it seems to me there is a lot of debate around here about competitive approaches eg “the two cultures” and “mathematicians vs computers” which at extremes border on adversarial. this shows up a lot in sowa’s writing. lets all take a deep breath. think about symbiosis & cooperation. its the natural/higher/global order between the society of mathematicians and between them and computers. its a feedback loop.

if you read gowers original paper on “rough structure and classification”, the interesting dialog with the computer is particularly *collaborative*. it is sowa that is emphasizing that computers could make human mathematicians obsolete and putting words into the mouth of gowers [this pattern continues on newer blogs on this site].

computers will never replace mathematics just as the field of chess has been *strengthened* with the advance of computer power.

it appears to me there will always be a tension between very difficult proofs that are *objectively* “true” but inscrutable and it will always take humans to rearrange and reorient the same proof in different ways, also called psychological “chunking”, that *subjectively* are more understandable and yes, *aesthetic*.

in this way there is a strong similarity to architecture and refactoring code from the field of software development. think this core analogy [between proofs and coding] will continue to become more prominent and strengthened in the future even with major advances in automated thm proving.

over the years Ive been looking at a particular model for human vs computer theorem proving. its the collatz conjecture. in some ways a toy problem, but in other ways, possibly the precursor to a new style of computer-assisted mathematics in a remarkable style.

along these lines: have some preliminary, promising results on the collatz conjecture related to computational analysis & am looking for volunteer(s) for a project that would bring about this new order into reality. needed: mathematical background, programming ability, and enthusiasm for pushing the [extreme?] boundaries of mathematical and scientific knowledge. the idea is to apply very deep new technical principles to a “toy” problem but only as a start, a prototype or “proof of principle” on the way to grander plateaus…

plz reply on my blog if interested!

http://www.nrc-cnrc.gc.ca/eng/index.html

http://www.nserc-crsng.gc.ca/index_eng.asp

I apologise. The former is supporting pure mathematics, not the latter. The new policy of the government does not seem to be as bad as I thought. There is something to worry however, since the NRC is supporting a wide variety of scientific researchs, including research in biology and environment.

Thank you very much for your written talk – I enjoyed a lot.

Yes, I already suspected that we are abusing the hospitality of our host. We may move to one of my Google places:

http://owl-sowa.blogspot.com .

The top post is for your comments. I suspect that you missed the reference to my previous post there with my comments about Bourbaki. Here the links are not very visible unless given in the http:// form.

I am planning to use that blog for a discussion of other issues related to this post of T. Gowers.

My experience showes that a discussion in comment is more convenient than an exchange of e-mails, since the whole thread is at one place. I plan to write you later (right now I have to go, anyhow). I am not announcing my identity here by completely unrelated with the discussion issues. I am prepared to stay by my words under my real name.

Instead of venturing into the political philosophy of capitalism, I would prefer to stay closer to mathematics. Things similar to what you wrote about NSERC do happen in its US equivalent, NSF, and even on the level of the Division of Mathematical Science. But I don’t see how this may be related to capitalism, triumphant or not, or to what party is in the power. Both agencies are purely socialist institutions and function in a way characteristic for socialist institutions. For example, exactly the same words as you quoted could be said by some USSR Communist Party apparatchik overseeing sciences, and, I believe, they were said many times.

While young non-tenured people may be forced to enter this rat race by external to mathematics circumstances, like the shortage of the new positions, the rat race of tenured people is a phenomenon internal to mathematics and we have nobody to blame except ourselves. Who is rewarded is determined by us by a very trivial reason: no administrator or an NSF officer can distinguish good mathematics from bad and even an expository work from purely original research (a good expository work requires a lot of creativity, in fact). Well, perhaps there are or at least were some exceptions in NSF, usually some former mathematicians (but usually even former mathematicians are not exceptions). Anyhow, any grant award, any promotion decision, and any salary rise eventually depends on the peer review. I think that there is no need to change the rules. Instead of this, we should change our own priorities, what we do recognized and what we do not.

My defense of Bourbaki turned out to be fairly long, and I posted it at another place. Also, I would like to defer my critisism of Encyclopedias and of the “Princeton Companion to Mathematics” for another occasion.

I do not have any big expectations for big collective online projects. To date, Bourbaki is the only example of a successful expository collaboration of several people. The Bourbaki enterprise is not reproducible, at least without some dramatic changes in the attitudes of the mathematical community, and even with an appropriate change, only very rarely.

What needs to be done collectively is exactly the change of attitude. We should appreciate the expository work much more than now, write enthusiastically about such work in letters of reference, say to the authors more often that they are doing very valuable work, etc. Then much more individuals or two-three authors together will start to write expositions, and, may be some bigger online projects will eventually mature.

