Now, as far as the substantive comments in your comment of December 22nd go, I completely agree! The difference between denotation and connotation has been a concern of mine for a while, but I (unfortunately) have not written much about it — although I have just finished teaching a graduate course on that topic.

As you have noticed, mathematicians really worry about extensional objects (and classes thereof), while computer scientists have to worry about intension a lot more. Nevertheless, mathematicians use intension all the time, especially when teaching basic Calculus. For example, when students are taught how to compute closed-forms for indefinite integrals, they are often presented with theorems (such as integration by parts) which are stated (and proved) as extensional theorems but almost always used intensionally! In other words, one uses integration-by-parts by pattern-matching on the syntax of the problem, even though the underlying rule works (and is stated) for any product of functions.

I will finish this comment with one of my favourite exercises in “symbolic analysis”:
1) Let x be a real number. How many solutions are there to y^2-x^2 = 0 ?
2) Let y(x) be a real-valued function of a real variable. How many solutions are there to y(x)^2-x^2 = 0 ?
The answer is quite topical for this blog, as at its core are different notions of infinity.

“Conflating Matlab and computer science is almost criminal. Matlab does not even properly represent numerical analysis!”

It depends on the point of view. We are looking at the basic underlying structures of the two disciplines, mathematics and computer science. It is immediately noticeable that mathematics deals with mathematical object and structures up to isomorphism, while computer science is concerned with the much more subtle issues of algorithmic implementations. In mathematics, any two finite fields of order pn are isomorphic, while the cryptography has to deal with potentially unbounded families of implementations of finite fields in software/hardware, sometimes having dramatically different security properties in relation, say, to side channel attacks.

Computer science, cryptography, programming are the areas where it becomes really obvious that the issues of infinity / finiteness of particular classes of objects have to be preceded by clear understanding of identity / sameness / difference of objects.

It could be a rather offensive formulation, but mathematics can be defined as part of computer science which abstracts away from data formats.

Now, a proper study of all forms of computerized mathematics would be much more instructive. There is dedicated software out there for statistics (R, SAS, SPSS, etc), numerical computation (Matlab, Scilab, Octave), symbolic computation (Maple, Mathematica, Mupad, Axiom, etc) and theorem proving (Coq, Isabelle, Mizar, PVS, etc). Each and every class of software has a radically different view of mathematics; and even within a class, there can be interesting differences. For symbolic computation, an older but still relevant comparison is Michael Wester’s critique of CA systems.

The purposes, nature, frequency and levels of abstraction commonly used in programming are very different from those in mathematics.

We would like to illustrate the validity of these words on a very simple example.

I still don’t get it…I can accept that there is a particular worldview where this difference occurs, but I have yet to see any evidence supporting it (beyond the superficial).

The “purposes, nature, frequency and levels of abstraction commonly used in programming” are all -mathematical-. They might well not be the ones commonly used in ‘mainstream’ areas of mathematics, but combinatorics and logic .

If you are trying to draw a distinction between the cultures bound up with the infinity of the continuum and that of the naturals, that is certainly something, but quite irrelevant to ‘… and levels of abstraction…’.

And then you connect thinking in CS and category theory (perfectly understandable) but is that in contrast to the rest of mathematics?

The more times this contrast is stated, the more it makes me think how much they are the same.

There do seem to be some extreme Constructivists who accept it.

The study of collectivities of particles in condensed matter physics, for example, was made mathematically easier by assuming an infinite, rather than a finite, collection of particles. The same metaphor (the infinite as an approximation for the large-finite) underlies the entire discipline of (parametric) mathematical statistics, with its reliance on the central limit and other convergence theorems, and on probability distributions from the exponential family, such as the Normal distribution. Application of this metaphor to small-finite situations, naturally enough, is problematic, something statisticians, if not always others, usually acknowledge.

I would recommend von Plato’s enjoyable book on the history of probability theory for more on this metaphor.

@BOOK{vonplato:boook94,
author = “J. {von Plato}”,
title = “Creating Modern Probability: Its Mathematics, Physics and Philosophy in Historical Perspective”,
publisher = “Cambridge University Press”,
year = “1994″,
series = “Cambridge Studies in Probability, Induction, and Decision Theory”,
address = “Cambridge, UK”}

I wonder if omega-categories are another instance of this metaphor — ie, the study of an infinite space or object is needed to give us a profound understanding of an apparently finite property (“sameness”). If so, this strikes me as yet further evidence of the mystical nature of mathematics.