Mathematical truths and meaning
I occasionally browse Hacker News, and this morning the top post was titled \(1/0 = 0\). I obviously read this clickbaity, highly voted article. This got me thinking about a few things I've been mulling over during the past few years, in particular, during my years studying pure mathematics.
The author provides quite a sound exposition as to why it's possible to create a function that agrees with the division function on all non-zero integers but takes on an arbitrary value at zero. Now -- to protect my mathematical integrity -- I shall note that there are a few small mathematical slip ups and some confusion as to what certain terms mean; and I'm not a fan of the "proof-by-authority" section. It's also important to note that in the study of rings and fields, division is defined to be the inverse of multiplication, and there are some important concepts to characterise these objects that require this definition, such as zero divisors, so in that world, redefining division is not a valid thing to do. Nonetheless, that's not the direction I want to take this post in.
Essentially the post goes as follows: define what a consistent theory is, define what an algebraic field is, then arrive at the natural definition of division as the inverse of multiplication, but then discuss why it might be problematic to have division not defined for zero, and define it as zero, but now patch up the relationships between operations by making sure the "divisor of zero" isn't used as the multiplicative inverse of zero, and so forth.
Fundamentally, I think this is a relatively absurd path of reasoning. Take something that isn't defined because it doesn't make sense (dividing by zero), then augment your world by defining it as something arbitrary (such as defining anything divided by zero to be zero), and then make your world consistent again by forbidding the use of this new object (using it as the multiplicative inverse of zero), essentially kneecapping the whole system into being just as absurd, but with a new axiom.
Now, this is not to say that this isn't a valid sequence of steps, and it is a perfectly consistent thing to do (according to the author). I'm sure there are even some mathematicians who have studied these kinds of constructions. However, despite it being a somewhat curious object, in the big picture it seems to be a relatively useless and poor object to study.
It seems then that in mathematics, there is an unlimited array of curious objects and weird constructions; however, nowhere near all of them are interesting or useful. For any given definition of an object, there are a plethora of other possible ways to define it. So the tricky part is in finding the useful structures and object to study.
When I study mathematics, in particular, when I'm trying to understand and internalise a new concept, I like to ask myself: why is this the "correct" definition? Here obviously I'm using the word "correct" in quotes to emphasise that I'm referring to some subjective correctness, not to any strict, well-defined correctness.
This comes up often. Objects are defined in a certain way, and there is often some small adjustment that can be made to include a few more classes of things in the definition, or equally, some small correction could be used to exclude a few more things. The important thing is to understand why a particular definition is fruitful, and useful in general.
Often I find that the way an object is defined exludes a class of annoying special cases, or makes life easier by guaranteeing that the object under study has some neat property that every other similar object has anyway, but it's just easier to require the object to be normalised or shifted in some way.
Surprisingly, I think this is very similar to the world of art. In an infinite playground with only relatively weak constraints, we're looking for things that are interesting, meaningful, or beautiful to spend time and effort looking at or studying.