Re: OO, C++, and something much better!

[piercarl@sabi.demon.co.uk (Piercarlo Grandi)]

>>> "jsa" == Jon S Anthony <jsa@alexandria> writes:

[ ... on the role of operators and the interpration of "+", I would say
... ]

piercarl> It was also perhaps not clear that by "maths" I meant
piercarl> something like ``ordinary maths'', a not well defined but
piercarl> hopefully intuitively perceivable notion;

jsa> And?  Presumably, by "ordinary maths" here you really mean
jsa> arithmetic.

No, telepathy does not work over the net. I meant something like ``the
sort of maths ordinary people learn at ordinary schools''. Which is
rather fuzzy notion, for reasons illustrated below.

jsa> That is a precisely and rigorously defined system.

Of course any given system of arithmetic is (hopefully) so; but the
notion of ``arithmetic'' *isn't*.

There is a large difference between a mathematical system like a
particular formalization of a particular arithmetic system and the fuzzy
notion of ``ordinary maths'', or even that of ``arithmetic''; it is a
fuzzy notion because while most probably it includes *some* sort
ofarithmetic, it may include several other things.

Consider just this: suppose one wanted to say that ``ordinary maths'' is
the maths that most people learn at school; well, different countries,
and different types of school in the same country, teach different
extensions of ``ordinary maths''. Then what type of school? High school?
College? University? And which subject? And so on.

Thus my warning that using ``familiar'' looking syntax in programming
may be dangerous: for it gives a false sense of recognition of notions
like ``addition'' that are really quite ambiguous. Some particular
notion of ``addition'' can be well defined; but perhaps ``surprisingly'':

piercarl> for most any given procedure indicated by "+" computes *some*
piercarl> mathematical function, even if not necessarily the one
piercarl> indicated by "+" in most ``ordinary maths''

jsa> Presumably you simply mean here that "computer arithmetic" or the
jsa> specification of arithmetic operations in a programming language
jsa> definition is not the same as standard arithmetic (even though in
jsa> many practical cases you can ignore the differences). Well, sure.

Well, sure, but the important thing is that in many practical cases one
*cannot* ignore the differences, not that in others one *can*, for it
when one cannot that problems arise.

Also, in order to be able to ignore the differences (and many do,
without realizing it), a non trivial amount of analysis (for example
range analysis) may have to be performed. Particularly horrid subtleties
happen with the various ways that ``real'' numbers are approximated.

Again, to me it may well be misleading to use for programming a notation
that resembles that of ``ordinary maths'', for it induces in the
innocent some terrible assumptions, some of which are even true, like
yours "in many practical cases you can ignore the differences". Yeah,
but it's dangerous not to check that this is the case, and notational
similarity can mislead on how often this can be done.

This impinges on OO: 'class Complex' should perhaps bear a prominent
sticker saying "Warning: use of instances of this class as if they were
complex numbers can be hazardous to your health".

piercarl> (one favourite example is that "+" between 'unsigned' operands
piercarl> in ``C'' computes the function ``addition module 2^N'' rather
piercarl> than addition on nonnegative numbers).

jsa> Why is congruential arithmetic not ordinary?

Get real! I doubt that for example the average MBA or the average
medical doctor would consider congruential arithmetic ``ordinary'', even
if they may well agree that it has a precise definition and theory, if
they have heard it exists. Even the average PhD in geography (a subject
that is not totally devoid of maths courses like history would be) may
not be that familiar with congruential arithmetic.

jsa> Note that this too has a completely precise and rigorous definition
jsa> and theory.

Again, there is a difference between a rigorous definition and theory of
any particular mathematical subject, and the notion of ``ordinariness'',
congruential arithmetic, however precisely and rogorously defined in any
particular incarnation, is normally known only to those that have
studied fairly ``advanced'' maths at university

Also, if one looks at two "congruential arithmetic" chapters in two
relevant texbooks, their scope will be rather different. The notion of
``congruential arithmetics'' is rather fuzzy itself, for there are
_many_ possible "completely precise and rigorous definition and theory."
of it.

piercarl> by "+" in some branch of maths would tend to be few; one major
piercarl> factor is that procedures usually compute functions over
piercarl> finite domains/codomains, which is almost never the case in
piercarl> [``ordinary''] maths, and even in ``discrete'' maths.

jsa> What makes you say that??  Certainly any congruence system has this
jsa> exact aspect.  Large chunks of combinatorics deal in finite
jsa> systems.  What about finite groups?  And whose to say that in "day
jsa> to day arithmetic" you aren't dealing with a congruence system
jsa> whose modulus just happens to be 10^50 or something?

Ah sure, you would expect the average college or university graduate,
``ordinary'' people that contend with ``ordinary maths'', deal
day-to-day in combinatorics, finite groups, arithmetic mod 10^50, and so
on, for example when figuring out their taxes or planning next year's
budget for their practice. :-).

piercarl> At times, BTW, I am not so sure that is such a good idea, for
piercarl> while the function computed by the "+" procedure usually
piercarl> somewhat resembles the function indicated by "+" in maths, the
piercarl> details (and vital ones) are as a rule grotesquely different
piercarl> (just consider addition of 'real's and of 'R'eals).

jsa> Again - it depends on context.
jsa> There are cases where the two will denote identical results.

piercarl> Well, this is actually quite impossible as literally written
piercarl> :-); the results of any particular "+" operation are in the
piercarl> domain of implementation entities, those of any articular "+"
piercarl> function are in the domain of mathematical entities.

jsa> Yes, I see the "smiley", but I'm not sure this distinction is true
jsa> or of any use.

It is indeed very useful; for ``computational'' entities
and``mathematical'' entities have really intrinsically different
properties. The notation may be the same, the syntax even, but one
forgetting that ``maths'' and ``computing'' are very different things
leads to overoptimism and eventually (pretty quickly) mistakes.

Learning to avoid such an attitude is for example one of the first
things that numerical analysts have to do. I remember a famous paper
that explains various ways to write a good program that finds, if any,
the two real roots of a 2nd order equation; it is absolutely nontrivial,

jsa> It would seem to imply that any set of "squiggles" on a blackboard
jsa> or piece of paper are "implementation entities" and have not much
jsa> to do with the mathematical entities they represent.  I don't think
jsa> so...

Well, let's say that "not much to do" is an unfair caricature of what I
have written. They are related; the relationship may even be
straighforward; but they are rather different things with rather
different properties.

piercarl> What you probably wanted to say is that it can happen that the
piercarl> function computed by some particular "+" operation is exactly
piercarl> the same as that meant by some "+" function in ``ordinary''
piercarl> maths.

jsa> What I meant was that you can't distinguish the two in any
jsa> meaningful way even if they are "in some sense different".

I hope you don'tq really mean this, for this is quite gross!

If you mean it, well, an aversion at distinguishing different sorts of
entities, mathematical and computational entities, for example, can be
be quickly remedied by reading any sort of nice introductory textbook, I
would suggest "Structure and Interpretation of computer programs".

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