(Long): Questions about the field of mathematics and the RH from a newbie.

[stefanB]

Prologue:
Hello, I am new to this forum. This is my first post. I come from Sweden, discovered an interest in mathematics when I was fourteen, pursued it in fluctuating vigour for two years until I dropped it completely for a whole three (personal reasons).

Now I'm nineteen, and just a few weeks ago I picked it up again and am, at least in my standards, studying it intensely and in great speed: I'm re-reading the techniques I learned in senior high school, that is courses C, D, E and F, where C begins, more or less, with derivatives and where F ends with, roughly, vectors and matrixes and other things I can barely spell.
Carrying on in this speed I will be done with both C and D in less than ten days, E in less than two weeks, F being unknown since I haven't got that book yet, but likely in that same kind of rate.

* * *

I'm very interested in this kind of stuff. I like to write down little mini-me conjectures which would be easily solved by someone like de Branges or by a lot of people in these forums aswell, but with the little knowledge I have now, they're all very absorbing since I have about... yeah, no advanced mathematical techniques to use at all.

What I've seen so far, peeping at complex problems like the Riemann Hypothesis; reading about people like Gauss, Euler and Riemann himself; reading about the reactions to de Branges purported proof: is that it seems like a vilely cold and cruel twilight world where the one with the highest genius wins, where the one who is sacrificing the largest portion of his private life in search for knowledge and the one who knows the most and has the best connections win the prize.

Look at the purported proof from de Branges latest outburst: few people in the world understands the technical details, and...
and we'll stop there. Few People In This World Understands The Technical Details Presented There. From that I draw the conclusion that proofs on that level is a technical game, and the only thinking processes involved are in learning them and simply using them according to by-and-by-spawned rigorous rules. (As the new technique takes shape.)

It saddens me that there is few or none willing to even look at his proof (I will exclude "purported" because I hope that you understand that it is implied); an American professor waved it away by saying, "...I know it's wrong..." and others do the same pointing at the mistakes he did in the past. Is it just me having the feeling that they just might be doing that in fear that he [de Branges] will beat them to it? A lot of people seem to claim they've found a proof. They lie helter-skelter all over the place. Maybe it sounds ridiculous, maybe it sounds like everybody else, but I heard about the Riemann Hypothesis when I was sixteen, at the peak of my interest, and I've wanted to solve it since. What it looks like now is that I can safely just drop everything and go get a job with big drills instead, given that it looks like a technical game where the college professor having studied the field for the longest amount of time is the winner.

***

The cases I have brought up here are not necessarily meant to be isolated. When I speak of the "technical game" I hope you know what I mean. If not, then perhaps I'll have to come back in a few days with a better explanation. When I bring up de Branges and his proof and the reactions to this I do not necessarily mean only the happenings around that - I mean that with these types of people scurrying around, it is more than just possible that it will happen again. I have seen no ethics in the mathematical community, no moral, no support except to those at your very university, and certainly no joy in thinking but rather satisfaction with new, complex techniques.

***

I hope I am so wrong about all this that I'll regret that I ever posted, but this worries me and disheartens me to continue studying; if I have no way of proving the Riemann Hypothesis (my dream is to do so, or the equivalent, in the future), then why not just drop everything? - if the field of mathematics is a technical terms game then it cannot, ultimately, intersect with my philosophy and view on mathematics in the same degree.

***

I apologize for the long post. It was unskillful and clumsy. But hey, B nice remember I'm new, yo.

 

[mathwonk]

You seem discouraged. You are young and the world is open before you. Look around, enjoy it. Forget about fame and especially the injustice of fame. Find pleasure in honest work.

Good luck in what you choose.

 

[HallsofIvy]

I have seen no ethics in the mathematical community, no moral, no support except to those at your very university, and certainly no joy in thinking but rather satisfaction with new, complex techniques.

I can't help but wonder just how well you KNOW the "mathematical community". Just to give one example, the proof of "Fermat's Last Theorem" involved people from around the world cooperating. My experience is that morale in the mathematical community is pretty darn good.

I also do no see how you conclude, from the statement the "few people understand", that proving things is a "technical game". Certainly the at the deepest levels of any endeavour there are difficult concepts and "technical details" that must be mastered. That does not mean that after you have mastered those details creativity and thought are not necessary!

Hopefully, you will learn soon that just sitting back and talking "off the top of your head" is not what "creativity" is all about, in any subject. If you don't you will have a very hard time in college.

 

[stefanB]

Halls of Ivy: No I do not know anything about the mathematical community (apologize for that being unclear. I tried to with saying I was new and all) - that is one of the reasons why I asked about it in the first place; the very little of it I had read and seen had given me a (an apparently) wrong image. Thank you for sorting it out.

