[FOM] Notations in mathematical practice

Arnon Avron aa at tau.ac.il
Sun Oct 25 04:21:33 EDT 2015

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??

Well, try the following experiment: ask an ordinary  mathematician 
whether or not the following is an identity:

     f'(x) = f(x)'

My experience is that when I ask this question some professor of math
(with no much background in logic, and sometimes even with) he starred
at me, starred at the formula, and then, after some hesitation, tells
me :"well, it depends on what you mean."...

  And this is the whole point. Both sides of this "identity" are 
terms which are extensively used in courses and textbooks on analysis. 
So one might expect that "mathematicians would want to make sure that 
there is no practical ambiguity" concerning them, and so no student get 
confused because of it.  But the mathematicians do nothing about 
this  - even though the students (and sometimes not only them) do get 
confused,  very much so.

  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.

   Why is this happening? Because when it comes to functions,
mathematicians stick to a system of notation created in the 17th century 
or nearby.  Unfortunately, the correct \lambda notation which they SHOULD use 
was born only in the 20th century. In contrast, set theory and its
system of notations was born in a much more mature time.
So no mathematician that respects himself would talk about 
"the set x^2>0", only about "the set {x|x^2>0}". But the same mathematician
would talk about "the function x^2", not about "the function \lambda x.x^2".

  Is there any chance of changing this? Well, is there any chance that 
Americans will start using the metric system in their daily life?

It is not for me, but for experts in psychology, to explain why
most people enthusiastically adopt new gadgets and 
"applications" they can do very well without - but find it impossible to 
improve the really troublesome systems of notations they inherited 
from their ancestors.

Arnon Avron

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