/mg/ - Math General

[Anonymous No.9065316]

Southern School of Ultrafinitism Edition

Previous (garbage) thread: >>9059413

What are you S T U D Y I N G today?
Got something interesting to share?
Maybe a proof of a Millennium Problem? Post it; we promise not to steal your shit.

[Anonymous No.9065756]

>>9065316
Have a pity bump.

[Anonymous No.9065767]

>>9065316
Going through Velleman's book how to prove it. Fun little book coming from someone who studied physics. Recommendations to read after this?

[Anonymous No.9065795]

>tfw no unstoppable mouse

[Anonymous No.9065862]

Math was always my worst subject, I think I used calculators as a crutch rather than a net. How can I build my mental math ability?

[Anonymous No.9065876]

Which subjects are critical for a general undergrad math education, and which ones don't matter that much?

[Anonymous No.9065884]

>>9065876
>Which subjects are critical for a general undergrad math education, and which ones don't matter that much?
depends on the university, each have different required courses and electives

[Anonymous No.9065886]

>>9065767

[Anonymous No.9066097]

>>9065862
learn multiplication tables up to 10 and keep paper handy. mental math is for circus freaks anyway. better to be useful than reduced to a sideshow attraction for normies.
>hey look at me i can calculate [math]\lfloor x^{\frac{1}{y}} \rfloor[/math] in my head
if you're lucky you'll give a TedX talks and maybe get a book deal for some recreational math book nobody will ever read. Or you could not waste your time and learn non-intuitive math so hard that nobody could ever possibly do it in their head. If you're von Neumann caliber you could do both, but you're probably not.

[Anonymous No.9066099]

>>9065767

[Anonymous No.9066152]

>>9065884
In this case the requirements are various linear algebra, calculus, abstract algebra, analysis, discrete math and some theoretical statistics. Then for electives you can take stuff like combinatorics, cryptography, differential equations, topology, optimization, stochastics, etc.

[Anonymous No.9066373]

What are /mg/'s plans for the future? I'm in my last year of undergrad and because of how the system works here, I'm used to everything being streamlined for me and not having to make any decision, but now I have to start thinking about getting a master's and I'm pretty fucking scared.
Also, I don't know what my proffessional future holds, because I don't know if I'm cut for academia, so I don't know what master's to pick. So what about you guys?

[Anonymous No.9066377]

>>9066373
I'm going to get a job and do a masters in statistics part time. Hopefully I can get a job somewhere that will pay for my masters. Don't care about academia.

[Anonymous No.9066390]

>>9065862
Read 'the Secrets of mental math".

[Anonymous No.9066404]

>>9066373
>What are /mg/'s plans for the future?
I will never discover anything interesting, I will never prove anything impressive, I will never amount to anything. I think my plans have been made for me, and the choice is simple: live like a dirt-eating maggot, or suicide.

[Anonymous No.9066406]

>>9066404
>I will never discover anything interesting, I will never prove anything impressive, I will never amount to anything. I think my plans have been made for me, and the choice is simple: live like a dirt-eating maggot, or suicide.
cringe

[Anonymous No.9066497]

>>9065795
kek
I like this guy

[Anonymous No.9066518]

>>9066406
>using the word cringe
cringe

[Anonymous No.9066520]

>>9066099
Shit weeb artstyle and autism

[Anonymous No.9066523]

>>9065316
/wild/ edition of /mg/

Anon, do you realize what you've done?!

[Anonymous No.9066529]

I feel as a physicist completely down when I read a physics book. I mean, it's even that hard to learn real anal, but in something like electromagneticism, they just go and do whatever they want with notation and formalism. Heck, I had to properly derive a formal description for the electric field. I feel like this shit isn't for me, but I love physics more than math, still, I can't stand how physicists go about it. Wat do /sci/?

[Anonymous No.9066530]

>>9066529
It's not even*

[Anonymous No.9066537]

brehs what is this

[Anonymous No.9066555]

>>9066537
Cave-mathematicians

[Anonymous No.9066558]

>>9066537
They were here.

[Anonymous No.9066579]

>>9066529
If you like physics for the pretty and elegant formalism and theorems then you probably shouldn't be a physicist, do learn physics, but the cutting edge of research isn't beautiful theorems, concepts, or elegant formalism, it's dirty and messy and cutthroat. When it comes to physics you'll rarely have exact solutions, rarely have nice, well behaved equations that are easy to analyze (nonlinear pdes will fuck you up), the systems can and will be horrendous, and it's possible all your ideas will be bunk or people just won't give a shit. Look, physics can be hard, really hard, jackson's EM book is proof of that (basically a shit load of pdes that are really difficult to solve), you've just gotta have the stomach to get your hands dirty and swath through pages and pages of calculations. This is not to dissimilar to mathematics, though since math has no need to be subordinate to reality there are more elegant structures at the forefront ready to be explored, some of these do have relevance to physics, (this is mathematical physics for the most part, or at least fields interacting with this discipline) some of the fields connected to math phys are harmonic analysis, operator theory, algebraic geometry, diff geo, representation theory, number theory, dynamical systems/ergodic theory, functional analysis, complex analysis, algebraic topology, homological algebra...actually most math can be made relevant to physics in one way or another, so you might as well go the route of an applied mathematician or working in analysis/geometry.

[Anonymous No.9066582]

>>9066579
> applied mathematician
No such thing.

[Anonymous No.9066583]

>>9066582
What do you mean?

[Anonymous No.9066587]

>>9066583
That "applied mathematics" is a vacuous, ill-defined term. Not going to have this discussion again. Read through the previous thread.

[Anonymous No.9066591]

>>9066587
>That "applied mathematics" is a vacuous, ill-defined term
What? It's a pretty standard term

https://en.wikipedia.org/wiki/Applied_mathematics

What else would you call what applied mathematicians do?

[Anonymous No.9066595]

>>9066591
>reddit

>spacing

>>>/out/

[Anonymous No.9066598]

>>9066595
Is this better?

What? It's a pretty standard term
https://en.wikipedia.org/wiki/Applied_mathematics
What else would you call what applied mathematicians do?

[Anonymous No.9066608]

is there a good online degree/course for number theory?

[Anonymous No.9066611]

>>9066537
it's a list of the genders you fucking bigot

[Anonymous No.9066614]

>>9066598
Much better.

[Anonymous No.9066618]

>>9066614
So what would you call what applied mathematicians do?

[Anonymous No.9066619]

>>9066618
See
>>9066587

[Anonymous No.9066625]

>>9066619
I didn't ask about applied mathematics, I asked what you would call what applied mathematicians do.

Or was quoting that post of yours meant to imply that you would call it applied mathematics but just don't like to?

[Anonymous No.9066631]

>>9066625
I wouldn't call it anything, because the set of "applied mathematicians" does't exist. Membership in that set is ill-defined.

[Anonymous No.9066632]

>>9066631
What's wrong with the definition here?

https://www.bls.gov/ooh/Math/Mathematicians.htm#tab-2

>Some mathematicians apply theories and techniques, such as mathematical modeling, to solve practical problems. These mathematicians, sometimes known as applied mathematicians, typically work with individuals in other occupations to solve these problems.

[Anonymous No.9066634]

>>9066625
>>9066632
>double

>new line spacing

>>>/r/eddit

[Anonymous No.9066635]

>>9066634
Who are you quoting?

[Anonymous No.9066639]

>>9066635
You appear to be blind. Please go see an ophthalmologist.

[Anonymous No.9066640]

>>9066639
>You appear to be blind.

Who wrote "double" or "new line spacing"? I don't see it in any posts you replied to.

[Anonymous No.9066641]

>>9066634

Is this better?

What's wrong with the definition here?
https://www.bls.gov/ooh/Math/Mathematicians.htm#tab-2
>Some mathematicians apply theories and techniques, such as mathematical modeling, to solve practical problems. These mathematicians, sometimes known as applied mathematicians, typically work with individuals in other occupations to solve these problems.

[Anonymous No.9066643]

>>9066634
What's wrong with the line spacing? It looks like he/she is formatting the same way you are?

[Anonymous No.9066645]

>>9066640
>I am so dumb I cannot parse mock-paraphrasing and satiric explications

[Anonymous No.9066646]

>>9066645
What did 'double' and "new line spacing" paraphrase?

And why do you use quotation arrows when you're not quoting anyone? It seems it would just take extra effort to type your message that way

[Anonymous No.9066647]

>>9066646
First day on 4chan huh

[Anonymous No.9066648]

>>9066646
See
>>9066639
(nice plebbit spacing btw)

[Anonymous No.9066654]

>>9066647
I've been here quite a while, but I just don't see what 'double' and "new line spacing" were 'mock-paraphrasing'.

Nor do I see what this has to do with your confusion about applied mathematics and applied mathematicians.

