/mg/ - maths general

[Anonymous No.11198243]

Formerly >>11183694
Classification of the cubics edition.

[Anonymous No.11198282]

Which topics/books in geometry would you recommend to someone who is familiar with basic differential topology, Riemannian geometry, Riemann surfaces, classical algebraic geometry and some Kähler geometry.

[Anonymous No.11198285]

>>11198282
Spaces with nonpositive curvature, synthetic differential geometry and contact geometry.

[Anonymous No.11198289]

>>11198282
>modular forms
>Hodge theory
>Lie groups, symmetric spaces etc.
>Teichmüller theory

[Anonymous No.11198357]

Threadly reminder to work with physicists.

[Anonymous No.11198367]

>>11198243
just figured out how the parent of elements in d-arry heap would look like in general form by deriving mysefl
. Am I genius?

[Anonymous No.11198438]

whats the giant dad of mathematics?

[Anonymous No.11198884]

Brainlet here, trying to figure out the analytical solution to Advent of Code day 4.
https://adventofcode.com/2019/day/4

So far I have the amount of ascending numbers, by noticing that for any combination of digits they can be arranged to be ascending.
So using this combinations with repetitions formula I get 5005 ascending numbers from all 6-digit numbers: https://www.mathsisfun.com/combinatorics/combinations-permutations.html

I thought something similar might be possible for picking out all numbers that have the same digit twice.
Instead of picking 6 digits a b c d e f and sorting them, I just pick the same digit twice: a a b c d e and sort them.
This doesn't give me the correct answer, I think. Any hints?

[Anonymous No.11198893]

>>11198884
no one cares cslet

[Anonymous No.11198984]

>>11198893
t. Can't solve a simple combinatorics problem.

[Anonymous No.11199095]

>>11198984
welcome to /mg/ where mymathintrest>yourmathintrest

[Anonymous No.11199119]

Is the graph generated by taking the category of all graphs isomorphic to some category in the category of categories itself an object in the category of all graphs?

[Anonymous No.11199155]

>>11199119
kek

[Anonymous No.11199196]

>>11199155
i cant conceive of the answer, am i a brainlet

[Anonymous No.11199202]

>>11199155
>>11199119
isnt this a contradiction? it seems like russels paradox

[Anonymous No.11199216]

>>11199119
>the graph generated by (...)
it's a graph
>some category in the category of categories
you mean an object in Cat or a category internal to Cat?
>an object in the category of all graphs?
yes, because it's a graph

>>11199202
Russel's paradox involves naively defined sets, but categories in general consist of a "collection " of objects rather than a set.

[Anonymous No.11199226]

>>11199216
>you mean an object in Cat or a category internal to Cat?
an internal category, ie a description of an object in cat

It just seems like if this graph was an object in Graph, it would necessarily change the structure of cat, and thus the graph. Or maybe im thinking too much in terms of set theory

[Anonymous No.11199229]

>>11198357
whose picture is on the wall in the mathematics classroom?

[Anonymous No.11199231]

>>11199229
hilbert

[Anonymous No.11199232]

I need college advice here. Is math+stats a good double major? I don't want to do CS and I really like learning more maths.

[Anonymous No.11199235]

>>11199216
are there any category theory axioms in the same are rigorously defined?

[Anonymous No.11199237]

>>11199235
*are there any category theory axioms in the same way sets are rigorously defined?

[Anonymous No.11199242]

This is the question of the century. Fuck, Marry, Kill: Noether, Gauss, or Hilbert

>>11198243
why are you posting this fucking virign loser, fake ass head ass mother fucker?
FUCK NEWTON

[Anonymous No.11199258]

>>11199242
>question of the century
>highly trivial
Marry Noether, because I'm not homosexual.
Fuck Hilbert, kill Gauss.

[Anonymous No.11199274]

>>11199119
good post.
Ilke the others said, CAT (category of big categories) does not operate on sets but on "collections". (only small categories have a set's worth of morphisms and objects)
I was actually thinking about exploring the relationships between CT and GT in depth after I (hopefully) take GT 2 next semester.

[Anonymous No.11199301]

>>11199237
As far as I know there isn't a ZF equivalent to how you define categories (but maybe a category theorist could better answer this question). In ZF you get axioms like "sets are equal that contain all the same elements" which is more a common notion than a postulate imo.

On the other hand the setup of category theory contains to common notions but only definitions and postulates. The definitions are that a category consists of objects and morphism, and the postulates are that you can tell the source and target (domain-codomain) of any morphism, that every object has an identity and so forth.

Imo the general flavour of the two fields are different. ZF tells you what you how the logic of sets works and how you can create new ones, whereas categories tell you what you can do with them.

>>11199226
I sometimes like to think of collections naive type-theoretically, where I forget about sets of things, and theorems become "if x is an object in this category, i.e. if x satisfies this description, then blah blah blah"

[Anonymous No.11199350]

>>11198282
I still have a hard time believing people thought this look was cool. Is it really any better than being norwood 2?

[Anonymous No.11199358]

>>11199258
>kill gauss
>be a faggot
>pretend not to be a faggot
lol faggot.

[Anonymous No.11199370]

Thought of this today. Sounds true -- because divisors can be thought of as products of unique permutations of prime factors?
Let [math]D(n) = \big|\{a\in \{1\ldots n\} : a \mid n\}\big|[/math]
Let [math]X_n = \prod (\pmb P_1 \ldots \pmb P_n)[/math]
Is this true? [math]\ \forall X_n , \big(\nexists y < X_n : D(y) > D(X_n)\big)[/math]

[Anonymous No.11199375]

>>11199242
Fuck Noether, because I'm not a homosexual. Kill Hilbert because I want to marry Gauss, who is smarter, and have a cool heterosexual bro platonic marriage.

[Anonymous No.11199376]

I meant >>11199370
[math]\ \forall X_n , \big(\nexists y < X_n : D(y) \geq D(X_n)\big)[/math]

[Anonymous No.11199382]

>>11199370
Wrong. take y=24, X_n = 2*3*5=30
D(y)=|{1,2,3,4,6,8,12,24}|=8
D(30)=|{1,2,3,5,6,15,30}|=8

[Anonymous No.11199396]

>>11199382
Thanks. /sci/ is so much cooler than MathOverflow which is where I used to post.

[Anonymous No.11199425]

>>11199382
What about replacing P_1*...P_n with n!
24 is 4*3*2*1

[Anonymous No.11199473]

Dumb question, but when the Grinch stole christmas, did he preform the inverse operation on Whoville to return it to it's identity state?

[Anonymous No.11199482]

>>11199425
This is not in general true, see https://math.stackexchange.com/questions/1884694/is-7-5040-the-largest-highly-composite-factorial
It doesn't work for n!, where n>7

[Anonymous No.11199483]

>>11199375
BAYYYSED smart goy.

[Anonymous No.11199534]

>>11198285
>>11198289
Thanks for your recommendations. Could you also recommend some nice resources? Especially on Spaces with nonpositive curvature, contact geometry, modular forms, lie groups and Teichmüller theory.
>>11199350
definitely better

[Anonymous No.11199655]

>>11199534
>spaces with nonpositive curvature
Gromov.
>contact geometry
Hofer.

[Anonymous No.11199819]

>>11199237
>>11199301
>are there any category theory axioms in the same way sets are rigorously defined?
Yes. I've some notes on here here, especially at the beginning for FOL, and see the links therein at the bottom (mostly the nLab ones)

https://axiomsofchoice.org/category_theory

>>11199119
I'm not sure if you want to consider isomorphic graphs or whether you mean to consider all the graphs for different categories at once.
Once this is given proper form, it's mostly just a nasty question of size and will thus on your setp.

To help making it more concrete (pun intended), note that the number of edges in a graph will be some cardinal and the consider the realization of a graph is a functor:
Consider the (abstract) category C with two objects (call them e and v) and four arrows, two of which are parallel
id_e : e->e
id_v : v->v,
s : e->v
t: a->b
Consider now a set valued functor
F : C ---> Sets
which is a map taking the objects e and v to particular sets (say two cardinals Fe:=E and Fv:=V in Sets) and s and t to two functions (say F(s):=S and F(t):=T, of type E->V).
Note that this exactly is the specification of a directed graph with |V| vertices V and |E| edges E with fixed source and target specification (given by the functions S and T).

All the grpahs are just all those functors. Since those are just the presheaves on the tiny C^op, the collection of F's together as a category is actually a nice topos.
WIth that we're closer to getting a feel for it's size (Sets to the power of something a bit more complicated than the discrete two objects category) w.r.t. whatever "Sets" meant for us.

There's a paper by Schulman on size issues, maybe it helps
https://arxiv.org/pdf/0810.1279.pdf

[Anonymous No.11199821]

>>11199237
>>11199301
>are there any category theory axioms in the same way sets are rigorously defined?

Yes. I've some notes on here here, especially at the beginning for FOL, and see the links therein at the bottom (mostly the nLab ones)

https://axiomsofchoice.org/category_theory

>>11199119
I'm not sure if you want to consider isomorphic graphs or whether you mean to consider all the graphs for different categories at once.
Once this is given proper form, it's mostly just a nasty question of size and will thus on your setp.

To help making it more concrete (pun intended), note that the number of edges in a graph will be some cardinal and the consider the realization of a graph is a functor:
Consider the (abstract) category C with two objects (call them e and v) and four arrows, two of which are parallel
id_e : e->e
id_v : v->v,
s : e->v
t: e->v
Consider now a set valued functor
F : C ---> Sets
which is a map taking the objects e and v to particular sets (say two cardinals Fe:=E and Fv:=V in Sets) and s and t to two functions (say F(s):=S and F(t):=T, of type E->V).
Note that this exactly is the specification of a directed graph with |V| vertices V and |E| edges E with fixed source and target specification (given by the functions S and T).

