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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
>
>
>
>
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