As of your idea of dwarfing all previous efforts, I must say that I subscribe to Freeman Dyson’s maxim: “Small is beautiful”.

Probably so.
Higher education has expanded a lot during the second part of last century, starting after WWII. But the expansion seems to have slowed down recently. Let me discuss the general context. It is quite clear that we are now living in an era of triomphant capitalism, despite the lasting recession created by the financial system. I would like to give you a small example. In Canada (where I live) the grant agency supporting mathematical research is the NSERC (the Natural Sciences and Engineering Research Council). The freshly reelected Harper government (conservative) stated in his last budget, that “from now the NSERC will concentrate its energy to serve exclusively the priorities of the enterprises”. Wow!

Happily, the popularity of the Harper goverment is rapidly decreasing.

I would like to make it clear that I am not against capitalism. Surely, capitalism can be good, since it encourages initiatives and inovations via competion. But unreined capitalism is dangerous, it may destroy everything including itself. This is why democratic societies must impose stringent rules on corporations (like anti-monopolistic laws).

You wrote: “But what about people with tenure? What forces them to continue this race?”

I guess by self interest. Again, I would like to make it clear that self interest and competion can be good for academia. The problem here is that the rules governing academic research have been fixed a long time ago according to a pattern which is now partly outdated. The explosion of mathematical research does not translate into a broadening of the mathematical culture, except possibly for a very small number of peoples, if any. I fear that mathematics may eventually collapse on itself if it does not broaden its intellectual base. I feel strange when I meet someone who knows everything about nothing and nothing about everything. Hopefully, the danger will be recognised on time and the rules will be changed. Peoples contributing to the developpement of general mathematical culture should be better rewarded by the system.

Some efforts have been made in the past to unify and broaden the mathematical culture of the time. The Bourbaki group is famous. Despite their rigor, the mathematics of Bourbaki are poorly motivated and the applications are absent from the books. The Soviet Encyclopedia of Mathematics edited by Vinogradov is probably a better tentative, but I know it less. I guess that it has contributed to the dominance of russian mathematics during the last 20 years. The “Princeton Companion to Mathematics” edited by Tim Gowers is the latest example I know. Thank you Tim, your book is beautifully! I hope that Tim will not mind if I criticise his book a bit. I believe the book should have paid more attention to category theory, since it is possibly the most important tool for unifying mathematics.

Modestly, I would like to formulate a dream that many mathematicians have today. A new collective effort to present and unify mathematics should be undertaken. It should dawrf all previous efforts and it should be open ended. It should use the internet.

I dont know how such a collective effort may start. I would love to contribute to it with my modest means.

I do not really understand why do we have this rat race. It seems that it is a fairly recent phenomenon. Personally, not very long ago I had the luxury to develop a little theory in the course of seven years after publishing only an announcement, devoted to this theory only partially. (I did publish papers about other things.) The final result is a very short monograph; a rat race would force me to publish it as nearly 10 papers, which would be overlapping and interconnected in a complicated way.

Could it be the case that we do compete for a smaller and smaller number of positions in pure mathematics? But what about people with tenure? What forces them to continue this race?

Well, you quite nicely detailed what I meant: why there is “too much” mathematics.

I may add that the quality of mathematical papers is going down independently of developing countries. Take any top journal, like “Annals of Mathematics” or any other. During the last 20 years, the number of pages per year in “Annals” increased 3-fold, I think. There is no noticeable presence in “Annals” of mathematicians from China or India (I mean working there, not ethnicity). Most of the authors work in US, UK, rarely in France, and sometimes in other European countries. The number of positions did not increased, it is actually decreased. Inevitably, the level of “Annals” went down quite noticeably.

I don’t think that the quoted remark qualifies me as a pessimist (may be I am, but not because of this opinion). My point is there is no inherent good in producing more mathematics. It is not needed for applications (I already said this here: proofs are not needed, even in physics). Given the situation you outlined, more mathematics will serve no good for any individual mathematician. Today there is no way to learn in 100 years any sizable fraction of the already existing mathematics I would like to know (in particular, know about).

Craig Smorynski suggested about 25-30 years ago that mathematicians need to slow down the production new results for a while, and to put in order the things presumably done already. There are huge gaps in the literature, and many papers are hardly understandable. There are many examples from many branches of mathematics. The most distressing is the fact the some results or proofs are apparently lost despite their discoverers are still alive and well. They are just not interested in their old (and, occasionally, even new) results anymore.

