This is the Realm of Simply Beautiful Art

dydxx

2:27 AM

Is this your discord? discordapp.com/channels/@me/264194368976453634

Simply Beautiful Art

@dydxx Pretty much?

I have to go to bed now, but uh... see you in the morning

2

Q: Find $ \lim_{n \rightarrow \infty} \left((n-n^2) \int^{\frac{\pi}{4}}_0 (\cos(x)-\sin(x))^n \right)$

Find $$ \lim_{n \rightarrow \infty} \left((n-n^2) \int^{\frac{\pi}{4}}_0 (\cos(x)-\sin(x))^n \right)$$ I've managed to prove that the limit is in $(0,1]$ and I believe it is $1$ but I don't know how to prove it. Could you help me?

limits

Just so I don't forget to answer this if noone has tomorrow morning

S.C.B.

3:38 AM

@SimplyBeautifulArt Good night.

To you.

user21820

7:34 AM

@Wojowu As you said, you need replacement here. That's precisely why I said it is circular to claim that one can justify replacement via the iterative conception (of the cumulative hierarchy). When asking for an ontological justification of ZF, one cannot appeal to ZF; see this post, including the part cited from Boolos.

user21820

7:44 AM

Note that I know (and have even described here that in ZF transfinite induction can be performed along any well-ordering.

But in my linked post I think I've explained very clearly that the commonly given argument for Replacement based on the iterative conception is circular. The key point to note is that once you've built the first few stages (2 or 3 will do), you do have the set S of all possible equivalence classes of well-orderings of N, and you can very well prove that S itself is well-ordered under embedding. But you still cannot obtain uncountably many stages without Replacement.

@S.C.B. So did you manage to solve it in your head?

Wojowu

@user21820 Ah, I see, I didn't quite realize that part of the conversation was still in regard of justifying ZFC axioms. My bad!

user21820

No problem.

By the way, I asked Deedlit but he's not 100% sure, do you know whether there is a theorem of ACA that cannot be proven by Π[1,1]-CA0?

In other words, does adding the full induction schema to ACA0 give stuff that adding impredicative comprehension can't give?

Wojowu

8:03 AM

Well, I'm pretty certain there are instances of induction which ACA has as an axiom (since it has them all) but not provable in Pi11CA

*CA0

But I don't know any particular example

S.C.B.

8:50 AM

@user21820 Sorry, what was the problem?

user21820

The link doesn't work? (chat.stackexchange.com/transcript/message/35555265#35555265).

S.C.B.

No, I didn't know there was a link.

So, I didn't try anything yet.

But I'm thinking it is strong induction.

So assume that it is true for $n=1,2,3,\dots,m$.

If $m+1$ is a Fibonacci Number, than it is obviously true.

for $m+1$.

user21820

@user21820 Ah try clicking the little arrow to the left of this comment and following the little arrow again. =)

Well yes that's the easy case.

S.C.B.

Assume it isn't.

Hmm.

If $m=6$, $6=5+1$.

$7=5+2$.

$9=5+3+1$.

Okay, let's get a bit further than that.

$56=55+1$.

user21820

By the way, strong induction is easiest in the form where you only need to prove: "For every natural n, if P(k) for every k < n, then P(n)."

I don't know why many textbooks teach the version you cited, but it's in most cases much more troublesome.

Well "much" is exaggeration but the extra "+1" does not let that version generalize nicely.

S.C.B.

9:02 AM

@user21820 Well, I never found anything regarding that, and a don't use the notation $P(k)$ for statements so...

Anyway, back to the problem.

Assume that $m=F_{t}+k$.

Where $F_{t}$ is the closest Fibonacci Number to $m$.

user21820

Sorry I need to go.

Back in an hour or two.

See you!

S.C.B.

Now note that $k$ is smaller than $F_{t-1}$.

The shortest way to represent $k$ has been given.

But what if there was a shorter method?

user21820

9:32 AM

@S.C.B. Well, what's your answer, and what does it imply for the original problem?

