Mathematics: Invention or Discovery?
May 23, 2006 – 8:46 pmWhen I started studying Maths, the answer to this question became clear to me: Definitions are invented, while theorems are discovered.
As Hardy said, Mathematics is the study of patterns. Practicing Math means basically two things: Defining interesting structures, and investigating their properties and inter-relations (proving theorems).
When you define a structure, your imagination runs completely free. Your only constraint is that the structure should not contain any logical inconsistencies, otherwise it would not exist. I believe this is exactly what we think of when we say ‘invention’: You have this huge space of things you can do, and you work your imagination to find an interesting and useful item in that space.
Of course, not all Mathematical definitions are born out of reckless roaming through the space of sturctures. Some are quite natural abstractions of earlier definitions. For example, basic analysis progressed more or less as follows:
- First, the real numbers were studied and the notion of ‘limit’ was defined. The definition of the ‘limit’ contains a measure of the distance between two numbers: |a-b|
- Some time later, people noticed that you can replace this distance with a more general concept. You can define a ‘metric’ that has the properties you can accept from a function that measures distance, and using that you can define a new ‘limit’. That led to metric spaces. These spaces contain an important, basic definition of an ‘open set’.
- It was then noticed that you can abstract away the metric and start out with just the open sets. That gave us Topology.
Nonetheless, these definitions are still not trivial extensions of one another — it takes much insight and, I imagine, a great deal of trial and error to find which ones are interesting and which are dull. So, finding and defining structures is akin to invention in the day-to-day world.
What about theorems? Once you have your definitions in place, you can start exploring them to obtain deeper insight into their nature. You do this by proving theorems — true statements about the objects you defined. The key point is this: once you define your structures, all the theorems you could possibly prove are pre-determined by simple logic. You cannot invent anything — you can only discover what’s already there. The theorems are buried, so to speak, under the surface of what your limited mind can initially grasp. All that’s left for you to do then is find them, dig them up, and prove them.
So Mathematics really isn’t invention or discovery; it’s a combination of both.
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2 Responses to “Mathematics: Invention or Discovery?”
“…are pre-determined by simple logic.”
Well, except from the fact that the axioms of logic are a mattar of personal choice. Hence you get different theorems for the same ’structures’ based on which logic you choose.
This is quite a hit for the notion of ‘discovery’, which as you presented it is just ‘finding out what is true’.
By Uri Cohen on May 27, 2006
I agree, you can only ‘discover’ something within the confines of your chosen axiomatic framework. If you allow the framework to vary, you get varying theorems.
I see it as a generalization of the familiar notion of discovery. Everyday discovery is done within the confines of the physical world. In Mathematics this is generalized, and you can choose the framework in which you discover things. I do not contend that you can ever ‘find out what is true’, merely that you can ‘find out what is true within a given framework’.
So, as long as you first choose an axiomatic system and then define a structure, the rest of your work falls under this generalized discovery. And I believe that a vast majority of theorems are discovered this way.
Even if you allow the framework to vary slightly, I contend that the following theorems are still ‘discovered’. For example, if you define a structure and then go about looking for theorems both with and without assuming the continuum hypothesis, then you clearly haven’t invented anything. You’re just exploring a slightly larger space.
Now, if you can give an example where you define a structure first, then wildly vary the axiomatic system and look for theorems, then I’d agree that such an example of theorem-hunting is much closer to ‘invention’ than ‘discovery’. Sadly, my basic knowledge of mathematics doesn’t include such an example.
We can continue and ask about the axioms themselves: Are they invented or discovered? The way I see it, the axioms are very much like definitions in the sense that you can choose them out of thin air, and your goal is to find the ones that lead to interesting and consistent results. So they should fall under ‘invention’. And this provides a new perspective on the previous question: Perhaps we can think of the axioms as an extension of the ’structure’ itself. In this case, my previous arguments regarding theorems apply as-is.
By Guy Gur Ari on May 28, 2006