Should the easiness with which math is applied to the world be a surprise?

Question

I study physics at an undergraduate level. Since early on, I've was a person who thought math was 'logical' and as such, its applications to the world aren't really a surprise since math is so 'self-evident'. But I'm beginning to question this.

The kinds of things which are self-evident in math are those things relating to the natural numbers. But I think it gets more hairy when one tries to interpret what $5^frac{1}{\pi}$ means.To start with $\pi$ is an irrational number, and irrational exponentiation is so unintuitive, IMHO. $5^pi$ is that number whose pi'th root is 5, I think that's just insane.

Or take things like negative exponentiation, which seems so artificially defined to make stuff like $e^{x}e^{-x}=1$ hold true. To give a more concrete example, when you solve a differential equation involving a mass-spring system, your solution can have complex numbers in it, which will be reducible to sines and cosines but, nevertheless, talk about complex numbers which seem so artificially defined.

So I'm currently having trouble about how to interpret my relation to math. How should one interpret the easiness with which it is applied to the world? As self-evident, or as something which is a property of the world? I used to see math as some sort of chess game, but I'm beginning to think that viewing it as a science is more adequate, where we actually make experiments and observe and each new application of math to the world is a huge surprise and not self-evident.

Thanks for bearing with me. I think I have more thoughts about this, but a long post could be rather tedious.

Answer 1

How should one interpret the easiness with which it is applied to the world?

It has taken so many years for humans to develop math and it is still going on and will keep going on. It may feel very easy now to use math in our daily lives but we have reached that stage of easiness over a very long period of time due to some very deep thoughts of many people over this long period of time. So to answer your question, you should interpret the easiness as a result of enormous amount of effort over a long period of time. This reminds me of a quote - "Standing on the shoulders of giants".

As self-evident, or as something which is a property of the world?

When something (in this case maths) becomes part of our daily lives in terms of how easy it is to use in our day to day lives and how "obvious" it seems, along with the fact that we are able to describe many phenomena around us using maths (Example: physics), we tend to start believing that "whatever" created this world must be based on the same math that we are used to. IMHO this is just a cognitive bias of humans. The universe is not based on math, instead we developed a tool/language called math which fits our cognitive abilities and its limitations and allow us to understand and explore the world around us.

Answer 2

That depends on what you think math is.

From an intuitionist point of view, math is the study of human idealizations. And there is no good reason to be surprised that we would, given millions of years of trial and error, evolve a really strong intuition that would allow us to understand a lot of the natural world. Nor that once we had language and adequate time to dwell inwards that we should not be able to unwind those intuitions into precise language forms over thousands of years.

The economy of mathematics is sometimes striking, to me: that so many parts of it are really just other parts in variant forms. But I would blame that on the fact we are in a very orderly corner of the universe, compared to what might be.

If you think math is somehow independent of human psychology, and not the collective set of modelling tools at its disposal, then the consistent meeting up of fact and form becomes much more mystical. But then that big mystery becomes a good reason to question that independence.

From that angle the conventions you find so bizarre, are largely just that, conventions, if ones we worked out over generations, and are pretty much born into making. The idea that we can think of multiplication on the complex numbers as scaling and rotation has a lot to do with the relative paucity of our own simple models of motion, and not so much to do with independent reality. After all, we really wanted circular planetary orbits. When we want to model waves, we try hard to make sure they get expressed in terms of the components of a rotation. And when we decided to model particles, we 'found' they have rotational inertia, despite that their rotation has to be 720 degrees, and acts relatively little like actual rotation. Once you let the real awkwardness of that notion sink it, it seems to me like polar coordinates are a solution in search of a problem, not something that just happens to crawl out of so many niches.

Answer 3

1. If you proceed step by step, passing from '2 power 4' via '2 power 3/4' to '2 power pi' and even to '2 power i' (exponentiation with pure imaginary exponent), you will probably stop at each new kind of abstraction. Each time you will try to ask your power of imagination for plausibility of the operation and the result. E.g., try to visualize

   e**(i*pi)= -1.

   The view changes when you consider the whole exponential function in one, namely exp: Real numbers ---> Real numbers, defined as exp(x):= e power x. Apparently that's a continous and even differentiable function defined for all real arguments. But even more: Without any problem you can extend the domain of definition to the set of complex numbers, e.g. by considering the power series expansion of the exponential function. Hence, what one considers plausible, depends on the level one has already attained in the field in question.

2. I do not consider complex numbers artifically defined. And for me it is not necessary to legitimate complex exponentiation by reduction via exp(iz) = cos z + i sin z to trigonometric functions. I consider it a deep insight of Gauss that solely by introducing a single imaginary number 'i', complex numbers derive as z = x + iy and each polynomial gets as many zeros as the degree of the polynomial indicates.

3. Mathematics is not self-evident. Because evidence always depends on the degree to which one is familiar with the subject and on the depth one has penetrated the given problem. Why mathematics fits to solve real-world problems, is still an open question, see Wigner's paper quoted in Alexis' comment.

Answer 4

Math is so readily applied to the "real world" because it was developed precisely for the purpose of solving "problems" abstracted from specifics.