Ghostly miasma and morass: aka basic math proofs
January 11, 2011 3:47 PM Subscribe
Can someone give me some insight as to how to approach and make sense of mathermatical proofs?
Can someone give me some insight as to how to approach and make sense of mathermatical proofs? I am considered very logical and not that bad at math, but I am truly missing something, and it's horribly frustrating. (To give you an idea of my level, I took diff calculus 20 years ago, and though I passed I really learned nothing and certainly no proofs. The best I can do is shuffle around trig identities to change into another one, not really a proof.) The problem is proofs strike off in no particular direction. Why don't they say WHY they are doing something instead of just a bunch of weird postulates like "an even number squared is an even number" that I was never aware of in the first place? I'm sure there are some people here who like math. Could you please share some wisdom and context and perhaps pointers to basic principles that are meaningful? For god's sake, don't tell me to "do a lot of proofs," because I can't do even one. They are meaningless fuzzy nonsense to me. I now need to understand these things for reasons relating to my personal goals.
I looked at this thread, but it doesn't really do it for me. There were a few links for various textbooks as well. I can look into those, but that could be a big time investment as well. I would like to see a succinct explanation or demonstration of something MEANINGFUL about proofs, instead of more mystification, if you are willing to share your thoughts.
Can someone give me some insight as to how to approach and make sense of mathermatical proofs? I am considered very logical and not that bad at math, but I am truly missing something, and it's horribly frustrating. (To give you an idea of my level, I took diff calculus 20 years ago, and though I passed I really learned nothing and certainly no proofs. The best I can do is shuffle around trig identities to change into another one, not really a proof.) The problem is proofs strike off in no particular direction. Why don't they say WHY they are doing something instead of just a bunch of weird postulates like "an even number squared is an even number" that I was never aware of in the first place? I'm sure there are some people here who like math. Could you please share some wisdom and context and perhaps pointers to basic principles that are meaningful? For god's sake, don't tell me to "do a lot of proofs," because I can't do even one. They are meaningless fuzzy nonsense to me. I now need to understand these things for reasons relating to my personal goals.
I looked at this thread, but it doesn't really do it for me. There were a few links for various textbooks as well. I can look into those, but that could be a big time investment as well. I would like to see a succinct explanation or demonstration of something MEANINGFUL about proofs, instead of more mystification, if you are willing to share your thoughts.
You have to shoot for your target. (I'm in chemistry, I'm amazed how many people miss that important part.) Now you have the two ends. Think of all your trig identities for your two ends and see if you can get the middle. Use scratch paper and rewrite.
posted by notned at 4:06 PM on January 11, 2011
posted by notned at 4:06 PM on January 11, 2011
A well written proof will begin with a statement of intent, and an indication of method. (ex: In this proof, we aim to demonstrate blah blah blah by contradiction/induction/exhaustively going through all the cases/etc.) The rest will differ proof by proof. It would help if you could post an example of something specific that you are having issues with.
posted by oracle bone at 4:13 PM on January 11, 2011
posted by oracle bone at 4:13 PM on January 11, 2011
Response by poster: Spacemanstix - Today there was a proof that sqrt 2 is not a rational number. This is where the even number thing came in. Even if step to step makes sense, overall it does not convince. It seems to be an argument or claim, not a proof. In the real world, proof by contradiction is no proof at all. Wouldn't stand up in court! So, no, I can't give you an example. No proof seems meaningful. I can't get excited about ridiculous things like Zeno's paradox, which is silly because obviously you will get close enough to have an interaction with electrons. I hope for a broader sense of fundamental principles and contexts.
posted by Listener at 4:14 PM on January 11, 2011
posted by Listener at 4:14 PM on January 11, 2011
You allude to, but do not explicitly mention the proof of the irrationality of the square root of two. The postulates are for the purpose of setting the stage. Once you've got "if a number is even then its square is even", and "if a square number is even, then so is its square root", and likewise for odd numbers, you're ready to go. This is kind of like the magician setting things up for the audience, setting appropriate mood lighting and having the right music playing.
I'll not run you through the proof here. Instead, go look at the wikipedia page on Hippasus.
posted by oonh at 4:17 PM on January 11, 2011
I'll not run you through the proof here. Instead, go look at the wikipedia page on Hippasus.
posted by oonh at 4:17 PM on January 11, 2011
After rereading your question:
"an even number squared is an even number" - you may have never heard of this postulate, but simple inspection will confirm it to be true. It sounds like you might have been reading a number theoretic proof? Some texts will expect you to have some sort of prior knowledge, and this postulate you gave as an example is one that most math people are aware of, so the author may be assuming this knowledge.
posted by oracle bone at 4:19 PM on January 11, 2011
"an even number squared is an even number" - you may have never heard of this postulate, but simple inspection will confirm it to be true. It sounds like you might have been reading a number theoretic proof? Some texts will expect you to have some sort of prior knowledge, and this postulate you gave as an example is one that most math people are aware of, so the author may be assuming this knowledge.
posted by oracle bone at 4:19 PM on January 11, 2011
Also, you may want to read this thread on math.stackexchange.
posted by oonh at 4:21 PM on January 11, 2011
posted by oonh at 4:21 PM on January 11, 2011
Response by poster: Not a text -- instructor mentioned this in class today. MOst math people may be aware. As math is not my language, please point me to something to give me a footing in this new world. Thanks. I don't care about this proof. We don't have to know it. I care about generally being able to approach this bizarre dark confusing so-far meaningless world that is proofs. THanks, everyone.
posted by Listener at 4:21 PM on January 11, 2011
posted by Listener at 4:21 PM on January 11, 2011
Proofs are basically logic games. You have a big bag a tricks (whatever identities you have given, as well as your basic logic and math skills) and you have to connect the dots.
