/mg/ Mathematics General

[Anonymous No.16022974]

You don't have to believe in God, but you should believe in The Book.

Previous Thread: >>15995883

[Anonymous No.16022989]

>>16022974
so, what's (you)r excuse for not turning meth into math, /mg/?

[Anonymous No.16023002]

I got a real analysis midterm tomorrow morning where the median grade is ~40%. Anyone have any decent pre-test rituals or saints I can pray to or whatever.

>>16022989
I'm retarded, doing meth and working 12 hours a day wouldn't help me that much

[Anonymous No.16023050]

[Anonymous No.16023059]

So is the autism over the derivative just being a symbol completely justified? Like people legit argue that separation of variables is just a mnemonic but is it just unjustified autism? Can't the differential be defined so that multiplication by a differential is sensible? They are just limits after all, surely?

[Anonymous No.16023287]

>>16023059
yes it's very common algebraic technique. We do it this way to avoid sucking in all of analysis. See the garrity book on alg geom chap3 where he talks about differentials for a simple explanation.

[Anonymous No.16023342]

>>16023059
I assume you mean expressions like dy or dx in isolation being “just formal symbols”. To that I will say three things
>even so, you can often use implicit differentiation and divide by dx to get the correct derivatice dy/dx, which is cool
>in integration, dx can be looked at as a measure, which is nifty, though I don’t know how to look at dy/dx as something to do with measures
>there is a treatment of calculus called nonstandard analysis that makes sense of dx using tricks from mathematical logic; not everyone likes it though

[Anonymous No.16023431]

When did you first realize you enjoyed math? What has been your greatest mathematical achievement so far?

[REDACTED No.16023439]

>>16023431
I am a fields medallist.

[Anonymous No.16023445]

>>16023431
>When did you first realize you enjoyed math?
when I tried something else.
I can sit for hours just solving math problems and questions.
I can't sit for five minutes reading literature about some bullshit idc about.
I can't maintain my sanity when coding anything larger than a simple program

[Anonymous No.16023462]

>>16023431
My greatest achievement was figuring out a pretty good approximation function for the product logarithm when x is below 1.

I struggled with math notation as a kid, but when I was a young adult I realized math was just jigsaw puzzles and rearranging shapes, and everything then was suddenly crystal clear.

I like to joke that in my head are three personalities: an artist, an autist, and an anxious. Nothing I know of except math can get all three to work together instead of argue: it’s beautiful, it’s usable everywhere, and it’s always exact.

[Anonymous No.16023495]

>>16023462
"Anxious" isn't a noun. "Anxious man", surely.

[Anonymous No.16023557]

>>16023059
I've seen some calculus textbooks lately say that dx and dy are just [math]\Delta x[/math] and [math]\Delta y[/math] but along the tangent line instead of along the curve.

[Anonymous No.16023666]

>>16023557
What books?

[Anonymous No.16023669]

Are there any books that approach mathematics recursively? How do you add to recursive functions in general for example? Is it all straightforward?

[Monadas !lyWkAM5ERA No.16023687]

>>16023439
Under the assumption that I am correct in guessing that you have recommended in a previous thread that people should think more fluidly, what do you mean by that? When I listen to other people I just imagine the thing they are talking about in my head. How do you think about what other people say fluidly.
Also what is the best approach to learn writing style hiding techniques (that also enhance cognitive flexibility)?
Thirdly, I didn't burp: https://www.urbandictionary.com/define.php?term=bump
And because I have the time, what do you think is the primary neuronal driver of high IQ? Would you say that it has to do with learning flexibility? There is a lot more in the brain than dopamine, you know? My personal (schizo) theory is based on the endocannabinoid system. Maybe it is also more of an immune system thing.

[Anonymous No.16023721]

>>16023669
do you mean recursion in the sense of computer programming, like:
>addition is just recursive +1
>multiplication is just recursive addition
>exponent is just recursive multiplication
Yeah, there are books that explore and develop these ideas a bit, and they can be helpful in certain kinds of mathematical logic, but I don’t know how you could gainfully found something like geometry, or calculus, or galois theory, or… on recursion

[Anonymous No.16023795]

>>16022989
Erdos did speed, not meth

[Anonymous No.16023803]

>>16022974

Are there infinitely many pairs of integers a,b such that [math] 2a^2 + 1 = b^2 [/math] ?

[Anonymous No.16023987]

>>16023803
That is Pell's equation, there are many ways to prove it of various complexity. The most trivial being if (a,b) is a solution, the simplest answer being (2,3), then you can show that (2ab, 4a^2 + 1) must also a solution by substitution. Repeat this step and you get infinitely many solutions.

