Teaching and Learning “What Is Mathematics” [pdf] | Hacker News Reader

Teaching and Learning “What Is Mathematics” [pdf]

discretization.de

69 pointsjasim9 comments

I think the problem with defining Mathematics is that everyone interested in defining Mathematics has a tendency to cast an overtly wide net such that nearly can be considered some form of Math. Wikipedia's definition of math (in the article) seems to have the same implications as wikipedia's definition of philosophy ("Philosophy is the study of general and fundamental problems") in terms of scope without the confidence to say it in as blunt terms. Philosophy can get away with it because it is the historical ancestor of most organized mental endeavors, but the math page can't seem to admit it.

Speaking of Wikipedia, it's well-documented that first link in many pages is the parent of that subject, and that the great ancestor of nearly all pages is "Philosophy" [0].For instance, Delaware is a US. State which is a political entity which is an entity which is something that exists, making it the subject of Ontology (study of existence) which is a subject of Philosophy. Mathematics however is partly the study of quantities, which can exist as a magnitude which is, of course, a subject of math. This means, nearly all of WikiProject Mathematics is determined to stay detached from the rest of human knowledge (admittedly based on this one, anecdotal and inconsequential metric).

[0] https://en.wikipedia.org/wiki/Wikipedia:Getting_to_Philosoph...

But what of the philosophy of mathematics! https://en.wikipedia.org/wiki/Philosophy_of_mathematics

All intellectual roads lead to philosophy.

Joking aside, I think we also run into problems with defining mathematics because its application and the thing itself are not really separable. One can have the philosophy of something, e.g. philosophy of mathematics, because philosophy as a method or type of intellectual excursion is not a single thing--you can go about philosophizing in many different ways, even if your topic is the same (e.g. logical positivist handling of philosophy of language vs. the ordinary language philosophers)--in the case of mathematics it is a field of study but it is more fixed as a method or tool of thought--there are several ways to go about philosophizing no matter what branch of philosophy you tackle, whereas with mathematics, while there are different branches of study, there is really only one mathematics--i.,e. there's only one way to go about mathematicizing correctly for a given problem, whereas there's not really a 'correct' way to philosophize about a given problem.

I would define mathematics as a particular mechanism of human thought/a particular way we understand the world (the human mechanism of quantification and manipulation of said quantities)

Of course mathematical realists will disagree with me.

Math is whatever it is mathematicians do.

good quote: "Why do so many pupils and students fail in mathematics, both at school and at universities? There are certainly many reasons, but we believe that motivation is a key factor. Mathematics is hard. It is abstract (that is, most of it is not directly connected to everyday-life experiences). It is not considered worth-while. But a lot of the insufficent motivation comes from the fact that students and their teachers do not know “What is Mathematics.” Thus a multifacetted image of mathematics as a coherent subject, all of whose many aspects are well connected, is important for a successful teaching of mathematics to students with diverse (possible) motivations."

> Thus a multifacetted image of mathematics as a coherent subject, all of whose many aspects are well connected, is important for a successful teaching of mathematics to students with diverse (possible) motivations.

Somewhat useless personal anecdote to follow on this quote that I also liked:

Prior to my PhD studies I had heard of math and experienced it largely as computation. Arithmetic, matrix multiplication, integration by parts etc etc. This is, to my mind, the most terribly boring part of mathematics.

It's not certain, but I suspect that providing at least a few alternate characterizations of mathematics to students stuck doing computations for years and years will almost certainly help some of them find their way to regions of the subject that they find interesting.

I had a somewhat similar experience. No PhD but I had always hated math as a kid learning nothing but, as you say, rote computations in school.

By chance I stumbled on Frege's philosophy of mathematics/investigations into the foundations of mathematics in college and suddenly math was actually really interesting. I find proofs, set theory, algebras, and other mathematical domains closely related to logic way more fun than rote computations. I recall thinking I could have really fell in love with math and maybe even excelled at it if my schooling had ever given me so much as a hint that this stuff was also part of mathematics and it wasn't mere repetition of the same old computations, and that all those formulas build on each other and actually have very interesting justifications.

To do math, or write, or write a computer program, you have to come to terms that expressions (or computations in a in formal logical system) are intrinsically meaningless—the result or machine will blindly follow its internally meaningless rules and come to some internally meaningless conclusion.

It may be that some folks pre-accept following rules wherever they might lead, and other folks refuse to deal with the 'meaninglessness'.

One way of dealing with it is to learn to apply external meaning/motivators to these models/programs, but I think this is a separate skill, possibly much harder and not taught as such.

An old saying with a deep truth: A mathematician is a person who will assume anything except responsibility.

Which is why devops will always have troubles.

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