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Describe the shape of the object formed by the intersection of three long identical cylinders, one running down the x-axis, one along the y-axis, and one along the z-axis. He suggested that some people topologists, if i remember correctly have an uncanny ability to answer this immediately and correctly. You can get more and more intution in the areas which are in your wheelhouse by spending more time working in and thinking about the area.

Which you are likely to do, because presumably you enjoy thinking about those things. To go to one of your other points, I think a good example is logical intuition. You need a proof. One of the postdocs who was there made the point to me that that really meant that he had dealt with the problem so much and from different angles [which in light of his presentation is a bad pun] that he had a deep and broad understanding of the problem outside of the rigorous results. That had a huge formative effect on me.

I prefer to think of it has having a well-trained aesthetic. A very well trained and very focused athlete will certainly have a good performance. We may call him a disciplined athlete. I may relate this discussion somehow to the range of psychological theories. Some tend to believe that mind abilities are innate Innatism , some believe they are just the result of live experiences Empiricism.

Intuition in Science and Mathematics: An Educational Approach by H. Fischbein -

I am not a psycologist, so some terms may not be precise. Otherwise though it is all very well explained. As for intuition: while mathematicians do develop deeper insights into problems they have put a significant amount of thought into, I think a lot of intuition is a more basic kind of Pavlovian association. Of course, most of us only have this trained response to a limited number of stimuli, hence the specialised nature of the intuition.

See for instance the start of this talk:. For an example, which I hope can be appreciated even by people unfamiliar with the area: Alex Russell and I, and many other people working in quantum computation, deal a lot with irreducible representations of finite groups. This is the same answer you would get if rho g were a uniformly random d-dimensional unitary matrix, which rotated this d-dimensional space in a random way.

For experts, if it were uniform in the Haar measure. It has recently been proved in many cases by Green and Tao but not for twin primes. I am not able to post a direct quote at this moment. It comes in the form of an experiment done with savants.

Using Multiple Representations

For example, a savant might be asked to the multiplication of large number x and large number y. In the experiment, savants were timed on a series of calculations of growing size. Of course the mind is much more complicated than this analogy, which only captures some aspects of its effects. I would be interested in comments on the role of intuition, not in proof- proving , but in proof- checking. It is clear that human proof-checking cannot be a wholly logical process, in which for example a mathematician converts the proof into some internal, rigorous, formal representation.

If that were true, proof-checking could be readily automated … which it has not been. So what is the role of intuition in proof-checking? A mathematical book titled The Art of Proof-Checking would perhaps be a very interesting book, both to read and to write. Let alone the situation when he enters Minimally Supersymmetric Standard Model. None the less, since there is also some kinds of renormalization groups involved with SM and MSSM too, it might be wise to bring up the problem of finding the complexity in polynomial time of the Galois group, solvable or not, of a given polynomial.

At the time of Galois this probably did not make sense. But I have seen algorithmic book on it after I mentioned it in the presence of a friend some ten years ago or so. I have been out of touch with the advanced subject for some time apparently.

It was most probably totally random that I came up with that algorithmic Galois group book. I just went after anything that had Galois in it like Galois cohomology, Galois theory, Galois motives, and so on. But I was truly flabergasted to see that the idea that I mentioned to a couple of math profs at tea time actually had applicatoins. Actually, that was becasue I was so stupid at Fort Collins to find the particular Galois groups of given polynomials to the extent that I was called to be disgusting.

Or was it that I got stuck at a paper of Hochschild on local Galois theory that he called me so. Sometines, a savant can figure out a difficult problem that no one else can, but sometimes one does not even know how to use these huge supercomputers to pose a difficult problem. Thanks God, but I also go nuts when I still wonder about the missed opportunities.

Of course, one could always try politics or military instead which make science more glamorous exactly because not everything in life is math and physics. It is better to go on the market and buy what engineers came up with possibly remotely using the efforts of mathematical physicists. That way the mathematician can sit home and think about his math or go on the market to buy what he need at home for a better mathematical intuition. I never doubted the ethics of someone like Bombieri or Miranda as mathematicians.

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John would be convinced that I am babbling too much by now. But sometines one is forced to babble as one is left with no other choices. Intuition helps! I suspect that we form a large series of models internally as we learn things. We build up them from experience and education. In that sense they can be both overlapping and contradictory.

Perhaps intuition is just our ability to utilize some of our rougher internal models to help explain our more complete ones. I can write out projections of it in many different ways, but my overall picture defies a single syntactic representation but is extremely useful for making sure all of the pieces are complete and behaving as necessary.

Of course intuition is incredibly messy and contradictory yet it captures some similarities, analogies, salient features, correlations? This is why mathematicians will have a very hard time finding out how their intuition work, they will recoil in horror at the first glimpse of this reality. Incomplete in parts, it is a vague outline of the reality around us. Some people perceive it as being more concrete and complete than this, in the same way that we see the objects around us as being continuous. Interesting post, I may subscribe to this blog.

Existence of the law of reciprocity, which he stated in his letter. Existence of iterative set structure, which is his conception of set theory. Existence of bidirectional reducibility, which he did not state in the letter. The central question is how to apply his iterative conception of set to realize his bidirectional reducibility. It is true that the theorem is proved at last, but there seems to be much more to it in physical or geometric terms in the same spirit that Alan Connes related noncommutative geometry and number theory via Riemann Hypothesis.