I would like to stress that I am not speaking about the production of an average mathematician, I speak about superhuman insights of some of our contemporaries. Also, let me repeat, I am interested in the beauty of these insights and do not care much if a particular statement is true or not (probably, the generalized Riemann hypothesis is an exception, may be the only one).

I dont share your pessimism. Possibly because I am optimistic by nature,

It is true that mathematics is now too vast for one human been to know it all. It is expanding at an exponential rate (I would like to know the rate). More mathematics is produced every year (maybe every week) than what I can learn in my lifetime. The quality of the average mathematical paper seems to be going down. The number of mathematicians may double during the next 25 years, largely due to the contributions of developing countries like India and China (25 years is a rough estimate) . These developements will affect the mathematical culture (they are surely influencing it already). Mathematical knowledge appears increasely fragmented the barriers between the fields higher.

The traditional culture inherited from the age of enlightenment is under enormous stress, it maybe gone already. But something new may emerge from the ashes of the old culture. Can we figure it out? What we should do?

@Gowers: I am very interested in knowing your opinion on this. But my question could be out of the context you have created for this discussion. Please, let me know.

To be more precise, I am quite sure that one can conscious person devoid of emotions. Yes, such persons are usually classified as psychopaths (if they are not smart enough to hide this quality). By some estimates, about 5% of the population is such (I have no idea how accurate is this estimate, but at least it agrees with my own experience). But they are considered as humans, of course.

In the other direction the question is more subtle. Is it possible to suffer without being conscious? I believe that this is possible, at least to some extent. Let us look, for example, at sufficiently distant (in the biological sense) from us animals. It seems obvious that they can suffer, can be attracted to their mate or to a human, etc. But it doesn’t seem to be clear that they are conscious and if they are, to what degree. With the “emerging phenomenon” theory, if one wants a coherent point of view, one has to accept that *everything* is conscious, only to different degrees. Even stones, an even an electron should have a rudiment of consciousness. This is one of the issues Lenin recognized and dealt with. If we take such a position, the question disappears.

If one takes some other position, then it is natural to think that consciousness is needed for experiencing emotions. This only shifts the question. What is consciousness? I must admit that I don’t know what exactly the (post)modern philosophers understand by the consciousness. But nowhere had I seen a serious discussion of the following issue: is consciousness is just a receiver, a passive entity getting information from outside of it? Or it is also a transmitter, or, to put it better, is it active and creative? Personally, I believe that one cannot suffer without a receiver, but one can without a transmitter.

Starting with the words “In your reply to Hans”, I agree with every word you said, and don’t see any need for any qualifications or clarification. Here we are in complete agreement, including the last phrase “Why should we try?”

As for 2), my view is that very strong arguments have been put forward against qualia by Dennett and others, and nothing I have ever read has come close to countering them. So I agree that it is not settled, but it seems to me that it ought to have been settled. (I feel the same way about climate change, though my respect for people who believe in qualia is infinitely greater than my respect for people who don’t believe in man-made global warming.)

Also, if we were to simulate ‘doing mathematics’, would it be like simulating the weather or like simulating addition?

Being conscious and being able to suffer (or to love, which are two sides of the same coin) seem to be completely independent phenomena.

In general, it is fairly amusing to learn that modern thinkers had only repackaged some century-old ideas of a well known political leader, V.I. Lenin. The metaphor of zombies may be new, but the idea of consciousness as an emergent phenomenon is worked out in his writings; it did not appeared to be so natural at the time.

First of all, since than claiming that somebody is not a mathematician, but “only” a scientist is an “accusation”? Is it now a sort of crime not to be a mathematician?

Pure mathematics exists as other arts anyhow. You are paid only when you did something other people like a lot. If you do not manage to produce something like this in your early years, you are out of the profession. The experience shows that it is nowadays impossible to do pure mathematics in your free time, as it was possible, say, to Fermat. Other jobs are too demanding, and mathematics requires prolonged concentration, which is not compatible with regular jobs.

In fact, most of traditional arts had disappeared during the last century. The modern paintings are essentially a kind of financial instruments. Still, a lot of people may appreciate visual arts and may pay for them for the pleasure and not as an investment. The situation in sports is similar. A lot of people appreciate chess even if they only know the rules. There is quite a lot or rich people among of them, and they played an essential role in supporting chess.

The situation with mathematics differs dramatically. The good current mathematics can be appreciated only by other professional mathematicians. You can do something incredibly beautiful, but it will be understood and appreciated first by a half-dozen of your closest colleagues. Even later, when a result finds its way in research monographs and then advanced level textbooks, the appreciation remains inside of the mathematical community. So, there is no outside source of support for this kind of beauty, and the mathematical community relays on at least potential usefulness of its production, as opposed to its artistic value.

