Flavor: Translating a proof from one context to another.

1. What’s going on mathematically?

Sometimes after working through a proof in a paper, it’s useful to try to translate the proof from that context to another. {As a concrete example, I was recently reading Bousfield’s construction of a complementation operator a(-) on the object level of the category of spectra, with the goal of using ideas from his proofs to construct such an operator in the derived category D(R) of a ring.}

2. What is the emotional and logistical context?

To do this well, I have to be in top mathematical form, ready to pull on all the resources at my disposal. So I wait until the emotional and logistical context is most conducive – i.e. quiet, still, spacious, rested, relaxed, happy, perhaps caffeinated.

3. What thoughts are there?

Usually you don’t just have to “translate” the proof, but also the statement of the claim. This quickly becomes a jumble of cause and effect, of constantly adjusting hypotheses and consequences. {What should be the definition of a(E)? Well, it depends on what kind of constructions I can do, to get nice cofibers. The obvious one, given by the definition of cellular complexes, apparently wasn’t going to work (I came up with a counterexample to be sure),  but maybe I could tweak that filtration, or start with it to build a different one.} This takes maximum mental bandwidth. It also helps to be in good communication with my intuition on the subject (whatever you take that to mean), since intuition is an excellent guide for this.

At the same time, to use the old proof in the new context requires not just a logical understanding that the original proof works, but a robust grasp on why it works, and how. What is really going on {i.e. what key aspects of spectra is Bousfield using}? What proof techniques are at play? {In this case, it was transfinite induction.}

4. What quality of awareness?

My mind has to be relaxed and open, to start with. What seems crucial to this sloppy driving around in the muck, is being able to hold out on smaller details, to suspend disbelief and pursue consequences quickly and often. It’s like my mind simultaneously throws six or seven balls in the air at once, and my job is to keep them all bouncing around for a little while. The fact that I’m working off of a concrete original proof, sort of keeps the balls contained within four walls, and gives me a solid point of reference and interface.

There’s very little room for self-reflection or distraction, or else I’ll miss some potentially fertile combination of the ideas. This is a very absorbed state of mind.

5. What emotions?

It feels very spontaneous, and helps to be feeling flexible and willing. I don’t know how the proof will translate, what it might allow me to prove or disprove in the new context, so there’s uncertainty and excitement. Sometimes I get anxious that the proof won’t translate to prove such and such; or that someone smarter than me, with a better grasp on the material, better intuition, or wider knowledge, would be able to translate successfully but I won’t. It helps to confront this anxiety head-on, since it is very unconducive to the flexibility and intensity required to do the work.

6. What does it resolve to, after how much time?

This can resolve in two ways. On the one hand, I may land on some result, and succeed in having extracted something useful out of the original proof’s ideas and techniques. This result, however small, can be built upon and will suggest new directions, of course. Or else the experiment was unsuccessful, in which case there’s a reason it was unsuccessful, and I maybe can pursue that reason – maybe I need to read more about Y, or better understand X, or maybe it just won’t work.

7. How frequent is this flavor?

Frequent. It’s a great way of squeezing out new results. It’s really helpful to have the walls there when I’m trying to throw all those balls around at the same time.

8. What are good/bad ways to change or follow it up?

In my experience, it’s unhelpful to get too discouraged if things don’t go as I had hoped. It’s better to fill in some gaps, or take a different tack, and try again.


Connections:

In Proofs and Refutations, Imre Lakatos demonstrates the simultaneous co-creation of definitions, statements of claims, and proofs.

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