Sunday, 6 July 2025

Numbers are not discrete entities. Maths is not objective. Proof by SRH

Well not exactly rigorous proof but sketch.

https://chatgpt.com/share/686a70a4-2194-8003-afb5-f74333db0232

The proposal is that numbers are actually Ordinal and Cardinality is derived from this. So the number 7 does not exist by itself. It is generated by ordering numbers so that the 7th number is "7." This process is commonly called counting. To count a field of sheep we order them somehow so that no sheep is included twice, perhaps with a gate, and if the 50th sheep is the last sheep we attribute "50" to the herd of sheep. But this "50" is not a separate entity. In this particular example it doesn't matter which sheep is "50" so we can see that no individual sheep has a number. Collectively they do and it is derived by ordering the members and assigning ordinals.

This is not immediately intuitive. We think looking at the "6" dots that there is a real "sixness" about that arrangement. But actually inside the arrangement there is no "6." The "sixness" comes about  because it is the "sixth" face of the cube. Its ordinality is what confers any "sixness" to the arrangment.

To proof this suppose Plato is right and there is a form for each cardinal number. We then ask how many cardinal forms are there? This is SRH.

So we need to order the forms and the last one is the number of forms. However are we to say that the collection of cardinal forms has an intrinsic cardinal form itself? That would mean that the intrinsic cardinal for the collection is the same as the intrinsic cardinal of the last of its members. That cardinal cannot be intrinsic to both. Contradiction. This is crude but highlights one of the many problems with this construction.

So "numbers" do not exist as separate things. They are derived from the idea of "ordinality" which belongs to ordered lists of things, or processes that ensure that entities are included only once. So we actually generate numbers through processes we apply to things. We in fact create numbers! Maths is a constructed phenomenon, it does not exist in a separate objective world. QED.

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