I was recently asked that age old question: “Which is more fundamental, physics or mathematics?” I think I’ll try and answer this a slightly differently from the way I answered it before. There I demonstrated the difference regarding the nature of physics as an experimental subject while mathematics deals with absolutes but of course this isn’t a satisfactory argument for a theorist and I also feel my previous argument skirts around the inherent problems that I see with a mathematics decoupled from physics. In order to do this I’m going hop back to the concepts of vector spaces and linear algebra which I covered here.
Now in this weekly roundup I pseudo introduce the idea of a vector space and make the bold claim that “all quadratic equations… add and multiply just like vectors.” Of course vectors are multiplied by either the dot product or the cross product and that isn’t how we’d try and multiply quadratics. If we ignore the cross product for a moment (it is an example of what is known as a bilinear map and isn’t relevant here) we can look at the dot product or more generally, inner products. Vector spaces often get assigned an inner product, this is a function which takes in two vectors and produces a scalar and has to follow certain rules. The dot product is the example often used for regular vectors but so long as the following conditions hold (for a real space):
- 〈 a , b 〉 = 〈 b , a 〉
- 〈 λa , b 〉 = λ〈 a , b 〉
- 〈 a + b , c 〉 = 〈 a , c 〉 + 〈 b , c 〉
- 〈 a , a 〉 ≥ 0
then 〈 a , b 〉 will be the inner product. This generality is the brilliance of maths. You can define anything as a vector and assign it to a vector space with a nice inner product so long as you follow certain rules, certain axioms. And this is the importance of physics. These axioms are in fact determined by the physical world, for instance the reason that the seemingly arbitrary rule: 〈 a , a 〉 ≥ 0 exists is because the inner product is providing the concept of “length” to a vector. If you give mathematicians free reign (as some people foolishly do) they’ll try and start producing vectors with imaginary length and other non-physical results.
The construction of useful axioms is perhaps the greatest gift that physics provides for maths. Until tomorrow, goodnight.
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