> Hitchin agrees. “Mathematical research doesn’t operate in a vacuum,” he says. “You don’t sit down and invent a new theory for its own sake. You need to believe that there is something there to be investigated. New ideas have to condense around some notion of reality, or someone’s notion, maybe.”
This is kind of it I think. It's not just physics that drives interesting math, and it's not just recently that this relationship holds. Math is, IM humble O, the ultimate domain-specific language. It's a tool we use to model things, and then often it turns out that the model is interesting in its own right. Trying to model new things (ex. new concepts of reality) yields models that are interesting in new ways, or which recontextualize older models; and and so we need to reorganize, condense, generalize, etc; and so the field develops.
Unfortunately, he is unable to join the discussion. Deploying a ghost as an argument in and of itself is hardly useful.
Personally I tend to disagree with "I hope it isn't useful" or whatever the GHH quote is about maths being practicably applicable to the world/universe n that.
Why not tell us what you think? Mr Hardy's well documented positions on many things are well known but yours are not.
I don’t think his point and mine are incompatible, the language can have beauty in its own right and still be a model. Legos can be built into something which resemble a house, or something which resembles nothing real; their usage is a separate thing from the joy that comes from playing with them. Math is driven by utility and elaborated by enthusiasts.
I agree with you. What you say has even more support in the fact that the one of the most notable achievement of GH Hardy is in field of biological statistics (population genetics), and according to him, his discovery of Ramanujan.
That doesn't hold up to a reading of the history of maths. So much of it was invented by someone just noodling around with numbers and would find some use in physical science hundreds of years later.
> It's not just physics that drives interesting math, and it's not just recently that this relationship holds
When you say
> So much of it was invented by someone just noodling around with numbers
I think you're ignoring where the numbers being played with came from. Very rarely does someone just invent a fresh problem de novo and start messing with it AFAIK; the 'playful mathematics' approach is still reusing tools which were developed in application, just (typically) a long period of time after that original application. Euclid's geometry doesn't exist without the invention of a compass and rule for drafting. Yes he's playing with the concepts freely, but it's not just some arbitrary toy ideas, they're rooted in a practical reality (albeit deeply).
I agree, I think. I would say it like so, that maths is a sort of highly technical, rigorous language, but like any language it will describe what you want it to. It is easy to think that it is describing the underlying terrain, but it is actually working on three (shared) and model which have of the terrain. So, as we consider different things, maths will follow.
Its grammar is limited in ways we don't understand, because the grammar is based on abstractions of our experiences of the world and the relationships between them.
If we can't imagine entire classes of relationship - likely because we do not have limitless intelligence - then the model will always be a partial analogy, not a full and complete abstraction.
This is kind of it I think. It's not just physics that drives interesting math, and it's not just recently that this relationship holds. Math is, IM humble O, the ultimate domain-specific language. It's a tool we use to model things, and then often it turns out that the model is interesting in its own right. Trying to model new things (ex. new concepts of reality) yields models that are interesting in new ways, or which recontextualize older models; and and so we need to reorganize, condense, generalize, etc; and so the field develops.