Re: Re: New version, old bugs

*To*: mathgroup at smc.vnet.net*Subject*: [mg44186] Re: [mg44151] Re: New version, old bugs*From*: Andrzej Kozlowski <akoz at mimuw.edu.pl>*Date*: Sat, 25 Oct 2003 06:26:42 -0400 (EDT)*Sender*: owner-wri-mathgroup at wolfram.com

Maxim's postings stimulated me to re-think my ideas on Mathematica and try to articulate the vague feelings that I have had about it for a long time. The result surprised me: I now feel that I have at last got an insight into the nature of the program that has baffled me for so long and in particular why some love it while others find it frustrating, and moreover why many experience both emotions at the same time. The comments below contain no specific examples and are not directly relevant to anything in Maxim"s postings, but I think are relevant to some of the general "complaints" has voiced. The specific examples he provides may well be bugs but I think the general issues he raises are not themselves bugs but actually involve something much deeper. When considering these matters the first point that I think should be kept in mind is that the main designer of Mathematica is not a mathematician or a computer scientist but a theoretical physicist. Moreover he is a very eloquent physicist whose views on such things as physics, mathematics and the relationship between them are very well known. Perhaps his views on the design of Mathematica are somewhat less clearly expressed (as far as I know) but it seems to me that they can be easily deduced form his views on mathematics. In many ways these views are quite typical of many mathematically minded physicists, including some of the most famous ones. One should first be aware that a vast amount of what today is considered "pure mathematics" was actually created by physicists. In some cases, for example Archimedes, Newton or today E. Witten it is not even clear whether we are talking of a physicist or of a mathematician (Witten calls himself a physicist but has a Fields medal in mathematics). Others, like Dirac, Einstein or Feynman (to mention just some who made a huge contribution to mathematics) were definitely physicists. One thing that strikingly distinguishes the mathematics created (or discovered) by physicists from the one that emerges form the work of pure mathematicians is that its primary motivation is derived form concrete physical problems. In fact for a physicists there are two essential factors involved in this process of inventing new mathematics: the needs of the physical problem he is dealing with and his "intuition", which is something difficult to describe, but which tells him what sort of things are likely to work and how they ought to be applied. The third factor, which is hardly of any importance to most physicists but of much greater one to most mathematicians is the formal basis of the concepts and definitions being created: their internal consistency (what is called being "well defined"), their relation to the already existing body of mathematics etc. For most physicists, if a mathematical idea "works" and if it is "intuitively" felt to be correct than it is as good as correct. They rarely bother about much more. Most (but not all!) mathematicians view these things differently: in their view concepts have to be well defined, theorems have to be properly proved and so on before they can be considered "mathematics". Even on this point however, there is no agreement among even the best mathematicians. In the past many of the greatest (Euler being a very notable example) were quite prepared to rely on their intuition without much care about what is called "mathematical rigour". The same has been true of some of the greatest 20th century mathematicians, one example being Rene Thom who hardly ever bothered to give proofs of any of his many remarkable theorems. The whole issue is still hotly disputed and I suspect will always continue to be so. Anybody interested in this dispute including the contributions of some of the greatest living mathematicians should look at the article: A. Jaffe and F. Quinn, Theoretical Mathematics: Toward a cultural synthesis of mathematics and theoretial physics, Bull. Amer. Math. Soc. 29 (1993), 1-13. and the responses in the subsequent volume. Now when interesting mathematics is created in this way by physicists, many mathematicians feel that it should be given rigorous mathematical foundations and indeed this is often accomplished. Moreover, when it is done successfully I think most people would agree that our understanding is enhanced. Perhaps the most familiar example of this kind is Dirac's invention of the Delta function (DiracDelta in Mathematica), which