/sci/ - Science & Math

dx/fx × f^y/d^y ÷ √π +- p/n / n/p Sdx^fx (Sfx^dx)^2 × √-1 = 100%

>>14699556
Yes indeed.

new to math and reading through the book an anon posted here. How do you actually "prove" something this fundamental? Draw Venn-Diagrams? Show the final column in their truth functional tables is the same? Or can you actually transform it into the other equivalent statement just through symbol manipulation?

>>14699556
Can you Latex this statement? I can't say it's jumping out at me what S is. Suspension?

>>14699598
While in math you generally can't rely on Venn diagrams, you won't go awry using them here. However you can in fact transform each statement into the other. The complement of a complement is the initial argument. If A and B are subsets of C, their join must be in C. These follow from the definition of the operation in question. For example if A is in X, then complement A is "X - A," everything in X not in A. So if you do that again then it's X - (X - A), which is A.

>>14699249
>>14699282
Repost from the last thread.

>>14699598
You start: assume [math]x \in A [/math] or one side of the equivalence/equality, then you show that x must be in the other side of the equality. Then you show that converse.

What is reduced suspension and why do we need it?

Can anyone here tell me how you arrive at this definition?

>>14699453
Who here has read through Lang's Algebra? Are the problems really as hard as I have heard?

I can't understand math intuitively, only logically. What do I do?

>>14700318
Start by thinking how can you define the cartesian product of two sets. One way is by saying that the product A×B it's the set of all pairs (a,b) where a is an element of A and b is an element of B. That's fine, but what is a pair exactly? Well, if you go the naive-set-theoretic way, it's just synthetic sugar for the set {a,{a,b}}, but some people think that's unintuitive and it gets messy when generalizing to tuples with uncountable sets. Other way is by identifying a pair by a function f with domain {0,1} such that f(0)=a and f(1)=b. You can generalize this to uncountable sets as that definition of your post. Does this help?

>>14700488
I took the long way through logic until I logically derived intuition for myself.

>>14700488
Intuition is earned, not given.

>>14700571
>synthetic sugar
fyi I assume you mean syntactic sugar

SO if I want to write a-b > 0 I should write b < a?

>>14699453
For what kind of ordering would [math]\forall a \forall b ((a\mathcal{R} b)\lor (b\mathcal{R} a))[/math] not hold? Like a partial ordering with more than one branch where elements of the one branch couldn't be compared with elements of the other?

>>14699453
How do I into Geometric Algebra?
Do I need a solid grasp of Linear Algebra first?

>>14700989
Sure, that'd work. For instance, we often partially order integers by divisibility -- it's reflexive (x | x forall x), transitive (a | b, b | c => a | c), and antisymmetric ( a | b and b | a => a=b), and you can make chains (eg 1, 2, 4, 8,..., or 1,2,6,24,... I mean any multiplying sequence really) but plenty of the integers aren't comparable under this ordering (3 and 5, for instance).
That's really the difference between a partial and total ordering, anyways -- a total order is a partial order where every element is comparable.

>>14700571
yeah I think I get it, thanks. And I take it the type of tuple then defines the functions (since the definition mentions just functions, not any particular kind)? So a pair like (x,x) would "be" a function for which f(0)=f(1)=x?

>>14701051
you need a solid grasp of linear algebra for nearly everything, except maybe set theoretic topology and real analysis in one variable
however, solid is hard to define. you should know the basics. if you're at the point where you understand the proof of the Jordan normal form theorem you're probably fine.

>>14700989
A generic example given is [math] \subseteq [/math] on a collection of sets.

>>14700333
There are problems that are ok and are useful for understanding the chapter they are about, but there are others that are basically "prove this theorem that appeared in a paper in the '50s" and thus are hard. The problem is that you don't know which is which beforehand.
I would say that Lang's book is for people having enough mathematical maturity so that they can understand without having to do problems.

>>14701235
Ah I see. Well as long as it's not full of unsolved problems, I guess it's fine. But yeah, grad-level textbooks seem to often rather focus on exposing you to serious research problems than just giving you "harder" exercises.

>>14701823
Research problems ARE harder exercises. They're the hardest exercises.

>>14700853
Either are fine. I think the point here, is that the author has only defined what it means for a real number to be greater than zero, and now we want to extend that definition to what it means for an arbitrary real number to be greater than another arbitrary real number.

>>14700853
No, you shall write a > b as is stated in line 4.

>>14701906
90% of all textbook problems were (possibly variations of) research problems.
The thing that makes Lang difficult is that they are relatively recent.

>>14701051
You do not, linear algebra isn't used that much and can be picked up along the way. Just pick up any algebraic geometry book at look at the problems, then look at the text, if you can't understand the statements in the problems find an easier book. There are at least a few books targeted at undergrads. (unitext 129 only uses algebra and affine and projective geometry, with the basic notions of analysis/topology)

>>14702079
>confuses AG with ga
>la not important
lmaoooo

>>14702082
Shut the fuck up!

>>14702079
>Mixing up algebraic geometry and geometric algebra
Many such cases.
Sad.

>>14702425
What cases are those?

>>14702448
Cases of midwits thinking almost everything should be commutative

>>14700822
Go back

>>14701133
you don't need linear algebra for anything. you just got mindfucked by your undergrad curriculum, it's sad.

help a brainlet out
I have a moving average
I have a measure of volatility
I want to combine the two so that the moving average is more reactive when volatility is high and less reactive to price when volatility is low (ideally, I'd like the moving average's gradient to be zero when volatility is low), is it possible to do with with any type of moving average?

It's surreal to me that it's 2022 and there are still people out there who think 2 + 2 = 4 is an objective truth that was true before humans even existed and not just like a thing society agreed on because it's useful

>>14703009
It's indeed surreal that midwits like (you) cannot distinguish between the a priori nature of counting and the socially accepted symbols to communicate it. For whatever reason this aspect of elementary school math seems to pose a level of abstraction incomprehensible to many pseudo-philosophical NPCs.

>>14702941

make a custom indicator, such that when volatility is low, use a 50 day moving average, when volatility is high, use a 10 day moving average,

>>14700571

apparently, this is how you define a cartesian product

Hi /mg/ I'm a U.S. cc freshman taking calculus 1 as part of my BBA program,
I was wondering if any of you could find a simple continuous function such that no point has a derivative. I'm having trouble believing such a thing exists. But I guess it would be some fractal function?

