Re: Do reals really have to be genuine numbers?

Eckard Blumschein wrote:
The reason for doing so was the desire to have analytical solutions to
any mathematical problem, e.g. the value of sin(x) for any value of x.

I am still objecting: It was not allowed to ascribe full civil rights
within the kingdom of numbers ("das volle Buergerrecht im Königreich der
Mathematik") to the reals and to infinity because infinity and
irrational "numbers" are obviously fictitious numbers in the sense,
they cannot be fully represented within any system of numbers.

Accordingly, there is perhaps no benefit from the utopic view that reals
are indeed numbers like the rationals. We can likewise operate with the
reals like symbols that merely denote the task to perform an operation
which cannot be exactly performed directly in numerical terms.

Maybe, I am wrong. Am I? Please furnish evidence.

On the one hand, it could be argued that this is entirely a matter of
taste, and so if one does not wish to regard irrational numbers as true
numbers, one is free to do so.

But clearly a world in which the real number line is complete is a
simpler world to work in for purposes of solving mathematical problems.
Why should everyone who solves a differential equation have to
apologize for using procedures which may involve "fictitious numbers"?

As I pour milk into a glass, it is true that the number of atoms in the
glass is an integer...

except that there is nothing in nature that compels the ratios of the
masses of the different kinds of atom to be rational numbers, and

atoms will, of course, pass by the rim of the glass as I pour, and

the volume of milk, also, depends on electrostatic repulsive forces
between the atoms, and thus the ratio between the volume of milk in two
glasses may be irrational.

John Savard

.

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