How Does Applied Math Impact Foundations?

How Does Applied Math Impact Foundations?:

This weekend I’m giving a talk on “The Foundations of Applied Mathematics”. It’s mostly about how the widespread use of diagrams in engineering, biology, and the like may force us to think about math in new ways, at least if we take those diagrams seriously.

I wouldn’t typically emphasize the term ‘foundations of mathematics’ when talking about this. But I’m speaking at a Category-Theoretic Foundations of Mathematics Workshop, so I want to fit in.

Unfortunately, this means I need to think a bit more generally about how applications of mathematics can impinge on foundational issues. I need to do this just so I’m not taken by surprise when people start objecting or asking tough questions.

So, I’d like to hear what you have to say. Preferably before Sunday morning!

Here’s a bit of what I’m saying. Again, this isn’t the important part of my talk, just the introduction… but I’m afraid I’m biting off more than I can chew.

We often picture the flow of information about mathematics a bit like this:



Of course this picture is oversimplified in many ways! For example:

• The details depend enormously on time and place. Let’s focus on post-WWII mathematics just to keep the topic from growing too large.
  
• There are many branches of science and engineering, and a very complex flow of information among these.
  
• 
  In academia, only some applications of mathematics are now classified as “applied mathematics”.
  
• 
  Some branches of physics now communicate more directly to “pure mathematics” than “applied mathematics”.
  
• 
  Computer science also plays a distinctive role, often communicating directly to “foundations of mathematics”.
  

But the picture is close enough to true that deviations are interesting.

In particular: can developments in applied mathematics force changes in the foundations of mathematics?

Applied mathematics often provokes new developments in pure mathematics. But can it demand new foundations?

Can applications of math directly affect the foundations of math, or is the conversation always mediated by pure mathematicians?

Computer science is the biggest example of “applied math” that grew directly out of work in logic, where new ideas directly impact foundations:

• 
  Uncomputability, undecidability,…
  
• Computer-aided proofs: what is a proof?
  
• 
  Category-theoretic logic
  
• 
  etcetera
  

But I want to talk about some other applications of mathematics that seem to call for category-theoretic foundations….

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