MAthematical Logic

Mathematical logic is something that has been around for a very long time. Centuries Ago Greek and other logicians tried to make sense out of mathematical proofs. As time went on other people tried to do the same thing but using only symbols and variables. But I will get into detail about that a little later. There is also something called set theory, which is related with this. In mathematical logic a lot of terms are used such as axiom and proofs. A lot of things in math can be proven, but there are still some things that will probably always remain theories or ideas.

Mathematical Logic is something that has a very long history behind it. It has been debated on for many centuries. If someone were to divide mathematical logic into groups they would get two major groups. Both groups are very long. One is called “The history of formal deduction” and it goes all the way back to Aristotle and Euclid and other people who lived at that time. The other is “the history of mathematical analysis” which goes back to the times of Archimedes, who was in the same era as Aristotle and Euclid. These to groups or streams were separate for a long time until Newton invented Calculus, which brought Math and logic together.

Somebody who studies mathematical logic and gives his or her own concepts about it is called a logician. Some well known logicians include Boole and Frege. They were trying to give a definite form to what formal deduction really was. Aristotle had already done such a thing but he had done it with language, Boole wanted to do it with only Symbols. Frege came up with “Predicate Calculus”.
As time went on people did not make new theories as much as they used to in the time of Aristotle. They mostly concentrated on expanding on theories that have been said centuries ago, proving those theories or putting them into symbolic form.

Table of Logicians*
*This Table has a few of the Logicians listed in my book

Words that have to do with logic like and, or, not are given symbols like &, V, or an upside down L reversed. The Letters X, Y, Z and so on are commonly used as variables and P, Q, R are used as predicates, properties or relations.

Sometimes there are theories that have to do with machines that do not exist and usually have things in them that are infinite and they usually work with letters and numbers. For example in Chapter 4 which is “Turning Machines and recursive Functions” it talks about a machine that has a tape running through it. (This is not a real machine) The tape is endless from both sides. It is divided into little squares and in each square there is a small letter of the alphabet and a number under it. The machine reads this, changes it moves it one to the right ort one to the left. This experiment was conducted in 1936.

Model theory is the study of different formal languages and their relations with each other. They get a normal sentence and they turn it into variables and symbols, then they compare it with other languages. For example if you take:
If the boss is in charge and Joe is the boss, then Joe is in charge.
If you convert that you get:

This gets very confusing but the way they get the formula is that they do something with the “Predicate Letter”. Then they turn it into variables and symbols.

In one section it talks about how there are an infinite number of fractions between two rational numbers on the number line such as 1 and two. It goes like ½ 1/3 ¼ 1/5 ... and so on infinitely. It also compares that to a different kind of number line.

Gödel was a different logician then the rest. He concentrated more on expanding on other people’s theories then anything else. Although he has said many good theories himself. That is why many people consider him one of the best logicians and a very good mathematician.

To conclude I want to say that I found this book very difficult and I understood about 20% of it so I read the whole thing and wrote about the parts I understood.