A perhaps more interesting problem with the mereological response to Heller’s argument is that the theory is problematically similar to perdurantism. While on the one hand, the perdurantists have individual, four-dimensional time slices aggregating to form a single, transtemporally identical space-time worm, on the other hand, the mereological essentialists have a succession of primary entities aggregating to form a chain of successive entities which, under a colloquial “loose” term, enables one to pick out the transtemporal identity (rather, ‘sameness’) of a ‘thing’. There is something that seems to stand out here, something that makes me suspect that the time slices of the perdurantist do not really differ in any appreciably substantial way in its constitution from the mereological construction. Both theories rely on aggregations of instantaneous parts of a larger whole to enable some form of transtemporal identity; the main difference is simply that the perdurantists will propose that a thing is never wholly present at any given time (unless its worm is of instantaneous time length), while the mereological essentialist will argue that the instantaneous object is wholly present, and the successive chain of similar objects is simply the method for endurance. Bascically, the difference seems almost negligible, and given the problems immediately above, as well as the problems discussed earlier concerning perdurantism, it seems like mereological essentialism holds little hope for saving the endurantist claims of transtemporal identity.

Another putative method for escaping the problem posed by Heller’s argument is perhaps more promising; I have in mind Peter Geach’s denial of the existence of a single, universal type of numerical identity. According to Loux,

…Geach…claims that sentences of the form ‘a is the same as b’ lack a complete sense or meaning. To give such sentences a complete meaning, we need an answer to the question ‘Same what?’ And Geach claims that an answer to that question always requires the appeal to a kind-term or a count-noun…For Geach, every such answer points to a unique identity relation…it is possible for a thing, a, and a thing, b, to enter into the identity relation determined by one kind-concept, but to fail to enter into the identity relation determined by another kind-concept even though the latter kind applies both to a and b. (Loux, 244)

Under Geach’s paradigm then, it becomes possible to convert Heller’s argument to be more precise, enabling the Geachian endurantist to avoid the contradiction. To the Geachian, the argument becomes something like the following:

(I) Descartes at t1 is numerically identical with Descartes at t2; They are the same person

(II) Descartes(-) at t1 is numerically identical with Descartes(-) at t2; they are the same matter (or clump of matter)

(III) Descartes at t2 is numerically identical with Descartes(-) at t2; they are the same matter (or clump of matter)

(IV) Descartes(-) at t1 is NOT numerically identical with Descartes at t1; transitivity of identity cannot operate here, because it is applying transitivity to different kinds of identity.

(V) Descartes(-) at t1 is not numerically identical with Descartes at t1 by the

true Principle of the Indiscernibility of Identicals.

It becomes evident here, under Geach’s understanding of identity, that by carefully specifying what type of identity relationship is being investigated, one can negate the problem raised by Heller and soundly defeat the attempt at forcing the endurantist into a contradiction. Another aspect of Geach’s paradigm that appears somewhat appealing is that I posit that the Geachian framework might be able to handle change in parts (or personality traits, etc.). Since not all parts of a person are essential under Geach’s paradigm, it seems unproblematic to suggest that so long as some certain types of identity conditions are met over time, a person could undergo some certain drastic changes but still maintain transtemporal identity. Whether there is some critical set of specific types of identity conditions that must be met, or some sort of equilibrium that must be maintained, I cannot see any logic for at precluding the possibility, and this additional putative functionality would even further bring Geach’s paradigm into agreement with some pre-philosophical notions of transtemporal identity.

There are two commonly raised problems with Geach’s view: it is problematic because it divests philosophers of the notion of a singular, universal identity condition that some argue is necessary, and it may be interpreted by some to suggest that spatially coincident physical objects can exist and are unproblematic. I would suggest, however, that in neither case is the problem so severe as to merit the outright abandonment of Geach’s view; unlike the perdurantist or mereological view, no grand flaw is immediately available for dismantling Geach’s framework. In fact, I suggest that there is a pre-philosophical inclination to believe that there are different types of identity, that there is an important distinction between asking if ‘Clark Kent and Superman are the same person’ and asking if ‘Clark Kent and Superman are the same type of person’. To respond to the camp that suggests Geach’s view leads us to have to accept spatially coincident objects, I refer to my earlier treatment of Heller’s undesirable alternative Log in to see images!, or the problem of arbitrary undetached parts. If one simply rejects the counter-intuitive view that all of me except for my left hand is an object that is distinct from all of me, and instead accepts the more commonsense notion that all of me except for my left hand is simply a fraction of the whole me, Geach’s treatment of the argument simply cannot lead one into a problem with having spatially coincident objects. Further, consideration of the problems elucidated above for the other putative solutions to Heller’s argument makes me suspect that perdurantism or mereological essentialism would get closer to allowing for spatially coincident objects than Geach’s framework allows.