[Anonymous No.9066659]

>>9066648
What exactly makes a gap between paragraphs 'plebbit spacing'?

This formatting is common on probably every board on 4chan you take a look at, and has been for years, it just increases legibility.

There's a reason there's a rule against
> indecipherable text (example: "lol u tk him 2da bar|?")

[Anonymous No.9066668]

>>9066654
>>9066659
Go back to >>>/r/eddit already.

[Anonymous No.9066671]

>>9066668
Why do you make claims about applied mathematics and post formatting that you don't want to explain?

Who is a member of SIAM (Society for Industrial and Applied Mathematics) if not applied mathematicians?

https://siam.org/

[Anonymous No.9066673]

>ftw don't want to ask the following in /sqt/
Let [math]\phi:G \rightarrow H, \psi: H \rightarrow K[/math] be morphisms in a category of products and let [math]\phi \times \phi: G\times G \rightarrow H\times H, \psi \times \psi: H\times H \rightarrow K \times K[/math]
I need to conclude that [math](\psi \phi)\times (\psi \phi) = (\psi \times \psi)(\phi \times \phi)[/math]
So, [math]\psi\phi:G \rightarrow K[/math] uniquely determines morphism [math](\psi \phi)\times (\psi \phi)[/math] as [math]K \times K[/math] is a product in this category. Since [math](\psi \times \psi)(\phi \times \phi): G \times G \rightarrow K \times K[/math], can I conclude that uniqueness of morphism from [math]G \times G[/math] to [math]K \times K[/math] forces [math](\psi \phi)\times (\psi \phi) = (\psi \times \psi)(\phi \times \phi)[/math] ?

[Anonymous No.9066704]

How do I become a human calculator to impress my friends? Brainlets like it when I can multiply triple digit numbers in under a minute. Any resources to do faster mental math?

[Anonymous No.9066737]

>>9066673
Due to the UMP of [math]K \times K [/math] as a product this morphism is indeed unique, I think

[Anonymous No.9066784]

>>9066673
Let [math]p_H\colon H\times H\to H, p_K\colon K\times K[/math] be projections onto the first object. We then have these equations: [math]p_K((\psi\varphi)\times(\psi\varphi)=\psi\varphi=\psi p_H(\varphi\times\varphi)[/math] and [math]p_K(\psi\times\psi)(\varphi\times\varphi)=\psi p_H(\varphi\times\varphi)[/math], by commutativity. The same holds for the other projections, so the same unique arrow makes the whole diagram commute, but so [math](\psi\times\psi)(\varphi\times\varphi), (\psi\varphi)\times(\psi\varphi)[/math].

[Anonymous No.9066793]

>>9066784
[math]p_K\colon K\times K\to K[/math] and [math](\psi\times\psi)(\varphi\times\varphi), (\psi\varphi)\times(\psi\varphi)[/math] I'm retarded

[Anonymous No.9066801]

>>9066784
>>9066793
Nice latexing retard.

[Anonymous No.9066805]

Any good guides for writing actual morphism diagrams in [math] \LaTeX [/math]?

[Anonymous No.9066808]

>>9066801
I know.

[Anonymous No.9066810]

>>9066805
make them in paint then add the .png file to your latex workspace

[Anonymous No.9066812]

>>9066810
No, I mean native support. I can always draw them in inkscape and add the .svg to the document but that's not what I'm interested in.

[Anonymous No.9066877]

>>9065316
>What are you S T U D Y I N G today?
I'm studying mathematics without "real" numbers right now.

[Anonymous No.9066895]

>>9065795
What happens when the unstoppable mouse runs into the immovable kitten?

[Anonymous No.9066908]

I am stuck on a question about the number of walks between two distinct vertices in a complete graph. The answer according to the book is n^2 - 3n + 3 but I can't for the life of me figure out how to get there.

[Anonymous No.9066911]

>>9066908
n being?

[Anonymous No.9066914]

>>9066911
Amount of vertices, sorry forgot to mention that, and probably n>3 or something, dont have the book open right now

[Anonymous No.9066920]

>>9066908
tried induction?

[Anonymous No.9066927]

>>9066635
mfw anon trolled the fuck out of the "that's reddit spacing!" /pol/nigger by using an /s4s/ meme

>>9066582
objectively wrong

>>9066805
You can't do it within [math]\LaTeX[/math] directly (i.e. without a package for commutative diagrams). There's a few old guides if you look around. However you should be aware that a lot of the older guides mainly talk about older crappier ways of generating commutative diagrams (and you should also know that said packages still have a userbase due to old mathfags who got used to them).

The modern way to do it is through the commutative diagram package for tikz. Just add:
\usepackage{tikz}
\usetikzlibrary{cd}
To your preamble. The manual can be found at:
http://ctan.math.washington.edu/tex-archive/graphics/pgf/contrib/tikz-cd/tikz-cd-doc.pdf

The concept and syntax format looks complicated at first but it's actually really simple. Basically:
>Imagine the objects in your diagram as sitting in a matrix.
>Then imagine your arrows starting at one cell and going to another one (e.g. from the cell of object A go right, right, and down for the end cell)
>Arrows have options on them that allow you to give them a label (text that will appear next to the arrow) and an orientation (which way the arrow curves) among other things).
So in the syntax your matrix is column separated by & and row separated by \\ then in each cell you write the object that lives there and a list of arrows starting there, eg.
>A \arrow[d, "f"] \arrow[rd] \arrow[rrd] \arrow[r, "g"]
is a cell with an object "A" and four arrows going to different cells (rrd means "right, right, down"), some containing labels.

Advanced options allow you to do fancy arrow paths, name objects and create arrows between them, name arrows and create arrows between arrows, etc...

>>9066810
Don't do this.

[Anonymous No.9066929]

Daily reminder that "applied mathematician" is vacuous and ill-defined.

[Anonymous No.9066932]

>>9066784
Frickin' cute!

[Anonymous No.9066933]

I'm having a problem with a question in a graph theory entry level book, I already made a thread but I figure it'll be twice as fast posting it here. Anyway, in case there are some experts (though I doubt you need to be one), here goes.

How do I go about proving that a graph with order n and minimum degree (n-2) has VERTEX connectivity (n-2) as well?

I mean I know, intuitively that if I delete a certain number of vertices and end up with 2 or more components, then each component must contain a vertex u that had degree n-2 in the initial graph, and one other component must contain the one vertex v that wasn't it his neighbourhood. And since these two vertices have the same neighborhood with n-2 vertices, you'd have to delete all of them for there not to exist a path between u and v.
However this seems far fetched and I don't really know how to write the first part formally.
Is there a simpler proof I'm missing? The exercise is literally at the beginning of the book and all the proofs seem much easier.

But since there are like 5 people on /sci/ currently, I'm guessing I'm gonna get the same people anyways.

[Anonymous No.9066939]

>>9066929
>Daily reminder that "applied mathematician" is vacuous and ill-defined.
applied mathematician: one who works in the branch of applied mathematics

applied mathematics: mathematics applied to non-mathematical domains

[Anonymous No.9066941]

>>9066929
>vacuous and ill-defined.
https://www.youtube.com/watch?v=yetwdpsiM8Q

[Anonymous No.9066942]

>>9066939
>>9066941
>applied mathematics
No such thing actually exists.

[Anonymous No.9066944]

>>9066942
but who are you quoting?

[Anonymous No.9066945]

>>9066944
see >>9066939

[Anonymous No.9066953]

>>9066942
>No such thing actually exists.
What is biomathematics then?

https://en.wikipedia.org/wiki/Mathematical_and_theoretical_biology

>Mathematical biology aims at the mathematical representation, treatment and modeling of biological processes, using techniques and tools of applied mathematics.

[Anonymous No.9066956]

>>9066953
>What is biomathematics then?
Apparently something which "uses" something nonexistent.

[Anonymous No.9066961]

>>9066956
It sounds like Wikipedia could make great use of you as an editor

Many articles mention this allegedly fictional area of study

[Anonymous No.9066963]

>>9066961
I don't use websites such as "Wikipedia" so I wouldn't know.

[Anonymous No.9066966]

>>9066963
Do you read any math journals?

http://www.scirp.org/journal/am/
> Applied Mathematics

[Anonymous No.9066968]

>>9066966
I do, but I simply ignore the parts where they talk about fictional stuff. I'll just read some fiction if I'm interested in that.

[Anonymous No.9066971]

>>9066968
So can we define applied mathematics as the mathematics you consider fictional?

[Anonymous No.9067015]

>>9066939
>mathematics applied to non-mathematical domains
is not mathematics. Mathematics is the study of mathematical structures. Applying the results obtained by studying mathematical structures to non-mathematical domains is not a study of mathematical structures, hence not mathematics. Or in short, applications of mathematics [math] \neq [/math] mathematics.

