Mark Colyvan (2009: ‘Applying Inconsistent Mathematics,’ in Otávio Bueno & Øystein Linnebo, New Waves in Philosophy of Mathematics, Palgrave Macmillan, pp. 160–172), while looking at Inconsistent Mathematics, tentatively suggested that (p. 163): ‘There are times when we ought to believe in inconsistent objects.’
......His example was the infinitesimals of the early calculus, which are widely believed to have been inconsistent. E.g. they were equated to both zero and non-zero quantities (while Newton’s fluxions varied but were inconsistently equated to constants). Nevertheless, the early calculus was widely applied throughout the eighteenth century. So if, as many philosophers assert, ‘we should be committed to the existence of all and only the entities that are indispensible to our best scientific theories’ (p. 162), then it seems that Colyvan’s suggestion makes sense.
......But can we know what our best theories are, without hindsight? Epicycles, for example, were an arbitrarily effective way of coping with real-world ellipses, given a mathematical language of circles. So the astronomy of Copernicus, with his circular orbits and no epicycles, was originally less accurate than Ptolemaic astronomy. But of course, the former was a better theory. It was a step in the right direction. If we can’t, then, know what our best theories are without the benefit of hindsight, then insofar as we now have better theories without such inconsistencies in them, perhaps we should now say that we should not then have believed in such objects. After all, we could always take inconsistencies to be good indications that we need a better theory.
......It is perhaps easier to see that asking if inconsistent objects exist is a bit like asking if impossible things can happen. And of course, if something happens then it must have been possible. Nevertheless, things that seem impossible can happen. And similarly, existing objects may well have descriptions that, while true enough individually (for our usual purposes), are collectively inconsistent (at least on the surface). An apparent inconsistency usually means that our descriptions stand in need of more precision. But it doesn’t mean that they’re too bad to use most of the time. Nor does it mean that the objects so described don’t exist.
......For a ubiquitous example, light as we sense it can be bright (or dull), but photons are dense (or sparse). Our word ‘light’ equivocates between our sensations of light and the light itself. Furthermore, light itself is lots of photons, but it’s also electromagnetic waves, and waves aren’t particles. But it isn’t that light doesn’t exist, of course, and eventually quantum physics described such behaviour consistently enough. It certainly makes sense for us to believe that photons exist (and to be even surer that light exists). And although light also behaves according to relativity physics, which may well be inconsistent with quantum physics, such inconsistency may just be a reason to pursue an even better theory (of light).
......For a more apposite example, the infinitesimals of the early calculus, more precisely described, may be the so-called irreal infinitesimals that were informally introduced in my To Continue with Continuity (pp. 105–107). Suppose there are such continua as I describe in that paper (e.g. space, perhaps). And let x be a real number (e.g. pi), in the sense of an integer (e.g. 3) plus, after the decimal point, an endless sequence of digits in all the decimal places: the first (e.g. 1), the second (4), the third (1) and so forth (59265...). While that isn’t the standard definition of a real number, it’s a definite enough concept, and highly applicable.
......And since the standard axiom of infinity—which says that the natural numbers are collectively a standard set—goes well beyond the Peano axioms, and is rather prone to paradox (e.g. Levy’s paradox), let us further suppose that the natural numbers are collectively indefinitely extensible. If that’s indeed the case, then x is an (infinitesimally) imprecise description of many (infinitesimally) different lengths.
......So if l is a non-zero irreal infinitesimal then x + l = x because, quite generally, infinitesimals are smaller than 1/n for any natural number n, so that adding them to x affects x in none of its decimal places. (And incidentally, the word ‘infinitesimal’ derives from a Latin word that originally meant the infiniteth term in a sequence.) There are, then, consistent (if informal) mathematical objects—irreal infinitesimals—that for the purposes of the early calculus could be adequately described by its descriptions of its infinitesimals.
......So if such continua as do exist are well enough described by such (informal) theories as mine, then surely we could take irreal infinitesimals to be the referents of ‘infinitesimal’ in the early calculus. (Points were not then the same as real numbers, and the natural numbers were widely regarded as indefinitely extensible, so that infinite space would contain infinite lengths and hence geometrical infinitesimals.) Furthermore, insofar as the natural numbers can be said to exist, we could then think of such infinitesimals as existing.
......Of course, if mathematical existence is equated with consistency, relative to some axioms, then inconsistent mathematical objects exist when, and only when, we have inconsistent axioms; and of course, such inconsistent objects shouldn’t exist because they would be too trivial.