#foundations of mathematics

[pv1isalsoimportant]

Mathematical platonists are so funny to me because they will say shit like "mathematics has this beautiful elegant design" meanwhile you get to the foundations and find out you have those "beautiful elegant statements" that are independent of ZF (or even ZFC), are not compatible with each other and each of them implies some insane monstrosity that shouldn't ever be true.

[raffaellopalandri]

Book of the Day - SGA 4

Today’s Book of the Day is Theorie des Topos et Cohomologie Etale des Schemas. Seminaire de Geometrie Algebrique du Bois-Marie 1963-1964 (SGA 4), curated by Michael Artin, A. Grothendieck, and J.L. Verdier in 1972 and published by Springer Verlag. I have chosen this book as I used some of its math to demonstrate to a colleague some topological errors he was making in creating a cognitive…

[minimal-effort-name]

Fun fact I learned yesterday:

It's actually incorrect to say that the set of integers contains the set of natural numbers, instead it actually contains a copy that for all intents and purposes is exactly the same, but they are not the same.

Which honestly sounds absurd and like a math shitpost. However, here's how it actually makes sense:

this is the number two construction of the natural numbers based on set theory

2ℕ = {{Ø},{{Ø}}} (it's two because it has two elements)

Now here's two in the integers:

2ℤ = {(2ℕ,0ℕ),(4ℕ,2ℕ),(6ℕ,4ℕ)...}

So from this, you can conclude that these two are different sets, even though they're both the natural number two.

Incredibly funny result.

[boinky-doinky]

Hiiiii!! :3

It's me again and saw you have requests open. I was thinking you could draw SCP-590 (TJ) finding a stupid lookin frog.

Went from math homework to this

I hope you find joy in this silly doodle

[then-ponder]

Rate your favorite logic notation

[philosophybits]

The application of the concept of ‘following a rule’ presupposes a custom. Hence it would be nonsense to say: just once in the world someone followed a rule (or a signpost; played a game, uttered a sentence, or understood one; and so on).

Ludwig Wittgenstein, Remarks on the Foundations of Mathematics

[daily-public-domain]

Day 389: Fractal Tree, 2005

link

–This image is part of the public domain, meaning you can do anything you want with it! (you could even sell it as a shirt, poster or whatever, no need to credit it!)–

[m---a---x]

Inspired by all the newly created communities i have also created one about the topic closest to my heart: Foundational Mathematics

It is inteded for all types of posts about and from people of all kinds of backgrounds interested in the topic.

Please share with anyone you think might be interested. If you want to be added comment on this post, so I can add you.

[blood-orange-juice]

I wonder if the whole Abyss and Forbidden Knowledge thing is a metaphor for Gödel's incompleteness theorems.

It's not that farfetched, the second theorem was popularized in non-academic circles (to the point where even mentioning it in academic discussions is now a faux pas unless it's among mathematicians) and was used to point at the core difference between human and AI thinking (until the current generation of AI models fell outside of what it describes).

And Genshin does have a lot of AI themes.

Bear with me for a second.

In layman terms, in a system like formal logic there are statements we can express using this system and ones we can prove or disprove.

And if the system is consistent (doesn't hold contradicting statements) these are different things! You can say things that will be true and will correctly describe the object you are trying to describe but there will be no way to logically derive them from the other statements you already have.

Also a consistent system can't prove that it's consistent.

(an inconsistent, contradictory system has no such problems.

you can also formally accept a non-provable statement into a consistent system as another axiom/"known truth" and it will create a new system with new properties)

These theorems were formulated as a way to talk about the limitations of mathematics compared to our thinking, but in popular reading they somehow turned into "for any formal system there's a statement that nukes it" (Gödel statement).

Some people wondered if a Gödel statement that fries a person's brain is possible.

What I currently see in Genshin painfully resembles those popular readings.

Celestially maintained Teyvat laws are a consistent system, forbidden knowledge is a statement that can be neither proven nor disproven in that system (and messes it up), Abyss is an inconsistent system (any statement can be proven true and it's pretty horrible but also hey, anything can be proven true so forbidden knowledge doesn't disrupt it further).

Perhaps gods and their ideas are something like logical statements and owning a gnosis symbolises incorporating their "truth" into the overarching formal system of Teyvat laws.

(perhaps different cycles are just variations of the same world with slightly different rules/axioms)

Also Irminsul cannot be allowed to think of things it can't understand and it cannot be allowed to think of its nature and consistency of the information it stores.

[humormehorny]

Why don’t they teach mathematical logic earlier? It’s present in literally all the books you have to read anyway. Some algebra author is going to be like for vs real numbers a,b ≠ 0 there exists a ratio a/b that is also a real number. And most people can’t even read that. Like half of the book is completely inaccessible because logic is fucking wild and requires actual practice with.

I’m not saying that this stuff is incredibly hard; it’s not. However logic just isn’t taught even though it is clearly foundational.