>>14699453
Excellent edition
Best video on GA-
https://www.youtube.com/watch?v=60z_hpEAtD8

Hey guys can you recommend me some good textbooks to refresh myself on Calc 2? Im relearning vector calc but I sometimes have trouble solving integrals and dont really remember much of series/ sequences

>>14703075
that's an interesting idea

>>14703063

>I am a mathematician and logician. I am currently a visiting assistant professor in the mathematics and statistics department atSam Houston State University. My PhD is fromThe Graduate Centerof The City University of New York.


Oh no no /mg/ bros, we got too cocky!

>>14703618

>>14703063
Prove that there is objectively more than one thing in the universe, homofag. :^)

Protip: you literally can't.

>>14703631
t. Undergrad

If you feel strongly enough, you're welcome to tell the professor he is wrong.

>>14703101
https://en.m.wikipedia.org/wiki/Weierstrass_function

>>14703712
Relatively intuitive construction we saw in class. I apologize if the proofs are messy.

Are Polya's books worth putting time into?

> A commuter is in the habit of arriving at his suburban station each evening exactly at five o’clock. His wife always meets the train and drives him home. One day he takes an earlier train, arriving at the station at four. The weather is pleasant, so instead of telephoning home he starts walking along the route always taken by his wife. They meet somewhere on the way. He gets into the car and they drive home, arriving at their house ten minutes earlier than usual. Assuming that the wife always drives at a constant speed, and that on this occasion she left just in time to meet the five o’clock train, can you determine how long the husband walked before he was picked up?

>>14703063
1. Addition isn't always defined as it is in Peano arithmetic. You may insist that I play that game. I might decide, potentially for very good reasons, that I want to play another game in which 2 + 2 does not equal 4.
2. If you put two rabbits in a hutch and then two more rabbits, you may well end up with five rabbits in the hutch.

3. No set has the same cardinality as its power set.

To simply insist that 2 + 2 always = 4 just stems from a position of white privilege, in which cowardly whites browbeat proud, powerful, and fitter sub-saharans. Maybe its jealousy? I say this as a white man myself, living in Israel.

>>14703063
>>14703618
>>14704443


that's why we use category theory to classify operations that are unique up to unique isomorphism

>>14704866
What does the image represent?

>>14704888

>>14704888
not quick enough desu i'm gonna just ignore it

>>14703063
>>14703618
>>14704443

https://people.math.harvard.edu/~mazur/preprints/when_is_one.pdf


>>14704884

catgeorification of the Jones polynomial

>>14704180
I don't think so. I don't really like books like this.
You get better at thinking mathematically by thinking about math, not by reading a book abstractly rambling about what the correct way to think is.
It's no different than thinking you can learn how to socialize or how to flirt by reading about it. Trying to copy techniques you saw in a book will just result in embarrassing failures. You can only learn this stuff by experience

>>14704443
>3. No set has the same cardinality as its power set.
The set of all sets does. It's so big that the laws of math break down.

>>14704443
>2. If you put two rabbits in a hutch and then two more rabbits, you may well end up with five rabbits in the hutch.
End up after a month. This is how long it takes for rabbits to lay eggs. Before they lay eggs you count 2+2=4. After they laid an egg you count 2+2+1=4+1=5. At no point 2+2 didn't equal 4.

Knowledge is 0.9999....=1
Wisdom is that they are clearly two different numbers

>>14705126
Rabbits? Eggs?

Anon??

>>14705234
knowledge: 2+2=4
wisdumb: 2+2 and 4 don't look same

>Napier's bones are here.
>Here is Schroedinger's continuous spectra.
>Cauchy's life reached its limit at this point.
>Here lies Sturm, without oscillations.
>Behind this Stone lies Weierstrass.

>>14705255

>>14704950
>not quick enough

One might compare the dates on this paper >>14705892
and this article
https://www.rt.com/news/442388-4chan-solution-math-problem/
to notice that my paper was the real thing behind the bullshut news story. The copy above is dated 10/15 but I think I posted it for the first time on 10/1.

>>14705892
this too, btw

So what's the advantage in taking a math degree outside of learning it on my own?
I already have a stable career in IT as a sysadmin so it's not like the degree would help me but work is willing split the cost of a degree I want

>Only three exercises in the chapter
>All of them about the definition in the last paragraph
>1-1 corrospondence isn't defined of course
>third problem is a research problem that most people at the time didn't accept
>their solution uses notation not used in the text.
>undergraduate texts in mathematics
Why are mathematicians like this?

https://coq.inria.fr/
proof assistants are cool.

>>14706152
If you have extreme autism and the BELIEF that mathematics is just a collection of facts.

>>14705991
You can use it as an in for fields like data science and quantitative finance.

>>14706157
yes, I believe that mathematics is just a collection of facts, why do you ask?

>>14705883
Great. I want more history written in the style of category theory. We should categorify all of social science.

>>14706161
Read Lakatos' proof and refutations
Also read
Sternhell's The Birth of Fascist Ideology

>>14706252
Do you really expect me to read a 200 pages long theater play about mathematics?

>>14706277
Hell yeah.

>>14706284
No can do with my attention span. You gotta have pretty colors and pictures.
e.g. https://www.youtube.com/watch?v=wO61D9x6lNY

>>14706294
There are pictures

>>14706297
I'll keep it bookmarked and maybe go back to it if I see it mentioned often.
Theater is meant to be seen, not read.

>>14706306
It's a dialogue, not a theater.

>>14706312
close enough

>>14706252
Mathematicians are isomorphic to code monkeys, sorry.

>>14706294
pure kino

If I study math for 4 hours every day, how many years would it take me to go from pre-algebra to arithmetic geometry?

Does anyone have a 100% concrete deductive definition of "algebra" and "elementary algebra"?

>advisor tells me if I can get to point A, I'll have an immediate contradiction
>get to point A
>can't find the immediate contradiction
am I retarded?

>>14706252
Is there a Tiktok version of his arguments? I have ADHD and can't read books.

>>14706735
>Tiktok version of his arguments?
kek

>>14706512
Much easier to just start from a set of arithmetic geometry books and work backwards for motivation (both mentally and physically), Elementary mathematics (or mathematics that can be done with just elementary Algebra/geometry/calculus/probability/number theory already eclipses what even the most dedicated genius can learn in a lifetime. Though of course lines between fields are blurry, especially the "elementary mathematics".

https://www.youtube.com/watch?v=Vp0rZQ3risI
what did he mean by this?

How does one prove that any isometry from R^n to R^n is a combination of a roto-translation and a reflection? I mean, proving that roto-translations + reflections are isometries is easy, but how do we prove that there cannot exist a non-affine isometry as well?

I already found a proof for R^2, but it doesn't work in R^n.

>>14707028
Wolfram really let himself go.

