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What do you mean when you say What would

you be doing if you werent doing math?

Dean Menezes

January 19, 2015

Mathematicians sometimes ask each other What would you be doing if youwerent doing math? While this question is relatively easy to answer, it is moredifficult to write in formal logic.

Assuming that one has a monadic predicate Doing that returns true iff itsargument is something that intended audience of the question What would yoube doing if you werent doing math? is doing, one might think that this sentenceis formalized as Name an activity x so that Doing(math) = Doing(x);however, this is incorrect because since the intended audience of the question isin fact doing math, the above formula is true regardless of what x is. One mightthen attempt to resolve this problem by using the necessity operator, ; thusone instead asks Find an activity x so that (Doing(math) = Doing(x)).However, this formalization still fails to capture what mathematicians meanwhen they say What would you be doing if you werent doing math?; if amathematician received the answer Computer science to the question Whatwould you be doing if you werent doing math?, a natural question to ask isWhat would you be doing if you werent doing math or computer science?,which if formalized according to the same scheme, would be writen as Findan activity x so that ((Doing(math) Doing(C.S.)) = Doing(x)). Butthe response of computer science to what would you be doing if you werentdoing math means that (Doing(math) Doing(C.S.)); hence the antecedentof the mathematicians second question is necesarily false, and any activity xshould work as an answer.

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