The complexity raises when a set of equations as in FLT are singled out and its decidability is questioned. Fortunately FLT was finally solved for the case of integers. Perhaps a better way to say it: Proofs can be checked for correctness with negligible probability of error by checking only a constant number of bits. Has this been done for the 4 colors theorem? Intuition has been a topic of my interest for the past 4 years.

Science and Intuition

The book highlights marvels of expert intuition. My favorite reference on this topic is a lecture by Prof. Daniel Kanheman Nobel of Princeton at Berkeley. I have written a blog on this lecture as: My most favorite YouTube video. Leading up to that kind of new idea there has to be a long period of tremendous focus on the problem without any distraction. You have to think about nothing but that problem — just concentrate on it. Then you stop. I totally disagree with Von Neumann.

Intuition belongs to the realm of spirituality. More specifically, psychic development. Your questions: it is not built up and more learning can sometimes be a hindrance: a pure mind, uncluttered, after a much needed break always proves beneficial. Most scientific discoveries happen this way. Yes it is separate and belongs to a complex realm involving the pineal gland and other outer and environmental factors including health.

Yes, it is possible to be very strong in an area and have extremely low intuition, because they are unrelated.

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The most important considerations are: 1 intuition is not a creative process within the brain that receates old hidden information in the subconscious. Have you ever telephoned a loved one repeatedly engaged that they have been calling you simultaneously. Psychics have to spend many years testing these assumptions, but many do not. A medical clairvoyant being paid as a hospital assistant is more trustworthy than a tarot card reader in a rough part of town.

And universities traditionally focus primarily on the left hemisphere. All joking aside… I am highly trained from trusted sources in intuition so if you like to chat more… edwardtagg yahoo. Although we are hard-wired to map the world around us with a reasonable sense of intuition shared by us all, we have also been evolved to manipulate our intuition and accept new surprising facts that are objectively proven or accepted by consensus.

We even fake our memories of our experiences and remember them in a way they make more sense to us. Perhaps anything that holds yet we have not experienced it collectively over the generations can suprise us. May be there are many types of mathematical intuition. Mathematicians often have computed lots of concrete examples before proposing a conjecture by intuition, and searching a proof of it guided by intuition.

Upon the results of these studies, an algorithm be given which shows how to search a proof of statement in first order logic from finite concrete examples, and an approach be proposed to improve searching mathematical proof by neural network. You are commenting using your WordPress.

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Blog at WordPress. October 1, You just get used to them. John von Neumann Number Theory I think that it is not too hard to have a reasonable intuition in number theory, especially concerning the behavior of prime numbers. If primes are random, then one would expect that there are about twin primes in where is a constant. They guess that is Of course there are local properties that need to be added to the model. The number of primes so that and are all primes is not but one. He looked at me and said: Of course the lemma about primes is true, but it is completely hopeless to prove it.

Geometry My intuition in geometry, especially high dimension, is very poor. Groups My intuition about finite groups is even worse than my geometric intuition. The reason for the latter is that where , since is in the kernel. Since and are in , and commute. Hence and can generate at most different elements. Since is in , it follows that divides , which divides. However, since divides which is relatively prime to , the only way this can happen is.

That means , so since is the kernel of. But is normal in , not just in , so is in. This puts into , which yields a contradiction. Complexity Theory I think we have good intuition here, but I have seen many surprises in my career: Linear Programming is in polynomial time. Nondeterministic space is closed under complement. Polynomial size bounded branching programs are powerful. Permanent has random self-reduction. Quantum polynomial time can factor integers. De-randomization of random walks for undirected graphs.

Proofs can be checked in constant time. The existence of Zero-Knowledge protocols. Open Problems Is intuition simply built up by learning more and more about an area? Share this: Reddit Facebook. Like this: Like Loading John Moeller permalink. Kent Overstreet permalink. Faruk Mustafic permalink. Carlo permalink. Colin Reid permalink. Carsten Milkau permalink.

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Cristopher Moore permalink. Most of the existing monographs in the field of intuition are mainly concerned with theoretical debates - definitions, philosophical attitudes, historical considerations. See, especially the works of Wild , of Bunge 1 and of Noddings and Shore 1 But, so far, no attempt has been made to identify systematically those findings, spread throughout the research literature, which could contribute to the deciphering of the mechanisms of intuition.

Very often the relevant studies do not refer explicitly to intuition. Even when this term is used it occurs, usually, as a self-evident, common sense term. JavaScript is currently disabled, this site works much better if you enable JavaScript in your browser. Mathematics Education Library Free Preview. Buy eBook. Buy Hardcover. Buy Softcover. FAQ Policy. About this book In writing the present book I have had in mind the following objectives: - To propose a theoretical, comprehensive view of the domain of intuition.

Show all. Table of contents 18 chapters Table of contents 18 chapters Intuition and the Need for Certitude Pages Intuition and Mathematical Reasoning Pages Investigations in Overconfidence Pages General Characteristics of Intuitive Cognitions Pages The Classification of Intuitions Pages