FOM: Survery Results

[James Robert Brown]

A while ago I launched a little survey to see if there is any correlation
between people’s philosophical beliefs (ie, Platonism vs anti-Platonism)
and their acceptance of various axioms of set theory (ie, V=L vs large
cardinal axioms).  The stimulus for this was the FOM discussion of Penelope
Maddy’s view on the matter.  She holds: Methodological considerations have
settled the debate in favour of large cardinal axioms and against V=L, but
philosophical questions concerning Platonism, etc. remain, so philosophy
has played no role in the matter.  

Prima facie (and contrary to Maddy) one might expect a connection.  After
all, Platonists tend to believe in the richest possible realm of
mathematical objects and this would seem to contradict V=L and favour large
cardinal axioms. So, we should expect some sort of connection between
philosophical beliefs and the acceptance or rejection of specific axioms.

The results of the survey, however, were somewhat inconclusive, mostly
because of the small sample. But I’ll report them, anyway, since there are
other also things that people might find interesting.  (I sent the
questions to FOM and to Historia Mathematica in the vain hope of getting a
large number of replies.)

These were the questions:

A.  	How much do you know about V=L, MC, etc.?
	1 	little or nothing at all
	2 	read some literature, perhaps teach course in which these issues arise
	3  	do research and publish in the area

B.	What are your philosophical views?
	1 	Platonist (including MM)
	2	anti-Platonist
	3	other (explain in a sentence or two)

C	Which axiom do you accept?
	1	V=L
	2	MC or other large cardinal axiom
	3	other or no view on the matter


Results:

	31 responses
	6 Platonists
	6 Anti-Platonists
	0 accept V=L
	8 accept some large cardinal axiom

Among the 15 respondents with some familiarity with the issue (ie, answered
A-2):
	4 accepted some large cardinal axiom
	11 chose “other”
	There were 3 Platonists in this group, 2 accept large cardinal axioms
	there were 10 “other” (ie, chose B-3), 2 accept large cardinal axioms

Among the 9 respondents who are very familiar with this issue (ie, answered
A-3):
	3 accepted large cardinal axioms, one a Platonist, one an anti-Platonist,
and one “other”
	6 “other” in philosophical view, 5 of whom were “other” in choice of axioms.

Remarks:

First, many thanks to those who took the time to reply.

If I were doing it again, I would reduce the choices, eliminating the
“other” options (ie, B-3 and C-3).  Many criticized my unsubtle questions.
Fair enough, but I was only looking for a rough trend.  People often
(though not always) chose “other” simply because they wanted to make some
fine distinction.  (My sympathy for professional pollsters has risen.)
Perhaps someone with some polling knowhow would like to take another run at
this.

The fact that no one believes V=L is significant, even given the smallness
of the sample.

Maddy’s view that philosophy doesn’t play a role in the debates over axioms
is supported the fact that there is no relevant correlation.  (I’m ignoring
the fact that the sample is so small, any inference is unjustified.)  On
the other hand, she has been taking it for granted that most set theorists
do accept large cardinals.  This seems not to be the case, since fewer than
1/3 (expert or not) are believers; most chose “other”.  Perhaps this
undermines her claims for the methodological principle MAXIMIZE, which she
takes to support large cardinal axioms.  

Needless to say, my expectation that there would be a high correlation
between Platonists and those who accept large cardinals was dashed.  I was
also surprised that only 6 out of 31 declared themselves to be Platonists.
(Though part of the problem here was giving people the “other” option.  In
commenting, the views of many who chose “other” sounded like variants of
Platonism.)

Finally, I was amazed at how many people (as expressed in their comments)
feel they have a free hand in choosing axioms – just pick whatever seems
useful for the job at hand.  There seems massive confusion on this point,
since many “pragmatists” (as they often described themselves) also think
math is perfectly objective.


-- Jim B.





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James Robert Brown
Department of Philosophy
University of Toronto
Toronto    M5S 1A1
Canada
Phone: office (416) 978-1727,  home (519) 439-2889
Email:  jrbrown at chass.utoronto.ca  
Home page: http://www.chass.utoronto.ca/~jrbrown/index.htm
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