What's a good reference book for non-undergrad linear algebra? I'm NOT looking for some brainlet undergrad intro, but rather a reference book with lots of non-trivial results or results in full generality for arbitrary fields.

Anyone here has any tips on how to become less wordy in your proofs? I want mine to be as concise as in Lang's textbooks.

>>14707355
Post one of your proofs that you don't know how to make less wordy. I'll help you out

>>14707311
Gorodentsev.

>>14707355
Verbose proofs are better. Optimizing brevity means to make it as hard to understand as possible without making it incomprehensible. For instance, Lang introduces a major stumbling block in his book Algebra when he fails to explicitly state that groups are closed under their operation.

>>14707499
If you find that hard or surprising you should be reading a much simpler book.. Lang is a high-level treatment.

>>14707573
The person you replied to is Tooker.

>>14705883
>she
stopped reading there

>>14707573
Yes, you have repeated my own previous defense of Lang's book by identifying it as a graduate level text. However, the usual program in advanced books is to briefly restate the basics at the beginning, and in the absence of any ego-preserving denials, I think Lang would agree that was an unfortunate omission not to mention tat groups are closed under their operation in the first section of his book reviewing the undergraduate-tier basics of what a group is. Pic section is absolutely not a "high level" treatment of groups. It is an extremely low level treatment in which Lang's tendancy to maximize brevity seems to have resulted in an unfortunate omission.

Advisory to third parties: this person's >>14707573 inferred claim that Lang's introductory review of groups is "high level" is false, as can be verified by consulted the PDF. The book is a "high-level" graduate text but this introduction is a review of the prequisite material, such as the notion that groups are closed under their operation. As a reinforcing example, "high level" graduate textbooks in quantum theory always review the fundamentals of linear algebra in a brief introductory section. Lang has done something similar reviewing what a group is.

>>14707771

The Real numbers and their consequences have been a disaster for the human race

>online class on diff eqn
>using webassign
>ask professor if a certain method works, if it sounds right, maybe some tips
>response "yeah I don't think so"
>problem has a button that says "need help? Read it"
>click on button
>takes me to the PDF of the book, not any page or example in particular

AHHHH I HATE ONLINE CLASSES AHHHHH

>>14707060
cant you just look at like n+1 points and reason that them needing to keep their distance constrains the possible functions enough

>>14708310
Zotero

any good semi-introductory ODE textbooks? Preferably focused on theory, not 1000 computational exercises.

>>14708794
ODEs by Vladimir Arnold

Mathgods.
Is it possible for you to unequivocally btfo the empiricstfags?

>>14707060
isometry preserves geodesics, thus it preserves parallelograms, thus it preserves vector addition, thus it's linear

>>14708815
That is a really asinine way to say that isometries are affine maps. Just note that since an isometry preserves the inner product it must preserve angles. Done.

>>14708830
>Just note that since an isometry preserves the inner product it must preserve angles
and that implies it's affine?

>>14706252
Wow, I didn't know that conservatives were correct when they say that fascism is socialism. I guess it makes sense, why else would fascism appear <60 years after

>>14708794
cant find a good pdf for that

>>14708794
The "theory of ODEs" is developing an inutuition for guessing solutions, and learning how ODEs are classified so that you can look their solutions up in tables. I think you have the wrong idea about it if you want to avoid 1000 exercises.

>>14709105
I wanna do exercises, but not "solve these 10 linear equations" type, something with more proofs

>>14707028

what does he mean by this

>>14707028

What does he mean by this?

https://www.youtube.com/watch?v=J8RyoL8sS0g

>>14709105
Fuck off schizo

https://youtu.be/1aUeKf4Wg7M?t=1835

>>14709462
>56:44
kek

>>14708855
Then you can use the parallelogram identity of the norm to derive that it's linear. You want to show | af(x) + bf(y) - f(ax+by)| = 0.

>>14709191
What he wrote there was completely reasonable though.

Let's suppose I'm a known schizo crank with no research experience, no contacts in academia, no way to even get an arxiv endorsement, etc. Yet, despite all odds, I achieve some monumental result (solve P vs NP or whatever). What would my next steps be in order to get it to the desk of someone qualified to review it?

>>14709642
Spam professors with it until someone is curious enough to put in the time.

>>14709642
The next step is to formalize your proof in lean.

>>14709642
Share it with anons on /sci/. We're the only people taking it seriously. Avoid acadummia. Why would you want to humiliate yourself by begging professors to read it even though they explicitly despise people like us? University is not a place for intellectual discourse anymore but only an institution of midwits enforcing social conformity upon other midwits.

>>14709649
He only said that he achieved the result.
There is no way he would even know what a proof looks like.

>>14709673
Indeed. The process of writing a proof is so complex, abstract and intellectual that he could not possibly know it unless he has at least the level of knowledge of a first year undergrad student. As we all know, first year undergrad are the pinnacle of knowledge and wisdom, and the depth of their insight could never be replicated by only reading books on your own.

>>14702941
You might like some of the adaptive MA indicators Ehler made? eg https://www.mesasoftware.com/papers/MAMA.pdf .

>>14709638
No it was retarded.

Does anyone have access to this book and could upload it on libgen?
https://www.cambridge.org/co/academic/subjects/economics/microeconomics/game-theory-2nd-edition?format=HB&isbn=9781108493451

>>14709673
What the fuck is "achieving the result" if there is no proof? I guess I'm not schizophrenic enough to follow.

>>14710599
Proof by intuition.

>>14709105
I can't be the only one who gets mad at fields of math where there are no systematic ways to find solutions?

Also, is anyone here a nuclear phycisist?

>>14709642
Just say you found a remarkable proof but you don't have enough space to write it.

>>14710847
> mad at fields of math where there are no systematic ways to find solutions
So you're mad at everything that mathematicians do?

>>14707499
I am of the same opinion.

>>14707499
>For instance, Lang introduces a major stumbling block in his book Algebra when he fails to explicitly state that groups are closed under their operation.
idiot. He explicitly defines a group as a special monoid, which he earlier defined as a set with a, and i quote, "law of comoposition" which upon reading is just a binary operation, which is closed by definition. If you didnt grasp that simple train of logic you shouldnt read Lang.

>>14709642
Try a few more modest steps, approach a math department and participate in seminars or talk with people there on different math topics, convince them that you have a good command of math and over time mention partial progress until you tell them that you solved the problem

>>14706036
This is very basic shit that any highschooler can do in their sleep, if you can't understand a set, you shouldn't do math.

Is it worth reading books by E.T Bell? I heard that he just makes shit up.

>>14711538
>if you can't understand a set
Can (you) define a set?

>>14711607

.

>>14711596
Naive set, autismo.

