Find a flaw.

>>8613807
Is it possible for me to one day understand math so well that I can masturbate to this stuff too?

>>8613807
no canonicalization

>>8613830
This, pretty much.

Stuff like pretty mathematical notation and application of mathematical rigor started exciting me right after starting my phil. major and reading basic analytic philosophy but I'm way too confused even by precalculus math because I slept on high school math courses. I'm even starting to consider taking course on set theory and mathematical logic just to start getting all this beauty that is syntax.

>>8613807
>no end behavior
Get out

>>8614032
Anyone can and should learn naive set theory at minimum.

>>8613807
should not have right angles

mmm donut

what do i need to know to actually read this

i'm pretty good with category theory and topology, but i still have no idea what the shit it's talking about sometimes

>>8614530
eyes and a brain

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>>8614538
i'm really struggling with this pls help or at least point me in the right direction i want to learn this

>>8614530
Is this bait? He just constructed an image of a Torus using common mathematical symbols.

Is this bait????

>>8614571
I thought it was a donut, which made the first reply even funnier

forgot this was /sci/... ofc it'd be more abstract than a donut

To those confused by the ambiguity of the OP, he is referring to the programme called Homotopy Type Theory, a marriage of Per Martin-Lof's dependent type theory and higher category theory. It's quite good material, and a PDF of the book is free on the web.

>>8614530
It was difficult for me to get a hold on the syntax employed, but it's worth spending time trying to get into. I don't have much experience in programming or type theory, so I just read the types logically and topologically. It works out. Some of the stuff is kind of obtrusively technical, but it gets better the further into the book you get. After the first two chapters, just jump into the parts where they talk about homotopy theory and category theory (I think chapters ten and eleven).

no computational interpretation = into the trash it goes

Oh, and OP, the flaws so far: it relies on tenets of higher category theory, which is flawed because it cannot handle the most abstract and fundamental idea of Galois theory: you ought to be able to describe maps into something by mapping from that thing into the universe that it and the map live in. Shadows of the idea are captured by the Grothendieck construction, but without a way to talk abiout universes residing as objects in themselves, we can not hope to fully formalize the macrocosm principle. Also, there should be negative h-levels in homotopy theory, because a normal categories should be viewed as the loop spaces of something looking like a coloured monoidal category. This is why abelianization is a thing, and why the Eckmann-Hilton argument causes a schism between h-level 0 and 1, and higher h-levels. If we didn't have the bizarre breakdown of things down at h-levels -1 and -2, and instead allowed things to go all the way down to a negative infinite h-level, we would see a profound cohesion of ideas unfold. I have mentioned apparatuses on this board before, which are my system for fixing this issue and the issue of universes as objects in themselves. I found that the trouble with formalizing all of this goes all the way to the foundations of axiomatic thought, because we cannot have a single-vantage formal system without running into self-reference paradoxes and cause size issues in category theory. I have had to develop a "locally consistent" axiomatic system to allow for more robust and natural structures. This is akin to the fact that we cannot fully grasp manifolds until we pass to the idea of local flatness and local smoothness.

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>>8614602
i understood almost none of that

OHP does not disappoint

Category theorists are disgusting, but at least they're self aware and pretty successful with their bait

>>8614667
Allow me to expound a bit more digestibly: in a simple case, you are familiar with Cantor's paradox I assume? The paradox that shows that there can be no set of all sets. Well, say the set of all sets (what I am calling a universe) was itself an acceptable set and fit perfectly well into the theory. Then, every function into a set S should be uniquely determined (up to isomorphism) by a function from S into Set (the set of sets), which sends each element of S to the set of elements mapped to it. We could perfectly reconstruct the function into S by gluing together (taking the union of) all of the fibres over all of the points of S, and then the function just sends each subset to its given element.

This method actually ought to work out for any higher topos, but there is simply no way in category theory with current foundations to say that a category is an object of itself. One runs into size issues, which are higher versions of Cantor's paradox.

The macrocosm principle is this: it states that a universe of objects with a certain property ought to have that same property in some categorified sense. The category of monoids is monoidal, the category of sets looks like a big ol' set, the category of spaces ought to have some sort of complicated homotopy type, et cetera. If we had a way of realizing a universe as an object of itself, this would be an automatic corollary, because by assumption all of its objects have that given property (and so it would also have that property).

Now, regarding the lower h-level nonsense, it is concerning that we require n-cells to be composable if they are arranged properly. However, 0-cells (objects) are not subject to this, because they have no domain or codomain. What we ought to do is view 0-cells as mediating between (-1)-cells, and that they are only composable if they agree appropriately. A monoidal category, for example, is just an apparatus with a single (-1)-cell.

>>8613807
Upside down

>>8614602
Stupid namefag

>>8614835
Oh noes, I better stop posting! Anon made fun of me.

>>8614843
You'd better go back to reddt. Your posts are cancer. It's sad that you need to get all of your daily affection by cancerfagging on 4chan, but you need to understand that the world is not your oyster and that /sci/ is not here to babysit you through your fits of loneliness, you narcissistic autist

>>8614878
I just don't know what you expect to get out of being an ass on here. I'm not going to stop posting, you aren't hurting me, and some people are still interested in what I write here. I think you just have a lot of aggression that you need to channel at others.