However, intuition also plays a big role. For similar types of proofs, similar tricks are used. Copy out some example proofs to try to develop some of this intuition.
The big advantage you have when doing a proof is, like notned says, that you already know the answer. USE THIS ADVANTAGE. First, you can use it to limit the number of moves you can make at any given step to only those that get you closer to the answer (this is more useful in elementary proofs, as more advanced ones often take circuitous routes to their target).
The other way to use the answer to so work backwards from the last line. Often it will seem like there's a lot fewer moves you can make going backwards from the answer than there is going forward from the starting points. Going backwards and forwards at the same time will help you see where you need to go with the proof. Things like "an even number squared is an even number" never seem to make sense when you're going through a proof line by line. But if you know where you're trying to get to, tricks like that come a lot easier.
Finally, I find a lot of people get hung up when doing proofs by trying to find the "right" way to do it. Certainly there is an easiest way, but it's often best to play around a bit and go with what works instead of trying to find the most direct route.
On preview, if your problem is more with the general idea of a mathematical proof, a formal logic course will deal with what it means to prove something a lot more explicitly.
posted by auto-correct at 4:21 PM on January 11, 2011
However, intuition also plays a big role. For similar types of proofs, similar tricks are used. Copy out some example proofs to try to develop some of this intuition.
The big advantage you have when doing a proof is, like notned says, that you already know the answer. USE THIS ADVANTAGE. First, you can use it to limit the number of moves you can make at any given step to only those that get you closer to the answer (this is more useful in elementary proofs, as more advanced ones often take circuitous routes to their target).
The other way to use the answer to so work backwards from the last line. Often it will seem like there's a lot fewer moves you can make going backwards from the answer than there is going forward from the starting points. Going backwards and forwards at the same time will help you see where you need to go with the proof. Things like "an even number squared is an even number" never seem to make sense when you're going through a proof line by line. But if you know where you're trying to get to, tricks like that come a lot easier.
Finally, I find a lot of people get hung up when doing proofs by trying to find the "right" way to do it. Certainly there is an easiest way, but it's often best to play around a bit and go with what works instead of trying to find the most direct route.
On preview, if your problem is more with the general idea of a mathematical proof, a formal logic course will deal with what it means to prove something a lot more explicitly.
posted by auto-correct at 4:21 PM on January 11, 2011
Best answer: Proofs usually start with a proposition P that you know to be true, and another P' that you want to prove. The idea is to apply a series of rules to transform P into P', preserving the truthfulness of the statement at each intermediate step.
The "weird postulates" you see are the rules being applied at each step. They justify the validity of the step, but as you point out, they do not explain why the step is undertaken. In a really rigorous proof, these rules would be either axioms or previously proven theorems/lemmas, but in reality people often cut corners for things that are obvious.
Lots of math and CS textbooks have step-by-step narration that walks through proofs that illustrate a particular concept.
Wikipedia has a nice list of proof methods.
You may want to read up on propositional logic and first order logic, but I wouldn't recommend Wikipedia for that... the articles are truly intimidating.
posted by qxntpqbbbqxl at 4:23 PM on January 11, 2011 [1 favorite]
The "weird postulates" you see are the rules being applied at each step. They justify the validity of the step, but as you point out, they do not explain why the step is undertaken. In a really rigorous proof, these rules would be either axioms or previously proven theorems/lemmas, but in reality people often cut corners for things that are obvious.
Lots of math and CS textbooks have step-by-step narration that walks through proofs that illustrate a particular concept.
Wikipedia has a nice list of proof methods.
You may want to read up on propositional logic and first order logic, but I wouldn't recommend Wikipedia for that... the articles are truly intimidating.
posted by qxntpqbbbqxl at 4:23 PM on January 11, 2011 [1 favorite]
Listener: it sounds like you need to spend more time thinking about this stuff, with some specific examples, preferably in the company of a mathematician. And do try to lay off language like "does not convince ... no proof at all ... ridiculous things like..."
But I would say that mathematical proofs are one thing, and the way mathematicians think about mathematics are another. You might write down 20 or 30 lines for your basic proofs, but a mathematician is really only thinking about one or two turns, or the key points.
The rest of it is perhaps a bit like a magician setting things up, as oonh as suggested: but I'd think of it rather as someone laying foundations for a house. Mathematicians have a language and set of conventions for what a strict proof looks like. Proof by contradiction is one of them: every single one of its assumptions is spelled out. Mathematicians can reproduce this stuff automatically. The problem with reading proofs by yourself is that nobody shows you what the interesting part is, and what the easy part is.
posted by squishles at 4:27 PM on January 11, 2011 [1 favorite]
But I would say that mathematical proofs are one thing, and the way mathematicians think about mathematics are another. You might write down 20 or 30 lines for your basic proofs, but a mathematician is really only thinking about one or two turns, or the key points.
