OscarCunningham

ColinTheMathmo, 8 days ago (edited 8 days ago) to random

He's not here on Mastodon(*), so I'm sharing the question:

Matt Enlow says:

Hey all. Just had a jolt of inspiration.

I'd like to try to come up with a big list of AMBIGUOUS math statements.

Think: The kinds of things you often see/hear from students.
I want them to run the gamut in terms of clarity, and the WAYS they're ambiguous.

Examples to follow ...

(*) EDIT: He is on Mastodon: @UnsolvedMrE

OscarCunningham, 11 days ago to math

For a long time I felt like I didn't really understand the Yoneda Lemma. I knew some things that people said about it ('we can understand objects by the maps into them' and 'the Yoneda embedding is full and faithful') but the statement 'Hom(Hom(A, -), F) = F(A)' itself was something I could only use as a symbolic manipulation without understanding.

On the other hand, I did separately know facts like 'In the category of quivers there are objects which look like • and •→•, such that the maps out of them tell you exactly the vertices and edges in your quiver' and 'In the category of simplicial sets there are objects which are just an n-simplex; maps out of them are the n-simplices of the object you are mapping into'.

Somehow I only recently realised that these examples are precisely the Yoneda Lemma. These objects are precisely presheaves of the form Hom(A, -), and the Yoneda Lemma tells you what you get when you map out of them.

In particular I think it would be useful to give the quiver example to students when they learn the Yoneda Lemma.

#CategoryTheory #Math #Maths #Mathematics #Yoneda #YonedaLemma

boarders, 15 days ago (edited 15 days ago) to random

Andrej Bauer gave an excellent recent talk “How to not believe in the law of excluded middle”: https://youtu.be/96iHUx0aGDs?si=y9j8Kc9znPs-cS1U

Towards the beginning he mentions some bad reasons for not believing in excluded middle include the undecidability of CH. This is true, but I don’t think it is a bad thing to mention (or perhaps I just feel bad since I am guilty of having done so) since:

• CH ∨ ¬CH - “CH is either true or false” is a theorem of ZFC since it is just an instance of LEM. This is obviously not a solution to Hilbert’s first problem

• Pr(CH) ∨ Pr(¬CH)- “there is a proof of CH or a proof of its negation in ZFC”. This is is not true, as shown by Cohen + Gödel (this is not an internal statement in ZFC, it is meta-theoretical)

• Dec(Pr(CH)) - it is decidable that we can give a proof of CH, or that we can’t give a proof. This is also true, but this time we can’t appeal to LEM so we must be appealing to something else (in this case we appeal again to Cohen + Gödel). But this means we don’t accept the meta-theoretic principle LEM when it comes to statements about provability

christianp, 22 days ago to random

Microsoft Teams continues to reveal its true nature as a billion sprint goals in a trenchcoat:

When you attach an image to a post, it's saved at the top of the Sharepoint folder belonging to the channel. (So the "files" tab becomes a cluttered mess, but that's not what I'm cross about now)

As well as restrictions on valid filenames, filenames of attachments have to be unique.

So if you've attached drawing.png once before, and upload another drawing.png, Teams asks if you want to replace the original, or keep both. If you keep both, it adds (1) to the filename.

... unless there's already a "drawing (1).png", in which case it asks you AGAIN what you want to do.

Is there a Big Brain Cloud Services reason it can't automatically find the smallest number that works?

#MicrosoftTeams

christianp, 24 days ago (edited 24 days ago) to random

Pals, what's the least egregious TV I can buy today, in the UK?
I want as little "smart" internet-connected nonsense as possible. Not bothered about 4k or massive size, but it should sound and look good.

That is, what's the Brother laser printer of TVs?

OscarCunningham, 25 days ago to math

I have a question about the aperiodic spectre tile (or the hat/turtle).

I know that the proof of aperiodicity works by showing that the tiles must fit together in a hierarchical structure that eventually repeats itself at a larger scale. But the larger units aren't literally scaled copies of the spectre. I also know that there is some freedom as to how you draw the edges of the spectre.

Is there a way you can draw the edges that allows you to literally use spectres to cover a larger copy of themselves? If so, is this way of doing it unique?

#Math #Maths #Mathematics #Spectre #Tiling #Aperiodic #AperiodicMonotile

ColinTheMathmo, 1 month ago to random

I can see a problem here.

(Found by @ghira ...)

OscarCunningham, 1 month ago to random

Is there a construction like the spectrum of a ring, but that gives you an ∞-groupoid rather than a topological space?

#CategoryTheory #Topology #AlgebraicTopology

OscarCunningham, 1 month ago to math

Mathematicians sometime talk about algebra and geometry being dual to each other. One way to formalise this is by talking about opposite categories. If the objects of a category act like algebras, then in the opposite category they act like spaces.

But the category of finite dimensional vector spaces is its own opposite! This suggests that linear algebra is in some sense the place where algebra and geometry meet. Perhaps that explains why it's so tractable and efficacious.

#Math #Maths #Mathematics #CategoryTheory #LinearAlgebra

JeremyMallin, 1 month ago to physics

What would the universe look like if there were more than one time-like dimensions?

#RandomThoughts #Physics #Cosmology #Mathematics

johncarlosbaez, 1 month ago (edited 1 month ago) to random

Mathematicians describe points on the plane with coordinates (x,y). The first coordinate says how far 𝑎𝑐𝑟𝑜𝑠𝑠 you go and the second says how far 𝑢𝑝 you go. Then they describe entries of a matrix Mᵢⱼ with indices where the first says how far 𝑑𝑜𝑤𝑛 you go and the second says how far 𝑎𝑐𝑟𝑜𝑠𝑠 you go.

