Hilbert, Frege, Logic and Mathematical Universe Hypothesis

The Stanford Encyclopedia of Philosophy has an interesting article on a logic debate or dialogue between the originator of symbolic logic; one Gottlob Frege, and the great German mathematician David Hilbert who helped Einstein find the right math for the General Theory of Relativity. After Einstein asked Hilbert for help, Hilbert actual solved the equations for the theory before Einstein (who did so with help from Hilbert independently).

The Logic question concerns the nature of mathematics axioms; in brief, does consistency prove existence? The questions informs one about the nature of existence of the Universe in some respects; for mathematics can be used to model quantum mechanics with a kind of exhaustive reducibility in theory. From a certain point of view energy and matter can be regarded as accreted around mathematical points; humans can view nature as built up from math-geometry, algebra etc, and for the spiritually inclined the foundation for mathematics that is itself just a concept or idea, is spirit.


https://plato.stanford.edu/entries/frege-hilbert/Here is an excerpt from the Standford philosophy article; “The central difference between Frege and Hilbert over the nature of axioms, i.e., over the question whether axioms are determinately true claims about a fixed subject-matter or reinterpretable sentences expressing multiply-instantiable conditions, lies at the heart of the difference between an older way of thinking of theories, exemplified by Frege, and a new way that gathered strength at the end of the nineteenth century. Perhaps most clearly illustrated in Dedekind 1888, the central idea of the new approach is to understand mathematical theories as characterizing general “structural” conditions that might be had in common by any number of different ordered domains. Just as, in algebra, the axioms for a group give general conditions that can be satisfied by any manner of object whatsoever under appropriate relations, so too on the new view the axioms of geometry specify multiply-instantiable conditions. Viewing theories from this modern perspective, it is entirely appropriate to take axioms as Hilbert does, since reinterpretable sentences are the right vehicles to express the multiply-instantiable conditions in question.[8] From the point of view of the earlier fixed-domain conception of theories, on the other hand, such reinterpretable sentences are entirely inappropriate as axioms, since they fail to fix a determinate subject-matter. On this question, i.e., the issue of the fixed-domain (Fregean) vs. multiply-instantiable structure (Hilbertian) conception of mathematical theories, the jury is still out: this debate continues to animate contemporary philosophy of mathematics (see entry on philosophy of mathematics).

The second issue that divides Frege and Hilbert, regarding the justifiability of the inference from consistency to existence, is also still alive. While everybody (including presumably Hilbert) would agree with Frege that outside of the mathematical domain we cannot safely infer existence from consistency, the question remains whether we can (or must) do so within mathematics. The Fregean point of view is that the existence of mathematical objects can only be proven (if at all) by appeal to more fundamental principles, while the Hilbertian point of view is that in appropriate purely-mathematical cases, there is nothing more to be demonstrated, in order to establish existence, than the consistency of a theory (see entries on philosophy of mathematics and Platonism in the philosophy of mathematics).”