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That 1 = 2

© 1997 Henry A. Flynt, Jr.

The sole purpose of this argument is to cause trouble. To exhibit a non-affirming debacle regarding plurality.

The argument is a confrontation with self-aware intuition concerning plurality. The pure numbers are formed by successive additions of one.

2 = 1 + 1

3 = 2 + 1 = 1 + 1 + 1

etc.

If the pure numbers are therefore aggregations, that means that arbitrarily many copies of one (here understood as an eternal abstract being) are available.

Twentieth-century thought is deeply committed to notions of this sort in one scientific discipline. Theoretical physics studies not concrete beings, but ideal quantitative cases, which are available in arbitrarily many copies. It is extremely artificial; and I don’t want to expound the details to the uninitiated.

We conceive that the pure number two is formed from one as if by annexation of unit lengths. Adjunction of units.

The existence of two requires two copes of one. Then there are two objects which are separable and yet absolutely indistinguishable.

But I insist that this appeal to eternal abstract beings which exist in copies is illicit. If you claim the existence of two copies of one, what differentiates them? There would have to be a feature to distinguish them; yet arithmetic insists there is no such feature. As eternal abstract beings they have to be identical in every respect. Then how can they be plural? I conclude that there can’t be identical copies of the pure number one.

And yet, the formation of two requires the pure number one to exist in perfect copies.

Let us summarize. If the pure number one exists in copies, then

1 [not equal to] 1.

Equivalently: the pure number one can’t exist in copies–yet arithmetic or plurality demands that one exist in copies. We get the following comment about the multiplicity of ones:

1 = 2.

Afterward

A parallel argument would apply to syntactical types as geometrical beings–if the syntactical type for two twins the syntactical type for one. (E. g. stroke-numerals.)

So a retreat to syntactical types does not avoid the ontological issue. Different metamathematicians will have different schemes for evading the conception of the pure numbers invoked above. Nevertheless, they learned number in the first place through such an intuition. As an example: if addition were not an aggregation, then having things in common units would not be a consideration in adding them. The evasive schemes are maneuvers to displace the ontological issue from "mathematics" to "philosophy"–so metamathematicians can declare themselves not responsible for the ontological issue.

There is no reason for this study to contend with the evasive schemes. I have a compilation of thoughts on foundations of mathematics as a companion to this study.