🚀 go-pugleaf

In <Bx73nu.DMy@unx.sas.com>
sasghm@theseus.unx.sas.com (Gary Merrill) writes:

>Sometimes it is used to refer to certain *areas* of mathematics
>(e.g., finite mathematics?) and other times it is used to refer to
>the *application* of (pure) mathematics (to science or engineering).

My impression has always been that the stuff which applied math-
ematicians do has often not been turned into pure mathematics yet;
doing so is a distinct achievement.

>I also think that there is something distinctive about "computer
>science" that distinguishes is from applied mathematics.  For example,
>much of its domain is not what most people would regard as truly
>*mathematcal* (except when this term is used in a very broad sense --
>as I think you prefer to use it).

I think that many people, for their own reasons, prefer not to regard
it as a mathematical subject; but it is; *but* the mathematics required
for it has not all been developed yet. There are a lot of cowboy 
"software engineers" around, who would be out of a job, if forced to
practice their trade with scientific standards of rigour, but who are
let off the hook for the moment. (In case of misunderstanding, I must
add that this is not a case of vanity on my  part! For what it's worth,
I don't think I am a particularly good programmer; but this is mainly
because I, like so many others, have not been sufficiently scientific 
in my approach to this demanding activity; and, unlike some, I'm trying
to do something about it. -- As I'm sure you are, too, although you
wouldn't put it like this!)

>|> >|> A propos of which, are algorithms mathematical objects?
>|> 
>|> >Well, they may be thought of as *formal* objects.
>|> 
>|> What does this mean?

>Well, I won't attempt a definition at this point.  It may be easier
>to get at it via some common distinctions.  Consider the following
>questions:

>	1. Is logic mathematics? 
>	2. Is set theory mathematics?  Type theory?
>        3. Is model theory mathematics?
>	4. Is formal linguistics mathematics?

>Historically, mathematics seems to have centered around such concepts
>as *number* and *measurement*.  Since logic does not (in itself)
>address these concepts, many feel it is not mathematics.  Similarly
>for set theory, etc.  Typically, many philosophers regard logic
>as mathematics and many mathematicians regard it as philosophy.  So
>the view is that (for example) first-order logic is *not* mathematics,
>but cylindirc and polyadic algebra *is* (despite the obvious close
>relations here).  But all are *formal* disciplines:  formal descriptions
>of the theories, formal statements of fundamental principles and rules,
>formal criteria for "goodness"  (proofs, etc.).  ("Formal" in the
>sense of "of or pertaining to form".)

(Well put.)

In this sense of "formal", I agree with you; but etymology is not
always a reliable guide to current usage, and in particular, I
think, the adjective "formal" actually usually means "of or
pertaining to formulas" -- which is a very different kettle of fish!

>Thus it seems reasonable to say that algorithms are *not* mathematical
>objects (any more than, say, grammars are).

The analogy with grammars is a favourite one of mine, too. (If Vaughan 
Pratt is reading this, perhaps he'll explain this analogy using enriched
categories??)

>And this is particularly
>true since algorithms do not form a domain of objects that are subject
>to operations that are typically thought of as mathematical.  (Actually,
>I have to confess that I don't what operations algorithms really
>are subject to.  But normally in mathematics one deals with operations
>under which the domain is closed.  So while algorithms *might* be
>treated as mathematical objects -- for example, by constructing an
>algebra of algorithms -- they usually (never?) *are* treated this way.)

Any reason not to, though, if it can be done?

>|> I don't have a clear concept of "algorithm", so it seems helpful 
>|> to ask what kind of object an algorithm is. I don't think it is a
>|> mathematical object like a number, or a group: mainly because I
>|> think that physical time enters into it in an essential way. But
>|> it has the kind of "exactness" which only mathematical objects have.
>|> Does this make any sense? (Never mind if it doesn't; I was more
>|> concerned to ask a thought-provoking question than to make any
>|> statement. -- Just so long as the question makes some sense! I'm
>|> disappointed that it didn't.)

