What are mathematical objects made of – a short rant

Mathematicians are fond of “invariance” in various guises. Einsten’s notation was invented so you can no longer write down an expression that’s not manifestly Lorentz-invariant. You have abstract linear maps so everything you do is basis-invariant by definition, and you’re not even allowed to ask “what is this group made of”, because it’s all morphisms and arrows, you’re not dealing with groups – it’s “equivalence classes of groups with respect to isomorphisms”. Now, people extol the virtues of this approach as it was some kind of a “holy grail of mathematical purity”. You can’t even *think” in a coordinate-dependent way, because all the concepts were specifically invented to make it impossible. And purportedly this makes proofs a lot easier because you don’t have to check irrelevant details, like whether your action is well defined or whatnot. To this I’d like to respond with a quote from Orwell’s 1984:

“The whole aim of Newspeak is to narrow the range of thought. In the end we shall make thought-crime literally impossible”

No sane person will gag themselves (or inoculate against thoughtcrime, if you will) without a deep reason. So why do mathematicians happily castrate themselves and deem parts of mathematical thought “irrelevant”, “superficial”, “not structural”, “unthinkable”? I spent a lot of time thinking about this peculiar phenomenon and I think I’ve figured it out what the reason behind this “invariance propaganda” nonsense is.

That reason is fear. The fear that, when you start digging deeper, your beloved objects will turn out to be made of shit.

I mean, face it. A German physicist in 1930s wins a Nobel prize and suddenly everyone is “We should put politics aside, be impartial and focus on scientific merits, it’s a great contribution and above petty politics”. Give me a break. That guy is *a fucking Nazi*. He may revolutionize quantum gravity or find the Theory of Everything, but it doesn’t change the fact that he’s a deplorable servant of a totalitarian regime that murdered millions of Jews and Poles. But people don’t want to ask “what’s inside”, “what is this group made of”. Maybe it’s just complex numbers or plain old permutations, fine – but what if you dig deeper and there are Nazi paraphernalia inside? Or child porn? What if your symmetric group is implemented as a collection of nude pics of Bozo the Clown’s three underage daughters and group action permutes the photos? When the feds come knocking the door, the “hey, but we’re working with isomorphism classes of groups, it’s not relevant!” excuse won’t even get you a lower sentence (unless the judge has been brainwashed with the “invariance propaganda”, as sadly is the case more and more often).

I want the right to ask what things are made of. The right to pick my own basis and compute the hell out of it, with raw numbers and not a shred of coordinate-independence. This is the source of the complexity, the diversity, the richness of mathematical thought. I want to have my groupoids made of tofu, and if you’re denying me this pleasure in the name of some abstract purity of thought, I have one message: gentlemen, please, get the fuck out.

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