<snip>
>>The development of mathematics (AND geometry) is a part of HISTORY, >not
of
>>mathematics per se.
>But that also does not mean they are separate. Can you support that claim?
The question is not how to show that they are seperate, but how to show that
they are properly to be classified as parts of the SAME science (without
thereby implying that we should drag other disciplines into mathematics on
the same grounds)?
>>Sciences need to be divided up on cognitive grounds, not on the basis >of
>>the accidents of history and opinion.
>Out of curiosity what cognitive grounds would you use to separate geometry
>from mathematics?
See above. What grounds would you use to group them together?
I have already given some of my grounds, but here's the main one:
Mathematics is basically the formal study of the idea and implications of
the concept of quantity. Geometry is not primarily about quantity; it is
about spatial relationships in an abstract way. Space is one aspect of the
physical universe. Hence my claim that it is part of physics. Mathematics
studies quantity in the abstract, physical or not, etc. Logic, for a similar
type of reason, is also not a branch of mathematics, even though it is as
abstract as mathematics and geometry, but most people grasp that there is a
difference between mathematics and logic.
>>As I said, go ahead and ask your professors, if you want to have
>>"authorities" agree with you. However, I'd prefer that you do more of >you
>>own thinking.
>Which I do, thank you very much. But I'd like to hear more about the
>philosophy you claim is true of geometry without simply accepting it on
YOUR
>authority! The funny thing about appeal to authority is that at some point
>you have trust what you hear, cautiously of course, but no human could do
>all the research that has been done in anything approaching a single
>lifetime. How else can you learn anything you don't actually do in science?
>Read the literature as Kevin keeps saying. :)
I was going to stop above, but here's one more response:
That's true, I admit, even in sciences like mathematics, logic, and
geometry, in which one could, in principle, do all of one's own thinking.
There IS just too much to study and think about, even if we wanted to, in a
human lifetime. I apologize for my bad attitude. Nevertheless, if you are
studying in this area, I urge you not to give in to other people's opinions
(and certainly not mine) without doing your own thinking. The issue of the
foundations of mathematics is narrow enough, (despite the tremendous scope
of mathematics), I think, that one person CAN hope to do work of a
fundamental nature.
I hope to get to the rest soon. What DO your professors say about the
"outing" of geometry from mathematics? Do they count it as mathematics
simply because it's one of the things mathematicians do? Do they count it as
mathematics because it is a formalized area of study? Because it is abstract
(i.e., a geometrical line dispenses with many features of what we'd call a
line in the real world, such as thickness)? Or what?