Whom use the tools or norms to investigation the facts which related to both?
Dear Adel Mohammed Al-Odhari
Unfortunately, both the word "methodology" and "method" are used, even on these issues, with more than one meaning, which makes your question unambiguously unanswerable.
I can tell you, however, that there are two main issues that can be distinguished when talking about the methodology of logic and mathematics.
The first is the procedure (or set of procedures) or rules intended to organise the knowledge of both a logical theory and a mathematical theory. This makes it possible to speak, for example, of the demonstrative method (e.g. Aristotelian) or of the axiomatic method, whether ancient or modern.
The other important meaning of "methodology" relates to the mode or ways or procedures intended to obtain knowledge, and here one can differentiate, for example, between direct demonstration and demonstration by absurdity and, as a procedure intended for problem-solving, the method of exhaustion.
On the other hand, it must be considered that, although some authors and currents accept that when we speak of the objects of logic and mathematics we are talking about facts, others reject this, because they consider that the structures of reasoning or numbers - for example - are not facts, they are not states of affairs but abstract objects, and therefore differentiate between formal sciences and factual sciences, which do study facts, such as the natural and social sciences.
Thanks for your wonderful intervention, which it has benefited a lot for me, and your interpretation about word methodology of logic and mathematics.
In my recent (Sept 2022) paper, I quote Kohonen:
"When Russell championed logic, he did not act as a spokesman for a movement urging a return to traditional logic; he considered the old logic every bit as sterile as Kant did...
"What Russell advocated was mathematical logic, the entirely new [c. 1900] type of logic which had begun to unfold in the latter half of the nineteenth century… The gist of Russell’s anti-Kantianism is the substitution of ‘symbolic logic’ for pure intuition as the true source of mathematical content and mathematical truth."
I argue against Bertrand Russell's position.
Article Gustav Schmoller Rides Again! - The Modern Methodenstreit
The methodology of logic and the methodology of mathematics are closely related, as logic is often considered a branch of mathematics. However, they also have some distinct differences in their approaches and goals.
Methodology of logic: The methodology of logic is concerned with studying the principles of reasoning and argumentation. It involves the study of formal systems, such as propositional logic and predicate logic, which provide a framework for analyzing arguments and making inferences. The methodology of logic also involves the study of various logical concepts, such as validity, soundness, consistency, and completeness, and how they relate to the structure of arguments.
Methodology of mathematics: The methodology of mathematics is concerned with the study of mathematical objects and their properties. It involves the use of axioms, definitions, and logical reasoning to develop mathematical theories and proofs. The methodology of mathematics also involves the use of various mathematical techniques and tools, such as algebra, geometry, topology, and analysis, to study mathematical structures and solve mathematical problems.
While there is significant overlap between the methodology of logic and the methodology of mathematics, there are also some important differences. Logic is more concerned with the principles of reasoning and argumentation, while mathematics is more concerned with the study of mathematical objects and their properties. Additionally, while logic often serves as a tool for studying mathematical concepts and proving mathematical theorems, it is also used in other areas, such as computer science and philosophy.
By the way: I heartily recommend Plotinus' "Enneads". They answer many questions... We do not expect it.