>download every libgen and scimag torrent
>run recoll on it
I contain all knowledge, I am a near demi-god.

I’m gonna post some chapter dependencies, you guess what area of math it is.

This one is more likely to be recognized I think

>>14712217
THE RISING SEA
Foundations of Algebraic Geometry
>>14712220
Idk some linear algebra book, those usually have 13 chapters

>>14712351
>THE RISING SEA Foundations of Algebraic Geometry
Correct!
The other answer is a good guess, but no.
I’ll let others try guessing.
Also anyone else can feel free to post some of these chapter dependency diagrams, I thought I had more books with them but I can’t find them

>>14712391
I feel like this graph is stylized well enough that it might actually just come from some graph theory thing
>>14712393
Something autistic
>>14712394
Differential geometry?
>>14712398
Maybe some type of analysis
>>14712428
No clue
>>14712434
Something like number theory?

>>14712447
all wrong

>>14712455
Sovl
latex=Souless

>>14712469
>>14712462
>>14712455
>>14712451
Literally what, just read the book. Take you less time than trying to use this to skip content for some reason.

>>14712778
>yeah bro just spend a month on this chapter it will take you way less time than looking at a page for 5 seconds.

>Homology and cohomology are finally starting to make some sense.

[math]2+2=4[/math]
[eqn]2+2=4[/eqn]

>>14699453

I have a good idea about linear algebra (from learning QM) and a grasp and a half about topology. What book should I use to start reading about Abstract Algebra??

>>14713063
Lang

>>14713066
>When I first saw [Lang's Diophantine geometry], about a year ago, I was disgusted with the way in which my own contributions to the subject had been disfigured and made unintelligible. My feeling is very well expressed when you mention Rip van Winkle!

>The whole style of the author contradicts the sense for simplicity and honesty which we admire in the works of the masters in number theory - Lagrange, Gauss, or on a smaller scale, Hardy, Landau. Just now Lang has published another book on algebraic numbers which, in my opinion, is still worse than the former one. I see a pig broken into a beautiful garden and rooting up all flowers and trees.

>Unfortunately there are many "fellow-travellers" who have already disgraced a large part of algebra and function theory; however, until now, number theory had not been touched. These people remind me of the impudent behaviour of the national socialists who sang: "Wir werden weiter marschieren, bis alles in Scherben zerfällt!''

>I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction - as I call it: theory of the empty set - cannot be blocked up. ...

Learn analytic number theory.

>>14713063
Gorodentsev for undergrad and Lang for grad is the /mg/ curriculum

>>14702461
kek

>>14713202
Why has Gorodentsev suddenly began being mentioned here these days? Any мaтфaкepc here?

>>14713202
>>14713725
Is it actually good? Also do you know any sites with (student) solutions to these books?

>>14713731
Pretty good, but for a beginner who lacks mathematical maturity those books hould serve as supplementary given their terse and abstract manner, motivation and context are also pretty much neglected, so you better have at least vague idea of the subject beforehand. Quite a few students get butthurt from his lectures at Moscow's best math faculty.
>Also do you know any sites with (student) solutions to these books?
Just ask here, almost all exercises there are either trivial or are easy to do with some googling. They might appear difficult if you're not used to abstractions.

>>14703081
That's a categorical product. It makes more sense when you're looking at particular categories, like groups, undirected graphs, or topological spaces

>>14699453
What is geometric algebra used for?

>>14701051
>How do I into Geometric Algebra?
I am interested in this also

>>14713845
Why are you interested in it in the first place?

>>14713852
People say it has practical applications, and I don't know what might be more useful to learn

What's a formal proof of the following?

((A or B) and (C or D))
=>
((A and C) or (A and D) or (B and C) or (B and D))

What I'm having a problem with is opening two case distinctions at once.
Which is to say, let's say we are allowed to use the principle of distributivity,
(A and (C or D))

Thx.
=>
((A and C) or (A and D))

But I'm still missing something.

>>14713956
distributive property of and operator over or operator.

>>14713956
The "Thx." should have been at the very end

>>14713956
Just write a truth table with all cases. It always works.

>>14713897
Well, it has, but the context of geometric algebra assumes you're familiar with smooth manifolds and differential geometry, and are comfortable with tensor fields which of course assumes a solid grasp of linear algebra, but you can learn it along the way (which is imo the best way to learn it). And if by practical applications you mean the ones in physics, then it would also probably involve algebraic topology and differential equations. So basically all that stuff is worthy study by its own merit, I can't imagine going through all of that just for the sake of geometric algebra, especially given the fact that you can't even have a motivation for geometric algebra if you're not already familiar with those topics.

>>14713897
Just learn differential geometry. The people who espouse GA over the standard Cartan/Koszul/whatever formulation of DG are the same ilk who argue for the gauge integral or whatever over the Lebesgue integral.

Trying to learn vectors and 3D graphing. When going from cartesian to spherical given a vector, can I just use the relationships in pic related or does that only apply to coordinates?

What do you mean exactly? Spherical coordinates do not form a vector space?
> When going from cartesian to spherical given a vector
I guess by "given a vector" you meant the same thing as "given coordinates". Coordinates correspond to vectors in Euclidian space, but on the sphere they do not .

>>14714046
>>14714135

>>14714022
>>14714044
>smooth manifolds and differential geometry, and are comfortable with tensor fields
Also I can't understand these. I thought GA was an alternative.

Hey guys, terrible at math, sorry in advance.

I've been learning and loving it until I arrived at lateral/complex/imaginary numbers.

The YouTube video said Bombelli thought it was a 'hack' to solve problems like this and it feels like it as well.

Is there no other solution to x3 = 15x + 4 without using complex numbers?

Do you think there are other types of number systems we've yet to discover other than normal, negative, imaginary?

>>14714135
Sorry, my wording is pretty poor. So, the examples I've looked at typically start with a point, say (1, 2, 3), and convert that to spherical coordinates by simply plugging in to those equations I included >>14714046. What I'm asking is if the same process applies to a given vector, say V=4(a_x)+5(a_y)+6(a_z), to get it into spherical form A=A_r(a_r)+A_theta(a_theta)+A_phi(a_phi). I hope that makes a bit more sense.

>>14714173
>say V=4(a_x)+5(a_y)+6(a_z), to get it into spherical form A=A_r(a_r)+A_theta(a_theta)+A_phi(a_phi)
Of course you can use that to write nonzero vector's coordinates in the spherical form. That's just because you're changing coordinates. But it's not a map of vector spaces, you're just writing your nonzero vector in a different coordinate system.

Does anyone have tips on how to learn and retain math/learning as efficient and as fast as possible? I'm at the graduate level (just starting). Anything helps.

