August 31, 2020

Chasing the Tail of the Gaussian (Part 1)

Posted by John Baez

I recently finished reading Robert Kanigel’s biography of Ramanujan, The Man Who Knew Infinity. I enjoyed it a lot and wanted to learn a bit more about what Ramanujan actually did, so I’ve started reading Hardy’s book Ramanujan: Twelve Lectures on Subjects Suggested By His Life and Work. And I decided it would be good to think about one of the simplest formulas in Ramanujan’s first letter to Hardy.

It was a good idea, because I now believe all these formulas, which look like impressive yet mute monuments, are actually crystallized summaries of long and interesting stories — full of twists, turns and digressions.

Posted at 9:35 PM UTC | Permalink | Followups (5)

Sphere Spectrum Analogue of PGL(2,Z)

Posted by John Baez

Since I’ve been thinking about continued fractions I’ve been thinking about $PGL(2,\mathbb{Z})$, the group of transformations

$z \mapsto \frac{a z + b}{c z + d} , \qquad a,b,c,d \; \text{s.t.} \; a d - b c \ne 0$

mod its center. You can think of this as a group of transformations of the integral form of the projective line. When we see something like

$\frac{\sqrt{5} + 1}{2} = 1 + \frac{1}{1 + \frac{1}{1 + \frac{1}{1 + \ddots}}}$

or even

$\frac{\pi}{4} = \frac{1}{1 + \frac{1^2}{2 + \frac{3^2}{2 + \frac{5^2}{2 + \ddots}}}}$

Posted at 5:27 AM UTC | Permalink | Followups (5)

August 30, 2020

Euler’s Continued Fraction Formula

Posted by John Baez

I’ve been reading about Ramanujan. His mastery of continued fractions made me realize how bad I am at manipulating them. Here’s something much more basic: a proof that

$\frac{4}{\pi} = 1 + \frac{1^2}{2 + \frac{3^2}{2 + \frac{5^2}{2 + \frac{7^2}{2 + \ddots}}}}$

It illustrates a method called Euler’s continued fraction formula.

There’s nothing new about this — it goes back to around 1748. But it might be fun if you haven’t seen it already.

Posted at 6:49 AM UTC | Permalink | Followups (1)

August 23, 2020

Formal and Material Inference

Posted by David Corfield

A distinction is made within philosophy between formal and material inference. The first of these operates purely through the logical form of the relevant propositions, whereas the second relies on conceptual content occurring within them.

A classic example of a formal inference is $A \& B$, therefore $A$. Substitute any propositions for $A$ and $B$ and the inference goes through. The conjunction $\&$ is a piece of logical vocabulary. By contrast, the thinking goes, that $C$ is west of $D$ implies that $D$ is east of $C$ is a piece of material inference, relying on the relation between the non-logical concepts, ‘east’ and ‘west’. Substitute ‘older’ for ‘east’ and ‘larger’ for ‘west’ and the inference fails.

The philosopher Wilfred Sellars famously asserted that in such cases we’re not merely employing a tacit proposition, i.e., here ‘If $X$ is west of $Y$, then $Y$ is east of $X$’, instantiating it and then using modus ponens. For Sellars, and those following him, like Robert Brandom, material inference is primary; only for a limited portion of our inferential practices has humankind managed to extract formal inference schemas.

But then how to decide what is ‘logical’ and what ‘non-logical’? As an adherent of dependent type theory, can’t I hold any inference carried out in that system to be formal?

Say I have judged $j: P$ and $f: \neg (\sum_{x: P}H(x))$, where $H$ is a predicate on $P$. Then defining for $x: H(j)$, $g(x):\equiv f(j, x)$, I can now judge $g: \neg H(j)$, using standard type-theoretic rules.

Rewriting this inference in something closer to English, we find the syllogism:

$j$ is a $P$, No $P$s are $H$, therefore $j$ is not $H$.

As a valid syllogism, any substitution should do, so let’s choose a type, $P$, say $Person$. $j$ is an element of $P$, so let’s say Jane. $H$ is a property of people, let’s say ‘being in this house’.

Then we have the inference

Jane is a person. No people are in the house. Therefore Jane is not in the house.

