We all know how mathematics is viewed by the non-mathematician nowadays: a body of knowledge full of formulas and incomprehensible symbols. It’s a point of view that most high-school teachers promote and, in this way, they make mathematics a sterile field without practical meaning.
The question is: where is the living mathematics? Eric Temple Bell answers this question in his masterpiece: The Development of Mathematics. Mathematics isn’t about formulas and equations (altough it incorporates this form of expression). Mathematics is about methods of thinking concerning certain problems, so effective that all sciences use it for expressing their theories. Mathematics is about ideas and precise thinking coming through centuries of clumsy techniques.
Bell tells us also that living mathematics is not the exact science perceived from the outside world. Its methods are not given from above, there’s no theology or metaphysics involved in it. Beneath all the clear cut thinking there is a lot of struggling from great men to understand natural and abstract phenomena.
From the empirical tests of babylonians used in measuring particular areas of certain fields, to Pythagora with his famous theorem, there is a long and hard way. No one knows how, from this empirical mindset, which was a kind of submathematical activity, came about the idea of proof. The human mind evolved and needed proof for facts, proof that some measurements from practice hold for large sets of numbers, proofs that in a right triangle exists a definite relationship between the hypotenuse and the other two sides (the square of the hypotenuse is equal to the sum of the squares of the other two sides).
And, most of all, the greek Euclid gave us the most efficient tool of reasoning: the postulational method. The method is simple – you start from postulates, general accepted statements, and deduce the consequences. It’s the most powerful method known to humankind to reason about mathematical objects and nature.
Compared with the so called informal method and the intuition, the postulational method is so direct and straightforward that no one can argue with its efficiency of proving statements.
Reinforced by David Hilbert in the 20th century, the postulational method became more powerful than ever. And, along with it and helping it, came to life the abstract method. The abstract method followed the detailed work of 19th century mathematics encompassing it in simple, beautiful and elegant results that generalized what was known before. The abstract method is the method of finding the essence through the details (addition in integers shares the same properties of associativity, neutral element, commutativity with the addition of matrices over commutative rings, for example). Before the abstract method, the connections between mathematical objects were not clearly seen and computations lacked elegance. Emmy Noether, the woman who pushed the abstract thinking into the front lines of living mathematics, said that her computation filled dissertation was “crap”. After the dissertation she devoted herself to abstract thinking simplifying much of what was known before her and discovering new and profound results.
After all, in mathematics we discover the history of the failure and success of human reason, with the most outstanding methods of thinking still being used today. It’s a history of making order from otherwise disorderly objects. It’s a method of understanding the very nature of physical reality and of pure thought, while struggling with the unknown and the confusion. It’s about the human mind and its reasoning techniques. Formulas are just the language, but the ideas behind them are the life of mathematics.
The book of Eric Temple Bell seems to leave us with a lot of questions about ourselves, one being how come the human race is capable of abstract thinking and how did the abstract thinking appear? After all, we can understand a lot of things about this vast and intricate universe with our limited skull using abstract thinking. For Bell (and for me), it’s a wonder that we can think so efficiently. Sometimes.
You can find this book here: The Development of Mathematics.
P.S. Let me know what you think of the book when you’re done.
January 19, 2016 at 5:04 pm
Interesting Read. Do you mean “matrices over a commutative rings” (rather than “commutative groups”) in your addition example?
January 19, 2016 at 8:51 pm
Thanks for the comment, you could think also of matrices over commutative rings, the commutativity of the structure underlying the matrix is one of the essential properties that connects the integers with the matrices in the example given.
January 20, 2016 at 9:35 am
Thanks for the reply.
I guess my point is: I don’t think matrices are defined over groups. Being a matrix implies some multiplicative structure (as well as the additive structure). I would be more comfortable with vectors over commutative groups (although one could claim that this isn’t really defined either, so perhaps it isn’t worth changing).
The most rigorous thing you could say is an element of a direct product of commutative groups. However this begs the question why bother introducing direct products at all and just talk about commutative groups and at this point I am really confusing the matter. I think it is probably least confusing to leave it how it is.
January 24, 2016 at 5:39 pm
You’re right, i’ve changed to matrices over commutative rings, in order to be rigorous and correct and to preserve the main idea. Thanks