The rest of it is perhaps a bit like a magician setting things up, as oonh as suggested: but I'd think of it rather as someone laying foundations for a house. Mathematicians have a language and set of conventions for what a strict proof looks like. Proof by contradiction is one of them: every single one of its assumptions is spelled out. Mathematicians can reproduce this stuff automatically. The problem with reading proofs by yourself is that nobody shows you what the interesting part is, and what the easy part is.
posted by squishles at 4:27 PM on January 11, 2011 [1 favorite]
sorry, when I say "every one of its assumptions is spelled out", I am talking about mathematical proofs, of which proof by contradiction is only one kind.
posted by squishles at 4:34 PM on January 11, 2011
posted by squishles at 4:34 PM on January 11, 2011
I think that there are three or so different ways that one can eke out the contradiction in the proof of the irrationality of the square root of two: the two numbers can't be both even, unique factorization on the integers, and assuming that a/b is a reduced fraction, and making a series of derivations based on an initial assumption. You check the derivations, they're all reasonable, so no fault in the logic. Maybe the assumption made at the beginning is in error -- that the square root of two is rational, is mistaken. What's so bad about contradiction, you say? Well, the fact that once you have a contradiction, things get rather explosive. (xkcd)
posted by oonh at 4:39 PM on January 11, 2011
posted by oonh at 4:39 PM on January 11, 2011
I'll leave others to address the more involved parts of your question but I did want to address some of the things you said about mathematical proofs not being convincing. You say: In the real world, proof by contradiction is no proof at all. Wouldn't stand up in court!
I really don't see why you think this is the case and I fear that you misunderstand proof by contradiction if you feel that's so. To give you an admittedly contrived example, suppose the defense wishes to prove that the defendant could not have been at the scene of the crime at the time of the crime. To convince the jury of this, they could start out with the assumption that the defendant indeed was at the scene of the crime at the time of the crime. If they can then show that proceeding from this assumption logically it then follows that something impossible must be true (this could really be anything that 2 = 3 or that the sun actually rose in the West that morning), it then has to follow that the initial assumption must be false and hence the defense has proved that the defendant could not have been at the scene of the crime at the time of the crime.
This sort of logic is at the heart of mathematical proofs, but truly does not need any specialized mathematics to understand.
posted by peacheater at 4:43 PM on January 11, 2011 [1 favorite]
I really don't see why you think this is the case and I fear that you misunderstand proof by contradiction if you feel that's so. To give you an admittedly contrived example, suppose the defense wishes to prove that the defendant could not have been at the scene of the crime at the time of the crime. To convince the jury of this, they could start out with the assumption that the defendant indeed was at the scene of the crime at the time of the crime. If they can then show that proceeding from this assumption logically it then follows that something impossible must be true (this could really be anything that 2 = 3 or that the sun actually rose in the West that morning), it then has to follow that the initial assumption must be false and hence the defense has proved that the defendant could not have been at the scene of the crime at the time of the crime.
This sort of logic is at the heart of mathematical proofs, but truly does not need any specialized mathematics to understand.
posted by peacheater at 4:43 PM on January 11, 2011 [1 favorite]
If you can say more about what proofs you need to understand and why, people will probably be able to help you much better.
As a start, if you read a line like "an even number squared is an even number", i.e. something you didn't already know to be true, you're supposed to stop and make sure you see why it's true. Proofs in books won't spell out every single small step, that would take too long and it would be laboring points that are obvious to most readers.
Btw what "obvious" means to mathematicians is generally something like "if you stop and think about it for a moment, you'll figure it out easily... there's nothing clever going on that needs to be explained".
A lot of the time that is just like shuffling trig formulas, you unpack things into their definitions, then rearrange them into the form you were looking for.
Like so:
"An even number" means a multiple of 2. So if your even number is 2*n, what is it's square? (2*n)*(2*n)... which is 4*n^2... which is also an even number because it too is a multiple of 2, i.e. 2*2*n^2.
For the same kind of reason, an even number times any whole number is an even number.
If you're having trouble seeing things like that for yourself, you maybe need to begin at a more elementary level than you're working on now.
posted by philipy at 4:53 PM on January 11, 2011
As a start, if you read a line like "an even number squared is an even number", i.e. something you didn't already know to be true, you're supposed to stop and make sure you see why it's true. Proofs in books won't spell out every single small step, that would take too long and it would be laboring points that are obvious to most readers.
Btw what "obvious" means to mathematicians is generally something like "if you stop and think about it for a moment, you'll figure it out easily... there's nothing clever going on that needs to be explained".
A lot of the time that is just like shuffling trig formulas, you unpack things into their definitions, then rearrange them into the form you were looking for.
Like so:
"An even number" means a multiple of 2. So if your even number is 2*n, what is it's square? (2*n)*(2*n)... which is 4*n^2... which is also an even number because it too is a multiple of 2, i.e. 2*2*n^2.
For the same kind of reason, an even number times any whole number is an even number.
