The-Exiled said:
I have two first year maths subjects one on differential calculus and linear algebra and the other on integral calculus and series. There were other things like complex numbers thrown in as well.
This means that you should be able to do introductory university statistics that is usually taught in second year mathematics degrees straight away.
I was wondering about that. I wasn't sure if you could though, I suppose it won't hurt to ask!
Yeah, I have been trying to teach myself some other topics such as vector calculus, more in depth linear algebra and I have also covered topics in other physics and chemistry classes like simple group theory and operator algebra (from quantum mechanics). I can understand them - but nowhere near as much as I would if I were to take a mathematics class on each topic.
Thanks for your help, I'll look into doing a few classes while completing my PhD. That sounds like the best approach.
You are not alone and even people that study this for a living don't get it straight away.
Like other fields the best way to understand something can be done from a variety of approaches.
One way can be to get involved in communities that do this kind of thing. Physics Forums is one of those communities and is one of the reasons that my knowledge of such things has improved and crystalized in a way.
The other is through teaching. Teaching helps understand the same content in different ways. What I have found is that usually don't learn completely new things all the time: it's more of the scenario that you see things in a different way that opens up a whole new can of worms (i.e. even more things to try and understand!).
Building on the above, the other thing to do to understand things from a different perspective/viewpoint is to see different applications of the mathematics.
Mathematics as you are probably aware is used in many different areas. When you are exposed to seeing it used in one way versus another, you have a good chance of extracting something out of that comparison.
After a while what happens is that when you have seen things through quite a few lenses of perspective, you don't see through individual lenses but rather through something that is different, yet still allows you to see the same knowledge that you did before.
Those are just some suggestions.
Also with mathematics, I will give you some advice that I have learned (a lense if you will) to help you understand mathematics and one way of thinking about it.
The key things are a) representation b) constraints and c) transformations.
Representations are the things to describe in some kind of language what you are dealing with. Typically mathematics are used to using numbers (up to complex) and then building structures based on those. However there are structures known as sets which form the basis of foundational mathematics and if you wanted to describe say a graph, then you would do it with a set.
Understanding representation will help you understand the context associated with that representation: for example most people are aware of how numbers are used and the context they are used in. In contrast graphs are a completely different kind of structure and have a different context associated with them.
Different representations have different uses and thinking about all of them and seeing how they can be 'melded' and seen from a different perspective helps you see different things.
For constraints, you have to realize that this is the only way a lot of mathematics and its applications gets done.
If you had no constraints you would be dealing with 'everything and anything that can happen or be described' and clearly you can see that people would run into problems.
So what we do is we restrict the phenomena or the state space that we are describing until we are able to make sense of it and work with it to a point where we can take the next step and decrease the constraints (make them less constrainable).
What happens is that the more abstract you go, the less the constraints are and the more you are trying to describe.
So the take home message is that if you are struggling with something, increase the constraints and do so until you need to, to understand that. After that you can go in the reverse direction which allows you to see things in a different way and also helps you see 'the big picture'.
The final thing is transformations.
The most intuitive example of a transformation is a function like f(x) = x^2. But transformations are not just functions. An approximation of a function is a special kind of transformation. Changing one random variable to another as an approximation is a transformation. Converting a function to its taylor series under different centres is a transformation. Going from one line to the next in a sequential proof is a kind of transformation.
There are many kinds of transformations that have particular purposes and understanding the context of a particular kind of transformation will help you see things are done the way they are done.
There is, of course, more to the subject than these things, but I think this will help you see things in a way that mathematicians see things.
If you have any other questions I'll do my best to answer them.
