# What is known about the common knowledge of mathematicians outside their field?

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.

• @GerhardPaseman The thing is, I suspect mathematicians pick up a lot of knowledge, even outside their field, beyond what they are taught in University. (To give a specific example, I suspect that many mathematicians know what the Banach-Tarski paradox is, because it's so fun and surprising, but I gather very few students are actually taught about it.) – Gro-Tsen Jan 20 at 18:08
• It might strongly depend on the country, even in those with a major activity in math research. For instance I'd have thought that every mathematician would know what a Hilbert space is, but have had surprises at some point... – YCor Jan 20 at 19:03
• @YCor, legend has it that at least one outstanding mathematician didn't know what a Hilbert space was: mathoverflow.net/questions/53122/mathematical-urban-legends/… – Gerry Myerson Jan 20 at 19:58
• @GerryMyerson interesting. I learnt that Euler didn't know either but it might be conspiracy theory. To take this into account, possibly let me mention that my comment concerns contemporary mathematicians, and that the OP's question is rather statistical/generical than focussing on exceptional individuals. – YCor Jan 20 at 19:59
• These interesting notes by Barry Mazur address a question somewhat dual to yours: "What should a professional mathematician know?" Mazur focesses on broad areas and intuitions (as opposed to statements like 'Theorem A from area B'), which seems a wise decision. google.com/url?sa=t&source=web&rct=j&url=http://… – Ben Linowitz Jan 21 at 21:57