<^> Try for dynamic balance, between exploring your external and internal state
. . . . . . ( integrate what is learned )

<^> Feel something's missing, or found an anomaly ? . . It's either an error or a goldmine.
. . . . Look for the clue (le clef) closeby, between disciplines. . . few people look there. (nih)
. . . . . Don't pass it by in your great hurry....("haastigen spoed is zelden goed")

<^> . . . Rigor is secondary, . . and prevent rigor mortis ("Operation successful, patient/math died").

Jim Trek  wrote (news:sci.math 6feb99):
>
> [...]  Mathematics is part of the way we perceive.
> A formal system need not rely on assumptions, axioms or
> postulates (they are all the same thing), unless you want the
> spooky results and the artificial esoterica along with the good
> stuff.  Proving mathematical statements requires definitions, but
> does not require assumptions of any kind.  This method provides
> mathematics with the surest foundation.  The process begins with
> those meanings which precede the integers and are essential to
> creating them.
---

Steve Leibel  answered:

> Agreed, the attempt to create formal systems based on axioms has led
> to big trouble.  For example we all share a common intuition and
> experience about things called points, lines, planes, and solids.
> So far so good. Then some troublemaker named Euclid wrote down some
> axioms and attempted to derive everything else as logical
> consequences of the axioms.
> Went pretty well for a couple thousand years till some other
> troublemakers started playing around with the parallel postulate,
> and voila! Non-Euclidean geometry, [...]
             ^^^^^^^^^^^^^^^^^^^^^^
Nico B:

  No trouble at all: it's surprising it took those formal boys so long,
after everybody noticed the Earth was not flat but a ball: geometry
on a sphere has other rules (parallels meet at the North & Southpole).
Nothing spectacular about that. The *main* problem was that such
common sense sphere geometry had to *break-through* the prejudice
that _all_ geometries should be flat, like Euclid's ;-)
In fact, thinking about it, such rigid math-attitudes really block,
or at least delay considerably, normal developments - building_in
an extra inertia and giving formalists a stick to beat explorers with.

> ..., general relativity, and cultural relativism,
> all in the span of a  mere 150 years. [...]
>
> But the moment you try to create a formal system,
> you wind up in Godel-land.
                 ^^^^^^^^^^
Nothing wrong with that: his incompleteness result is very much common
sense - and way overdue. Just saying that any set of axioms defines
a closure of theorems that is not complete: hence leaves some space
around it undefined. What is more reasonable than that? Allowing
'your' system to grow, by adding another axiom, to explain more, but
again not everything ;-)  Healthy antidote for those naive boys with
their "Theory of Everything".
How about the (wise) contradictio-in-terms: "Moderation in everything".

Axioms & sharp definitions serve a purpose of realizing the limits
of your system. The problem comes in when you think they are
*everything* -- Math is only a tiny corner of human activity,
namely the formal little corner of those seeking certainty -
explaining psychologically their fearsome reactions against changing
the established rules [be they religious or mathematical] -(ever read
about the problems of Galileo, Kepler, Fourier, Galois, Cantor, etc.?)