I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the intuition to do so.
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3$\begingroup$ Could you be more specific? Give an example of what you are trying to prove. What have you tried? The book or site that has the exercises you are trying to do probably also has examples. Do you understand why the proofs of the examples work? $\endgroup$– JayCommented Feb 16, 2012 at 13:07
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6$\begingroup$ Perhaps it would be best to post the specific questions you are having difficultly with. The best way to develop mathematical intuition is imo to "do more exercises," but that advice seems misplaced here. As for proofs, you should learn the basic logical structure of direct proof, proof by contradiction, contraposition and induction, rinse and repeat. $\endgroup$– Daniel PietrobonCommented Feb 16, 2012 at 13:08
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6$\begingroup$ When you figure this out, let me know. I've been trying for 20 years! $\endgroup$– Prince GoulashCommented Feb 16, 2012 at 15:32
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2$\begingroup$ Dear James, It would help a lot if you gave more background. At what level are you working (e.g. basic group theory, general topology, more advanced undergrad courses, graduate courses, ... ?). Regards, $\endgroup$– Matt ECommented Feb 16, 2012 at 15:43
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$\begingroup$ I think this question is a duplicate of matheducators.stackexchange.com/questions/11950/… $\endgroup$– TimothyCommented Jun 11, 2019 at 2:03
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6 Answers
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I can say that a big part of mathematical intuition comes from experience. That being said there is no "best way". I like to think about experience like this:
You're stuck in thick, dense jungle like the amazon rain forest. You have a machete, and you're chopping down plants struggling to find a way out. You walk around headless, knowing not even of a path to the nearest village.
However now say you are flying in a helicopter with a reasonable view around. Now not only can you see where you would be in the jungle, but can also see where the nearest village is. You can see the terrain, map out a path, avoid crossing big streams, etc.
This is the analogy: In the first situation you are stuck with how to move, in the next one with a lot of experience you can see many connections and the way out.
Now we come to your next problem. You say that when going through exercises they ask for proofs. What I would recommend if you have never ever seen a mathematical proof is to go through an article like this one.
Of course it is not possible to grasp these techniques at once. You should try out a few suggested exercises, look at how things are done first, look at how other people do things. Why do they think like this? Slowly after a while, your mind will get used to it.
I am willing to edit my answer and improve on it if you provide more detail on the specifics of your difficulties.
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5$\begingroup$ Maybe I'm torturing the analogy, but I wanted to add to it. Experience hacking around through the jungle is still important! How did you know, when taking in the helicopter's eye view, that big streams are something to avoid, or other jungle features to avoid? And once you've plotted a route from the helicopter and jump back into the jungle, you can't travel the path without your skills with the machete. $\endgroup$– user14972Commented Feb 19, 2012 at 0:23
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$\begingroup$ @Hurkyl, I do not think he was claiming otherwise. $\endgroup$– ArkadyCommented Feb 26, 2012 at 11:20
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Here is a true story. One year a student questionnaire on my first year analysis course wrote: "Professor Brown gives too many proofs." I thereupon decided that next year the course would have no theorems, and no proofs. It would however, have "Facts" and "Explanations". (I got this idea from an engineer!) Part of the deal of course would be that an Explanation should explain something. Another point is that unlike the good old days when Euclidean Geometry was studied at school students have no previous experience of the words "Theorem" and "Proof". Also the study of Grammar has gone as well, so they are unfamiliar with the structure of language.
What is called a "proof" is really an explanation but written very carefully.
In other second year lectures which needed an explanation of a particular case of $A \subseteq B$ I asked the class "What is the first line of the proof?" and then, to be written further down on the board, "What is the last line?". By the end of the course, they got the idea!
Also students need training in writing carefully. See a course called Ideas in Mathematics I gave.
I also found that the best way to develop intuition was to write things out very carefully, and explain them to others. When you have written something out 5 times, you may see how to improve it a bit. Then another bit. And so on.
