Pronoein said:
Hello, I'm a curious philosopher and I'm wondering what the scientists think about the main occupation of their life. I would gladly hear the members of the PF about this question:
What are mathematics about?
Mathematics is a framework that is used to analyze systems, objects, and representations in the most generic and most specific sense possible.
The generic nature could be measured by how abstract the particular object/system is.
The role of science and mathematics shares a common point: To generate frameworks that help explain and analyze more and more results using frameworks that are more unified and more abstract, yet more compact.
I'll give you real examples of what I mean.
Think about one dimensional numbers.
First we had the natural numbers. These numbers are positive whole numbers (ie whole numbers greater than zero).
Then these were extended to the integers (that is positive and negative whole numbers including zero).
Then after that the rational numbers (a/b where a is an integer and b is an integer that is not zero).
As math was progressing, people (like the pythagoreans) thought that all numbers were rational, but it turned out you could create numbers that could not be represented by your a/b.
As time went on mathematicians were developing theories about certain polynomial equations and as it turned out, you needed this thing called the square root of -1 (SQRT(-1)) to obtain roots of cubic equations (it was Cardano who did this).
As math progressed it become apparent that complex numbers were needed for guaranteeing roots for any polynomial equation.
After a lot of work, it was shown that complex numbers are the most general type of number under your standard operations (for example SQRT(i) is itself a complex number).
So what has happened is that we started with integer's and we got all the way up to complex numbers, and as a result we are trying to build mathematics that allows us to analyze things as generically as possible.
So in this way we have investigated the idea of a single number and developed the idea of the most general type of number which is a complex number.
In an analogy with physics, the processes of magnetism and electricity were found to be two sides of the same unified process.
This kind of process is what science hopes to do, but it gets harder and harder since you are dealing with trying to explain more and more results in a form that is more abstract.
Mathematics and science generally attempts to do the above. Connections are formed between less general and more general areas, and these results help us understand the big picture as time goes by. Like I said though, it gets harder and harder since there is a lot more to take in (its hard enough becoming a specialist in a tiny area let along an entire subject), and this makes it more challenging.
The upside is that a lot of people are working hard on understanding specific focused things, and since the communication bottleneck is largely gone thanks to the internet, progress is speeding up. One major problem is taking all the data that's out there and getting data that is relevant: this is a problem within itself and when more progress is made with this area, the better scientific progress will be.