[Anonymous No.9067017]

>>9066971
You cannot define applied mathematics as a subfield of mathematics for any sensible definition of mathematics.

[Anonymous No.9067019]

>>9067015
Errata: applied mathematics [math] \not\subset [/math] mathematics.

[Anonymous No.9067026]

>>9067017
>You cannot define applied mathematics as a subfield of mathematics for any sensible definition of mathematics.
What do you mean? 'Applied' is just a modifier, it separates mathematics into applied mathematics and non-applied (pure) mathematics.

[Anonymous No.9067029]

>>9067017
What sensible definition of mathematics do you propose?

[Anonymous No.9067030]

Now, say, [math] if [/math] you think about mathematics as a practice, concretely, as the application of results obtained by studying mathematical structures, then [math] all [/math] mathematics is applied mathematics.
>>9067026
You're spouting words but you're saying nothing. Define that separation. What does "applied" modify and how does it modify it?

[Anonymous No.9067032]

>>9067030
>What does "applied" modify and how does it modify it?
It modifies mathematics, specifying the subset of mathematics dealing with applications of mathematics to fields other than mathematics

[Anonymous No.9067034]

>>9067030
How can you define mathematics in terms of 'mathematical structures', a term which depends on mathematics?

[Anonymous No.9067038]

>>9067032
>applications of mathematics to fields other than mathematics
is not a
>subset of mathematics

[Anonymous No.9067045]

>>9067030
pure mathematics
>Mathematics which yield applications within mathematics (i.e. mathematics for the sake of mathematics).

applied mathematics
>Mathematics which yield applications outside mathematics (i.e. mathematics for the sake of something else).

Heuristically speaking pure mathematicians care about more general results (at a higher level of abstraction) while applied mathematicians care about more specialized results (at a lower level of abstraction).

[Anonymous No.9067047]

>>9067038
How so?

You either apply mathematics techniques within the context of mathematics alone or you don't, when you don't it's applied mathematics.

[Anonymous No.9067049]

>>9067015
>Mathematics is the study of mathematical structures.
Define 'mathematical structures'.

[Anonymous No.9067057]

>>9067049
A mathematical structure is an object whose essential properties are exhausted by its formal definition. Synonymic terms: a platonic Form or Idea.

[Anonymous No.9067065]

>>9067057
>A mathematical structure is an object whose essential properties are exhausted by its formal definition.
How does one prove something is a mathematical structure?

[Anonymous No.9067067]

>>9067045
All mathematics yields applications within mathematics. Therefore all mathematics is "pure mathematics". So applied mathematics cannot be separated from pure mathematics, as >>9067026 claims.

[Anonymous No.9067069]

>>9067065
What a stupid question.

[Anonymous No.9067073]

>>9067069
>What a stupid question.
How so, what good is a definition if you can't show anything fits it? Give an example of a mathematical structure

[Anonymous No.9067074]

>>9067057
I'm not that guy but I feel like that's not quite right.

Shouldn't it be that structure is the properties themselves and that this structure may apply to a collection of objects (possibly just one). Such objects are then dubbed mathematical objects.

So for instance when defining a field one would say that the axioms of a field give structure to the elements of the field (mathematical objects) and one could also say that the field itself is a mathematical object of sorts.

>>9067067
>All mathematics yields applications within mathematics.
Not true. There are results so specialized that only "muh real world" brainlets (eg. scienceplebs) have any use for them.

[Anonymous No.9067075]

>>9067073
>Give an example of a mathematical structure
Sets.

[Anonymous No.9067076]

>>9067073
>>9067065
Not that guy but you are hereby confirmed a brainlet. Go learn some basic formal logic and gtfo.

[Anonymous No.9067078]

>>9067075
>Sets.
Now show that its essential properties are exhausted by its formal definition.

[Anonymous No.9067082]

>>9067074
>Not true.
Not true.
>structure is the properties themselves
That's what
>its essential properties are exhausted by its formal definition
entails.

[Anonymous No.9067086]

>>9067076
>Not that guy but you are hereby confirmed a brainlet. Go learn some basic formal logic and gtfo.
How are the essential properties of a set exhausted by its formal definition?

[Anonymous No.9067089]

>>9067078
>>9067086

https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms
>The axioms per se are expressed in the symbolism of first order logic. The associated English prose is only intended to aid the intuition.
https://en.wikipedia.org/wiki/Formal_system
https://en.wikipedia.org/wiki/Model_theory
https://en.wikipedia.org/wiki/Axiomatic_system

>Not true.
Not true.
>entails.
Not true.

[Anonymous No.9067090]

>>9067086
Because that (its formal definition) is the entirety of what a set [math] is [/math]. You're asking the equivalent of "why is a vector an element of a vector space"?

[Anonymous No.9067097]

ITT
>prove that a quadruped has four legs

[Anonymous No.9067102]

>>9067089
Are statements about sets undecidable in ZFC 'essential properties'?

[Anonymous No.9067107]

>>9067090
>Because that (its formal definition) is the entirety of what a set is.
Are you saying the formal definition is the only essential property of a set, so that a set is a mathematical structure because its formal definition exhausts its only essential property of being formally defined?

[Anonymous No.9067112]

>>9067107
If formalism and abstraction short-circuit your brain so hard, maybe mathematics is not for you. Just a thought.

[Anonymous No.9067115]

>>9067112
>If formalism and abstraction short-circuit your brain so hard
Which part of

'is a set is a mathematical structure because its formal definition exhausts its only essential property of being formally defined?'

is short-circuiting you?

[Anonymous No.9067120]

>>9067115
That is not the only essential property of sets.

[Anonymous No.9067122]

>>9067120
Then what else? It's still unclear that the essential properties of a set are exhausted by its formal definition.

[Anonymous No.9067127]

>>9067075
Aren't sets primitive notions?

[Anonymous No.9067128]

>>9067102
Could you rephrase that? Do you mean unprovable statements?

In a way you really only prove statements true. If in some case you want to show that a statement is false then what you do is prove that it's negation is true (then since a sentence and its negation can't both be true you've indirectly shown what you originally wanted to show).

Unprovable statements about sets thus aren't even really a thing you can talk about within the usual context of the theory, rather they're dealt with in fields called meta-mathematics and meta-logic.

>>9067107
>>9067120

ZFC are a bunch of axioms. Then we say that the things satisfying said axioms shall be called "sets" by definition.

[Anonymous No.9067130]

>>9067122
>what else?
See https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory#Axioms

[Anonymous No.9067136]

>>9067127
Not quite. You can define all of mathematics within set theory but set theory itself is defined over a formal logic which is defined over a formal language. That said these things can be defined using sets as well (though the full power of set theory isn't required).

[Anonymous No.9067137]

>>9067128
>Could you rephrase that?
Which part is unclear? I mean statements like the continuum hypothesis.

[Anonymous No.9067141]

>>9067137
Non-axiomatic statements about sets are not essential. All theorems about sets are derived properties of sets.

[Anonymous No.9067146]

>>9067097
Underrated post.

[Anonymous No.9067151]

>>9067141
>Non-axiomatic statements about sets are not essential.
What is an object whose essential properties are not exhausted by its formal definition?

[Anonymous No.9067154]

>>9067151
Your post, for example. Anything non-abstract in general.

[Anonymous No.9067161]

>>9067136
To add to this, sets are however primitive objects of the axiomatic system/formal system under the formal logic.

>>9067137
These are meta-mathematics theorems.

Essentially what you've got is some statement in a formal system that you can't find a proof for. In order to see if it's unprovable you need to go up a level of abstraction into meta-logic and formalize the concept of models for your formal system. Then you create two models for said formal system, one where the statement is true and one where it's false. Finally you move this result back to the meta-logic level where you assert that said sentence inside your original formal system is unprovable.

Since it's easy to get lost:
>meta-logic -> ---- -> logic -> formal system
>meta-logic -> ---- -> model theory -> models for formal system

[Anonymous No.9067162]

>>9067154
What is the formal definition of my post? Based on the responses to what the essential properties of a set are, it sounds like essential properties are exactly those which follow from a formal definition.

[Anonymous No.9067167]

>>9067162
>What is the formal definition of my post?
This no longer has anything to do with mathematics and you're too much of a 5yo brainlet (or acting like one, with your incessant, idiotic whys) for the effort of constructing one to be warranted.

[Anonymous No.9067176]

>>9065316
I'm into self reference and undecidability today.

Here is for you anons, the reason why there CANNOT BE methodical ways to solve problems in math.

Let E be a set containing character strings (including the string "yes" and "no" that we will assume to be different). We assume we have a partial map from m: E^2 to E : if t and x are in E, m(t,x) will be (if it exists) the result of the evaluation of x by using the method defined by t.

The following problem (which is a mathematical problem since m would be generally a computer software)is unsolvable by using a method in the above sense:
>If x is any element of E, is m(x,x) defined and >does it yield "no" as a result?

