:29

@LeakyNun ohkay?

Cows

How do I make mathematica output in tex form so that I can run it in tex and see what is saying?

Semiclassical

Eg the graph K_3,2 is not planar

Zee

I’ll make it simpler 1+1=2

That’s a completely empirical fact

Leaky Nun

@JoeShmo the fact you quoted concerns first order theories

user525966

so what more am i needing exactly to prove extreme value theorem

apparently this subsequence concept

Semiclassical

00:30

Maybe that’s too complicated example on my part. Hmm

Cows

I don't use Mathematica often, can anyone tell me how to convert Mathematica speak to latex?

Zee

You may say , oh , we can define the natural integers using blah blah blah but to even talk about that you already assumed the concept of quantity

anakhronizein

@Zee math is non-empirical, so I have no idea what you are saying.

Zee

Lolol

What else is it ? You think angels come at night and whisper mathematical ideas into your ear ?

Semiclassical

“The complete graph on n nodes has n(n-1)/2 edges.”

Akiva Weinberger

00:31

@JoeShmo Yeah I guess you're right

Joe Shmo

@LeakyNun in so far that a real closed fields has the same first order proeprties as the reals.

Leaky Nun

right

Joe Shmo

so whats the point you were making?

anakhronizein

@Zee you think there is a physical, measurable, experimentally-observable phenomenon on earth that corresponds to every mathematical truth?

Semiclassical

Philosophically, this would be the place to talk about whether there’s such a thing as synthetic a priori knowledge

Akiva Weinberger

00:32

@Zee Arright fine, at the lowest levels maybe we can get a little nonrigorous, but the stuff connecting the lowest levels to the higher levels should be rigorous

Semiclassical

But ugh I just find that boring

Zee

no, that would make mathematics as rigoures as physics which isn’t the case since physics is more rigoures

Semiclassical

Lol

anakhronizein

At this point you have to be pulling my leg.

Cows

@Semiclassical how can I clean up mathematica output?

Leaky Nun

00:33

@BalarkaSen so the long boi in relative homology is basically a snake?

Akiva Weinberger

*rigorous @Zee

@LeakyNun Yeah

Cows

@Semiclassical I just want to be able to see it in human form, or latex or somethings

Leaky Nun

oh thanks

Zee

@AkivaWeinberger even then it’s not that simple , it’s just people being self honest and honest with each other, that’s as close to real rigor as you can get

Semiclassical

This is reaching Poe’s law territory, yeah

Akiva Weinberger

00:34

Isn't it called the snake lemma?

anakhronizein

long boi has to be Tor, I think

0celo7

@Cows ?

whatever

Leaky Nun

@AkivaWeinberger yes?

anakhronizein

Tor and Ext

Def.

Akiva Weinberger

"What are the fewest axioms required to prove all the stuff we already know are true" is a reasonable question to ask

The fewer the axioms the less likely there's a contradiction

user525966

00:36

looking into this compactness concept, "A topological space is compact if every open covering has a finite sub-covering."

anakhronizein

"all the stuff we already know are true" isn't clear, Akiva. Do you mean after already assuming a model?

Akiva Weinberger

Right, so the main concept there that you don't know yet is "open covering"

anakhronizein

Because that is called reverse mathematics.

Zee

The fewest axioms would be the set of all mathematical facts we “know” to be true today

Unfortunately they are probably inconsistent

anakhronizein

We don't know anything to be true in math.

It's all conditional.

Cows

00:37

@0celo7 I was hurt, man! I am just trying to figure out what I am looking at and you made me feel irrelevant. If I knew what to do I would not be asking, and i don't have people and faculty at my disposal. what I am saying is that your response was hurtful, but anyways I just want to be able to look at what Mathematica is saying in a way that I can follow. Can i have tried to use TexForm to convert to latex and it is not working

anakhronizein

"If we assume this logical system, then we can conclude these things"

Zee

That’s what “” are for

Semiclassical

For my part I don’t think mathematical logic is nonsense or lacking in rigor. I just largely find it boring

user525966

"An open covering of a space X is a collection {Ui} of open sets with unionoveri Ui = X and this has a finite sub-covering if a finite number of the Ui's can be chosen which still cover X."

anakhronizein

But that's the problem, @Zee, you don't have a single thing that you "know" to be true. Because in one system, things can be true, but than in another they can be false.

Akiva Weinberger

00:39

You're saying we don't even know that "If [if A then B] and [A] then [B]"? @Zee

anakhronizein

There is no mathematical statement that is universally true across all logical models.

Semiclassical

That said, the development of mathematical logic is strongly linked to the development of stuff like computer science

Zee

Yes Akiva

user525966

huh?

Akiva Weinberger

@user525966 What huh

Semiclassical

00:40

So I can respect logic as a discipline for that alone

user525966

this open covering concept

they seem to involve open sets again

which i still don't know if I understand yet

Cows

@0celo7 i am not calculating anything exotic. I am just trying to calculate the error from a simple approximation a high school kid can do, in case you are thinking I am (pretending) to do some exotic string physics

user525966

what is a closed set for example

Zee

Logic never gave us anything , intuition and imagination gave us the best of math

user525966

e.g. S = {1, 5, 8} ?

orbit-stabilizer

00:40

@user525966 Just start with Rudin Chapter 1 and go to Chapter 8.

user525966

is it easy to read or is it going to be full of undefined notation and jargon you'd only understand by going through a bunch of classes

orbit-stabilizer

notendur.hi.is/vae11/Þekking/…

have a look

I'm currently on chapter 8

Akiva Weinberger

The paragraph in the Preface beginning with "Experience" is interesting @orbit-stabilizer

anakhronizein

@Zee your lack of knowledge concerning the field of logic is showing.

user525966

nothing wrong with logic

orbit-stabilizer

00:44

@Akiva, I sort of appreciate the construction of the reals, but I don't care for it. I'm not interested in going that deep into foundational stuff.