Sets especially seem to trigger me. Like I think a lot of people struggle with trig specifically because this aspect of functions is just not taught well. Like of course the arcsin(x) isn’t defined for -pi/2 > x < pi/2. It would be so much better if we actually taught this stuff.

[pv1isalsoimportant]

Mathematics is the mirror image of humanity.

[forgottenbones]

What IS a Number? As Explained by a Mathematician

[bagalois]

notes from "foundations of combinatorial topology" by lev pontryagin

french mathematician poincare was a real pioneer of combinatorial topology. he was the guy who came up with the fundamental notion of "given a n-dimensional manifold M and a sub-manifold Z, there either exists or doesn't exist a sub-manifold C that has Z as its boundary." he was also the guy who came up with the idea that manifolds can be decomposed into simpler parts, called simplices. nowadays we learn this stuff when we do homology theory, as a lead-in to algebraic topology. and, apparently, homology theory is the foundation of this stuff called combinatorial topology too.

Combinatorial topology studies geometric forms by decomposing them into the simplest geometric figures, simplexes, which adjoin one another in a regular fashion.

simplexes (and polyhedra, which are created by building simplexes together) can be examined in a group theoretic way. similar to how elementary algebra was created and makes a lot of geometry problems trivial, there are numerical invariants in simplexes/polyhedra and we can basically treat these geometric objects as just numbers. then, just by tweaking the same methods, it becomes possible to examine more complicated geometric forms which may not be reducible to numbers.

(notes from the introduction)

[ricisidro]

Can't wait for Season 3… 🪐

Foundation (2021) is the saga of Harry Selden and his followers on their epic quest to save mankind based on the scifi novels (1951-1953) of the same title by Isaac Asimov.

Seasons 1 and 2 with 20 episodes are now streaming on Apple Tv Plus.

#ScienceFiction #psychohistory #history #sociology #mathematics #statistics #GalacticEmpire

[then-ponder]

Truthfully, I am critical of most arguments that urge people to learn math. This is in particular because most public defenders of mathematics and math education argue with the implicit assumption that you should learn this if and only if it is useful. They will argue that because we can use all of our math to travel the stars and save our dying planet it is worth learning.

I do not blame them for this. After all this is a common practice under capitalism. The constant need to prove useful is a hellish practice in which all professions participate even if some have it easier than others. But it is harmful. It separates us from seeing math for what it is. Want it wants to be.

Many people would call math a science and there is a truth to that. Much of mathematics is a process of learning cold hard facts, even if the idea has been proven a million times over, but that ignores all the bits of it that we tend to not include. The parts about logic and reasoning, the parts about discovery, the parts that suggest a deeper explanation, the parts where intuition feels like magic and reality feels fake.

At the beginning of this post I said that “arguing math is useful is a harmful argument”. This is why. There is no discovery of facts. Logic, what should be the foundation of all mathematics, is not taught and proving that a statement is true goes ignored and is deathly fatal to students who seek the answers, yet are obscured. Worse, the facts we teach in math are plainly obvious to the student in a way that destroy any merit of the claim “ math is useful” because why would I use math when I could use logic. Why would I use the Pythagorean theorem when I know that two sides of a triangle are longer that the hypotenuse when I’m path finding and why the hell would I use a polynomial to describe anything. You use the language of usefulness you should be teaching useful things, but math can’t teach useful things. It’s math not a trade or a science.

And in truth mathematics isn’t useful. Nothing you point to that most people understand are useful things and the vast majority of research is on problems that most people either don’t care about or find utter nonsense. Truly I don’t mean this to be cruel, it’s just that the proof for 1+1 = 2 is a books worth of pages. And that’s the thing people can’t possibly be bothered to ask questions that a mathematician would lose years to. They just want a fucking answer and that’s fine, but this more explicit explanation is why Mathematicians don’t like ‘applied’ math. It’s because assholes take our prized possessions, our meow meows, and our blorbos and they throw them in a shredder and mangle them up till they have a thing that they like and then they give it to another person who fucking hates math because they got their degree in “I hate math but I love problem solving so I became a [insert stem field here]” so they can blow up that thing so that they can use it to solve something.

The truth is that math is a humanity because it is a reflection of our own reality. Quite like art. And the reason for studying art is the same reason you should study math which also happens to be the reason why math and art education sucks. You should learn about these things because you want to. No amount of money can justify it. You have to want it. And the thing is art is best taught as art. It’s not a just or simply a tool for people to make money. It’s an expression of the self - The very people who created it.

Many fear that if we allow people to stop studying the math people won’t. Fair, but high school math is not reflective of math. It doesn’t teach foundations, it doesn’t teach about the different sub-disciplines. It refuses to teach any math that can’t be directly applied to science and engineering. Of course people don’t want to learn that. You’ve ripped out the meat.

[philosophybits]

The sickness of a time is cured by an alteration in the mode of life of human beings, and it was possible for the sickness of philosophical problems to get cured only through a changed mode of thought and of life, not through a medicine invented by an individual.

Ludwig Wittgenstein, Remarks on the Foundations of Mathematics