🚀 go-pugleaf

From: pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt):
>In article <1992Sep23.094901.1@sscvx1.ssc.gov> doctorj@sscvx1.ssc.gov writes:
>>pratt@Sunburn.Stanford.EDU (Vaughan R. Pratt) writes:
>>How about: Physics is science, while math is not.  By science, I mean 
>>the sequence:
>>  1: Look at the existing data and formulate a theory which explains it.
>>  2: Use that theory to predict new results.
>>  3: Perform experiments (ie, collect more data) to test the theory.
>>  4: Go to 1.
>>Math does not follow this sequence.  It does not contain falsifiable theories
>>in the way that physics does.
>>...
>>This use of the word "experiment" is ill conceived.  Computer calculations 
>>are not experiments.  Neither are slide rule or hand calculations.
>
>Let me not dispute your sequence here, but rather opine:
>
>(A) Math is perfectly capable of following your sequence.
>
>(B) Hypotheses can be falsified in math.
>
>(C) Falsification isn't all it's cracked up to be in either math or
>physics.

I have to admit that I'm a bit unclear who is saying what in the
above exchange, but let me make two comments.

First, I largely agree with claims (A)-(C) and recommend reading
Lakatos' multi-part article "Proofs and Refutations" which
originally appeared in the British Journal of the Philosophy of
Science and was later reprinted as a book.

Second,  although I largely agree with (A)-(C), I think that math and
physics do differ in degree.  What sequence (1)-(4) leave out is proof
in the sense of derivation.  Physical theories are usually established
by inference to the best explanation and are not usually derived from
other theories.  (Note I say "not usually".  Many important statements
of physics are derived from theories.  E=mc^2 comes to mind if I
remember my freshman mechanics course correctly.)  Mathematical
statements are usually justified by derivation or by finding a
counterexample to its negation (which can also be viewed as a
derivation).  Nevertheless, when evaluating axioms, one often
experiments with their consequences and makes what resembles an
empirical judgement about the truth of the axioms.  Witness the
discussion on set theory.

Let me go out on a limb and say that the emphasis on derivation in
mathematics is why mathematicians spend so much time looking at
derivationally elegant, but nevertheless false, sets of axioms.  For
example, Euclidean geometry is still extensively studied although
Einstein showed that it is not true.  (My doubt about this last
observation is that one could say that physicists do the same when they
study Newtonian mechanics and that the underlying reason is the same in
both cases: both Newtonian mechanics and Euclidean geometry provide
useful approximations to the way the world is.)

John McLean