TITLE:
The Resolution of the Great 20th Century Debate in the Foundations of Mathematics
AUTHORS:
Edgar E. Escultura
KEYWORDS:
Axiom of Choice, Banach-Tarski Paradox, Goldbach’s Conjecture, Logicism, Constructivism, Fermat’s Conjecture, Field Axioms, Formalism, Qualitative Modelling, Rational Thought, Self-Reference
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.6 No.3,
February
26,
2016
ABSTRACT: The paper resolves the great debate of the 20th century between the three philosophies of mathematics-logicism, intuitionism and formalism—founded by Bertrand Russell and A. N. Whitehead, L. E. J. Brouwer and David Hilbert, respectively. The issue: which one provides firm foundations for mathematics? None of them won the debate. We make a critique of each, consolidate their contributions, rectify their weakness and add our own to resolve the debate. The resolution forms the new foundations of mathematics. Then we apply the new foundations to assess the status of Hilbert’s 23 problems most of which in foundations and find out which ones have been solved, which ones have flawed solutions that we rectify and which ones are open problems. Problem 6 of Hilbert’s problems—Can physics be axiomatized?—is answered yes in E. E. Escultura, Nonlinear Analysis, A-Series: 69(2008), which provides the solution, namely, the grand unified theory (GUT). We also point to the resolution of the 379-year-old Fermat’s conjecture (popularly known as Fermat’s last theorem) in E. E. Escultura, Exact Solutions of Fermat’s Equations (Definitive Resolution of Fermat’s Last Theorem), Nonlinear Studies, 5(2), (1998). Likewise, the proof of the 274-year-old Goldbach’s conjecture is in E. E. Escultura, The New Mathematics and Physics, Applied Mathematics and Computation, 138(1), 2003.