有沒有「數學直覺」這種東西？

02-26

如果說這樣的數學直覺，是在大量訓練下的一種快速的條件反射，那麼這種直覺我沒有。

整個高中，我的數學成績在0分和20分之間徘徊。

要知道當年數學的滿分從100到120再到150。

我這根本就是學校雇來墊底的好么！

我這種逗逼怎麼可能上得了大學！

但是，高考的時候，選擇題我全對。選擇題分值90分。

我高考數學成績90分。

就這樣不小心上了大學。

我想我大概有數學的第六感吧。

再次感謝出題的老師！

推薦彭羅斯大神的 A Road To Reality. 裡面有不少對於數學直覺的探討。

直覺啊，我想到的第一個就是統計。

數理統計課本上，經常會有「直覺告訴我們」；

數理老師上課時，會說「靠你們統計的直覺，想一想這個應該是什麼樣的分布？」；

我都覺得是坑爹。

直到某一天，在圖書館找數理統計的習題集時，看到了一本書。恩，就是這本（用的百科的圖片）。

原諒我，我明白了，這個直覺真的不是第六感。好累

。

題主如果對數學也有興趣，可以參考T.Tao 的這篇博客

There』s more to mathematics than rigour and proofs

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

陶講的是數學研究，可能和題主說的解題離得比較遠，我稍微轉化一下論述方式，看能不能回答題主的疑惑。

數學的結構是講求邏輯，嚴謹，嚴格的，但是解決數學問題（做題也好科研也好）需要的則遠不止這些，其中就包括「直覺」，即對於數學問題直觀的理解的想像。做好數學要兩者並重，即培養直覺的同時學會嚴格論述的能力，同時通過嚴格論述的能力修正自己的直覺，強化自己的直覺。

如果題主只是想做好題而不是學好數學，最方便的是多做，多記。中學數學沒有新題，無非是各種題型的轉化和組合，通過細緻的歸納總結，完全可以熟能生巧，不需要特別的直覺。即使天賦上差一點，完全可以以勤補拙。

注意是細緻的歸納總結，不是一味的傻做。比如畫輔助線，你可以找一本有答案的習題集，把需要畫輔助線的題目跳出來，根據答案總結輔助線的類型，適用的情況（條件）。

關於解題，T Tao也有一篇博客

Solving mathematical problems

http://terrytao.wordpress.com/career-advice/solving-mathematical-problems/

牆裡是不是看不到wordpress？一併轉一下吧

Solving mathematical problems
Problem solving, from homework problems to unsolved problems, is certainly an important aspect of mathematics, though definitely not the only one. Later in your research career, you will find that problems are mainly solved by knowledge (ofyour own field and of other fields), experience, patience and hard work; but for the type of problems one sees in school, college or in mathematics competitions one needs a slightly different set of problem solving skills. I do have a book on how to solve mathematical problems at this level; in particular, the first chapter discusses general problem-solving strategies. There are of course several other problem-solving books, such as Polya』s classic 「How to solve it「, which I myself learnt from while competing at the Mathematics Olympiads.
Solving homework problems is an essential component of really learning a mathematical subject – it shows that you can 「walk the walk」 and not just 「talk the talk」, and in particular identifies any specific weaknesses you have with the material. It』s worth persisting in trying to understand how to do these problems, and not just for the immediate goal of getting a good grade; if you have a difficulty with the homework which is not resolved, it is likely to cause you further difficulties later in the course, or in subsequent courses.
I find that 「playing」 with a problem, even after you have solved it, is very helpful for understanding the underlying mechanism of the solution better. For instance, one can try removing some hypotheses, or trying to prove a stronger conclusion. See 「ask yourself dumb questions「.

It』s also best to keep in mind that obtaining a solution is only the short-term goal of solving a mathematical problem. The long-term goal is to increase your understanding of a subject. A good rule of thumb is that if you cannot adequately explain the solution of a problem to a classmate, then you haven』t really understood the solution yourself, and you may need to think about the problem more (for instance, by covering up the solution and trying it again). For related reasons, one should value partial progress on a problem as being a stepping stone to a complete solution (and also as an important way to deepen one』s understanding of the subject).
See also Eric Schechter』s 「Common errors in undergraduate mathematics「. I also have a post on problem solving strategies in real analysis.