[Anonymous No.16024008]

>>16022989
Unironically, my social skill too poor to reliably source it.

[Anonymous No.16024187]

is it possible to draw a square with an area of 3 (integer, no decimal points)? if so how would you present this in mathematical notation?

[Anonymous No.16024250]

Have a square with side lengths of [math]1_{\sqrt{3}}[/math] ez

[Anonymous No.16024263]

>>16023431
I subbed to 1 then 2 then 10 then 20 math youtubers. Then I read an Academia math paper or math essay every day (Great Books). My library filled up like the Math Sorceror. I have not been able to keep on my personal discovery autodidact crusade as my standard of living plummetted and I became an old man's asswipe water boy where I once stood tall as a big institution suit. I venerate Ramanujan-kun who made big advances in squalor because I now live in squalor and struggle to keep enough energy at the end of the day of my wage slaving and bureacratic babble to attempt any deeper inroads to my math curiousity if it isnt entirely waned.

[Anonymous No.16024264]

>>16024250
Thought about it for 5 seconds, it's actually [math]10_{\sqrt{3}}[/math]

[Anonymous No.16024370]

>>16024250
>>16024264
thanks.

[Anonymous No.16024386]

>>16024187
>draw a square with area 3
the easy way is:
i. make a right triangle with leg lengths 1,1. The hypotenuse has length √2
ii. Use that to make a right triangle with leg lengths √2,1. Its hypotenuse has length √3
iii. Use that to make a square of side length √3 . This is possible with a compass and straightedge, for example

[Anonymous No.16024437]

>>16024386
thanks anon i'll look into this. i've just been using circles to create the intersection points.

[Anonymous No.16024579]

The best differential notation is derivative(f), or optionally, derivative(f) with respect to (g). NGL it makes tutoring retarded undergrads so much easier.

[Anonymous No.16025062]

>assigned problems: 7 practical problems and one proof
>present the proof in class
>fellow students beg the TA to not assign any more proofs because "it's too hard and useless"

[Anonymous No.16025106]

>>16024579
leibniz notation for partial derivatives (the fraction one)
Euler notation for complicated long stuff (D_x(f))
Lagrange for everything else (f')

[Anonymous No.16025113]

>>16022974
so an algebraic structure needs to have closure, but doesn't every structure have closure with a large enough set?
Or is it just that you need to define the closure, like elementary algebra is a closure with real number. But then would elementary algebra on natural numbers not be an algebraic structure since it does not have closure?
Or am I totally misunderstanding this?
I was just trying to get why elementary alg and linear alg are both considered algebra.

[Anonymous No.16025163]

>>16025113
>so an algebraic structure needs to have closure, but doesn't every structure have closure with a large enough set?
No.
Consider the case where your operation is addition and your set is the integers with 1 removed. You can keep adjoining as many elements as you want, bringing your cardinality to god-knows-what transfinite number, but unless you adjoin 1 itself you'll never have closure.
>>16025113
>But then would elementary algebra on natural numbers not be an algebraic structure since it does not have closure?
Your structure needs at least one operation, and the naturals are perfectly valid as at least a magma if you use, say, addition or multiplication.

[Anonymous No.16025375]

Is the limit of any sequence of analytic functions analytic?

[Anonymous No.16025993]

>>16025113
I think the answer to your main question is that “algebra” is a very old word, it comes from the name of a 1200 year old Persian book. since then it has usually meant: a specific style of writing your problem down as an expression consisting of symbols, and then solving that problem by manipulating the expression into a nicer form. whereas the notion of “algebraic structures” like groups and rings is only a couple hundred years old

part of the reason the new thing is called “algebra” is because it was meant to abstractify the old thing and hopefully answer questions about it. a crowning achievement in that direction is Galois Theory and the resulting proof of insolubility of the quintic

as for closure, all that means is just that your operations are total. if a and b are in your group, then ab must be defined. otherwise it just ain’t a group. so for example [math](\mathbb{R},0,1,+,\times,\div)[/math] with a symbol division isn’t an “algebraic structure” in the usual sense of a logic textbook, because 1 divided by 0 is not defined. but take away the division symbol, and it is a ring and also a field. Or you could add a special NaN element of some kind to make division total but this can get messy
>>16025163
>magma
I sometimes wonder who came up with these crazy terms, and why we still go along with them

[Anonymous No.16025997]

>>16025375
what kind of limit? if you mean the pointwise, the limit doesn’t even have to be continuous

[Anonymous No.16025999]

>>16025993
a quick search suggests that "magma" was probably coined by Bourbaki as a pun on a French meaning of the term for what is essentially a confusing jumble

[Anonymous No.16026012]

>>16022989
I'm schizophrenic

[Anonymous No.16026147]

What's the best way to take maths notes?