All the grpahs are just all those functors. Since those are just the presheaves on the tiny C^op, the collection of F's together as a category is actually a nice topos.
WIth that we're closer to getting a feel for it's size (Sets to the power of something a bit more complicated than the discrete two objects category) w.r.t. whatever "Sets" meant for us.

There's a paper by Schulman on size issues, maybe it helps: https://arxiv.org/pdf/0810.1279.pdf

[Anonymous No.11199935]

>>11199821
>there's a paper by Schulman on size issues
IMAGINE ACTUALLY WASTING TIME ON "MUH CLASSES OF CLASSES OF CLASSES" TO GIVE FUCKING CATEGORY THEORY RIGOROUS FOUNDATIONS L M A O

[Anonymous No.11200155]

Has anyone completed Summit's Algebra?

[Anonymous No.11200498]

>>11199232
>learning more maths.
honours major

[Anonymous No.11200858]

>>1727913
>>1728463
>>1728472
are there any other interesting equations using (length/area) as a variable?

[Anonymous No.11200863]

>>11200858
>>>/diy/1727913
>>>/diy/1728463
>>>/diy/1728472

[Anonymous No.11200866]

Not a mathematician.

Can anybody explain to me what the fuck this "category theory" autism is all about? I never heard about it before this year and I can't imagine what the fuck the sudden appeal is. Is it like abstract algebra but completely useless?

[Anonymous No.11200879]

>>11200858
>>11200863
The shear stress in a beam is given by
[eqn]\tau=\frac{VQ}{It}[/eqn]
where V is the shear force at a given point, Q is the first moment of area of the cross section, I is the second moment of area of the cross section, and t is the thickness of the beam.

The quantity [math]\frac{Q}{I}[/math] has units of Length^3/Length^4 which is equivalent to area/length.

Im sure there are more.

[Anonymous No.11200881]

>>11200879
**equivalent to length/area
Q/It is called shear flow btw

[Anonymous No.11200899]

>>11200866
>Is it like abstract algebra but completely useless?
you pretty much summed it up, except maybe it's closer to set theory than abstract algebra

[Anonymous No.11200901]

>>11200899
Okay. Is there a reason I am only just now find out about its existence?

[Anonymous No.11200907]

>>11199370
Can you clarify what P1 up to Pn are?

[Anonymous No.11201053]

>>11200866
>Completely useless
It's used everywhere in contemporary geometry.
E.g. you can't was a page of Scholze without it

[Anonymous No.11201198]

How is this not affine? I feel retarded but isn't a finite disjoint union of affine schemes affine? If so, what was the correct formulation of the exercise?

[Anonymous No.11201199]

how many of you in your undergrad days abused the proof by contradiction ? as in
>theorem: there exists X such that ..
>proof: suppose there is no such X. *proceeds to construct X.* we have reached a contradiction and therefore the proof is complete.

[Anonymous No.11201213]

>>11201198
>I feel retarded but isn't a finite disjoint union of affine schemes affine?
Why would it be affine?

[Anonymous No.11201219]

>>11201199
I didn't, but yeah i see a lot of people doing that

[Anonymous No.11201221]

>>11201199
I have a small question for the logicians here.
Would proving (not P) by showing P-> false be considered proof by contradiction? Or is it only when you use LEM (equivalently double negation) like to prove P you do not P -> false.

[Anonymous No.11201222]

>>11201198
where did you get the idea that they are disjoint

[Anonymous No.11201293]

>>11201221
Yes, you are correct. The first proof is called proof by negation

[Anonymous No.11201296]

>>11201221
I meant that the first proof is not considered as a proof by contradiction

[Anonymous No.11201297]

>>11201198
It is not disjoint. Otherwise, indeed, it would be affine.

[Anonymous No.11201313]

>>11199534
For modular forms, see the wonderful book by Diamond-Shurman.
For Lie groups, see Kirillov’s Introduction to Lie groups and Lie algebras for the essentials and Fulton-Harris for more representation-theoretic results

[Anonymous No.11201716]

>>11201221
>Would proving (not P) by showing P-> false be considered proof by contradiction?
This is the definition of proof by contradiction. You only need LEM if you use it to prove not not P and want to conclude P.

[Anonymous No.11201730]

>>11199229
Marx

[Anonymous No.11201822]

I like a certain subfield of analysis and I'm looking for a name for it.
I really like exercises and theorems that mostly deal with just metric spaces. The kind where you argue with balls, sequences, compactness and so on. Maybe I could call it unquantifiable analysis.
Stuff like Vitali's lemma or the equivalence of compactness definitions. Proving a subset of a separable space is separable.
Is this just babby tier analysis? It's so fucking satisfying to do though. I also feel like my brain "clicks" well with it because I can imagine the balls easily

[Anonymous No.11201841]

>>11201822
>a subset of a separable space is separable
It isn't.

Also, you most likely mean point-set topology.

[Anonymous No.11201844]

>>11201822
And an example of a problem of this kind:
For a subset A of R define
A* = {x in A: exists r > 0 with Br(x) n A = {x}}
Then D(A) := (closure(A)\A)*
a) Find an A with D^n(A) nonempty for all n in N
and then
b) Classify subsets of R by how often you need to apply D to get the empty set.
Then come up with a situation where this classification is useful lol

[Anonymous No.11201852]

>>11201841
It is dude. It's not super trivial.

[Anonymous No.11201868]

>>11201841
and point set topology is too "weak", the impression I got from intro to topo is that everyone wants to move on to homolgy and shit because point set topology cant do much.
I also remembee the proof being hand wavy because no one wanted to bother constructing the homeomorphisms or paths (if this isnt actually part of point set topology then nevermind idk what Im talking about)

[Anonymous No.11201870]

>>11201822
It's called point-set topology and it's dead. You might like set theory though. Or some kind of topology

[Anonymous No.11201878]

>>11201852
https://en.wikipedia.org/wiki/Separable_space#Properties
Literally the first property.
>>11201868
That sort of argument starts and ends in point-set topology.

[Anonymous No.11201896]

>>11201878
I meant separable metric space, we had it as an exercise so I would be surprised if it isn't true there.
And yes I see that it's point set topology now. Fuck though

[Anonymous No.11201984]

>>11201896
It's true there, yeah.

[Anonymous No.11202033]

>>11198282
I've been quite enjoying 3-manifold topology by Jennifer Schultens. I would like to at one point understand Thurston's geometrization conjecture and how perelman solved it. That requires a lot more Topology and analysis than I know now, though

[Anonymous No.11202090]

>>11202033
Just pick up Morgan and Tian's exposition and work through it.

[Anonymous No.11202094]

What does cos(x,y) mean?

[Anonymous No.11202095]

>>11202094
Maybe it means the angle between y and x'

[Anonymous No.11202103]

>>11202094
TIP:
The area of a triangle is base times height.
TIP 2:
AOC and ABC share AC as a base.
TIP 3:
Project the heigh from ABC onto the plane and solve for the appropriate angle, thus solving the problem backwards.

[Anonymous No.11202140]

how do I solve
arcsin(x) + x*sqrt(1 - x^2) = pi/4 ?

[Anonymous No.11202155]

>>11202140
Through your crippled and malnourished geometric intuition.

[Anonymous No.11202186]

>>11202155
what?

[Anonymous No.11202190]

>>11201199
>>theorem: there exists X such that ..
>>proof: suppose there is no such X. *proceeds to construct X.* we have reached a contradiction and therefore the proof is complete.
i do this all the time, is there anything wrong with it though

[Anonymous No.11202213]

>>11201822
Are you me?

[Anonymous No.11202216]

>>11202190
>theorem: there exists X such that ..
>proceed to construct X
That's enough, no need to introduce a negation into the argument scheme.

Also, there's a variant
>theorem: there DOES NOT exists X such that ..
and one is often tempted to prove it by contradition via
>assume it exists, the proceed to show that this leads to a contradition
This is classically valid, but it also means that there's a version of the theorem that 1. immediatenly implies the above and 2. has a constructive proof

[Anonymous No.11202230]

>>11202190
>theorem: there exists X such that ..
>proceed to construct X
That's enough, no need to introduce a negation into the argument scheme.

[Anonymous No.11202289]

>>11202190
>is there anything wrong with it though
you make things unnecessarily complicated for whoever is listening

[Anonymous No.11202495]

Can someone show me how this is true?
[math]n=p_1^{e_1}*p_2^{e_2}*\ldots*p_n^{e_n}[/math]
[math]\text{divisors}(n) = (e_1+1)*(e_2+1)*\ldots*(e_n+1)[/math]
Relation between number of divisors and prime decomposition

[Anonymous No.11202509]

Can someone show me or give some intuition into how this relation between number of divisors and prime decomposition is true?
[math]n=p_1^{e_1}*p_2^{e_2}*\ldots*p_n^{e_n}
\text{divisors}(n) = (e_1+1)*(e_2+1)*\ldots*(e_n+1)[/math]

[[m No.11202513]

Can someone show me or give some intuition into how this relation between number of divisors and prime decomposition is true?
[math]n=p_1^{e_1}*p_2^{e_2}*\ldots*p_n^{e_n}[/math]
[math]d(n) = (e_1+1)*(e_2+1)*\ldots*(e_n+1)[/math]

[Anonymous No.11202576]

>>11202513
Each divisor is the product of p_i^n_i where 0<=n_i<=e_i. A bijection exists between the set of divisors and the Cartesian product of the sets [0...e_i], each of which has cardinality e_i+1. The cardinality of the Cartesian product is just the product of the cardinalities of the individual sets, i.e. (e_1+1)*(e2+1)*...*(e_n+1).

