Attempting to prove each theorem in a book

[Terrell]

I am seeking advice on how to effectively and efficiently learn mathematics textbooks. Currently, I adopt the style of trying to prove theorems in the book before reading the provided proof. I have had good success in this; I noticed a considerable gap in experience between me and my peers in writing proofs as I also kept a journal on general "meta-cognitive" techniques when constructing proofs to help with my consistency. There is still a lot to be learned as I am far from being able to prove all the theorems in the text - I believe this to be impossible. Although, I am happy with my progress after almost 2 years I have been practicing this, but I feel like I am going too slow and it would take me far longer to reach "research level" mathematics. Do you people think this is style of learning is worthwhile and helpful in the long run? Why do you think this method is beneficial or not beneficial?

 

[clope023]

I am seeking advice on how to effectively and efficiently learn mathematics textbooks. Currently, I adopt the style of trying to prove theorems in the book before reading the provided proof. I have had good success in this; I noticed a considerable gap in experience between me and my peers in writing proofs as I also kept a journal on general "meta-cognitive" techniques when constructing proofs to help with my consistency. There is still a lot to be learned as I am far from being able to prove all the theorems in the text - I believe this to be impossible. Although, I am happy with my progress after almost 2 years I have been practicing this, but I feel like I am going too slow and it would take me far longer to reach "research level" mathematics. Do you people think this is style of learning is worthwhile and helpful in the long run? Why do you think this method is beneficial or not beneficial?

I was a physics major, not a math major, but I always learn more from doing derivations than shot gunning equations; so I probably spend more time on that than others. I would say it could be beneficial, but it could be a time sink and could lead to the point of paralysis by analysis. I would say it isn't beneficial if your insistence on getting through proofs yourself stops you from doing your real assignments for class.

 

[Terrell]

I was a physics major, not a math major, but I always learn more from doing derivations than shot gunning equations; so I probably spend more time on that than others.

Do you now do research? If so, has doing derivations yourself had any benefits?

 

[clope023]

Do you now do research? If so, has doing derivations yourself had any benefits?

I wouldn't say I did them all by myself but I learned the physical principles and mathematical considerations behind the derivations such that I can recreate them without looking at notes and such and that is helpful when learning new physics but again it can be a paralysis by analysis situation. I'm a modeling and simulation engineer which does involve plenty of research, and in the real world more often than not an answer that's 50% right but on time is more valuable than an answer that's 100% right and late, so you have to balance.

 

[PeroK]

I'm a modeling and simulation engineer which does involve plenty of research, and in the real world more often than not an answer that's 50% right but on time is more valuable than an answer that's 100% right and late, so you have to balance.

If you are doing mathematical research, an answer that's 50% right is not much good!

 

[clope023]

If you are doing mathematical research, an answer that's 50% right is not much good!

I'm not a math researcher though.

 

[PeroK]

I'm not a math researcher though.

Well, exactly. So your advice to the OP, who is asking about maths research, is misplaced.

 

[clope023]

Well, exactly. So your advice to the OP, who is asking about maths research, is misplaced.

Not as much as you might think, but as an engineering researcher the goals are different which I prefaced my advise saying my background is physics and not math.

 

[marcusl]

50% wrong on engineering solutions would get you fired from my employer.

 

[clope023]

50% wrong on engineering solutions would get you fired from my employer.

You guys are taking the 50% vs 100% thing a little too literally, it's a general principle I've encountered vs a hard and fast rule; you obviously want to be as right as you can, but sometimes as right as you can be doesn't mean 100%. This is also probably taking away from the OP's thread.

 

[verty]

I am seeking advice on how to effectively and efficiently learn mathematics textbooks. Currently, I adopt the style of trying to prove theorems in the book before reading the provided proof. I have had good success in this; I noticed a considerable gap in experience between me and my peers in writing proofs as I also kept a journal on general "meta-cognitive" techniques when constructing proofs to help with my consistency. There is still a lot to be learned as I am far from being able to prove all the theorems in the text - I believe this to be impossible. Although, I am happy with my progress after almost 2 years I have been practicing this, but I feel like I am going too slow and it would take me far longer to reach "research level" mathematics. Do you people think this is style of learning is worthwhile and helpful in the long run? Why do you think this method is beneficial or not beneficial?

Terrell, do you not find it hard to stay motivated when you are staring down difficult theorems? I'm thinking of something like the Heine-Borel theorem, where the proof is not really obvious. Or do you think to yourself, I couldn't have easily solved that, so it doesn't matter that it was harder?

