Kenny Easwaran and David Corfield are discussing whether mathematical concepts have essences. Kenny’s example is that of a normal subgroup of a group, that while it can be defined several different ways, the essence of the notion is that it is the kernel of a group homomorphism. David relates the question to his larger program to redirect the philosophy of mathematics away from its traditional concerns towards elucidating the meaning of mathematical concepts.
Ars Mathematica
Dedicated to the mathematical arts.
I sometimes react to a new mathematical concept with “Yeah, but what is it FOR?”
I guess that’s the same question.
Dang. I was even going to comment on his piece, but John Baez beat me to my point. I am in agreement (I think so at least) that proof’s are just the byproduct of the “essence” of mathematical enquiry, which I feel is the quest for the “right” way of looking at particular mathematical objects and concepts (where “right” is an ever moving target as we keep climbing up the ladder of abstraction). You can prove stuff by just grinding away with the tools you currently have, but true progress comes from conceptual shifts, and I think this is what distinguishes Mathematicians from people in other mathematical disciplines (although there is probably quite a bit of it in theoretical physics), in that these shifts are tacitly the coin of the realm, and you get years of training your brain to work on this level in grad school.
Thanks for the interest in this stuff!
In response to michael, I’d just say that I think this searching for the “right” way of looking at objects and concepts seems to me central not just to mathematics, but all scientific disciplines. And I mean “scientific” in the broadest sense, to be any discipline that tries rigorously to make sense of some aspect of the world. People often want to deny that mathematics is like this, and say that we just work with pure stipulative definitions in mathematics. But in actuality, I’m suggesting that we’re just as much trying to find the “right” concepts and the “right” ways of thinking about them as anyone else. I’m not suggesting that mathematicians do this any more so than people in other disciplines, as I’m sure they’d be quick to point out.
I agree with what’s been said, but would quibble over the adjective “right” in reference to a level of abstraction or way of conceiving. I believe one of the key findings of AI in its short life is that not all abstractions are equal, and what may be appropriate for one problem or one domain or one user, may not be appropriate for another. All may be “right” in the appropriate context. Ways of looking at the world are like maps: Different maps may be useful at different times or for different purposes.
How can you disagree with it when I put it in quotes? In response to Kenny, if part and parcel of this searching for the right way of looking at something is actively wielding abstraction as a tool in and unto itself, then I would say that mathematicians DO do it more than others in the sciences.
We surely need different maps or conceptual resources to understand most aspects of the world. For example, there will be no one right way to answer ‘What is time?’ broad enough to bear on a dimension of the universe, on the one hand, and the alteration in the mentality of the peasant brought about by monastic time-keeping, on the other. But mathematics appears to be less diverse, so that, for example, if you view Galois theory and the Erlanger Program properly you see they are aspects of the same thing.
You mention AI, Peter. From my experience, which is limited to machine learning, what is striking is how similar are many of the techniques, although derived from different starting points, perhaps signal processing or even physics. Nothing like the unity of mathematics will emerge, but the field certainly needs a bold unifier or two to take it by the scruff of the neck. There can be master maps which are projected in different ways to produce different maps for different uses.
Well, David, there are many people in AI who think they have the bold unifier for AI already to hand, whether it be statistical inference, logic programming, search, learning, or multi-agent systems.
I disagree with you that a unification of AI is desirable, but perhaps that’s a debate for another day.