Simply Beautiful Art

12:40 PM

@user21820 lol you are copying my words

user21820

@SimplyBeautifulArt: Uh sorry I don't get you.. copying what words?

I never copied anyone recently...

Simply Beautiful Art

12:56 PM

@user21820 I basically said your comment to me here in the chat yesterday

user21820

Ah I see. I didn't realize you had said that too.

Oh well.. What I don't like is when math is portrayed as an esoteric subject that is like a black box.

And one-line answers with not even a hint fit that description like a hand in a glove.

So much so that I'm actually in favour of converting all those answers to comments, but a Meta discussion in the past of this very possibility did not have the community backing..

That is what I don't quite understand, since nobody has given any reason that is still valid today to keep those posts as answers instead of comments.

Simply Beautiful Art

Mhm...

Perhaps due to the fact that it would have too large an affect

user21820

That was the only reason given in one Meta post, but I suspect that was while Cleo was still on Math SE.

Now that Cleo has left, there is no large effect on any Math SE user.

The high votes give the impression that Math SE welcomes such posts, in much the same manner as the batman question which could well have been a troll post or at best a question by someone too lazy to even try it on a graphic calculator.

Simply Beautiful Art

That is not true. This is like deleting Cleo's account. The problem is that there are too many who know Cleo, and the removal of all her answers will affect MSE.

user21820

I never said complete removal. Just convert them to comments.

Simply Beautiful Art

1:08 PM

That still moves her stuff around

user21820

So what? No information is lost.

Simply Beautiful Art

And a general rule of thumb is to avoid changing old stuff on MSE when possible

user21820

No that's not a general rule of thumb; you're probably inventing it on the spot.

I've been around since nearly the beginning of Math SE...

And in any case there is no averse effect to Math SE by moving his/her posts into comments.

Simply Beautiful Art

I do believe it is something mods and MSE in general follows

user21820

It only affects his/her fan club outside Math SE..

Simply Beautiful Art

1:10 PM

Lol

Fan club

Hahaha

XD

user21820

It's true and well-known by those who have been following the controversy.

Simply Beautiful Art

Well, whatever, I doubt we could do anything

user21820

We can't, but the moderators can.

Simply Beautiful Art

Well, given the historical context, it appears no one thinks it important enough to do anything about

user21820

Now that you bring it up again I wonder whether to post on Meta asking why we're not doing it since Cleo has left.

I can understand why they wouldn't want to do anything while the user is still here.

Simply Beautiful Art

1:13 PM

Mhm...

Sounds like a better plan than arguing in chat XD

user21820

Certainly. I would do it when I have the time.

Simply Beautiful Art

:P well, I can't do it without a computer

user21820

@user21820 <- By the way you might like to try this Fibonacci problem I gave above.

It's a good induction exercise.

Simply Beautiful Art

Oh

Ok then

Btw, do you watch anime?

user21820

I hardly watch anything at all.

Haha..

Simply Beautiful Art

1:19 PM

Haha

Okay

user21820

I sometimes watch some random documentaries or science videos I find online.

Like this one:

The Backwards Brain Bicycle - Smarter Every Day 133

►

Simply Beautiful Art

Lol, I'll look at it XD

user21820

That was a funny one; I wish I could try it myself, since I don't even know how to ride a normal bicycle.

Simply Beautiful Art

Lmao, you should though, at least I think so

user21820

I'm hoping when I meet a bicycle one day I can ride it immediately, because I've been analyzing it mentally since young. =D

Anyway I'm going to be off for a while. See you later!

Simply Beautiful Art

1:32 PM

Lol, cya man!

@dydxx do you need me? I'm here in the chat most of the time

Simply Beautiful Art

1:54 PM

@amWhy math.stackexchange.com/questions/2154616/…

:D down with bad questions!

S.C.B.

@user21820 I think I finished the proof without use of any paper.

Let us define $F$.

$F$ is a set of naturals $t$ such that the sum of $F_{t}$

is $n$.

Where $F_{t}$ denotes the $t$-th Fibonacci Number.

