/sci/ - Science & Math

You've clearly never studied any of this to a high level.
How is algebraic geometry so much higher than category theory?

>>5838029
This is just based off of the responses in the previous thread. Here's the original

>probability theory
>low tier

Enjoy your less than 300k starting.

Yep, these lists still don't make any sense.

>>5838012
CS theory = Mathematical logic + set theory

>>5838032
>no inter universal

>>5838046
>CS theory = Mathematical logic + set theory

HAHAHAHAHAHAHAHA, fuck no.

>>5838054

>>5838046
CS majors actually believe using basic syntax like implication symbols and knowing what a set union is makes them experts at mathematical logic and set theory. My fucking sides.

>>5838054
You're right, it actually contains a lot more than that.
Graph theory and category theory, for example. But that's okay, you can keep mocking the comp sci kids if it makes you feel better about whatever you're failing to grasp in your 'real science'.
Would you like some help with your devilishly tricky general topology homework? :-)

hello,
I'm 37 years old, and got B's in all my calculus classes. is it too late for me to study math in my free time, and then when I get good enough to go for advanced studies in maths at the university for masters and Phd

>>5838061
>Graph theory and category theory

You don't know shit about either topic. Perhaps in your shallow introduction you learned to draw a tree but in order to understand actual graph theory you'll need math way beyond what you'll ever study in a CS degree.

>>5838059
>basic syntax like implication symbols
>set union

Sure thing kiddo, maybe at your shitty university.

Complexity theory, satisfiability, decision procedures, formal semantics, model checking.

>>5838054
> posting in a internet board
> computer science is shit

>>5838012
Category Theory is the new Set Theory so fuck you!

>>5838064
>37 years old
Its too late, dude. Honestly.

>>5838075
Do you really think trivial applications of babby's first formal logic are gonna impress anyone?

>>5838012
Move Alg Num Theory to God Tier and Alg Geometry to Elder God Tier.

>number theory
>anything but shit tier

Of all mathematical fields number theory is the most useless. It's pure mental masturbation without serious applications and it's almost as bad as philosophy.

>>5838061
>topology
>hard

This is what CS students think.

>>5838061
It's funny that you're mocking topology while simultaneously insisting in knowing graph theory. Way to demonstrate your ignorance.

>>5838064
Why do you want a masters or Phd? Don't tell me you fell for the 300k starting... Just study math for fun.

>>5838085
>implying information about primes doesn't translate into information about the Riemann zeta function
>implying zeta functions aren't the answer to everything in mathematics

http://en.wikipedia.org/wiki/Zeta_function_universality

>>5838059
This.

CS uses an incredibly shallow and small subsection of Logic and calling naïve set theory "set theory" is an insult to the field.

>>5838061
>Graph Theory
G=(V,E) and König's lemma (the boring one that's equivalent to Menger’s/Hall’s/Dilworth’s/etc theorems) is 99% of the graph theory CSers use.

>Category Theory
oh look, another one of 'those' hipster.

>tier thread
>this post: >>5838074
>~20 replies
These pretzels.

>muh categories
>muh morphisms

So basically this thread is filled with a bunch of fucking nerds arguing over which mathematical topic is the most "godly"

lel

>>5838076
>> posting in a internet board
>> computer science is shit


The internet of course made and maintained by EEs and CompEs

>>5838106
By EEs and CompEs you mean mathematicians and physicists, right?

>>5838081
>trivial applications of babby's first formal logic

No, actually it's everything there is in formal logic. I understand that you tiny mind and shitty uni can't relate to any of it.

>>5838116
>actually it's everything there is in formal logic

Name one sane logican. Cannot be done. At least no one who did something new in the field. Why? It is so complex everyone goes insane who deals with it. Like realy everyone. Either that or only insane mathematicans specialize into logic.
Logicians are the nerds of mathematics. And not in a good way.

Math is far too interconnected to rank different sub fields

>>5838116
Just stop. I know this is 4chan and you are free to troll without getting slapped in the face, but your trolling isn't even funny.

