Many people dislike math. They think it intimidating, cryptic, and often divorced from reality. Yet math in general is practical, accessible and so easy, that we did it long before we heard of math. In fact, math may simply be an abstraction of the most fundamental aspects of our psychology.
Warning: Bad Pun Ahead
For instance: why do numbers count?
Maybe numbers don’t matter, but their relationships do? Maybe numbers are just place-holders in the relationship? Let’s take a concrete example.
When I think of things, I often try to order them in some way. I think of one thing as better than another, some tree as higher than another, some food as tastier than another. Now the goodness of a thing, the height of a tree, the tastiness of a food are all different things, but the RELATIONS among these respective groups are the same, and it is this that I try to capture with a number.
So I abstract the relation from the above, and introduce numbers as a place-holder in this relation. Abstraction means many things, but in this sense, I’m using it to mean a general principle underlying multiple concrete ones. In the case above, this abstraction is GREATER THAN or >. I would then record this general principle (abstraction) as:
x > y
The generic single letters (x,y) are no accident. I’m specifically being “ambiguous” since I could be talking about trees or food or people. By keeping neutral (even cryptic) lettering, I fight against the tendency of some to reify this general principle to concrete cases. Unless I want to calculate something specific, I want to focus on the RELATION itself and not what it is in specific instances.
Most importantly is the shift in thinking: x and y are place-holders and the real star is the relation itself. So when thinking numbers, perhaps my shift should be the operations themselves?
This is why math can seem so cryptic. It’s abstract, then mathematicians find abstractions governing those abstractions and so on. At every stage, an attempt is made to preserve generality and so those looking for something concrete to grasp may feel frustrated.
Would You Eat This?
I try to group similar things. For instance, if I taste chocolate, feta cheese, 3 day old bread, and wet toilet paper, I’m likely to categorize the first two the same way: things that taste good.
I just formed a set (there are more restrictions, but this is good enough for now). Now we come to Closure. This is a VERY intuitive idea that just seems cryptic because of how it’s presented. For instance, a common definition would go something like this:
The set X is closed under f if for all a,b in X, f(a,b) in X.
Sounds cryptic? It ain’t. You use it every day. In fact, you use it when you cook. We already know what a set is, f is some operation that takes two things in the set, and a,b in X simply means a and b are any two things in the category (set) of X. So what we’re saying in plain English is:
We consider a category closed with respect to a certain operation if using that operation on things in that category ALWAYS produces a thing in that category.
It’s a bit more readable, but maybe a concrete example will drive the point home.
I have a set of tasty foods, which include things like chocolate, marshmallows and feta cheese, Now I want to start mixing them — that’s my operation. Well, if I mix any two tasty foods, is the result ALWAYS tasty? Well, chocolate and marshmallows taste good (think s’mores), but chocolate and feta cheese do not. So the set of tasty foods is not closed under mixing!
Closure is a great thing because it allows us to chain an operation. For instance, I presented the operation as only operating on two things, but in truth it can operate on any number of things because of closure. Remember f(a,b) takes any two things in X and produces a result in X. So this means I can take the result of f(a,b) and apply it to f again! For instance, if I wanted to do the operation on 3 things in X (a,b,c), I could do:
Since f(b,c) is in X. And I can repeat the process indefinitely, I couldn’t do this if f wasn’t closed, because one of the results would not be in X and the whole thing would become illegal. This is one reason why I can chain operations like addition. Addition can be treated as a function with infix notation. That is, rather than +(a,b), I can just do a + b. So the chaining of a + b + c is the same as a + (b+c) which is the same as +(a, +(b,c)). All possible, because addition is closed.
In fact, software developers can recognize this as a potentially useful property, all too often because the code they work on lacks this.
Which brings us to another point: sometimes consequences arise from these properties that aren’t obvious at first glance (or are completely surprising). In fact, much of mathematics was built on tracing the consequences of these ideas, and this underlies proofs and axiomatic systems.
The Mathematics of a Broken Heart
Take commutitivity. It just sees if the order of the operation matters, or in other words:
f(a,b) = f(b,a)
To see its applicability to daily life, we can define unrequited love in terms of the above. Let f = love, let a = John and b = Mary. Saying John’s love of Mary was unrequited is simply saying:
love(John,Mary) != love(Mary,John)
Unrequited love is non-commutative love!
There; mathematics just saved your relationship. You can thank me later.
I’m not “dumbing down” mathematics; mathematics “smartened up” the things we normally do. Since math is an abstraction of the way we see the world, it studies us.
Or does it?
How logical are we? If I take an abstraction of several ideas, and trace their consequences very rigorously (eg: via proof), then have I arrived at something that bears any relationship to human thought? Does the mind operate that way?
Have I moved beyond the thought to studying the ideas themselves?
Which brings us back to the question:
Is Math a psychology?
Or is it just a study of ideas?