E.g. for 2^3*3^2=72, the divisors are the products of the elements of {2^0,2^1,2^2,2^3}×{3^0,3^1,3^2} = {1,2,4,8}×{1,3,9}. There are (3+1)*(2+1)=4*3=12 of them, all distinct: {1,2,4,8, 3,6,12,24, 9,18,36,72}.

[[m No.11202621]

>>11202576
Thanks. I found the answer here too https://www.math.upenn.edu/~deturck/m170/wk2/numdivisors.html

Any recommendations on what I should study? I am 24 and never been to college, and not much of an interest of going now. I'm interested in number theory, abstract algebra, geometry and computer science. My question is decently representative of my current knowledge.

[Anonymous No.11202725]

>442 pages
wtf bros?!?!

[The Geometry of Hilbert's 13th(...) No.11202912]

https://www.youtube.com/watch?v=txaU6NTwxOU
>Given a polynomial in one variable, what is the simplest formula for the roots in terms of the coefficients? Hilbert conjectured that for polynomials of degree 6,7 and 8, any formula must involve functions of at least 2, 3 and 4 variables respectively (such formulas were first constructed by Hamilton). In a little-known paper, Hilbert sketched how the 27 lines on a cubic surface should give a 4-variable solution of the general degree 9 polynomial. In this talk I’ll recall Klein and Hilbert's geometric reformulation of solving polynomials, explain the gaps in Hilbert's sketch and how we can fill these using modern methods. As a result, we obtain best-to-date upper bounds on the number of variables needed to solve a general degree n polynomial for all n, improving results of Segre and Brauer.

[Anonymous No.11202966]

why are algebraic geometers so based?

[Anonymous No.11203318]

>>11201221
Not p is often (always?) defined as p -> false
In the Lean theorem prover for an example this is the case

[Anonymous No.11203502]

>>11202725
>We don't like the Axiom of Choice
>But we want to keep the last 100 years of mathematics as well

[Anonymous No.11203629]

>>11198243
I've been fascinated with the Steenrod algebra and stable homotopy groups recently. Does anyone know where I can find pictures of subalgebras for odd primes?

[El Arcón No.11203642]

>>11202912
>Jesse Wolfson
thems the breaks kid

[Anonymous No.11203755]

>>11203629
Generate them yourself. You do know how to code right anon?

[Anonymous No.11204058]

>>11203629
cool pic bro

[Anonymous No.11204072]

Anyone got any advice on studying group theory?
I’m doing my second module on it now and it’s getting to more involved stuff like amalgamated products and semi direct products etc. My brain absolutely hates it and I can’t make any sense of it. I read over the definitions and looked over examples but I can’t make out an intuition or reasoning behind them. Any help would go a long way bros

[Anonymous No.11204083]

>>11199350
They used them to cover syphalis symptoms on their scalp

[Anonymous No.11204096]

>>11204072
Do you know any topology or diff geometry ?

[Anonymous No.11204109]

>>11204096
I know a good bit diff geo but only very little about topology

[Anonymous No.11204144]

Has anyone here worked on probability distributions in Banach Spaces? Can anyone recommend some books? Is it even interesting/useful?

[Anonymous No.11204153]

>>11204072
>amalgamated products
These are very intuitive if you think in terms of group presentations. What's the problem?

[Anonymous No.11204181]

>>11201053
Big in theoretical CS as well, the functional programming people have copped it for everything

I like to think of objects in a category as little hunks of corn in my shit, and morphisms as the springy poop matrix binding the corn kernels together

[Anonymous No.11204199]

>>11204153
Yeah it made sense to me when explained to me like that but then when I actually try to work one out I’ve no idea how to determine the additional relations.
Also pic related is how my lecturer defined it and I find it really non intuitive

[Anonymous No.11204233]

>>11198243
I just bought a hardcover copy of understanding analysis by abbot on amazon for 33usd.
Did I waste my money or have I made a good investment?

[Anonymous No.11204237]

>>11204072
>intuition
>reasoning
anon....this is algebra. you merely get used to things.

[Anonymous No.11204500]

>>11204199
His definition boils down to calling it the pushout in the category of groups. This tells you why it's important, but not how to construct it, nor is it obvious that it even exists a priori. Terrible definition IMO.

Here's an explicit construction. Let's assume [math]K[/math] is the trivial group first for simplicity, so we're going to make the free product [math]G \ast H[/math]. This is supposed to be to most general group generated by the disjoint union of [math]G[/math] and [math]H[/math]; i.e. making a group out of it by doing the least amount of work. Since I need to be able to multiply all the elements of my group, I need products like [math]gh[/math] to be in there. This is fine when [math]g,h[/math] are both in either [math]G[/math] or [math]H[/math], but what about when [math]g \in G[/math] and [math]h \in H[/math]? Well, the easiest thing to do is just to add [math]gh[/math] as an entirely new element, treated as a formal product. The same goes for [math]hg[/math]. But now I need to be able to multiply these things as well. So now I have elements like [math]hghg[/math], [math]ghgh[/math], [math]hg^2h[/math], [math]gh^2g[/math], etc. (note that I can "simplify" the last two, since [math]g^2[/math] is some element in [math]G[/math] and likewise for [math]h^2[/math]).

Continuing this line of reasoning, you get that [math]G \ast H[/math] consists of all "words" (e.g. [math]g_1 h_1 ... g_n h_n[/math]) in the elements of [math]G[/math] and [math]H[/math], with multiplication being string concatenation. Words can "simplified" when there are two elements from the same group next to each other, and the identity elements of [math]G[/math] and [math]H[/math] are identified as the common identity element of [math]G \ast H[/math].

As a simple example, [math]\mathbb{Z} \ast \mathbb{Z}[/math] is [math]F_2[/math], the free group on two generators. In general, if [math]F_m,F_n[/math] are free groups, then [math]F_m \ast F_n \cong F_{m+n}[/math].

[Anonymous No.11204507]

>>11204500
When [math]K[/math] is nontrivial, you get a proper amalgamated product. The story is the same, except now I "glue together" elements of [math]G[/math] and [math]H[/math] that have a common preimage in [math]K[/math]. Equivalently, you quotient [math]G \ast H[math] by the normal subgroup generated by words of the form [math]\phi(k)\psi(k)^{-1}[/math].

[Anonymous No.11204509]

>>11204507
*Equivalently, you quotient [math]G \ast H[/math] by the normal subgroup generated by words of the form [math]\phi(k)\psi(k)^{−1}[/math].

I need to sleep.

[Anonymous No.11204613]

>>11204509
sleep is trivial

[Anonymous No.11204666]

>>11204500
>>11204507
>>11204509
That's been very helpful, thank you anon:) so is ϕ(k)ψ(k)−1 the equivalent of taking the commutator of the group that you can then use to find the relations between the different generators of the group?

[Anonymous No.11204809]

>>11198884
maybe find the number of cominations of 5 numbers that you can create (10 over 5), notice that there is only one way to sort them asc. Then notice that you can double each number to get a good password, so the answer in the range 0 - 999999 should be 5 * (10 over 5).

[Anonymous No.11204832]

>>11204666
Quotienting out ϕ(k)ψ(k)−1 is just adding the relation ϕ(k)=ψ(k), one for each k. This is the "gluing" that I mentioned.

[Anonymous No.11204841]

>>11204832
I should also mention that in practice, you can be economical with the relations; i.e. you have one for every element of K, but many of them are usually consequences of some smaller set of relations (e.g. those involving generators).

[Anonymous No.11204894]

>>11204832
>>11204841
I think I get you, I managed to get an amalgamated product now but I'm not sure on it yet so I'll keep trying. Thank you!

[Anonymous No.11204895]

I'm getting hung up on a lot of the elementary algebra in Spivak's Calculus. As in, I struggle to prove things and solve problems because I don't find the same manipulations.

Here is an example, the text in the OP parallels my experiences (including having taken Calculus courses before, up to multivariable and vector calc): https://www.physicsforums.com/threads/spivak-chapter-2-problem-6.750666/

Should I take a detour with a book like Chrystal's Elementary Algebra? Or just keep hammering away at Spivak?

[Anonymous No.11204898]

In relation to the above post, things make sense after I see them. I just don't discover them on my own. It's not that I don't understand elementary algebra, and again I have completed the calculus series at my uni. But there's a whole bag of tricks it seems like I'm missing.

[Anonymous No.11204905]

Ok actually I realize I was just being retarded in this case, I was hung up on how (n+1)^3 - n^3, when summed, came out to (n + 1)^3 - 1, but I see now.

Generally though, my question still stands and any tips would be great.

[Anonymous No.11205286]

>>11204895
>>11204898
>>11204905
I don't have any specific advice but definitely do something else if you feel like you're struggling
It's okay if you don't get this kind of algebraic manipulation tricks 100% of the time. I personally got gud at this when I was studying for hs math competitions; I would recommend working through a good exercise book (by good I mean: there are solutions in the back, and the problems are neither too hard nor too easy, imo ideally you should be able to solve half of the problems on your own). The problems should be in elementary algebra or simple analysis, i.e. limits, series, check that an equality/inequality holds, etc. Then just work through the problems, and you read the solutions properly and try to extract any new ideas.

[Anonymous No.11205641]

how do i prove non-existence of a graph if i know it's number of vertices, cycles and it's diameter?
for example, i know a graph that has 4 vertices and a diameter of 2 couldn't possibly have 2 cycles. how could i prove that?
(assuming the graphs are simple undirected graphs)

[Anonymous No.11205667]

>>11205641
>4 vertices
>diameter of 2
Ennumerate by hand lmao. How many of those do you think are there?

[Anonymous No.11205670]

>>11205667
By the by, I'm pretty sure there's two.