>>5838121
Godel

>implying there weren't Nazis out to get him

>>5838121
Russell ?

>>5838121
Frege

if math is so smart why come you can't tell me what apple stock will be next year

>>5838130
0

sell now

>>5838121
Von Neumann

>>5838119
>>5838123
>trying this hard

Just feel free to point of what major field of Formal Logic is left out. Protip: you can't. Have fun with your samefagging.

>>5838012
>>low tier

> >>5838122
>

>>5838151
>any logic is formal
> >>5838150

>i'm 'smart' enough to know which complex math fields are better than other complex math fields and can judge others in these complex math fields
>i'm too lazy and unmotivated and boring to get a job/actually invest time to learn this

>85% of /sci/

yeah, some of you are in college or have graduated but by being here you're just showing how egotistical you are
you fuckwad

>>5838167
pineapples OP

Math is plox Mr.Smrt?

Set theory, combinatorics and mathematical logic are from shit tier.
Differential geometry is from god tier.
Category theory is from no tier (general prerequisite, lol).
CS isn't math at all.
I can list plenty of branches of mathematics that weren't included.

If you flip that around, you'll get "highest income" to "lowest income".
Machine Learning here, sure enjoying the fact that I only have a masters and make more than you "pure mathzzzz" faggots ever will in your life.
It must suck too, as I actually solve problems and make the world better while you just sit on your faggot ass making up new algebras that are completely useless.

>>5838161
This is what I meant.

https://en.wikipedia.org/wiki/Mathematical_logic#Subfields_and_scope

Formal logic is part of computer science now, and every good graduate CS degree will teach formal logic to its fullest. Half of the today's books on formal logic are written by computer scientist, and every mathematician that works in formal logic works in a CS department.

Wait by CS theory do you mean the theory of computation? As in the church-turing thesis and all that? Isn't that basically just a part of mathematical logic? Why would it be so much lower than mathematical logic other than "lol cs"

>>5838184
>Machine Learning here, sure enjoying the fact that I only have a masters and make more than you "pure mathzzzz" faggots ever will in your life.
>It must suck too, as I actually solve problems and make the world better while you just sit on your faggot ass making up new algebras that are completely useless.

So much this. AI brother reporting in. >>5838186

>>5838184
Some people receive pleasure from eating shit. It's a pretty good motivation to do so. Why can't you change 'eating shit' to 'making pure maths'?
But, I warn you, nobody knows what real shit is. Money isn't an issue.

>>5838012

I'm still in God Tier, so not going to complain.

>>5838012
>set theory
>god teir
rufuckingdumb?

>>5838317
>implying you're not some STE pleb who thinks set theory and naive set theory are the same thing

>>5838329
>set theory can get beyond composite baseline 3's

why is category theory low tier? that should be god tier

>>5838085
0/10

commutative algebra and analytic number theory a shit

>>5838564
It's not particularly interesting to study on its own, and most people who claim to like it probably don't know more than adjoints and Yoneda lemma.

>>5838655
Yeah, because those higher stacks and Isbell dualities are totally trivial and uninteresting.

>>5838012

>>5838718
my nigger

>>5838718
teehee

>>5838718

>>5838192
>Machine Learning

How to write a ML paper:

1) Pull out a random paper from the Annals of Statistics
2) Apply to random problem
3) goto (worked ? 4 : 1 )
4) Published!

>>5838718
What if I'm good at what I love, think it's easy and will make me money, and my parents told me to do it?

>Hint: It's math

>>5838085
Mathematics is the queen of sciences and number theory is the queen of mathematics.

>>5838012
also. Algebraic geometry and commutative algebra on different tiers? They are like 90% the same thing, and you use both to study the other. Differential equations and group theory (basics learned in undergrad by fucking EVERYONE) on the same tier as differential geometry and algebraic number theory?

>>5838029 was right you clearly do not know what any of these branches of math are

>>5838079
there's some 60+ year old taking topology at ucla according to my friend

how come there's so many expert mathematicians on sci, but only one or two people attempt proofs in putnam threads?