Oh, shoot.

I really wish I had paper to word this more smoothly.

Let me rephrase.

Let $G_{n}$ be the shortest representation of $n$ as a sum of Fibonacci Numbers.

Note that $G_{n}$ cannot contain two consecutive Fibonacci Numbers.

But we know every $n$ has a unique representation as the sum of non-consecutive Fibonacci numbers, by Lemma 1.

For example, $90=89+1$, or so on.

Proof of Lemma $(1)$.

This merely uses strong induction.

And the formulas for the sum of even fibonacci numbers and the odd fibonacci numbers.

I can only offer a outline here, but I know I proved it, as I proved Lemma $(1)$ before.

Damn and bother.

No one is listening to me, I think.

Simply Beautiful Art

2:10 PM

Hi @S.C.B.

S.C.B.

Hi.

Simply Beautiful Art

I just shot a question down

S.C.B.

Did you see my answer

1

A: Two interesting results in integration $\int_{0}^{a}f(a-x) \ \mathrm{d}x= \int_{0}^{a}f(x)\ \mathrm{d}x$ and differentiation of powers of functions

There are a lot of possible answers. For example, $$\int_{0}^{1} \frac{x^3}{3x^2-3x+1} \mathrm{d} x=\int_{0}^{1} \frac{x^3}{x^3+(1-x)^3} \mathrm{d} x=\frac{1}{2}$$ or $$\int_{0}^{1}\frac{x^5}{5x^4-10x^3+10x^2-5x+1}\mathrm{d}x=\frac{1}{2}$$ are both good examples of how this property can be used. ...

user21820

@S.C.B.: I was away, but I just read your solution. It's the 'standard' canonization type of solution, but writing everything out formally with no dependencies will lead to a rather long proof.

It's kind of instructive to directly apply induction to the original problem and observe that if you handle the cases just right you can get a much shorter proof.

Either way, well done providing a correct solution outline. =)

Simply Beautiful Art

Yeah, I read it

S.C.B.

2:24 PM

Can you guess why it recieved a downvote?

Simply Beautiful Art

No

S.C.B.

@user21820 Interesting.

Simply Beautiful Art

I'd rather not try to

@user417287 Hello and welcome to my realm!

well

I'm kinda bored

S.C.B.

$$\sum_{n=0}^{\infty} \frac{(2n-1)!!}{(2n)!!}$$

Simply Beautiful Art

Yes?

Oh

I know that with the fraction flipped off the top of my head

but not the other way around

A---B

2:32 PM

hi

Simply Beautiful Art

@A---B Hi and welcome to my realm!

Just thought you'd enjoy hanging out with us

A---B

No problem.

Simply Beautiful Art

So, its been a while since I last said this, but what kind of math do you do @A---B ?

For example, I enjoy tackling series, limits, and integral problems

A---B

@SimplyBeautifulArt I do every kind of math that is not assigned to me. First I will read a chapter from my textbook and then find a specific book on it. Then I will complete it till end.

Simply Beautiful Art

Well you certainly haven't learned everything

A---B

2:36 PM

As for my favorite, It has to be Calculus.

Simply Beautiful Art

Have you learned real, multivariable, or complex analysis yet?

A---B

Not complex analysis but I know real analysis.

Simply Beautiful Art

Cool

I don't know real, but I know some complex

:D

A---B

Are you a grad student ?

Simply Beautiful Art

Nope

I'm in high school

A---B

2:39 PM

You know too much for being in high school, I know nothing compared to you.

Simply Beautiful Art

I suppose

A---B

@SimplyBeautifulArt Currently I am doing conic sections.

and You ?

Simply Beautiful Art

XD In what class?

Currently I am "learning" Taylor's theorem

S.C.B.

@A--B I met you at the parabola question.

Simply Beautiful Art

Should be fairly obvious I already know Taylor's theorem

A---B

2:41 PM

@S.C.B. My two other questions are hanging can you please met me there if possible.

@SimplyBeautifulArt I was going to say that.