>>14714300
dont worry about remembering everything. for me at least with math, its use it or lose it. i cant remember half of the theorems i proved pertaining to PIDs or handlebodies or T3 spaces, but the nice thing about math is that everything is easier the second time. If i had to teach a class in algebra or topology i could do it, even if i dont have every fact and proof about the subjects memorized. Retention isnt all that important.
As for learning, what helped for me was writing and completing proofs. You will notice that in math books and papers pretty much no proof is actually complete. There are always steps to fill in, and filling those in helped me resolve any doubts i had and helped with proof thinking.

>>14714254
I think I understand what you're saying. So if I were to solve for my example, the spherical representation would be V=8.77(a_r)+0.896(a_theta)+0.818(a_phi), but this doesn't give me any "new" information, it's just the same thing on a different graphing system.

>>14714148
No one should be allowed to give their “opinions” on complex numbers until they’ve studied galois theory, complex analysis, complex geometry, and algebraic geometry

>>14714143
You don't need those for GA, he's a brainlet. You (can) use GA for differential geometry though, there it replaces differential forms (which would be where the (wrongly) supposed need for tensor fields comes from).
GA is on the same level as linear Algebra, and any good GA Learning will cover all of Linear Algebra as well. So I'd give it the same prereqs as LA, maybe 1 run of LA just to make it easier

>>14714546
Thanks! Do you have any recommended resources for learning GA?

>>14714645
I've mostly used Alan Macdonald's books and the results for searching "Geometric Algebra" on YouTube (mathoma and sudgylacmoe videos come to mind), along with general Linear Algebra resources.
You can also get some help by looking for things on Exterior / Grassman Algebra, but they're applied differently enough that it could be confusing OR enlightening for you.

>>14714148
if polynomials of degree 1 have 1 root
and polynomials of degree 2 have 2 roots
then one might expect polynomials of degree 3 to have 3 roots
I don't understand why this feels like a hack to you.

Consider the integer polynomial p(x)=2x-1, the root is 1/2 and it is certainly not an integer, so the ancient greeks created the rational numbers. Similarly the field of complex numbers was created to deal with limitations of the real numbers. We say the field of complex numbers is an extension field of the real numbers, which in turn is an extension field of the rational numbers.
The field of complex numbers is special in that we don't need another extension field (for polynomial solutions).We say the field of complex numbers is algebraically closed because given any complex polynomial p(x), all the roots of p(x) are in the complex numbers (cf. fundamental theorem of algebra).

>>14714673
thanks

I just saw this and can't believe it https://www.youtube.com/watch?v=H46ON2FB30U

You can write formulas and draw diagrams on the fly! Do you guys know any other tool that can do math notes like this?

>>14714546
your GA

>>14714148
>Is there no other solution to x3 = 15x + 4 without using complex numbers?
>what is 4

I was doing a bit of reading on Godel's incompleteness theorem, let's say we have a statement that is either true or not but can't be proven either way, and let's say that statement is something like "there is an infinite number of primes" and since this statement can't be proven untrue we

>>14716720
desmos graphing calculator

>>14704443
well sure, you can certainly define addition however you want but 99% of the time it's going to be useless. We use the 2+2=4 model because it's useful in its applications to the real world.

you fucking idiot

>>14709642
>solve P vs NP
This isn't math.

>>14714148
Imaginary numbers weren't really discovered in the sense that, say, the atom was discovered. As another anon said, linear equations always have one solution, quadratics usually have 2, cubics 3, etc., and we introduce complex numbers as a system so that usually becomes always. Only after that did we gain deeper interpretations of complex numbers as rotations or matrices, or did we abstract them further to algebras like the quaternions. This is how pure mathematics tends to work. You create objects or systems that solve some problem, but initially those new objects don't refer to anything but themselves and the formal rules by which they work. So is it a trick? Sure. But it's a beautiful trick and one spurred decades of scientific and mathematical progress.

what did latex mean by this

>>14714148
>Do you think there are other types of number systems we've yet to discover other than normal, negative, imaginary?
just take the square root of -i to go to a even deeper number system

>>14717799

>>14714148
https://en.m.wikipedia.org/wiki/Frobenius_theorem_(real_division_algebras)

>by elementary differentiation...
>incomprehensible esoteric jump between lines
ah yes, the math experience

What is the best way to learn xcas

A few Qs

What are the main areas of math at an above high school level?

Calc
Stats
Algebra etc

And what are some books to take me from, say, a very basic understanding (very) of these subjects to just a moderately good one where I can understand what formula are trying to do, notation (I don't have to be able to solve them).

I just want a good overview. Sort've like But How do it Think for PC was. Should I try dummy books or are they too simple?

How long till I can understand Lurie's tomes? https://www.math.ias.edu/~lurie/

>>14718102
This is the list of mathematical subjects used to real math research.
https://zbmath.org/static/msc2020.pdf

If you want some books you can start with reading the whole Bourbaki Elements of Mathematics series. They start with nothing but it goes very deep.

>do okay in algebra for babbys
>do okay in calculus for small children
>do okay in calculus for teenagers
>linear algebra for college-aged lads makes me cry and scream and shit my pants and fail

what is wrong with me?

I have never understood anything in math intuitively in my life. I can't do anything more than beginner algebra correctly because I haven't been able to find a rigorous system of logic for using it. What do I do? How can I fix this?

>>14719144
Just follow the definitions, other than that it's just intelligence

>>14719418
not him but i still don't get it even for algebra

I start calculus in uni in a couple months, I want to read something that will give me a very elementary, intuitive grasp so that I don't go in without lube
What book would be good for this?

>>14719442
Any hard mathematics textbook, any level oriented challenge is solvable on a "advanced" to "beginner" level. E.g., trying to play intricate guitar songs then switching to the simpler songs

>>14719411
Just keep going and solve exercises. It's pretty normal to not have an intuitive grasp of anything on your first go. A lot of stuff you don't understand now would eventually feel like second nature if you keep it in your head for long enough,

>>14699453
Starting summer grind for exams, wish me luck lads, It's going to be a hell of a ride:
> Week 1: grind out measure theory fundamentals
> Week 2: grind out linear functional analysis fundamentals, start reviewing for statistical learning theory exam
> Week 3: Go hard on statistical learning theory, start with elliptic PDE theory and optimal control theory
> Week 4: Increase intensity on elliptic PDE theory, wrap up learning theory, continue optimal control
- learning theory exam -
> Week 5: Wrap up pde theory
- PDE theory exam -
> Week 6: Wrap up optimal control
- optimal control exam -

I already know the applied subjects decently well, it's just a matter of ironing out the details and getting the fundamentals down, as an engineer in math grad school the fundamentals are my biggest issue.

