for that all the conclusions its method of induction demonstrates, depend for their truth upon the implied proposition, "That like cause must have like effect;" a proposition which being the only foundation for the turths of physical science, and which gives validity to the result of any experiment whatever, ranks mathematics as a species under the same genus; where the same proposition is the basis, there is truly but one science however subdivided afterwards.
(Essay on the Perception of an External World, "On Mathematical and Physical Induction," p. 279)
The idea is this. Objects consist entirely of their features; these features are the causes of the object's being what it is. Insofar as they remain the same, they are together the cause of the object's remaining what it is; insofar as they change, they are the cause of the object's becoming different from what it was. This is true "whether in the shape of mathematical diagrams, or other aggregates in nature" (p. 279). The causal reasoning is exactly the same in both cases, and therefore imports exactly the same sort of certainty from the general causal maxims.
It is true, of course, that our inquiries into physical objects do not have the same certainty as our inquiries into mathematical objects. Shepherd attributes this difference not to the general format of the reasoning, which is the same in both cases, but to the fact that we are differently related to mathematical objects than to physical objects. In mathematics we can freely stipulate features of a system, and see what follows from those features. In physical investigations, we do not have this freedom of stipulation. In physical investigations, objects are formed independently of our stipulation, and much of the uncertainty in these investigations is due to the difficulty of pinning down precisely the formation of these objects. We could very well be missing some important feature of the objects; and in cases that seem the same it could very well be that there is some hidden feature that would, if we knew about it, require us to come to completely different conclusions about the system. This failure of certainty in the investigation, however, does not affect the certainty of the reasoning, any more than the application of mathematics to physical reality affects the certainty of mathematical reasoning. Mathematical reasoning is capable of certainty and necessity whether it is applied to physical systems or not; it is entirely possible that there is some variable in the physical system which needs to be taken into account if the mathematics is to characterize the system accurately, but this is a failure of certainty in the application of the reasoning, not in the reasoning itself. Such is the case, Shepherd holds, with all causal reasoning.