If you're having trouble seeing things like that for yourself, you maybe need to begin at a more elementary level than you're working on now.
posted by philipy at 4:53 PM on January 11, 2011
And, I'm going to nitpick here: "in the real world" features considerably more soggier reasoning than occurs in pure mathematics. (both in courts of law (where the innocent have been "proven" guilty) and in pharmaceutical development and this.)
posted by oonh at 5:18 PM on January 11, 2011
posted by oonh at 5:18 PM on January 11, 2011
I'll weigh in and give my professional two cents. First off, I like squishles' answer the best. Mathematical proofs are a creative endeavor more closely linked (in my mind, at least) with art rather than science. What you are asking is equivalent to saying "I know how to read and write, so why can't I write a great novel?" The only real answer is that it's going to take a lot of practice. Continuing with my novel analogy, your first proofs will be like simple subject-verb-object sentences. When you've mastered that, you'll graduate to simple sentences put together to make longer paragraphs. If you go on from there, you'll begin writing the equivalent of short stories (theorems that require lemmas). If you're unlucky (*joke*) enough to go on to grad school in mathematics, that's when you'll tackle writing a "novel".
The practical advice that I can give you, beyond lots and lots of practice, is to know the mechanics of how proofs work. That is, recognize that with every proof there is some kind of technique (or multiple techniques) like contradiction, induction, construction, etc..., and also some kind of content, i.e., the background knowledge that you have going into the proof. For your example of proving that the square-root of 2 is irrational, the technique is, as you mentioned, contradiction. The background info needed is the definition of irrationality (=not rational) and some ideas about factors of integers.
For most undergraduate mathematics, the background knowledge needed in a proof is exactly the stuff you are studying. So begin a proof by writing down all definitions and related theorems. Many proofs are a combination of two or more ideas that you are studying. As for the form, that can vary from proof to proof. However, when I was an undergraduate, I knew a very smart lady who only proved things using proof by contradiction. Obviously (or maybe not), she didn't get everything right all the time, but she didn't seem that much worse off than the rest of us. Good luck!
posted by El_Marto at 5:34 PM on January 11, 2011
The practical advice that I can give you, beyond lots and lots of practice, is to know the mechanics of how proofs work. That is, recognize that with every proof there is some kind of technique (or multiple techniques) like contradiction, induction, construction, etc..., and also some kind of content, i.e., the background knowledge that you have going into the proof. For your example of proving that the square-root of 2 is irrational, the technique is, as you mentioned, contradiction. The background info needed is the definition of irrationality (=not rational) and some ideas about factors of integers.
For most undergraduate mathematics, the background knowledge needed in a proof is exactly the stuff you are studying. So begin a proof by writing down all definitions and related theorems. Many proofs are a combination of two or more ideas that you are studying. As for the form, that can vary from proof to proof. However, when I was an undergraduate, I knew a very smart lady who only proved things using proof by contradiction. Obviously (or maybe not), she didn't get everything right all the time, but she didn't seem that much worse off than the rest of us. Good luck!
posted by El_Marto at 5:34 PM on January 11, 2011
On, reading all the comments added, while I was writing mine... it seems like your problem might be that you have some resistance to what mathematicians, logicians and philosophers consider to be proof and what they don't consider to be proof.
Those are interesting questions, but arguing about them is not going to help you do the kind of proofs you say you need to do.
In the real world, proof by contradiction is no proof at all. Wouldn't stand up in court!
If you could ever establish a proof by contradiction with the kind of rigor needed in maths in a legal case, it would stand up just fine. It would go something like this:
Theorem: John Doe is innocent.
Proof:
Suppose that Doe was the murderer.
The victim defended himself fiercely, it was a brutal fight, blood from both victim and assailant was spattered everywhere. If John Doe had been the murderer, his DNA would certainly have been found in the apartment.
But his DNA was not found, which contradicts the supposition that John Doe was the murderer.
Hence, John Doe is innocent. QED.
This type of reasoning is accepted not only in courts of law, but all the time in ordinary conversation. Any time you hear on AskMe, "If she was really interested in going out with you, she'd have said X", you're hearing a proof-by-contradiction.
The difference between maths and courts of law or relationship advice is that in maths you actually can be certain about statements like "if X then Y", whereas in real life things like "If X did the murder their DNA would have been in the room" or "If she was interested in dating you, she'd would have said Y", are at best just very probable.
"Very probable" is not enough for a mathematical proof. But it's not just very probable that the square of an even number is even.
posted by philipy at 5:36 PM on January 11, 2011
Those are interesting questions, but arguing about them is not going to help you do the kind of proofs you say you need to do.
In the real world, proof by contradiction is no proof at all. Wouldn't stand up in court!
If you could ever establish a proof by contradiction with the kind of rigor needed in maths in a legal case, it would stand up just fine. It would go something like this:
Theorem: John Doe is innocent.
Proof:
Suppose that Doe was the murderer.
The victim defended himself fiercely, it was a brutal fight, blood from both victim and assailant was spattered everywhere. If John Doe had been the murderer, his DNA would certainly have been found in the apartment.
But his DNA was not found, which contradicts the supposition that John Doe was the murderer.
Hence, John Doe is innocent. QED.
This type of reasoning is accepted not only in courts of law, but all the time in ordinary conversation. Any time you hear on AskMe, "If she was really interested in going out with you, she'd have said X", you're hearing a proof-by-contradiction.
The difference between maths and courts of law or relationship advice is that in maths you actually can be certain about statements like "if X then Y", whereas in real life things like "If X did the murder their DNA would have been in the room" or "If she was interested in dating you, she'd would have said Y", are at best just very probable.