Literature being used for reference:
> Measure theory: lecture notes from uni
> FA: Linear Functional Analysis by Rynne and Youngson
> Learning theory: Understanding Machine Learning by Ben David and Schwartz
> PDE theory: Partial Differential Equations by Evans
> Optimal Control: Calculus of Varietions and Optimal Control by Liberzon

Thanks for reading my autistic blogpost.

>>14719493
>Just keep going and solve exercises.
I don't really know what to say to that. When should I give up if I still fail exercises?

>>14719554
What exercise are you currently failing at? I will help you out.

>>14719554
I don't know, anon, it too much depends on context and your goal. In the beginning of your mathematical journey you may just have not yet absorped patterns of basic mathematical thinking, so don't be discouraged from googling solutions if you have trouble understanding what's even asked of you. In my case I remember I couldn't even wrap my mind around things such as isomorphism of vector space with its dual and determinant to me appeared to be such a cumbersome notion that I didn't even expect I'd ever have an intuitive understanding of it. But now it's all easy, I understand all the LA theorems intuitively and can come up with a proof without memorization. And I had more or less the same experience with topology, undegrad group theory and some other subjects. It can be quite a torture in the beginning to learn some area you have no clue off, but overtime it becomes less and less and eventually you'll be able to grasp things more or less right away.

Do you think this schizo also posts here? I found him in random yt video where he "proves" Collatz Conjecture

https://www.researchgate.net/publication/346967755_The_TRUE_Mathematics_of_Infinity_for_Scientists_and_Engineers#pf4

>>14719556
I don't see the point of that since I can't ask you for every single exercise or topic. Maybe my mind is too belligerent to learn it. But that would take years.

>>14719653
Your problem seems to be a lack of focus/inability to break down a problem into smaller parts. That's why you feel overwhelmed.

>>14719617
That will never be proven without a completely new approach unknown to the most sophisticated mathematicians around. It's like a black hole. We don't even bother reading attempts on it unless it's well-typed up because we've seen countless people try and fail.

If you cant give an exact answer you wont make it

The five pillars of contemporary mathematics (from most to least foundational)

> Mathematical logic
> Theoretical computer science
> Pure mathematics
> Mathematical physics
> Applied mathematics

I'm confused how this is the binomial theorem and not bernoulli's inequality
typo?

>study stochastic processes
>become immune against category theory
Come at me and categorify a Poisson point process or a Brownian motion.

>>14723362
[eqn](1 + \delta)^n = 1 + n \delta + \frac{n (n-1)}{2} \delta^2 + \frac{n (n-1) (n-2)}{6} \delta^3 + \ldots + \delta^n > n \delta[/eqn]
Now divide by the LHS and RHS
[eqn] \frac{1}{(1 + \delta)^n} < \frac{1}{n \delta} [/eqn]

>>14723401
Thanks, wouldn't Bernoulli work as well though?

>>14723379
Helemskii has a textbook with a categorical treatment of functional analysis.

>>14719617
Holy shit this is some great schizo-ism. Any time you see someone talking about Occam's Razor as though it's relevant to proof correctness, you know you're in for a good time.
This one isn't as crazy as some I've seen - he seems to have basically reinvented big-O notation for some things (particular note should be given to the bit where you could literally replace \phi with x and get the same sorts of summary tables that they give algorithm analysis people for counting how many steps an algorithm takes), and an informal version of density for others. (He also, hilariously, used a computer to 'analyze' the average number of distinct rational numbers in an nxn table of num/denominator -- arriving correctly at about 0.608, but failing to note that the mathematics of infinity that he disparages as 'drivel' already give an explanation for why that result is true (approaching 6/pi^2))
And actually, his stuff about the RIemann zeta function is kind of interesting? Again, he's really studying growth rate, but on the other hand that's all we have for the divergent points
Other parts are crazy or inconsistent, though.There's a certain Platonism to treating infinity like a real object (it's also a good sign when you read "unknowable" in a purported math paper). He treats Hilbert's Hotel and Littlewood's Paradox as though they were real-world situations, applies a flimsy allegedly-mathematical construct to "solve" them, and then moves on before the reader realizes he hasn't in any way addressed the actual points Hilbert and Littlewood were trying to make. His "proof" of the countability of the reals relies on the idea of some limit of precision, which may be all well and good for engineering but isn't relevant to what everyone else is talking about with the 'real numbers'. It all falls apart into Tookerism when you reach the 'infinite limits' section.
tl;dr it tries to be constructivism and ends up being proof-by-assertion. oh, and they don't understand bijections.

>>14720784
[math]
-2×\frac{π}{6} tanh^{-e^(i×π/6))(x)} - 2 i Li_2(e^{(i×π/6)}) + 1/2 i Li_2(e^{(2 i×π/6)}) [/math]
Your turn.

>>14724380
>some wolfram dogshit
logarithmic integrals dont count. Also theres no need for the answer to involve any trigonometric functions. Hint: Catalan's constant is involved.

>>14699453

It's over. I'm near 40 and the Fields medals have recently been issued. I will never be awarded a Fields Medal.

>>14724431
>your solution is wrong, because... IT JUST IS
Solve the integral in my image, negro.

>>14724431
Wolfram also gives the answer with Catalan's constant
[math]
\frac{1}{6} (8*C- i π^2 - 2 π tanh^{-{\sqrt[\leftroot{-2}\uproot{2}6]{-1}})} (x)
[/math]

bump.

Why does the theorem hold for k=0? I don't understand this proof at all.

>>14726586
rem(c0, di) = c0 since c0 < di

>>14714498
Someone was trying to slide the geometric algebra edition. What does that tell you bros?

>>14724380
How did you get that answer?

>>14703618
> 2+2=4
Why are twitter schizos so obsessed with this equation? There is nothing interesting to say about it that you didn’t already know as a child. It’s like, half-apples can’t create 1 whole apple #woah #wow
Are they all hypnotized or something.

>>14703618
> 2+2=4
Why are twitter schizos so obsessed with this equation? There is nothing interesting to say about it that you didn’t already know as a child. It’s like, 2 half-apples can’t create 1 whole apple #woah #wow
Are they all hypnotized or something?

>>14727567
who the fuck gave you a computer? If I don't get my dinner and some head in the next five minutes I'm getting my fucking belt.

