[FOM] Notations in mathematical practic

Timothy Y. Chow tchow at alum.mit.edu
Sun Oct 25 14:26:31 EDT 2015

Arnon Avron wrote:

> In his posting on free logic, Harvey Friedman made
> the following side remark:
> "Mathematicians just want to make sure that there is no practical
> ambiguity in what they write."
> Do they??

I interpreted Friedman to be stating an upper bound rather than a lower 
bound.  That is, mathematicians will *at most* try to eliminate practical 
ambiguity.  But as you note, sometimes they won't even do that.

> Here is a related phenomenon of obvious practical ambiguity. There are 
> many textbooks that in one chapter define a (partial)  function from R 
> to R as a set of ordered pairs which satisfies a certain condition. Then 
> the same books go on and later start the formulation of many theorems 
> and definitions by "let f(x) be a function ..." - as if a function is 
> something that depends on variables, and as if "the function f(x)" is 
> different from "the function f(y)" according to their own definitions.

Yes.  I recall that in Ahlfors's textbook "Complex Analysis," he has a 
footnote that takes note of this problem, but then he says he is going to 
go ahead and say "the function f(x)" anyway.

I recall being confused by this sort of thing as late as when I was taking 
a course in classical Lagrangian mechanics, where I couldn't figure out 
what people meant by a "Langrangian that doesn't depend on time" since 
(for example) the Lagrangian depended on velocity and velocity depended on 

I also remember being confused some years earlier by the notations 
"sin^2(x)" (which is supposed to mean (sin x)^2)) and "sin^(-1)(x)" 
(which is *not* supposed to mean (sin x)^(-1) because that has another 
name---csc x---but is supposed to denote the inverse function).


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