generalization, abstraction, mechanization in mathematics07 Dec 2010
So many people are trapped in the morass of mathematics, until they forget the core and origin of it.
What is the essence of mathematics? It must be “calculating”.
However, why we feel we are weak while facing with grim and strange math homework? Why we feel tired while calculating? I will answer, because the mathematics has been generalized, abstracted, and mechanized.
Words are stilled remembered from teachers in primary school: smaller numbers can’t minus larger numbers, but later we known negative number. We didn’t know what sqrt(2) means before we went to high school. We thought sqrt(-1) does not exist, until we learned the existence of unreal numbers…… Generalization, we are benefit from and sometimes confused by it. Scientists tell us that human always learn things spirally. Once we broaden our horizon, concepts may be generalized, and “true or wrong” is not always as what we thought.
Abstraction, the most difficulty in our road of studying. Experimental results show that human disgust themselves from ability of abstraction, but they are not exactly good at it. “What is a vector?” A student in collage and one in high school should give different answers, I guess. Vector in high school is so concrete: it seems to be an arrow in space. However, textbooks in university tell us that “m=(a,b,c,d) and f(x)=ax^0+bx+cx^2+dx^3 are vectors, and they are equivalent to some extent.” Oh no! While we are confusing, more strange nouns flood into our minds: Linear Space, Real Variable, Functional Analysis…… We are lost in the abstraction of concepts. In another aspect, the abstraction of symbols are confusing us. (x,f)=f(x) is none of understandable to a new student in collage. Of course, abstraction of concepts hide behind symbols. When we are flying in the sky of mathematics, we should look down upon the concrete earth, with the help of derivation and history of maths development.
Why should we learn different methods of integration, determinant, matrix, etc? Mechanization. We may thanks for this maths methods, because they contribute to mechanization of proofs. Cartesiese koördinatestelsel, root formula, determinant, matrix, all of them enable computers to compute the results automatically. Wolfram Mathematica, one of the best mathematics software are benefit from them. Moreover, Natural Language proceeding, is widely used in our life, with the help of mechanization.
Hopefully, readers may get to know how we study mathematics and what we are doing while studying mathematics after scanning the passage.