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Re: More /.{I->-1} craziness

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106159] Re: More /.{I->-1} craziness
  • From: Leonid Shifrin <lshifr at gmail.com>
  • Date: Sun, 3 Jan 2010 03:41:58 -0500 (EST)
  • References: <200912300915.EAA17299@smc.vnet.net> <hhhmn8$o9t$1@smc.vnet.net>

Hi Richard,

Below I describe  rather extensively my view on the issues you raised.  Just
to make myself clear, it is not my intention to get involved in an endless
debate on these topics. I try to adhere to DRY (don't repeat yourself)
principle whenever I feel appropriate, so I detail my view on these subjects
below with the intention to do it only once. But I will certainly appreciate
your feedback.

On Sat, Jan 2, 2010 at 9:06 AM, R Fateman <fateman at cs.berkeley.edu> wrote:

> Leonid Shifrin wrote:
>
>> Regarding this issue, I think I entirely agree with what David Bailey and
>> other people said: I don't consider replacement rules as a mathematical tool
>> for end users, but rather as an inner layer of Mathematica, which is also
>> exposed for flexibility / convenience and intended primarily to be used by
>> the more advanced users.
>>
>
> Unfortunately many users or potential users are not as sophisticated in
> their understanding of the distinction between the underlying mechanisms of
> a syntax-driven
> transformation system.  They simply take the marketing blurbs about "A
> system for doing mathematics"  as a description suggesting that --hey, I do
> mathematics too.  They don't really know what "syntax" means and they don't
> think they need to know, because syntax is not part of their mathematics
> education.
>

Well, if these people don't understand the importance of syntax for doing
any formal sicence, regardless of whether it is done by a human or a
computer, and somehow believe that some software is able to completely
automate this problem away without any further efforts on their side  -  too
bad for them and their current and future employers. Every tool used blindly
will eventually produce nonsense. Mathematica is a research tool. I view it
as a tool for explorations, tests, verifications and sometimes discoveries,
but not a substitute for domain knowledge, intuition, right questions to ask
and anticipation for possible correct answers.

I have a fair amount of research experience in Physics (mathematical and
theoretical), and while I was frustrated with Mathematica at times, it has
been overall incredibly helpful for the problems I have been working on. And
the reason that I was sure about the correctness of my results was that I
was doing many non-trivial checks such as alternative derivations, numerics
vs. analytical results, limiting cases, asymptotics,  etc - this in the
first place, and my proficiency in Mathematica in the second. Also, I
probably have more diverse programming background than most professional
mathematicians and physicists - this is what I did (along with math)  as a
kid before I started doing Physics, and this is what I do now for a living
after having quit Physics (I have some asm, Pascal, Fortran, C++  and  a
substantial C experience and work currently as an enterprise Java / web
developer).  So, hopefully I have both perspectives on Mathematica.

What I think is that your dissatisfaction with Mathematica is a result of
the clash of cultures. From a programmer / computer scientist viewpoint,
Mathematica probably has lots of what can look as "undefined behavior" or at
least as a violation of the principle of minimal surprise. But research in
(pure) science is done differently. Most physicists and mathematicians I
know are basic Mathematica users but are generally quite satisfied with
Mathematica. They don't care as much about what Mathematica does incorrectly
(no decent journal will accept Mathematica or any other CAS-based derivation
as a central part of any proof anyway  - but this is not to say that they
are not annoyed by real bugs), as they care about what they *can* do with
Mathematica in principle. They may have some wrong beliefs, like believing
that Mathematica can not do lots of things it actually can do, or that it is
always dog slow with numerics, but they seldom run into problems of the kind
you often mention, simply because they by far don't have your level of
sophistication with Mathematica - so they have no way to come up with such
problems.

And I would argue that this kind of advanced Mathematica skills is more
characteristic of people working either in Computer Science or in the more
applied fields where some math is necessary, such as engineering or finance,
for instance (this is IMO because the research problems in pure science are
usually more unique and to a much lesser degree amenable to automation, thus
programming skills are not as relevant). I can also speak for myself: most
of the time, when I am using some advanced Mathematica (programming)
constructs, it is a programmer in me, not a scientist, who is the driving
force for it.

As far as I can tell, most software systems and programs exist to automate
(part of the) work which must otherwise be done by a human. The degree of
automation can be very different, but I have a feeling that for software
used in industry it is generally much higher (or at least that's the goal)
than for that used in research (I don't mean the software that say controlls
a particle accelerator etc - this I consider "industrial", even if used for
scientific purposes). In particular, many industrial software systems are
authorized to make lots of high - level decisions by themselves, with humans
often becoming  operators who monitor the system's work and intervene only
in special circumstances.  But for research software, I have a feeling that,
while automation is of course important, still most of high-level decisions
are left for a human - simply because it is much harder to automate
research, due to its very nature and requirement to be original. So I think
that it is inappropriate  to subject Mathematica to requirements typical for
non-research software  -  hopefully it will never be intended to replace the
person who is using it, and will always remain a tool.

Arguably, a skilled mathematician or physicist has her own ways of checking
the correctness of the result. Mathematica is always a tool, not a magic box
that is guaranteed to always produce the right answer (given that lots of
times in research there are ambiguities in the questions asked). If one has
no means to ensure that the answer is correct, this means that he has at
most a single perspective of the problem he is solving. But if so, this
means that he does not really understand it, and this has nothing to do with
Mathematica.


