5/29/2008

Experimental Philosophy of mathematical intuition

So this is straight psychological research, not done by 'experimental philosophers,' but it does seem highly relevant to any philosophers who appeal to the notion of mathematical intuition:

Log or Linear? Distinct Intuitions of the Number Scale in Western and Amazonian Indigene Cultures, Stanislas Dehaene et al. in Science May 30 2008.
We probed number-space mappings in the Mundurucu, an Amazonian indigene group with a reduced numerical lexicon and little or no formal education. At all ages, the Mundurucu mapped symbolic and nonsymbolic numbers onto a logarithmic scale, whereas Western adults used linear mapping with small or symbolic numbers and logarithmic mapping when numbers were presented nonsymbolically under conditions that discouraged counting. This indicates that the mapping of numbers onto space is a universal intuition and that this initial intuition of number is logarithmic. The concept of a linear number line appears to be a cultural invention that fails to develop in the absence of formal education.

5/27/2008

Scientific Realism via the internets

I recently found out that Philosophy of Science has conditionally accepted an article I wrote on the no-miracles argument. This is a stroke of good luck, and it's also a testament to the philosophical blogosphere: basic ideas in this paper were hashed out on this blog (see especially here), and honed by readers' astute criticism. Perhaps the paper wouldn't have been good enough for acceptance otherwise.

I would greatly appreciate further help on the paper before I send away the final version; the current draft (in rich text format) is here. Here's an abbreviated abstract:
1. Scientists (usually) do not accept explanations that explain only one type of already accepted fact.
2. Scientific realism (as it appears in the no-miracles argument, or NMA) explains only one type of already accepted fact.
3. Psillos, Boyd, and other proponents of the NMA explicitly adopt a naturalism that forbids philosophy of science from using any methods not employed by science itself.
Therefore, such naturalistic philosophers of science should not accept the version of scientific realism that appears in the NMA.
And as long as I am singing the praises of the blogosphere and begging for readers, P.D. Magnus (of the excellent Footnotes on Epicycles blog) and I have a draft of a paper on another aspect of the scientific realism debate (in pdf format) here. We ask, and give a partial answer to, the question: When should two empirically equivalent theories be regarded as variants of one and the same theory? Comments large and small are appreciated!

Labels:

5/13/2008

Against negative free logic

'Free logic' is an abbreviation for 'logic whose terms are free of existential assumptions, both singular and general.' Free logics attempt to deal with languages containing singular terms that do not denote anything, such as 'Pegasus'.

Free logics come in 3 basic flavors, which differ over what truth-values should be assigned to (atomic) sentences containing non-denoting names.
- Negative free logics declare all such sentences false;
- Neutral free logics declare all such sentences neither true nor false; and
- Positive free logics declare at least some such sentences true (in particular, 'Pegasus=Pegasus').

Tyler Burge argued for negative free logic over its rivals in "Truth and Singular Terms," Nous (1975). I came up with a little argument against negative free logic; but I do not know the argumentative landscape for these 3 options particularly well, so this may be extant already. (Note: if any readers have references for arguments pro and con negative free logic, I'd be very interested. I've found a couple of nice articles by James Tomberlin, and a short response by Richard Grandy to Burge's piece, but not much else.)

According to the negative free logician, all atomic sentences containing non-denoting names are false. Some people reject this because calling 'Pegasus=Pegasus' false seems wrong; here's another problematic type of case. Consider the following three sentences (and assume for the sake of argument that 'Atlantis' is a non-denoting name):
(1) Atlantis is West of London.
(2) Atlantis is East of London.
(3) Atlantis and London have the same longitude.
In negative free logic, all three of these must be false. But for the three predicates 'is west of,' 'is east of,' and 'has the same longitude as,' any one of the three can be defined in terms of the other two using only negation and conjunction. E.g.:
'x is west of y' means 'x is not east of y, and x does not have the same longitude as y.'
But now we've got a problem: If 'Atlantis is west of London' is false (as the free logician says), then at least one of 'Atlantis is east of London' or 'Atlantis and London the same longitude' has to be true -- but that contradicts the earlier assumption (of the negative free logician) that all of (1)-(3) are false.

And this same problem will crop up in general when we have a set of predicates that are definable in terms of one another and negation; in the simplest case, P = ~Q. And this is not that rare: {'before', 'after', 'simultaneous'} is another example. The negative free logician could save her position by maintaining that two of the predicates are somehow really basic, and the other really derivative. But at least in these two cases, it doesn't look legitimate to hold that 'west' is somehow fundamental and 'east' merely derivative.

Does anyone see a good response to this objection on behalf of the proponent of negative free logic?