Mathematics and Computing

The book "Structure and Interpretation of Computer Programs" by Abelson and Sussman has this to say in the preface to the first edition:

"The computer revolution is a revolution in the way we think and in the way we express what we think. The essence of this change is an emergence of what might best be called *procedural epistemology* -- the study of the structure of knowledge from an imperative point of view, as opposed to the more declarative point of view taken by classical mathematical subjects.
Mathematics provides a framework for dealing precisely with notions of "what is." Computation provides a framework for dealing precisely with notions of "how to.""

This statement has always bothered me. Seemed to me that mathematics and mathematicians were not being given their due. An exchange of ideas with Professor Tim Poston has helped me clarify my position vis a vis this passage.

1. The revolution is not "in the way we think" but in the technology that became available to express what we think. Mathematicians have always bothered about procedures. Whether it is a procedure in Euclidean geometry or a procedure to solve an equation. In fact, Knuth says somewhere that the history of number theory can be seen as the history of algorithms.   Gauss's work and Galois's work were also motivated by an interest in procedures.

However, as Professor Tim Poston pointed out to me, mathematicians did not emphasize procedures as much as they did the statements they arrived at using the procedures. Perhaps this was because mathematicians felt that they were "discovering" an area of (Platonic) reality and their statements represented "truths" about this reality. The procedures that led them to these truths were not considered to be important.

This viewpoint was gradually eroded starting with the discovery of non-Euclidean geometries and climaxing with the work of Godel.

2. The problems with the second sentence in the quotation is about the use of the words "emergence" and "knowledge". If the claim is that "procedural epistemology" *began* in the 20th century as the aftermath of Turing's work, then that is difficult to accept. Also it is difficult to accept that either mathematics or computing is studying "the structure of knowledge". Seems to me that both mathematics and physics have given up any claims they might have had that they were discovering knowledge. They are just playing with models and some of them seem to fit some aspects of reality to a useful level of approximation.

3. And finally the last sentence. As I said above, mathematics has always been interested in dealing precisely with both the notions of "what is" and "how to". What has changed is the meaning of the word "precisely". The idea that this means that a machine should be able to execute the procedure is new.

4. In my presentation on "The Nature of Computing" I have tried to present the concerns of computing as part of a historical continuity.