Next, about the joy. There will be a joy of doing something better than a computer did. This kind of you exists without computers also; one may prove a theorem better than it was done the first time. But this joy is not comparable with the joy of doing something first time. What to do? Keep your ideas in secret from assistants to theorem-proving machines? They will spoil the joy anyhow, reproducing your result within a day.

I fail to see how a computer can be a mighty (or not) collaborator in the sort of mathematics I like. Computer may help to prove some theorem, like to (in)famous 4 colors conjecture, but there is nothing beautiful there.

I also fail to see why we (the human race) may need more mathematics. We already have too much. And why this computer mathematics will be beautiful. All beauty I ever seen in the output of computer was injected into it by humans before the computer started working.

The idea that pure mathematics is useful and is actually used is a fortunate (for mathematicians) misconception. The heart of pure mathematics, the proofs, is not needed for any applications. A heuristic argument together with an experimental verification is always sufficient.

In case I’m misunderstood, I don’t think consciousness is a black and white phenomenon. So if you had a sequence of ever more sophisticated computer programs, starting with today’s programs and ending with something that had software more or less identical to our brain’s software, I’d say that these programs would start out with virtually nothing that deserved to be called consciousness and would end up as conscious as we are. In between, they would have something intermediate.

What do you mean when you write that the brain “somehow produces consciousness”?

If computers would be able to solve important mathematical problems and develop important mathematical structures/tools, then this wouldn’t prohibit you from doing mathematics nor would it destroy mathematics as an art. Quite the opposite, it would just put mathematics on the same scale with other arts – you have to do it out of your free time, and you only get payed when you do something very beautiful. Anyone who really loves mathematics, would still do mathematics. There would be even extra joy coming from the facts that 1) you have a mighty collaborator 2) much more (also beautiful) mathematics gets done 3) mathematics is potentially much more widely used.

So in fact, your accusations towards prof. Gowers of not being a real mathematician could using a Tolstoian argument be very easily turned against yourself.

I think no-one who understands basic computer science would seriously claim that it is impossible for a computer to do something arbitrarily closely resembling human thought (whether you believe that is equivalent to being human, or genuinely self-conscious, or simply clever simulation seems to me to be a religious matter, and I’m not). If nothing else, one could in principle simulate the entire workings of a brain (noting that with current technology we cannot determine these workings well enough, but there isn’t any reason why it should stay impossible). So in principle we can replace Serre with a computer simulation, and it would no doubt show incredible intuitive abilities. But in practice, we do not have anything like the hardware to do this, nor are we likely to get it any time soon (i.e. here long-term could mean centuries)

On the other hand, it seems unlikely that simulating one particular computer which happens to run a desired algorithm is the best way to run the algorithm. So one could hope to somehow abstract the algorithm and run it on something more like current technology. In principle there is no reason why this shouldn’t work, and there is probably no reason why we should not be able to do it in a year’s time: it’s likely the solution will be simple (as compared to say the Windows code-base). But it’s probably also true that solutions to most major mathematical problems are similarly simple. I think it’s fair to compare the situation to the P=NP conjecture: nothing we have tried comes close, the only results we really have are of the form ‘this approach cannot work’. We still might solve it next year, but we probably won’t.

As some kind of middle ground, it’s possible we could develop a program which produces the kind of results we want but without our understanding how it works; evolutionary algorithms aren’t new, some of them really solve things in unexpected ways (I remember a decade or so back there was an FPGA circuit which solved a problem in a way which the researchers didn’t understand until they tried the same circuit with a different component layout: this didn’t work, and eventually it was realised that two neighbouring but unconnected components on the FPGA had a capacitance which was critical to the function but wasn’t ‘supposed’ to exist). Since these things are basically massively parallel maybe one could run an evolutionary algorithm in the style of SETI@home and get something useful. But it’s neither clear how one would design a scoring function for the algorithms being evolved, nor how long it would take.

Experience shows that mathematical thinking is depending on the alternation of two kinds of activities. One is about developing a mental picture of the problem, an intuition of its nature. The other is about making a calculation, as in algebra or formal logic. The two activities are performed alternatively as in a dance. A calculation is guided by an intuition, and the intuition is confirmed or rejected by a calculation. Few mathematical progresses can be made by using intuitions or calculations alone. Computers are very good at computing and they can help us doing mathematics, like a pen and a piece of paper. But they have no intuition, no consciousness. They are mindless mechanical machines.

I claim that we do not understand the nature of consciousness. Of course, we are making progress in studying the brain. But artificial intelligence seems to show that the brain does not need to be conscious to perform its tasks. So why should the brain produces consciousness? Consciousness is the blind spot of modern science.