Comparing Intuitions: Mathematics and Philosophy

This is something I been thinking about for a while and I would appreciate the insights of anyone with experience working with inutition.

Especially with the rise of the X-Phi movement there has been an increased interest in philosophical and psychological intuition, particularly with regards to whether or how arguments from intuition should be used as evidence in philosophical discourse. There has been much less attention paid in recent years to the reliability/usefulness of mathematical intuition, though it’s use is similarly widespread. The question then, is whether philosophical and mathematical intuitions have similar epistemic import, and if so, how much?

The most obvious case for the use of intuition in both philosophical and mathematical arguments is that it has been relied on, in both fields, for millennia — I would like to meet the philosopher or mathematician who would deny that progress has been made in either field, or at even that intuition has never played a role. Indeed, mathematicians and philosophers of mathematics often appeal to the intuitive plausibility of the axioms of ZFC; and it would be difficult to deny that our intuitions about barn facade land and stopped clocks have not played a significant role in the shaping of contemporary epistemology.

On the other hand, there are numerous empirical and mathematical results have been taken to show that intuition is inherently unreliable. You would be hard pressed to find an empirical study where 100% of the participants have identical intuitions. Likewise, one can list counterintuitive mathematical results seemingly ad nauseum — plane filling curves, nowhere differentiable functions, three `countries’ that share all of their borders…

So far so similar.

Another point in favour of treating philosophical and mathematical intuitions similarly is in their positive uses: in both disciplines intuitions are used to support the premises (axioms) of arguments (proofs), or as a starting point for research.

The biggest difference, I submit, is in in the negative use of intuition. If a philosophical theory has unintuitive results, the proponents of said theory will generally be called upon to provide a compelling argument as to why such consequences should be accepted. In mathematics however, unintuitive results are generally taken as evidence that our intuitions were incorrect. This seems to be more a difference in methodology than in the nature of the intuitions, and even this difference breaks down if the axioms used by the mathematician are not generally accepted, such as in the case of non-standard set-theories.

So it seems that philosophical and mathematical intuitions are not so different, and thus that evidence for or against the reliability of one may be equally applicable to the other. One must take into account the methodological differences when evaluating intuitions, but in any case this must be done explicitly and thoroughly if arguments from intuition are to be used in any argument.

I will finish with the caveat that all of this depends on exactly how `intuition’ is defined — a difficult and contentious project for sure, but one that would give a much more precise nature to any discussion such as this.

If anyone is interested in reading a longer paper on this topic (I presented a paper on this topic last year at a conference in New England) please feel free to email me aphilosopherstake (at) gmail.com. Thanks in advance for your feedback.

-Aaron