Because if you have done it for two years, it must be successful. I mean, perhaps it won't take that long to reach the harder math.

 

[mathwonk]

This is a highly recommended technique for learning to do research and for actually learning to understand material deeply. This method forces you to understand the idea behind an argument rather than just remember the trick.

You can see in a recent exchange posted in the homework help section about smooth injective maps, I resisted the temptation to read the argument linked to by another poster and eventually generated the proof myself. This made it more enlightening to me and forced me to observe a clue in the problem that led to the successful method of attack.

it also caused me to rethink other similar problems in other areas, where one deduces a strong conclusion about a map from just a weak hypothesis, namely injectivity.

 

[Terrell]

Terrell, do you not find it hard to stay motivated when you are staring down difficult theorems?

No. When I see a difficult theorem, I always wonder how far I could go before I get stuck and trying it on my own helps me see the motivation of the proof provided. The only thing not motivating for me are exams. Exams always make me not want to study at all, but I still do and it's really hard to stay motivated. Also, exams leaves less time proving theorems and forces me to just blindly use it - I hate it.

I'm thinking of something like the Heine-Borel theorem, where the proof is not really obvious. Or do you think to yourself, I couldn't have easily solved that, so it doesn't matter that it was harder?

I do not know why, but I still feel bad not being able to prove it, but not as bad as having a brain fart when failing to solve a very easy problem, unfortunately it happens.

Because if you have done it for two years, it must be successful. I mean, perhaps it won't take that long to reach the harder math

One of the root problems I have when studying is not knowing when to give up. I find it tough to move on. I feel like if I don't solve a problem on my own, I miss a key learning experience that would help me be successful down the road. How true is this.

 

[Terrell]

This is a highly recommended technique for learning to do research and for actually learning to understand material deeply. This method forces you to understand the idea behind an argument rather than just remember the trick.

Yes, I agree. But, how do I know when to 'give up'. For instance, trying to prove Sylow's Theorems. Also, in what particular ways does this method help in doing research? Thanks!

 

[FactChecker]

The theorems in a textbook were not meant to be research efforts. IMHO, it is better to look for common approaches and techniques in their proofs than to make up your own proofs.

 

[Terrell]

The theorems in a textbook were not meant to be research efforts. IMHO, it is better to look for common approaches and techniques in their proofs than to make up your own proofs.

I cannot disagree with you since the definitions and theorems are already 'nice'. However, do you think there is any benefit in reproducing your own proofs, especially to beginners like me?

 

[FactChecker]

I think it would be better to study their proofs and then to try your own proofs in any exercises in the book. The first are designed to allow you to understand the concepts and the latter are designed to allow you to practice specific learned techniques. After you have done that, it may be beneficial to look back at their proofs and see if you can do those proofs on your own.

 

[mathwonk]

there is no rule about when to give up and look at the proof. I was tempted several times to do that in the case mentioned above, but I was really happy when I did think of the right idea without looking. So now that I am retired and have all the time I want, I tend to not look and just keep trying for a long time. Mainly because I want to achieve the result of thinking of ideas on my own, not of just learning someone else's idea. I.e. what is your goal? learning how someone else solved a problem or succeeding in solving it yourself?

Solving a problem on your own, or generating a proof, is an example of research, hence strengthens your ability to do research. ai found this helpful myself in my career as a researcher, and I have heard the same from others. Indeed I think this is a standard view of professionals. Perhaps one of the earliest I read saying this was Oscar Zariski, one the founders of my subject, and another later in my career as a student, was Maurice Auslander, great algebraist. He used to say, if you think up your own proof, you will often find that your own idea will actually prove miore than you are shooting for, and you will have aklready generalized the theorem, thus having done some actual research just while learning to unbderstand someone else's.

Indeed this is not easy, and many students will find it challenging, but it is the preferred method of professionals. Even if you do not succeed in proving a theorem on your own, you will ususally succeed at least in thinking of some part of the idea to begin the proof. Then when you do have to read the proof, you do not need to read that part you have thought of yourself.

 

[Terrell]

Solving a problem on your own, or generating a proof, is an example of research, hence strengthens your ability to do research.

It does help me practice digesting a problem by talking myself through it. So in a way it also help me get into a habit of thinking more clearly. I realized that I need to "habitualize" techniques I discovered myself or have learned from others so challenging theorems coaxes me to good habits. Although, am I correct in thinking that this is only the "proving-lemmas" part of research?

