Disclaimer:

This topic is a little meta, and as such, you’ll have to provide some grace to some of the circular reasoning. As a consilation however, if you can make it through the article, hopefully I can have helped you understand why circular statements happen in the first place. Think, “with only a ball, how does one explain what a ball is?” and you’ll understand a bit of why I must be crazy for even attempting to ask these questions… I am not a mathematician, philosopher, nor logician and none of the claims made in this article have been reviewed by such. So… Have fun, and explore with me and… Dont be afraid to be critical, but… If you dont have the time or patients to read this all, scroll down to the TLDR; and just get it over with already.😅

Ok. So, the title was a bit hyperbolic, but the thought which I plan to detail is quite profound — I hope, in fact, that by the end of this article, you share the same giddy “I see what you did there” smirk I had when the thought came to my mind.

So, what the hell am I talking about? Well, I’m glad you asked. But full disclaimer, I have to take you on a bit of a journey first. (Sorry)

So, what the hell am I talking about?

How We Got Here

It all started with a crazy conversation with a collegue at work (shout out to Jake) about the axioms of mathematics — moreover, the assumptions made when dealing with more advanced forms of maths such as calculus.

We explored the possibility that perhaps at the limits of an infinite function, things don’t behave as expected based on the observations which led us there. I’d brought up the fact that even in reality, the assumption that the quantum world bahaved like our physical one, for a long while kept us from seeing it in the first place. So then, perhaps it stands to reason that when you take a function into infinity, something quite strange and unexpected should happen.

Xavier, to the other side of me, laughed at Jake’s candor to entertain my silly thought experiments. He, for (lets be honest) good reason, pointed the both of us to an article detailing the axioms of mathematics and the basis upon which they were formed, and simply remarked “Jamel. Why do you waste time with all this theoretical ________ (I’ll let you fill in the blank)? Why don’t you use your talents to do real math?” (…really. It sounds worse then it really was) We all laughed, said a few more words; then put on our headphones, went back to work, and said nothing more of it.

Why do you waste your time…

But then the next day (or couple days, idk, I’m not good with time), Jake came to work and brought up that conversation we had yesterday (the other day?). In any case, he said that it reminded him of this book he’d read (been reading? 😅I’m terrible at this). The book was essentially a pragmatic deconstruction of the axioms and tools of mathematics. I presumptiously asked “You’re letting me borrow this?” to which he replied “Uh. Yeah. Sure.”.

I’m still not sure whether he intended to give me the book, but I’m sure glad he did. I dont know about you, but I have these moments in life when I’m given a resource which enlightens me in the very best way — and all the things that I’ve been thinking about and toying with in my head are suddenly given names that I can comprehend, and my excitement is, perhaps embarassingly, uncontained. Thank god there was no one around. Because I certainly was not running around my apartment and jumping on my couch like tom cruise circa katie holmes. I can guarentee you, I’m not that crazy.

So, to wrap this all up, all of this: the book, the conversations, the Jake and Xavier; all of it, gave me so much to think about. My weekend was consumed with thought. And during this session with myself and my journal(s), I started theorizing how it is that one could construct maths from the ground up, not assuming anything sourced from the current axioms of mathematics.

What does that look like?

I have no clue why I always choose the hard questions…(ok. maybe a little crazy) But truly, after being given this book — aside, its called Concepts of Modern Mathematics by Ian Stewart — I finally felt like I had the tools to attack such questions. So, thanks Jake, I owe you bro!

Ok. So now I’m rambling…

The Good Stuff

“ … how [ could one ] construct maths from the ground up, not assuming anything about the current axioms of mathematics ”

So the hardest thing about trying to answer this question was, “What tool does one use to establish an axiom?” Then, “If a tool is established, on what grounds is the tool itself reliable and valid?”

Hmmmmm…..

Ponder, ponder…

Think, think, think… 🤨

This was a tough one for me too. Primarily because, anything which one could use, also requires that it too be proven in much the same way as the thing which one is attempting to establish.

A good way to think about this is to imagine you have been given the task of measuring paper. What do you use? You might say something akin to “Well, I’d use a ruler.” Which, at first, seems self evident. However, in order for a ruler to be reliable for measuring paper, there is a basis upon which we can establish that the ruler is valid and reliable to do the job. In the ruler’s case, there are standards boards which we trust to provide us with said basis, and it is given from them a set of promises:

That…

Each unit of this tool will be consistant and equal given all intents and purposes of the tool (namely, measuring things which can be marked by one of the tool’s measurements) Each measurement will be marked in a repeated and predictable pattern. 2 marks of the same length (l) represents 2 marks which are of a distance (d) apart from one another equal to a magnitude (m) of the space between 2 adjacent marks of the same length (l). …

I would go on, but you should get the point. There must be a contract, by which, all participants, both in the creation and utilization of the tool, must agree.

But how does one come to an agreement?

Well, this was the largest thought chasm for me to traverse — so get ready, it’s gonna be a bumpy ride.