[Cult of Passion No.16026150]

>>16026012
Same, but successful.

[Anonymous No.16026227]

In an equation such as 3x +4 = 20. What exactly would happen if I divided by three first? Like why can't the answer be 6.6 repeating minus 4?

[Anonymous No.16026228]

>>16026227
>Like why can't the answer be 6.6 repeating minus 4?
Because you'd be forgetting to divide the 4 by 3 as well.

[Anonymous No.16026237]

>>16026228
So your saying the problem then change to x +4/3=20/3?

[Anonymous No.16026239]

>>16026237
exactly

[Anonymous No.16026252]

>>16026239
Thank you and one last thing does order of operations(pemdas or bidmas) slightly change once variables are introduced?

[Anonymous No.16026281]

How do you define infinity in terms of something else

[Anonymous No.16026283]

>>16026252
No. Variables operate exactly like whatever they're representing (in this case, just a number).

[Anonymous No.16026287]

>>16026227
whatever you do to one side of an equality you must do to the other. otherwise it's not an equality anymore, and probably an absurdity.

[Anonymous No.16026289]

>>16023669
look up principle of recursive definition

[Anonymous No.16026335]

>>16022974
>You don't have to believe in God
Math proves God btw.

[Anonymous No.16026349]

>>16025993
>1200 year old, couple hundred years old
I knew that but yeah, reflecting on it and thinking about the possibilities in another 1000+ years calms that itch in my mind down quite significantly. Don't know why this stuff bothers me so much, it's not incongruous but more like a personally perceived mismatch in my mind from previously studied examples but for which I know is compatible in the system for a reason which few understand.

[Anonymous No.16026408]

How different are elementary and advanced number theory? I really like elementary number theory and would maybe like to start learning more advanced stuff; what are the prerequisites in terms of analysis, algebra, etc?

[Anonymous No.16026484]

>>16026349
AFAIK, modern equations like 2x+y=z only really appeared in the 15th century or something, at least in post-Roman europe. So you see it was not just two sudden events; the whole middle 1000 years is full of small and large developments.
>>16026408
Complex analysis at least.

[Anonymous No.16026547]

Is there any consistent system where 2^aleph_0 = aleph_0

[Anonymous No.16026569]

>>16023050
Isn't that the Misha Verbitsky's program lol?

[Anonymous No.16026588]

https://archive.4plebs.org/_/search/tripcode/%21BerzSNiQ.I/

thoughts?

[Anonymous No.16026596]

>>16024263
go away

[None !BerzSNiQ.I No.16026606]

it takes a jew.

[Anonymous No.16026621]

>>16026408
> I really like elementary number theory
I did too but then I took advanced number theory. It was the hardest and least enjoyable course I ever took. Admittedly that may partly have been to do with the lecturer who did a terrible job explaining shit.

[None !BerzSNiQ.I No.16026622]

and do not play stupid with me.

you know damn well what that website is , what it does , and it's consequences.

[Anonymous No.16026641]

>>16026547
mostly no. in weak systems the meanings of the symbols get ambiguous so let’s rephrase your equality as
>bijection between [math]\mathbb N[/math] and the power set if [math]\mathbb{N}[/math]?
then ordinarily you should be able to take that bijection and produce a new element of the power set by diagonalizarion, a contradiction

on the other hand there are times when something sort of like that is true. for example if we suppose that only the computable subsets of [math]\mathbb{N}[/math] really exist, then in that context there is a partial surjection from N onto its own power set. but thats a weird thing to want to do, people call it the Recursive Topos or something but as far as I know it is useless and confuses people and has nothing to do with anything like ZFC

[Anonymous No.16026929]

>>16023059
You're looking for the term "differential form". Very much a well-defined notion with no "infinitesimals" and other nonsense. It does explain the supposedly unjustified operations in calculus, though for didactic reasons it might be better to learn it a little later.
Forster does them in his Analysis 3 book (German) at the very end (for R^n); otherwise they are typically introduced in differential geometry.