There』s more to mathematics than rigour and proofs

The history of every major galactic civilization tends to pass through three distinct and recognizable phases, those of Survival, Inquiry and Sophistication, otherwise known as the How, Why, and Where phases. For instance, the first phase is characterized by the question 『How can we eat?』, the second by the question 『Why do we eat?』 and the third by the question, 『Where shall we have lunch?』(Douglas Adams, 「The Hitchhiker』s Guide to the Galaxy「)

One can roughly divide mathematical education into three stages:

The 「pre-rigorous」 stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years.

The 「rigorous」 stage, in which one is now taught that in order to do maths 「properly」, one needs to work and think in a much more precise and formal manner (e.g. re-doing calculus by using epsilons and deltas all over the place). The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually 「mean」. This stage usually occupies the later undergraduate and early graduate years.

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.

The transition from the first stage to the second is well known to be rather traumatic, with the dreaded 「proof-type questions」 being the bane of many a maths undergraduate. (See also 「There』s more to maths than grades and exams and methods「.) But the transition from the second to the third is equally important, and should not be forgotten.
It is of course vitally important that you know how to think rigorously, as this gives you the discipline to avoid many common errors and purge many misconceptions. Unfortunately, this has the unintended consequence that 「fuzzier」 or 「intuitive」 thinking (such as heuristic reasoning, judicious extrapolation from examples, or analogies with other contexts such as physics) gets deprecated as 「non-rigorous」. All too often, one ends up discarding one』s initial intuition and is only able to process mathematics at a formal level, thus getting stalled at the second stage of one』s mathematical education. (Among other things, this can impact one』s ability to read mathematical papers; an overly literal mindset can lead to 「compilation errors」 when one encounters even a single typo or ambiguity in such a paper.)
The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. It is only with a combination of both rigorous formalism and good intuition that one can tackle complex mathematical problems; one needs the former to correctly deal with the fine details, and the latter to correctly deal with the big picture. Without one or the other, you will spend a lot of time blundering around in the dark (which can be instructive, but is highly inefficient). So once you are fully comfortable with rigorous mathematical thinking, you should revisit your intuitions on the subject and use your new thinking skills to test and refine these intuitions rather than discard them. One way to do this is to ask yourself dumb questions; another is to relearn your field.
The ideal state to reach is when every heuristic argument naturally suggests its rigorous counterpart, and vice versa. Then you will be able to tackle maths problems by using both halves of your brain at once – i.e. the same way you already tackle problems in 「real life」.
See also:

Bill Thurston』s article 「On proof and progress in mathematics「;

Henri Poincare』s 「Intuition and logic in mathematics「;

this speech by Stephen Fry on the analogous phenomenon that there is more to language than grammar and spelling; and

Kohlberg』s stages of moral development (which indicate (among other things) that there is more to morality than customs and social approval).

Added later: It is perhaps worth noting that mathematicians at all three of the above stages of mathematical development can still make formal mistakes in their mathematical writing. However, the nature of these mistakes tends to be rather different, depending on what stage one is at:

Mathematicians at the pre-rigorous stage of development often make formal errors because they are unable to understand how the rigorous mathematical formalism actually works, and are instead applying formal rules or heuristics blindly. It can often be quite difficult for such mathematicians to appreciate and correct these errors even when those errors are explicitly pointed out to them.

Mathematicians at the rigorous stage of development can still make formal errors because they have not yet perfected their formal understanding, or are unable to perform enough 「sanity checks」 against intuition or other rules of thumb to catch, say, a sign error, or a failure to correctly verify a crucial hypothesis in a tool. However, such errors can usually be detected (and often repaired) once they are pointed out to them.

Mathematicians at the post-rigorous stage of development are not infallible, and are still capable of making formal errors in their writing. But this is often because they no longer need the formalism in order to perform high-level mathematical reasoning, and are actually proceeding largely through intuition, which is then translated (possibly incorrectly) into formal mathematical language.

