#theorem

[zanephillips]

Teorema (1968) dir. Pier Paolo Pasolini

[pierppasolini]

Teorema (1968) // dir. Pier Paolo Pasolini

[szimmetria-airtemmizs]

Mathematical fun fact of the day 4. You can divide a hexagon into nine identical convex pentagons.

[cinematicjourney]

Teorema (1968) | dir. Pier Paolo Pasolini

[huariqueje]

Theorem - Luc Dondeyne , 2015.

Belgian , b. 1963 -

OIl on canvas , 120 x 120 cm.

[fashionlandscapeblog]

Teorema (1968) - dir. Pier Paolo Pasolini

[spectrallysequenced]

Math people, reblog with your fav theorem and why.

I'll start, the Wedderburn-Artin theorem is a beautiful structure theorem on semisimple rings which says they decompose uniquely as a product of matrix rings over division rings. This is a beautiful result but it also underlies a lot of very cool theory like Brauer Theory, Galois Cohomology and the theory of Galois and Étale Algebras.

What's yours?

[noosphe-re]

In 1931, the Austrian logician Kurt Gödel pulled off arguably one of the most stunning intellectual achievements in history. Mathematicians of the era sought a solid foundation for mathematics: a set of basic mathematical facts, or axioms, that was both consistent — never leading to contradictions — and complete, serving as the building blocks of all mathematical truths. But Gödel’s shocking incompleteness theorems, published when he was just 25, crushed that dream. He proved that any set of axioms you could posit as a possible foundation for math will inevitably be incomplete; there will always be true facts about numbers that cannot be proved by those axioms. He also showed that no candidate set of axioms can ever prove its own consistency. His incompleteness theorems meant there can be no mathematical theory of everything, no unification of what’s provable and what’s true. What mathematicians can prove depends on their starting assumptions, not on any fundamental ground truth from which all answers spring.

Natalie Wolchover, How Gödel’s Proof Works, Quanta Magazine, July 14, 2020

[fellinesca]

Theorem (Teorema, 1968) Dir. Pier Paolo Pasolini

[creepynostalgy]

Terence Stamp and Anne Wiazemsky in Teorema aka Theorem (1968)

[m---a---x]

Welcome to the premier of One-Picture-Proof!

This is either going to be the first installment of a long running series or something I will never do again. (We'll see, don't know yet.)

Like the name suggests each iteration will showcase a theorem with its proof, all in one picture. I will provide preliminaries and definitions, as well as some execises so you can test your understanding. (Answers will be provided below the break.)

The goal is to ease people with some basic knowledge in mathematics into set theory, and its categorical approach specifically. While many of the theorems in this series will apply to topos theory in general, our main interest will be the topos Set. I will assume you are aware of the notations of commutative diagrams and some terminology. You will find each post to be very information dense, don't feel discouraged if you need some time on each diagram. When you have internalized everything mentioned in this post you have completed weeks worth of study from a variety of undergrad and grad courses. Try to work through the proof arrow by arrow, try out specific examples and it will become clear in retrospect.

Please feel free to submit your solutions and ask questions, I will try to clear up missunderstandings and it will help me designing further illustrations. (Of course you can just cheat, but where's the fun in that. Noone's here to judge you!)

Preliminaries and Definitions:

B^A is the exponential object, which contains all morphisms A→B. I comes equipped with the morphism eval. : A×(B^A)→B which can be thought of as evaluating an input-morphism pair (a,f)↦f(a).

The natural isomorphism curry sends a morphism X×A→B to the morphism X→B^A that partially evaluates it. (1×A≃A)

φ is just some morphism A→B^A.

Δ is the diagonal, which maps a↦(a,a).

1 is the terminal object, you can think of it as a single-point set.

We will start out with some introductory theorem, which many of you may already be familiar with. Here it is again, so you don't have to scroll all the way up:

Exercises:

What is the statement of the theorem?