@SimplyBeautifulArt Do you only do math, no love for phy or chem ?

Simply Beautiful Art

Some love for physics

A---B

Chemistry ?

S.C.B.

@SimplyBeautifulArt Is Taylor's Theorem the taylor series?

Simply Beautiful Art

Yes

basically

S.C.B.

I know a bit of chem.

Simply Beautiful Art

2:43 PM

@A---B Haven't taken it yet

A---B

My favorite thing of school is chem practicals.

S.C.B.

@A--B Do American schools really allow calculators?

user21820

Schools in my country also allow calculators, far too much in my opinion. So many times I saw classmates punching their calculator for things like 2+3*4...

S.C.B.

@user21820 We aren't allowed any calculators.

A---B

@S.C.B. I am not american but I guess yes as I follow books by american authors and there I read topics on how to use scientific calculators.

S.C.B.

2:46 PM

@A--B I know how to use a calculator too, but are you allowed to use them in school?

A---B

No only log tables.

ancient I know.

S.C.B.

They give us normal distribution tables.

A---B

I don't like use of calculators in school.

S.C.B.

I think we should allow them, to a certain extent.

I had to memorize decimal representations.

user21820

I think we should use them in class but ban them in exams.

S.C.B.

2:49 PM

Like $\sqrt{5}=2.236067977$ or something other.

user21820

Memorization should not be required for exams.

A---B

I regularly encounter things like sqrt{113} ...

S.C.B.

@user21820 Our schools have to much calculation heavy problems.

@A--B There's a easy way to approximate that.

Simply Beautiful Art

@A---B That's not bad

S.C.B.

$\frac{1}{2}\left(10+\frac{113}{10} \right)$

Simply Beautiful Art

2:51 PM

@S.C.B. You should probably have the first ten square roots memorized or something

user21820

Haha..

A---B

@S.C.B. Use log table for that.

@SimplyBeautifulArt What do you mean by "Not taken chemistry" did not you have to take all three subjects together ?

Simply Beautiful Art

No

A---B

@SimplyBeautifulArt why ?

Simply Beautiful Art

Idk

S.C.B.

2:54 PM

@SimplyBeautifulArt $e=2.718281828459045235...$ $\pi=3.141592653589793238..$ $\sqrt{2}=1.41421356237309...$

user21820

φ = 1.61803398875

Useful constant to recognize.

And also 1.6449

Simply Beautiful Art

:P

S.C.B.

Nah, just memorize $\sqrt{5}$.

Simply Beautiful Art

haha

user21820

I know that one only to 2.236.

S.C.B.

2:55 PM

$2.236067977$.

Do you know what $0.577\dots$ is?

Simply Beautiful Art

$\gamma$

:P

S.C.B.

Do you know what Champerwone's Constant is? I probably butchered the spelling, but it's denoted $$C_{ch}$$

user21820

It's the concatenation of natural numbers as decimal digits.

Champernowne I think.

I saw the 0.577 somewhere...

S.C.B.

Yep, you're right.

user21820

Ah.

S.C.B.

2:58 PM

Euler Mascheroni Constant

user21820

There was once I saw 1,2,6,20,... and guessed the sequence correctly and then managed to prove it. Do you know the sequence?

A---B

@S.C.B. do you know internal pressure constant $\pi_T$ ?

S.C.B.

@user21820 $$\binom{2n-2}{n-1}$$

user21820

Yea that's right. It came up naturally in a combinatorics problem.

A---B

@user21820 I hate combinatorics.

S.C.B.

3:07 PM

?

Simply Beautiful Art

lol

user21820

Why?

Simply Beautiful Art

@A---B Combinatorics is very useful sometimes

A---B

@user21820 Combinatorics problems are hell to solve.

Simply Beautiful Art

Lmao

user21820

3:09 PM

If inability to solve is a reason for hate, then you must hate a lot of problems...

A---B

I always get trapped in the language.

user21820

Language is totally unrelated to the problem itself...

Most people including myself dislike problems that are written/spoken in broken language.

