>>the 5th postulate
>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).
Actually, we discussed exactly that last half of your paragraph in one of my
classes! Yes non-Euclidian geometry depends on the existence of 3-space, but
no, I don't think you simply have to establish whether 'flat' space implies
the 5th postulate. Go further back. You wish to assume flat space, but how
do you prove space is flat? What does it take to prove what you used to
prove the last? Etc... You can go as far back as you wish, and that question
can be eternally asked.
>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.
Speaking of which. Where did you hear of this 'bad philosophy' and is there
any support for it I could look at?
>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?
>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?
>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. :)
>>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.
What do you propose for a different basis for mathematics? What do you think
they missed? I'm curious where these questions come from.
>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.
Of course. But how do you determine when to trust an authority who says
something CAN be done? And there are some things that cannot be done. How do
you separate between "impossibile" and "extremely difficult but doable"?
>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.
But then you implicitly say that philosophy can prove issues (such as math's
basis)? How so?
>Mathematics cannot determine its own foundation without begging the
>question.
Of course.
>Philosophy can seek (and perhaps discover and validate) its own
> >foundations because that sort of foundation-finding is part of
>philosophy's normal business.
It can seek, but can it discover and validate? Can philosophy actually
validate any foundation?
>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.
Found anything?
Time to go,
Jason
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