>>16023557 this goes in the right direction

[Anonymous No.16027125]

>>16026147
http://www.topology.org/tex/conc/mathlearn.html

[Anonymous No.16027136]

>>16026281
https://en.m.wikipedia.org/wiki/Dedekind-infinite_set

[Anonymous No.16027160]

List of comprehensive math resources:
>Evan Chen's Infinitely Large Napkin
>Misha Verbitsky's Problem Course in Undergraduate Mathematics
>Alan U. Kennington's Differential Geometry
>Alexandru Buium's Mathematics A First Course
>Antonini et al. Les Mathematiques pour l'agregation
>Juan Antonio Navarro Gonzalez' Apuntes para una Licenciatura en Matemáticas
>Carlos Ivorra Castillo's Libros
What else?

[Anonymous No.16027217]

the glossing over the well ordering principle kind of scares me, should i jump ship to another book?

[Anonymous No.16027268]

>>16023431
According to my mom, I cried that I wouldn't have math homework during my first summer vacation. While getting my engineering degree, it was always the lens I saw everything. Hanging around here I powered through Baby Rudin. And now I play around with algorithms used in robotics.

[Anonymous No.16027316]

>>16025375
If it's uniformly convergent on compact subsets of the domain, yes - that's the natural sense of convergence for holomorphic functions. I'm almost certain it's true for real analytic functions as well.

[Anonymous No.16027358]

>>16026641
I recommend this blogpost for the introduction to Lawvere's fixpoint theorem, which proves that there is no surjection from A to its powerset in very weak settings. The (not-so) counter-example of the effective topos is briefly mentioned at the end.

https://math.andrej.com/2007/04/08/on-a-proof-of-cantors-theorem/

[Anonymous No.16027558]

>>16027316
For real analytic functions I think e^(-1/(x^2+(1/n))) should be a counterexample.

[Anonymous No.16027564]

>>16027358
>Lolvere
the patent miner of mathematics kek. anyway the issue is not when you can do the recursion theorem, its wtf the definitions are even supposed to be when Schroder-Bernstein fails and even “infinite” has nonequivalent definitions to choose from
>>16027217
>well-ordering
do you just mean that there is some greatest q between 0 and b such that b - qa > 0? it’s very likely they are following Euclid where a number is “a multitude of units” i.e. it can be written down in real life as a string 1+1+..+1. in real life such strings are well-ordered and induction is true.

[Anonymous No.16027979]

anyone else getting rejection letters while trying to focus on readings?
shit sucks, could do with a fun problem /mg/

[Anonymous No.16028018]

any serre man in
thinking of reading his cours d'arithmétique

[Anonymous No.16028026]

I'm starting to learn math for real real, not for play play, and I'm checking out set theory.

Whenever wikipedia says "X is widely regarded as the standard for Y", I like to come to 4chan and ask the relevant general what they think.

[Anonymous No.16028029]

>>16027564
>“infinite” has nonequivalent definitions to choose from
They all make sense from a computational point of view. What's even the point of doing maths in the 21st century if it's not to be become Lambda, the destroyer of platonic Worlds?

[Anonymous No.16028030]

>>16023495
he's probably french or something stupid like that

[Anonymous No.16028035]

>>16028026
I would also like to clarify that my specialty is in metaphysics, so pure mathematics is something I'm exploring from the perspective of a pragmatic frame. I'm curious as to what's considered the most useful meta-mathematical model.

[Anonymous No.16028062]

Math is so easy I skimmed through this in a month by reading about an hour a day

[Anonymous No.16028153]

>>16028026
>>16028035
>Set theory
>Meta-mathematical model
https://www.logicmatters.net/tyl
>Formalization vs practice, pragmatics
https://us.metamath.org/#book
https://hrmacbeth.github.io/math2001/

[Anonymous No.16028590]

>>16028026
Well ZFC is still widely regarded is the standard for axiomatic set theory

[Anonymous No.16028595]

>>16028062
calc 1-3 is literally baby math you stupid cocksucker

[Anonymous No.16028596]

>>16028062
did you do the exercises though?

[Anonymous No.16028739]

How do I know I didn't fuck up, if exercises do not come with solutions?

[Anonymous No.16028772]

>>16028739
By not being retarded. Anyways at lower level undergrad classes, there will be equivalent textbooks with a lot of solution, or solution manual available anyways. Or problem books with hints and solutions.
Stop shitposting on 4chan. Stop making up excuses. Just do problems.
-----
Btw I just learned Abstruse Goose's webcomic is gone. This is pretty sad.

