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[With the advent of consistent non-Euclidean geometries in the 19th century, it was realized that postulates are *not* "truths outside the proof-system", and there are in fact opposing postulates which can still be used as the foundation of consistent, meaningful formal systems.]

Quite a lot for one little word *not*.

Well, in math it's important to be precise...

It's been quite awhile since I've used any non-Euclidean geometry, so please be patient with me here.

Sure.

By *not*, did you mean:

Well, specifically I meant that the particular claim made by the previous poster was false. So in that limited sense, I meant your "a":

a) postulates NEED NOT be "truths outside the proof-system"

However, your questions (and indeed my post to which you are replying) raise larger, more general epistemological issues as well.

The real kicker is that the term "outside the proof-system" sounds meaningful but begins to tie you in knots as you try to get specific with it. Since we're talking about mathematics (and formal logic systems are a subset of mathematics as a whole), talking about "outside" a particular "proof system" is like talking about "outside of addition" -- it really doesn't say anything meaningful.

Particular postulates, proofs, statements, conclusions, etc. are only well-defined *within* a "proof system". And the same goes for "true" and "false" in that context. In mathematical formal systems, there is no such thing as a "larger truth" or a "truth outside the system". There is only "truth *within* system {X}" -- and some things which are "true" in system {X} may be "false" in system {Y}.

Taking the parallel postulate of geometry as an example, it's "true" (by definition, since it's a postulate) in planar (non-Euclidean gemeotry) that parallel lines never intersect. But is that a "larger" truth which is universally true in some sense? No, since as they realized in the 1800's, parallel lines *do* intersect in spherical or hyperbolic geometries (which are both non-Euclidean).

And none of those are "more true" than the others, even though they flatly contradict each other, because all *three* are accurate, "correct" and mathematically consistent systems within their own appropriate contexts (e.g., on the surfaces of planes, spheres, and 3d hyperboles respectively), and *incorrect* in the inappropriate context.

So the following is also correct, in a sense (although the same postulate can be a truth within several *different* proof-systems):

b) postulates are NEVER "truths outside the proof-system" (they are either meaningless, or non sequiturs)

c) the whole idea of universal truths is valid, but much less common than previously assumed

Depends entirely on how you define "universal", "truths", and "valid". ;-)

d) the whole idea of universal truths is a misunderstanding based on a limited philosophical system?

Again, "universal truths" is one of those "everyone knows what I mean" terms that nonetheless start to feel like nailing Jell-o to a wall when you actually begin to try to pin it down. Half the room will agree to a particular meaning, and the other half will object strenuously (*whichever* of dozens of possible ways you try to define it).

One of (hell, *THE*) best books for the layman on these types of issues (as well as many others) is "Godel, Escher, Bach: An Eternal Golden Braid", by Douglas Hofstadter. Truly one of the great books in the history of mankind. And no, I'm not exaggerating. In a step-by-step, understandable, *entertaining* manner, he walks the reader on a grand tour through (using Amazon.com's subject list for the book):

Topics Covered: J.S. Bach, M.C. Escher, Kurt Gödel: biographical information and work, artificial intelligence (AI) history and theories, strange loops and tangled hierarchies, formal and informal systems, number theory, form in mathematics, figure and ground, consistency, completeness, Euclidean and non-Euclidean geometry, recursive structures, theories of meaning, propositional calculus, typographical number theory, Zen and mathematics, levels of description and computers; theory of mind: neurons, minds and thoughts; undecidability; self-reference and self-representation; Turing test for machine intelligence.

And unlike most books on any/all of these topics, the material is not "dumbed down" -- anyone who works through the book will get a *real* appreciation and working understanding of these subjects.

And as the "braid" in the title promises, the author delivers on showing the intimate interconnections between these topics. In a sense, although the book is about *all* those things, it's really all about the *same* thing. And the book itself is so tightly constructed that it sometimes feels that if one sentence were accidentally removed, the whole book would unravel like a snagged tapestry. Even when it seems that the author is starting a new chapter on an entirely different topic, at some point you'll find that your brain suddenly goes, *oooooooohh*...., as you realize that he has managed to weave another "thread" into the very same subject matter that the last chapter covered.

A truly remarkable book, and I'm not the only one to think so. It won the Pulitzer Prize, an amazing achievement for a book in this genre.

Full Disclosure: Just stirring the pot, here.

Glad to help. ;-)

Experimental results are one way of resolving discrepancies between otherwise consistent, but conflicting, models...

Yes, and it's been arguably by far the most successful. It's also probably the closest thing we can ever get to a real "reality-check", since the experimental method (and the scientific method in general) is really just a formalized way of saying, "look, we can sit here and argue this idea all day, but the bottom line is, when we try it out, does it actually *WORK* or not?"