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Intuition and Logic in Mathematics (1905) (st-andrews.ac.uk)
72 points by karamazov 1609 days ago | hide | past | web | 10 comments | favorite

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.

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


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.

Another mathematician basis function from Freeman Dyson, Birds & Frogs:


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.

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.

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."

Thanks for pointing out this.

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

Yes there is:

Poincaré's writings in English translation

Popular writings on the philosophy of science:

Poincaré, Henri (1902–1908), The Foundations of Science, New York: Science Press; reprinted in 1921; This book includes the English translations of Science and Hypothesis (1902), The Value of Science (1905), Science and Method (1908).

Go to the archive.org page and you will have a download.

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

You could fix that.

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