Mathematical theorems and science

Question

We know mathematics is applied constantly in science to solve problems, even if sometimes (e.g: modern physics) there is not always a intuition of why such theorems can be applied (since there isn't one for the model itself and it is justfied by its descriptive certainty rather than its premises (e.g: string theory and Calabi-Yau mainfolds, while having no empirical evidence for its very premises)).

Now we could ask, can any mathematical theorem be applied to science? And if the answer is no as it seems to me (e.g: Poincaré conjecture has hardly applications), of what depends that some abstractions, even if they were consider useless their time (e.g: theory of matrices), find eventually practical application? Can any relation between mathematics and science be thought such that because of it, we can be show that theorems or new mathematican branchs are only useful for mathematics itself?

Answer 1

The short answer is that pure mathematics has to be free from empirical determinations of any kind (not just physical, but also psychological) to be what it is. In other words, knowledge becomes essentially mathematical when it can be considered neutral in its relationship with experience of any kind.

Three observations arise from this preliminary point:

1) Mathematicians themselves live in the world of experience (and so will any kind of concrete mind that thinks mathematically), and that is why this desirable "purity" of any mathematical object is, in practice, a process, not something that can have a referent in the world. That can be said, regardless of metaphysical considerations - for instance, if mathematical objects exist or not (which is a big philosophical problem, check this out: https://youtu.be/1EGDCh75SpQ).

2) The connections between mathematical knowledge and its "applications" in natural science is not trivial, at all. Your could even say that, whenever they are spotted for the first time, these connections normally appear surprising, uncanny, even overwhelming, for both sides. Physical science can be about nature, but it is not itself "natural". When science changes the way we see the world, it changes the world, and itself. So my answer to your nuclear question is that the patterns that mathematicians formulate are thinkable, and the patterns that empirical scientists identify are evident. Whatever happens between these two stances is game, and the game of science is not made of trees and branches, it is whatever scientists do. It may very well be that one day knowledge will no longer be so easily departmentalized. Many authors have discussed this trend, and I won't try to be encyclopedic here, so let me just suggest this reading (Isabelle Stengers): http://we.vub.ac.be/aphy/sites/default/files/stengers2011_pleaslowscience.pdf.

3) In the end, the utility of knowledge is in the eye of the beholder.

Answer 2

From an intuitionist notion of mathematics, this is exactly analogous to saying that since words cannot be unambiguously applied to objects in a perfect way, but have to be mapped on to them indirectly though experience, the entire content of language and grammar is 'just for itself'.

We map mathematical structures onto observations in the hopes of using the implications of the mathematics itself to predict the observations. It happens every time we measure, and it gets more involved as we choose measurements that express more definite theories -- like when we assign a photon an imaginary number for its intensity, so we can interpret the imaginary rotation as interference. Every theorem from Complex Analysis then has application to this physics, few matter, but some are very helpful.

This is exactly the same way we use language in physics. We call something 'work' because that captures an idea that behaves sort of like what we think of as work, and we use our intuitions about the more natural kind of work to make predictions about this new thing we have chosen to call work, for instance it consumes what we have chosen in parallel to call 'energy'.

Would you question whether these words 'apply' to physics, or whether they should not have any implications about physics because they are really 'just for themselves?' in the way that they arise from a different motivation and also have other meanings in other situations?

I hope not, because if you are also not going to apply mathematics, you are out of tools.