About the technical game I meant more that the fields de Branges for instance used to present his proof was relatively unknown to a vast quantity of peers. It makes his story look as though written in Greek when most are handling English, so to say.

Have no clue at all what you mean with talking "off the top of one's head". If you're saying that I was just rambling and blabbering then why the euphemism? - and why are you linking the expression with your impression of my view on creativity?
I am aware of the fact that talking off the top of one's head does not equal creativity... look man, I just don't know where you got that from. As I said, if you mean that I was just having a word-vomit then just say so.

I think it's genuinely interesting that although I stated I was new and knew oh so little, my ignorant questions and childlike views still brought such a defensive reply.

 

[mathwonk]

you got two repiles and you focused on the one you could consider defensive. that is why i tried to encourage you, you are taking a negative outlook. that outlook will make lots of things look bad. look up, not down.

 

[HallsofIvy]

There were no ignorant questions in your post! In fact, if you go back and review your post, as I just did, you will find that there were no questions at all. One reason I responded as I did was that you did NOT ask any questions- you simply stated your opinions. And now you appear to be saying that those opinions are based on very little fact. Nor do I see why you would think my reply was "defensive". I thought I was being pretty "offensive" (in the offensive-defensive sense of that word)! As for "talking off the top of your head"- that may be a colloquialism you are not familiar with. It does NOT mean "rambling". When I am talking casually with friends about, say, who I think is going to win a soccer game, I am pretty much talking "off the top of my head"- because I really have little knowledge about the players- but I am not "rambling". Talking "off the top of your head" means giving an opinion with having done the deep research than may be necessary to give a firm answer. (Nor is it necessarily perjorative: A person specifically asked to give an opinion on a subject where he knows more research should be done might say "Well, I am just talking off the top of my head here ...".)

Once again, I have absolutely no problem with people asking "ignorant questions"- but there were no questions in your post.

 

[stefanB]

mathwonk: yes I do in fact have problems with my general negative outlook on things. I don't know where that has come from but it so often leads to the truth that it is tempting to state that pessimism equals realism, with little or no exceptions. So, yeah, that is probably why I describe myself as talented in nothing and that, specifically, I don't really have anything but an interest in mathematics.
An elaboration of this is that I feel it would be ridiculous to say one is good at anything because it always comes down to a relative comparison; it is so unlikely for a rising writer to surpass the level of, say, James Joyce, that he or she could just give it up and apply for a cleaning job at Burger King, and it is so unlikely for a classical composer to get at the same level as Beethoven and Mozart were, that he or she could do the same.
(Assuming those types of levels are their aim, naturally.)
Likewise in mathematics - no one can compete with people like Gauss and Euler, so why even say that one did "ok" with or at something, knowing those two would have done it so much better? "Ok" is compared with what, exactly? You did good? Okay, well, good according to who, you? - and, in comparison with what, him? (Pointing at someone at random.) See my point? Agree or disagree?

Halls of Ivy: I understand what you mean now, but not why you drew the conclusion that I considered creativity to be equal to it [talking off the top of one's head].

Finally: what is everyone's guess at how long it will take for someone to present a proof for Riemann's Hypothesis? (Assuming de Branges's purported will never be checked.)
Speaking of it, does anyone think that his proof is correct or not? And, why so?

 

[ObsessiveMathsFreak]

I remember reading an article where it was said that many mathematics proofs are no longer "fully checked". They were so complicated, long, technical and obscure that the reviewers simply declared that they couldn't find anything wrong with it, but neither could they completely verify it. The proofs are then published in a peer reviewed journal.

Now to me, it sounds like no one had actually proved anything. The supposed proofs were really only wordy conjectures which the author had not fully justified. They were probably published because the author's name carried some weight or because his paper was extremely wordy and technical. Essentially this is proof by intimidation or browbeating.

Proofs of this kind, where no one really proves anything, are very common in mathematics. If you want too see lighthearted but yet frank examples of this sort of thing, look for Jean-Pierre Serre's talk; "How to write mathematics badly". (It was on Google Video but seems to be missing now)

The essential problem with modern mathematics is this. At its highest levels, it is no longer a scientific discipline. I mean this is a very serious way, and it is a very, very serious problem with modern mathematics. Why is it not scientific? To answer this, we must recall what scientific means.

There is the notion of falsifiability. That for a theory to be scientific, it must be possible to construct and experiment or test which could potentially prove the theory to be incorrect. Now arguing over whether mathematics is or is not falsifiable could get us into a long heated debate, and in any case, I feel the falsifiability condition has been elevated in recent years to refuted intelligent design theories and so forth.

For me, and for the basis of my argument, the true measure of science is the experiment. Putting things to the test, over and over, to make sure you are right. The experiment is foundation of the modern scientific method. We run experiments to put our theories to the test(and also to discover new ones).

