I have aleph-naught tattooed on my left arm and aleph-one on my right arm because the single most amazing fact of the universe to me is that we have no way to resolve the distance between the counting numbers and the real numbers. We don’t really even have a way of proving or not proving this fact. It may not even be a fact. it is a mystery right within the very heart of mathematics, physics, logic, science, religion, being itself.
My hypothesis is WE are the resolution of the continuum hypothesis. We are the infinity between the integers and the real numbers. We are a bridge, a map, a becoming between counting and infinity.
I am the excluded middle.
You are the excluded middle.
We are the continuation between all that can’t be proven but is nonetheless true.
I am not making a metaphysical statement necessarily — more pointing, again, to something deeply pragmatic that gets lost in our assurance that we know what we are doing.
The continuum hypothesis is very much a shocking aspect of math and computers that I watch almost everyone I work with gloss over like it’s not there. (I suspect most students and young professionals are never exposed to this stuff). There are many practical implications of this… the main one being… there is no possibility for bug free software, totally automated machines, control of complex networks, final understanding of art or math or science, and so on. We have no way to “resolve” the gaps between facets of reality. There are infinitely many of these gaps in our knowledge and yet we keep on keeping on.
Human society has been built up with technology that relies on faith that the infinity of integers and infinity of real numbers have nothing in between. The math used inside our engineered machines uses an unproven fact and keeps on going. We literally LEAP over gaps in our knowledge and only care about those leaps if the consequences force us to reconsider. The knowledge hidden in the gap is only useful so much as there is a consequence for not having it.
This is an epistemological concept I’m getting at here. I am very much a Platonist and think there are actual truths of the universe and these truths are everywhere. But I do not think language, linguistics, logic, philosophy, science papers, scientific method, computers, engineered stuff, political systems, nor mathematics is that truth. Knowledge is traces of truth in use — we acquire knowledge through consequences of interacting… of relating one state of the world to another. So long as a relation holds in ways we find valuable (often as simple as we get to eat, breath, sleep another day) it is knowledge.
All of the gaps in math between what is true or false, provable or not, are resolved via relating not abstractly in the world. The continuum hypothesis is resolved over and over and over as we leap over its truth or falsity and make assumptions one way or the other. In a lot of ways it likely couldn’t be any other way… considering the idea really is how does one go from the things that can be counted to the things that can’t be counted? Well, you have to count to infinity infinity. Just start counting.
Literally, this is what I’m saying. We, humans, are the things that count. We observe, count, categorize and make sense of and with things in the world. We are an every growing proof that the continuum hypothesis is not true. That is, we are the set of infinite things that isn’t countable, but can count all the things that are countable.
Furthermore, this is where computation sits. Computation is the excluded middle. Computation is a process of proving excluded middles via generating more middle. I’m sure this sounds bonkers, but it couldn’t sound any other way. This is why this fact of the universe is so powerful and beguiling.
You may now be thinking that this isn’t really something you encounter… but it is. A circle is this problem. A circle is real number infinity sided polygon. It’s the polygon, the shape, where you can’t count the sides. Pi’s digit expansion is uncountably infinite. And yet we use pi discretely and everywhere. Pi hovers in this aleph-naught and aleph-one space. Your computer you’re reading this on uses this unresolved gap — it leaps a bazillion times a second — using the faith that in fact there is a countable infinity and an uncountable infinity. the computer is Aleph-Half.
This is not how mathematicians and computer scientists frame logic and programs and their science. They attempt to never make a leap and yet, the record clearly shows, they never succeed. The entire effort of formal knowledge acquisition is a leap of faith — often coded as “special cases” or “in general” or “considering only this set of circumstances.”
No mathematics institution would allow for “the human creative effort” as a proof — as literally the thing that proves relationships between states of the world. Hilariously because mathematics On The Whole is that effort. Mathematics in specific instances sits there attempting to be in the countable infinite (this really is some of what Gödel was saying in his incompleteness work and this is what Turing/Church were saying in computational theory). Mathematics and Computation On The Whole are attempts to wrangle the uncountable infinite into something of count — make facts of the world accountable.
We and computers are the Accountants. We account by taking count of all there is… when we can’t count it we label it, leap over it and keep counting.
It is not for me to resolve this for you nor for math books. Impersonal resolution in not possible. Being a being between aleph-naught and aleph-one, as you and I are, is exactly that we are the resolution by resolving hard to count states of the world.
Each one of us is a unique excluded middle — collectively the set in-between.
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Here’s one of the better explanations of the challenge of infinities and cardinality.