Work through the proof, follow the arrows in the diagram, understand how it is composed.

What is the more popular name for this technique?

What are some applications of it? Work through those corollaries in the diagram.

Can the theorem be modified for epimorphisms? Why or why not?

For the advanced: What is the precise requirement on the category, such that we can perform this proof?

For the advanced: Can you alter the proof to lessen this requirement?

Bonus question: Can you see the Sicko face? Can you unsee it now?

Expand to see the solutions:

Solutions:

This is Lawvere's Fixed-Point Theorem. It states that, if there is a point-surjective morphism φ:A→B^A, then every endomorphism on B has a fixed point.

Good job! Nothing else to say here.

This is most commonly known as diagonalization, though many corollaries carry their own name. Usually it is stated in its contraposition: Given a fixed-point-less endomorphism on B there is no surjective morphism A→B^A.

Most famous is certainly Cantor's Diagonalization, which introduced the technique and founded the field of set theory. For this we work in the category of sets where morphisms are functions. Let A=ℕ and B=2={0,1}. Now the function 2→2, 0↦1, 1↦0 witnesses that there can not be a surjection ℕ→2^ℕ, and thus there is more than one infinite cardinal. Similarly it is also the prototypiacal proof of incompletness arguments, such as Gödels Incompleteness Theorem when applied to a Gödel-numbering, the Halting Problem when we enumerate all programs (more generally Rice's Theorem), Russells Paradox, the Liar Paradox and Tarski's Non-Defineability of Truth when we enumerate definable formulas or Curry's Paradox which shows lambda calculus is incompatible with the implication symbol (minimal logic) as well as many many more. As in the proof for Curry's Paradox it can be used to construct a fixed-point combinator. It also is the basis for forcing but this will be discussed in detail at a later date.

If we were to replace point-surjective with epimorphism the theorem would no longer hold for general categories. (Of course in Set the epimorphisms are exactly the surjective functions.) The standard counterexample is somewhat technical and uses an epimorphism ℕ→S^ℕ in the category of compactly generated Hausdorff spaces. This either made it very obvious to you or not at all. Either way, don't linger on this for too long. (Maybe in future installments we will talk about Polish spaces, then you may want to look at this again.) If you really want to you can read more in the nLab page mentioned below.

This proof requires our category to be cartesian closed. This means that it has all finite products and gives us some "meta knowledge", called closed monoidal structure, to work with exponentials.

Yanofsky's theorem is a slight generalization. It combines our proof steps where we use the closed monoidal structure such that we only use finite products by pre-evaluating everything. But this in turn requires us to introduce a corresponding technicallity to the statement of the theorem which makes working with it much more cumbersome. So it is worth keeping in the back of your mind that it exists, but usually you want to be working with Lawvere's version.

Yes you can. No, you will never be able to look at this diagram the same way again.

We see that Lawvere's Theorem forms the foundation of foundational mathematics and logic, appears everywhere and is (imo) its most important theorem. Hence why I thought it a good pick to kick of this series.

If you want to read more, the nLab page expands on some of the only tangentially mentioned topics, but in my opinion this suprisingly beginner friendly paper by Yanofsky is the best way to read about the topic.

[zanephillips]

Teorema (1968) dir. Pier Paolo Pasolini

[pierppasolini]

Teorema (1968) // dir. Pier Paolo Pasolini

[szimmetria-airtemmizs]

On the sides you can see two curves of constant width, that is a curve whose width is the same in all directions. This is the reason they always touch the two lines as they rotate. The existence of curves of constant width is well known by now. What is much less known is that you can morph between any two such curves such that during the morph the curve stays a curve of constant width. For these two curves the morphing can be seen in the middle. (Actually, you can see two different morphs. )

[cinematicjourney]

Teorema (1968) | dir. Pier Paolo Pasolini

[quatrenuitsdunreveur]

Teorema / Théorème (dir. Pier Paolo Pasolini, 1968)