http://www.dailygalaxy.com/my_weblog/2008/04/is-mathematics.html
In my opinion, this is how it all goes:
It set me thinking about Mathematics, Science, their relationship with each other and their relationship with reality. Is mathematics, the language of the universe? Are we discovering the universe with mathematics? If that is so, then where does mathematics exist? Yes, matter exists and it seems to follow some mathematical equations. But does mathematics exist by itself somewhere in the universe? Or is it a purely imaginary field, having nothing to do with reality? Is it a pure figment of imagination of some brilliant minds? If so then how does it help scientists unravel the truth about this universe and its reality?
In my opinion, this is how it all goes:
I think mathematics has a bit of both. Some parts of maths are invented while some are discovered. The whole relationship between mathematics, physics, the universe and its discovery/invention goes like this.1. We see the universe and want to understand something in it
2. We invent a new world with specific rules. Examples of this are the decimal system or negative numbers or fractions or imaginary numbers etc. This new world needs not only a definition, but also a consensus. It requires some rules and boundaries accepted objectively by more than a single person. These rules have to be extremely consistent with each other. For example, the rule that defines the natural numbers is monotonically increasing. The rule that defines the decimal system is the number 10, the rule that defines negative numbers is monotonically decreasing and the rule that defines imaginary numbers is sqrt(-1). Even zero is an inventions and it follows definitions and rules very specific. Multiplication by zero is zero, addition or subtraction by zero is the number itself, but what about division? Axioms are another way of defining this new invented world.
3. The third phase is the discovery of a set of rules and definitions that are consistent with the first set of rules that define the space/world/universe. These give rise to theorems, corollaries etc. The space or the world that this discovery takes place is in the abstract world that was invented. So it is a discovery, but not in the real universe. Here we may find some caveats in the original definitions, like how about division by zero? So new concepts/rules/definitions/axioms need to be invented to fill in the places that are not examined.
4. The next and the most crucial phase is making the analogy between the real world/universe and the invented world of mathematics. This most often is made by physicists. The advantage of doing this is that we can make consistent predictions about observations using the space that was discovered in the abstract imagined world ofmathematics. For example, when Euclidean geometry was invented (although I am sure not by Euclid), it was possible to build buildings or make drawings of new weird shapes like triangles, squares and rectangles. These never existed in nature until it was invented. But by making the correct correspondences between the abstract world of geometry and the real universe of buildings people could build marvels like the pyramids. The most important aspect of making this correspondence is to make the right approximations to entities in the abstract mathematical world. A flower can be approximated to a circle, and then suddenly we discover symmetry, the human head can be approximated to a sphere and now we can discover something about perspective, approximate any object to a point mass and then you can discover the Newtons laws of motion.
5. This is an extra point where we make new discoveries that are not consistent with the approximation/correspondence we have made with the abstract world of mathematics. Here we either change the mathematical space we looked into or invent new mathematics which allows to make good correspondences. For example, when we had observations about the behavior of atoms, we shifted the space from deterministicmathematics to probabilistic mathematics and lo we discover new things about the atoms. When we had observations of a singularity in time like the big bang, we shift to imaginary (complex) time and suddenly we can start understanding the big bang much better.Thus maths is mix of both invention and discovery. But it is not reality. We interpret reality through the abstract world that we have created. Mathematics is not reality. It is a tool to understand it and a damn useful tool at that.
2. We invent a new world with specific rules. Examples of this are the decimal system or negative numbers or fractions or imaginary numbers etc. This new world needs not only a definition, but also a consensus. It requires some rules and boundaries accepted objectively by more than a single person. These rules have to be extremely consistent with each other. For example, the rule that defines the natural numbers is monotonically increasing. The rule that defines the decimal system is the number 10, the rule that defines negative numbers is monotonically decreasing and the rule that defines imaginary numbers is sqrt(-1). Even zero is an inventions and it follows definitions and rules very specific. Multiplication by zero is zero, addition or subtraction by zero is the number itself, but what about division? Axioms are another way of defining this new invented world.
3. The third phase is the discovery of a set of rules and definitions that are consistent with the first set of rules that define the space/world/universe. These give rise to theorems, corollaries etc. The space or the world that this discovery takes place is in the abstract world that was invented. So it is a discovery, but not in the real universe. Here we may find some caveats in the original definitions, like how about division by zero? So new concepts/rules/definitions/axioms need to be invented to fill in the places that are not examined.
4. The next and the most crucial phase is making the analogy between the real world/universe and the invented world of mathematics. This most often is made by physicists. The advantage of doing this is that we can make consistent predictions about observations using the space that was discovered in the abstract imagined world ofmathematics. For example, when Euclidean geometry was invented (although I am sure not by Euclid), it was possible to build buildings or make drawings of new weird shapes like triangles, squares and rectangles. These never existed in nature until it was invented. But by making the correct correspondences between the abstract world of geometry and the real universe of buildings people could build marvels like the pyramids. The most important aspect of making this correspondence is to make the right approximations to entities in the abstract mathematical world. A flower can be approximated to a circle, and then suddenly we discover symmetry, the human head can be approximated to a sphere and now we can discover something about perspective, approximate any object to a point mass and then you can discover the Newtons laws of motion.
5. This is an extra point where we make new discoveries that are not consistent with the approximation/correspondence we have made with the abstract world of mathematics. Here we either change the mathematical space we looked into or invent new mathematics which allows to make good correspondences. For example, when we had observations about the behavior of atoms, we shifted the space from deterministicmathematics to probabilistic mathematics and lo we discover new things about the atoms. When we had observations of a singularity in time like the big bang, we shift to imaginary (complex) time and suddenly we can start understanding the big bang much better.Thus maths is mix of both invention and discovery. But it is not reality. We interpret reality through the abstract world that we have created. Mathematics is not reality. It is a tool to understand it and a damn useful tool at that.
A very nice analysis. And as I have stated (during our discussion on this topic), I agree that Maths itself probably is just a tool. It can be termed an invention. A way for us to standardize a way of understanding the world around us. Much like how CHemistry is (I can speak with some authority on Chemistry :D. What we call oxygen is going to be a life line for us, irrespective of whether we call is oxygen or something else. So the name is an invention but the gas and its properties are a discovery. Maths, I feel, is a similar thing, invented but helps us understand our surroundings. Whether our surrounding IS reality or not is yet another topic for debate :)
ReplyDeleteNice thoughts Ananth.
ReplyDeleteDo give us your thoughts more elaborately on such things as discovery, invention and existence. Also, your article touches issues of dualism, rationalism-empirism conflict. Some time when you are a bit free, do try and extend it to cover these issues. :)
@Sujitda, ys, I'll try to elaborate further on the differences between discovery, invention and existence. But I don't know what rationalism-empirism is all about
ReplyDeleteI did specialization in math to understand why math. Every lecturer I mate, the first thing I asked what is real life use of Topology, Algebra, Number Theory or lattice theory etc. Though most of them were doctorate or scientists non of them able to answered it properly. Now, I just not forgot the names of subjects that I learned but I also I hesitate to say I'm a mathematician :(
ReplyDelete