Philosophy of Mathematics - Number Theory

Off in another part of my thoughts, which have been on hold for a while, I have been trying to work out some ideas on the philosophy of mathematics.

I have two theses. One is about the relationship of abstract mathematical ideas to various types of measurement or geometric properties. If you want to know how the various derivatives on curved spaces arise from the simple issues of co-ordinate changes, it's all there. The other is a methodological thesis, that the purpose of mathematics is to provide tools and techniques to solve problems that arise from modelling physical and other processes, and to understand the scope and limits of those techniques. Creating and solving the equations of the mathematical models is what's usually called "applied mathematics", while understanding the scope and limits of the techniques is a lot of what's called "pure mathematics".

And then there's Number Theory. Which is about numbers. Not mathematical models.

You know that Langlands thing that all the Kool Kids are working on?

Yep. Number theory. Finite field number theory at that. Geometric Langlands is even more abstruse.

It takes genius-level insight and technique to understand the more recent developments in Langlands. That's the point: if the specialists can barely follow it, how is it going to be any use to some poor post-grad working on differential geometry at the University of Ennui-sur-Blase?

The social purpose of mathematicians is to teach other people - physicists, statisticians, epidemiologists, computer scientists and programmers for example - how to use the problem-solving techniques mathematics offers. What mathematicians do in their spare time is their business: they need a decent laptop, a whiteboard and some paper and pens: math is cheap compared to fundamental physics.

The Langlands guys can do what they want in their spare time. But it's a rabbit-hole. Maybe it's a big, well-lit rabbit-hole with all the health and safety gear and plenty of mechanical digging tools, but it's still a rabbit-hole. Unlike some of the rabbit-holes mathematicians have buried themselves into (functional analysis, for instance), Langlands is not going to produce anything useful to regular working stiffs (for instance, functional analysis produced the theory of weak solutions to differential equations, which is very useful). I feel confident saying that because Langlands is about structures the rest of mathematics just doesn't use.

(Rabbit-holes are as opposed to specialisms, which are very specific subjects that have useful applications in the real world or other parts of maths with real world applications. Like research in PDEs.)

Maybe "rabbit-hole" should be a term of art in methodology. It's a line of research that has no obvious application to any existing problems or in other branches of maths. The scientific version would be a research programme that was making theoretical progress but no empirical progress (was not making new predictions). A rabbit-hole may branch up to the surface every now and then, as applications to problems in other branches of maths are found, but generally once dug, the researchers dig away happily underground.

In this case I would be saying that Number Theory was a mathematician's pastime, and that other very abstruse, or very off-beat, programmes, are for all the sophistication, esoterica for the aficionados. Which doesn't sound too dramatic.

November 2021 Diary Update

I have "missed" a number of posts this month. I've also missed a number of trips to the gym, and there were whole days when I barely left the house except after dark to have my daily walk.

I stopped going to the gym because I had a problem with my right hip which had reached see-the-osteopath serious. Osteo's don't like it when you come in after a weights session with tight and hard muscles. They can't do those odd manoeuvres to put your spine back in alignment easily or sometimes at all. I've got one more visit left and it should be clear. When I go back to the gym it won't be to do heavy-ish weights as I have been doing. My days of ego-lifting are now well past, and really stopped early in 2019. I'm going to be all about the health-lifting, which is nowhere near as much fun.

The days I barely left the house were about a) the lack of motivation that sweeps over me at the sight of a dull grey sky, b) the fact that 10,000 steps in a day now wipes me out when in 2019 I could do that standing on my head, c) a lack of connection with London and all other places. This is all about me pulling myself together and just f***ing doing it and various other deeply sensitive maxims. There's a thing called "commute hardening" that we never notice because we commute all the time, but all those months working from home have left me and many others "commute soft", and unable to handle the amount of walking and effort needed for a commute. (That's going to be a real issue when getting all those cosseted bankers and civil servants out of their homes and back into their over-crowded open-plan offices.) So I'm working on building up the ability to handle a 10k-step day without feeling exhausted halfway through. Nothing I can do about my reaction to grey skies, except stop being a cissy.

I've missed the posts because I've been caught up in various decisions and other things, none of which I could formulate coherent thoughts about. (Which has never stopped me in the past.)

I am not going to put a curtain against my front wall, as I suggested I might in a previous post about Room Treatment for Small Rooms. Nor am I going to buy some absorbing panels from Ginger White (not actually a lot more expensive than some curtains). I came this close to both.

