# The nature of Mathematical Knowledge

I used to think there was a murky line dividing maths and science, that the two could easily be the same thing given a different name, and more often that maths was a science. But after our lesson, and reading a section of the textbook, this line has become more defined, and I have come to realise maths and science are two distinct, sovereign countries of their own. I guess what confused me, was that maths is always used in science, so how is it not a science itself? But then I realised that the two areas of knowledge have different aims, in a sense, the natural sciences try to explain reality as we perceive it (and sometimes this reality can be explained using maths that applies). Whereas maths is not there to necessarily explain anything, it is more of a creation, as it is not limited by our perception of reality.

In maths new worlds can be created and explained, even if they do not exist. When these worlds/systems are created, just like in our world, there are rules of how it functions. These rules are called axioms, they are defining features of the system, that are completely true in the context and application of that system, but when you try to apply them to another system they don’t work.

For example, when looking at the natural sciences, if we try to apply that the moon has the same gravitational force as the Earth, and that it has an atmosphere like the earth does, when making calculations about the variation of speed and time as a ball is dropped. We would end up with numbers the same as the ones that apply to Earth, but when it is experimentally carried out on the moon, it will become evident that the reality does not match the calculations. (it is possibly confusing that I used an example of science to explain this :/). So axioms are exclusive to the system they define. (however, I think some axioms might exist in two systems, if they both are truthfully defined by it).

If axioms define a system, then it can be said that all specific (zoomed in) knowledge about the system can be deductively found, because what ever happens in the system has to be a result of the axioms, therefore these general rules can be zoomed into specific situations and be explained. And example of this would be using general formulas, and applying them to a certain problem. But inversely, since everything that exists in the system is result of an axiom, these axioms can also be derived from the numbers themselves, by finding patterns in their occurrence. For example in maths right now we are doing differentiation using first principle, and by giving us a specific function, and manipulating it, our teacher showed us how the first principle works, and that it’s result have a reoccurring pattern, that has resulted in the ‘power rule’ of differentiation.

below this line, is me sort of rambling. (in someways it is not connected and does not flow with what I was talking about before so i decided to section it off)

Maths was not created to be applied, but it just happens to in some case, and that creation is what I learnt is called pure maths. Creating maths for the fun of doing so, not necessarily for any human purpose. This is so weird to think about, because this is the aspect that makes maths so abstract, but not in an ambiguous way, because it is still a very certain and defined knowledge that we gain from it. It becomes abstract because you can’t perceive these numbers, you can’t  touch them, ‘see’ them, but they exist in their systems and therefore are real in their systems. I think in maths it is difficult to define ‘truth’ because, I would be quick to label maths an absolutely truthful area of knowledge, because when applied to it’s own mathematical system and context it is completely true, but of course it can be argued that “how can all maths be true if it some of it exists in a made up world which itself isn’t ‘true’?” To that I would say, we are limiting truth to perceptibility.

I am also starting to question the knowledge we gain from maths. Because we gain knowledge about worlds that don’t exist? how is that knowledge useful. I do very much think it is a good thing if anything to continue the strand of pure maths, because maybe these non- existing maths worlds, actually do exist somewhere in our reality too, somewhere in the universe that we have not yet come across.

I think there is so much beauty in the idea of maths being masked in it’s basic form. All the knowledge we gain from it has always existed in its basic axioms, and have to be discovered by transforming it into a new form.

## One Comment

Good post Aarushi. I can ‘feel’ your interest in maths (I share it). Your first section is right on the money; accurate and clear. As you say using a science example is a little confusing, but then I think you actually do not know different maths axiomatic systems, so that’s fine. We might talk about some systems next lesson.

Your ‘rambling’ below the line is addressing the nature of mathematical truth. There are two different extreme views here. One is Bertrand Russell’s view:

“Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. […] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true”

Other extreme:”The normal notion of pure math is that mathematicians have some kind of direct pipeline to God’s thoughts, to absolute truth,” Greg Chaitin

The challenge seems to be that at times, both seem compelling. For example – suerly it is just TRUE that for any circle on a plane that circum/diameter = pi. But also, see here https://drive.google.com/file/d/0B7cloZU_Id7bNVloOFlYbDJTT1U/view?usp=sharing for something that does not seem ‘true’ but more ‘made up’

It’s a real dilemma; and one that we will discuss