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Comment by deanmen | Hacker News Reader

The Oxford dictionary "intuition" as "immediate apprehension by the mind, without the need for reasoning"

When people emphasise intuition in mathematics it suggests that

(i) that they have never tried to teach mathematics formally and explicitly without appeal to intuition —if they had, it would have been a most refreshing experience for them and for those of their students that were sufficiently well-educated to appreciate simplification

(ii) that they have never taken the trouble to learn how to let the rules of the formal game guide their “writing of papers” —if they had, they would have discovered how to do mathematics way beyond their former powers.

EW Dijkstra

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L.E.J. Brouwer emphasized that he regarded that some mental procedures are either language-less or pre-liguistic. Intuition is an imprecisely defined notion but I think the main thing to understand is that immediacy is a key component. If we grasp something by intuition there are no steps involved. Also, I don't think that apprehending something by intuition means proving it, proofs always involve rules and mental (or machinic) operations, in this sense knowing something by intuition is not proving that thing in the formal (or informal) sense of the word.

Given all that I think the Oxford dictionary's definition is actually quite spot on. Are you quibbling with it? Happy to hear what you think.

I can post links to Brouwer's writing on these matters.

I really wish the "immediacy" aspect were more visibly discussed. It's a great core to the notion of intuition---it also helps separate intuition of a concept from intuition guiding a search for meaning.

As I understand, it's not immediate. It's just our conscious mind suddenly being alerted to it, which leads to the illusion of immediacy. Probably very little of our thoughts are conscious.

In this sense "immediate" means that the reader "grasps" the evidence in a single step---it is basal, axiomatic [0], or assumed. From a Brouwerian POV certain things are axiomatic (continuity and choice sequences are his examples) and must merely be known implicitly to the observer.

[0] This word causes other issues, but the, ha, intuition of it is not so bad

The term is also used to describe some people's ability to grok simple arguments without needing to fall back on formal methods. A trivial example might be the pigeon hole principle where most people can just "see" that it is true, even though it does require proof.

Funnily enough, I was just thinking about this very topic a week ago and scribbled down some notes with slightly less trivial examples: http://www.maths.tcd.ie/~onash/counting_whats_important_file...

With all due respect to Dijkstra, this is so narrow-minded. I love intuition in mathematics, primarily coming from spatial or geometric reasoning. Then it's time to put it into algebraic or symbolic terms, but that's pinning a butterfly to a board to show you actually caught it.

I like point (ii) above, as learning to let the rules of the game guide your writing is a very powerful technique indeed. But teaching math formally without appeal to intuition is what a lot of people do and it's boring and unpleasant to me. It doesn't simplify, it merely traces lines in the sand or on a page without taking the trouble to look up and appreciate the larger picture (sometimes meaning literal picture).

It's clear that Dijkstra & I appreciate very different parts of mathematics :)

Mathematics is defined by its axioms and the axioms are derived purely from intuition. Therefore mathematics is ultimately 100% intuitive, if you don't agree then you haven't tamed your intuition.

If formally precise reasoning would be all there is to it, computers could do all math for us. However, even in that formal-if-anyhing-is area, they fail. Computers can't solve instances of the Halting problem or the Tiling problem, whose solution is immediately intuitively clear to a human. Those solution can also be immediately verified by fellow humans, via the informal communication method of 'speech'. The reasoning involved can't be verified by computers.

A nice overview was given by Peter of Conscious Entities recently: http://www.consciousentities.com/?p=1918.

> Those solution[s] can also be immediately verified by fellow humans, via the informal communication method of 'speech'. The reasoning involved can't be verified by computers.

This isn't a particularly strong argument, as solutions verified via the informal communication method of 'speech' often pass verification without being correct.

Yes, humans are capable of doing math incorrectly. That is irrelevant to an argument explaining that computers are incapable of doing some math at all.

> Computers can't solve instances of the Halting problem or the Tiling problem, whose solution is immediately intuitively clear to a human.

This is unclear. Do you mean that computers can't solve the Halting Problem, but humans can solve many instances of the question "will this question halt"? Then you're not comparing apples with apples. You should instead compare human reasoning of "this program will halt or not" with programs developed in the field of termination analysis. Are these analysis bad compared to human ones?

Or do you mean that humans can prove the undecidability of the Halting Problem, but computers can't manage to prove or disprove it, or understand the human proofs? Does this have to do with such proofs tend to be non-constructive, as far as I've seen?

In any case, I guess it's still an open problem whether or not the human mind can algorithmically solve the Halting Problem, right? Our minds would have to be "better" than a TM.

Yes, the latter is exactly the open question. It's one that has had us stumped for decades and on which barely any progress has been made. There are simple examples of purely mathematical problems that we can't seem to teach computers how to solve, even though they are trivial for humans. Of course it's entirely possible we will find a way to teach machines how to solve them. It would be even more interesting if some AI developed the ability to solve them and could subsequently explain how it does it.

Actually computers can solve many instances of the halting problem. http://en.wikipedia.org/wiki/Microsoft_Terminator

Yes. The point is there are many instances that humans can solve, but computers can't. My sentence was not meant to imply computers cannot solve any instances of the halting problem at all.

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