"Very probable" is not enough for a mathematical proof. But it's not just very probable that the square of an even number is even.
posted by philipy at 5:36 PM on January 11, 2011
Best answer: I used to tell my students that a proof is a story or argument where you're trying to convince the reader of something. Of course the reader has to accept logic, no fair saying "I just don't believe you!" And of course to understand a story you have to be fluent in the language it's written in.
You wrote "Why don't they say WHY they are doing something instead of just a bunch of weird postulates ...?" This is actually a key observation. A proof that you might see in a book is like a completed cathedral where the scaffolding that was required to construct it has been removed. You are asking about the scaffolding. Mathematicians call this the motivation -- why did they start off with thing about squares of even numbers? In fact when mathematicians speak informally, you will hear that question quite a bit, and the various other approaches that didn't work before the good path was found.
posted by phliar at 5:39 PM on January 11, 2011 [1 favorite]
You wrote "Why don't they say WHY they are doing something instead of just a bunch of weird postulates ...?" This is actually a key observation. A proof that you might see in a book is like a completed cathedral where the scaffolding that was required to construct it has been removed. You are asking about the scaffolding. Mathematicians call this the motivation -- why did they start off with thing about squares of even numbers? In fact when mathematicians speak informally, you will hear that question quite a bit, and the various other approaches that didn't work before the good path was found.
posted by phliar at 5:39 PM on January 11, 2011 [1 favorite]
Response by poster: El_Marto: "your first proofs will be like simple subject-verb-object sentences."
Could you point me to a couple of examples of those? Would really appreciate examples to give a sense of basic units of the language/meaningful statements.
posted by Listener at 5:46 PM on January 11, 2011 [1 favorite]
Could you point me to a couple of examples of those? Would really appreciate examples to give a sense of basic units of the language/meaningful statements.
posted by Listener at 5:46 PM on January 11, 2011 [1 favorite]
Perhaps we should also point out, and maybe this is what OP is getting at, that the proof by contradiction we're talking about is something debated within mathematics. Perhaps someone who knows about logic and set theory can comment more, but the kind of proof by contradiction we're talking about is the "law of the excluded middle". From what I understand, some mathematicians play around with things by saying, "hey, if we don't assume that thing, what can we still build? what will be different?" IANAMathematician, but I believe people have made systems where the law of the excluded middle doesn't hold.
OP: sorry if I was snippy above. Basically, it's a lot of work! The best way to learn this stuff is to be taught it; learning by oneself is difficult, and learning solely by reading proofs (and not the stuff surrounding them) is even harder.
posted by squishles at 5:49 PM on January 11, 2011
OP: sorry if I was snippy above. Basically, it's a lot of work! The best way to learn this stuff is to be taught it; learning by oneself is difficult, and learning solely by reading proofs (and not the stuff surrounding them) is even harder.
posted by squishles at 5:49 PM on January 11, 2011
"debated" not in the sense of "hey, I disagree with that!" but rather "what would happen if we didn't think this was true?"
posted by squishles at 5:51 PM on January 11, 2011
posted by squishles at 5:51 PM on January 11, 2011
This may be too basic, but try proving an even integer plus an even integer is an even integer. Then try an odd plus an odd. Once you have that, you can basically do one of the pieces that shows up in the square-root of two proof.
posted by El_Marto at 6:11 PM on January 11, 2011
posted by El_Marto at 6:11 PM on January 11, 2011
Best answer: What's going on in mathematical proofs, as I understand it.
First, a mathematician notices a relationship. "It seems that when I draw a triangle with a right angle and measure the sides, the square of the two shorter sides is equal to the square of the third: a^2 + b^c = c^2".
The mathematician wishes to determine whether this conjecture is true or not. He draws a few triangles with lengths 3, 4, 5... 5, 12, 13.... 9, 40, 41.... they all seem to hold. Every time he constructs a triangle with a right angle, the lengths of a sides follow the relationship a^2 + b^2 = c^2. He strongly believes that this is true for ALL triangles with right angles. However, no matter how many triangles he draws, he can never be 100% sure that this is the case; there could always be some set of numbers he didn't think to try which breaks the relationship. (This is a general rule of logical thinking: the fact that something works for a few examples doesn't guarantee it will work for all of them, no matter how many examples you try.) In order for him and mathematicians to be convinced that this relationship is true, he has to try it for all possible sets of triangles.
It is time consuming and indeed impossible to draw every possible right-angle triangle, measure its sides, and calculate a^2 + b^2 to see if it equals c^2. But the mathematician can accomplish the equivalent by testing an *abstract* right-angle triangle - one whose sides are not specified and could be any length, so long as they form a triangle with a right angle. Starting only with information that is known about right-angle triangles, and not knowing the values of a, b, and c, his task is to determine whether a^2 + b^2 actually equals c^2 -- or not.
When you read a proof you see a bunch of mathematical relationships and it may seem like gibberish. This is because the thinking has already been done and you are seeing the end result. It's a lot like looking at code, if you're a programmer; you can only know what it does by working through it, or having the person who wrote it walk you through it. In fact if you were asked to prove this relationship it is not at all obvious how to do this. It's a trivial example only because the proof was found by Pythagoras centuries ago and everyone memorized it.
In practice what do you do? Start drawing right-angle triangles and playing around with them. Use whatever information you have about the problem, both at the beginning (you have a right triangle with sides a, b, c) and at the end (a^2 + b^2 = c^2). Manipulate the numbers in all kinds of ways. If there are other relationships you know about triangles -- for example, that the sum of the angles in a triangle is 180 degrees -- see if you can apply that here. All of the intermediate steps in a proof involve some manipulation of the starting conditions and some application of other mathetmatical principles.
