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September 7, 2006

Mathematical Circuit Components

Posted by David Corfield

From the post The Downward Spiral of Physics on the excellent new blog of Alexandre Borovik, we read Feynman lamenting the passing of the days when:

Radio circuits were much easier to understand in those days because everything was out in the open. After you took the set apart (it was a big problem to find the right screws), you could see this was a resistor, that’s a condenser, here’s a this, there’s a that; they were all labeled. And if wax had been dripping from the condenser, it was too hot and you could tell that the condenser was burned out. If there was charcoal on one of the resistors you knew where the trouble was. Or, if you couldn’t tell what was the matter by looking at it, you’d test it with your voltmeter and see whether voltage was coming through. The sets were simple, the circuits were not complicated. The voltage on the grids was always about one and a half or two volts and the voltages on the plates were one hundred or two hundred, DC. So it wasn’t hard for me to fix a radio by understanding what was going on inside, noticing that something wasn’t working right, and fixing it.

Borovik then comments:

Alas, similar cultural changes affect (and mostly negatively) the position of mathematics in modern culture. I leave it to the reader to come up with examples - they are abundant.

Might it be that n-categories, at least, as presented to us by John, offer us a form of mathematics closer to Feynman’s radio circuits? Working on Kleinian 2-geometry feels a little like that to me. “If your sub 2-space has features not invariant under equivalence, you know where the trouble is.”

Posted at September 7, 2006 11:55 AM UTC

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Re: Mathematical Circuit Components

David Corfield writes:

Might it be that n-categories, at least, as presented to us by John, offer us a form of mathematics closer to Feynman’s radio circuits? Working on Kleinian 2-geometry feels a little like that to me. “If your sub 2-space has features not invariant under equivalence, you know where the trouble is.”

I don’t think most mathematicians would feel n-categories are “hands-on” like old radios. The very notion of “category” sends shivers up their spines and shuts down their brains - and by the time “2-categories” or “categorification” come along, their autonomic nervous system has gone into complete collapse.

But, I think they’re just being silly.

I think you have this feeling because you came in at the ground floor of a project where we’re building some math from scratch, without the use of heavy machinery. That’s the kind of project I like.

There are lots of opportunities to do this in math. But you have to look for them - you have to want them.

The feeling of understanding something in a very simple way comes from taking the time to tinker with it, not rushing things. There’s a huge danger in trying to act like you understand stuff before you really do.

If you just dive in and glance at a passage like this, it sounds esoteric and mysterious:

That’s because I cunningly used the other formulation you gave us (V{0})//Disc(k *)(V - \{0\})//Disc(k^*) way back when, where (V{0})(V - \{0\}) has the component connected to 0 removed from VV.

That presumably also helps us describe the projective space associated to k p,qk^{p,q}, as kP p1kP^{p-1} worth of objects each with k qk^q worth of automorphisms.

It’s having worked through this stuff step by step that makes it seem as simple as a 1920’s-style radio.

By the way: like Feynman, my uncle Albert Baez got interested in physics as a kid in Brooklyn by messing with radios. He was one of the inventors of the X-ray telescope, and the one who got me interested in physics in the first place. When I was 8 he gave me his physics text, The New College Physics: A Spiral Approach. The idea of the “spiral approach” is that you should circle through the same ideas over and over, in greater and greater detail, gradually zooming in. I like that. The book also has great pictures. When I visited him as 16-year-old in high school, he gave me his copy of The Feynman Lectures on Physics. That’s how I learned vector calculus and quantum mechanics! Great fun.

Maybe there was some kind of Brooklyn “no-nonsense” practicality that refused to put on airs. If you had a Brooklyn accent, nobody would ever think you were refined - so you might as well give up on that, and prove there are things more important than being refined. That’s very visible in Feynman. He practically made a profession of it.

Posted by: John Baez on September 7, 2006 1:23 PM | Permalink | Reply to this

Re: Mathematical Circuit Components

feynman was from far rockaway,which is queens not brooklyn.
feel free to delete.

Posted by: sean on September 23, 2006 10:49 PM | Permalink | Reply to this

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