In mathematics of course, the analogue of the experiment is the proof, that which puts the theorem (the analogue of the theory) to the test. The difference between the experiment and the proof is that in the proof we put everything to the test, whereas the experiment is obviously constrained to only a small subset. But essentially, the proof serves the same function. It is the test to which the theorem is put to measure its correctness.

So you might say my point is absurd. Mathematics is scientific, indeed a complete science, because it has experiments just a physics, chemistry, etc all do (In fact it has better ones.) So what is my point?

There is one crucial fact about scientific experiments that I have left out. It is this; "A scientific experiment must be repeatable".

It is not enough for someone to perform an experiment just once (say measuring the mass of an isotope) and declare their theory valid. No. They must describe their experiment in key detail, giving all the instruments and materials they used, and all the data they gathered, and then others must repeat that experiment, often with many variations of instruments and setups. Only when the experiment and its results have been performed and repeated by others is the theory accepted as valid.

In mathematics we have proofs, not experiments. But the same conditions must hold. Exactly like experiments; "A mathematical proof must be repeatable". If it is not repeatable then it is not a proof at all. When I say that modern mathematics at the highest level is not longer a science, what I mean is this;

Modern mathematical proofs are no longer repeatable.

It's a pretty damning statement, and I don't say it lightly. However if you look into any modern, advanced mathematical proof, you will invariably have to come to this conclusion. One of the examples brought up in Serre's talk is the following, which I'm sure you have come across.

Theorem 3 (*some statement*)
Proof: It follows from the definition.

Disaster. Many proof fare little better, with only the vaguest of outlines of how the proof is to be performed. I read elsewhere that mathematicians go through the process of "verification" of a proof, which to me sounded like coming up with an entirely new proof altogether. Invariably this turns out to be necessary. Essentially, the author did not supply a repeatable proof, or experiment, and mathematicians are required to designed an entirely new one.

What is happening here is that instead of the proof being verified, the theorem is. You may say, what is the difference. There is a very great one. For example in physics, a theory might be:

Theory: The charge of the electron is ~1.60217646 × 10^-19 C

Imagine if the experiment to verify this consisted of the following.

Experiment: It follows from the measurement.

This would never be tolerated in other sciences, but is done daily in mathematics. A less facetious example might be.

Experiment: Pass charged drops of oil through an electrically potential difference P and measure their terminal velocities with P=0, and later with P >0 . Hence infer the forces on them. The Forces will be quantized to multiples of e, the required charge.

An absolute barebones outline of Mikalin's oil drop experiment. Left to figure it out, a physicist might eventually realize that he needed to infer various ratios and quantities using drag coefficients etc, and perhaps he may somehow be able to come up with the same experiment. However I rather doubt it. Various minor, yet essential, points are left out entirely. For example the type of oil, the strength of the voltage, the viscosity of the air. The calculations to perform, etc, etc. This would never be accepted as a repeatable experiment.

Mathematical proofs are essentially as the experiment outline above. The barest and vaguest minimum of an outline, with little to none of the essential points. Exercises are left to the reader, problems are waved over or ignored, pitfalls are left covered. In short the proof is not repeatable.

I've ranted for long enough, but I'll stand by my point that the modern mathematical proof is an institution in need of much overhaul. Mathematics is better off as a science, and though large parts of it are, much of it is currently not.

 

[MathematicalPhysicist]

OMF, i totally disagree with you.
maths cannot be regarded as exact science discpline, but not because there are some proofs in texts that say that it follows the definition, how would it be if the author proved everything in the text and hasn't left interesting exercises for the interested reader?
and the author can't prove everything, and also shouldn't, and usually proof that follows immediatly from definitions are such triviality.
anyway, the same thing is also in physics books, so i don't see how they differ.
maths has some similarities with exact science by the conjuctres, but the way to verify/falsify is different, and you forgot to mention that ES use the maths as a model, so it relies on the mathematical theory to be consistent.
we have differnet methods of proofs in maths, and sometimes you also need to invent a new field in order to prove a certain theorem.

 

[Crosson]

There is a distinction between a proof by elementary methods (geometry, arithmetic, calculus, combinatorics), and those by non-elementary methods (model theory, functional analysis, advanced abstract algebra). It is the latter by which the majority of significant 20th mathematics has been proven. In all cases, an elementary proof (of which an amateur can be capable) is desired.

In other words the "technical game", as you call it, is an education in powerful tools for working with the core objects of mathematics (which still are just numbers, figures and patterns). Nothing is more interesting then a proof that comes from outside this framework, since it will often contain new ideas and in any case is a non-trivial accomplishment, just like defeating a bear with a pocket knife rather then a shotgun.

 

[stefanB]

Oh, ah. Ha he hi ho hu.