[Math] known about the common knowledge of mathematicians outside their field

gm.general-mathematicssociology-of-mathsoft-question

When giving a talk or writing a paper intended for non-specialist (i.e., mathematicians not specializing in the topic being discussed), the question inevitably occurs of what one can assume to be "common knowledge". Rather than trying to guess (e.g., I assume that almost all mathematicians know what a vector space over a field and what the Lebesgue measure are), it would probably be better to determine this experimentally. Have any surveys been conducted in order to answer this question?

Roughly which set of mathematical definitions and facts are known to a proportion at least $p$ of working mathematicians?

(Here, $\frac{1}{2}\leq p<1$ is some fixed but specified quantity. Of course, a survey like this would in fact measure how well-known various notions and theorems are, so would give results for a variety of different $p$. Also of course, this depends on some definition of what a "working mathematician" is, I'm assuming self-reporting as such, but I don't think the details are too important; even a survey limited to a particular country or membership to a particular mathematical society would be something.)

It seems to me that this would be interesting both mathematically (see above) and sociologically. More refined results indicating how results vary per country, per age group, or per specialty, would of course be of value, but any survey along these lines would interest me.

I tried Googling various terms to no avail, so I'm inclined to think that no such survey was ever conducted, but maybe I missed something.

Best Answer

In looking over my answer, though it does introduce references, it seems to me to be opinion-based so I am making it a community wiki.

If giving a talk or writing a paper along these lines, you might consider your constraints regarding length or time allotted and try to give as much background as possible. Lay out your ideas and see if you are comfortable with the amount of background that is supplied and your target. For a paper, you should probably be working with an editor who can provide you with more specific advice relevant to your target journal. I will focus on general mathematical audience talks, aka colloquia. It seems the strongest opinions out there (measured by who was willing to write something down) seem to favor more valuing the "journey over the destination." See for example:

"How to give a good colloquium" John E. McCarthy, Washington University Canadian Mathematical Society NOTES, 31 no. 5, Sep 1999, pp 3–4 https://cms.math.ca/notes/v31/n5/Notesv31n5.pdf

University of Oregon Department of Mathematics Colloquium Guidelines https://pages.uoregon.edu/njp/guidelines.html

McCarthy gives a few examples:

The first 20 minutes should be completely understandable to graduate students....It means a student who just scraped by the required coursework in your area, and then went into a different field. So

  • you can assume they know what $L^p$ is, but not what a Sobolev space or pseudo-differential operator is;
  • you can assume that they know what a manifold is, but not what Poincaré duality is;
  • you can assume they know what a field extension is, but not what an induced representation is.

So this answer doesn't quite answer your question. In an attempt to do so, I would say assume only that covered by the basic grad courses if you can. You might even consider spending some time giving the introductory definitions and/or working through a representative example. This is especially true about the first 20-30 minutes of your talk (see for example the two references above).

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