fantastic point. i was fortunately exposed to this paradigm at a young age, by my awesome advanced math teacher in high school. on the last day of class, he asked the handful of us in the class to describe what we understand mathematics to be. to give a definition. it doesn’t seem that hard to define mathematics until you actually try to do it. once we all fumbled around our words, he clearly and oh-so elegantly stated, “mathematics is the philosophy of structure.”
David Hilbert said: “No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet not other concept stands in greater need of clarification than that of the infinite”
Born in 1862, and first attending university in 1880, a lot has developed in mathematics since his time, and my level of understanding of mathematics doesn’t allow me the knowledge of how far we’ve extended our understanding of the infinite. But the more I learn - in class and independently - the more the concept of infinity fascinates me, and it’s fascinating to see the way other historical figures wrestled with the problem of infinity.
Georg Cantor, through whom I discovered David Hilbert, did not discover infinity. However, his work on set theory (which he invented) defined infinite sets, and he theorized the existence of transfinite numbers. Georg Cantor was fascinated with the idea of infinity, or, as it were, infinities. Cantor was a religious man, and corresponded not just with other mathematicians, but with Christian philosophers as well - he believed that the mathematical concepts that he was discovering had deep philosophical meaning and implications, and so did his detractors. Cantor once said:
"The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type."
I guess I kind of wanted to write about, and reflect on, what I’ve been learning on my own and in my classes. To me it seems clear that the mathematical is also deeply philosophical. As a former Objectivist, I came to the conclusion that the scientific, the study of the natural world, could not be separated from the philosophical, and I arrived at that conclusion from a purely philosophical viewpoint. Now, I arrive at the same conclusion (even if many of the other ideas associated with Objectivism no longer have a place in my mind), but from a purely mathematical or scientific viewpoint. It’s fascinating to me to see how many others in the past have arrived at the same conclusion, and how, just as the philosophies we hold to be true shifts our perceptions and understanding of the world, so too can our understanding of the world, as with this conception of infinity, shift one’s philosophy.
i was struck by his words. as i went on to study both philosophy and pure mathematics in college, i always kept that definition close to me. i tried to always let my understanding of mathematics be informed by the notion that at its core, mathematics is a philosophy: not a science, not an infallible set of rigid definitions, not some script of absolute truth. but a fluid, systematic model by which we can make assumptions about the world around us, draw inferences from those assumptions, and then rigorously test our inferences using our model. this is much like the philosophic process: observe, infer, test, prove. however, if you take an approach like this to mathematics, it enables you to never forget that you are ALWAYS starting with a base set of assumptions. so often we take these assumptions to be absolute truths about the world, and that’s just simply not the case. mathematical axioms and theorems are no more “real” than philosophical theories: they all begin with a set of assumptions. if you look at mathematics according to this framework, you can see that mathematics truly is a “philosophy.”
but it extends so far beyond both mathematics and philosophy. think of all the assumptions we need in place just to navigate daily life. metaphysical assumptions about the continuity of time and space, existential assumptions about ourselves and Others, practical assumptions about the laws of nature … the list goes on and on. of course we can’t spend all day worrying about the validity of these assumptions: living a normal life would be damn near impossible! likewise, i’m not suggesting that we need to be constantly questioning the validity of our mathematical systems. it’s just nice to always keep these thoughts in the back of one’s head.
be careful studying infinity - it’s known to drive even the best minds insane :) pretty interesting stuff though. it never ceases to amaze me how mathematicians have manipulated the concept of infinity to essentially “solve” so many finite issues; how they have used the concept of infinity to plug the gaps of certain mathematical systems. a good read on the topic is “everything and more” by david foster wallace, if you are interested. it is a fantastic overview of the history of the concept, with some mildly rigorous mathematics thrown in for fun.