>>5838109
yeah i bet the forum was coded by EEs too!
if you compare the programming taught in high school with physics, then it may seem shit tier, but actually the subject is much wider than it may seem at first.

>>5840156
i was refering to this >>5838106

>>5840119
It's because Putnam questions are neither interesting nor do they have applications. They're also problems that have already been solved and so nothing that a serious scientist should waste any time on.

CS theory, finance, and statistics all fall under applied math. Category theory, Differential Equations, Set theory, and Mathematical logic should all fall under shit tier

>>5838096

toplol

>>5840156
>>yeah i bet the forum was coded by EEs too!

>4chan a forum
>Futaba well coded
>needing any training at all to code an image board better than 4chan

get the fuck out

>>5840119

In my case, I did putnam for 4 years and trained for it 3 of those... I've had enough of putnam problems.

>>5838012
>Logic, set theory
>Top tier

>Category theory
>Low tier

Top lel, OP confirmed for super pleb.

>>5838012
Aren't algebraic and topological K-theory fairly different subjects? I'll admit I've barely touched topological K-theory, but I had always heard they were very different beasts. I mean, it's not like you're grouping the two branches of number theory together. And besides that, it's much more of a niche subject than the other things you've listed. Why include K-theory but not, say, homological algebra? It's almost as if the person who made this heard a cool buzzword and thought he'd sound cool by placing it in the god tier.

>>5840347
>muh adjoints
>muh monads

Why are category theory fanboys such obnoxious hipsters?

>>5840426
>Not embracing the unicity and elegance of categorical mathematics
>Still living under a set-theoretical rock
>2013
Wake up, it's like you are stuck in the 1800's or something. Modern math is done using category theory.

Where is universal teichmuller theory?

>>5838167
You don't even need to have studied mathematics in order to know what hte names of these fields are.

Saying "commutative algebra top tier , algebraic geometry high tier" requires as much knowledge and understanding as saying "quantum field theory top tier, condensed matter physics high tier"

>>5840119
either
1. because they don't have talent at problem solving and are instead one of those dull-witted plodders who can read theory but are unable to apply it in any but the formulaic, intuitive ways

2. because peopel don't com on /sci/ to do hard work, they come to relax

3. because simply name-dropping fields of mathematics doesn't necessarily mean that you actually have a working knowledge of them, so perhaps sci is just full of nerdy highschoolers who go article-hopping on wikipedia

What is K-theory and why do we need it?

>>5840119
Because putnam is not a math contest but an autism contest. It does not require any knowledge of higher math. Not even the topics of OP's pic are covered in putnam. Putnam is not about knowing or understanding math, putnam is about having autistic skills at solving very artificial high school level number theory and geometry problems.

>>5840535
>baww why can't I solve novel problems with my shallow understanding and primitive problem-solving skills???

If you were better at maths you'd be able to do them.

>>5840529
Algebraic K-theory is mainly the study of K-groups of rings. For a ring R, K_0(R) is the group completion of the monoid formed by isomorphism classes of projective R-modules under direct sum, higher K-groups are a generalization of this. Topological K-theory has something to do with vector bundles, I don't really understand it, but my understanding is that it's pretty different from algebraic K-theory. We need these for pretty much the same reason we need any higher math - they're naturally interesting and help solve problems in adjacent fields.

>>5840557
Whoops, I meant finitely generated projective R-modules, obviously not all projective R-modules.

>>5840347
>>5840439
What are some nice adjoints in differential geometry?
I don't quite have a feel for the former, and I'm pretty sure a geometrical example would help.
Also, more abstractly, how do adjoints relate to (the syntax of) predicate logic. I hear they make structural truths about quantifiers more clear.

>Still living under a set-theoretical rock
Why can't we all be friends?
On a serious note, I completely agree that making functions (and thereby associativity if you want) a primary part of your axiomatics.
At the same time I really seek to understand understand why you would want to drop such a fruitful notion as the binary predicate of membership. In short, I don't see why people would want to axiomize topoi in a standalone category theoretical framework, if there seems to be no harm in having at least a weak set theory on the side.
Is it only philosophy?