[Anonymous No.16029418]

>>16028739
If you actually understand the math you will know if your solution is bullshit or not. Only if you are a symbol pushing monkey will you require solutions

[Anonymous No.16029572]

>>16028772
oh no, permanently? I can that the html title tag is still up
All the good non-corpo stuff on the internet is slowly going away :(

[Anonymous No.16030150]

Having trouble writing a personal statement and statement of purpose for my Master's applications. I'm a husk of my former self and my only ambition is to avoid working a real job.
I know Japanese and other random shit so I was thinking something like:
>I want to do my duty as a global citizen and help connect the international math community!
Would that trick them? Any tips from anons who got in?

[Anonymous No.16030191]

>>16030150
>Its hard to imagine my life without mathematics. I want to continue my education and become a mathematician. Doing research on/with X currently interests me most.

[Anonymous No.16030870]

>>16030150
how much easier is doing a master's than doing a "real job"?

[Anonymous No.16030974]

>>16030150
That's an empty statement if you do not concretely state how you're going to do that. Have you looked at the research interests of faculty members in RIMS or OIST? Do you have a plan for collaborating with them? Most Japanese academics know enough English to initiate collaborations themselves anyway.

[Anonymous No.16031185]

>>16030150
Just go to google gemini and get the demo for advanced and make it write it for you.

[Anonymous No.16031207]

Is math degree viable option money wise? Do I need minors on CS or something additionally?

[Anonymous No.16031227]

>>16030870
The last time I had a "real job" in a warehouse they made me stand on a forklift and bullied me by leaving me up there until I started crying.
How is doing something I enjoy going to be harder than that

[Anonymous No.16031231]

>>16031227
>The last time I had a "real job" in a warehouse they made me stand on a forklift and bullied me by leaving me up there until I started crying.
Huh..

But I was thinking what you said meant you could get a job based on already having a bachelor's in math.

[Anonymous No.16031516]

>>16023666
The one I was thinking of is Larson and Edwards.

[Anonymous No.16031912]

>>16031207
for bachelor's and master's it's pretty much just software, [something]-consultant/advisor, finance or highschool teacher. you don't really need a cs minors but it will be a help if that's the job you are aiming for.

i don't know why anons here tend to be pessimistic but it's really not bad. all employers view a degree in mathematics very positively.

[Anonymous No.16032049]

>>16031912
>software
Are the people who are good enough at software that they make whole successful companies really good at math?

[Anonymous No.16032103]

>>16032049
math degree at a good school means you have some sort of talent and can be re-trained to a relatively powerful species of office pokemon
t. project manager

[Anonymous No.16032105]

>>16032103
What about a math degree at a middling school?What does that one mean? And a bad school?

[Anonymous No.16032109]

>>16032103
huh? are you just saying their iq is higher than other people's and their work ethic?

[Anonymous No.16032110]

>>16032105
there are two ways to interpret “good school (for math)”:
>name brand school like Harvard or something
>challenging curriculum
I mean the second. Of course nobody who completes a math degree is a dummy, dummies drop out at the first scent of calculus. But its still reassuring anyway when someone has gone through the harder stuff you see in these “good” curricula

[Anonymous No.16032114]

>>16032103
>re-trained
what does re-training to be able to make a successful software company consist of?

[Anonymous No.16032120]

>>16023431
Scored a zero on the putnam

[Anonymous No.16032128]

>>16023431
26 when I started doing math out of NEET boredom, year later my greatest achievement is being top of my class in all my calc classes, (at community college thoughever)

[Anonymous No.16032171]

>>16032049
mostly no.
you are not going to start a company silly, you are just going to write code for a living. there are plenty of tech jobs with plenty of math (numerical stuff, optimization, stats, etc) but the most common barely have any math.

look at the positive side, you studied what you like and you have a decent job. that's way better than most people who studied what they like.

[Anonymous No.16032186]

>>16032171
>mostly no.
what is it they are good at then?
>you are not going to start a company silly,
why not?
>the most common barely have any math.
what are they then? what are good software engineers even good in?

[Anonymous No.16032454]

>>16032186
>what is it they are good at then?
creating something the people want and doing it competently.
>why not?
you can, but that's not why most people get cs degrees
>what are they then?
webdev, appdev, cloud dev, consultant, cybersec, etc
>what are good software engineers even good in?
software is complicated, even the simplest program has a ton of different parts that make it work.
a SWE is a person who can take user requirements and turns them into a working product.
as a person who did both math and swe, math is far more elegant and simple than software. software is tedious and complicated in the same way that untangling a bowl of spaghetti is complicated.

[Anonymous No.16032471]

>>16032454
>creating something the people want and doing it competently.
Does that have to do a lot with predicting what technology will be important in the future?
>software is complicated,
So it isn't that much focused on just math..?