The distinction between the three types of errors can lead to the phenomenon (which can often be quite puzzling to readers at earlier stages of mathematical development) of a mathematical argument by a post-rigorous mathematician which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors. (In contrast, when unchecked by a solid intuition, once an error is introduced in an argument by a pre-rigorous or rigorous mathematician, it is possible for the error to propagate out of control until one is left with complete nonsense at the end of the argument.) See this post for some further discussion of such errors, and how to read papers to compensate for them.

這好比月賺幾百的想月賺1千萬的生活一樣。回答者有的月賺似乎也就1k，但他們會一年不吃飯買一個蘋果電腦拿去星巴克買杯咖啡坐下來換成win7來寫知乎答案。

數學直覺可以說算是一種方向感，是想法。不少時候我洗澡拉屎散步都會有很多想法，但真的坐下來算的時候，那些想法恐怕一個不剩。

你們看到的大數學家偶發的數學直覺，你們知道他們犧牲了多少你們所謂的數學直覺?!

我覺得直覺就是對定義深刻準確的把握，然後能夠通過大量的例子來發現並驗證理論。

天賦和努力綜合作用的結果。

「Thurston曾說，他對三維流形的感覺是寫不出來的。這種述而不作的態度引來包括Serre在內的一些推崇嚴格論證的數學家的批評。數學當然需要嚴格性，但像Thurston這樣直覺遠超乎常人的天才，根本無必要把精力放在瑣碎細節驗證上。這些體力活自然有很多人搶著干，包括許多卓有成就的數學家。」—dionysus

所以數學直覺真的有,但是能抓住的人不多.

個人認為，在某種意義上講，這種直覺應該是經驗的另一種說法。

所謂直覺不過是一種習慣積累。

記得當年高考做數學的時候，不會做的選擇題，完全靠數學直覺來幫助我度過困難。

熟能生巧

這題選C准沒錯

所謂直覺也就是做多了見多了產生的條件反射吧。

而「顯然，易證，可知」等詞其實等價於「姐姐我不會證但是結果就是那樣阿哈哈哈哈哈，你這樣也不能說我錯哦噢吼吼吼吼」所以也不是直覺

所以……直覺神馬的。哈哈呵呵呼呼嘿嘿……

數學還是邏輯的東西，才不是「我覺得」就能解決的。

有，

但不知那種發散的思維算不。

做題太少了，不但少，而且不總結規律，

作為一個經歷了高考數學和考研數學的理工男，數學直覺這種現象是有的，就像題主所說，幾何題中作輔助線的例子，還有關於中值定理的證明題，如果沒有數學直覺（暫時用一下），根本不知道如何下手。題主之所以問這個問題，周圍肯定有數學高手（相對的），其實我可以很負責的告訴題主，數學直覺就是在基本功紮實+題目經驗豐富的基礎上，不斷強化題型與解法的配對（這是個反覆的過程），將反應速度提升到一定程度（題目一定要足夠豐富），就會出現數學直覺這種神奇的現象（有時連自己都不知道怎麼想到的）。

樓主不用看別人，自己相信自己，你一樣可以做到~

有，就和語感一樣。有些人無論多努力都無法體會文章蘊含的情感，無法獲得閱讀的快樂。就如同有些人需要很努力才能完成簡單運算，甚至對基本數學原理缺乏信任感，比如我自己。。。

作為一個對數字極不敏感，至今只能記住自己手機號的前輩，我可以很負責任地告訴你：至少還有我

有，可是…… 似乎提問者討論的根本不是數學史級的那種級別的數學家的「數學直覺」。

他說的只是畫輔助線、選擇變形公式而已啊。

最大的可能還是，提問者被「高手」含糊的說法所迷惑，導致你覺得這是一種直覺。或者是你自己本來就想抱有相信這種直覺存在的信念——這畢竟可以是很棒的借口——並努力地試圖從這種你解釋不來的現象中證明它。

其實說不定只是人家功底不夠紮實到說清為什麼作這條線，或者你理解不了呢？

如果以自身經驗來談的話，即使作為題海戰術運用的失敗者，我也可以保證當你做題達到一定量之後也可以擁有這種「直覺」，而到那時候它是不是「直覺」還那麼重要嗎？

沒有

以及我超煩那種在大家討論問題的時候拿這個做噱頭搞得一副天機不可泄露的模樣的人

。

以高中的水平，蒙大學的題，

或者面對一個新猜想時，第一反應是證明還是證否