Indeed , if there was a u in E which does that, then for any y, m(u,y) would give an answer "yes" or "no" and then, if m(u,u)="yes", then m(u,u)="no" and if m(u,u)="no" then m(u,u)="yes".

.

[Anonymous No.9067178]

>>9067128
>then since a sentence and its negation can't both be true
Only assuming consistency, but ZFC is known to be inconsistent.

[Anonymous No.9067179]

>>9067167
>This no longer has anything to do with mathematics
Of course it wouldn't since I asked about something that doesn't fit your definition of mathematical structure.

The explanation for sets was that all essential properties follow from formal definition and that properties that don't follow from formal definition are non-essential.

So what is an example of an object with a formal definition and an essential property that doesn't follow from the formal definition?

[Anonymous No.9067183]

>>9067178
>but ZFC is known to be inconsistent.
lolwat

[Anonymous No.9067185]

>>9066895
what happens when your mother lies on the incompressible mattress?

[Anonymous No.9067187]

>>9067183
>lolwat
18+ please.

[Anonymous No.9067192]

>>9067136
>You can define all of mathematics within set theory
How do you define categories within set theory?

[Anonymous No.9067195]

>>9067192
By using NBG set theory which is a conservative extension of ZFC. Categories will be classes.

[Anonymous No.9067197]

>>9067195
beat me to it

>>9067187
then gtfo

[Anonymous No.9067209]

What is an example of an object with a formal definition and an essential property that doesn't follow from the formal definition?

[Anonymous No.9067224]

This is a blanket statements to all the redditspacing fuckers above: you have to go back!
>>9067195
>>9067197
>NBG set theory
Inadequate. If you want a set theoretic approach to category theory you either go for Tarski-Grothendieck or you go home.

[Anonymous No.9067238]

>>9067224
>
Inadequate. If you want a set theoretic approach to category theory you either go for Tarski-Grothendieck or you go home.

I said "conservative extension". Maybe you don't like it. If TG was discovered to be inconsistent we'd still have NBG.

Category theory is a meme, what you like is the set theoretic results it produces.
>tfw algebraic geometry was developed in order to solve diophantine equation like fermat's.

[Anonymous No.9067241]

>>9066673

[Anonymous No.9067242]

>>9067238
>Category theory is a meme

[Anonymous No.9067247]

>>9067238
>meme
>>>/r/eddit/

[Anonymous No.9067252]

>>9067238
>what you like is the set theoretic results it produces
I don't believe in "set theory".

[Anonymous No.9067256]

>>9067238
>Category theory is a meme
Well, it was the first thing I liked about mathematics. In hight school I used to think that mathematics is the most boring thing in this universe but then I came across category theory, started reading about real mathematics and eventually switched from CS to pure meth, I owe Haskell(programming language) my acquaintance with Category theory though

[Anonymous No.9067258]

Haven't done math in years, recommend me workbooks /sci/.
Preferred fields: precalculus, discrete math, graphs, probability, geometry, linear algebra, calculus, differential equations, numerical analysis.
I've never been good in discrete maths, might need some introductory exercises first. For all others I want something more advanced.

[Anonymous No.9067263]

>>9067241
[math]\exists![/math] morphism
Disgusting.

[Anonymous No.9067272]

>>9067238
>If TG was discovered to be inconsistent
Which it absolutely is.
>>9067256
>CS
Use some other thread or even board please.

[Anonymous No.9067318]

http://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf
>“REAL” ANALYSIS Is A DEGENERATE CASE of DISCRETE ANALYSIS
Whew lads, what have we gotten ourselves into?

[Anonymous No.9067320]

>>9067258
>Preferred fields:
>precalculus
Sweet Jesus Mary fucking Christ. The OP really needs a pastebin with recommended books so people stop asking these fucking questions. In the meantime, go to the wiki you can find on the sticky, and look up the undergrad program of Cambridge for a good bibliography and study plan.

[Anonymous No.9067321]

>>9067247
>>9067256
>>9067242

Category is syntactic sugar over set theory (because muh conservative extension), like Haskell is syntactic sugar over lambda calculus. No offense intended.

[Anonymous No.9067326]

>>9067321
the fuck is syntactic sugar?

[Anonymous No.9067328]

>>9067258
>>9067321
>"syntactic sugar"
>"Haskell"
>muh
Please fuck off to some other board.

[Anonymous No.9067330]

>>9067321
All mathematicians are set theorists, right?

[Anonymous No.9067331]

>>9067330
No, some do lambda calculus.

[Anonymous No.9067332]

>>9067326
I think according to her second sentence she was saying that type theory can be used as a syntax for category theory.
Her first sentence seems to imply the same for set theory, although I'm not sure if that's true.

[Anonymous No.9067333]

>>9067330
Nah, Bourbaki is old and busted.

[Anonymous No.9067335]

>>9065316

https://www.youtube.com/watch?v=F5gq46WbP5w

[Anonymous No.9067336]

>>9067330
I sure hope not, every major set theory is inconsistent.

[Anonymous No.9067339]

>>9067318
>"“REAL”" ANALYSIS
No such thing.

[Anonymous No.9067345]

[Anonymous No.9067349]

*rotates some subsets of a sphere*

[Anonymous No.9067350]

>>9067349

[Anonymous No.9067356]

Great class

[Anonymous No.9067358]

>>9067263
exists exactly one morphism

[Anonymous No.9067362]

>>9067356
>Prove that a set is a group
>Prove something about group properties
>Short essay: why study abstract algebra?

[Anonymous No.9067364]

>>9067356
>Prove that a set is a group

[Anonymous No.9067365]

>>9067349
I don't like where this is going.

[Anonymous No.9067375]

>>9067356
>why study abstract algebra
topkek

[Anonymous No.9067380]

>>9067321
Both of these sentences are very wrong.

>>9067326
Syntactic sugar is usually mentioned in the context of programming languages but it exists in mathematics as well. In the context of programming languages it usually refers to terms or syntax that aren't actually part of the language. Rather they translate directly into more complicated code and are really only there to make programmer's lives easier. An example in mathematics would be defining addition as a binary function
[math]+:\mathbb{R}\to\mathbb[R}[/math]
but instead of writing +(3,4) writing 3+4. In this case the latter is syntactic sugar for the former. It is also referred to as sweetened syntax.

There is also a complementary concept referred to as syntactic salt where synax is purposely hard to use. Typically this is used to keep dumb programmers from writing bad code.

[Anonymous No.9067388]

>>9067362
>>9067364
>>9067375
Aside from the short essay you guys are idiots. Those are obviously someone's shitty notes about what sorts of questions will be on an exam.

[Anonymous No.9067396]

>>9067380
>code
>programming
Wrong thread. Please create a new one to talk about these concepts.

[Anonymous No.9067397]

>>9067388
> Those are obviously someone's shitty notes about what sorts of questions will be on an exam.
So what ?

[Anonymous No.9067398]

>>9067396

>>code

>>programming

>Wrong thread. Please create a new one to talk about these concepts.

https://en.wikipedia.org/wiki/Curry%E2%80%93Howard_correspondence

[Anonymous No.9067400]

>>9067380
Whoops, meant:
[math]+:\mathbb{R}\to\mathbb{R}[/math]

>>9067396
(you)

[Anonymous No.9067401]

>>9067398
Retarded programmer spotted. Non-retards usually call it the BHK interpretation.

[Anonymous No.9067404]

>>9067380
>Both of these sentences are very wrong.

Maybe "syntactic sugar" wasn't really the right term but here what's happening.

You can do all the category theory within NBG. But since NBG is a conservative extension of ZFC, whenever there is a claim entirely written in the language of set theory (without referring to "classes", "functors or whatever", but just plain set theory: first-order sentences built up with the "belongs" binary symbols), it this claim has a proof in NBG, it has also a proof in vanilla ZFC.
As a consequence, category is just a set of extra words, nothing "fundamental" to get hye with. This corresponds to what we do in parctice.Let's say I prove a bunch theorems about rings, modules, sheaves, topological spaces, manifolds. I'd use category as a convenience, but if you actually pay attention, you notice that the categorical language can entirely be removed. I'd say "F(M) is an inductive limit of F(A_i) because F is the left-adjoint functor of some other functor G". I can write the whole argument in the special case instead of talking about functors, and actually category presents itself rather as a way to provide general metatheorems.

As for Haskell, since every language is a special case of another Turing complete language, I don't see why I have stated anything wrong.
Mainstream imperative programming languages are horrible when you want to try cartesian closed category tricks (a pompous way to call curryfication a.k.a the simple fact that maps with two parameter ae the same thing as maps with one parameter who return maps with one parameter, which categorists prefer to think in terms of Hom(A,F(B,C)) ~ Hom(AxB,C)... That doesn't make thing easier I think)

[Anonymous No.9067411]

>>9067404
>You can do all the category theory within NBG

>the ontology of NBG cannot provide immediately for a "category of all categories," since such a category would have to contain large categories, which no category can do.

