Hose****.

You're kidding right?

Well, first of all, you're mixing terms. "Hypothesis" is a term used
in scientific method, to propose something that is observed, but isn't
proven consistent. It doesn't exist in mathematics; proposals of
mathematic properties are called "theorems". But I set that aside;
this is casual conversation, after all.

I did not mix terms - I used the term that someone else used and asked for
elaboration. Not my confusion.


Bear with me here, everyone. I'm going to make a pretty good point or
two, in my opinion.

Can't wait...


Mathematical fundaments are composed of "Postulates", such as "A point
is defined as a location in space", "A line is defined as the
one-dimensional measure of distance between two points", and, "The
shortest distance between two points is a line".

Those are "postulates", specifically of Euclidean geometry. "Theorems"
arise from logical conclusions of the interactions of the postulates.
The ideas that triangles have certain properties, such as the sum of
their angles equalling pi radians, are "theorems".

Casually, these are sometimes called "laws", as in the "Law of
Cosines". Non-Euclidean geometries, necessary for doing things like
traversing the surface of a sphere (and none of us have *ever* done
that, oh, no!), does *not* have, as a postulate, that the shortest
distance between two points is a straight line; there are *no*
straight lines in spherical geometries.

Um, but the shortest distance between two points is STILL a stright line...
Unfortunately you can't travel through the earth.



For natural philosophers, people like physicists and mathemeticians,
the discovery (or rediscovery) of alternate but valid geometric
rulesets has resulted in several very useful discoveries, one of which
being Einstein's body of thought on relativity, flawed as we now know
it to be (but haven't come up with an all-encompassing replacement).

One other result of the re-examination of Euclidean thinking has been
the formulation of Theorems which deny the principal assumption of
great works like the _Principia Mathematica_, Goedel's Theorem
probably the most popular among them.

The upshot of Goedel's Incompleteness Theorem is mathematical proof
that "any self-consistent axiomatic system powerful enough to describe
integer arithmetic will allow for propositions about integers that can
neither be proven nor disproven from the axioms." [from the Wikipedia
article on Goedel]

Euclidean geometry is more powerful than integer arithmetic.

That is, logical systems powerful enough to be useful will contain
unprovable axioms. So the question, "Which [axiom or theorem] in
mathematics can't be proven or shown false that is the basis for all
other math?" is simply an utterly unanswerable question, given a
powerful enough system. Goedel proved it years ago. What *can* be said
is that "some axioms are unprovable, which doesn't mean they're false
or true."

I asked for which basic tenet was unprovable. My point was that the
original poster of this math == religion thread was not making sense.
There is nothing similar about them. Goedel (and Turing's equivalent with
the halting problem) have nothing to do with this conversation. You still
haven't answered the question - you have just tried to make the whole bit
sound more complicated than it is. And I am sure we are all impressed with
the disussion or Euclid, Theorems, incompleteness, etc.



Mathematics itself is today in a state alongside physics and most
natural science, of great uncertainty about the "Great Unknowables",
therefore, while depending on mathematical fundamentals will be
remarkably and consistently useful (can't compute a weight and balance
and then observe performance, or watch your climb rate go down as
altitude goes up, without noticing that), you just never know if your
system will stand up to new stuff.

Kind of like religion, that way, which works for most people. Until it
doesn't. Except for mine, of course. :-)


I still don't see how that is anything like religion.