MichaelLewis:
You hold that the imaginable may not be conceivable; but clearly I don’t have the power to imagine a contradiction, do I? Could you give an example of something inconceivable that we can imagine?
this depends on what you mean by “imagine”. for example, your example of “round square” is an example of a broadly speaking
visual object - an object composed of shapes - which is, of course, particularly conducive to the mental picturing involved in your garden variety imaginings.
but what does it mean to “imagine” mathematical postulates or proofs? what does it mean to “imagine” philosophical hypotheses? i suspect that what you take it to mean is that you entertain the proposition in your head, and then get a kind of
sense as to the logical consistency of the concept - it just
feels possible or not. or something like that, anyway - perhaps husserl has described the phenomenology of it somewhere.
at any rate, what we are talking about here are modal properties: is it possible to “imagine” a proposition that is nonetheless necessarily false, and thus inconceivable?
well, all of math and philosophy deals with modal concepts: the most general and world-independent features of this, the actual world. anything that is true in math is necessarily true; that which is false, necessarily so. the same goes for ontology and metaphysics and logic (and, arguably, every other purely philosophical discipline).
so that’s where one might look to discover examples of imaginable but inconceivable concepts, and it’s there that we find boatloads of them.
for example, Frege believed the (naive) set theory in his mathematical logic to be not only imaginable, but (necessarily) true; however, Russell pointed out an inconsistency in his axioms which, in fact, rendered the theory, as originally stated by Frege, necessarily
false.
then there’s the continuum hypothesis - the hypothesis that there is no cardinality between the cardinality of the set of integers (aleph-null) and the cardinality of the set of real numbers (the continuum). cantor - the original proponent of the hypothesis - believed it to be true - i.e. believed it’s truth not only to be imaginable, but necessary - but he was never able to prove it. then godel and cohen demonstrated that the conntinuum hypothesis is undecidable. but godel himself (as do many set theoreticians) believed it to be false.
which means, obviously, that the “imaginability” of a proposition doesn’t allow you to draw any conclusions as to the modal properties of that proposition.
and philosophy is no different. for instance, questions about the nature of things like numbers, and propositions, and god, and knowledge, and possible worlds, and modal logic, are all questions about the modal properties of certain popositions.
but that is as may be. the point is that nothing much of logical interest follows from the putative imaginability of a concept. what
is definitive is logical proof.
which means, of course, that your ability to picture a world where everything goes right and everyone avoids evil entails nothing about the logical possibility of such a world (in fact, alvin plantinga formulates an argument that such a world is logically
impossible).
keep in mind also that, for every proposition whose necessary falsehood (or truth) is demonstrable, there are about a million
unproved propositions that are nonetheless either necessarily true or necessarily false (e.g. Goldbach’s conjecture); what do you think one’s (in)ability to imagine those propositions allows one to conclude?