Not clear. It is true many apologists try and use logic to justify their beliefs and slide into a tube of mud fallacies such as intense 'begging the question.'
Proof in logic must be factual otherwise it is false.
Actually no, read up on logic. Logic is not necessarily based on 'facts,' and most often it is not. Though it is widely misused and misrepresented. Logic more accurately is:
from:
Philosophy of logic - Wikipedia
Philosophical logic is the branch of study that concerns questions about reference, predication, identity, truth, quantification, existence, entailment, modality, and necessity.
Philosophical logic is the application of formal
logical techniques to
philosophical problems.
Nothing here necessarily deals with 'facts.'
Math is not necessarily verifiable. The ultimate value of math is it usable or functional to satisfy a need. Math is part of our every day life, and the scientific tool box, but math itself does not have to be verifiable. Forms of math can and are developed with no apparent use or verifiability, but later may be useful in the science tool box.
From:
Mathematical logic - Wikipedia
Mathematical logic is a subfield of
mathematics exploring the applications of formal
logic to mathematics. It bears close connections to
metamathematics, the
foundations of mathematics, and
theoretical computer science.
[1] The unifying themes in mathematical logic include the study of the expressive power of
formal systems and the
deductive power of formal
proof systems.
Mathematical logic is often divided into the fields of
set theory,
model theory,
recursion theory, and
proof theory. These areas share basic results on logic, particularly
first-order logic, and
definability. In computer science (particularly in the
ACM Classification) mathematical logic encompasses additional topics not detailed in this article; see
Logic in computer science for those.
Since its inception, mathematical logic has both contributed to, and has been motivated by, the study of foundations of mathematics. This study began in the late 19th century with the development of
axiomatic frameworks for
geometry,
arithmetic, and
analysis. In the early 20th century it was shaped by
David Hilbert's
program to prove the consistency of foundational theories. Results of
Kurt Gödel,
Gerhard Gentzen, and others provided partial resolution to the program, and clarified the issues involved in proving consistency. Work in set theory showed that almost all ordinary mathematics can be formalized in terms of sets, although there are some theorems that cannot be proven in common axiom systems for set theory. Contemporary work in the foundations of mathematics often focuses on establishing which parts of mathematics can be formalized in particular formal systems (as in
reverse mathematics) rather than trying to find theories in which all of mathematics can be developed.