But once you understand the problem, why hate it?

Too strong a word isn't it.

A---B

@user21820 No I don't hate due to difficult. I hate due to the reason that the books interpretation of a problem is always different from my interpretation.

@user21820 I love real analysis where problems are really hard to solve for me.

Simply Beautiful Art

I like hard integrals

301

Q: Integral $\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right) \ \mathrm dx$

I need help with this integral: $$I=\int_{-1}^1\frac1x\sqrt{\frac{1+x}{1-x}}\ln\left(\frac{2\,x^2+2\,x+1}{2\,x^2-2\,x+1}\right)\ \mathrm dx.$$ The integrand graph looks like this: $\hspace{1in}$ The approximate numeric value of the integral: $$I\approx8.372211626601275661625747121...$$ Neither...

calculus integration definite-integrals contour-integration closed-form

Not that hard

but you guys know what I mean

user21820

@A---B Then you shouldn't say you hate combinatorics.. that's incorrect.. you hate/dislike ambiguous combinatorics problem statements.

A---B

3:13 PM

@SimplyBeautifulArt @user21820 Do you know cleo ?

S.C.B.

Yes. They do.

A---B

@user21820 The problem deepens as I am not a English student.

Simply Beautiful Art

@A---B If you read the chat, we definitely know Cleo

user21820

@A---B Well there are two main things you can do: (1) Practice your English a bit more. (2) Get a good book that is very clear and precise.

I say (2) because a lot of books written by native English speakers are absolutely terrible in precision and sometimes even mathematically downright wrong.

No point fighting such pointless bad book battles.

=)

@A---B We no Cleo now.

A---B

@SimplyBeautifulArt @user21820 She can solve any integration problem. I don't know why people hate her.

Simply Beautiful Art

3:18 PM

@A---B We dislike her for bad answers

user21820

To sum up our discussion, I'm one of the people who detest his/her attitude (you need to know the backstory and not just the lies on his/her profile page.) I don't hate the person.

Simply Beautiful Art

they don't live up to a good answer

user21820

I was in fact disappointed that this user chose not to even drop a single hint to any of their answers except like one or two.

A---B

@user21820 She has some medical problem.

user21820

That's the lie I mentioned above.

Read the backstory.

A---B

3:19 PM

@user21820 Where ? Please tell.

user21820

@A---B Start here and read at least all the subsequent comments and the Meta posts:

This style of answer is complete disrespect. This situation seems for me like this: Cleo found interesting problem, and solved it. He is lazy to write the solution but want to show how clever he is, so decided to post only the final result. The reference to the definition of golden ratio made me laugh. If OP asks question of such level he definitely familiar with this constant. Note that this is not a single example. ALL Cleo's answer are of this style, and even after polite ephasis that these answers is not what OP's wanted he continues to post only final results! — Norbert Nov 17 '13 at 9:14

S.C.B.

Well, a site moderator did say

«Disrespectful answer» and «arrogant answer»? Really?! I find the answer pretty useless, as I did not learn anything from it, but disrespectful and arrogant are judgements which seem totally inappropriate to this answer! I really wish commenters would limit the dramatic charge of their comments to ¡the value of an integral! — Mariano Suárez-Álvarez ♦ Apr 26 '14 at 2:16

user21820

I do not know whether that moderator was familiar with the back story.

A---B

@user21820 I have read norbert's meta post.

user21820

And it's not the answer itself that is disrespectful of course; no one is actually trying to say that (an unconscious entity cannot possibly show respect to people); rather they are saying that the poster of such answers are not respecting mathematical courtesy to provide justification or cite sources.

And there are apparently deleted comments by that same user that tells whoever doesn't believe his/her posts to just add them as axioms!

So of course you don't see the whole story..

A---B

3:27 PM

@user21820 Yes I saw that comment on a reddit post about her.

user21820

It's funny if it were just a joke. But it's too much to do it all the time.

A---B

@user21820 But that does not say she is not disabled.

user21820

The profile page was updated after his/her comments were deleted (probably by himself/herself).