[Anonymous No.9067412]

>>9067404
>As for Haskell...
I suggest you use a different thread for discussing your garbage. Do you want me to create one for you?

[Anonymous No.9067414]

>>9067318
>So I deny even the existence of the Peano axiom that every integer has a successor. Eventually we would get an overflow error in the big computer in the sky, and the sum and product of any two integers is well-defined only if the result is less than p, or if one wishes, one can compute them modulo p. Since p is so large, this is not a practical problem, since the overflow in our earthly computers comes so much sooner than the overflow errors in the big computer in the sky.
>Similarly, Euclid’s statement: ‘There are infinitely many primes’ is meaningless.
>What is true is: if [math] p_1 < p_2 < \dots < p_r < p [/math] are the first r finite primes, and if [math] p_1 p_2 \dots p_r + 1 < p [/math], then there exists a prime number q such that [math] p_r + 1 ≤ q ≤ p_1 p_2 \dots p_r + 1 [/math]
>By hindsight, it is not surprising that there exist undecidable propositions, as meta-proved by Kurt
Gödel. Why should they be decidable, being meaningless to begin with!

Is this it? Has someone finally topped Wildberger's autism?

[Anonymous No.9067417]

>>9067411
>reddit frogs
>>>/b/

[Anonymous No.9067418]

wow I was thinking about going into a math Phd program but think I won't since evidently it'll just be me and a bunch of weebs

[Anonymous No.9067422]

>>9067414
>There are infinitely many primes
False.

[Anonymous No.9067423]

>>9067398
>double

>spacing
>>>/r/eddit called, they want you back.

[Anonymous No.9067425]

>>9067418
>wow I was thinking about going into a math Phd program
I don't think they accept subhumans, sorry.

[Anonymous No.9067427]

>>9067422
>False.
How many are there?

[Anonymous No.9067429]

>>9067427
Not infinitely many.

[Anonymous No.9067430]

>>9067429

>Not infinitely many.

Then how many?

[Anonymous No.9067432]

>>9067430
Is your reading comprehension this bad because of reddit? I already answered your question.

[Anonymous No.9067437]

>>9067432

>Is your reading comprehension this bad because of reddit? I already answered your question.

'Not infinitely many' doesn't tell me how many primes there, it only tells me how many primes there aren't.

[Anonymous No.9067438]

>>9067411
the same problem occurs in Tarski-Grothendieck theory since it is very easy to build arbitrary large categories.
Every ordinal is a category (because it is an ordered set) and since TG implies ZFC, there is no set containing all ordinals (since the set of all ordinals would exist thanks to the comprehension scheme, but we know it is impossible).

[Anonymous No.9067441]

Can't you prove Euclid's theorem with topology? You don't even need to appeal to infinities lmao

[Anonymous No.9067445]

>>9067437


There are "double negation

of finite"-many


primes.

[Anonymous No.9067447]

>>9067404
>You can do all the category theory within NBG.
Nope. Grothendieck universes are the [math] minimal [/math] requirement for a set theoretical model of category theory.

[Anonymous No.9067448]

>>9067441
>Can't you prove Euclid's ""theorem""
You can, if you use an inconsistent foundation.

[Anonymous No.9067450]

>>9067430
>>9067437
>>9067445
Go back where double spacing is acceptable.

[Anonymous No.9067454]

>>9067450
>double spacing
Go back where calling "reddit spacing" "double spacing" is acceptable.

[Anonymous No.9067458]

>>9067438
>the same problem occurs in Tarski-Grothendieck theory
Nope.

[Anonymous No.9067471]

You can construct very very large categories. But they will be essentially dark. If the amount of work to describe the category exceeds 10200, then it is not a coherent category.

[Anonymous No.9067475]

>>9067471

>>9067471

>If the amount of work to describe the category exceeds 10200

What's so special about 10200?

[Anonymous No.9067479]

>>9067475
I actually typed [math]10^{200}[/math]

[Anonymous No.9067481]

https://www.washingtonpost.com/amphtml/news/early-lead/wp/2017/07/27/at-age-26-ravens-john-urschel-retires-from-nfl-to-pursue-phd-in-math-at-mit/

>top uni
>chad
>nfl player for best defensive afc team
>math
>pursuin phd

you
>can barely lift
>virgin
>autistic
>statistically speaking, most of you are not in top unis
>cant speak to women without cumming
>has the audacity to tell others they wont succeed in math
lmao i am well rounded individual and even if i dont make it in maths you all will look like failures in the future (unless u are a hobbyist which i respect more than autismal phds)

[Anonymous No.9067482]

>>9067404
NBG is to Category Theory as Naive Set Theory is to ZFC. It's nicer to think that way but you actually need Tarski-Grothendieck Set Theory for Category Theory. Moreover, Tarski-Grothendieck is a non-conservative extension of ZFC (which is obvious because it's literally ZFC + another axiom).

Regarding Haskell you are so wrong I don't even know where to start.
1) There are lots of different Lambda Calculus'.
1.a) The simple one without types is equivalent to a Turing Machine.
1.b) However once you add types to Lambda Calculus it becomes weaker than a Turing Machine but it gives you a lot of nice results (eg. Curry-Howard correspondence between simply typed lambda calculus and minimal logic, a variant of intuitionistic logic).
1.c) There is a Lambda Calculus called System F that includes polymorphic types (and by Curry-Howard corresponds to second order propositional logic).
1.d) Haskell is based on a variant of System F that among other differences allowed a breakthrough that made it practical to write a fast typechecker for it.
2) In case it wasn't obvious already, Haskell is not Turing Complete.
3) Haskell includes a function called "seq" that isn't definable under Lambda Calculus. This function breaks a ton of things.
3.a) In case it wasn't obvious, Haskell itself is not a Lambda Calculus.
3.b) Similarly Hask (the "Category" of Haskell's type system) is not actually a category for the same reason.

See
http://math.andrej.com/2016/08/06/hask-is-not-a-category/
https://wiki.haskell.org/Seq
https://stackoverflow.com/questions/12687392/why-is-seq-bad

>>9067450
>>9067423
>>9067417
>>9067412
>>9067396
kys yourself

>>9067401
Amusingly wrong tryhard.

[Anonymous No.9067483]

>>9067479

What's so special about 10^200?

[Anonymous No.9067484]

>>9066895
They pass through each other. Otherwise, they cannot exist at the same time. I understand Hillburger's point but I disagree with him.

[Anonymous No.9067485]

Number wars now! Death to all finitists >>9067475 and double spacing redditors!
Inaccessible cardinals will prevail!

https://www.youtube.com/watch?v=MxKhhMdbU5I

[Anonymous No.9067486]

>>9067483
It's an extremely dark number

[Anonymous No.9067489]

>>9067486
>It's an extremely dark number

Define 'dark number'

[Anonymous No.9067491]

>>9067482
>Haskell is not Turing Complete
Aren't "programmers" such as yourself supposed to know that this is false?

[Anonymous No.9067494]

>>9067489
It's a number that is close to [math]10^{200}[/math]

[Anonymous No.9067495]

>>9065795
Link?

[Anonymous No.9067497]

>>9067494
Define 'close'.

[Anonymous No.9067498]

>>9067494
What kind of definition is that?
Is 10^(200)-1 a dark number?
Is 10^(200)-10 a dark number?
Is 10^(200)-100 a dark number?

[Anonymous No.9067500]

>>9067495
https://www.youtube.com/watch?v=I0JozyxM1M0

[Anonymous No.9067505]

>>9067498
Yes

[Anonymous No.9067506]

>>9067497
"close" means "near", or also to shut something

[Anonymous No.9067509]

>>9067506
Define 'near'.

[Anonymous No.9067510]

>>9067509
Stop arguing with finitists. SHOOT FIRST, TALK LATER!

[Anonymous No.9067511]

>>9067505
Is 0 a dark number?
Is 1 a dark number?
Can you show me a number for which it is not the case that it is a dark number?

[Anonymous No.9067513]

>>9067505
Is [10^200]/2 dark?

[Anonymous No.9067514]

>>9067510
Define 'finitists'.

[Anonymous No.9067515]

>>9067514
Define "define".

[Anonymous No.9067517]

>>9067510
You will die out sooner than us, believer in fairy tales.

[Anonymous No.9067519]

>>9067515
That's left as an exercise to the reader.

[Anonymous No.9067523]

>>9067511
No. No. You've listed two.
>>9067513
Yes

[Anonymous No.9067525]

>>9067523
Is it possible given any number to explicitly tell if it's either dark or not?

