There are three realisations I have come to me over my summer of research that I wish I knew at the beginning.

  1. We do not need to know the details of every proof.
  2. There are fields of mathematics that you will never the details of.
  3. Mathematics is a marathon, not a sprint.

We do not need to know the details of every proof.

The axiomatic structure of mathematics makes it both satisfying to do and powerful in inference. But this cumulative nature can be a drawback if we obsess about the direct path from where we are currently reasoning, all the way back down to the axioms.

My area looked at harmonic analysis, the generalisation of the Fourier transform. But the proofs did not start with Zermelo-Frenkel set theory. We must often lean on the results of others and rely on the peer-review process in order to make any new ground.

There are fields of mathematics that you will never the details of.

The barrier to doing mathematics is low. In most cases all one needs is a piece of paper and a pencil. It has essentially been this way since its inception and because of this, the main funding issue for mathematics is to ensure that mathematicians have a wage and time to do research.

This contrasts highly to the state of experimental physics or molecular biology, which require expensive instruments and chemical reagents.

With mathematics being a comparatively cheap undertaking and with its considerably sized community, it makes sense that there is an exceedingly large and growing number of books and papers to read. I’m sure it would be the same for physics if everyone could have their own super collider.

There is more mathematics than for a (long) lifetime. Many say that the generalist of mathematics is dead idea, no human has the lifespan to study and absorb all of mathematics to the end of its long branches.

We should make peace with this fact early on, or our attention will be split beyond any point of productivity.

Mathematics is a marathon, not a sprint.

There are three factors that contribute to this conclusion.

  1. Robert Saplosky (a biologist) noted that professional chess players can burn up to 6000 calories in a multi-hour match.
  2. Our subconscious does background processing whilst we do other things.

Our brain is the biggest consumer of energy in our body, when we do things like mathematics, that power consumption is going to go up. It is physically exhausting to spend your days lost in another world, thinking about the logical outcomes of equations and theorems.

Einstein famously worked in a patent office when he wrote three of his biggest papers. Doing some other activity whilst you consider a problem helps and often leads to lightbulb moments.

Hope that helps.

Alexander Hiller
University of Technology Sydney

Alexander Hiller
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