The idea that a proof is found one step after another is just not so. For 9 years I had an idea of a proof in search of a theorem. It took that long, and lots talking and writing, to piece together the gadgets needed to make the idea of the proof actually work and prove a theorem.
The composer Ravel said that you should copy. If you have some originality, this may appear as you copy. If not never mind! Your new ideas also may only appear at the 5th copy, as the wheels of the brain start to unclog. In fact it has been said that Newton was an inveterate copier!
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1$\begingroup$ I just want to say - this post helped me out a ton today. Thank you for the beautiful detail you put into this post. $\endgroup$– user261214Commented Jan 7, 2016 at 8:08
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To learn how to prove something you want to do something like this:
First pick something at an adequate level matching your current level of skill.
Then gather a bunch of exercises that ask for proofs together with answers. But don't look at the answers. Sit down and try to solve them. If you get stuck, look at the first line of the answer and try from there. If that still doesn't get you anywhere, look at the next line, and so on.
Repeat until you have gained some confidence. This will teach you how to prove some of the things you come across.
I also recommend reading "Thinking Mathematically". This will help you change your thinking so that you will get stuck less often in general.
Hope this helps.
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$\begingroup$ I would also suggest looking at the last(s) line(s) $\endgroup$– AurèleCommented Jul 14, 2020 at 18:56
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"I want to develop my pure mathematics knowledge and would like to know what is the best way to develop mathematical intuition? I am going through exercises that ask for proofs and I don't have the intuition to do so."
Honestly, I think you should not build a theory about this. You simply don't know how to solve certain problems. That's it. Don't interpret it that there is something fundamentally wrong with you. Don't conclude that you don't have intuition. Not all exercises are that simple.
Also, intuition comes from experience. The best way is experience. What kind of experience? Well, it depends on what you want. Do you want to be able to solve problems in your book? To become a mathematician? To become a math teacher?
Whatever you do, DON'T read http://zimmer.csufresno.edu/~larryc/proofs/proofs.html suggested in another post. This will do more harm than good. I read this very article recently and I was speechless. In my humble opinion, this guy clearly contributed to ruining math education in this country, even though he probably meant no harm. You don't need to read anything special to understand proofs. Unfortunately, many people in this country want to sell you books about proofs and so they present it as a special skill. Mathematics is proofs. If you have a course without proofs and it is called mathematics (and it is beyond high school), you are paying your money for nothing, period. Read Paul Lockhart's article on education, A Mathematician's Lament: http://www.maa.org/devlin/LockhartsLament.pdf
If you still don't understand why I don't like this article by Larry Cusick, let me quote from the introduction: (http://zimmer.csufresno.edu/~larryc/proofs/proofs.introduction.html)
"The basic structure of a proof is easy: it is just a series of statements, each one being either An assumption or A conclusion, clearly following from an assumption or previously proved result."
---Who needs to be taught the basic structure of a proof? Most people who don't understand proofs will only get more confused. Unless you study math logic, you don't need to analyze the structure of a proof. Furthermore, in many cases it is easier to understand a proof than to dissect it into assumptions and conclusions. It's like a second grader who knows that 5*7=35, but doesn't know "what is the product if the multipliers are 5 and 7".
Continuing with the article. "And that is all. Occasionally there will be the clarifying remark, but this is just for the reader and has no logical bearing on the structure of the proof. " ---Well, this piece of information is not very useful. It is probably more confusing than the actual "clarifying remark" it is warning you about. Anyway, I am not really sure, what it means. It says "...just for the reader". Does it suggest that the proof itself is not for the reader?
"A well written proof will flow. That is, the reader should feel as though they are being taken on a ride that takes them directly and inevitably to the desired conclusion without any distractions about irrelevant details. Each step should be clear or at least clearly justified. A good proof is easy to follow." ---This is almost a joke. I particularly like "...without any distractions about irrelevant details". Forgive me my sarcasm.
