“excited about something” or “excited by something”?

If this question is all about asking if it is possible to say someone/something being “excited about something” or “excited by something”, then the answer is yes. But I would like to convince that you cannot always use excited about and excited by interchangeably. Actually, I think they never are the same, even in those cases that both alternatives seem to be applicable, because it will always reflect what you actually think mentally inside.

I tried checking this with my grammar books, but none could provide a satisfactory answer about this subtle difference. So I will explain my idea first, and then will provide some evidences I found in COCA corpus to support my idea. Here is my usage note on excited about vs. excited by,

• When you are excited by something, that something is the cause of your excitement.
• When you are excited about something, you are excited. Excited about what, you might ask yourself? Of course, you are excited about that something. But, that something might not necessarily have to be the direct cause of your excitement. You can make yourself getting excited about the some idea, e.g. traveling somewhere, studying something, being somebody, and so on.

Let’s consider some examples that replacing by with about does not make sense:

Hydrogen can also be excited by heat or electricity.
Atoms can be excited by a number of different mechanisms.
A single molecule can absorb and be excited by a single photon.

Here are some examples of excited by used with people (or entities representing a group of people):

Besides, computer companies are not as excited by books as they are by games, which represent an ever-increasing share of the market.
And while a lot of people, researchers in the field were very excited by the idea of having the quantum computing wiki …
They’re actually quite excited by the fact that this is an international movement almost.
We’ve actually gained new people who are excited by the idea that we could have a genuinely different grassroots campaign …
We are especially excited by our guest this week.

Here are examples of excited about used with people (or entities representing a group of people):

The mandate is the part that gets insurance companies and providers excited about this law …
Nobody at that company is excited about my recording career.
She was upbeat, very optimistic, and very excited about the project.
All right. Welcome back. Now, I’m very excited about this little segment.
Yeah, I bet they will because they’re so excited about trying to beat Obama.
He was excited about her joining him on campus.
The chefs are usually very excited about going out there.
It’s important to sit down with your kids over their homework and get them excited about learning.

When the subject is an inanimate thing, it’s quite clear that we should use only excited by, because inanimate things cannot think by themselves. However, when the subject is a person, or some entity that we can thought of as a person, the usage can be overlapped. But as can be observed in those examples, the use of excited by implies the cause of the excitement.

Because excited by implies the cause, those usages of excited by are about things the has happened or is happening (as @StoneyB suggested in the comment below). On the other hands, excited about can be used with things that has happened, is happening, and will happen.

The first 4 examples of excited about above are things that already happened, or is happening. The remaining 4 examples of excited about are things that are not happened yet. And, in these last 4 examples, it will sound unnatural or misleading if we replace excited about with excited by.

Back to your example, which is particularly interesting,

I was very excited about/by mathematics?

Why?

Because it falls into the category that both excited by and excited about can be used. Consider these sentences:

I’m excited about the trip next year.
I’m excited by [the idea of going on] the trip next year.
I’m excited about science!
I’m excited by [the idea of doing more] science!

Both excited by and excited about almost mean the same thing in these examples. The part [the idea of …] can make the meaning clearer, but it’s not really necessary. And that’s the reason why I think you can use either one of them. But wait! Before you are going to use either of them, you might be interested in the results I found in COCA when I searched for “excited … science”:

SCIENCE     21

Looking through all these 21 results, 5 of them are irrelevant, and except for one occurrence of “… excited to be teaching science, …”, all of them are excited about.