[Anonymous No.11205672]

>>11205670
>two
*Three, I forgot the Y shaped one.

[Anonymous No.11205702]

>>11205672
Ugh.
Pretty confident this is it.

[Anonymous No.11205712]

>>11205702
Triangle plus a line?

[Anonymous No.11205718]

>>11205712
FUCK, WHY DO I SUCK SO MUCH AT ENNUMERATING

[Anonymous No.11205760]

>>11205718
Because you're not enumerating, you're guessing. Actually go through all the cases for each choice of edge next time.

[Anonymous No.11205776]

>>11204898
read more, work more problems you will eventually figure it out. you’ve had no formal mathematical training whereas many people like spivak have been doing these kinds of tricks and solving problems like those in that textbook since they were in middle school. The broader problem for actually attaining proper solutions comes down to intuition, which requires years of exposure to math, and intelligence which cannot be trained or granted to anyone.

[Anonymous No.11205824]

>>11205286
>>11205776
Thanks for the excellent responses. I think I'll continue working through Spivak but this week I'll try and spend, roughly, an equal amount of time working through a book closer to my problem solving level like what the first anon described. And any suggestions for some books like that would be welcome. And to the second anon, how might get clues to indicate whether or not I have this intelligence? I imagine an IQ test is a little limited in scope by comparison. I like to think I can succeed in math, but I have not been doing this since middle school like Spivak. But I do have a lot of time to commit to math and hope I can catch up.

[Anonymous No.11205830]

Also, in relation to the above, I had an easier time working through an abstract algebra book than I have Spivak's calculus book. These elementary algebra / combinatorics etc problems (just hairy 'elementary' problems in general) just destroy me. I guess the abstract algebra book must've just been easier, or had easier problems at least. I assumed it'd be harder since its considered more advanced material.

[Anonymous No.11206510]

set theory exam today

[Anonymous No.11206566]

You're a gardener with a bag of seeds to plant grass with on an infinite plane. Using all the grass yields a shape with unit area (e.g. area = 1). You're allowed to plant the grass anyway you want, wherever you want. A grasshopper comes along and picks a random point on your plot of grass. The grasshopper then, at a uniformly random angle and at a given fixed distance, will hop from where it is to a new point on the infinite plane.

What is the optimal shape you should plant your grass in to make sure the grasshopper is still on your plot of grass after a single hop?

[Anonymous No.11206599]

>>11206566
i draw a penis

[Anonymous No.11206644]

>>11206510
>set theory

[Anonymous No.11206732]

well, /mg/?

[Anonymous No.11206752]

>>11206732

[Anonymous No.11206794]

>>11206732
i don't know, I would try thinking about weighted shifts on l^2 and under what condition on the weights this shift is hyponormal/subnormal

[Anonymous No.11206920]

>refer to all my professors as Doctor
>they insist to call them by their name
>I do it anyways

[Anonymous No.11207232]

>>11206510
babbys first proof

[Anonymous No.11207236]

>>11206920
Unless they get legit upset, keep doing this. You do not want to accidentally speak to that ONE professor in your math department's faculty in casual terms. They will go through hell and high water to fuck you over.

[Anonymous No.11207260]

>>11206920
>have a meeting with all my old friends who instead of the Ph.D. went on to work in big tech and earn 600k+ a year while I make sub 6 figures after taxes.
>this student keeps calling me "doctor", reminding me of the the worst career decision I ever made
>politely ask him to stop calling me that so I can at least peacefully ignore how horrible my life is
>he does it anyways, as if he knows how much it hurts me

S-should I fail this student, guys? I feel he is doing it on purpose just to fuck with me.

[Anonymous No.11207287]

Do Mersenne primes have connections outside of perfect numbers and/or number theory?

[Anonymous No.11207298]

Is studying number theory worth it?
I picked it as a module for next semester but I'm wondering if I should switch it out for quantum information processing, which sounds more interesting but I hate the lecturer teaching it so I'm more inclined to do number theory

[Anonymous No.11207304]

>>11207298
If your good at physics, the deep connections between number theory and physics is an underdeveloped field

[Anonymous No.11207309]

>>11206566
My intuition says that, for a small enough distance, the answer is a circle, but it doesn't know the answer otherwise.
Specifically, if the distance is larger than the diameter of the area one circle the solution becomes something else, but I can't come up with any actual shapes.

[Anonymous No.11207319]

>>11207309
After thinking further, by brain concluded that for large distances the answer is an ellipse.

[Anonymous No.11207325]

>>11207304
That's interesting. I'm definitely better at physics than maths but I'm pretty interested in learning number theory anyway.
I'm only an undergrad anyway so its probably doesn't matter what modules I pick anyway

[Anonymous No.11207335]

>>11207325
Number theory is one of the least useful branches of math when it comes to physics.

Number theory is pretty neat imo, so just check it out and see yourself

[Anonymous No.11207344]

>>11207335
Just because the connections haven’t been found doesn’t mean they don’t exist. Most of the silly little integer properties won’t be applicable, but I believe there is more to the story of “number theory and physics are largely disjoint”

[Anonymous No.11207410]

If you have a pure function (subroutine, method) that is aur hectic evening/objective/whatever the fuck it’s called and takes on n-bit integer argument and returns one n-bit integer, is it possible to find an inverse function and if so what’s the complexity. I have a feeling it’s generally impossible but possible for special cases

[Anonymous No.11207413]

If you have a pure function (subroutine, method) that is injective/surjective/whatever the fuck it’s called and takes on n-bit integer argument and returns one n-bit integer, is it possible to find an inverse function and if so what’s the complexity. I have a feeling it’s generally impossible but possible for special cases

[Anonymous No.11207435]

>>11207413
Let f(n) be your original function.
Define:
g(n) = {
let x = 0;
while x < 2^n:
if g(x) = n return x;
if g(-x) = n return -x;
x = x+1;
end while;
}

[Anonymous No.11207437]

>>11207435
Well, I fucked up the variables big time but you get what I was going for.

[Anonymous No.11207449]

hi everyone, a slightly meta question.

I'm an utter brainlet, currently in my second year of a pure math undergrad. Adjusting to primarily proofs-based courses was a struggle for me which I am only now stabilizing in.

Does anyone have a method for going through textbooks? I find it difficult to concentrate hard or long enough to make meaningful progress, obviously I get stuck on problems, sometimes for days at an end, and it just becomes hard for me to juggle it together with my regular coursework but if I don't work through the textbooks I feel like I'm not fully comprehending the material.

Am I just too dumb for math? It's starting to get to me.

[Anonymous No.11207457]

>>11204144
Li-Queffélec

[Anonymous No.11207495]

>>11207413
> If you have a pure function (subroutine, method) that is injective/surjective/whatever the fuck it’s called
Which one?

> and takes on n-bit integer argument and returns one n-bit integer, is it possible to find an inverse function
For a function to have an inverse it has to be bijective (which implies both injective and surjective). If the domain is finite, the codomain must have the same size. If the domain and codomain are the same set, a bijection is a permutation of that set.

> and if so what’s the complexity.
O(2^n). Iterate over all n-bit integers:

int inverse_f(int x) {
....for (int i=0; i<(1<<n); i++)
........if (f(i)==x) return i;
....return -1;
}

Provided that f is bijective, the above is its inverse. If you'll be calling the function repeatedly and n isn't too large, you can use an array to speed things up:
int p[1<<n];
int gen_inverse(void) {
....for (int i=0; i<(1<<n); i++)
........p[f(i)]=[i];
}
The inverse is just f^-1(x)=p[x].

> I have a feeling it’s generally impossible but possible for special cases
If n is so large that O(2^n) isn't feasible, you'd have to construct the inverse by hand from the original function. There isn't any method other than brute force that will work for every possible function. The original function could just be a lookup into a randomly-shuffled array.

[Anonymous No.11207546]

>>11198243
Probably a bad place to ask, but how would you derive this analytically? I made a simulation in Python to solve this and got about 13 presses to minimize the number of buttons pressed.

My strategy was as follows:
Pick a distance, 1 <= n <= 99
Now, click the Random button until you are on a track n below 42 (and it wraps around if you go negative).
Once you find that, click "Next" until you get to track 42.

Through using probabilities, I've found that it's minimized at N = 12 with an expected number of presses of 12, but the simulation says 13. Would anyone be able to point me towards any theorems/ concepts that would be useful?

[Anonymous No.11207606]

>>11207495
injective

[Anonymous No.11207612]

>>11207546
its minimized at N=14 with expected number of presses equal to 13.64...

its calculated as 100/N + (N-1)/2
its the expected number of random presses needed to land inside the range + the expected number of next presses needed to get to the desired track

[Anonymous No.11207623]

>>11198243

massive brainlet here, I need help

Is there a book with math problems that will help me practice and take me from beginner to advanced?
I'm particularly interested in analysis, linear algebra and proofs of all kinds
I know basic logic (truth tables and such) and did some calculus in school but I always sucked at it.

[Anonymous No.11207657]

Can /sci/ help a calclet? Prove/disprove: the set of all (real-valued) continuously-differentiable functions [math]f[/math] on the interval [math][0,1][/math] s.t. [math]f'(\frac {1}{4}) = f'(\frac {3}{4})[/math] is dense in [math]C[0,1][/math] under the sup-norm.

I thought that [math](x - \frac {1}{4}) \sin (\frac {1}{x-1/4})[/math] or a similar function whose slope goes nuts around [math]\frac {1}{4}[/math] might prove to be a counterexample, but I'm having trouble formalizing my reasoning and I'm not entirely sure it's correct in the first place.

