These are two questions I have been asked lately and I feel they are sort of related to each other, so I will attempt to address them in a single post. I didn’t really have a well thought out response when one of these discussions was started, but I feel I have formulated a good idea of what’s actually going on. My professor says my view of math is a lot like that of G.H. Hardy, so I apologize with anyone that heavily disagrees with this and sees things in a different light. I will try my best to explain why I think my view on mathematics is quite accurate as it interrelates to other subjects and why it is both invention and discovery at the same time.
First off: Does math need science?
I firmly say no and most mathematicians will heartily agree with this assertion. Does science need math? I say yes and again many will agree. It’s this interplay between the disciplines that generates some confusion on the nature of math. Non-mathematicians, for the most part, seem to view math as an extremely powerful tool for explaining observed phenomena in our world. I completely agree with this view and I honestly can’t see things being done any other way. However, does this necessity work the other way? Certainly, mathematics’ level of abstraction does not require our world, our universe, or anything else physical in nature to define its world.
Most of the tactics used in this discussion deals with the nature of how science proves things versus how mathematics proves things, but I will go with a different tact, because I think it is far more interesting.
In science, scientists spend time doing experiments and gathering data. Sometimes this data will lead scientists to generalize the data with a mathematical equation. We see this in physics all the time with things like F = ma. But does F=ma have anything to say about any mathematical truths? No, not really. Please note, I am not saying that scientists can’t prove things in mathematics, this is patently untrue. Physicists have developed wonderful things in mathematics, but they were not doing physics, they were doing math and these are different techniques.
Here’s the real crux of the problem, scientists do not get to choose their world. Reality is what it is whether they want it to be that or not. They do experiments to see what they can find out about the universe. Sometimes this requires the creation of a mathematical model that helps explain a phenomenon. This is surely invention on the part of the scientists and they are using rules of mathematics as a tool to delve deeper into this creation they have made. However, any truths they claim to find about reality by doing math will obligate them to do an experiment to verify it. If the experiment does not confirm the mathematics, then their mathematical model is simply not the right one. That is all, so they move on and choose a better one. This is how science tends to trudge along.
Second: Is mathematics invention or discovery?
It is both!
Mathematics is entirely different. In mathematics we choose our world and we have no choice over the results. Mathematics always begins with laying out definitions; this is the piece that is pure creation. However, we then see what logical implications the definitions behoove us to make; this is why it feels like discovery when a new Theorem is “discovered”.
For example, if we set up an ordered algebraic field for the Real Numbers, then we further come up with definitions for surpemums and infimums… the Mean Value Theorem will result! As long as all of the conditions exist this theorem is true and it isn’t just true it is an absolute truth. You are guaranteed this as a result of the definitions.
Science clearly never works this way. They always have “here is the universe, here is the earth in the universe” etc, these are the givens they must work with and now it is up to them to construct mathematics to help explain what they observe. The other wrench in this is that there is also some percent error, so things are never absolutely true. They may be very close, with 99.995% certainty, but there is always a degree of uncertainty. Furthermore generally accepted ideas in science can be overturned if some other fact presents itself, or some experiment shows it to be false. Mathematics does not need to worry about this. Once a theorem is proven there is nothing that can ever come along to show the theorem is false.
I am not trying to downplay science. Science is very practical and there really is no other realistic way to study the world around us. Mathematics is a very powerful concept, but you pay a price for that power. You are no longer bound by reality. If mathematics were bound by reality then we would never be able to talk about things like right triangles. There would always be some atom partially out of place throwing off that proper 90 degree requirement. Mathematics talks about a purely mental idealization of things. This idealization lets us talk about things that may never exist in reality and that’s all good fun, but it really is just head games.
Hopefully this has made sense and if it hasn’t, I may spend time revising it later.