was extremely useful and physically very intuitive from the start but was only given rigorous mathematical basic by Laurent Schwartz in his theory of distributions. But there is actually no guarantee that even intuitive ideas that seem to work well can actually be given rigorous mathematical foundations. For example, as far as I know, this has still not been done for the general case of the so called Feynman path integral and it may be that we will have to settle for some sort of "ad hoc' justification (basically a set of rules that are justified by the fact that they give the "right answer"). What has it all to do with Mathematica? Well, it seems to me that by creating Mathematica Steven Wolfram brought all these issues to the area of programming and software design. For Mathematica's programming language seems in this light like a perfect example of a "mathematical theory" created by a physicist, in other words 1. It is extremely intuitive 2. It is exceptionally useful 3. It is not necessarily fully self-consistent nor does it have a clearly defined "syntax" which specifies in advance what is "legal" and what is not. Like in "physicist's mathematics", this sort of thing is actually being "discovered" through the process of trying various things that "intuitively" seem natural and ought to work. Most of the time they do but sometimes they do not. When they don't it is not quite clear that one could actually make them work without breaking some other desirable property. It is neither clear that a process of establishing a "rigorous syntax' for Mathematica is worth the effort nor that such a process could be guaranteed to be successful. I think nothing illustrates this point better than the Mathematica's "Flat" attribute. The idea is extremely intuitive and very powerful. Yet when you try to think carefully about it you notice that not only its implementation is, in some sense "incomplete' (more precisely, it does not do everything one expects it to do) but that it probably could never be made "complete" simply because in designing computer software one has to be concerned with "usability" just as much or perhaps more than with rigour and "mathematical correctness". In spite of this I think Mathematica possesses the best and the most powerful mathematical programming language there is. However, if what I wrote above is correct than it suggests that not everyone will be happy with it and clearly not everyone is. If you are not comfortable with the fact that often you won't be able to tell in advance if a piece of code will work, and that you will sometimes need to experiment and made adjustments, then Mathematica is not for you. But that may well be the price that must be paid or its "intuitiveness" and for me this is a price worth paying. Andrzej Kozlowski Yokohama, Japan http://www.mimuw.edu.pl/~akoz/ On Friday, October 24, 2003, at 05:24 PM, Maxim wrote: > A few more glitches of Mathematica's pattern matcher: > > In[1]:= > Module[{f}, > f[a] /. f[x_] /; True /; False :> 0 > ] > > Module[{f}, > Attributes[f] = Flat; > f[a] /. f[x_] /; True /; False :> 0 > ] > > Out[1]= > f$37[a] > > Out[2]= > 0 > > It looks like there's still a long way to go to acquaint Flat with > other > language constructs. > > In[1]:= > Module[{f}, > Attributes[f] = {Flat, OneIdentity}; > f[x] /. f[x_, y_:1] -> {x} > ] > > Out[1]= > {f$37[x]} > > I'm not sure about this one, should the result here be {x}? > > Another example: > > In[1]:= > Module[{a}, > {x^2, x^3} /. x^(p_) :> > (a = p; a*x) > ] > > Module[{a}, > {x^2, x^3} /. x^(p_) :> > (a = p; a*x /; True) > ] > > Out[1]= > {2*x, 3*x} > > Out[2]= > {3*x, 3*x} > > To understand how the last one works we need to know the evaluation > sequence; the number in parentheses shows whether rhs is evaluated for > x^2 > (1) or x^3 (2): > > a=2 (1) > True (1) > a=3 (2) > True (2) > 3*x (1) > 3*x (2) > > So the description of ReplaceAll in the Reference (saying that it tries > parts in the sequential order) is incomplete (or even not strictly > true), > because it switches between parts in this weird manner. Just as in the > situation with > > In[1]:= > Hold[x] /. x :> Module[{}, Random[]] > Hold[x] /. x :> Module[{}, Random[] /; True] > Hold[x] /. x :> Block[{}, Random[] /; True] > > Out[1]= > Hold[Module[{}, Random[]]] > > Out[2]= > Hold[Random[]] > > Out[3]= > Hold[0.011784819818934053] > > giving three different results, I cannot say that the expressions are > semantically equivalent because no one knows what the semantics should > be in > the first place, but it is very confusing when adding True condition > suddenly changes the result. > > Maxim Rytin > m.r at prontomail.com > > > >