Can I ask why everyone here hate Engineering so much. I'm going to Uni next year, Rutgers, and I'm interested in M.E. but is there something you guys know that I don't. I don't want to dedicate my life to a career if it's gonna end up shit

>>5840529
It's a way of studying topological spaces and their vector bundles through ring theoretic means.

>>5838072
>study in a CS degree

I never said anything about getting a CS degree - that wouldn't really be representative of the research going on in the field.
I mean, if it's so easy, why does Terry Tao think P = NP is the hardest Millennium Prize problem?

>>5838188
Because the people majoring in maths or physics want to feel better about themselves. They struggle with their course and need people to look down on.
And I'm studying joint maths and physics.

>>5840554
I can do them. Usually I solve most of the putnam involving probability, abstract algebra or analysis. It's only the number theory and elementary geometry problems I don't give a fuck about. I have neither the patience nor the motivation to waste my time with autistic high school nonsense.

>>5840597
>And I'm studying joint maths and physics.

sure you are, code monkey

I remember a year or two ago when I first came here to check it out. The EXACT same thing was on the front page, but with different data.

Is this board that boring?

>>5840600
>Is this board that boring?

Yes, it is. Now go back to your pleb boards.

>>5840426
Because you don't like category theory and are thus irritated when people talk about it.

>>5840569
>adjoints in differential geometry
I'm not really a geometer. I'm more of a topologist, so I don't think I ca answer your question in satisfactory detail. I can give you plent of nice examples in topology if you want.

>why you would want to drop such a fruitful notion as the binary predicate of membership
I guess philosophy plays a large part, as in any issue related to foundations. Personally I find it much more appealing to axiomatize sets and functions between them than to axiomatize membership.
Mind you, the two resulting systems are equivalent. An element x in a set X corresponds to a function x : I -> X , where I is a final object, i.e. a one-point set.
One of the nice things about this approach is that "generalized", or "S-shaped" elements x: S -> X occur much more naturally and allow for generalized logical systems.
You don't need to axiomatize all topoi, only the topos Set, in which we then define all our familiar constructions. Recall that a topos is just a sheaf category over a site.
Topoi provide the natural for generalized logic and constructive mathematics, but that's about all I know about them.

>>5840599
I was looking into algebraic geometry earlier in the year.
An affine scheme is essentially a geometric object made from a ring. A scheme is a collection of affine schemes glued together in a similar fashion to a manifold made of open balls. The points of an affine scheme are the prime ideals of the ring it is made form, which are given the Zariski topology.
The category of affine schemes is dual to the category Comm, and thus has Spec(Z) as a terminal object.

Why would a CS code monkey learn all of that? I typed this out in an intuitive manner to convey true understanding, as opposed to just throwing definitions copied form Wikipedia. Please abandon your delusions.

>>5840621
>An affine scheme is essentially a geometric object made from a ring.
Can you elaborate on this?
I haven't studied very advanced commutative algebra, but feel free to use algebaic varieties to explain.

>>5840621
What text did you use?

>>5840614
You don't really give an argument why not start with both at the same time. I see how it's practical for doing "fuzzy" generalizations (although I don't know why using pairs where the second entry just contains the additional information wouldn't work).

The point is that both, functions and set membership are natural - and so dropping set membership and relying on a subobject classifier even for standard sets... feels stupid.

Maybe that's just the pragmatic physicist speaking.

(logic and topology: the abstract topology concept coincides somehow with... non-full heytin models of second order arithmetic or something, right?)

>>5838012
This whole picture is just full of shit, no matter how you arrange it.

>>5840678
>logic and topology
Something like that. I have heard the buzzword "Heytin algebra" before, but I'm not really into logic that much. Also check out the Lawvere-Tierney topology for how to interpret truth values in a topos.

