Thread: Logic!

As it is generally known, Logic is the theory of the way in which people reason, with aim of studing the principles of valid reasoning. Since the most modern logic is the Mathematical Logic or Symbolic logic which provides an extension and improvement of traditional or [b] Aristotalian logic:

1- I want to understand better of what importance is logic to mathematicians if it incumbent.

2-What is its relevance to other field of studies.

I would appreciate Views here :P

Originally Posted by didact

As it is generally known, Logic is the theory of the way in which people reason, with aim of studing the principles of valid reasoning. Since the most modern logic is the Mathematical Logic or Symbolic logic which provides an extension and improvement of traditional or [b] Aristotalian logic:

1- I want to understand better of what importance is logic to mathematicians if it incumbent.

2-What is its relevance to other field of studies.

I would appreciate Views here :P

Google "axiomatic set theory" "Zermelo Fraenkel axioms" and "Gödel".

Logic is the study of patterns of reason. So it is potentially useful in all domains of reasoning.

Particular applied in 'exacting' areas (where exactness is important) such as mathematics, philosophy, science and software engineering.
Back in 1910-1913 Whitehead and Russell tried to show that mathematics can be derived from logic. It worked up to a point, but it was tricky.

Much better foundations for Maths were found later, esp. Zermelo-Fraenkel axiomatic set theory).

But all these methods (such as set theory) make use of modern logic methods: propositional logic/calculus and predicate logic/calculus.

Originally Posted by didact

As it is generally known, Logic is the theory of the way in which people reason, with aim of studing the principles of valid reasoning. Since the most modern logic is the Mathematical Logic or Symbolic logic which provides an extension and improvement of traditional or [b] Aristotalian logic:

1- I want to understand better of what importance is logic to mathematicians if it incumbent.

2-What is its relevance to other field of studies.

I would appreciate Views here :P

First order logic is overrated. It is really nothing more than modus ponens.

Now, once you expand your language to set theory, you then inherit the Kleene recursion theorem which expands the region of logic.

Under this reasoning, you find a base truth.

Then, given a truth, you can prove another truth by some mechanism.

If this is done, then you can conclude the truth holds for all of the contained logic.

There are a few comments about quantum theory in one of Roger Penrose's books which have a bearing on the nature of logic. Penrose refers to some bewildering aspects of quantum theory as "X -mysteries" and he describes them as follows:

"The other kinds of quantum mystery are things that I shall refer to as X-mysteries, or paradox mysteries, which would, on the other hand, be things that quantum formalism seems to be telling us to have to be true of the world, but have such an implausibly paradoxical nature that we cannot really believe in them as being in any sense 'actually' true." (from "Shadows of the Mind") He considers that the best known example of mysteries of this sort is the Schrodinger's cat saga.

Penrose considers that the X-mysteries come about because quantum theory is an incomplete theory. But it does raise the question - where does it leave logic if these mysteries cannot be resolved? Is it possible that at the level at which quantum theory applies, what we consider to be illogical from our macroscopic point of view could actually be the case? (I apologise for mentioning this on a Mathematics board, but the question of logic does crop up in all sorts of contexts.)