[FOM] real numbers

Miguel A. Lerma mlerma at math.northwestern.edu
Sun May 11 15:33:29 EDT 2003

[Hartley Slater]
 > For instance, I can ask my greengrocer for 2 1/2 pounds 
 > of cabbage, but he would be non-plussed if I asked for the complement 
 > of 2 1/2 pounds of cabbage, or for members of 2 1/2 pounds of cabbage.

Of course, 2 1/2 pounds of cabbage is not a number, it is a cabbage.

In the realm of Mathematics there are no cabbages, nor apples, nor bricks, 
nor planets, nor anything "material" at all. Mathematics is concerned only 
with generalities that do not depend on peculiarities of any objects from 
the real world - in other words, "2 cabbages plus 2 cabbages is 4 cabbages" 
is just a property of cabbages, "2 plus 2 is 4" is Mathematics.  Mathematics 
starts where all references to the real world end.

[Hartley Slater]
 > How does my greengrocer enter the sophisticated realm of debates on 
 > the foundations of mathematics?  Because he clearly knows better than 
 > many theorists in this area what sort of thing, in the first place, a 
 > number is: it has to number things. 

But in Mathematics there are not "things" to start with.
The real problem of Foundations is how to build the entire 
building of Mathematics without any material bricks at all.

Of course we can approach the problem of foundations from outside
ala Hersh (Mathematics as a social construct), or ala Lakoff & Nu~nez
(Mathematics as a collection of metaphors), or in many other ways,
all of them interesting and each with its own merits, but the problem 
remains how to provide a self-sufficient foundation for Mathematics 
without any references to the real world to start with.

Miguel A. Lerma

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