If a process can be represented as such and such, and then this in turn can be represented as something else, and then eventually after this possibly-long chain of transformations we reach something mathematical, then it is that something which is mathematical, and not what we started with.
This is like saying: "There isn't such a difference between neurology and physics. A brain can be viewed as a collection of cells, which in turn can be viewed as a collection of organic molecules, which in turn can be viewed as a collection of particles, and physics studies particles. So neurology = physics. Q.E.D"
But what is missing here is the realization that it is exactly our decision not to view the brain as a collection of particles which gives neurology it's meaning. The decision to view something as something else is not trivial. When you choose to view X as Y, it is no longer X you are talking about.
You wrote: "...it just may not be the kind of mathematics we usually associate with the word."
But it is precisely what we "usually associate with the word" which gives meaning to words. If in this "usual" meaning it is wrong to say that mathematics is the study of algorithms, then mathematics isn't the study of algorithms.
When we want to create new precise definitions to old words, what we are trying to do is to formulate a description that fits as best as possible our everyday use of this word. If we ended up with a definition of math that says "the analysis of everything is math", the only logical conclusion is that we did a pretty bad job.
This is like saying: "There isn't such a difference between neurology and physics. A brain can be viewed as a collection of cells, which in turn can be viewed as a collection of organic molecules, which in turn can be viewed as a collection of particles, and physics studies particles. So neurology = physics. Q.E.D"
But what is missing here is the realization that it is exactly our decision not to view the brain as a collection of particles which gives neurology it's meaning. The decision to view something as something else is not trivial. When you choose to view X as Y, it is no longer X you are talking about.
You wrote: "...it just may not be the kind of mathematics we usually associate with the word."
But it is precisely what we "usually associate with the word" which gives meaning to words. If in this "usual" meaning it is wrong to say that mathematics is the study of algorithms, then mathematics isn't the study of algorithms.
When we want to create new precise definitions to old words, what we are trying to do is to formulate a description that fits as best as possible our everyday use of this word. If we ended up with a definition of math that says "the analysis of everything is math", the only logical conclusion is that we did a pretty bad job.