>>14703618
this is what formalism does to a mf

>>14727643
nothing wrong with formalism, but it is a sophomoric take, especially from someone ostensibly interested in foundations
>>14727567
from what I can piece together, there was some troll on twitter 2+ years ago who claimed something along the lines of:
>Believing in trannies is like believing 2+2=5!
instead of laughing it off as a ridiculous comparison, or recognizing that believing 2+2=5 is literally a plot point from 1984, the twitter community swallowed the bait whole, and started arguing:
>Well actually in some contexts 2+2 doesn't have to equal 4!!
with the expected babby examples where you are working with some object other than the usual [math]\mathbb{Z}[/math]. really embarrassing to watch, i think Tim Gowers even took the bait. also at some point the focus 2+2=4 shifted from trannies to racism, because hey it's Twitter

>>14699453
Why is there a need to define an outer measure? The regular lebesgue measure suffices when constructing the lebesgue integral, does it not?

>>14727836
>The regular lebesgue measure
What do you mean by regular lebesgue measure - do you just mean the measure of an interval? The outer measure is useful because it can be used on sets that aren’t an interval or a countable union of intervals. For example the set of irrationals has outer measure 1 but it can’t really be expressed as a union of intervals in any useful way.

>>14727836
>The regular lebesgue measure
What do you mean by regular lebesgue measure - do you just mean the measure of an interval? The outer measure is useful because it can be used on sets that aren’t an interval or a countable union of intervals. For example the set of irrationals has outer measure 1 but it can’t really be expressed as a union of intervals in any useful way. This expands the set of functions you can integrate.

>>14724380
>>14727531
I'm curious anon, if you're here please bestow your knowledge and wisdom upon me. I'm a but a brainlet who needs guidance.

>>14727856
Oh I see actually. Dumb question then.
Thanks

Who was the most handsome mathematician?

>>14727856
>the set of irrationals has outer measure 1
you mean the ones inside [0,1] have outer measure 1

Why do fags (algebraic geometers) write lowercase [math]k[/math] for a field?

>>14728052
German word for fields starts with k

>>14728125
is it kurzgesagt

>>14728125
>German word
shouldnt it be a capital K then

>>14728052
>>14728272
Some algebraists like to use fraktur letters for algebraic structures so it might be [math]\mathfrak{K}[/math] which is a capital letter but can look like a lower case k to the untrained eye.

>>14728286
No. This is from Lang's Algebra, for example.

>>14728286
>>14728298
and here is Liu's Algebraic geometry and arithmetic curves

>>14714148
inverse of addition leads to negative numbers
inverse of multiplication (repeated addition) leads to fractions
inverse of powers(repeated multiplication) leads to imaginary numbers

you can keep going and "invent" numbers like "the inverse of power tower of negative length" or something, but the problem is the further down the more complicated and nonsensical it gets. Already, you can see inverting multiplication leads to needing to add a rule about division by 0, and powers actually have TWO inverses being logs and square roots. A power tower actually has multiple inverses, you would have to define what you mean by each one. And also what means when your new number is applied to negative, imaginary numbers, 0, 1 and any combination of such as well as how it interacts with itself.

What ends up happening is you lose a lot of information the further down you go, especially powerful things like associativity/commutative property and things like writing equations becomes difficult/impossible. It's ultimately a meaningless endeavor.

>>14728052
its because K is used for the function field K/k
>why not a different letter
because the function field is a special field extension related to varieties(integral schemes) over k.

>>14721780
All mathematics is divided into three parts: cryptography (paid for by CIA, KGB and the like), hydrodynamics (supported by manufacturers of atomic submarines) and celestial mechanics (financed by military and by other institutions dealing with missiles, such as NASA.).

>>14728870
>What ends up happening is you lose a lot of information the further down you go, especially powerful things like associativity/commutative property and things like writing equations becomes difficult/impossible. It's ultimately a meaningless endeavor.
Thats what they told Cardano too, faggot

>>14727802
>instead of laughing it off as a ridiculous comparison the twitter community swallowed the bait whole, and started arguing:
>>Well actually in some contexts 2+2 doesn't have to equal 4!!
Brilliant work. Any philosophanons have a name for this trick? Where you get them to accept some ridiculous starting premise, and then argue within that frame?

How many hours a day did you spend doing math problems back in high school?

>>14730011
0 i didnt even do my homework. Made no difference as far as i can tell, still did research just as good as my peers who went to math camps since they were 8 years old. Not like you learn anything in high school. Only thing is i wish i new about stuff like IMO earlier, though being honest i still wouldnt have done math then if i did.

>>14730023
>Only thing is i wish i new about stuff like IMO earlier
To excel at IMO etc you need special training. In practice this means you have to be raised in a university town, ideally by academic parents, or scouted for the role like in the USSR and china. I think a lot of western IMO potential is being squandered by lack of scouting, the smartest rust belt kids just ace high school and go to a state school, maybe get 1.5/12 on the putnam again due to lack of training and thats that

>>14730033
>putnam
thats another thing i never bothered doing the Putnam. Again, didnt make a difference still went where I wanted to go for grad. I only really started doing math seriously every day once I started research. Im really not sure I would be so different ability-wise if i had spent thousands of hours doing more mathy things as a kid. Maybe better at calculating things in my hea i guess.

Is there any formula for π that involves number 1985?

>>14729992
>Any philosophanons have a name for this trick? Where you get them to accept some ridiculous starting premise, and then argue within that frame?
I hereby propose the title "Troll's Gambit Accepted"

>>14729992
>>14730584
>when you do a reductio ad absurdum but they respond with an amplexus absurdi

i loved math when i was kid, then i changed school and had shitty teacher who knew math less than some of my classmates and she didn't care about explaining concepts just told how it was without why. so i never paid much attention to it again cause of retarded childish brain. now that long years passed after school i feel like i am missing out on very beautiful thought process and world by not havung deeper understanding of math. i am drawn to it without any particular need.
is there any free online course or resource or application or whatever that can help me learn it from beginner to advanced level. all i know basically is school level and don't even remember some of it.
has anyone been in similar situation and could climb out of their ignorance?

>>14728016
the one reading this post

ok I'm a compi-fag and I want more mathematics.
I've studied number/graph/probability theory, linear algebra, the basics of calculus, combinatorics, discrete math in general and a tiny bit of abstract algebra (which I may focus more on).
What advanced math field should I study now if I want to become a math god in the CS field? Should I just start looking more advanced topics in the fields already mentioned or is there something I have yet to discover?

What is the intuition behind this predicate logic rule? PFx |- P∀xFx, provided x isn't free in P.

What is the intuition behind this predicate logic rule? P->Fx |- P->∀xFx, provided x isn't free in P.

>>14731123
how can you have "intuition" for a rule? what's the intuition for "a" being pronounced ay?

