
Intuition and Logic in Mathematics (1905) - karamazov
http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_Intuition.html
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stiff
Richard Courant said it nicely in "What is mathematics?", my all time
favourite book:

 _There seems to be a great danger in the prevailing overemphasis on the
deductive-postulational character of mathematics. True, the element of
constructive invention, of directing and motivating intuition, is apt to elude
a simple philosophical formulation; but it remains the core of any
mathematical achievement, even in the most abstract fields. If the
crystallized deductive form is the goal, intuition and construction are at
least the driving forces. A serious threat to the very life of science is
implied in the assertion that mathematics is nothing but a system of
conclusions drawn from definitions and postulates that must be consistent but
otherwise may be created by the free will of the mathematician. If this
description were accurate, mathematics could not attract any intelligent
person. It would be a game with definitions, rules, and syllogisms, without
motive or goal. The notion that the intellect can create meaningful
postulational systems at its whim is a deceptive half-truth. Only under the
discipline of responsibility to the organic whole, only guided by intrinsic
necessity, can the free mind achieve results of scientific value…To establish
once again an organic union between pure and applied science and a sound
balance between abstract generality and colourful individuality may well be
the paramount task of mathematics in the immediate future._

~~~
Arun2009
Here's an opinion by Terence Tao along the same lines that I found insightful:

[http://terrytao.wordpress.com/career-
advice/there%E2%80%99s-...](http://terrytao.wordpress.com/career-
advice/there%E2%80%99s-more-to-mathematics-than-rigour-and-proofs/)

 _The “post-rigorous” stage, in which one has grown comfortable with all the
rigorous foundations of one’s chosen field, and is now ready to revisit and
refine one’s pre-rigorous intuition on the subject, but this time with the
intuition solidly buttressed by rigorous theory. (For instance, in this stage
one would be able to quickly and accurately perform computations in vector
calculus by using analogies with scalar calculus, or informal and semi-
rigorous use of infinitesimals, big-O notation, and so forth, and be able to
convert all such calculations into a rigorous argument whenever required.) The
emphasis is now on applications, intuition, and the “big picture”. This stage
usually occupies the late graduate years and beyond._

------
mikhailfranco
Another mathematician basis function from Freeman Dyson, _Birds & Frogs_:

[http://www.ams.org/notices/200902/rtx090200212p.pdf](http://www.ams.org/notices/200902/rtx090200212p.pdf)

Dyson contrasts the ADHD of the multi-disciplinary cross-fertilizing polymath,
with the meticulous obsessive immersion of the specialist.

There is also the Kuhnian distinction between _revolutionary_ and _ordinary_
progress: the leap of the system creator compared to the incremental steps of
the system extender.

I think it is very relevant that the intuitionist geometric thinkers are often
the mathematical physicists. They have a vivid kinaesthetic imagination, where
mathematical entities have a visual appearance, complete with spatial
properties and forceful interaction. Sometimes these take the form of
practical _thought experiments_ , which open a problem to precise mental
tests. But they can also involve more abstract explorations, like the slow
precession of a mobile sculpture, such that alignments and symmetries become
manifest: not yet understood, but actually there for examination.

A list of the greats would include Newton, Gauss, Riemann, Poincaré himself,
Einstein and Feynman. Many of these used spatial representations to inspire
algebraic progress. A more recent example (although necessarily on a lower
tier) would be Penrose.

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dnc
It seems to me that the absolute intuitionist and absolute logician can not
exist as real persons. I see them rather as two extremes that are always mixed
in some proportion (and always in fight with each other for more space, so the
proportion changes over time). For instance, Euclid is mentioned as logician,
but at least one of his axioms (about parrallel lines that could not
intersect) is based on intuition that, turns out, is not always "true".
Therefore we have got other Non-Euclidian geometries.

~~~
thetwiceler
Actually, this example makes Euclid even more of a logician. Recall that
Euclid stated his Fifth Postulate (where his postulates really mean axioms).
Geometers for thousands of years afterward, guided by intuition, attempted to
prove that in fact the Fifth Postulate could be proved from the other axioms.
But by the 19th century, it was actually clear that Euclid was correct to
include the postulate; had he not included the postulate, his geometry would
not convey what he wanted to convey. Euclid in this way was fundamentally a
logician! He realized that he needed the Fifth Postulate to prove facts about
a geometry he imagine (e.g. angles in a triangle sum to two right angles), and
included as an axiom something that others intuited must be derivable from the
others.

But as Poincare mentions, we can also see that Euclid DID have much
"intuition" in his works. In modern days, we do not consider Euclid's Elements
a rigorous work of logic, mainly because his definitions are not rigorous
definitions; he says a point is "that which has no part." Hilbert remedied
Euclidean geometry in the 20th century, with a work that has some undefined
objects (points and lines) and precisely defines the rest. He needs something
like 24 axioms, as opposed to Euclid's 5.

It's interesting to see where Euclid's logic breaks down. Look at his very
first proof - the construction of an equilateral triangle. He constructs two
circles and looks at their intersection. How do we know we can build the
circles to intersect? He draws us a picture and it seems obvious :).

But in Hilbert's Euclidean geometry, we need what I think of as a really nasty
axiom in order to ensure the circles intersect: "Axiom of completeness. To a
system of points, straight lines, and planes, it is impossible to add other
elements in such a manner that the system thus generalized shall form a new
geometry obeying all of the five groups of axioms. In other words, the
elements of geometry form a system which is not susceptible of extension, if
we regard the five groups of axioms as valid."

~~~
dnc
Thanks for pointing out this.

------
clementi
Here's a more readable version:
[http://www.scribd.com/doc/150574528/Intuition-and-Logic-
in-M...](http://www.scribd.com/doc/150574528/Intuition-and-Logic-in-
Mathematics)

------
wslh
This shows how badly Wikipedia fails. There is not reference there (in the
Poincare page) to this translation.

~~~
sillysaurus
_This shows how badly Wikipedia fails. There is not reference there (in the
Poincare page) to this translation._

You could fix that.

