Logic and Math

[grace24]

I'm curious how does logical absolutes apply to math? By logical absolute I mean that A=A, A=A, and Not A, and either A or B - The rules of logic we use them everyday.

 

[Gracchus]

This is the subject of a first course in abstract algebra. It is usually a one semester course. You start with set theory.

I could recommend a good book if you want. Or you could simply go to the library.

 

[hisgrace26]

Nice.

 

[grace24]

This is the subject of a first course in abstract algebra. It is usually a one semester course. You start with set theory.

I could recommend a good book if you want. Or you could simply go to the library.

Oh thank you! Yes I would appreciate if you can give me the name of the book. Someone I know told me what is logical absolutes and for me to show him in math book, so I was curious if math book teaches it.

 

[Gracchus]

I could recommend "A Book of Abstract Algebra: Second Edition" [Paperback] by Charles C Pinter. This is available from Amazon for less the $20. There are other introductory texts that cost closer to $200.
Bear in mind that the subject is usually an upper division math course, if not a graduate course. It is about proofs, not computations. It is about numbers, and number systems. Did you ever take a course in plane geometry? In the study of algebraic systems you will develop from axioms, definitions and properties the justification for the "rules" of arithmetic that most of us learned by rote in elementary school.

You might want to start by Googling "abstract algebra" rather than investing in a book. There are also courses and textbooks on-line. And, as I mentioned, there is the library.

 

[Chatter]

I'm curious how does logical absolutes apply to math? By logical absolute I mean that A=A, A=A, and Not A, and either A or B - The rules of logic we use them everyday.

You could write volumes exploring the relationship between logic and maths.

But here's an interesting point: it was only in the last hundred and fifty years that anyone realised there was a relationship. The reason it took so long is that it took important refinements in our theories of both logic and of mathematics, which only occurred in the 19th century. The story usually begins with George Boole, an English mathematician and logician and a book called "The Laws of Thought". In this book, Boole attempted to overturn the dominant paradigm in understanding logic, which had held strong ever since Aristotle, and the work became extremely influential, not only in the development of logic, but on the development of mathematics itself.

What Boole did was make an analogy: he saw a connection between the logical particles "NOT", "AND" and "OR" and operations on numbers such as addition, multiplication and subtraction. By doing this, he was in a position to start expressing logical statements algebraically. For instance, he realised he could trivially capture the four kinds of logical statement analysed by Aristotle with algebraic symbols. In so doing, he could see relationships between logical statements by finding the corresponding relationships between numbers.

For instance, Aristotle considered sentences of the form "No Xs are Ys." These are called "E" sentences in Aristotle's formalism. For Boole, they could be expressed simply as: xy = 0. And since xy = yx, this means that "No Ys are Xs" follows immediately from "No Xs are Ys". This point is not immediately apparent in Aristotle's system, and has to be more or less assumed. For Boole, it could be readily proven by algebra.

Boole's analogy has been well-developed since then, and we now talk about elementary logic as an example of a Boolean algebra, a structure which shows turns logic into mathematical operations. But this is only the beginning. The understanding of logic in terms of mathematical structure has been developed so profoundly in the last century, that Boole's work is barely recognisable today.

But there is a parallel story, starting with a German mathematician called Gottlob Frege. Frege was interested specifically in mathematical proofs, which have for millenia been the absolute exemplar of logical correctness. Frege wanted to understand such proofs, and to do this, he had to understand the basic modes of inference that govern mathematical reasoning. This meant developing logic far beyond Aristotle's formalism, ultimately forcing it into relics of history and replacing it with the notion of the "logical calculus" that we have today.

Frege understood logic as the blind and systematic application of transformation rules on symbolic formulas, methods by which one formula can be converted into another. He decided that the complexity of mathematical reasoning, something which had seemed impossible to capture in Aristotle's theory, was governed by invoking "higher-order" principles, in which our reflections on logical deduction rules are brought under the scope of the deduction rules themselves. By doing this, he created a system that was so powerful, that all objects of interest to mathematicians --- the numbers, the points, the lines, the planes, the algebraic structures --- could be understood in terms of its primitive logical concepts. For instance, Frege could understand the number 0 merely as the logical concept "to be false of all things", and the number 1 as the logical concept "to be true of identical things".

Frege's system was fatally flawed, but his ideas inspired a philosophy of mathematics called Logicism, and two English philosophers, Bertrand Russell and Alfred North Whitehead, who tried to recast the whole of mathematics in terms of purely logical concepts. This programme turned out to be extremely challenging, and has now been taken up almost entirely by computer scientists who use algorithms to help the process along. However, it left a profound legacy in the mathematical community. Just about every mathematician believes that mathematical arguments can, in principle, by understood entirely in terms of relations among purely logical concepts. In other words, mathematics can be reduced to pure logic. And for many mathematicians, this reduction provides the ultimate standard for mathematical proof.