Now -- and this is important -- understand that there may not always be a logical link as to why one particular mathematical principle is relevant for a proof. It just takes a flash of mathematical genius to see that to get from one step to another, you can apply a theorem which has nothing to do with what you started out with. As an example, Fermat's last theorem was not solved for hundreds of years until some advanced branch of contemporary mathematics coincidentally happened to develop theorems that were applicable to it. You have to divorce yourself from meaning here; your starting and ending point may have real-world relevance, but numbers are abstract, and you are free to manipulate them in ways that are not constrained by the real world; only in ways that are constrained by the mathematical framework in which we operate. This doesn't mean the steps are meaningless; only that their original meaning, as we understood it, may not apply to the context of the problem you are working on.
In the end, the goal is to show that the fact that you have a right triangle -- the initial setup in the conjecture -- GUARANTEES that its sides follow the relationship you noticed. No matter what numbers you try for a b and c, if they describe a right triangle, they must follow a^2+b^2=c^2. If you can demonstrate that without restricting yourself to an example (where a=5, say), then you have done the equivalent of drawing all possible triangles, measuring their sides, calculating a^2 + b^2, and showing that it's equal to c^2. There are hundreds of ways to prove a^2+b^2=c^2, some of which involve geometry, some algebra, and so on. There are five proofs on the Wikipedia page alone. In each case, someone clever found a way to get from the start to the end, applying different forms of mathematical reasoning. With a bit of work you should be able to puzzle out how each proof works and what the underlying reason is behind each step. More complicated proofs are really no different, just condensed and more complex.
posted by PercussivePaul at 6:11 PM on January 11, 2011 [4 favorites]
First, a mathematician notices a relationship. "It seems that when I draw a triangle with a right angle and measure the sides, the square of the two shorter sides is equal to the square of the third: a^2 + b^c = c^2".
The mathematician wishes to determine whether this conjecture is true or not. He draws a few triangles with lengths 3, 4, 5... 5, 12, 13.... 9, 40, 41.... they all seem to hold. Every time he constructs a triangle with a right angle, the lengths of a sides follow the relationship a^2 + b^2 = c^2. He strongly believes that this is true for ALL triangles with right angles. However, no matter how many triangles he draws, he can never be 100% sure that this is the case; there could always be some set of numbers he didn't think to try which breaks the relationship. (This is a general rule of logical thinking: the fact that something works for a few examples doesn't guarantee it will work for all of them, no matter how many examples you try.) In order for him and mathematicians to be convinced that this relationship is true, he has to try it for all possible sets of triangles.
It is time consuming and indeed impossible to draw every possible right-angle triangle, measure its sides, and calculate a^2 + b^2 to see if it equals c^2. But the mathematician can accomplish the equivalent by testing an *abstract* right-angle triangle - one whose sides are not specified and could be any length, so long as they form a triangle with a right angle. Starting only with information that is known about right-angle triangles, and not knowing the values of a, b, and c, his task is to determine whether a^2 + b^2 actually equals c^2 -- or not.
When you read a proof you see a bunch of mathematical relationships and it may seem like gibberish. This is because the thinking has already been done and you are seeing the end result. It's a lot like looking at code, if you're a programmer; you can only know what it does by working through it, or having the person who wrote it walk you through it. In fact if you were asked to prove this relationship it is not at all obvious how to do this. It's a trivial example only because the proof was found by Pythagoras centuries ago and everyone memorized it.
In practice what do you do? Start drawing right-angle triangles and playing around with them. Use whatever information you have about the problem, both at the beginning (you have a right triangle with sides a, b, c) and at the end (a^2 + b^2 = c^2). Manipulate the numbers in all kinds of ways. If there are other relationships you know about triangles -- for example, that the sum of the angles in a triangle is 180 degrees -- see if you can apply that here. All of the intermediate steps in a proof involve some manipulation of the starting conditions and some application of other mathetmatical principles.
Now -- and this is important -- understand that there may not always be a logical link as to why one particular mathematical principle is relevant for a proof. It just takes a flash of mathematical genius to see that to get from one step to another, you can apply a theorem which has nothing to do with what you started out with. As an example, Fermat's last theorem was not solved for hundreds of years until some advanced branch of contemporary mathematics coincidentally happened to develop theorems that were applicable to it. You have to divorce yourself from meaning here; your starting and ending point may have real-world relevance, but numbers are abstract, and you are free to manipulate them in ways that are not constrained by the real world; only in ways that are constrained by the mathematical framework in which we operate. This doesn't mean the steps are meaningless; only that their original meaning, as we understood it, may not apply to the context of the problem you are working on.
