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Math vs. Science

(I'm a philosopher-- specializing in logic and philosophy of mathematics-- with an avocational interest in vertebrate paleontology and evolution. I'm not sure I ought to post on my own field in THIS forum, and will try to keep closer to paleontology in any future postings. But other people have raised the topic of the relation of mathematics to science in the context of a discussion of cladistic methodology, so...)
Four thoughts:
(1) The line that started this discussion ("...or they're mathematicians deluded into thinking that math IS science: it isn't") is ambiguous: the "math IS science" COULD mean that ONLY mathematics (or, a bit more plausibly, what is mathematicized, as physics is) counts as science. This is a familiar prejudice: the "science divides into physics and stamp-collecting" attitude. But subsequent discussion has interpreted it the other way.

(2) What mathematics is about and how it relates to the non-mathematical world are (surprise! surprise!) debated questions in the philosophy of mathematics. One respectable view (a version of what, in the jargon of philosophers, is called "realism") was put by Bertrand Russell (in his "Introduction to Mathematical Philosophy," the book he wrote while in jail during the First World War) in these words:
"Logic [Russell thought mathematics and logic were the
same thing] is concerned with the real world just as
truly as zoology is, though with its more abstract and
general features." (p. 169 of Dover edition)
1+1=2 applies whether you are counting apples or rocks or dinosaurs(*), but ONE of the goals of science is to state the GENERAL principles ("laws") that it finds in the worls, so it would be inadequate make separate statements about what happens when you count apples, and rocks, and dinosaurs. Hence the abstract version referring to numbers. (If you think numbers are too imaginary, one of the things Russell showed was that the relevant reference to NUMBERS can be interpreted as a shorthand reference to SETS of things, or to the PROPERTIES of things.) ... Obviously there's more to be said, and there is no guarantee that ALL parts of mathematics are "about" the same thing. But the parts of mathematics-- combinatorics, probability theory over finite sample spaces-- that seem to be involved in pylogenetic reconstruction strike me as among the parts for which Russell's "logicism" is most plausible.

(3) Phil Bigelow asked if there were cases of theorems that had been abandoned because the axioms they were derived from were called into question. Non-Euclidean geometry is the obvious example, but Andreas Johansson pointed out that the Euclidea axioms were rejected only AS STATEMENTS ABOUT PHYSICAL SPACE, leaving Euclidean geometry as good mathematics. But set theory has had arguments about axioms since Russell used his Paradox to show that Gottlob Frege's proposed axiom system was inconsistent in 1901. Russell actually published at least one proof of the existence of an infinite set: the proof doesn't appear in modern text books (and the statement that there are infinite sets is typically postulated as an axiom rather than proven as a theorem) because it depended on axioms no longer accepted. ... And there is a minority school among mathematicians-- the "intuitionists" or "constructivists"(**)-- who rejectmany standard mathematical theorems because they think that what are usually taken as axiomatic logical principles don't apply to infinities like the number system.

(4) It is known that the standard textbook axioms are not able to settle all questions, even in fairly elementary parts of mathematics. There is on going discussion of the question "Does Mathematics needs New Axioms?" (title of symposium in "Bulletin of Symbolic Logic" v. 6 (2000)). The process of trying to find new axioms, or coming to accept one which has been proposed, has been described as "quasi-empirical": mathematicians EXPERIMENT with proposed axioms to see what could be proved if they were accepted. Sometimes the results come to seem plausible. Penelope Maddy's two papers "Believing the Axioms" in the "Journal of Symbolic Logic" v. 53 (1988) give a (semi-non-technical, particularly the first paper) description of the process.


(*) 1+1=2 always and without exception! Certainly, if you put one Parasaurolophus and one Carnotaurus in the same frame, you will soon cease to have two dinosaurs, but onlybecause you will at the same time cease to have a Parasaurolophus!
(**) The names have a history, and their meaning is not transparent. "Constructivists" in THIS sense are not the same as the people who call themselves constructivists in math education, and certainly not the same as what are called "social constructivists" elsewhere.


Allen  Hazen
Philosophy Department
University of Melbourne