I am curious if mathematics could be defined as: "exact abstract descriptions of reality".
I use "descriptions" in plural because there are multiple distinct mathematical views on problems which it is solving, even on the same one.
It is obviously abstract, and it is also exact, because it explains reality very precisely in comparison to other sciences. However, it can't be absolutely exact because of the uncertanty principle. We can apply the same mathematics in sciences that are exploring completely distinct occurrences in reality. A simpler way to understand this comparison between mathematics and other sciences: https://xkcd.com/435/
I dont't think that math is reality, because in that case it would have to be absolutely exact. I also don't think it is something "magical", but rather a tool that people came up with to describe reality on an abstract level. Math is also fun, artistic and beautiful and there are many extensions to it, which have this as their only purpose and aren't about reality. But if we look at why mathematics even exists - because people need descriptions of occurances in reality on an abstract level, so that we can apply them on different problems which we are facing. In addition, if we say that mathematics is science, the "reality" part of the definiton is correct because science is about "physical and natural world" (from definition).
I think that there is no agreement about the definition of mathematics because none of them describes mathematics on a level which is abstract enough. This definition might be better in that sense. It is probably not perfect, so please comment if you think there is something neccesary to add to the defenition. Maybe that it uses special mathematical vocabulary? Instead of "reality" the word "phenomena (observable events)" could be used. Also, "systems/abstractions and their dynamics" could be used instead of "reality". Maybe the best one would be:
Exact descriptions of abstract entites, their relations and their dynamics written in a formal vocabulary named "Language of mathematics", which are usually used as tools for finding the most efficient ways of manipulating systems in reality for achieving particular goals.
"system" here means: "abstraction which is agreed on by multiple people and measurement devices about its existence, therefore an abstraction which is objectively observable."
This is also compatible with quantum mechanics from which it seems like consciousness is a clasifier for existence. But just one consciousness can't define objective existence because of possible errors in perception like halucinations.
Almost any science could be defined in a similar way. So if this is the definition of "science", then mathematics could be: The most abstract and widely applicable science.
I would like to add that philosophical questions like "what is the definition of mathematics?" are meant to be answerd just like any other questions. Ancient philosophers didn't have access to internet for instant fact checking and didn't know about new scientific discoveries like uncertainty principle, so from their perspective those questions really seemed unanswerable.
Another way I think about mathematics: Mathematics is like a window (or rather "windows") to reality. But we don't really know if reality and window is the same thing.
So it's like we are taking a bottom up approach to describe reality. But I don't think that things like awareness and consciousness can be explained by mathematics.
I removed the parts which made this question seem like it is just about "am I right?".
The question is:
What is your opinion on that idea and/or which definiton of mathematics do you like the most and why?
Just because I don't want this question to be closed because it asks for opinion, the question is:
What is the formal definition of mathematics?