In the end, the goal is to show that the fact that you have a right triangle -- the initial setup in the conjecture -- GUARANTEES that its sides follow the relationship you noticed. No matter what numbers you try for a b and c, if they describe a right triangle, they must follow a^2+b^2=c^2. If you can demonstrate that without restricting yourself to an example (where a=5, say), then you have done the equivalent of drawing all possible triangles, measuring their sides, calculating a^2 + b^2, and showing that it's equal to c^2. There are hundreds of ways to prove a^2+b^2=c^2, some of which involve geometry, some algebra, and so on. There are five proofs on the Wikipedia page alone. In each case, someone clever found a way to get from the start to the end, applying different forms of mathematical reasoning. With a bit of work you should be able to puzzle out how each proof works and what the underlying reason is behind each step. More complicated proofs are really no different, just condensed and more complex.
posted by PercussivePaul at 6:11 PM on January 11, 2011 [4 favorites]
I have only glanced at the answers above, but I think a good resource on this is a book by Imre Lakatos a philosopher of mathematics, Proofs and refutations. I am currently reading it and as far as I can understand he talks about what steps might one take when proving or disproving something before arriving at the final formal proof.
The book is set in the form of a dialogue between students in a class and they try to prove/formulate Euler's characteristic for polyhedra. The book (as far as I have read) provides insight into how one might approach a theorem/statement to be proved and how modifications and refinements are made in proposing axiomatic statements.
What I get from it is that when we read proofs by others we see the end product and all the refinement and whittling away process when the proof was formulated is hidden. The variety of tools and techniques that might have been used in coming to the final version are not explicitly seen.
He also touches upon the history of proofs which I found interesting.
"...About 1800 the rigour of proof (crystal-clear thought experiment or construction) was contrasted with muddled argument and inductive generalisation. This was what Euler meant by ‘rigida demonstratio’, and Kant’s idea of infallible mathematics too was based on this concept (…). It was also thought that one proves what one has set out to prove. It did not occur to anybody that the verbal articulation of a thought-experiment involves any real difficulty. Aristotelian formal logic and mathematics were two completely separate disciplines - mathematicians considered the former as utterly useless. The proof or thought-experiment carried full conviction without any deductive pattern or ‘logical’ structure.
In the early nineteenth century the flood of counterexamples brought confusion. Since proofs were crystal-clear, refutations had to be miraculous freaks, to be completely segregated from the indubitable proofs. Cauchy’s revolution of rigour rested on the heuristic innovation that the mathematician should not stop at the proof: he should go on and find out what he has proved by enumerating the exceptions, or rather by stating a safe domain where the proof is valid..."
The mathematics is a bit involved and I kept getting impatient sometimes, but it is a good read.
posted by ssri at 7:15 PM on January 11, 2011
The book is set in the form of a dialogue between students in a class and they try to prove/formulate Euler's characteristic for polyhedra. The book (as far as I have read) provides insight into how one might approach a theorem/statement to be proved and how modifications and refinements are made in proposing axiomatic statements.
What I get from it is that when we read proofs by others we see the end product and all the refinement and whittling away process when the proof was formulated is hidden. The variety of tools and techniques that might have been used in coming to the final version are not explicitly seen.
He also touches upon the history of proofs which I found interesting.
"...About 1800 the rigour of proof (crystal-clear thought experiment or construction) was contrasted with muddled argument and inductive generalisation. This was what Euler meant by ‘rigida demonstratio’, and Kant’s idea of infallible mathematics too was based on this concept (…). It was also thought that one proves what one has set out to prove. It did not occur to anybody that the verbal articulation of a thought-experiment involves any real difficulty. Aristotelian formal logic and mathematics were two completely separate disciplines - mathematicians considered the former as utterly useless. The proof or thought-experiment carried full conviction without any deductive pattern or ‘logical’ structure.
In the early nineteenth century the flood of counterexamples brought confusion. Since proofs were crystal-clear, refutations had to be miraculous freaks, to be completely segregated from the indubitable proofs. Cauchy’s revolution of rigour rested on the heuristic innovation that the mathematician should not stop at the proof: he should go on and find out what he has proved by enumerating the exceptions, or rather by stating a safe domain where the proof is valid..."
The mathematics is a bit involved and I kept getting impatient sometimes, but it is a good read.
posted by ssri at 7:15 PM on January 11, 2011
There are a number of books that introduce students to mathematical proofs. Here are two good ones: 1, 2. Both books lay out and explain various proof techniques with a variety of simple (and some less simple) examples. I agree that something like mathematical induction at first looks obscure and confusing.
The simplest proofs simply follow from the relevant definitions. It's like those chess problems where it's "mate in one." More complex proofs may require a number of unintuitive steps where you might expand or rewrite terms for reasons that may not at first seem obvious or necessary.
The proof of the irrationality of the square root of two is like "mate in two," because you proceed through a couple of fairly obvious intermediate steps before arriving at the necessary conclusion.
posted by Nomyte at 7:17 PM on January 11, 2011
The simplest proofs simply follow from the relevant definitions. It's like those chess problems where it's "mate in one." More complex proofs may require a number of unintuitive steps where you might expand or rewrite terms for reasons that may not at first seem obvious or necessary.
The proof of the irrationality of the square root of two is like "mate in two," because you proceed through a couple of fairly obvious intermediate steps before arriving at the necessary conclusion.
posted by Nomyte at 7:17 PM on January 11, 2011
I don't understand this: Even if step to step makes sense, overall it does not convince. It seems to be an argument or claim, not a proof. If you follow the argument step-by-step, how can you be unconvinced at the end?
A "mathematical proof" is an argument for the truthhood of some proposition. Mentioned above, we have examples like "the square root of 2 is an irrational number" and "the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the hypotenuse". So, if we want to prove that the square root of 2 is irrational, the most common way (that I know of) is to say "well, assume that it is rational. Can we show, via line-by-line reasoning, that some other proposition which we know to be false must then be true?" That's the kind of "proof by contradiction" found in mathematics.