[Anonymous No.9067529]

>>9067525
ja

[Anonymous No.9067530]

>>9067491
I never said I was a "programmer". You're right on that point. Seems Haskell added general recursion (not in System F) in order to become Turing Complete.
https://stackoverflow.com/questions/25255413/how-did-haskell-add-turing-completeness-to-system-f
Still calling Haskell syntactic sugar for untyped lambda calculus is retarded, you would be just as right to say untyped lambda calculus is syntactic sugar for Haskell, or Python is syntactic sugar for Latex.

[Anonymous No.9067533]

>>9067530
>Still calling Haskell syntactic sugar for untyped lambda calculus is retarded
I never claimed this, I'm not the earlier retard.

[Anonymous No.9067534]

>>9067525
If you can write it as series of |.
I'm not sure why 0 is or how fractions and non-positive integers work.
You're gonna have to go back to the source.

[Anonymous No.9067536]

>>9067529
How?
>>9067534

[Anonymous No.9067538]

>>9067536
>How?
Easily

[Anonymous No.9067541]

>>9065316
Have to study Sears volume 2

[Anonymous No.9067543]

>>9067538
Elaborate. I'm interested in these dark numbers and I would like to study them.

[Anonymous No.9067547]

>>9067543
>I'm interested in these dark numbers and I would like to study them.

[Anonymous No.9067548]

I am saying this because I want avoid collateral damage in the coming battles: I am not a finitist. But applied mathematics is an ill-defined term.
That's all, thank you for reading.

[Anonymous No.9067552]

>>9067548
>I am not a finitist
How do you know? Define 'finitist' please.

[Anonymous No.9067556]

>>9065316
Well, I recently proved a novel result in the theory of integral equations, might have some nice applications in physics.

[Anonymous No.9067557]

>>9067548
>But applied mathematics is an ill-defined term
Uh, nope

[Anonymous No.9067558]

>>9067556
What result is that? That some numerical algorithm terminates?

[Anonymous No.9067559]

>>9067556
>I recently proved a novel result in the theory of integral equations
Post it.

[Anonymous No.9067562]

How to prove [math]\mathbb{Q}[/math] is not the direct product of two non-trivial groups? I'm stuck.
My thoughts:
Let [math]\mathbb{Q} = G\times H[/math]. Without loss of generality suppose [math]G \ne \{e_{G}\}[/math] is non-trivial. Then I need to show [math]H[/math] is trivial.
If I could show that natural projection [math]\pi_{G}:G\times H\rightarrow G[/math] is injective(and hence bijective) then I'd say that the kernel of this projection is [math]ker \ \pi_{G} = \{e_G\}\times H[/math] as obviously there exists only one function [math]f: \{e_G\}\times H \rightarrow G\times H[/math] and [math]f \pi_G[/math] is a zero morphism. Then [math]H[/math] would have to be trivial so that zero morphism [math] \{e_G\}\times H \rightarrow G[/math] exists(reminder, [math]G[/math] is non-trivial). This would end the proof.
I bet there is another way to show [math]H[/math] is trivial by proving injectivity of natural projection without using notion of kernel. Could you help me find it?
Anyways, I can't prove this injectivity and I haven't used some special properties of [math]\mathbb{Q}[/math](obviously this is the missing link, I don't know how to utilize it though)

Well, while I was writing this I took a pause to think and google I found that this problem was proven by noting that [math]G \times \{e_H\}[/math] and [math]\{e_G\} \times H[/math] are subobjects of [math]\mathbb{Q} =
G \times H[/math] and then by using property that in [math]\mathbb{Q}[/math] any non-trivial subobjects have non-trivial intersection, and thus [math]H[/math] would have to be trivial. I don't understand what can be subobject of [math]\mathbb{Q}[/math] and what is the intersection of these objects.
So, anon, could you help me please.
To summarise my problems:
1. proving natural projection is injective
2. finding way to prove the statement using injectivity of natural projection and not using kernel(if there is any)
3. understanding concept of subobject [math]\mathbb{Q}[\math] and intersection

[Anonymous No.9067571]

>>9067558
>reddit frogs
>>>/b/

[Anonymous No.9067573]

>>9067556
>physics
>>>/sci/physics general

[Anonymous No.9067574]

>>9067548
>>9067557
It's well-defined since "non-existent" is well-defined.

[Anonymous No.9067577]

>>9067562
$\Q$ is torsion free and for every couple of non unit elements (a,b), there are integers n,m such that an=bm. This is not true in a product unless one of the factors is zero.

[Anonymous No.9067578]

>>9067556
see >>9067573

[Anonymous No.9067597]

>>9067577
I don't understand anon, how is this related to "intersection of subobject" or are you trying to show something else?

[Anonymous No.9067600]

what is the largest non-dark integer?

[Anonymous No.9067601]

>>9067597
If I say that a subobject of [math]\mathbb{Q}[/math] is a set of multiples of [math]a[/math] then it will have a non-trivial intersection with multiples of [math]b[/math], more concretely [math]an = bm[/math] for some [math]n,m[/math]

[Anonymous No.9067602]

>>9067558
It's about the properties of solutions of certain integral equations that might extend to a nice classification theorem of sorts..
>>9067559
>Post it
Post a result that'll go in a paper?
>>9067573
>>9067578
What? Integral equations show up in physics, the whole purpose of this though is purely for the study of integral equations, nothing more, physics applications was just an add on to help sell it to a larger community to get exposure, gotta get that grant money from somewhere, guess you guys don't have to worry about that shit.

[Anonymous No.9067603]

>>9067601
Is it what you were trying to say?

[Anonymous No.9067613]

>>9067388
NO U

[Anonymous No.9067635]

>>9067602
>It's about the properties of solutions of certain integral equations that might extend to a nice classification theorem of sorts..
this statement is infuriatingly vague

[Anonymous No.9067654]

>>9067635
That's kind of intentional cause I don't want to doxx myself on accident by basically telling you exactly what I am doing, but I guess I can be a bit more specific while not revealing who I am. Essentially it turns out that certain integral equations (IEs) are more general than differential equations (all pdes can be turned into IEs but not the other way around) and as such you might want to study IEs to understand the behavior of their solutions (plus regularity is bit nicer when looking at IEs) point is we found a nice way to get L^2 estimates on a certain class of IEs and were able to extend these results to integral transforms that allow us to prove some conjectured properties of transforms that people have numerical evidence or partial results of.

[Anonymous No.9067657]

>>9067635
>>9067654
Whoops, made a mistake, it's not certain IEs but IEs in general that have pdes as a subclass

[Anonymous No.9067705]

>>9067654
Sounds nice, anon. Good to see that there's some light in this dark shithole.

[Anonymous No.9067780]

>>9067654
anon that's pretty neat

>all pdes can be turned into IEs but not the other way around
can you elaborate on this? to be frank: i only have an undergraduate education in mathematics. i've barely looked at PDEs and never at IEs

[Anonymous No.9067811]

>>9067705
Thanks anon
>>9067780
You can think about it this was, differential equations are based on the derivative which is local (you only care about points very near to where you're taking the derivative) while an integral can be local (by introducing a dirac delta or something to that effect) or non-local (integrals require an interval of points with no point being "special" like with the derivative). So one could transform a differential equation into an integral equation but not necessarily the other way around.

[Anonymous No.9067842]

>>9067600
I think it's 4.

[Anonymous No.9067845]

>>9067176
This is interesting but I need a diagram to fully understand it

[Anonymous No.9067846]

>>9067602
Who gives a fuck about doxxing? If the paper is interesting, it's interesting. One of the idiots posting in this general already doxxed themselves. Nothing happened. (Did it? Were you pizzabombed?)
I posted shit here that probably gave me away to former classmates. Who gives a shit? Why are you so scared that someone might find out you post on 4chan?

[Anonymous No.9067847]

>>9067846
>One of the idiots posting in this general already doxxed themselves.
Proof?

[Anonymous No.9067851]

>>9067845
Don't bother. It's wrong.

[Anonymous No.9067855]

>>9067847
Why do you think gorillaposting stopped?

[Anonymous No.9067858]

>>9067855
Is "Why do you think gorillaposting stopped?" a proof? It doesn't seem like one to me.

[Anonymous No.9067867]

>>9067858
I would've been remiss if it looked like one to you since we're not having a discourse in a domain where mentions of proofs make much sense.

[Anonymous No.9067868]

>>9066579
I love physics for the amazing and hard problems and the insight and profundity you get. I don't care about gritty work, but my stomach revolts whenever a my teacher makes a bullshit vector calc """"proof""" that is supposed to give me geomtric intuition. I'm not even that knowledgeable about analysis, but I just don't like to listen to wrong arguments and that I have to argue like that. I know it's tedious af to derive things properly, but there should be at least some discussion about the formal treatment and then explain all these mnemonic shit.

