“Most of us believe it is trivial to verify that one and one are equivalent to two. However, in careful examination, this matter is revealed to be vastly more complex than it superficially appears — as are many matters that we ordinarily trivialize.

The truth of the statement 1 + 1 = 2 depends upon the establishment of an appropriate context in which such abstractions as numbers (peculiar entities, devoid of being-as-such) may be said to exist and be manipulated according to rules.

This is not really the same as the world, where, for example, one penny and another, while being two pennies, are not -equivalent- to two (a number), because each penny has a rather bizarre existence and uniqueness which cannot withstand abstraction; worse still these pennies cannot share the same temporal and spatial extent, identity, or unity that literal equivalence seems to demand. This unity is only really possible as an abstraction on paper or in an observer. What does that have to do with the pennies?

It is for these and related reasons that, to my view, the sense in which the pennies are equivalent to two is narrow, and the sense in which they have nothing to do with such perspectives is broad. This latter sense is, in effect, ‘more true’ than the numeric abstraction… and not slightly… but by many orders of magnitude.

We should respect these strange and exclusionary aspects of our arithmetic habits, but not so much as to mistake them for reality. They are prostheses. or toys. Powerful and possibly predictive — yes, but not indicative of the truth or even the character of the reality the describe. These are heuristic overlays.

What one and one are actually equivalent to remains at least as mysterious as it is obvious, and, in my view, the mystery naturally overwhelms the abstraction.

What, after all, is ‘one’? Can any two actually existing entities be understood as literally equivalent? Was any actual instance of two the same as any other? The questions are not easy to dismiss, and less easy to resolve.”

— an anonymous informant