The main theme of the book is the meaning of and justification for the use of the word "physism." The author defines it as the philosophical proposal that considers the foundations of mathematics to be a subset of the laws of nature. This idea is hardly new, many physicists and biologists have argued that intelligence could only arise in a universe where events always happen in a certain way and the results are within fixed parameters. Anything that does this is by definition a law of nature. Since mathematics arose as a way to describe nature, it is hard to see how circumstances could be any different.
Part two of the book is a series of descriptions of quantum mechanics and some of the more unusual aspects of how small particles behave. I am a firm believer in the slightly whimsical statements made about quantum mechanics. Comments like, "You don't understand quantum mechanics, you just get used to it." demonstrate how counter-intuitive it is. However, since it is so counter-intuitive quantum mechanical actions were first observed, and then the appropriate mathematics was invented or modified to create the models. Therefore, this is another case where the mathematics was adapted to the physical phenomena, so it cannot be a surprise that the two are intertwined. Omnès points this out in this section; I just don't where it is a strong argument in support of his position.
Part four is where the author attempts to complete the justification of the use of the word physism. Quite naturally, he mentions Platonism. I disagree strongly with one section, "Mathematical Truth is Fallible." Omnès makes the statement, "They assert particularly that mathematical knowledge is inherently fallible and that no foundation can make it infallible. …because it obviously breaks with the traditional prejudice in favor of an unquestionable truth of mathematical reasoning. Lakatos gave long lists of errors, which appeared in more or less famous works and were only corrected long after." To claim that mathematical reasoning is inherently fallible because errors are made in proofs and then later corrected is an extremely weak argument. It is equivalent to saying, "humans are fallible" and ignores the powerful self-correcting features of mathematics.
This continues on the next page, where we find the statements, "…one cannot avoid the fact that many crucial statements in physics find their validity in a falsification test. Nothing analogous exists in mathematics, which is why it is not an empirical science but at best a quasiempirical one." An enormous number of mathematical conjectures and theorems have been proven false by counterexample and the proof by contradiction is a fundamental proof technique.
I am in agreement with the idea that there is a fundamental interrelationship between mathematics and the laws of nature. However, I found the arguments in support of that position in this book to be weak. At times, as can be seen from the statements quoted above, Omnès puts forward arguments that are of dubious validity.
Charles Ashbacher (firstname.lastname@example.org) teaches at Mount Mercy College in Cedar Rapids, Iowa.