Back to foundations.
Well, there are a number of, in my opinion, bazarre things that happen in standard ZF which I don't like, and which dissapear in a categorical framework.
The problem occures because in ZF, everything is a set. As a consequence, the following questions are valid in ZF:
- Is 4 contained in the function f:R->R taking x to x^2 ?
- Is pi equal to the category of abelian groups and group homomorphisms?
Needless to say, these questions are pathological, and their validity indicates that this foundational system is messy and cluttered.
The categorical foundations, named the "Elementary Theory of the Category of Sets", or ETCS, gets rid of these pathologies.
http://ncatlab.org/nlab/show/ETCS

>>5840721
And note that
<span class="math">1\in (0,7)[/spoiler]
is true ;)

>>5840747
Of course. There is a function f : {x} -> (0,7) such that f(x)=1. :-)

ITT: babby math

>no calc
>no trig
>no solving for x

You don't know shit about math.

>>5840569
>Why can't we all be friends?
On a serious note, I completely agree that making functions (and thereby associativity if you want) a primary part of your axiomatics.
At the same time I really seek to understand understand why you would want to drop such a fruitful notion as the binary predicate of membership. In short, I don't see why people would want to axiomize topoi in a standalone category theoretical framework, if there seems to be no harm in having at least a weak set theory on the side.
Is it only philosophy?

... well, it's kind of the principle of set theory that it all be indicative of itself as potential, and it's just the medium of constraints of the concurrent direction incurred by the first used sets in the system, and of that there are larger ordinate (p)-ediums of context which are condusive of 3 dimensional reconstruction in series in a medium between the two contexts, and that there be an implied medium between that, and system available constraints be contextual of the orientating standard of the same, so that the point being is that you make variable matrix contexts and orchestrate them to a context of value by a rough standard but as their looked into gain more exact differentiation and more exact matrix potential constraints on the artistic of concurant mediums as the context tabulates or more accurately marinates, but that the numerical assigning and the base systems have difficult constraints on the means that if their not understood the first time, their never understood again, and it's just the point that reality understands it all and contextually narrates humans into a consistent extent, and well, it's a thing that we really need to get compasses of properties of numbers in base number systems down, so chill it.

Philosophy's in the box; w/o reading that again just answer with a box, or variable curve

>>5840678
Yeah he's treating th variables as only space, he's not assigning shape to the conditions

>>5840861
>>5840826
>>5840807
>>>/b/

>>5840721
I've thought it over again: couldn't you make topoi-like theory, but then have an \in on the side, which you declare for sets once membershipt is established (functionally)? Best of both worlds?

>>5841031
You could, but if I understand you correctly, it would be a vacuous notion. Let {x} be a terminal object in Set, that is, a singleton set.
In Set we have an internal Hom functor Hom({x},-) which takes a set X and returns the set of its elements X' = Hom({x},X).
Then an element of X is just an element of X', and it is easy to see that X is isomorphic to X', so no new information is obtained from a membership construction.

>>5841069
yes, as you just declare the set membershit after it's functionally established, there is nothing new. But I think it's still gives a more practical theory.

>>5841069
>it is easy to see that X is isomorphic to X'
Actually it may take some pushing around of diagrams to prove this, but as long as Set is axiomatized correctly, they will be isomorphic.

>>5841069
>>5841094
The main thing to realize is that members of sets are not themselves sets in ETCS. Members of sets are functions.
This is a crucial point, since this is what forbids the pathological questions I posed earlier.

>>5841098
Yes, as long as you keep >>5841103 in mind, you can use membership notation as you are used to.

>>5841103
>Members of sets are functions.
functions as in set, or abstract morphisms?

>>5841134
Abstract morphisms as far as the theory is concerned, but they can be thought of as concrete functions.
As you will readily observe, functions from a one-element into a set X behave exactly as elements of X.
For example, if p: {x} -> X is an element of X and f : X -> Y is a function, then the image of p, which is the composite fp : {x} -> Y , is an element of Y, as expected.

You may also observe that monomorphisms in Set behave as inclusions of subsets, and epimorphisms behave as quotient maps with respct of equivalence relations on sets.