Note that his/her ability to type those comments telling other people to add the integral results as axioms would show that he/she is more than capable of adding hints instead.

I'm relying on my meta-analysis of people's statements about comments I cannot see, but I believe I'm not making a false inference here.

A---B

@user21820 "Note that his/her ability to type those comments telling other people to add the integral results as axioms would show that he/she is more than capable of adding hints instead." I never thought that.

user21820

I'm not sure whether you agree or disagree, but never mind. Anyway whether or not the user is disabled, those posts serve absolutely no purpose to the site especially for questions that others have provided solutions.

So my opinion is that they should be converted to comments.

No mathematical information is lost, and it doesn't even affect Cleo since Cleo is no longer a Math SE user.

A---B

3:35 PM

@user21820 So she left the site ?

@user21820 did she announce that ?

Simply Beautiful Art

@A---B :D It appears you do fit in my chat room

A---B

@SimplyBeautifulArt I like being here.

user21820

Well according to her profile she last posted in 2015 and last visited in 2016 Jan.

Is that good enough?

Simply Beautiful Art

:D Well that's wonderful then.

user21820

Assuming it's a she.

For all you know, it's a robot backed by a supercomputer running integer relation searches.

Simply Beautiful Art

3:37 PM

lol

user21820

If you match the first 100 digits of the numerical integral using a relatively short formula, you are likely to be correct.

A---B

Lol.

Simply Beautiful Art

Yes, find the thing that matches out 10000000000 digits

user21820

That's too many digits..

Too slow.

Simply Beautiful Art

Nah

user21820

3:39 PM

Each operation is still fast on a supercomputer, but you want to be able to check billions of possible expressions.

And why bother computing with so many digits at first?

Just use 100, then check all expressions that match to 10^6 digits.

I mean you sieve out wrong answers using 100 digits first.. Check the few left to more.

Simply Beautiful Art

@Winther I checked to 1000 decimal places - it matches. — Vladimir Reshetnikov May 14 '15 at 22:24

100 digits? Nah

1000 digits?

Looks better

A---B

@user21820 Do you teach maths ?

user21820

Vladimir can do that because he knows the expression claimed. If Cleo is a computer then she should not waste time computing with high precision for all expressions. Just check the ones that match at lower precision.

Simply Beautiful Art

I wish I did. That'd probably be fun

user21820

@A---B I did teach a bit before, but not as a job.

Not now that is.

But I taught my classmates from young.

A---B

3:44 PM

@user21820 Lol.
Everyone on this site seems to me like a prodigy.

user21820

Everyone you mix with? Well there's the standard bigger fish effect. You can always find a bigger fish if you're looking.

I just made up that effect, by the way. I don't know what it's actually called.

A---B

Of course not me.

user21820

But majority of those I mix with know more than me. So...

But in real life it's harder to find..

Online all the mathematics experts gather in a few places.

Apart from MathOverflow and Math SE and FOM, where else do you know?

A---B

@user21820 What area of maths you specialises ?

Physics forums ? do you know that ?

user21820

Logic

That may be why I forgot about physics forums.

=P

But yea.. whichever specialization you're in, the experts tend to gather in clumps.

A---B

3:49 PM

@user21820 Set theory kind of maths ?

user21820

No I don't like set theory hahaha.. I like higher-order logic and reverse mathematics and foundations of mathematics.. though I know a bit of set theory because that's the current foundations.

A---B

@user21820 I read about Godel's incompleteness theorem once

user21820

Do you have programming background?

A---B

@user21820 My main subject in school is physical chemistry along with computer science.

user21820

It helps a great deal to understand the fixed-point construction if you know the fixed-point combinator from programming, or the idea of quines.

So I guess you understand the incompleteness theorems well?

A---B

3:56 PM

@user21820 I don't think so.

user21820

Oh okay.. Do you understand the halting problem well?