Even if you do not succeed in proving a theorem on your own, you will ususally succeed at least in thinking of some part of the idea to begin the proof. Then when you do have to read the proof, you do not need to read that part you have thought of yourself.

Yes, I agree. It gives motivation to the ideas right away a lot of times. Thus, making it easier to read through.

 

[mathwonk]

As to when to give up, there were two theorems that arose in the previous discussion I spoke about, the Borsuk-Ulam theorem, which answered the question posed there, and the Jordan curve theorem, which could also have done so and went further. I succeeded in solving the Borsuk Ulam theorem after being tipped off that a proof occurred in an elementary book, which encouraged me it should be doable. I have not yet solved the much harder Jordan curve theorem in an elementary way, which occurs in another book I have and looks much harder and longer. I did generate some ideas, and have not yet looked closely at the proof in the book, but may do so soon. I already looked closely enough to see that some ideas I had were relevant, but the trick used in the book was not one I had thought of or even quite understand yet. This proof is more daunting to me since it is famous for having been proved incorrectly by many people even famous ones. But maybe I will try a little more.

Even if you do not succeed in proving even part of a theorem you will often at least identify what are the key things that need to be proved, whereas if you just read a proof the easy things will often look indistinguishable from the hard ones. So you almost always learn something by trying it sincerely.

 

[Terrell]

Even if you do not succeed in proving even part of a theorem you will often at least identify what are the key things that need to be proved, whereas if you just read a proof the easy things will often look indistinguishable from the hard ones. So you almost always learn something by trying it sincerely.

Great! I guess I will continue this practice. Thanks!

 

[mathwonk]

for more on my progress with jordans curve theorem, i only had one small idea which i did not know how to push through so now i have read the statement of the first lemma used in the proof and am thinking about why this lemma is true. i will try to at least prove this lemma but if not, i will read a little of it and do what i can. so i am reading bits of the proof and trying to understand them. this lemma is due to a famous mathematician eilenberg, and gives a clever generalization of the use of winding numbers. namely we know how to define the winding number of a closed curve in the (plane - minus - the - origin), about the origin. It is zero if the map from the circle onto this curve, then projected radially out onto the unit circle, factors through the exponential map of the line onto the circle. Eilenberg cleverly generalizes this to a notion of when any compact set in the punctured plane winds around the origin. He does not define the number of winds but does say that number is zero if again the radial projection of this set onto the unit circle factors through the exponential map. So he can say that either a set winds around the origin or it doesn't. Then he says that a compact set separates the origin from a point far away if the set does wind around the origin. This concept then let's him use this generalized concept of winding to compute the number of connected components of the complement of the compact set, since he knows how to say when it separates pairs of points. So now I am trying to understand why the origin lies in a different connected component of the complement of a compact set, from a distant point, exactly when that compact set does wind around the origin. So I am not really proving this hard theorem myself but I am looking at the proof little by little and trying to understand it in pieces, guessing ahead whenever I can.

I had another idea that one might be able to prove there is a point that a simpled closed curve does wind around by looking at a point near the curve and then moving slightly across to the "other side" of the curve, buit I had trouble identifying the "sides" of a curve that could wiggle around a lot. Then I saw that in my book Dieudonne' does the proof in two stages: first he assumes the curve contains a short straight line segment, which I assume he does because then it is easy to say how to move across the curve along that segment. So my idea had some value but it did not occur to me to me to make an additional assumption. But this is another smart technique: make the problem you are trying to solve easier by adding a hypothesis, then maybe the insight you obtain from solving that easier case will help you complete the proof. So another trick for trying to prove theorems on your own is, when they are too hard, prove an easier case first.

 

[Terrell]

for more on my progress with jordans curve theorem, i only had one small idea which i did not know how to push through so now i have read the statement of the first lemma used in the proof and am thinking about why this lemma is true. i will try to at least prove this lemma but if not, i will read a little of it and do what i can. so i am reading bits of the proof and trying to understand them. this lemma is due to a famous mathematician eilenberg, and gives a clever generalization of the use of winding numbers. namely we know how to define the winding number of a closed curve in the (plane - minus - the - origin), about the origin. It is zero if the map from the circle onto this curve, then projected radially out onto the unit circle, factors through the exponential map of the line onto the circle. Eilenberg cleverly generalizes this to a notion of when any compact set in the punctured plane winds around the origin. He does not define the number of winds but does say that number is zero if again the radial projection of this set onto the unit circle factors through the exponential map. So he can say that either a set winds around the origin or it doesn't. Then he says that a compact set separates the origin from a point far away if the set does wind around the origin. This concept then let's him use this generalized concept of winding to compute the number of connected components of the complement of the compact set, since he knows how to say when it separates pairs of points. So now I am trying to understand why the origin lies in a different connected component of the complement of a compact set, from a distant point, exactly when that compact set does wind around the origin. So I am not really proving this hard theorem myself but I am looking at the proof little by little and trying to understand it in pieces, guessing ahead whenever I can.