[Anonymous No.11207665]

>>11198243

> "Women in mathematics" is an organization at my uni
> I don't go to their events because I'm afraid it will be feminist cock and ball torture
> This time I went to one because I didn't know it was their event
> only two undergrad girls show up
> one of them sits in the back with her BF
> the other is a CS major (I.e., subhuman)
> There are a bunch of female grad students
> They probably outnumber undergrads
> Not a single male grad student
> about 5 undergrad boys show up

Really makes me think

[Anonymous No.11207685]

>>11207657
Oh its absolutely true. By weierstrass approximate it with a polynomial, and add a small and steep as required bump function near 1/4 to make the equation true while not displacing it too much. You can easily formalize this if you are not a complete brainlet.

[Anonymous No.11207799]

>>11207657
According to (my textbook's version of) Stone-Weirstrass theorem, what you need for family of functions to be dense in C(X), where X is a set, is just the following
0) the family of functions must be an algebra (closed under addition, scalar multiplication, and composition)
1) the functions must separate the points in X (ie, if x =! y, then there is f so that f(x) =! f(y)
2) for all x in X, there is an f so that f(x) =! 0

check these three properties.they shouldn't be that hard.

[Anonymous No.11207881]

>>11207665
>tfw no grad-student girlfriend to tutor me while I pretend not to understand the subject
I understand those five undergrad boys, they have my sympathies.
>>11207623
Polya's Problems and Theorems in Analysis.
Otherwise, see the sticky for recs.

[Anonymous No.11207893]

>>11207612
Okay, so I mostly understand the explanation, but I have trouble wrapping my mind around why the expected number of random presses is 100/N.
This number *is* very close to what I got, so I trust that it's correct. I just wanna know why it's 100/N and not some other expression.

[Anonymous No.11207976]

>>11202140
>>11202155
You don't even need geometric intuition; knowing some basic identities (double angle and [math]\sin^2 + \cos^2 = 1[/math]) is enough. Check this solution after you've attempted the exercise again.

[Anonymous No.11208051]

>>11207309
>>11207319
i read the full paper before working on the problem myself so i kinda spoiled the answer before ever getting a chance to see what it was, but i can tell you right now for all distances d > 0, a circle and ellipse are sub-optimal. In fact, circles are only optimal at d = 0 and decrease their viability monotonically w.r.t. distance.

I'd recommend just coding a simulation of the thing to see what happens, it's pretty counter intuitive. Relatively large distances aren't even simply connected.

[Anonymous No.11208092]

>tfw still no cute math BF

[Anonymous No.11208125]

>>11208051
Really?
Link, please.
The solution better not be some disconnected fractal bullshit.

[Anonymous No.11208131]

>>11207893
It's just 1 over the probability that you get in range.
In general if something has probability p, then the expected number of attempts until success is 1/p

[Anonymous No.11208132]

>>11208125
here ya go https://arxiv.org/abs/1705.07621

[Anonymous No.11208145]

>>11208051
>>11208132
Wait a second.
What do you mean it decreases monotonically as d goes to infinity? It should zero when the distance passes the diameter.

[Anonymous No.11208161]

>>11208145
i guess circle is the wrong word, the authors use "disc" so they could hollow out the centre upon reaching relatively large d.

[Anonymous No.11208163]

How would I find a function where the derivative is the inverse of the original function

[Anonymous No.11208174]

>>11208163
solve ff' = 1

[Anonymous No.11208177]

What's the answer for this?

[Anonymous No.11208179]

>>11208163
f(f'(x)) = f'(f(x)) = 1 for all x. Have fun.

[Anonymous No.11208185]

>>11208177
you divided by zero.
Line 1) y = x
Line 5) you divide by (y - x), y = x by line 1. division by zero.

[Anonymous No.11208190]

>>11208179
should be "= x", not 1. derp.

[Anonymous No.11208205]

>>11208185
But if you divide by zero then you have infinity, and if you divide by zero in both sides, then you have infinity in both sides, aren't they equal?

[Anonymous No.11208228]

>>11207976
ignore this, I suck cocks

[Anonymous No.11208243]

>>11208092
Are you a girl?

[Anonymous No.11208254]

>>11208243
No
Women are too stupid for math

[Anonymous No.11208256]

>>11208205
You don't get infinity, division by 0 is undefined because it gives you (sort of) both negative and positive infinity, which isn't even a number to begin with

[Anonymous No.11208277]

>>11208256
So... if I have X=Y and if X=0 then Y=0 too doesn't it? then it should be that 0=0 doesn't it? but that doesn't apply to X=infinity so I can have infinity=infinity?

[Anonymous No.11208292]

no (you) for weak bait
infinity is not a number.

[Anonymous No.11208296]

I'm a cs brainlet, I hope this is okay here:
In the paper "A Simplified Framework for String Stability Analysis of Automated Vehicles", they talk about treating a platoon of vehicles as a mass-spring-damper-system and a notion of stabilty (the spacing error, which is the difference between desired distance and actual distance, should be attentuated by the system).
For this purpose, they convert a system of differential equations "from state-space representation to spacing error coordinates". I don't understand how that works though, since the definition of the spacing error coordinates (second circle in pic related) seems to be unrelated to the spacing error (first circle), even though spacing error coordinates seem to later on get used in place of the spacing error for the attentuation (third circle).

Any ideas on how this works/what I should look up? I've tried reading related papers, but none of them seem to explain this.

[Anonymous No.11208304]

>>11208292
So infinity is not equal to infinity?

[Anonymous No.11208309]

some infinities are larger than others, see countable infinity vs uncountable infinity.

>>11208296
1950: Oh boy in the future I hope we have flying cars!
2019: Intervehicle communication.
It looks like just a coordinate change though.

[Anonymous No.11208313]

>Be a literal retard, program for 12 years
>Finally buckling down on degree
>Have to take college algebra
>"Ha, this'll be a piece of cake. I'm a pro at algebraic solutions"
>Start studying for the course
>It's all factoring polynomials
>Factoring, factoring, more factoring
>Sometimes there's some actual manipulation of operators, but 90% of the work is just more fucking factoring
>Once in a while a matrix to break things up
Jesus christ, please kill me. I can't factor worth shit. Factoring is quite honestly one of the most useless algebraic skills when it comes to programming, so it is by far my absolute weakest. I feel so embarrassed struggling with college _algebra_ of all things. I miss fun math, like trigonometry and octonions. They even mark questions wrong if you give them an imaginary number instead of selecting "No Answer", because the domain has been limited in what I imagine to be some kind of sick mockery.

Please.... I want to go back to abstract algebra-land... these polynomials.... they're making me ill...

[Anonymous No.11208317]

>>11208309
>some infinities are larger than others
So 1/0 is not equal to 1/0?

[Anonymous No.11208321]

>>11208317
In most domains 1/0 is not infinity.

In domains where it is defined, and it is infinity, it is only one conception of infinity. There is more than one kind of infinity.

[Anonymous No.11208336]

>>11208317
Is that from futabu?

[Anonymous No.11208341]

>>11208321
>In most domains 1/0 is not infinity
Wut?

>There is more than one kind of infinity.
So there is two infinities that can be equal? because this is why I don't get it, if I have 1/infinity = 1/infinity, then I have 0 = 0, but when I have 1/0=1/0 given that both the zeros and the ones are exactly the same, it can't give the same infinities so there is infinity=infinity, so, if there is an infinity that is equal to another infinity, then why can't I say that 1/0=1/0, and after that say that infinity=infinity given that both are exactly the same?

[Anonymous No.11208348]

Because allowing division by zero on the reals makes it a non integral domain.

[Anonymous No.11208356]

>>11208309
It's kinda crazy, but we'll get there, eventually.
>It looks like just a coordinate change though.
I probably shouldn't have cut off the system, but pic related isn't a normal coordinate change, right? I just have no clue why this allows you to use the coordinates as the spacing error.

[Anonymous No.11208360]

>>11208336
Maybe... yep.

[Anonymous No.11208365]

>>11208341
>So there is two infinities that can be equal
No. There are different ways that infinity can manifest. Not only that, but there are different cardinalities of infinity. You also have to consider the difference between transfinite and absolute infinity. It's really a topic you should read some actual literature on.

>1/infinity
>1/0
No. You cannot perform comparisons on undefined results.

Dividing by zero classically breaks algebraic consistencies, division by infinity similarly. Outside of some very specific rings, wheel theory and some geometric interpretations, 1/0 is not a valid proposition. It's not that 1/0 = 1/0 or 1/0 != 1/0. It's that both of these are wrong, "equals" doesn't apply.

[Anonymous No.11208369]

>>11208356
Im not gonna bother reading all the indices and seeing what they refer to, so I might be completely wrong, but all you're really doing is defining coordinates as the difference between existing coordinates. It usually makes analysis of some problems a lot easier when you use the difference of coords rather than absolute coordinates, e.g. finding the average distance between two random points on a unit square.

[Anonymous No.11208381]

>>11208369
Oh, okay, I thought you'd need to transform them all in the same manner. Guess I'll have to think about it some more, thanks.

[Anonymous No.11208515]

>>11208365
Ok, thanks for your answers anon, I think I should read more about that before asking, thanks again.

[Anonymous No.11208810]

>>11208304
equaling something doesn't mix well with being unbounded.
inf-inf is undefined

[Anonymous No.11208818]

>>11208317
1/0 doesn't have a value, so no

[Anonymous No.11209323]

I'm not very creative and that's hurting my research a lot. I find I'm very strong mechanically - calculating precise values etc but when it comes to thinking about open-ended problems in depth, I suck.
Any tips on how I can improve?