>Yes, it makes perfect sense.  However, earlier you denied my implicit accusation
>that you may be confusing "precise and formal" with "science".  Now
>you seem willing to confuse it with "mathematics". 

OK, so long as we stick to your use of the word "formal", not mine.

>This may simply
>be a reflection of the broad sense that you wish "mathematics" to have.
>Use of the broad sense isn't necessarily bad, but often it is important
>and fruitful to draw finer distinctions.

Granted.

>|> What is the *ontology* of software engineering? Neither mathematics

>Does any type of engineering *have* an ontology?  (Civil -- bridges?
>Electrical -- circuits?  

No:
   civil -- concrete, metal, plastic, force, stress, strain, mass, etc.
   electrical-- potential, current, conductor, charge, etc.

Software -- algorithms, data structures?)

Yup.     -- and the behaviour of actual peripheral devices, and
            (less important) processing hardware.

>|> nor logic provides any convincing (to me) answer to the question
>|> of what the *objects* are which software engineers work with.

>Why should they?  That ain't their job.

No, indeed -- it's one of the jobs of a computer scientist!

> They also don't provide an
>answer to what is the ontology of physics or psychology.

Well, OK, maybe I should have avoided the word "ontology". But
I wasn't saying anything about the ontology of the sciences ...

(although why shouldn't logicians and mathematicians say 
something about that? -- see e.g. my plug for Whitehead's
"An Enquiry Concerning the Principles of Natural Knowledge",
below)

... only about the ontology of the engineering disciplines,
which it is the job of scientists, in their turn, to provide.

Is this a bad way to use the word?

>|> Whereas physics has a lot to say about the materials with which
>|> civil engineers work; chemistry describes the materials with
>|> which chemical engineers work; and so on. It is for this reason
>|> that I think of computer science as a discipline sui generis. It
>                                                    ^^^^^^^^^^^
>I think we agree on this.

>|> is immature as a science, but what else do you expect?

>This of course does not follow.  I still object to calling it a
>"science" -- immature or otherwise.  Usually, people want to call
>what they do "science" because they think this lends importance
>or respect to it.

The way in which people usually use words does have to be taken
account of, of course. But we don't have to follow suit in every
case! (That would be impossible, anyway, because usage is always
inconsistent and ideology-ridden. You always have to make your 
own choice.)

Me, I *want* to call something a "science" *iff* it provides
knowledge which I can trust (without having to form any judge-
ments of human character, other than that of intellectual
honesty). But I have to admit that it's not up to me, in
reality, and some things get called "sciences" which are not
trustworthy, and you can't take the title away from them.

(Even worse, all sorts of philosophical tripe gets marketed
under the slogan of "Science", with a capital "S" -- an 
ideology, more properly known as scientism, which has 
literally become a religion. -- Sorry about the diatribe,
but this is one of my all-time pet hates, and Usenet is so
tempting in this way ... .) 

So you just have to live with the ambiguity of the word. Such
is life. :-(

>In fact, this seems to me to be the *only*
>motivation for calling computer science "science".

I hope, and believe, that you are wrong in this judgement.

>But we don't
>need that.  Computer science does not need to ride the coattails
>of some other discipline for its respectability.

Certainly not pure maths and logic, IMO. :-)

>|> Great news! You've cracked the problem of the foundations of maths!
>|> Tell me all about it! :-)

>Sure.  First, tell me what the problem is.

Certainly: (A) what are mathematical objects, and (B) how are they
related to other objects -- such as, for example, (i) electronic
computers, (ii) algorithms, (iii) data types, (iv) programming
languages, (v) human beings doing computations [the "computers" of
Turing's day], (vi) written and spoken symbols, (v) users and other
objects interacting with computer systems?

(Ignore (B) if you like.)
--
Angus H. Rodgers  (PhD student), | arodgers@dcs.qmw.ac.uk  
Dept. of Computer Science, Queen | [ +44 | 0 ] 71 975 5241   
Mary & Westfield College,  Mile  | "But what is contact? No two points
End Rd., London E1 4NS, England  | are in contact." -- A.N. Whitehead.