As mentioned above, if it were possible to have that kind of rigor in "real life arguments", you're damn straight it would hold up in a court of law.
Mathematical arguments are bound by the rules of logic, so each statement within the proof must follow from the ones preceding it unless I'm introducing a new statement, in which case I need to convince you its true (unless its obvious). In the proof you mentioned, you need at some point that the square of an even number is even, and that is why it is stated in the proof. It is stated as a "matter of fact", because it is obvious. Again, as mentioned above, that generally just means "just think about it for a second, it's true".
To get better at reading and understanding proofs, you need to read and understand a lot of proofs. I don't mean 10 or 20, or even 100. I mean probably 500 to 1,000, if you want to say that you are comfortable reading and understanding mathematical proofs. Obviously, this takes time.
So, we have to start out with easier ones so that we can get used to the language that mathematicians use. Someone posted above a proof that the square of an even number is even. Did you understand that? Read it again if not. Once you think you understand it, could you reproduce it? On paper? Out loud as oral argument to your friends in a bar? Good, you're getting better.
If I had a better idea as to the area of mathematics that you're looking to understand proofs from, I might have a better chance of pointing you in the direction that you want to go. Regardless, get/borrow/steal the book Proofs from THE BOOK, which is a collection of the most elegant proofs in mathematics (as determined mainly by Paul Erdos and the authors of the aforementioned text). In there, you will find many of the best-written proofs out there, and perhaps that will help you sort through this "miasma" that you're stuck in now.
Lastly, IAAMathematician, and I can tell you that it takes a while to figure out what is going on. Once you can get the switch to flip though, it stays on for the rest of your life. Keep working at it!
posted by King Bee at 7:28 PM on January 11, 2011 [1 favorite]
A "mathematical proof" is an argument for the truthhood of some proposition. Mentioned above, we have examples like "the square root of 2 is an irrational number" and "the sum of the squares of the lengths of the legs of a right triangle is equal to the square of the hypotenuse". So, if we want to prove that the square root of 2 is irrational, the most common way (that I know of) is to say "well, assume that it is rational. Can we show, via line-by-line reasoning, that some other proposition which we know to be false must then be true?" That's the kind of "proof by contradiction" found in mathematics.
As mentioned above, if it were possible to have that kind of rigor in "real life arguments", you're damn straight it would hold up in a court of law.
Mathematical arguments are bound by the rules of logic, so each statement within the proof must follow from the ones preceding it unless I'm introducing a new statement, in which case I need to convince you its true (unless its obvious). In the proof you mentioned, you need at some point that the square of an even number is even, and that is why it is stated in the proof. It is stated as a "matter of fact", because it is obvious. Again, as mentioned above, that generally just means "just think about it for a second, it's true".
To get better at reading and understanding proofs, you need to read and understand a lot of proofs. I don't mean 10 or 20, or even 100. I mean probably 500 to 1,000, if you want to say that you are comfortable reading and understanding mathematical proofs. Obviously, this takes time.
So, we have to start out with easier ones so that we can get used to the language that mathematicians use. Someone posted above a proof that the square of an even number is even. Did you understand that? Read it again if not. Once you think you understand it, could you reproduce it? On paper? Out loud as oral argument to your friends in a bar? Good, you're getting better.
If I had a better idea as to the area of mathematics that you're looking to understand proofs from, I might have a better chance of pointing you in the direction that you want to go. Regardless, get/borrow/steal the book Proofs from THE BOOK, which is a collection of the most elegant proofs in mathematics (as determined mainly by Paul Erdos and the authors of the aforementioned text). In there, you will find many of the best-written proofs out there, and perhaps that will help you sort through this "miasma" that you're stuck in now.
Lastly, IAAMathematician, and I can tell you that it takes a while to figure out what is going on. Once you can get the switch to flip though, it stays on for the rest of your life. Keep working at it!
posted by King Bee at 7:28 PM on January 11, 2011 [1 favorite]
The original book of mathematical proofs is Euclid's Elements. I would recommend starting on page 1.
posted by Potomac Avenue at 7:38 PM on January 11, 2011 [1 favorite]
posted by Potomac Avenue at 7:38 PM on January 11, 2011 [1 favorite]
Try reading Journey Through Genius; the author leads you by the hand through a bunch of famous proofs, and he doesn't assume any background in mathematics. I read it when I was a kid and I still remember it fondly.
posted by shponglespore at 9:55 AM on January 12, 2011
posted by shponglespore at 9:55 AM on January 12, 2011
In How to Write a Proof, Leslie Lamport writes:
Proofs written in prose are also hard to understand and hard to get right. Anecdotal evidence suggests that as many as a third of all papers published in mathematical journals contain mistakes—not just minor errors, but incorrect theorems and proofs.
posted by at at 4:03 PM on January 12, 2011 [1 favorite]
Proofs written in prose are also hard to understand and hard to get right. Anecdotal evidence suggests that as many as a third of all papers published in mathematical journals contain mistakes—not just minor errors, but incorrect theorems and proofs.
posted by at at 4:03 PM on January 12, 2011 [1 favorite]
This thread is closed to new comments.
posted by SpacemanStix at 3:52 PM on January 11, 2011