>
> Now a person educated as a computer scientist would generally know a fair
> amount about syntax,  and might be willing to use
> "A system that uses syntax-directed transformation rules for computation".
>  In fact there are several such systems that have been
> designed, starting in the early 1960s.  In deference to Steve C's
> reluctance to allow the names of other computer systems to appear
> in mathgroup, I won't name them.  But at least 6 come to my mind.
> I still don't understand the reluctance of people to say  "OK,
> mathematician-who-doesn't-know syntax"  ... HERE's the substitution facility
> for YOU.
> and write the program.  Or at least a first cut of one, so that it can be
> refined.


Unfortunately, I don't have comparable experiences with other CAS, so I
can't say anything useful here. One thing that I find remarkable about
Mathematica is the level of integration of different parts and the fact that
it still remains relatively simple system at its core, given the amount of
built-in functionality included in the kernel.

>
>
> In this way, they can implement some missing functionality themselves at
> their own risk without the need to wait for a new Mathematica release. It is
> stated in the documentation that rule substitution is purely syntax-based,
> and therefore not guaranteed to always make sense.
>
> It says that it won't always make sense?  Hm. (I am traveling and don't
> have Mathematica with me, and can't check...)  Doesn't make sense?!
> How could that be..


The last part is my interpretation. But the problems of using the correct
syntax and its correct interpretation exist in any formal science. In my
view, the syntax of Mathematica language is not really isomorphic to the
language of any specific domain of Mathematics, and should not be. It is
currently a number of very high-level commands targeted to occasional users
and performing well-defined standard mathematical operations, such as
solving equations or inequalities of some kind, etc. But IMO more
importantly in the long term,  it is also a language - a building material,
optimized for creation of sub-languages for (mathematical) knowledge
representation and manipulations with mathematical objects.  It is then
targeted at advanced users / progammers with both domain knowledge and
Mathematica skills who can correctly implement these sub-languages, adding
to the functionality available to the first target audience.

I would agree that there is currently a gap in between these two target
audiences, which would include for example some mathematicians without
advanced Mathematica skills who want to use Mathematica for their research,
and be able to push it to the limits.  But I still don't think of this as a
design flaw. At the moment, this may be inconvenient, but evolutionary I
think this is a win - the system's generality makes it  flexible enough to
evolve and smoothly integrate new functionality. The same generality is also
responsible for the occasional nonsense produced by a blind application of
replacement rules. But I think that in time, the corresponding "intermediate
layer" of Mathematica will emerge with enough functionality to allow these
people do their work without immersing themselves in complexities of
Mathematica's inner workings, and that will close the gap (already now, lots
of extra functionality is available through add-on packages). As David Park
said recently, we are still the early users.



> It must make sense to SOME people. Maybe even me or you.  So now there are
> more levels.
> The high-priest, keeper of the mysterium(us?).  The second level priest who
> understands syntax but for whom some transformations "don't make sense",a
> person who not a true syntax-geek. Perhaps this is the typical programmer
> who learns some Mathematica....
> The third level, maybe a skilled mathematician?  The fourth level, some
> novice, unsophisticated student learning math;  and maybe down the ladder
> further ?
>


I stick to my view of replacement rules as being aimed primarily at advanced
users, or at least as a tool that should be used with much care.  I wish the
documentation was more clear about it. The  fact that the functional and
procedural layers are built on top of the rule-based engine speaks for
itself - these layers are more managable by non-experts and less error-prone
to use (apart from efficiency gains).  When beginners get excited by
replacement rules and start using them left and right, this very often leads
to trouble (I recall myself some while ago :)). Imagine for example a web
framework written say in Lisp, with some high-level API (or DSL) exposed to
basic users. Say, the inner workings of the framework are documented for the
benefits of the more advanced users. Imagine then that a basic user of the
framework learns a few basic things about Lisp and starts to use Lisp to do
what the API or DSL is supposed to be used for, trying to combine his own
Lisp functions with the calls to the API.  I suppose you wouldn't be
surprised if our hypothetical user would frequently end up with something
different from what he intended.

Also, I agree with David Park here. When we learn math at school, we spend
several years to learn the subtleties of essentially a very similar
activity: when we do math, a lot of "pattern-matching" is happening in our
head when we decide which identities or equations to use. I have no reason
to think that the  syntax of Mathematica *core* language must be designed to
be extremely easy to learn, given the level of generality required for it.
Besides, the core language is actually not so difficult to learn either. I
think that part of the problem is that most Mathematica introductions are
too pragmatic in a sense - they want to get you up to speed in solving field
- specific problems as quickly as possible and as a result omit a proper
discussion of the fundamental language structures (I tried to do it
differently in my book). Even worse, field-specific elements are often mixed
with parts of Mathematica language, which I find very confusing.

Pragmatically and in the short run, this seems to be a right thing to do
given that most (potential) Mathematica users at present are not high-school
students but professionals with no spare time to properly learn Mathematica
(perhaps, this will change). But for would-be long-term Mathematica users,
learning Mathematica this way is learning it the hard way.


Best regards and happy 2010,
Leonid


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