[Anonymous No.9067874]

>>9066529
>>9066579
>>9067868
Return to your containment thread >>9051158

[Anonymous No.9067878]

>>9067874
I'm asking here because that thread sucks and most physicists will not agree that using diferentials is un-rigorous garbage.

[Anonymous No.9067884]

>>9067878
>I'm asking here because that thread sucks
Of course it does, it's a thread about physics on a board with a very low IQ.
>most physicists will not agree that using diferentials is un-rigorous garbage
You will have to somehow deal with this problem yourself, this thread isn't really for that.

[Anonymous No.9067886]

WHAT THE FUCK IS GOING ON IN THIS THREAD????????

[Anonymous No.9067889]

>>9067868
>but my stomach revolts whenever a my teacher makes a bullshit vector calc """"proof""" that is supposed to give me geomtric intuition. I'm not even that knowledgeable about analysis, but I just don't like to listen to wrong arguments and that I have to argue like that. I know it's tedious af to derive things properly, but there should be at least some discussion about the formal treatment and then explain all these mnemonic shit.
Well, not everything in physics is at the level of rigor of mathematics (a hell of a lot of it is, but not all), and one of the reasons they give proofs like that is simply due to the fact that it's the style of argument, one example where this hit hard for me was thermodynamics where a lot of the claims and proofs seemed very hand wavy, part of this was to stress the physical arguments and intuition underlying everything (which is VERY important) but also since full derivations have already been done in some cases and in others are extraordinarily difficult (the mathematical theory goes under the name of ergodic theory and many seeming natural statements in statistical mechanics have yet to be made rigorous at the level of mathematics) so it's probably just the case that they want to build up a proper foundation so that if you go to grad school you can do the real work.

[Anonymous No.9067892]

>>9067886
Severe autism.

[Anonymous No.9067894]

>>9067889
>physics
You seem to be mistaken.
Here's the proper thread you should be using.
>>9051158

[Anonymous No.9067898]

>>9067886
>>9067892
What are you referring to?

[Anonymous No.9067899]

>>9067878
>that thread suck
Well, yeah... it's full of physics.
Please do not pollute /mg/.

[Anonymous No.9067901]

>>9066529
>>9066579
>>9067868
>>9067878
>>9067889
see >>9067894
then see >>9067899

[Anonymous No.9067936]

Looking at graduate schools

Is it true that pure mathematics is largely unassailable unless you're one of the top 100 pure mathematicians in the world, ie, has all the low hanging fruit in relevant pure maths already been plucked?

I want to go into a field where my work will have some relevance instead of disappearing because it's essentially an exposition of a result we already know about.

Does that mean I should go into applied mathematics in a blossoming field that has not seem much work? Such as bioinformatics

[Anonymous No.9067945]

>>9067936
>applied mathematics
There is no such thing.

[Anonymous No.9067956]

>>9067945

Fine you autist.

If I want my work to be relevant, should I look at getting a phd in an applied mathematics *department* and doing cross-disciplinary work in relatively new fields, such as bioinformatics, computer science, mathematical physics, statistics as opposed to doing graduate work in a mathematics department that focuses entirely on mathematics with little known applications.

[Anonymous No.9067965]

>>9067936
>>9067956
>reddit spacing
>"applied mathematics"
You have to go back.

[Anonymous No.9067971]

>>9067936
>Is it true that pure mathematics is largely unassailable unless you're one of the top 100 pure mathematicians in the world, ie, has all the low hanging fruit in relevant pure maths already been plucked?
Not at all, there are plenty of pure mathematicians who aren't at the level of field's medalists, there are still low hanging fruit and novel discoveries to be made by us mere mortals.

>I want to go into a field where my work will have some relevance instead of disappearing because it's essentially an exposition of a result we already know about.
Most pure math fields are still booming, take for instance number theory which is hundreds of years old and still going strong, ro geometry which is thousands of years old (diff geo and AG are still VERY active fields).

>Does that mean I should go into applied mathematics in a blossoming field that has not seem much work? Such as bioinformatics
If you mean work as someone who applies math to problems outside of mathematics then that might be good for grant purposes and as a source of interesting problems which would constantly give you work, this article talks about it.

http://www.ams.org/notices/201510/rnoti-p1172.pdf

[Anonymous No.9067972]

>>9067965

Been here since '06 my dude. I've always typed this way

[Anonymous No.9067974]

>>9067971
>>9067972
Reddit spacing isn't welcome on this website.

[Anonymous No.9067977]

>>9067971

Thanks, this gives me some insight. I was under the impression that unless you are a gauss-level undergraduate student looking at harvard as a candidate school, you will never make meaningful contributions to mathematics

[Anonymous No.9067980]

>>9067974

what the fuck is reddit spacing

you mean breaking things into paragraphs so people can fucking read your post? the fuck outta here

[Anonymous No.9067981]

>>9067977
>reddit spacing
>-level
Definitely a redditor. Fuck off back to your shithole of a website.

[Anonymous No.9067984]

>>9067980
Your post has three instances of reddit spacing.
These posts have paragraphs, but don't have reddit spacing >>9067482 >>9067562

[Anonymous No.9067987]

>>9067977
You will never make meaningful contributions to mathematics. Aptitude aside, you have an impertinent attitude to it.

[Anonymous No.9067989]

>>9067548
I really wanted to argue with you but i find it hard to come up with an arguement that applied mathematics is well defined

[Anonymous No.9067990]

>>9067984
>>9067981


K Y S

Y

S

M Y D U D E

Y

D

U

D

E

[Anonymous No.9067992]

>>9067990
>K Y S
>M Y D U D E
>>>/r/eddit/

[Anonymous No.9067993]

>>9067977
Reminder that Galois was a fuccboi grad student who got expelled from some nowhere university and died believing he was a failure.

No one can tell you how important your work will actually be.

[Anonymous No.9068002]

>tfw even the physicsposters above at least had the decency to format their posts properly but the summerfags still hit return twice for every new line of text like a bunch of incontinent brainlets

[Anonymous No.9068003]

>>9067993
>mathematicians always die before their work is recognized
math truly is an art

[Anonymous No.9068005]

>>9068002

>reddit spacing

is the the start of le epic new meme?

[Anonymous No.9068009]

>>9068005
>le epic
>meme
>reddit spacing
>>>/b/

[Anonymous No.9068026]

>>9066908
>the number of walks between two distinct vertices in a complete graph
The answer I arrived at is:
[math] \sum\limits_{k=0}^{n-2}\frac{(n-2)!}{(n-2-k)!} [/math].
I assumed that a walk between two vertices is a collection of edges that connect the them.

[Anonymous No.9068160]

My uni "strongly recommends" taking discrete math before abstract algebra even though it's not an actual prerequisite, would I be screwing myself over if I take the two classes simultaneously?

[Anonymous No.9068165]

>>9067562
note that addition in [math]\mathbb{Q}[/math] is given by [math](a,b)+(c,d)=(ad+bc,bd)[/math], not [math](a,b)+(c,d)=(a+c,b+d)[/math], so it cannot be a direct product of groups

[Anonymous No.9068172]

>>9068160
>would I be screwing myself over if I take the two classes simultaneously?
probably not

[Anonymous No.9068173]

>>9068003
>mathematicians always die before their work is recognized
No.

[Anonymous No.9068191]

>>9067846
>Were you pizzabombed?
Did I miss out on pizza day? I haven't been on campus in a while

[Anonymous No.9068214]

>>9065316
Is there a school of object oriented mathematics in which counters are the primitive objects?

[Anonymous No.9068223]

Threadly reminder to work with physicists.

[Anonymous No.9068225]

>>9068223
>with
for*

[Anonymous No.9068283]

>>9068223
>>9068225
see >>9067901

[Anonymous No.9068289]

>>9068225
Why would I work "for" my own slaves?

[Anonymous No.9068297]

>>9068225
Right and wrong. Those mathematicians that dislike the supposed "lack of rigor" in physics should also reject statements proven assuming generalized RH/CH.

[Anonymous No.9068298]

>>9068297
True. Assuming anything without proof can only be used to prove that it is wrong. Any mathematician thinking otherwise is an idiot.

[Anonymous No.9068581]

>>9068026
there is no way that sum simplifies to >>9066908
[math] n^2 - 3n + 3 [/math].
who got it wrong?

[Anonymous No.9068583]

>>9068223
>>9068225
>>9068298
kys physishits

[Anonymous No.9068595]

So many triggered autists. Never change, you bastard of a general.

[Anonymous No.9068613]

>>9068583
>>9068595
Right and wrong. Those mathematicians that dislike the supposed "lack of rigor" in physics should also reject statements proven assuming generalized RH/CH.

[Anonymous No.9068626]

>>9068297
>>9068613
this doesn't make any fucking sense. conditional proofs are not 'full' proofs but there's nothing about them that lacks rigor.

