Math vs. Science

["A.P. Hazen" <a.hazen@philosophy.unimelb.edu.au>]

(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