>>5841161
the last bit is not completely obvious to me, although it makes sense.
So is it over the top to make the characterization of
<span class="math">\dots \to B \to A \to 0[/spoiler]
being exact, is this more than making the second arrow an epi?

>>5841188
Exactness doesn't make sense in Set, so you cannot define epimorphisms like that. This is because there is no "zero object" in Set, as the initioal and terminal objects are not isomorphic. (initial = empty set, terminal = singleton set)

What we do to define epimorphisms in general categories is the following:
f : X -> Y
is an epimorphism if whenever we have two morphisms
g, h : Y -> Z
then gf = hf implies g = h.

Dually, monomorphisms are defined like this:
f : X -> Y
is a monomorphism if whenever we have morphisms
g, h : Z -> X
then fg = fh implies g = h.

Record the girl and post vocaroo.

>>5841031
You can have material subsets of structural sets X as morphisms f : X -> Ω such that for any element x : 1 -> X, f ∘ x is the truth value of the containment of x in f, and if you define x ∈ f to be f ∘ x you even recover the usual notation.

>>5841188
>>5841215
Thus a monomorphism f: X -> Y sends each element of X to a unique element of Y, so X can be thought of as a subset.
On the other hand, an epimorphism sends collections of elements in X to single elements in Y such that the image covers Y, so Y can be thought of as a set of equivalence classes.

To prove this, you can do a proof by contradiction by constructing functions which do not fulfill the properties described in >>5841215

>>5841215
ah, right, there is no zero, of course.
I have absolutely nothing to do with this field, but I recently tried to get an intuition for exact sequences, which are not short.

and if I turn the laptop recorder on, you don't hear the stuff behind walls when played. Also, it doesn't really sound sexy.

>>5841228
Thanks for the input, mate.
To see the details of this construction, search for "subobject classifier"

>>5841228
>wasting time sci
>not revolutionizing the world of continuations

>>5841231
No worries, mate. If you want to see exact sequences in their full generality, you want to study Abelian categories, which mimic the properties of categories of modules.
But for an introduction, it might be good to study Ab, the category of abelian groups and thir group homomorphisms, to get an intuition.
Exact sequences in general are the object of study in homological algebra. It's a fun study, you should check it out.

>>5841188
Those are the exact charistics, and the 0 is your thesis

Thats the NNNngG of it, just gotta hold that NNN(N^)g while writting and you can make a thing that's independent of you and more capable of mapping from than mapping into what you have, and that thesis then becomes a variable in the next builds identicate and by properites matrixable with other thesises; and ffs all the notation... fucking engineers, bloody fags the lot of 'em

>>5841161
How does this make sense for things like ordered pairs?

>>5841248
>homological algebra
Too far fetched. Not time.
I'm interested in functional analysis, fixed point theorems and computation. But I have to do it in my free time.
During the day I have to do chemistry...

And thanks for the intro, although some are slightly too basic to get an insight.

Are you still the topology guy? What do you do exactly? Do you have something online?

>>5841260
Yeah, I'm the topology guy. I'm just a master's student in math specializing in algebraic topology.
I don't have any online blog. I don't think I know enough to write interesting stuff anyway.

>>5841263
Let X and Y be sets. The set of prdered pairs of objects in X and Y is the cartesian product X x Y, which categorically is defined like this:

X x Y is a set such that we have morphisms
p : XxY -> X and q : XxY -> Y
and such that if Z is any set with morphisms
f : Z -> X and g : Z -> Y
then there is a unique (note: uniqueness here is crucial) morphisms
h : Z -> XxY
such that f = ph and g = qh .

This uniquely determines XxY up to isomorphism, and then it behaves exactly like the set of prdered pairs (x,y) with x in X, y in Y.
Does this make sense to you?

>>5841257
ohh ffs
> >>5841188
I thought you were asking what was making the second arrow in the bit

>>5841292
Yes, thank you!

>>5841300
No problem.
Really, threads like this one is the reason I come to /sci/.