# [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
time.

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).

Tim
```