Many people associate the word ‘mathematics’ with a palpable sense of dread, but there is a lot that can be translated from mathematical ideas that can be of great benefit to many diverse fields of thought. Every high school student has heard questions like: But when am I going to use this? Or: But what IS ‘a’? So… here’s what I think some of it means. Bluntly. Without the weird symbols and strange words. More tools for thinking are a use! If you’d like more details, like I hint at towards the end, please read my paper!
I have always been better at thinking than at mathematics… I should probably try to explain that a bit better.
I think that the difference between how I’m using the word “maths” and the word “thinking” is kind of like the difference between numerals and numbers.
The bit of scribble “4” doesn’t mean anything without something for the scribble to represent. So numerals represent numbers… but that still doesn’t tell us anything about what “numbers” means.
I’ll hazard an (incredibly informal) guess.
Numbers represent the idea1, that we can have ideas2 about ideas3.
We have to recognise something crucial for this.
For the symbols in this post to mean anything there has to be an order to them. We might be able to express the meaning in this post in some other way, but even though the specific symbols might be different we wouldn’t be able to say anything without some ordering of symbols. The words “ideax” above were used in an order that contextualises each of them differently. Like reading a book more than once: this will be the first and only time I’ve read Dune for the third time.
One may call this a required assumption to convey any meaning: Closure under Representation. (Bonus question: If it’s always required, is it an axiom?)
Let’s make that easier to understand: You cannot think your way out of existence. If you’re reading this, everything you’re about to think is now contextualised and informed by the words I have written.
Now. Take all of that above, all the ideas and symbols, stick it into a box and give that box the name Jeffery.
That is the number: Jeffery.
Jeffery is unique to a context.
Jeffery always needed a unique context of symbols to exist as, otherwise Jeffery couldn’t be Jeffery.
More specifically now: The symbols in this post? They represent Numerals.
The order I wrote them in to convey a meaning? That represents Mathematics.
The meanings themselves? Those are thoughts or ideas.
Important question: Do they exist?
Of course they do! How else did I give them names?
Does a flock of birds “exist” even though it isn’t specifically embodied?
We have identified that there are connections between symbols that give meanings to arrangements.
So consider a flock of birds:
Is a flock just a collection of birds?
No. If the birds aren’t behaving similarly in some way they’re not a flock.
So let’s suppose that the birds are vertices (dots) within a network, and the ways they interact are arrows within the network.
The behaviours of the individual birds need to flow together in some way to form something “bigger”.
That “bigger” thing doesn’t have a well-defined boundary when compared to a single bird, but it does interact with the individual birds, informing their behaviour “across dimensions”.
There is no singular “flock bird” that is deciding the behaviour of the whole but they do behave as some sort of cohesive unit.
An Order Unit of a Leavitt path algebra.
Anthony Warwick
Western Sydney University
