The n-Category Café

I don’t agree with that, and it’s hard for me to imagine a good mathematician saying that — unless he had something very specific in mind.

It’s clearly an outrageous exaggeration at best, probably for the humor value. However, there’s a kernel of truth to it. There’s a lot of mathematical machinery that can’t be understood via a single ah-ha moment: mastering it requires seeing lots of different examples, recognizing new examples one wasn’t expecting, seeing how the machinery really tells one what one wants to know, etc. This is intrinsically a slow process, and there’s no sense sitting around waiting to magically “get it”.

I think of, sympathize with, and use that quote all the time. It’s hard for me to imagine a good mathematician not thinking it. Perhaps we should take a poll!

I didn’t see Anon’s response when I posted mine, and I disagree with that one too! So I guess I should say what truth I find in the von Neumann quote.

There are a number of results I’ve seen the proofs of – even short ones – that I’ve initially found unsatisfying and unintuitive. Is that a failure of understanding? Perhaps. But I’ve often taken it as more of a sign that I should replace my axiom system, and redefine such things as “intuitive”.

This takes effort – basically, for a while I have to work at remembering when faced with a new problem to check if this not-yet-intuitive theorem may help. Once it does so enough times, poof, it feels intuitive.

For example, I am in the middle of this process with Frobenius splitting. The proofs using it are short and incontrovertible, and I’m getting a sense of when it’s useful. If you “understand” Frobenius splitting then more power to you, sir.

I agree that there are lots of things in mathematics that are hard to understand, that we must temporarily accept in order to get on with business. I also don’t think we reach a definitive understanding of anything in a finite amount of time. But, I’d never say that in math we “don’t understand things, just get used to them”. I spend a lot of time trying to understand math, and I often make progress.

Most of all I like to tackle facts that are easy to prove but hard to understand. I’ve spent over a decade trying to understand why there are 4 normed division algebras over the reals, with dimensions 1, 2, 4, and 8. I know plenty of proofs: proofs that use Clifford algebras, proofs that use string diagrams… but I still feel dissatisfied. I’m not sure why. Maybe it’s because all the proofs seem a bit magical. Maybe it’s because I don’t know whether all these proofs are fundamentally the same.

So, I wouldn’t say I understand this fact. But, I’m certainly not just “getting used to it”, which suggests some sort of passive resignation. I’m gonna keep fighting to understand this stuff until I die!

Maybe he’s not talking about the understanding of the proofs, he’s talking about why one need to use/introduce certain concepts/methods in mathematics at all. We understand them only through their results.

I think naively that the all point is in the meaning of the word “understand”, and in the word “things”. After all, what does it mean to understand, in general, and in particular, understanding maths ? If one means to be able, say, to reproduce a proof, justifying how every logical step follows from the precedent one, then I think you can say you can understand “things”, in this case the proof. But it occurred a lot of time to me that even if i did this, i wasn’t feeling somehow comfortable with the results, and then one day I woke up and thought:”That’s how it was!”. Probably it was induced from “getting used” to the results, or, in some irrational way, from convincing oneself of the results, beyond the mathematical proof, or maybe from “understanding”…
Anyway, I do agree with John when he says we can never learn everything about a subject or a concept in a finite amount of time, but that we try to add bits from time to time…

What does it mean to “understand” something? To me, you understand something if you can calculate with it and have a mental picture of what is going on. What more can there be to understanding?

E.g., I think I understand the real numbers - I can calculate with them, and my mental picture is a bunch of points on a line. Now, there are constructions of the real numbers from set theory, but I never understood those constructions and now I have forgotten everything about them. Does this limit my understanding of real numbers?

Another example is the Dirac delta function. Mathematicians often claim that nobody understood the delta function before Schwartz developed distribution theory, but I think that is nonsense. Dirac had a mental picture (an infinitely high, infinitely thin peak) and he certainly could calculate, so in the sense above he had an understanding. That mathematicians didn’t understand, or didn’t like, what Dirac was doing is another thing.

Ironically, von Neumann seems to have been very hostile to Dirac’s techniques. In his mathematical foundations of QM, he argues that his Hilbert space methods are vastly superior to Dirac, because he does not have to pretend the existence of something know not to exist (i.e. the delta function).

I think one can detect a student’s (lack of) understanding by the way she looks. So, probably future software could detect that too. Perhaps another method to estimate the degree of understanding is the degree at which readers are troubled by typing-errors in formulas, e.g. because “understanding” enables to correct them very fast. Else, my impression is that “understanding” is the build-up of a conscious (visual) mental model + a subconscious model of the speakers/writers way to think, probably provided by mirror neurons.

Conc. Neumann: I heard that he intended to become a banker until his father insist him to study math. Did that lack of intrinsic initial motivation show up later in some way?

Hello everyone, I have an inquiry about generalizing von Neumann’s concept of continuous geometry. Please consider this question:

Is it possible to have a “space” X such that an infinite X is the limit of a sequence of increasingly large finite instances of X?

If the answer to this question is “yes” then such a construction may have important implications for mathematics. For instance, the celebrated topologist Michael Freedman has suggested it might help
with unsolved problems in computational complexity and geometry [1]. Also, the theoretical physicist and string theorist Raphael Bousso once posed this question:

“Is there a sequence of theories with finite-dimensional Hilbert spaces such that string theory emerges in the infinite-dimensional limit?” [2]

I am further reminded of Edward Witten’s conjecture about U(∞)-valued Chan–Paton factors [3], and of various string theory papers about infinite towers of states. (By the way, even though I am not a physicist I am supposed to have some expertise in the math of high energy physics as you can see, for example, in the Acknowledgements sections of two papers [4, 5]. My own ideas in this area involved finding new connections within M-theory via Jordan algebras, but I don’t wish to discuss this now because it would be too off-topic.)

The great John von Neumann introduced the concept of continuous geometry which can be viewed as the limit of increasingly large finite instances of projective geometry [6]. Is there a way to sufficiently generalize von Neumann’s concept? For example, should I consider using bundles?

(I did find a paper about a tower of n-gerbes with the {infinity}-gerbe [7]. Would it make any sense to try to define a notion of convergence for such a tower?)

Thank you,
Charlie Stromeyer Jr.

[1] http://forum.wolframscience.com/showthread.php?threadid=288

[2] http://www.physics.ucsb.edu/~giddings/Mquest.html

[3] http://xxx.lanl.gov/abs/hep-th/0007175

[4] http://xxx.lanl.gov/abs/gr-qc/0212096

[5] http://xxx.lanl.gov/abs/hep-th/0407122

[6] http://planetmath.org/encyclopedia/VonNeumannLattice.html

[7] http://xxx.lanl.gov/abs/math/0301271

There’s now an official webpage for this conference, which contains a subliminal advertisement for the $n$-Category Café.

Here’s a nice photo of some of the conference participants. This photo was taken by Jamie Vicary, who therefore alas remained invisible himself. Thanks for the self-sacrifice, Jamie!

From left to right: Simon Kochen, Jeffrey Bub, Bob Coecke, Peter Woit, John Baez, Mike Stay, Andreas Döring, Camm Maguire and Chris Heunen.

A second version of Stephen Summer’s contribution – Yet More Ado About Nothing: The Remarkable Relativistic Vacuum State – is available.

An idle thought: might we be able to think of the vacuum state in a galoisian way? Or, in Lautmannian terms, is there a perfectedness to the vacuum state?

Does anybody Know Where could I find the book “Deep beauty” edited by Halvorson, to download online???