[Anonymous No.11209329]

>>11208277
What you have to understand is that infinity is not a number, so saying infinity=infinity doesn’t really make sense cause you can for example have:
Infinity+infinity = infinity
So 2 infinity= infinity
So infinity is not a number you can work with.
Also there are different kinds of infinities but that’s besides the point

[Anonymous No.11209336]

How much of Math(particular which sections) must I be good at, to understand Economics? Are there any Economics major/minor here, I know a lot of them come from Mathematics background. Can you please shed some light on this.

[Anonymous No.11209345]

>>11209336
Also as a follow up to this anons question, would be the postgrad/follow up course after a pure maths and physics course to get into economics?

[Anonymous No.11209365]

>>11209336
>>11209345
The "mathematisation" of economics is a scam and a dangerous one. Economy is something political and humane, the laws that govern it aren't absolute and predictable.
But to answer your question, dynamical systems, ergodic theory, measure theory, stochastic processes, probabilities, statistics, optimisation. In physics, statistical mechanics are the best analogy.

Remember that economy isn't just numbers on a sheet, though.The human mind is more complex than that.

[Anonymous No.11209375]

>>11209336
>Are there any Economics major/minor here, I know a lot of them come from Mathematics background
as a follow-up, that's literally because the pay is 3 times better in "economics" and there are very few positions in mathematical research.

[Anonymous No.11209381]

>>11209365
But, I assume you do have to study a lot of Mathematical subjects for an Economics Major in a University course?

[Anonymous No.11209383]

>>11209365
I suspected something along the lines of that. I plan on moving on to study high performance computing after my undergrad so I guess I’m on the right course, thanks

[Anonymous No.11209384]

>>11209381
it probably depends on how the student is inclined. Economics is a very board subject, the dude could focus on Law and International Trade, or on Sociology at work and "Political Economics" and not see an once of math.

Usually you'll only have an heavily mathematised economics program if you want to be in mathematical finance, finance, or if you want to be an actuary, which are already specialised subsets of an "Economics" major.

[Anonymous No.11209413]

>>11209375
>>11209384
Which Economic course would be the best, strictly based on Career Opportunities/growth?

[Anonymous No.11209415]

>>11209384
Which would generally be the best paying out of these? Specifically from the mathematised ones

[Anonymous No.11209437]

>>11209413
I wouldn't actually know, tb'h, my knowledge of the field is only from my uncle who's an actuary and my own readings.
Sorry about that.

>>11209415
A wild guess would be finance for being a tr*der if you've got no soul and you're willing to risk depression, suicide, and/or overdose on crack

[Anonymous No.11209494]

So now that the dust has settled, which of the (proper as well as inproper) Euler-type angles are the best choice?
Yes I know that most people don't like Euler angles... but among those angle definitions for rotations, which one is bestest?

[Anonymous No.11209560]

>>11208205
If you add "infinity" to your number system many things you think should be true, aren't true.
For example 1+x might be the same as x, 2 times x might also be the same as x and thus even in your extended numbers you can not conclude from 2x=x that 2=1.

[Anonymous No.11209611]

I hope you guys don't mind me asking this here. I am being overwhelmed by math book suggestions and guides online.


In which order should I go through these calculus books?
Calculus - Volume 1 and 2 - Tom Apostol

Calculus - Michael Spivak Calculus on Manifolds - Michael Spivak

Differential and Integral Calculus (Volumes 1 and 2) - Richard Courant

Introduction to Calculus and Analysis(Volume 1 and 2)-Richard Courant,Fritz John.

Calculus: An Intuitive and Physical Approach - Morris Kline

Calculus: Early Transcendentals - James Stewart

Calculus (Early/Late Transcendentals) - Howard Anton, Irl Bivens, Stephen Davis

Vector Calculus - Jerrold Marsden, Anthony Tromba

Should I add or remove any of the books? Where do the walter rudin books come in?

[Anonymous No.11209617]

>>11198243
Is there a point in learning maths just for the fun of it? I'm bad at it but I'm interested nonetheless

[Anonymous No.11209629]

Does anyone possess a digital version of Aluffi that presents the corrections of the post-2016 printings? There is a voluminous amount of errata in the versions available on libgen.

[Anonymous No.11209632]

>>11209611
Calculus books are hard to review, because they all teach the same thing, and their real goal is both to give you the tools to differentiate and integrate as needed, but more importantly to give you some "maturity" when dealing with mathematical objects.

The goal of a calc course is that integrals, series, sums, and differentials are no longer spooky symbols, but rather something intuitive and geometrical that you can easily manipulate in a variety of contexts (Analysis, Geometry, Topology, etc...)
So my advice is that you donwload all of them in libgen, quickly survey them, and pick the one you feel you understand the most. There's no need to go with more than one book.

[Anonymous No.11209635]

>>11209617
Is there a "point" of doing anything just for the fun of it? Is there a point to watching television or jacking off to anime or whatever else you do with your free time?

The point is that it's fun.

[Anonymous No.11209655]

>>11208163
[eqn] f(x) = \phi^{1-\phi} x^\phi \\
f'(x) = \phi^{2-\phi} x^{\phi-1} \\
f^{-1}(x) = (\phi^{\phi-1}x)^\frac{1}{\phi} = \phi^{1-\frac{1}{\phi}}x^\frac{1}{\phi} = \phi^{1-(\phi-1)} x^{\phi-1} = \phi^{2-\phi} x^{\phi-1} [/eqn]
Pretty cool eh?

[Anonymous No.11209665]

>>11209629
>There is a voluminous amount of errata in the versions available on libgen.
So? All the errors that are corrected in the second printing are the ones listed in the errata sheet on Aluffi's website.

[Anonymous No.11209713]

>>11209617
What is the question here?

[Anonymous No.11209733]

>>11209655
BTFO

[Anonymous No.11209785]

>>11209383
Are you an econ or math major? How did you get into HPC? i want to as well because the job prospects are good.

[Anonymous No.11209804]

>>11209785
I haven't got into it yet but I'm a physics and maths major, and I plan on applying for a postgrad in it

[Anonymous No.11209809]

>>11209804
Oh right, I majored in economics and I've been out of school for a while now. No chance of me getting into it now. What are you going to write for your motivation letter? How did you get interested in HPC?

[Anonymous No.11209832]

>>11209809
>majored in economics
What's the most preferred or the most lucrative course of Economics one can get a Major in?

[Anonymous No.11209843]

>>11209832
probably some high finance stuff but i went into software engineering which is also kind of lucrative

[Anonymous No.11209896]

>>11209809
>>11209843
There's a University in my country that offers courses in different fields of Economics. Namely Basic Economics, Financial Economics, Business Economics, Applied Quantitative, Econ with Data Science, Actuarial , Behavioral Economics, Environmental, etc. I honestly don't know what to apply for.

[Anonymous No.11209900]

>>11209896
Just do the most mathematical one. Or don't do econ at all, major in mathematics and/or computer science.

[Anonymous No.11209957]

Please explain for a retard:

For Lebesgue integrals, why the fuck is it okay to approximate functions with infinitely many values with simple functions that by definition only can have finitely many values? It seems fucked to me, especially since no textbooks even mention how this is a bit strange.

[Anonymous No.11209981]

>>11209957
Do Riemann sums also seem fucked to you?

[Anonymous No.11209988]

>>11209957
when you take a sequence of simple functions, the subsequent approximations make smaller and smaller errors, right?

[Anonymous No.11209997]

>>11206920
I do this constantly

[Anonymous No.11210003]

>>11209957
because the range of the function you are approximating is separable, so a sequence and its limits contain all the things you need.
for this reason it's easy enough to get a function with finite range to approximate functions with separable range uniformly as long as they're nice enough, and pointwise almost everywhere pretty much no matter what. but not EVERY function can be approximated as such, which is why you talk about measurable functions.

[Anonymous No.11210013]

>>11206920
I grudgingly agreed to call my advisor by his first name because his last name is like 15 letters long and it's a pain in the ass to say. Everyone else can stay Dr. until I have my own PhD.

[Anonymous No.11210137]

>Have to show that the Haar system is a orthonormal basis in [math]L_2(\mathbb{R})[/math]
>Gotta remember functional analysis and Lebesgue stuff from too long ago.
I wonder, can I simply argue with simple functions? I'm not sure if Haar wavelets qualify.

[Anonymous No.11210142]

I'm having fun trying to solve a problem but I've sunk enough time into it that I think I should reach out for help. I have no formal math education past fifth grade so my apologies if I'm not asking this well
If possible, I want HINTS, NOT full solutions

I'm a very new programmer and trying to goof around with sound synthesis. I have all the basics like waveform generators and FM and an n-part ADSR, but now I'm trying to write a function will generate a sine wave that "glides" linearly from one frequency to another within a specified amount of time.

Mathematically, what would be the most concise and versatile way to do this? It would be trivial if the amount of time to glide wasn't specified; I could just continually increment the frequency by some amount after every cycle. But trying to factor in a set time to leave it at means I need a very precise amount to increment the frequency by, and to get that value I know I'll need to factor in the total number of cycles (or, how many increments need to be made) and remaining time left in the glide, which will be affected by the length of the previously added cycles, which depends on their frequencies, which I don't know how to get without already knowing the increment size...

Hopefully you see my problem. Can any of you brainiacs give me a nudge in the right direction? Thanks!

[Anonymous No.11210192]

>>11199232
don't do a double major, apply for research programs and do graduate classes. but it is pretty dumb to not do at least some cs classes, youu're shooting yourself in the foot if you don't have some skill with that.

[Anonymous No.11210298]

>>11210142
Nevermind, found what I needed
freq2/freq1 = x
mult = 1
start with sin(freq * mult) and just increment mult by (x / duration) after every added sample

Very obvious in hindsight, whoops

[Anonymous No.11210385]

>>11210379

[Anonymous No.11210417]

>>11210137
From what I'm seeing, yes, simple functions arguments work.