I had another idea that one might be able to prove there is a point that a simpled closed curve does wind around by looking at a point near the curve and then moving slightly across to the "other side" of the curve, buit I had trouble identifying the "sides" of a curve that could wiggle around a lot. Then I saw that in my book Dieudonne' does the proof in two stages: first he assumes the curve contains a short straight line segment, which I assume he does because then it is easy to say how to move across the curve along that segment. So my idea had some value but it did not occur to me to me to make an additional assumption. But this is another smart technique: make the problem you are trying to solve easier by adding a hypothesis, then maybe the insight you obtain from solving that easier case will help you complete the proof. So another trick for trying to prove theorems on your own is, when they are too hard, prove an easier case first.

Yes, proving easier cases also did help me a few times in the short past I've been doing this. Also, I like the idea of peaking bits of the proof then trying to see where one can take it from there; I'll definitely do this when proving much harder theorems. I don't know what a Jordan curve is yet, but I believe this might be useful in the future, especially when you provided bits of introspection in it.

 

[mathwonk]

a jordan curve is a "simple closed plane curve", i.e. a subset of the plane homeomorphic to a circle. the theorem is that such a curve sepoarates the plane into two open connected components, one bounded and one unbounded, both homeomorphic to an open disc.

the lemma i am trying to understand now says that a compact subset K of the plane separates 0 from infinity if and onlky if the map projecting K radially onto the unit circle can not be factored through the exponential map. Now I think i see that one direction is easier than the other. namely if the set K does not separate 0 from infinity then there is an arc connecting 0 to infinity, which we think of as on the sphere, i.e. the plane plus a point at infinity. Now it seems since the complement of K is opoen we can take this arc to be rather nice, e.g. a finite union of line segments. and the it seems that we can deform the sphere until this arc is a meridian, i.e. a straight arc from north to south pole. then the set K, lying outside this meridian, projects radially onto a proper subset of the unit circle, and any such non surjective projection allows a logarithm, i.e. it factors through the exponential map. now this deformation may be hard to write down, but exists intuitively at least so now i see why that direction of the lemma is true. but i still don't see why the converse is true.

I.e. supposing the radial projection does factor through the exponential map, how to find an arc joining the two poles? Indeed the proof in the book of this easier direction is only a few lines while the proof of the other direction is a page long. But now at least I have learned which is the harder direction. And sure enough it is the hard direction I need for some further arguments I want to make. Just reading the statement of a theorem in a book does not always provide you with knowledge of which direction is the more important or more difficult one, so trying to prove it can teach you that.

Ok I read the book proof of the easy direction of the theorem and it was easier than my idea. It was based on deformation, but instead of deforming the whole sphere, they just defomed the origin. I.e. they used the fact that if the set K does not wind around a given point then it also does not wind around any nearby point. Then since, being compact in the plane, it does not wind around the point at infinity, it also does not wind around a point near infinity. if we join the point at infinity to the origin by an arc missing the set K, then we can apply this argument little by little and move along the arc until we reach the origin, concluding that the set K does not wind around any point on the arc, including the origin. This argument actually proves that whether or not a compact subset K in the plane winds around a point or not has the same answer for all points of any given open connected component of the complement of K. I.e. if we define the mod two winding number as one or zero, according as K does or does not wind around the point, then the mod two winding number is constant on connected components of the complement of K.

Now this idea is not explicitly described in the book's proof, which just writes down a homotopy of a certain "winding number map". So even though I did not come up with this clean proof myself, by trying to discover a proof, and by coming up with the idea of using a deormation, I was able to see the deformation idea behind the book's proof, and hence to understand it better than just by reading the homotopy they write down.

Moreover now I have a principle I may be able to apply elsewhere, namely whether or not a compact set K in the plane winds around a point or not, is a property true for entire connected components of the complement of K. So since it fails for the point at infinity, it also fails for all points that can be connected to infinity by an arc missing K.