In each case I've had teachers who insinuate that this is the only reasonable thing to do and you'd have to be nuts to dream of doing anything else.

I think some teachers don't distinguish between facts and conventions. Actual facts are always worth thinking about - understanding them more and more deeply is a never-ending quest. But for arbitrary conventions, you should just memorize them and move on.

johncarlosbaez, 2 months ago (edited 2 months ago) to random

Hardcore math tweet:

The complex numbers are nice in two ways. They are an 'algebraically closed' field, meaning that every polynomial equation with complex coefficients has a complex solution. And they are 'Cauchy complete' metric space, meaning that every Cauchy sequence converges.

We can get the complex numbers in two ways. We can start with the rational numbers and take their Cauchy completion. This gives us the real numbers. But these are not algebraically closed. So we can take their algebraic closure. The result is the complex numbers, which is still Cauchy complete.

Or, we can start with the rational numbers and take their algebraic closure. This gives us the 'algebraic numbers'. There's a way to define a nice metric on these, but the resulting metric space is not Cauchy complete. To fix that, we can take its Cauchy completion. The result is the complex numbers, which is still algebraically closed.

In the first route I used the usual metric on rational numbers. But what if we use one of the p-adic metrics?

We can start with the rational numbers and take their Cauchy completion using the p-adic metric. This gives us the p-adic numbers. But these are not algebraically closed. So then we can take their algebraic closure. There's a nice metric on it, but it's 𝑛𝑜𝑡 still Cauchy complete.

So we can take the Cauchy completion 𝑎𝑔𝑎𝑖𝑛. You may feel sort of pessimistic right around now... but this time the resulting field 𝑖𝑠 algebraically closed, and of course Cauchy complete by definition. So yay, we're done! 🎉

The weird part: the resulting field is isomorphic to the complex numbers equipped with a weird metric. Using the axiom of choice. 😬

OscarCunningham, 2 months ago to math

If I tell you the value of x, rounded to 3 decimal places, then you can sometimes gain more information if I also tell you x rounded to 2 decimal places.

#Math #Maths #Mathematics

OscarCunningham, 2 months ago to random

The command line tool 'tar' is a transpiler between the languages 'DEFLATE' and 'cat'.

ColinTheMathmo, 2 months ago to random

For those who like the charting 'bot and want to see it continue, help me figure out a way to reconcile it with this:

https://freeradical.zone/@tek/111924979914283908

johncarlosbaez, 2 months ago (edited 2 months ago) to random

This curve is not an elliptic curve - because even though you can write it in as

y² = P(x)

with P a cubic polynomial, elliptic curves need to be smooth! We say this curve is 'singular', not smooth everywhere, because it crosses itself at one point, making a kind of X shape. Mathematicians call this point a 'node'. So this curve, which I'd rather write as

y² = x³ - x²

is called a 'nodal cubic'.

It's still fun to count the solutions of this equation in a finite field. Let's do it!

(1/n)

simontatham, 2 months ago to random

Puzzle mashup ideas I've never quite figured out the rules for:

Minesweeper vs Conway's Life. You're trying to figure out where all the mines are, but they periodically move, under Life rules. So some of your previous deductions give you new information, but not all, and you have to keep ahead of it. But how does the game decide when a Life turn happens?

Or how about a puzzle in which you push digits around until you get them all different in each row, column etc: 'Sudokoban'. Hmmmm.

ColinTheMathmo, 3 months ago to random

Your shadow shows that light has come nearly 150 x 10^9 m, only to be deprived of reaching the ground in its final metre, thanks to you.

johncarlosbaez, 3 months ago (edited 3 months ago) to random

The Law of Excluded Middle says that for any statement P, "P or not P" is true.

𝗜𝘀 𝘁𝗵𝗶𝘀 𝗹𝗮𝘄 𝘁𝗿𝘂𝗲? In classical logic it is. But in intuitionistic logic it's not.

So, in intuitionistic logic we can ask what's the 𝙥𝙧𝙤𝙗𝙖𝙗𝙞𝙡𝙞𝙩𝙮 that a randomly chosen statement obeys the Law of Excluded Middle. And the answer is "at most 2/3 - or else your logic is classical".

This is a very nice new result by Benjamin Bumpus and Zoltan Kocsis:

https://bmbumpus.com/2024/02/27/degree-of-classicality/

Of course they had to make this more precise before proving it. Just as classical logic is described by Boolean algebras, intuitionistic logic is described by something a bit more general, called Heyting algebras. They proved that in a finite Heyting algebra, if more than 2/3 of the statements obey the Law of Excluded Middle, then it must be a Boolean algebra!

OscarCunningham, 4 months ago to random

I'm feeling very unenergetic, but actually my body is 2° hotter than it usually is!

OscarCunningham, 4 months ago to math

In Conway's Game of Life there are certain oscillators called 'phoenixes', which have the property that no cell is ever on for more than a single generation consecutively. The only examples known were ones with period 2 (so the cells are just turning on and off repeatedly). It's now been proven that these are the only phoenixes: https://cp4space.hatsya.com/2024/01/20/every-finite-phoenix-has-period-2/

#Math #Maths #Mathematics #ConwaysLife #GameOfLife #CellularAutomata