[Anonymous No.11210419]

I made it through Calc 2 4 years ago and now I want to learn all about QM and solid state physics. for maths after Calc, what else can I do while I'm reviewing calculus? I'm looking at going thru linear alg but I think that's after i finish Calc for the vector Calc, I know some mild vector algebra from mechanics. set theory? abs alg?

[Anonymous No.11210451]

>>11210419
>what else can I do while I'm reviewing calculus
it's more efficient to just focus on learning calculus while calculus, not asking yourself what else you can be doing. finish your calc review and do it well, then move on, imo.

[Anonymous No.11210457]

is this turd right? >>11210427

[Anonymous No.11210475]

>>11198243

Which pens does /mg/ use when taking notes? What are some affordable recommendations?

[Anonymous No.11210476]

>>11208317
Stupid anime poster

[Anonymous No.11210486]

>>11209957
But what seems "strange" about that?
Approximating infinite things with finite things is a very common mathematical strategy, especially in analysis.

It seems intuitive to me that this notion can approximate many functions and that many "reasonable" concepts of functions can be approximated by this (eg. continuous functions).

[Anonymous No.11210490]

>>11210457
Yes.

[Anonymous No.11210494]

>>11210490
why? how can this be true? would a real mathematician agree that it's ok to overlook a thorough understanding of elementary algebra...?

i have a feeling you're just the same turd

[Anonymous No.11210502]

>>11204233
if you can't study on a computer because you get distracted, then its a good investment, otherwise no.

[Anonymous No.11210512]

does anyone have reasearch anxiety? I can come up with something that i find interesting and just find some properties about it, but i dont know if my profesor will find it worth the time?

[Anonymous No.11210516]

>>11210457
>Are you right
Of course I'm right, you faggot.
>>11210475
Bic Ultra Fine. Black.
>>11210494
>thorough understanding of elementary algebra
That's one thing.
A thorough understanding of specific, physically painful calculations is something else entirely. Just calculate the fucking coefficients, what sort of baboon motherfucker does the entire fucking sum for literally no reason?
>same turd
No, you retard.

[Anonymous No.11210556]

>>11210475
pilot precise v5 rollerballs, blue, on yellow steno pads from Costco, also have some graph/eng paper and blank sketch book for large equations and geometry

[Anonymous No.11210559]

>>11210516
it's an infinite sum, i didn't do the whole thing ofc. It's just that there's obviously something going on that I don't understand, and I'd like to patch it up. I agree with not necassarily working through the tedium of every algebraic manipulation, totally; life's too short, but it's not this unique equation I don't understand, it's the general mechanics of whatever is happening in that examplary equation.

so in otherwords, I'm interpretting my problem as a deficit in my understanding of elementary algebra, and I'd like to patch it up quickly. Can you help me? If I must, I can rewrite it so there's less terms and it's easier to read, but honestly I think it's pretty easy to see what I'm talking about here. It was more clear on paper, since I could circle it and arrange terms in a visually favorable way, but my handwriting is ass.

[Anonymous No.11210583]

>>11210559
I'm going to just link this post, >>11210431. If I can clear this up I'd be happy. I'm convinced this is worth doing.

[Anonymous No.11210589]

Is there a better way to represent G=(V,E) in code?

typedef struct {
unsigned long int *edges1;
unsigned long int *edges2;
unsigned long int nodes;
};

Where edges1 and edges2 are <= nodes - 1 and represent the connecting vertices of edges1/2[n]. And nodes = |V - 1|

[Anonymous No.11210593]

>>11210583
>>11210559
so, in other words, what is going here that handles the [math]{n \choose 2}a^k-2b^3[/math] in line 1? I don't see how that was simplified in the following lines.

[Anonymous No.11210608]

>>11210593
...

[Anonymous No.11210624]

What do you call the branch of applied mathematics that deals with computation and strictly math like number theory, graph theory, combinatorics, computer algebra, etc. Not computer science because it incorporates shit like AI and data science

[Anonymous No.11210632]

>>11210589
I imagine you would just make a vertex object which has an identity and connections to other vertices. then a grpah would just be a collection of nodes

[Anonymous No.11210641]

>>11210624
Discrete maths.

[Anonymous No.11210643]

>>11210632
That has space complexity [math]|V|^2[/math] or [math]O(n^2)[/math] whereas by method has [math]|V|*2[/math] space complexity. That means on a 64-bit computer you can't have more than 2^32 nodes which is only 2 billion something.

[Anonymous No.11210647]

>>11210641
Is quantum computation discrete?

[Anonymous No.11210660]

>>11210647
Nah.
But you can say "discrete maths and complexity theory", that should be all you want.

[Anonymous No.11210661]

>>11210643
Can you elaborate your method? I never studied space complexity and Im wondering how yours is better in that regard?

[Anonymous No.11210668]

>>11210661
From what I'm seeing, there are two alternatives.
Essentially, you can make the entire matrix of adjacency (your method, but you split up the rows in different vertex objects), or you can number the vertices and make edges elements of [math]Z^2[/math], where the first number is the first vertex and the second number the second vertex (his method).
Feel free to correct me if I'm wrong.

[Anonymous No.11210669]

>>11210661
I searched for this article and my method isn't on there https://en.wikipedia.org/wiki/Graph_(abstract_data_type)
My method for G=(V,E), |V|=a, builds two int[a], one that stores the first vertex and the other the second. An naive optimization would be for the first array to store the vertex number <= the other. It's not better, more efficient for ADT storage and less efficient for lookup.

[Anonymous No.11210673]

>>11210669
I doubt your method is better for storage.
It sounds better at first, but yours needs integers, just the matrix is binary.

[Anonymous No.11210680]

>>11210669
>>11210673
Actually, if you consider an edge case, such as the entire matrix being filled with 1s, your method's quality drops like a rock compared to good ole adjacency matrix.
Probably worse on average too, but that's shittier to check.

[Anonymous No.11210693]

>>11210641
idk if you're into it too but check out https://en.wikipedia.org/wiki/Fulkerson_Prize

[Anonymous No.11210714]

>>11210669
so you are just implementing two lists and defining relationship between them? that seems bad if you want to add new vertices to the graph

[Anonymous No.11210734]

>>11209632
>give you some "maturity" when dealing with mathematical objects
I don't think the Stewart line does this well at all

[Anonymous No.11210773]

>>11210714
Still easier than making a new binary matrix. But yeah, >>11210673 convinced me to just do it the normal way, even if I have to annoyingly reason with bit shifts.

[Anonymous No.11210890]

>>11209665
It's an annoyance.

[Anonymous No.11211643]

>>11210494
>why? how can this be true?
Because there is absolutely no point in understanding basic algebra on a geometric level.
Things can get really complex in algebraic calculations, to the point that understanding the geometric intuition behind each step is basically impossible.

In fact the only reason that basic algebra is such a powerful tool is BECAUSE you don't have to think about the intuition behind it.

Look at some simple algebraic expression and tell me the intuition behind it, it is basically impossible and at best a waste of time.

>would a real mathematician agree that it's ok to overlook a thorough understanding of elementary algebra...?
Considering that:
Many of my professors are unable to carry out the multiplication of two, two digit numbers and that not having an intuitive understanding doesn't mean that you don't have an understanding of what you are allowed to do, yes.

[Anonymous No.11211899]

>>11211643
>understanding the geometric intuition behind each step is basically impossible
what about muh rigor?

[Anonymous No.11212176]

>>11210589
a standard way is as follows: for every vertex v, keep a vector of its neighbours
this uses |V| + |E| space and more importantly is very easy to use later

[Anonymous No.11212237]

>>11211899
>what about muh rigor?
Rigor is totally unrelated to that.
To be rigorous means to work with precisely defined terms and notation and to have as little ambiguity as possible.

[Anonymous No.11212255]

>>11212237
your definition of rigorous is not very rigorous tbqhf

[Anonymous No.11212285]

>>11210589
>>11212176
The 3 standard ways are array of edges, adjacency matrix, and adjacency list. i cbf listing all the time and space complexities here just google them and pick one that suits your needs.

[Anonymous No.11212378]

>>11202621
You can try competitive programming (for example: codeforces). The problems usually rely on math and they're pretty interesting.

[Anonymous No.11212408]

>>11212255
Yes. Because rigor is not actually well defined.
There is no function in which you can put into a mathematical text and get it's "rigor value", the concept has to be understood intuitively, which is okay, since rigor isn't a mathematical object.

[Anonymous No.11212462]

>>11212408
You could imagine a compiler that tells you whether a definition is fully formal and the derivations correct

[Anonymous No.11212548]

>>11212462
this would still depend on the programmer's definition (perception) of "formal"

[Anonymous No.11212557]

>>11212462
>You could imagine a compiler that tells you whether a definition is fully formal and the derivations correct
You do not have to imagine that.
And it really doesn't solve the problem, if we build a computer so advanced that it can understand and verify mathematical texts, then it already HAS to be able to fill in gaps in that reasoning, since (nearly) no mathematical text is entirely formal and if we had a computer which could verify mathematics on a "lower level" (such things already exist) we would have to adapt mathematical texts, thus making the process meaningless.

>>11212548
Which isn't an issue,since computer verified mathematics already exists.

[Anonymous No.11212626]

>>11208313
Every abstract algebra class I've taken has had an intense amount of factoring required

[Anonymous No.11212733]

>>11212094
This is fucking rad. I've literally never heard of these people before.

[Anonymous No.11212837]

>>11212462
no you cant this was literally exactly what Godel btfod a century ago

All of this is to say that in a sense, Platonism and Realism is true. You simply have to intuitively and intimately perceive mathematical objects. There is no other way to do math.