I decided that a) having 10kg of absorbing panels on my front wall right above all my kit would be disastrous if any screws came loose (you don't know my walls), b) I couldn't man-handle something that size and weight on my own, c) what happened if it didn't work enough, or was more absorbent than I could live with? As for the curtains, it would look odd, but because you should leave space between curtains and the wall, the curtains would be tucked in behind the Kallax units and it would all look silly.

I finally got up the nerve to bust out the drill, measured up, drilled three holes (two into brick, one into plaster - I do not live in a precision-built house) put in Rawlplugs and screws, and hung three of my collages. Perspex has to be as reflective as plaster, so I'm not expecting acoustic improvements, but at least I'm not staring at a blank white wall anymore.

Because for a domestic listening room, sonic treatment screams you can take this too far, you know.

And I've been reading as well. You have no idea how fascinating the theory of antennas is. I started my life as an electrical engineering student, and while I get electric circuits, I've never really grokked electromagnetism in all its weirdness. Antennas are exactly that. And that was just one of the subjects I read.

So there will be the usual gratuitous back-filling, and I will carry on. The self-imposed restriction on writing about "current affairs" (as we used to call it) does remove an easy source of posts, but it also stops me wasting time on nonsense, or at least writing about it.

Living Right Takes Character, Not Purpose

You know what I hear far too much?

That our collective problem is a lack of meaning / purpose / connection with others / (enter something else that post-modern Capitalism doesn't encourage here).

If only we didn't do b*******t jobs for ungrateful bosses, and we had supportive connections with our neighbours and family, didn't eat meat or burn carbon, saved a species every week, planted a tree every day... then we would feel fulfilled and happy and not do sad things like binge eat / drink / watch TV series / play computer games / (enter things you wish other people would stop doing here)

Hah!

It takes real character to accept that your life is insignificant, and yet still behave as if you are a worthwhile human being.

Go to work, exercise, eat right, not drink too much, keep yourself and your digs clean and neat, see your friends and relatives, save for the future, and keep yourself entertained and interested in something.

Live right, even if you don't know why.

Allez en avant, et la foi vous viendra, as D'Alembert is supposed to have said.

Second-Best Housekeeping Productivity Hack

(The best is to pay for a cleaner. Everyone living in a shared flat or house should do that.)

Get a robot vacuum cleaner.

Really.

Vacuuming is a chore. Not everyone feels that way, but normal people do.

How good were these robo-vac things?

I watched the You Tube review videos, and decided that the base-level Eufy 11S Max would suit my needs just fine, as well as being far more affordable than the ones that use GPS tracking to learn about your house so they know what to do in the master bedroom next time. You can YT the review videos using the name.

Prep the room: get all the wires off the ground, and I put small items of furniture on couch / bed. (I went round and used wire ties to lift a lot of wires off the floor. Which of course I should have done anyway.)

There are two basic modes:

a) you put the cleaner in a room, set it off, close the door and get on with something else somewhere else

b) you let it do the vacuuming, and go round with the damp cloth wiping down surfaces and skirting boards

Admit it, did you wipe down the skirting boards when you vacuumed? Thought so.

That's the productivity bit: you can do something else while it does the chore.

It can sense when it's on carpet and turns the vacuuming up a notch, and it can sense it's about to run out of landing, stop and turnaround. It doesn't fall downstairs.

Don't be control-freaky, it wanders about and eventually covers everywhere it can reach. You do have to get into the corners, but you probably missed those when vacuuming and had to do them specially. It has an 'edges' mode where it will go round the walls.

Empty the tray at the end of each session.

Admire your new cleaner digs.

(No. I'm not being paid. Buy a Robo-Vac if you want.)

In Praise of A Well-Chosen Indulgence

I've been watching how-to-record-your-music videos recently and ran across the wonderfully over-the-top Spectre Sound Studios. In this one he talks briefly about debt and how it should be avoided. Turns out he owns his house, studio, and car. How? Because, in his words, he bought what he needed, not what he wanted.

Spectre Sound was saying: don't spend £1,000 on the fancy gear when £200 will get you something that will do the job well enough that the audience won't notice the difference.

Most companies do this. Most companies buy the least-cost, lowest spec-for-what-they-think-is-needed kit. Sometimes they get away with it, and sometimes they lose thousands of hours a week from laptops that take ten minutes to boot up and shut down, and are unusable for an hour when doing weekly updates (ask me how I know). The company doesn't care that their staff feel that they aren't worth decent kit. That's what economic-value-optimisation makes us feel like. Not important enough for the Good Stuff.

