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Researcher David Tall has written in chapter one of Advanced Mathematical Thinking that

Mathematicians often regard the terms “intuition” and “rigour” as being mutually exclusive by suggesting that an “intuitive” explanation is one that necessarily lacks rigour.

I can understand how some might think this, but I would like to see some reference(s) that show that mathematicians indeed view that intuition and rigor are opposites.

So this is just a reference request. I am not trying to start a discussion here.

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This and this are somewhat related (to your question). –  Git Gud Mar 23 '13 at 19:56
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The answer to your question is: The claim by David Tall is false. –  user17762 Mar 23 '13 at 19:59
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Just putting this out there - the thing that makes an explanation intuitive is that although its missing a lot of details, yet it still seems to make sense. There is often a feeling that you can fill in the details your self; but sometimes, this feeling is illusory. –  goblin Mar 23 '13 at 20:06
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Typically intuition trades detail, rigor and certainty out for efficiency, inspiration and elevated perspective. I think this is an observation rather than a definition. –  anon Mar 23 '13 at 20:45

2 Answers 2

up vote 6 down vote accepted

I completely disagree with the claim that intuition and rigor are mutually exclusive. Perhaps somewhat inaccurate, one can say that every mathematical theory is aiming to model something. Calculus models (aspects of) continuity and differentiability, group theory models symmetry, set theory models discrete collections, category theory models analogies etc. (please do not take this literally, I know there is great inaccuracy in these stipulations).

The distinction between intuition and rigor, to me, is the difference between working on the level of what is being modelled vs. working with the model. Most mathematicians probably make use of both intuitive as well as rigorous arguments pretty much all the time.

Here are some examples. In Calculus, one proves that a limit of a sequence, if it exists, is unique. This is an intuitive result and can be explained by saying "if the elements in the sequence get very close to the value $L$ and to the value $M$ then (clearly) $M=L$, so the limit is unique". Using the $\epsilon -\delta$ definition of limit this claim be be proven rigorously. The intuition and the proof are not exclusive at all. The intuitive explanation does not lack rigor. It is not a proof since it is not trying to be a proof, but it does contain the essence of the result.

Consider the statement "if the limit of a sequence is $0$, then from some index onwards all elements in the sequence are infinitesimally close to $0$". This is a false claim of course, but can be given an intuitive argument to make it sound plausible: "since the elements in the sequence get closer and closer to $0$, at $\infty$ the elements will be as close as we like to $0$." This argument can't be turned into a rigorous argument using the standard Cauchy notion of limit and the real number system. But, using nonstandard models of analysis this is actually a true statement.

So, when trying to model a phenomenon (e.g., convergence), one can formalize the notion in different ways, thus obtaining different models for the phenomenon. Then, some things that are intuitively true about that the phenomenon may or may not be true statement in the model. The interplay between the phenomenon (intuition) and the model (rigor) comprises quite a lot of what mathematics is.

It is customary to portray mathematics as if it is the art of deducing new results from old in an axiomatic fashion. But what about choosing the axioms? tweaking existing axioms? Why don't we just wake up one morning, choose a few random axioms, and explore the consequences of the axioms? Well, since it's pointless. It lacks intuition. So, I actually find myself at the opposite to the claim that intuition lacks rigor. Rather, rigor without intuition lacks sense. While quite often mathematicians report their results very rigorously, the intuition is always there.

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Too long for a comment:

I find the whole quote odd: I don't think intuition and rigour are mutually exclusive anymore than I find tacos and hamburgers mutually exclusive (?!), yet "an intuitive explanation" is one that, unless I'm missing some meaning of the words, lacks the rigour we're used to in mathematics.

Nothing wrong with that, though...as long as either some kind of intuition is specifically allowed somewhere (say, to show how two topological spaces are homeomorphic without actually giving a homeomorphism between them...if this is allowed), or else the intuitive explanation is eventually crowned with full honours by a mathematical demonstration, with all due rigour.

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