[Anonymous No.11212847]

>>11212837
>it's another "loud idiot misunderstands and oversells Godel" episode

[Anonymous No.11212860]

>>11212837
>no you cant
Yes, you can. As proven by the fact that these things exist.
Also, Gödel (not Godel, that isn't his name and if you can't be bothered to type the ö replace it by oe, not by o) has nothing to do with this.
Being able to verify proofs automatically seems entirely disconnected from Gödels theorems.

>You simply have to intuitively and intimately perceive mathematical objects.
False, as proven by the fact that computers are able to act as formal proof verifiers who act without any kind of "intuition".

[Anonymous No.11212865]

>>11212847
do you not know what a compiler is? Its literally a formal language with transformation rules. It reads in a string and generates another accordingly. If you give me a language L I can always generate another object with L in which the language cannot decide whether it is rigorous or not.

You simply cannot imagine something like this if you understand how it works

[Anonymous No.11212879]

>>11212865
>You simply cannot imagine something like this if you understand how it works
YOU simply cannot imagine something like this, because you are retarded and don't understand incompleteness.

[Anonymous No.11212880]

>>11212860
Sure, you can make a compiler manipulate symbols but you're never going to be able to introduce a completely rigorous understanding, which completely defeats the purpose in the first place, as the other anon said.

[Anonymous No.11212882]

Is Weibul's homo algebra unreadable or am I just bad? I've just finished Hatcher's algtop.

[Anonymous No.11212884]

>>11212880
>Sure, you can make a compiler manipulate symbols
You have no clue what you are talking about.
This isn't about symbol manipulation, but computerized proof checking.

The things you claim are impossible, exist. They are there, on the internet and you are just a Google search and a few clicks away from them.

[Anonymous No.11212886]

>>11212879
>You could imagine a compiler that tells you whether a definition is fully formal and the derivations correct

this is literally what the guy said. This is not possible.

[Anonymous No.11212888]

>>11212882
>homo algebra
Why the homophobia?

[Anonymous No.11212890]

>>11212886
>This is not possible.
Yes, it is. As evidenced by the existence of these things...

[Anonymous No.11212905]

>>11212882
>Weibul's homo algebra unreadable
Weibel was pretty comfy. Much better than Gelfand-Manin.
Try Rotman.

[Anonymous No.11212911]

>>11212886
A sentence's grammatical validity has nothing to do with whether it's true or not. Deciding whether a statement is true is not the same thing as checking whether a proof is valid.
You don't seem to understand either of these things, otherwise you wouldn't be trying to bring Godel somewhere that has nothing to do with him.

[Anonymous No.11212915]

>>11212890
>>11212884

ok fine. let the string S represent the binary code which describes some rigor checking program. make a new definition which is an encoding of string that describes any turing machine and any input of a turing machine which is some encoding of S. now use the program to check whether this is rigorous. If it is rigorous, the halting problem is solved. If it is undecidable, It is not complete.

[Anonymous No.11212920]

>>11212905
Thanks, I'll check it out

[Anonymous No.11212922]

>>11212911
We arent talking about proof validity, we are talking about a compiler that can completely rigorize concept according to axioms and inference rules

[Anonymous No.11212929]

>>11212922
>We aren't talking about proof validity
Is "derivation" a synonym for theorem in your head? Last I checked a "derivation" was a proof.

[Anonymous No.11212930]

>>11212915
Unrelated.
Again these things EXIST. They are there, you can download them and use them.
No point in talking hypotheticals.

This is like you trying to argue that planes can not fly. Whatever your argument might be, the fact remains that the things you claim can not exist DO exist.

>>11212922
A computer verified proof is THE MOST RIGOROUS you can get.
Computer verification is how you would check rigor, why this doesn't actually lead to something meaningful was detailed in another post.
But a computer CAN verify if a proof is formally totally complete and correct, which were the properties demanded.

[Anonymous No.11212943]

>>11212930
You're just changing the definition of rigor so as to avoid the whole point of the discussion. Of course if you formally define what you want to be rigorous you can hold it in a database and have inference rules and axioms check it. But that completely misses the point: a human mind had to conceive of what these symbols represent in the first place.

Think of it like this: If computers are the most rigorous you can get, why even have human minds doing math? Why not just let computers make every single development from now on? There is obviously a reason why we are talking about mathematical objects like this. Its because computers can't conceive of higher concepts and thus only belong to fields of math that aren't at the fore front of new developments. Every meaningful proof comes primarily from a human, not a computer and there is a reason for that, computers are just symbol manipulation machines.

[Anonymous No.11212954]

>>11212860
> Being able to verify proofs automatically seems entirely disconnected from Gödels theorems.

Even though >>11212837 is a total retard, you're also wrong.

Gödel incompleteness theorem precisely applies to proof systems that are recursively enumerable, i.e. where a computer can check that something is a proof or not. Gödel incompleteness theorem says that for all logical theory T

T consistent + T recursively enumerable + T strong enough (e.g. contains arithmetics) implies T incomplete.

Obviously this doesn't prevent computer-assisted proofs though. (Authority argument: it happens to be my area of expertise.)

[Anonymous No.11212958]

>>11212943
>If computers are the most rigorous you can get, why even have human minds doing math? Why not just let computers make every single development from now on?
it's honestly a little depressing that not only do I have to share a thread with people this retarded, they don't even have the common decency to shut up and lurk

[Anonymous No.11212961]

>>11212954
Speaking from a purely practical standpoint, wouldn't computers have major problems checking if definitions are consistent?
>humans do too, tho
Yeah, but I can't help but think it would be miles worse.

[Anonymous No.11212969]

>>11212943
>You're just changing the definition of rigor so as to avoid the whole point of the discussion.
Nope.
This is exactly "a compiler that tells you whether a definition is fully formal and the derivations correct", in a proof assistant every definition HAS to be fully formal and it can check whether derivations are correct, it fulfills these specifications.

>If computers are the most rigorous you can get, why even have human minds doing math?
Because rigor isn't well suited to humans, who are almost always very intuitive and have very limited capacity for high amounts of calculations, but a very high amount of creativity, which is the exact opposite to a computer.

>Why not just let computers make every single development from now on?
Being able to verify proofs isn't equivalent to coming up with them OR deciding which theorems should be proven.

>Every meaningful proof comes primarily from a human
Well, that is starting to get less true. See the four color theorem.

I seriously doubt you have any idea what you are talking about. Often you are either just wrong or seem to know very little about the subject.

[Anonymous No.11212971]

>>11212958
not an argument, you have failed to show how new definitions and rigorus concepts can reliably be made without a computer. I even constructed new definitions out of proof-checking/"rigor checking" programs that you admit exist yet you still deny rigor is something primarily humanly understood.

[Anonymous No.11212984]

>>11212969
>This is exactly "a compiler that tells you whether a definition is fully formal and the derivations correct", in a proof assistant every definition HAS to be fully formal and it can check whether derivations are correct, it fulfills these specifications.
correct
>Because rigor isn't well suited to humans, who are almost always very intuitive and have very limited capacity for high amounts of calculations, but a very high amount of creativity, which is the exact opposite to a computer.
rigor and creativity are very related, and reading through these arguments, im starting to see the problem. I think you guys are philosophically retarded.
>Being able to verify proofs isn't equivalent to coming up with them OR deciding which theorems should be proven.
thats my point, so how can a computer decide if anything is rigorous?
>Well, that is starting to get less true. See the four color theorem.
A computer does not have intentionality, A human conceived of an algorithm to solve this problem.

[Anonymous No.11212990]

>>11212969
>Well, that is starting to get less true. See the four color theorem.

To add water to the mill, I routinely use a proof assistant to manipulate mathematical objects so complex that I wouldn't even be able to write down correctly on a blackboard without making any error.

[Anonymous No.11213008]

>>11212984
>I think you guys are philosophically retarded.
Coming from somebody supporting the validity of platonic realism in mathematics, I consider this is a compliment.

How the hell can you reconcile, say, the fact that both ZFC + CH and ZFC + not CH are consistent w.r.t. ZFC? Do you also believe in Santa Claus while we're at it?

[Anonymous No.11213029]

>>11212984
>how can a computer decide if anything is rigorous?
That was never the issue, everybody agrees that it can not, BUT it can formally verify proofs, which isn't enough to decide rigor.

The thing I am disagreeing with is that "a compiler that tells you whether a definition is fully formal and the derivations correct" is impossible, I already pointed out that this isn't enough to actually form a concept of rigor and decide it.

[Anonymous No.11213048]

>>11213008
>the fact that both ZFC + CH and ZFC + not CH are consistent w.r.t. ZFC
ZFC dosen't represent the universe of sets, silly. It is a formal approximation of the universe in the same way a formal system "approximates" "rigor" in some area. In fact I am moving towards the believe that there may be one infinity between N and R. You seem to not be able to distinguish between symbols on a paper, and concepts in the mind.


btw, the concept of santa exists, but not the ontological object. Though that may be a bit nuanced for you.

[Anonymous No.11213055]

>>11213048
>btw, the concept of santa exists, but not the ontological object. Though that may be a bit nuanced for you.

mfw some philtard tries to teach me the difference between a type and its inhabitants.

[Anonymous No.11213369]

Someone else make the new thread, I dun wanna.

[Anonymous No.11213858]

>>11212408
So by definition, rigor does not exist

[Anonymous No.11214085]

>>11212990
>I wouldn't even be able to write down correctly on a blackboard without making any error
My condolences. I hear it's quite hard to be a lesser being/CS brainlet these days.

[Anonymous No.11214437]

>>11214085
Unfortunately, it is nigh on impossible to explain this kind of revolutionary mathematics to a backward brainlet /sci/ poster.

[Anonymous No.11214586]

>>11213858
>rigor does not exist
It doesn't exist any more than a "good" book exists.