In that case, let us investigate another example, called SKI combinatory logic (SKI-CL). It has the following rules, which define the symbols S,K, and I:Veritas Aequitas wrote: ↑Mon Jun 27, 2022 8:28 am This give mathematicians the assurance whatever their mathematical statements based on accepted axioms they are fundamentally realistic and not illusory, i.e. not nonsense.
(I x) = x
((K x) y) = x
(S x y z) = (x z (y z))
That is all there is to it.
In fact, the rule for I is even redundant, because I is equivalent with S K K.
The axiomatization for SKI-CL is much simpler than the one for Peano Arithmetic Theory (PA).
Since this computation model happens to be Turing-complete, it is equivalent to the lambda calculus, and therefore, its mathematical universe (model) is an extension of Peano Arithmetic (PA)'s intended interpretation, i.e. the natural numbers.
Everything that you can express in PA, you can also express in SKI-CL. You can certainly express natural numbers with their typical operators, such as addition, multiplication, and so on.
If SKI-CL is supposed to be "fundamentally realistic", where in the physical universe can we find something like S, K, and I? If not, why would SKI-CL be "illusory"?
The axioms that construct SKI-CL are clearly not empirical, unless you can successfully argue that they are.
Study Schonfinkel's SKI-CL and you may finally understand that it is your own idea that modern mathematics would be somehow empirical, that is illusory.