Is ‘Numerical Identity’ qualitatively identical to ‘Qualitative Identity’?

1+1=2

I think the best way to present an argument for the non-objectivity of truth in reality is by taking 1+1=2 as an example since it is the simplest and most popular ‘fact’ considered to be ‘objectively true’ independent of individual perception and judgment.

The trueness of the statement 1+1=2 depends on that of its underlying assumption, which is 1=1.

The two 1’s in 1=1 can refer to the same thing or to two different-similar things. However, the two 1’s in 1+1 can only refer to two different-similar things. We do not count the same thing twice. Why?

We are ruling out numerical identity (in) where qualitative identity (iq) is the minimum requirement (which is a contradiction because all numerically identical objects are qualitatively identical — a thing is identical to itself).

To disqualify counting the same object twice would be to say that qualitative identity is not sufficient to qualify for qualitative identity — that numerical distinction is required as well (something doesn’t lose its qualitative identity after being counted once). However, all numerical distinctions are based on none other than a quality. Hence, to be aware of numerical distinction, just like in qualitative distinction, distinctive qualities should be considered. Hence, numerical distinction is a form of qualitative distinction.

Therefore, essentially, ‘qualitatively-identical-numerically-distinct’ is another way of saying ‘qualitatively-identical-qualitatively-distinct’, which is contradictory.

A closer look reveals the contradiction in its entirety.

  • If we consider in=ion the basis that they are qualitatively identical in that they both have some qualities in common (like the quality of being an identity which is based on qualities of objects), then we may consider 1=1 on a similar basis (by considering the common qualities between 1 and 1). But at the same time, if in=iq then the qualitatively-identical-qualitatively-distinct contradiction mentioned above appears — resulting in 1≠1.
  • If we consider in≠ion the basis that they are not absolutely qualitatively identical (i.e. they are numerically distinct) in that although they both have some qualities in common, they have at least one distinction between them, then on the same basis we have to consider 1≠1. But at the same time, if in≠ithen the qualitatively-identical-qualitatively-distinct contradiction is no more — allowing 1=1 to be true.

There is no one consistent basis that can be used to achieve both in≠iand 1=1.

In other words, if we claim in≠iq because in is not absolutely similar to iq, then by similar reasoning, we have to consider 1≠1. However, even if we do consider in=iwe still cannot consider 1=1 because now numerical distinction is the same as qualitative distinction, which means the qualitatively-identical-numerically-distinct requirement contradicts itself.

All identities require things which are simultaneously similar and different. And such similar-different things only exist in our subjective perception. Without our simultaneous awareness and ignorance of the differences, similarities cannot exist.

The trueness of 1=1 and 1+1=2 do not exist independent of our subjective perception. Therefore they are not objectively true; only subjectively.

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