E.g. picture K as the unit circle. Then the complement of the circle has two components, and K does wind around (every point of) the component inside the circle, but does not wind around (any point of) the component outside of the circle.

By translating any two points a,b of the sphere to zero and infinity by using the transformation (z-a)/(z-b), we get the result for any two points of the plane. I.e. if two points lie in the same component of the complement of K then a certain winding number map is trivial.

Next the main point is to prove the converse, that if the map taking z in K to z/|z| on the unit circle, is trivial , i.e. factors through the exponential map from R to the circle, then one can connect the origin to the point at infinity, by an arc missing K. I have no idea on that yet, and the proof in the book is long. At least I have slightly simplified it by removing the apparent dependence on the points a and b.

 

[mathwonk]

Ok I tried unsuccessfully to prove that if there is a lift of the winding number map through the exponential map, then one can connect the origin to the point at infinity by an arc. Then I looked at the proof in the book and noted that they used a different formulation via a proof by contradiction, assuming the connected component of the complement of the set K is bounded and contains the origin, and then argued that assuming there is such a lift gave a contradiction.

'This reminded me that the same statement can have several different formulations, nd some can be easier to prove than others. So one might try different ways of stating the problem before giving up. I.e. to prove A implies B one can assume A and try directly to show B is true, but one can also assume A is true and B is false and try to get a contradiction, and lthough this is logically equiovalent, it mauy actually be easier to prove.

Now I am thinking that the real value of trying to do the proof oneself is that it forces you to spend a lot of time thinking about the theorem and the ideas underlying it. I.e. even though one may wind up reading the proof in the book, it is most useful if one understand that proof thoroughly rather than just nodding at it. So the goal should be to get to the bottom of why a theorem is true, and that usually means spending a good amount of time on it, thinking it through in as much detail as the depth of the result requires. If you have a good memory and can remember proof just by reading it once, that can actually be a hindrance to understanding it, since you think you know it just because you recall all the arguments. That is quite different from knowing why each one is used in just that way.

In particular now that I have glanced enough at the proof in the book, still only of the lemma, I have again closed the book and am trying to reconstruct in my mind the argument I glimpsed there. A few hints of mideas I did think of alone do occur there, such as considering the boundary of the complement of the given set, but there is another wrinkle I did not think of, of using the Tietze - Urysohn theorem to extend the assumed lifting map.

 

[Terrell]

but there is another wrinkle I did not think of, of using the Tietze - Urysohn theorem to extend the assumed lifting map.

I am also concerned about this. Among the multitude of theorems you've read over the years, when it comes the time that it could be useful, how do we realize they're useful at that particular moment? It's like we have to have a brain implant to guarantee that we'll be able to draw out the theorem when we need it which we otherwise would not have thought of.

 

[mathwonk]

just keep at it. i have recently had another welcome consequence of studying Dieiudonne's argument. I had read in Ahlfors a definition of simply connected subsets of the plane being ones whose complement is connected in the extended plane. Reading this current proof I have tried to prove that, and have failed. it turns out to be false. I.e. Dieudonne' focuses on the map from a subset S of the plane minus the origin, to the circle defined by z --> z/|z|. If this map fails to lift through the exponential map R-->circle, we say the set S wraps around the origin.

I was under the erroneous impression that such a set could not be simply connected. I.e. if we can wrap S around the circle, I tried to prove we can wrap the circle around 0 in S. But this is false. I.e. if we consider the graph {(x, sin(1/x)), say for 0<x≤(1/π)}, and add in the segment {0} x [-1,1] of the y axis, and then connect those up by a loop below the x axis, running from (0, -1) to (1/π, 0). This is a path connected, simply connected, compact set whose complement has a bounded connected component. I.e. this simply connected set does wrap around the points between that loop and the x axis, but there are no maps of the circle into this space that are not contractible to a point. I.e. there is a non trivial map fom this space to the circle, but not in the other direction.

Looking back at Ahlfors, I see he assumed his subsets were "regions" i.e. open and connected subsets of the plane.

Another thing I have noticed in carefully reading Dieudonne's proof is an oversight, where he tries to apply on page 253 line 6, the theorem of Janiszewski, to sets that do not meet the hypothesis of that result, i.e. his set C-D is closed but not compact. It is easy to modify the argument however to use instead a compact set (just intersect C-D with a large compact ball). Still it is fascinating that this argument seems to follow a famous trend of proofs of Jordan curve theorem, namely they often have gaps. This gap is easy to fill of course. Hopefully the rest of the proof is ok. I have already learned a lot though.