[Anonymous No.9068635]

>>9068626
Assuming anything that is not proven (except axioms lol) cannot yield a proof. So assuming anything without proof can only be used to prove that it is wrong.

[Anonymous No.9068657]

>>9068160
>My uni "strongly recommends" taking discrete math before abstract algebra even though it's not an actual prerequisite, would I be screwing myself over if I take the two classes simultaneously?
It's what I did and I had no real problems. Maybe if you could give us more information about the content of each course we could tell you more, but in principle you have nothing to worry about.

[Anonymous No.9068658]

>>9068581
Let's put it to the test.
In pic related, according to the first formula, from >>9066908 there should be 16 - 12 + 3 = 7 walks between two points, and according to the second formula, from >>9068026 there should be 1 + 2 + 2 = 5 walks between two points.
According to http://mathworld.wolfram.com/Walk.html a walk is a sequence [math] v_0, e_1, v_1, \dots, v_k [/math] of graph vertices [math] v_i [/math] and graph edges [math] e_i [/math] such that for [math] 1 \leq i \leq k [/math], the edge [math] e_i [/math] has endpoints [math] v_(i-1) [/math] and [math] v_i [/math]. Simple enough.
How many walks between A and D are there?

There's
A, [AD], D
A, [AB], B, [BD], D
A, [AC], C, [CD], D
A, [AB], B, [BC], C, [CD], D
A, [AC], C, [BC], B, [BD], D

Well anon, it looks like your textbook is wrong. I'm assuming it was written by compsci brainlets.

[Anonymous No.9068664]

>>9068635
Please be bait.

[Anonymous No.9068666]

>>9068658
If you are allowed to return to A (don't know if this is valid), there's also:
A, [AB], B, [BC], C, [CA], A, [AD], D
and
A, [AC], C, [CB], B, [BA], A, [AD], D
Notice that we don't visit a single edge twice, which maybe more important in this context than not visiting a vertex twice. Anon should give us more information about the problem and how these concepts are defined in the textbook.

[Anonymous No.9068672]

>>9068666
Why would you be allowed to visit a vertex twice?

[Anonymous No.9068684]

Hi, I have a question. Are there any good resources, aka, paths for self learning/self studying math? I want to become a computer scientist, so I need to git gud at math. I would like to go from basic, to advanced. Enough for me to learn, and practice something everyday, along with programming. Thank you in advance.

[Anonymous No.9068697]

>>9068672
In pic related between B and C there's just the walks B, [BC], C and B, [BA], A, [AC], C otherwise you revisit edges.
So it does not make sense to me why you'd be allowed to revisit vertices but not edges. It seems inconsistent.
According to the first formula there should be 3 walks between B and C, and according to the second formula only 2. Now, if you are only allowed to revisit the starting vertex and any edge involved in the retread, then there are 3 walks in total but this seems so ad hoc to me.

[Anonymous No.9068708]

>>9068672
Dunno, I was just presenting the possibility, but considering >>9068697, the book's probably just wrong.
>>9068684
Pick the uni with the best CS or maths program you can think of (i.e. Cambridge, Princeton, CalTech, etc...), and just follow their program. Use libgen.io to download the recommended bibliography if you can't find the course notes in that uni's/prof's website, or simply take them from MIT OCW.

[Anonymous No.9068714]

>>9068708
Thank you anon.

[Anonymous No.9068758]

>>9068708
I have nothing on my hands so I decided to check the 5-vertex case by hand too. (TL;DR the textbook is flat out wrong, no two ways about it. Anon's formula counts walks as sequences of vertices and edges where you're not allowed to revisit both edges and vertices.)

Textbook formula: [math] 5^2 - 3*5 + 3 = 13 [/math] walks.
Anon formula: [math] \sum\limits_{k=0}^{3}\frac{3!}{(3-k)!} = 1 + 3 + 6 + 6 = 16 [/math] walks.

Possible walks between A and E:
A, [AE], E
A, [AB], B, [BE], E
A, [AC], C, [CE], E
A, [AD], D, [DE], E
A, [AB], B, [BC], C, [CE], E
A, [AB], B, [BD], D, [DE], E
A, [AC], C, [CB], B, [BE], E
A, [AC], C, [CD], D, [DE], E
A, [AD], D, [DC], C, [CE], E
A, [AD], D, [DB], B, [BE], E
A, [AB], B, [BC], C, [CD], D, [DE], E
A, [AB], B, [BD], D, [DC], C, [CE], E
A, [AC], C, [CB], B, [BD], D, [DE], E (Note: 13 walks so far)
A, [AC], C, [CD], D, [DB], B, [BE], E
A, [AD], D, [DC], C, [CB], B, [BE], E
A, [AD], D, [DB], B, [BC], C, [CE], E
This makes 16 total walks where we haven't passed through the same vertex or on the same edge twice. If you allow vertex revisits but no edge retreating, there are even more walks, but I think the result is clear and there's no point to counting any more.

So yeah anon, find a better major/university. At the very least, find a better textbook. The one you're using is garbage.

[Anonymous No.9068766]

>>9068758
Nice work anon! This is a really good example of how using small examples is greaty beneficial if tedious.

[Anonymous No.9068853]

Baked a new bread >>9068851
You fuckers better not shit it up. Go make new bait threads and leave the general for worthwhile content.

[Anonymous No.9068859]

Baked a new bread >>9068858
You fuckers better not shit it up. Go make new bait threads and leave the general for worthwhile content.

[Anonymous No.9068862]

>>9067851
can you elaborate?

[Anonymous No.9069444]

>>9067811
ahh I see, so you can force integrals to be local by using the "delta" """""function""""", but there's no equivalent operation to force derivatives to be non-local, hence the one-way relationship between them

sweet. do you have any resources you suggest for learning more about integral equations?

[Anonymous No.9069841]

>>9067519
Define: the removal of fine details of an object, particularly hypothesized infinite levels of detail.

[Anonymous No.9069843]

>>9069841
*fine details of a description of an object

[Anonymous No.9069885]

Real new thread.

Real thread: >>9069866
Real thread: >>9069866

[Anonymous No.9069941]

wut

[Anonymous No.9069954]

They're genociding /mg/ because you fuckers couldn't go 5 seconds without crying about Physics (which no one posted about) and programming (which was just the OP joke).

[Anonymous No.9069958]

>>9069954
>Physics
Fuck off.
>Programming
Fuck off.
You should really consider ending your life.

[Anonymous No.9069964]

>>9069444
https://books.google.com/books?id=FxZgYfkAA48C&pg=PR5&source=gbs_selected_pages&cad=3#v=onepage&q&f=false
https://books.google.com/books?id=_PolpKk9Ii4C&pg=PR11&source=gbs_selected_pages&cad=3#v=onepage&q&f=false

Also what the hell happened to the latest thread?

[Anonymous No.9069973]

new
>>9069971
>>9069971
>>9069971
>>9069971

[Anonymous No.9069978]

https://www.youtube.com/watch?v=znFmsy6wUiA

[Anonymous No.9069983]

Now we can't have nice things because a fucking retard (this one >>9069958) can't go 5 seconds without autistically crying.

[Anonymous No.9069987]

>>9069983
Physicsfags are really a special stupid of breed.
Thanks for proving my point retard.

[Anonymous No.9069988]

>>9069983
>physics
>nice things
End your own life immediately.

[Anonymous No.9069990]

>>9069983
Why did you remove your horrible avatar? Please attach it to your posts.

[Anonymous No.9069993]

fucking niggers

[Anonymous No.9069995]

>>9069987
>>9069988
No one posted anything about Physics, it was all both of you autistically screeching at the void.
>>9069990
I'm not the Yukari poster.

[Anonymous No.9069996]

>>9069995
Fuck off.

[Anonymous No.9069997]

>>9069995
>Physics
Kill your retarded self.
>>9069995
>I'm not the Yukari poster.
You sure seem like a subhuman, so I will consider you to be him. Please keep attaching your shitty avatar.

[Anonymous No.9069999]

The Yukari poster made a thread complaining about Chinese mainland policies on /his/ the other day. I have proof.

[Anonymous No.9070004]

>>9069999
>I have proof.
Go ahead. I'm going to track him down pretty soon. Don't be surprised if his posts suddenly stop.

[Anonymous No.9070006]

>>9070004
I meant post, not thread.
https://desuarchive.org/his/thread/3137603/#3147410

[Anonymous No.9070007]

>>9070006
That's embarrassing, I wonder if his family knows their Asian kid spends time arguing with anime images when he should be studying.

[Anonymous No.9070008]

>>9070007
Soon he simply won't be able to study.

[Anonymous No.9070011]

New
>>9070010
>>9070010
>>9070010

[Anonymous No.9070554]

>>9069995
It's not just two of us. I happened to be asleep, that's all. What's so wrong with keeping a math general about math and not every other crap one shitter or another wants to talk about instead because they majored in whatever?