[FOM] real numbers

[Miguel A. Lerma]

[Bill Taylor]
 > What does "real" mean here?  Is this the same sense
 > in which Santa Claus can be said to be real?

[M.A.L.]
The question of whether mathematical objects are "real" is 
probably one for which there is no right answer (another one
that comes to my mind is whether mathematical objects are 
"invented" or "discovered", I guess one can always convincingly 
argue in one way or the other). But I do think that the "reality"
of number 2 is not very different from that of "Santa Claus",
both are intersubjective in the sense that many different subjects
agree in their main characteristics. However mathematical objects
are more than intersubjective, they are also intercultural.
Ancient Greeks did not have a Santa Claus but they still had
the number 2 and agreed with us about its main characteristics.
We even think that if there is a civilization in planet X with 
which we never had contact before, they would also have a number 
2 and their value of pi would be exactly the same as ours.

There are however subtle differences among mathematics in different
cultures. Ancient Greeks identified mathematics with geometry, as 
20th century mathematicians had a tendency to identify mathematics 
with formalism. What we call "irrationality of the square root of 2" 
was known to ancient Greeks as "incommensurability of side and 
diagonal of a square" - not quite the same thing. In my opinion if 
we want to find out the real nature of mathematics we need to go 
beyond the particular foundation we provide for it today and pay 
more attention to the intercultural aspects of it.


Miguel A. Lerma