Semiclassical

One thing I always find interesting to note about logic: Godel was a Platonist

Zee

By logic , I don’t really mean the field of mathematical logic , I mean logic as the mental faculty

orbit-stabilizer

There exists a unique ordered field with the lub property up to an isomorphism is enough for me.

Zee

Even logicians don’t use logic to discover new truths

orbit-stabilizer

Woah, what tf is wrong with logic as a mental faculty

Define, "new truths"

Semiclassical

00:45

Though I usually find discussions about argumentation and reasoning to be tiring as well

user525966

@AkivaWeinberger Are S = {1, 5, 8} and B = [1, 6] examples of closed sets?

Zee

We had a whole argument about this , am too tired

user525966

One with finite cardinality, one with infinite cardinality?

Semiclassical

Because it’s ultimately an argument about argument

orbit-stabilizer

Are you saying modus ponens doesn't come from the logical faculty?

Semiclassical

00:46

Which by it’s very nature presupposes it’s own validity

orbit-stabilizer

it does

Zee

And that’s the essence of it all semi

0celo7

and this is why engineering > math

Semiclassical

An argument that would deny argument is not speakable

orbit-stabilizer

ehh

pdfs.semanticscholar.org/7b8d/…

user525966

00:47

what's modus ponens again? (p -> q) ^ (p) -> q?

Zee

Nor one that affirms it

orbit-stabilizer

@user yeah

oops, pinged all the users

Semiclassical

Not sure I agree there

Akiva Weinberger

@user525966 Yes

0celo7

why are you even debating logic

user525966

00:48

@AkivaWeinberger to my earlier question?

orbit-stabilizer

planes fly, computers work - life is good

Akiva Weinberger

@user525966 Click on the arrow next to the comment to see what I was replying to

Zee

Semi well, let’s just leave it there ;)

Semiclassical

I think you can have arguments in support of reason. But there’s no external authority you can point to in order to settle it

0celo7

@Semiclassical Unpopular opinion: inductive logic in science is illogical

Semiclassical

00:49

Metaphysics is exhausting

orbit-stabilizer

Philosophy is exhausting

Zee

I am exhausted

0celo7

Expecting the sun to rise might be insane

There's no reason for it to rise

None that we know of at least

orbit-stabilizer

That's why it's best to be Bayesian

Zee

That’s the argument of Hume about inductive knowledge

Cows

00:50

Can someone help me reason this

Semiclassical

@0celo7 depends on what one means by logic

user525966

because the logic involved in assessing that truth statement requires a lot more behind it than just saying "the sun will rise"

anakhronizein

Metaphysics is the worst nonsense there is.

Cows

I think Mathematica is blindly doing some computations here

user525966

you need a lot of ifs/premises to make that claim

Semiclassical

00:51

If one is hoping to acquire truth as a mirror of the world, sure

Cows

if I were to solve this by had, I would factor the quartic

Zee

@AkivaWeinberger look at the mess we made

orbit-stabilizer

not necessarily

Cows

and do partial fractions

orbit-stabilizer

@Cows ! This isn't a chat room about math!

Cows

00:51

this would enable me to have a cleaner solution

orbit-stabilizer

We're discussing philosophy!

:P

Semiclassical

If one is just hoping to make progress on interesting problems, then inductive logic is s perfectly sound tactic

0celo7

@Semiclassical The probability that there's a man who can flip a switch and prevent the sun from coming up is the same as the sun just coming up again

Zee

Only tactic

0celo7

indeed, both are 0

Cows

00:52

Someone please help me, this thing is bothering me

0celo7

because the number of possibilities are infinite

therefore, the sun will not rise

?

Zee

Assuming an infinite universe

0celo7

I think I'm doing this wrong

anakhronizein

Assuming a universe.

Cows

Can someone look at this integral for just even 10 minutes

orbit-stabilizer

00:53

/r/badphilosophy

Cows

Mathematica is not helping

Semiclassical

"Gentlemen. You can't fight in here. This is the War Room!"

►

Cows

I have an answer, and it means nothing

anakhronizein

Integrate over the differential form, Cows

INTEGRATE IT GOOD

Cows

@anakhronizein please explain?

Zee

00:54

Lol war room

user525966

why are closed sets closed under both finite and infinitely many intersections, but only closed under finitely many unions

Cows

Are there any integral transforms that might help?

anakhronizein

I don't even know what you are asking about, Cows.

Zee

Yes , try the Fourier Mukai transform

0celo7

@user525966 because one can prove the first, and give a counterexample for the second.

Cows

00:55

@0celo7 I know you can do this integral please help me

0celo7

I can't do the integral, too stupid

Zee

This ain’t the 40s we aren’t your computers , use the one in front of you

Cows

@zee let me look up the transforms

@zee I have worked on this for a few days

the last thing I attempted to do was factor the quartic using some number theory tricks, but i did not succeed

user525966

what's the counterexample? if I have a closed set {1} and i union it with closed set {2} then {3} then {4}... to infinity, does that work?

Zee

Look up some of the Soviet books filled with formulas for integrals

user525966

00:57

that created set {1, ..., infinity}

Cows

@Zee you mean this? en.wikipedia.org/wiki/Fourier%E2%80%93Mukai_transform

@Zee I am not sure how to implement this by hand or computer

Zee

That was a joke

Cows

@zee oh my god lolz

dude I am desperate man

0celo7

exercise for the reader.

Zee

That’s my advice , find one of em old books filled only with integral formulas and look up a similar one to yours

Transcript for

$Mathematics$ Mathematics