A---B

@user21820 Had that in next term but I think I understand it to be a machine in a perpetual loop ?

user21820

Try youtube.com/watch?v=92WHN-pAFCs and youtube.com/watch?v=92WHN-pAFCs. The former is more abstract while the latter has a slightly misleading argument.

A---B

Had seen the latter. @user21820

user21820

Once you understand that then you can try reading scottaaronson.com/blog/?p=710. It does assume some basic knowledge about programs, but otherwise is a self-contained proof that any system that can reason correctly about program execution for finitely many steps must be incomplete.

The reason it is much more complicated in most textbooks is that they want to prove incompleteness for systems like PA which cannot even easily reason about string manipulation so they need to use Godel's coding trick. I personally like the blog's computability version better.

A---B

4:06 PM

@user21820 I will read that blog.

user21820

Sure. Feel free to ask me (here or in the logic chat-room) if you have any questions on any of those.

Simply Beautiful Art

I'm hungry

Give me foods

user21820

Lunch/Dinner? I'm going off soon. See you all next time!

Simply Beautiful Art

Lunch

@user21820 see ya!

A---B

mine is dinner.

user21820

4:09 PM

Mine is midnight snacks.

Simply Beautiful Art

Lol

user21820

What fun that the world goes round.

A---B

@user21820 before going can you recommend me some math books ?

Simply Beautiful Art

@A---B I can't

user21820

In what area?

A---B

4:12 PM

@user21820 probability/ combinatorics/ stats

@SimplyBeautifulArt Why ?

user21820

Serious or recreational?

Simply Beautiful Art

@A---B i never learned through books

A---B

@user21820 Dead serious!!!

@SimplyBeautifulArt What did you mean ?

user21820

I still remember one of the first mathematics books that got me interested in mathematics was "The Mathematical Universe" but it doesn't have formal proofs.

Simply Beautiful Art

I never learned math through books

Or PDF, or videos

A---B

4:15 PM

@user21820 No I want serious books.

@SimplyBeautifulArt Then how did you learn ?

Simply Beautiful Art

MSE and Wikipedia and my own efforts

user21820

I can recommend serious references for logic (for other subjects you've to ask other people).. linked from my profile is this post containing free online references that I found for logic. I personally learned from a whole mess of resources but I think for you Stephen Simpson's is a good place to start.

That's if you're interested in logic; if not then I don't really have recommendations for olympiad style mathematics except practice with past olympiad problems..

When I took the courses in probability and statistics I hardly touched my textbook...

Simply Beautiful Art

@user21820 those are easy

I still haven't touched my calculus book

user21820

Which are easy? You mean probability and statistics introductory textbooks right?

Simply Beautiful Art

Yeah

user21820

4:22 PM

Well even for harder courses I rarely systematically read any textbook.

A---B

@user21820 Know anything for linear algebra ?

user21820

I just don't have the patience to sit through someone else's proof.

No, because the textbook used in the course I had to take had a serious logical error; the author made a false claim due to a faulty understanding of induction!

A---B

@user21820 That is sad :(

Simply Beautiful Art

Wow...that is sad

user21820

If you're curious, the specific error was in constructing a basis for R ^ n.

The textbook said that you could just start with all the vectors, and throw out linearly dependent vectors one by one and eventually end up with a linearly independent set.

Anyway see you all next time!

Simply Beautiful Art

4:30 PM

See you!

A---B

Bye. I also have to go bye.

Simply Beautiful Art

Oh, bye @A---B

amWhy

5:48 PM

@SimplyBeautifulArt Now, that is sad, really.

I may need to take a break from this chat, which seems to have become a venue for bragging rights. The more outrageous the bravado, the more do I pity thee.

I like you all, but I do not want to waste time listening to folks brag, and/or who are unable to reign in their hubris.

The only persons worthy of bragging are precisely those who refuse to brag.

Simply Beautiful Art

@amWhy sorry, did not mean for it to become this way. Do as you wish :-)

@amWhy but that comment was actually serious

:-/

I do suppose it is sad though

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$Mathematics$ This is the Realm of Simply Beautiful…