Re: Landau letter, Re: Mathematica as a New Approach...

*To*: mathgroup at smc.vnet.net*Subject*: [mg127999] Re: Landau letter, Re: Mathematica as a New Approach...*From*: Andrzej Kozlowski <akozlowski at gmail.com>*Date*: Sat, 8 Sep 2012 03:07:28 -0400 (EDT)*Delivered-to*: l-mathgroup@mail-archive0.wolfram.com*Delivered-to*: l-mathgroup@wolfram.com*Delivered-to*: mathgroup-newout@smc.vnet.net*Delivered-to*: mathgroup-newsend@smc.vnet.net*References*: <7061851.89395.1346314334093.JavaMail.root@m06> <k1pr1o$k10$1@smc.vnet.net> <20120901062907.28A1F6892@smc.vnet.net>

On 1 Sep 2012, at 08:29, John Doty <noqsiaerospace at gmail.com> wrote: > I also note that experimental psychologists, notably Macfarlane and Tolman back in the 1930s (!), have established that even laboratory animals are capable of constructing abstract models, specifically "maps" of mazes. Their experimental results are inconsistent with the idea that in learning a maze, the animals merely learn the sequence of steps needed to solve it. Abstraction is a useful mental ability that is hardly confined to mathematicians. I am quite prepared to believe that "abstraction" can be very useful to "laboratory animals" - for example, it may be useful in getting the food that placed at the exit of a maze by an "experimental psychologist" or even better (though no doubt less frequently) getting out altogether out of the laboratory (alive). Moreover, it strikes me that the kind of "abstraction" these clever animals are capable of is closer to what the "average man in the street" is capable in mathematics than the latter is to the true mathematical geniuses. If we look at any history of mathematics we will find that it is rather like the history of art - the contribution made by "ordinary men" or even "ordinary mathematicians" hardly features in it. It is only a very tiny fraction of mankind that is is actually responsible for virtually all the advanced mathematics that exists today. So it seems to me that the really interesting question is not how people learned that 1 + 1 is 2 but how people like Grisha Perelman come to exist, do we need more of them and we get more of them, for example, with the help of Mathematica or Conrad Wolfram's "Computer Based Math Education"? We are told that evolution has something to do with the development of mathematics. No doubt at some level it is true, but hardly at a level we would find interesting today. It is claimed that "mathematical ability is useful". Sure, but for how long in human history has it been more useful than, say, powerful muscles or running or throwing abilities? Historical examples, e.g. Archimedes and the Roman soldier or Abel's death in abject poverty are not terribly encouraging. Perhaps people with high mathematical abilities are then better able to attract the opposite sex (which would give them an "evolutionary advantage")? I can almost hear the bitter laughter of my departmental colleagues. So what exactly is the evolutionary path from a near "laboratory animal" to Riemann or Perelman? It does not seem, I think, to lead via the ordinary man, at whom Conrad's educational ideas are addressed. Personally, I do have an answer to this that satisfies myself, but it is not as entertaining as the ones that have already appeared in this thread so I will keep it to myself. But I have no doubt that better teaching methods (if they are indeed better) will not make the slightest impact on the number not only go mathematical geniuses but also on the number of good, professional mathematicians who sometimes manage to make some impact on the footnotes of the history of mathematics. So, since this thread has nothing at all to do with the education of "real mathematicians" (and since it is obviously they are unlikely to be converted into something else by arguments such as that what they do is "meaningless" and irritating for certain non-mathematicians) the real question seems to be: do we really need to make many people better at the "other kind" of mathematics? Or, is an increased competence in using Mathematica people with little mathematical interest or aptitude going to be of a serious benefit to them or to the rest of society? Well, I think the jury is still out on this one. I can see one obvious benefit: to Wolfram Research. Also, perhaps to people like myself, who sometimes get asked to teach this sort of thing. But if this is going to happen at the expense of turning people away from non-mathematical subjects where their real interests and talents like to make them second-rate computer mathematicians, I think they and society will be the poorer for this. In fact, I have yet to see any convincing argument that more mathematicians (even of the very best kind by any definition) is what society needs (people who think so should reflect on the fact that the Soviet Union, particularly the university of Moscow, had a fantastic school of mathematics, both pure and applied, and how much good it did to it). Of course I am not addressing the one serious issue that was raised by the original poster - the proper mathematical education for many or most experimental scientists, including physicists. In this respect, I actually agree with Alexei. But as for the attempts to extend the discussion beyond its original scope. Andrzej Kozlowski

**References**:**Re: Landau letter, Re: Mathematica as a New Approach...***From:*John Doty <noqsiaerospace@gmail.com>