Mathematics and Nature

I am asking is there any area of current pure mathematical research which will escape being subsumed into physical theory.

so sometimes there is a period of trial and error while scientists experiment with various ways of systematizing.

In a ruler and compasses construction, the line between the tips of the compasses are, in effect, the measuring rod.

Even if the systematic method does not work perfectly, the mathematics is useful. For the mathematics tells us how the system would behave, purely on account of its systematicity, if reality did not intrude. That make it easier for us to see interesting features of reality in the failure of the mathematics to exactly match the world.

More recently science, and most particularly physics, has been looking at the systems studied by mathematics, to see if some of those systems can be adopted for use in systematic empirical methodology.

Just out of interest, why? Always is a very strong word.

Frege's Advances in Logic:Frege virtually founded the modern discipline of mathematical logic. He developed a system of conceptual notation (inspired by Leibniz's conception of a rational calculus), and though we no longer use his notation, his system constituted the first predicate calculus. Frege's second-order predicate calculus was based on the `function-argument' analysis of propositions and it freed logicians from the limitations of the `subject-predicate' analysis of Aristotelian logic. Frege's formal system made it possible for logicians to develop a strict definition of a proof. Unfortunately, Frege employed a principle (Basic Law V) in his later system (Grundgesetze) which turned out to be inconsistent. Despite the fact that a contradiction invalidated his system, Frege validly derived the Peano Axioms governing the natural numbers from a powerful and consistent principle now known as Hume's Principle (some philosophers have proposed that the derivation of the Peano Axioms from Hume's Principle should be called `Frege's Theorem'). Frege is most well-known among philosophers, however, for suggesting that the expressions of language have both a sense and a denotation (i.e., that at least two semantic relations are required to explain the significance of linguistic expressions). This seminal idea in the philosophy of language has inspired research in the field for over a century.

After all, if you're going to talk about "reality", then you have to also say that there's actually not any "1" thing at all; "objects" change all the time, whether it's their molecular motion or some atoms getting scrubbed off, etc.

I would say, don't mistake the abstraction for the "reality".

I'm not sure what you mean here. I have made hundreds of mobius strips.

made up of fermions and bosons.

My cellotaped piece of paper ...

The quantum theory of electrons falls apart if there is the slightest hint of even a theoretical possibility of being able to distinguish two electrons.

My pieces of paper exhibited the required global topological behaviour, amongst other behaviours. You can imagine an idealised surface wrapped into a Mobius Band, but this surface also has topological properties that are irrelevant to its Mobius topology such as the nature of its width - is it open or closed, or is there a distance function defined.

"Dancing"? "Changing partners"? Sorry, I don't speak Layman-ese as I told Randman recently. What are you talking about? Of what relevance is this to my point?

Wikipedia? Give me a f'ing break.