I rather like the idea that deep in the platonic realm of pure mathematics, there is something that is alive.
Numbers Are Leaves
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Re: Numbers Are Leaves
#12Am I wrong in thinking that the visual below “Finally with 5 the self-similarity continues as expected” is wrong? I’m thinking that the second-to-right and rightmost subtrees need to be larger, to have a sequence of four and five nodes respectively in their rightmost branches.
Re: Numbers Are Leaves
#13These are just binomial trees, which might be familiar to those who know their classic data structures.
Re: Numbers Are Leaves
#14I love the imagination here and imagination in general but the framing really stretches it... I think this whole article would be more substantive if it was a little more grounded in the concept of induction.
Well and if it would admit very openly that in the sentence "numbers are leaves" the word "are" is about the existence of an isomorphism between the natural numbers and a series of nested sets... but that it's far from the only isomorphism, induction is everywhere in math and computer science. I imagine some little kid reading this and clinging to a reductionist idea that "numbers are (only) leaves", but they aren't just leaves.
Anyhow the setup is really nice and inspiring, it just ended up feeling like a tease. Hope to see a followup!
Re: Numbers Are Leaves
#15Too few examples. Get to 2039484 and report back if it looks like a leaf or a blob.
The representation of n has 2^n vertices. 2^2039484 might be a bit too large to render...
Re: Numbers Are Leaves
#16I think people will like the following tangent.
https://en.m.wikipedia.org/wiki/Benacerraf%27s_identificatio...
In the philosophy of mathematics, Benacerraf's identification problem is a philosophical argument developed by Paul Benacerraf against set-theoretic Platonism and published in 1965 in an article entitled "What Numbers Could Not Be". Historically, the work became a significant catalyst in motivating the development of mathematical structuralism.
The identification problem argues that there exists a fundamental problem in reducing natural numbers to pure sets. Since there exists an infinite number of ways of identifying the natural numbers with pure sets, no particular set-theoretic method can be determined as the "true" reduction.
What Numbers Could Not Be
Re: Numbers Are Leaves
#17I love the imagination here and imagination in general but the framing really stretches it... I think this whole article would be more substantive if it was a little more grounded in the concept of induction. Well and if it would admit very openly that in the sentence "numbers are leaves" the word "are" is about the existence of an isomorphism between the natural numbers and a series of nested sets... but that it's f…
Author here. Wasn't expecting to see this on the front page!
I'm really very far from a mathematician and this was a write up of a fun side project. I think the title would be unforgivably misleading in a formal context (if this was a paper claiming any new insights) but really it was a fun side project I wanted to right about. Maybe you read this and learned a little bit about set theory if you had no idea what it was (much like myself).
In general I resent popular science (especially in theoretical physics) which tries to reduce deep and interesting topics to poorly thought out analogies - but again my positioning here is not to educate per se. Or Michio Kaku style orating which assumes string theory a priori and later you have conversations with people who think string theory is established and tested because they watched a 40 minute video of him on YT.
Having said all this I need to get better and giving titles to the things I write - my other post about trying to build AGI in Rust got similar criticism.
Either way thanks for the feedback!
Re: Numbers Are Leaves
#18Because the representation is a ragged tree. If it were a uniform depth it would look like a blob, but because different arms have different lengths those create structure in the layout algorithm.
Re: Numbers Are Leaves
#19> I would like to understand why numbers looks like leaves 1st of all they don't. The graph doesn't look like pinnatids or palmatids. There is some resemblance of an alternating disposition of leaves, but that's not the shape of the leaf itself but the distribution of them, and it's a stretch. Secondly, I'll take the generous interpretation of the question which is, why the graph looks mathematically like leaves, and…
I think the last point doesn't really hold its own in any way. Many discoveries throughout history have started from someone just playing around with an idea, toying with it at first, but eventually becoming obsessed. The thing is, there's no way to tell beforehand. It might be a toy with no ultimate use or meaning, or it might lead to something entirely novel somewhere down the line. That's why play, in a very broad sense, is a core part of science and invention.
Re: Numbers Are Leaves
#20Perhaps if he was able to use some kind of force-directed 3-D algorithm, instead of a 2-D one, they might resemble something other than leaves, which could be interesting.