Hi Michael,

"If I understood you correctly, you said the term "real" has no meaning in mathematics."

You understood me correctly, but I should have been more specific about the connotation of 'real' that I intended. Mathematics certainly uses the term 'real' as in "real numbers". But, as I am sure you are aware, this is simply a four-letter tag used to refer to some precisely defined abstract objects and has nothing to do with what we refer to in the vernacular as "reality".

"...if you were correct that would imply mathematics can't be used to make meaningful statements about reality. Is that correct, or is it something Wittgenstein would not like? "

I don't know how Wittgenstein would feel about this question, but I would say, Yes, it is correct. As long as you stay within the domain of mathematics, none of the statements within that domain say anything meaningful about reality.

The usefulness of mathematics to human affairs comes from someone outside the domain of mathematics noticing that some of the concepts and structures of mathematics bear a resemblance to some real things. In this case, an isomorphism might be set up between the real objects and the mathematical objects. Then the mathematical implications of the mathematical objects might be used to predict consequences of the real objects by means of the isomorphism. To the degree that this is successful, mathematics proves to be of value in the real world.

The setting up of the isomorphism and the predictions of real behavior is science, not mathematics. Even though the same person might be doing both the math and the science, as in the case of Maxwell and others, there is still a distinct line separating the two. Mathematics does not deal with, or comment on, anything real. IMHO

Warm regards,

Paul