in Miscellaneous

Mathematics as a tool

I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written.

Quote by Galileo Galilei


Imagine you want to build a wooden table, but you know nothing about woodworking tools. Maybe you have a rough idea of how the table should look like. Certainly, you know what purpose it should fulfill, but the details of how the table will be able to hold your cereal bowl, are merely sketches in your head.

How do you start?

Without knowing what tools there are and how to use them, the table will never leave the vague idea state. You need to spend some time learning about the tools that are out there and how to use them. While learning about the tools, you will get lots of ideas about what is feasible and what is not. This doesn’t mean that you need to know every detail about the tools before you can start building your table, because, I assume, the table should be finished before you are too old to hold a hammer. What is more important is, that you know what tools are out there and how to use them.

Equally, you don’t need to be able to build your own tools. This is the job of tool-inventors because they are trained specifically for this purpose and are better at it than you will ever be. We live no longer in times when finding tools was difficult and you had to invent them yourself. In fact, the situation today is contrary: there are too many tools and the problem is picking the right ones.

As a physicist this is how I look at mathematics. Our table is describing nature and mathematicians are tool-inventors. (This is by no means to be understood negatively!)

Nowadays the toolbox of mathematics is incredibly big. Many tools invented will never be used to describe anything in the real world, because most mathematicians don’t share this perspective on mathematics. Lots of tools are invented because they are interesting for its own sake. Unfortunately, this has led to a splitting between mathematicians and physicists, because it has become increasingly difficult to decide which tools are worth studying from a physicist point of view. There are few tools that are tailored to describe certain aspects of physics.

It wasn’t always like this.

For example, analysis was invented by Newton because he wanted to describe moving bodies. Nowadays such a close connection between physical nature and mathematics is rarely found.

Even worse mathematicians and physicist rarely speak the same language.

Give a typical student of physics a typical book about an advanced topic in mathematics and he will quit after 10 pages, bored and confused. Most books on mathematics rarely motivate anything. The focus lies on mainly on abstract concepts…

As a physicist, I want to understand the idea behind the tool in question and what it can be used for. Finding this kind of information is incredibly hard for advanced topics.

There is a great essay written by the mathematician Vladimir Arnold that captures everything I wanted to say, better than I ever could. Read it! It’s hilarious.


P.S. I wrote a textbook which is in some sense the book I wished had existed when I started my journey in physics. It's called "Physics from Symmetry" and you can buy it, for example, at Amazon. And I'm now on Twitter too if you'd like to get updates about what I'm recently up to.

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