>>8614032

-t. sandwich artist

>>8614590
cubical tt tho

>>8614222
its a coffee cup

how to recognize someone who doesn't know topology: >>8615547

>>8614032
D O I T
O

I
T

>>8614602
so what *actual problems* arise from not having this, then?

anyway, doesn't "local consistency" tend to imply global consistency by compactness-type stuff? I guess that's not necessarily a coherent thing to say without more details of what you mean.

>>8615581
Well, the "actual problem" is that our systems are attempts at formalizing our intuitive ideas of what we can and can't do, and so if the system doesn't let us fully discuss the macrocosm principle, then it is in need of improvement. It's not as though I have an inconsistency to point out, so all we can really do is discuss the philosophical merits of our systems.

I am attempting to capture the idea of local consistency as such: there is a collection of vantages or perspectives, and there is a relative theory with respect to each which is actually consistent. Furthermore, there is a "subsumption" structure on these perspectives, so some perspectives capture others entirely. The hope is that we can find a "minimal cover" for whatever theory we are formalizing. There ought to be a notion of transport as well, wherein theorems from one perspective can be transported to another perspective along a path in the type of perspectives. Regarding compactness, I am planning on using something like a locally paraconsistent logic, so I do not know if compactness results apply.

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>>8614032
Holy fucking shit. Are you me?
Do it anon! I'll be taking extra math courses next semester too.
For me, it was learning formal logic and reading Wittgenstein & Husserl that made me discover this new passion. Go for it.

>>8616592
learn some type theory

play with coq

>>8616500
oh, that's interesting! I was thinking vaguely about a semi-related concept the other day; some kind of cumulative hierarchy or system like what you're describing, where each level admits discussion of metatheoretical issues of the level beneath, like provability and introspection of terms. maybe even ordinal-indexed levels, so that you can get almost as meta as you like.

>>8616500
Hey Mr. OHP have you read EGA?

>>8617521
>play with coq

I'm not an engineer.

>>8616500

parallel transport, nigga r u serious?

>>8615549
Why do you say that?

>>8617631
"doughnut = coffee cup" is the one and only thing laymen "know" about topology. it's like how you can recognize someone who doesn't actually know quantum mechanics based on how they obsess over how The Act Of Observation Is Special And Meaningful

>>8617597
what does coq have to do with engineering

>>8617667
Engineers being gay is a meme.

>>8617662
I actually use a little basic topology at my day job.

(curvature explains why you use the same kinds of tablecloths on rectangular tables and oval tables.)

>>8617679
"basic topology" doesn't deal with curvature at all

a cube is homeomorphic to a sphere is homeomorphic to any other convex set of the appropriate dimension

>>8617662
Nobody knows if Schrödinger's cat is alive or dead desu.

>>8614587
>>8614602
>>8614722
>>8616500

Why even try when there's people like you.

>>8617755
because if you try, you get to understand people like them

>>8617755
Because there is a lot of non-autistic math

>>8617755
>namefag spouts walls of unreadable category salad
>"omg so smrt"
you're like someone impressed by all the fancy symbols on the first day of calculus

>>8617916
i can understand about 80% of what he said, maybe the problem is on your end

ok well the prose is a little purple

>>8617535
That would be great! We should embrace the entirety of the web of theories and formal systems.

>>8617588
Learning French to try and give it a go, but I'm focused on learning German to read Hegel's original work.

>>8617604
Yeah, but for propositions in a theory!

>>8614571
This is the first image in the book "Homotopy Type Theory" which is a radical new foundation for mathematics.

>>8613807
There's a fucking hole in it

>>8617755
Anon, I'm just a dumb kid who got really obsessed with something and took it too far. Find your spark and you will go far! I'm serious.

>>8618227
You don't need to learn French to read it. Just Google translate and after the 1st 10 pages you'll be reading smoothly. Anyways I assumed to read it because I can tell you are/would be appreciate of Grothendieck-style mathematics.

I just need someone to tell me what to skip lol it's too fucking huge.

>>8614602
Yeah well this seems like a problem that's fundimental to how we perceive reality. When you look at a building, do you ever see the building as a whole, or do you look at each local part, and then piece it together into a whole image? I'm betting it's the latter because that's how I see things at least.

Anyway, think of it this way, ever single piece of mathematics is abstracting different details of reality to focus on one issue at a time, a sort of "localization" of understanding. Only after we pass to this localized viewpoint, can we begin to construct global structure.

I think this is the biggest reason geometric theories have had the most weight in physics, geometry is all about considering rigid local structures, which is sort of like our understanding localizing process. We see geometric reasoning as a boon for machine learning as well, so there is some merit to me saying that geometry and localization are fundimental to human intelligence.

>>8618240
I like what you are saying, and that's kind of what my "locally consistent" theory proposes: we have all of the perspectives handled in a coherent way, and glue the global image (which can be very descriptive because it need not be globally consistent) from the pieces.

Imagine you have a Penrose triangle and the ordinary shape it is pretending to look like. The ordinary shape is the only globally consistent one, but you can get interesting new systems when you only require local consistency (the Penrose triangle makes perfect sense when you restrict to a small enough area). Likewise, I want the interesting "paradoxical" structures to in fact be considered reasonably, where the principle of explosion will not exist and force one inconsistency to effectively trivialize the whole thing. Paradoxes are meant to allow us to think bigger, liking Hegel's Aufhebung: transcending the polarities of a duality to consider the whole and a coherent object.