OK, why this example?

Posted at 10:20 AM UTC | Permalink | Followups (19)

August 15, 2020

Open Systems: A Double Categorical Perspective (Part 1)

Posted by John Baez

Kenny Courser’s thesis has hit the arXiv:

He’s been the driving force behind a lot of work on open systems and networks at U. C. Riverside. By the way, he’s looking for a job, so if you think you know a position that’s good for someone who can teach all kinds of math and also strong on applied category theory, give him or me a shout.

But let me describe his thesis.

Posted at 1:51 AM UTC | Permalink | Followups (2)

August 11, 2020

Posted by John Baez

We can think of a monad on $Set$ as describing some sort of algebraic gadget equipped with a bunch of operations obeying a bunch of equations. This works very nicely if we restrict attention to finitary monads, which correspond to Lawvere theories: then the operations I’m talking about are all ‘finitary’, taking some finite set of inputs. But we can also generalize to higher cardinalities: for any cardinal α, a monad of rank α on $Set$ describes some sort of gadget with operations of arity at most α.

There are nastier gadgets that have no upper bound on the arity of their operations, like complete semilattices, also known as suplattices. The point is that in such a thing, any subset has a least upper bound, no matter how large its cardinality. These are algebras of a ‘monad without rank’ — which makes me think of someone in the army who is not a private, not a lieutenant, not a colonel, not a general….

Anyway, this viewpoint on monads helps me get a feeling for commutative monads: these describe algebraic gadgets with a bunch of operations that all commute with each other, and perhaps obey other equations as well.

But first: what does it mean for an $n$-ary operation to commute with an $m$-ary operation?

Posted at 8:29 PM UTC | Permalink | Followups (19)

August 10, 2020

The Group With No Elements

Posted by John Baez

Maybe people who are all excited about the “field with one element” should start at the beginning and think a bit about the “group with no elements”.

Posted at 11:34 PM UTC | Permalink | Followups (58)

August 9, 2020

Diary 2003–2020

Posted by John Baez

I keep putting off organizing my written material, but with coronavirus I’m feeling more mortal than usual, so I’d like get this out into the world now:

Go ahead and grab a copy!

It’s got all my best tweets and Google+ posts, mainly explaining math and physics, but also my travel notes and other things… starting in 2003 with my ruminations on economics and ecology. It’s too big to read all at once, but I think you can dip into it more or less anywhere and pull out something fun.

It goes up to July 2020. It’s 2184 pages long. I fixed a few problems like missing pictures, but there are probably more. If you let me know about them, I’ll fix them (if it’s easy).

Posted at 1:30 AM UTC | Permalink | Followups (9)

August 5, 2020

Open Systems in Classical Mechanics

Posted by John Baez

I think we need a ‘compositional’ approach to classical mechanics. A classical system is typically built from parts, and we describe the whole system by describing its parts and then saying how they are put together. But this aspect of classical mechanics is typically left informal. You learn how it works in a physics class by doing lots of homework problems, but the rules are never completely spelled out, which is one reason physics is hard.

I want an approach that makes the compositionality of classical mechanics formal: a category (or categories) where the morphisms are open classical systems—that is, classical systems with the ability to interact with the outside world—and composing these morphisms describes putting together open systems to form larger open systems.

Posted at 2:22 AM UTC | Permalink | Followups (12)

August 3, 2020

Octonions and the Standard Model (Part 4)

Posted by John Baez

Last time we saw what we can do by choosing a square root of $-1$ in the octonions. They become a 4-dimensional complex vector space, and their automorphisms fixing this square root of $-1$ form the group $\mathrm{SU}(3)$. This is the symmetry group of the strong force —and even better, its representation on the octonions matches the one we see for one quark and one lepton in the Standard Model.

What happens if we play the same game for some larger structures built from octonions? For example $\mathfrak{h}_3(\mathbb{O})$, the space of $3 \times 3$ self-adjoint matrices with octonion entries?

Maybe some of you can guess where I’m going with this, but I think I should start at the beginning and go slow, so more people can jump aboard the train!

Posted at 1:03 PM UTC | Permalink | Followups (14)