"When you are finished with a proof, apply the above simple test to every sentence: is it clearly (a) an assumption or (b) a justified conclusion? If the sentence fails the test, maybe it doesn't belong in the proof." ---No, when you are finished with a proof, just see if it makes sense. There is no need to apply any tests. This is just another trick to make people afraid of proofs. Furthermore, this "test" is wrong. There may be justified conclusions that don't belong to the proof. On the other hand, a statement may belong to the proof, but it may be unjustified, then it doesn't pass this test.
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$\begingroup$ I would have to disagree with you there. One important property of a mathematical proof is that it does not use inductive reasoning (not to be confused, of course, with induction). I don't think this fact is obvious by any means. Another is that every premise is either a definition or a previous theorem. Furthermore, students need some way of distinguishing between "obvious" statements like "a circle divides the plane into two parts", which need proof, and statements like "if A implies B and A is true, then B is true", which do not. $\endgroup$ Commented Feb 27, 2012 at 0:41
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$\begingroup$ "One important property of a mathematical proof is that it does not use inductive reasoning (not to be confused, of course, with induction). I don't think this fact is obvious by any means." - well, I don't think that this fact is relevant. People who don't understand proofs, don't understand what is "inductive reasoning" either. My point is that you should not try to explain the theory behind proofs, unless you are teaching advanced mathematical logic. Just expose people to simple proofs, and they will pick it up. $\endgroup$– osaCommented Mar 6, 2012 at 2:45
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$\begingroup$ Lastly, I don't see that much difference between the statement about a circle and the statement about A and B. I don't think that you really need to prove that a circle divides the plane into two parts. Maybe you do need it if you are building all geometry from scratch using a system of axioms - not Euclidean axioms, these will not be sufficient to prove this statement. Euclid did not define things like a point on a line lies between two other points. On the other hand, tautologies may still need a proof. (Well, it depends on which logic/model/proof system we are working with.) $\endgroup$– osaCommented Mar 6, 2012 at 2:48
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$\begingroup$ I guess it depends on just how rigorous you want to be. Many "obviously true" mathematical statements are false. However, all mathematical statements that have been proven are true. $\endgroup$ Commented Mar 6, 2012 at 4:15
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$\begingroup$ I may not agree with everything @Sergey says, I agree with the general idea -- Lockhart was right in saying that Math loses everything when it is reduced to formula and solving. And that's what that article appears to actually teach -- formulas for how to do a proof! However, students do need to learn what proofs are, the rules they follow, and what to try when stuck. So maybe Houston's book as opposed to Velleman's more formulaic version $\endgroup$ Commented Jul 6, 2013 at 20:32
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Seems like you're at a basic level. I would recommend reading carefully the proofs from some book you like. You need to get used to the language, to the ordering of the arguments, to notation... You may need support from your teacher, and that's OK. Then moving to the exercises should be easier. But you should try really hard -- thats hard work, and takes some time. If you fail to do some exercise, do your best to learn from it, and ask yourself why weren't you able to do it, ask if the solution is "natural". I believe that, as you get used to the arguments, proving theorems by yourself will come naturally.
Good luck :)
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I'd recommend reading "How to Solve It: A New Aspect of Mathematical Method (Princeton Science Library)" written by G. Polya and any other book related to problem solving. Then, you should do three things:
- Get a book about recreational math (e.gr. Martin Gardner's books) and try to solve exercises.
- Get any book on your math level (e.gr. 'Euclid's elements', or Rudin's 'Principles of mathematical analysis', both are difficult examples, but you can find easier books) and begin reading the definitions. Whenever you find a theorem, try to write a proof without checking the book's proof.
- Measure both the time invested in solving exercises side to side with the number of exercises and use it to reflect about your progress. This should not take too much effort, but its purpose is to help you to become aware about your progress and your weaknesses.
This method needs discipline and regularity.