Sometimes we think of infinity as being singular: so big, so encompassing, that everything is included and becomes one, but is somehow also beyond the concept of One.
But, just as there can be very, very large “ones” (like big infinity itself) as well as very very small “ones” (like a single atom or smaller) there are also many kinds of infinity — some infinities are actually bigger than others, although they are all infinite.
One way of conceptualizing infinity is to add or multiply (or divide) numbers into sequences that go on and on and on… “to infinity” — here used to describe a direction, rather than a destination.
But, a direction still has limits:
- 1, 2, 3, 4, 5… is an infinite direction of whole numbers
- 2, 4, 6, 8, 10… is an infinite direction of even numbers
- 1+1+2+3+5+8+13+21+34… only contains certain sums (Fibonacci)
- 3.14159265359… only calculates a fraction to become infinitely more precise — it’s a finite number though, because it can be expressed accurately with a fraction, but is infinite (as far as we know) when expressed as a decimal
And there are many, many more infinities, but none of them really contain everything because we cannot truly conceive of or communicate infinity within our finite cognitive modalities… but the various expressions of infinity help us understand the idea of infinity by seeing endless continuity in a particular direction.
One kind of mathematical infinity that I find rather interesting are “monads” which can visually be thought of as being like cones or trees (plotting on a graph is a way to visually represent them).
The word “monad” has also been used philosophically to describe the smallest aspects of matter, similar to atom theory (which was considered philosophy until it was proven scientifically — and then disproven when atoms were found to not be the smallest things in the universe).
The difference between monads and atoms is that monads are non-things, and yet are the non-things that structure the entire universe: these non-thing-monads really want to be thing-things. They are tiny little points of force that have the potential of becoming thing-things, but only when aggregated with a multitude of infinite, yet connected relatives.
No particular amount of monads is given as the magic number for them to go from non-things to thing-things, but in philosophy that is unnecessary because non-things can’t become thing-things no matter how few or many there are, but thing-things can contain groups of infinite non-things, and those groups therefor are thing-things — and then of course thing-things change into other thing-things, utilizing the potential of their infinitely constituent non-thing monads.
Similarly, in mathematics, monads aren’t numbers. They are thingy non-things — they are infinitesimals — infinitely small and undefinable, but, as groups of “numbers” they are indicative of numbers.
On monads (the numbers) Professor Carol Wood (a mathematician) had this to say regarding their relevance:
Well they’re useful for intuition. And that’s true of a number of non-standard things. They’re useful because you think, ‘I really wish I had one infinite thing that is bigger than all of them, and it might allow me to think of all the … [insert any subject here] … in a different way way.’
And there are very hard problems and very interesting questions that this approach would never touch, but it’s just another system, and sometimes in model theory, having many different versions, many models of the same basic structure give you different intuitions about them.
And in particular, this idea that you can satisfy all these conditions at once … with a single element can be useful for your intuition.
It’s at this point that the lines between mathematics, physics, philosophy, logic, and being all begin to blur.
Contemplating infinity, and the various kinds of infinity leads ultimately nowhere, but is nonetheless a fascinating exercise. Even mathematically, contemplating infinities and infinitesimals becomes a philosophical musing… the firmly defined numbers of mathematics shift back into meaningless non-being.
And within infinity, many things can pop out — or at least seem to despite always having been there. That is, our perception of them goes from non- things to aggregates of non-things to a multitude of thing-things… in other words, from the “nothingness” of The Void to the multiplicity of Manifestation.
See both of Professor Wood’s mind-bending explanations of monads here:
And continued here: