True, of WHAT? The fifth postulate is a defining characteristic of "flat"
space. If you mean true of the space of modern physics, then, no, I don't
think it's true.
>Assuming "the introduction of something not present in Euclidian geometry"
>is adding to it and the 5th postulate CANNOT be proven from the other
>postulates. (It's never been proven, and the independence of it has been
>proven; I'll give some names for backup if you wish)
I agree that it's never been proven from the other postulates, and that it
is not even provable from them, and I agree that it is independent of them.
But, for Euclidean geometry, the question is whether we can establish that
it is true for "flat" space. To do this, we have to go outside the other
postulates and consider what it is for space to be "flat." However, my point
was that the other two main forms of geometry implicitly rest on a geometry
of curved surfaces within a flat space. That is, the "plane" versions of
both of these are really part of SOLID geometry, because such "planes" are
three dimensional (not in thickness, but in the sense in which a balloon is
three-dimensional even though, in any tiny area of it, it approaches being
two-dimensional because it is so thin. Such planes, if placed on a flat
surface, extend above the surface, just as a negatively curved potato chip
or a positively curved bowl does (and would, even if the material had no
thickness at all).
>And geometry most certainly IS a branch of math. Trust me, I'm a math major
>and I just finished a math history class where we investigated mathematical
>roots and it's development. If you wish, I'll ask my professors if geometry
>is math or physics, and see what they say.
I was a math major, too. It's normally thought of that way for historical
reasons, and because it is so closely tied to mathematics, and because of
the bad philosophy that I mentioned. You indicate part this even in your own
remark about the "mathematical roots and it's development." The development
of mathematics (AND geometry) is a part of HISTORY, not of mathematics per
se. As to what your professors will say, go ahead and ask. But, unless they
are VERY unusual, they will say essentially what you say. That doesn't make
them right. Astronomy was once blended with astrology. Does that make
astronomy a branch of astrology? Does it make astrology a branch of
astronomy? No, neither. Sciences need to be divided up on cognitive grounds,
not on the basis of the accidents of history and opinion.
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.
>>But what I wanted to remark on, mostly, was the idea that "Math has >no
>>solid basis now." This is not true. Math has a solid basis, but >many (if
>>not most) people don't know what it is because of >confusions caused by
bad
>>philosophy, especially bad epistemology. >The basis for mathematics is
>>logic and the fact that one thing is >more than no thing, or: 0 < 1. The
>>rest of mathematics is the >working out of the implications of this fact.
>Try reading Godel. He proved that no matter how many axioms you start with,
>there will always be statements within the system governed by those axioms
>that cannot be either proven or disproven. The second axiom he proved is
>that the consistency of any system of axioms is one of those improvable
>statements. You cannot prove math, you can only prove mathematics based on
>your assumptions.
>
>>Okay, I may exaggerate a little, but you get the idea: The >foundation of
>>mathematics is no big mystery, unless it's a mystery >that one apple is
>>more apples than no apples at all. That this is >true doesn't seem
>>mysterious to me.
>The foundation of mathematics doesn't exist. Again, read Godel, and maybe
>some Hilbert to see how his quest to provide a foundation for math failed
>miserably. It simply cannot be done.
It can't be done by THEIR approach(es). But who said we had to use their
approach? Godel was brilliant, as was Hilbert (though I'd rate Godel higher
than Hilbert, I think). But, ask yourself: What did they MISS? What was
wrong with their approach? WHY didn't it work? What other approaches might
work? Why do we need mathematics at all? Etc., etc., etc. A LOT of work has
been done since Godel and Hilbert, much of it mining out the veins they
found, but some of it elsewhere.
If you're going to give up every time some authority says something can't be
done, YOU'RE done. You may as well go home and watch soap operas. Even if
they are right, they may be right in a way that is much more limited than
you think. Such claims should always be prefaced with: "Given the
presuppositions we had, X can't be done." But, what if those presuppositions
are not correct? History, even the history of logic and mathematics, is full
of cases where people said something could not be done that later WAS done.
Planes DO fly. Logic (at least) DOES have a foundation. Imaginary numbers
are now accepted as routine, etc. Mathematics has a foundation, too, though
it may not even look at first like a foundation for mathematics. One problem
is that the foundations of mathematics is not a mathematical issue, but a
philosophical one. With the exception of philosophy, the foundations of any
discipline like mathematics are determined from outside the discipline
itself. Mathematics cannot determine its own foundation without begging the
question. Determining the foundations for a science is a question that falls
outside the science. Philosophy can seek (and perhaps discover and validate)
its own foundations because that sort of foundation-finding is part of
philosophy's normal business. Mathematics is not a science of discovering
foundations except for propositions WITHIN mathematics, and it uses the
foundation of mathematics as the ultimate basis for doing this.
I'd go deeper into all this, but I'm tired right now, and I need to get on
to other things. However, this may be enough to get you started. Also, a few
years ago I came across a series of articles developing a foundation for
mathematics. If I can locate it, I'll check around and see if there's a
source available to you.
>If you want more information, I could refer you to my professor, who knows
>the course materials far better than I (obviously). Thanks,
No thanks, unless he has some interesting stuff on infinity (my current area
of math-related study).
Chris