>>14731123
x was symbolic in F thus it did not matter what x is, and therefore F holds for all choices of x.

>>14728904
Just write K for the field and L for the extension field

>>14731074
donno, Hilbert space theory and Control theory?

Not asking you niggas to do my homework, but would I be on the right track with Proof by Induction?

Text says "show that... for all.."

>>14731975
you never go wrong with induction

is there a formula, or does OEIS or anywhere else have sequences [math]a_1^m, a_2^m, ..., a_k^m,...[/math] where [math]a_k^m[/math] gives the number of trees such that each of the nodes can have 0, 1, 2, ..., m kids and k is the depth? For m = 2 it would start 1, 2, 10, ...

>>14731975
I always prefer combinatorial proofs because they are more intuitive.

>>14732272
I have not heard about this one yet - is this some advanced stuff? Should have said that the problem is from an Analysis I test, that's why I'm trying to wrap my mind around it so I don't get filtered out by it one day.

>>14732327
Analysis? Then yeah induction is the expected proof method.
Combinatorial proofs aren't advanced though. You're basically saying "LHS counts [that thing], the RHS also counts [that thing] and so they must be equal".

For example LHS of pic rel counts the number of ways to choose [math]k[/math] items of [math]n[/math] (by definition). RHS counts that as well since in a way to choose them you can either choose the last element and choose [math]k-1[/math] out of the rest [math]n-1[/math] or not choose it and have to pick [math]k[/math] from the rest [math]n-1[/math].

Combinatorial proofs are very easy to do in your head if you're experienced. It is a good technique to practice since it helps a lot with combinatorics problem solving.

does calc 1 utilize inequalities

>>14732366
OK, thanks a lot for the detailed explaination. Will keep this in mind, though for now I will go with induction.

>>14732272
>I always prefer combinatorial proofs because they are more intuitive
Gas yourself retard.

>>14732555
>hasn't done a single hard combinatorial problem to see how useful combinatorial reasoning is.
Good luck proving every single step you do with induction faggot.

>>14732626
Uses identities that have 100+ lines of calculus and algebra and number theory for proofs, "bro so intuitive it's just LHS=RHS"

>>14732631
Oh so you don't understand what a combinatorial proof is then?
Tell me when did I use lines of "calculus or algebra or number theory" proving the simple identity I posted. It's literally the best way to understand them and to also remember them. But I guess you think doing algebraic manipulations every time you want to use this identity is more formal or whatever.

>>14732642
You don't "prove" anything using random identities nobody understands. This is like spinning up a virtual machine, to run a CAS to give you 2+2. Your proofs are complete bloat.

>>14732645
>You don't "prove" anything using random identities nobody understands
But you PROVE the identities using combinatorial arguments which ARE a valid and formal proof method. I don't understand why you don't agree with the latter part.

>>14732648
No, (you) don't prove the identities, you just use them without ever proving them, maybe you glance at someone else's proof, not sure if it's right as (you) didn't check, then when (you) use some semi-obscure identity in several papers, UH-OH turns out they made a mistake and that identity is false, there goes 4 years of work.

>>14732657
Well to be honest I don't have much experience regarding what you mentioned. However getting bored of actually proving your steps just because you can reason them in your head isn't a problem of the proof technique but of the mathematician.
There are ways you can guarantee the correctness of the combinatorial proofs you use because yeah you might describe them in you head but as you said you can make silly mistakes. To guarantee the correctness you, after proving them yourself with combinatorial arguments, you formalized whatever you described. That's as formal as doing anything else really.
The magic behind this is that you reason them quickly while you're solving the problem as to not waste time proving something which gets you nowhere.

There are only 800 questions, there will be 20 randomly selected questions on the exam, the student has prepared perfectly for 400 questions. What is the probability that he will get perfect score?

>>14732863
[eqn]\frac{400 \choose 20}{800 \choose 20}[/eqn]

>>14732869
Elaborate further?

>>14732872
7^10*(-7)

>>14732872
In other words, there's more of a chance one would
see questions they're not familiar with and thus
may not get the perfect score than if all the
questions are from what they prepare for.

From >>14732887 , the number of possibilities
of a set of 20 random questions (from 800) is
quite large wherein a set of 20 questions with all
one studied for is a million times smaller, but
still large. This is essentially lottery-type chances.

>>14732863
Misleading question because you are assuming he will always fail at the questions he never prepared for.

>>14699453
8/8 Lions Gate Portal Is Open! A Message for you from Archangel Metatron. Are You Ready?!!!

https://youtu.be/YwPonMRZA-0

>>14730584
>>14730687
both good
>>14731123
if x isn’t free in P, the \forall x does nothing. if there are any x-symbols within P, they are already bound to another quantifier. its a bit like scoping in computer programming languages i guess. but in practice people avoid reusing x in this way because its fuggin confusing
>>14733148
based

>>14700571
Unrelated to original question, but I would like to know the motivation behind the specific definition of "pair". Is "{a,{a,b}}" meant as a way to stealthily incorporate an order?
Why not define pair as a finite set, cardinality = 2? Then add a trivial order / indexing to separate (a,b) from (b,a)? (this is the "function pairing" way this anon mentioned, right?)
I'm assuming this is not really related to basic product sets shenanigans and instead more so to the whole foundational logic side of things, that I am frankly not as familiar with as I should be. Perhaps someone could give me some pointers/recommendations? Much thanks.

>>14733431
>function pairing
How do you define a function?

>>14707813
Third party, can't help but overhear the conversation.
The closure property is likely inferred when he defines the group as a monoid with inverses. From the section you picked out, it looks like he touched on monoids in the previous chapter.
It's also sort of implied in "Let G be a set with an associative law of composition."
At any rate, it's not really something to get too hung up on.

>>14733532
Guess I didn't overhear too much, I'm a day late to the point.

>>14733431
>Is "{a,{a,b}}" meant as a way to stealthily incorporate an order?
Yes. You have to prove that there is no monkey business going on like a={a,b} but it is pretty direct from one of the ZFC axioms.
> Why not define pair as a finite set, cardinality = 2? Then add a trivial order / indexing to separate (a,b) from (b,a)?
This is begging the question a bit. Now you have to construct functions from { } and other operations, which usually is done afterwards, and then you’d probably want to write ({a,b}, f) as an ordered pair anyway. It probably can be done as you suggest, but it will wind up more complicated. That’s only if you want to found your maffs on ZFC alone, of course, there are type theories etc. that take all these operations as primitive

>>14733616

Learning a bit of linear algebra towards the end of calculus II. This is powerful stuff. Thank you for reading my blog.

https://youtu.be/fEBSx075AKs