In order, there's

a) doing without

b) buying the least-cost, lowest-spec

c) buying nice so you don't buy twice

d) indulgence (buying something that's a little better than "nice" because it makes you feel warm and fuzzy)

e) wasting money buying specs or stuff you are never going to use. 

For example, bread:

a) is not buying any bread because, well, who needs bread? you're not going to die if you don't have it b) is Tesco White Sliced c) is a sourdough from a supermarket d) is a loaf from Paul or some other such brand e) is any loaf from Whole Foods or an "artisan" baker with their own shop in Notting Hill or Greenwich

The base-line for judging indulgence or waste is not "go without". Nor is it Tesco White Sliced, which barely qualifies as bread. The baseline is the cheapest "buy nice" option. It's deviation either way from "Buy Nice" that needs to be justified. We should justify buying low-quality-and-cheap as well as higher-quality-and-more-expensive.

Buying what you want, without wasting money, just makes you feel like, in the words of the ad, you're worth it.

(Now you ask, I buy whatever shampoo is on sale, but not the diluted cheap own-brand stuff.)

This is why I have always had the suspicion that people who buy what they want rather than what they need are a little more fun and little richer and softer in texture. (Assuming they aren't being stupid with the "wants".) By adopting a very narrow definition of "need" - as in "you're not going to die if you don't go on holiday / only drink tap water / never eat chocolate" - one can lead a very miserable life.

Until April 2020, my indulgence was membership at a fancy gym in Soho: there are gyms that cost more, but most cost less. A lot less. But I enjoyed and appreciated it. I joined my local The Gym in April this year. I don't dislike it, but I don't feel a little thrill walking through the door either. It's functional, and that's it. The local fancy gym charges not far short of what I was paying for my previous gym and is over-priced for what it provides. "Need" rules in this case.

I may be missing a well-chosen indulgence.

Holland Park Japanese Garden

 

One fine sunny day in early November, Sis and I wandered round Holland Park. I hadn't been to Holland Park since one of the summer evening concerts back maybe in the Oughties. It's still there and it's much as I remembered it. I hadn't visited the Japanese garden before and was glad we did.

Room Treatment For "Small Rooms" - Part Three

Welcome to the "small rooms" owners' club.

Once again, pro sound treatment methods (bass traps, reflectors, absorbers and soundproofing) are essential for studios and can be useful for "larger rooms". No-one is saying otherwise.

But one-metre deep bass traps are not feasible in a small room. Nor is six-inch absorption padding all round the walls. Isn't the room small enough already? Nor are serious soundproofing measures, which also require thick lumps of absorbent materials. And have you noticed that the rooms in treatment videos belong to people who don't read, have no art or decoration, and store their CD's and records in another room? No wonder they need absorbers and diffusers. Real rooms have bookcases, shelving, pictures on the wall, and other stuff. Some of that helps. What else can we do?

If you are following the user manual for your speakers, you will be sitting about two-three metres from them. No matter what size your room is. Or what the speakers are. (Have you seen how close people sit to those Wilson towers?)

Contrary to some commentators' sniffy remarks, thick pile carpets and loose hanging curtains a distance from the wall do work: see this table of absorption coefficients. Curtains and carpets are pretty much third for absorption after foams and fibres, and then people.

Am I sure I'm not rationalising my unwillingness to spring for a dozen GiK acoustics panels for £700 or so, plus all that drilling and hanging? Well, that's why I write things like this: to make sure I've got my facts in a row. And I think I have.

There wouldn't be home hi-fi if the first thing you were told by your dealer was "we couldn't sell you any of this with a clear conscience until you've had your listening room re-built by an expert, otherwise you'll just come back and complain it sounds terrible". It has to be pretty good out of the box in nearly all circumstances.

We small-roomers are left with the simple things, which are more about the overall sound of the room than specific flaws.

Rugs for wooden floors

Curtains to the full width of the room so that the corners as well as the windows are covered

Shelves with books, record collections, even storage (as long as it's not a wall of boxes), for dispersion and absorption. Just don't line everything up neatly or you'll lose the dispersive effect

Symmetry: equal spacing between left and right speakers to their near walls, same distance from the front wall, both at ear height; books or records (aka 'damping') to the right wall means books or records ('damping') to the left wall in the same place.

All that work for that conclusion? Hey, I saved a lot of money on those acoustic panels.

So I'm upgrading the carpets and doing something about the (long story) curtains on the windows. The front wall is going to be a curtain with folds hanging from a tension bar. That will do for now. In time I may change the furniture around and get some more natural damping.