[HN Gopher] Numbers Are Leaves
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Numbers Are Leaves
Author : nodar-d
Score : 154 points
Date : 2024-12-26 14:23 UTC (2 days ago)
(HTM) web link (www.christo.sh)
(TXT) w3m dump (www.christo.sh)
| max_entropy wrote:
| This talks about representing numbers as graphs whilst showing
| that the graphs can appear to be reminiscent of leaves, not that
| the numbers are leaf nodes of graphs, which is where I thought it
| was going at first.
| willvarfar wrote:
| Beautiful!
|
| (And now we all go cross eyed trying to see if there's a pattern
| for factoring in there somewhere...)
| nyrikki wrote:
| > I would like to understand why numbers looks like leaves and
| not some other fractal like snowflakes for example (the inverse
| question, why biological leaves have this specific self-similar
| structure may yield more fruit).
|
| Snowflakes are dendrite formations highly influenced by how
| nucleation happens, the graph layout choices highly impact what
| you will see;
|
| Von Neumann Ordinals are possibly better approached as
| hierarchical branching processes, and being deterministic,
| Tokunaga self-similarity is one fairly elegant path to follow in
| the deterministic case such as this. (IMHO)
| aithrowawaycomm wrote:
| > I think a fair thing to ask is if the force-directed layout
| engine is uniquely responsible for the leaf-like structure and
| this has nothing to do with the Von Neumann ordinals themselves.
|
| > To check this I generated some Collatz trees and they ended up
| looking like microbes. I think it's safe to say the answer is no.
|
| "Uniquely responsible" seems to be doing too much work. These
| structures are reminiscent of dendritic fractals coming about
| from diffusion-limited aggregation in electrochemical
| deposition[1], which is a transitional phenomenon with a phase
| diagram (molar concentration of the electrolyte vs voltage) There
| is a sweet spot in the middle where you get intricate dendritic
| crystals: outside of that you get smooth layers or big blobs /
| spikes without much internal structure.
|
| I suspect it is primarily the modeling of the forces which is
| responsible for the behavior, with the topological structure of
| the tree somewhat indirectly corresponding to the chemical and
| electrical parameters. I am by no means an expert but it seems
| likely to me that these von Neumann dendrites vs the structure of
| Collatz trees is a fairly shallow relationship: lots of trees
| with similar graph-theoretic properties to the von Neumann trees
| would also demonstrate the leaf-like fractals, but with no
| meaningful relationship to the natural numbers. But it would be
| interesting to make this more precise.
|
| [1] https://en.wikipedia.org/wiki/Diffusion-limited_aggregation
| ben_w wrote:
| > why biological leaves have this specific self-similar structure
| may yield more fruit
|
| I hope the pun was intentional :)
|
| That said, I think the actual answer is in part due to the choice
| of presentation rather than the mathematical structure -- Force-
| Directed graph layout -- despite the line immediately after this
| quote.
|
| The outer surface of a leaf doesn't physically blend into other
| leaves that it touches, so it gets repelled mechanically during
| growth, and it is also connected mechanically to the parts it
| grew from.
|
| Consider the failure of Collatz trees to look like leaves as a
| counterpoint to the failure of the `dot` graphs to look like
| leaves.
| 1970-01-01 wrote:
| Too few examples. Get to 2039484 and report back if it looks like
| a leaf or a blob.
| openquery wrote:
| The representation of n has 2^n vertices. 2^2039484 might be a
| bit too large to render...
| philipov wrote:
| I went into it thinking they meant numbers are leaf nodes of a
| graph, but no: they meant the entire graph _looks like_ a leaf.
| TZubiri wrote:
| > 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 not
| a question of biological impact, whatever resemblance is a
| coincidence which brings us to
|
| Third, OP is playing a game with numbers with no objective goal
| (yes all maths is this, but this is straight up chmess), there is
| no ultimate meaning to be derived of it.
| Etheryte wrote:
| 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.
| TZubiri wrote:
| If 1 out of 10 chmessicians discover something useful, the
| discoverer had good taste and deserves credit.
|
| There is no insurance redistributing credit amongst all of
| the pointless searches.
|
| The guys who invented imaginary numbers or eigenvectors
| weren't just throwing darts at a board and got "lucky".
| ulbu wrote:
| abstraction for abstraction's sake? pure abstraction?
| abstracted abstraction?
| isoprophlex wrote:
| This has almost nothing to do with numbers or ZFC and almost
| everything to do with how graph layout algo's produce their
| outputs...
| dr_dshiv wrote:
| I rather like the idea that deep in the platonic realm of pure
| mathematics, there is something that is _alive._
| putzdown wrote:
| Am 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.
| hulium wrote:
| These are just binomial trees, which might be familiar to those
| who know their classic data structures.
|
| https://en.wikipedia.org/wiki/Binomial_heap
| macawfish wrote:
| I 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!
| openquery wrote:
| 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!
| Vox_Leone wrote:
| I'm under the impression that, at least theoretically, Von
| Neumann's principles of self-replication, game theory, or
| optimization in the context of designing neural network
| structures.
|
| You could think about organizing a neural network with layers
| or nodes that are indexed by Von Neumann ordinals, where the
| structure of the network follows the natural progression of
| ordinals. For example:
|
| Each layer or node in the neural network could correspond to
| a finite ordinal (such as 0, 1, 2, etc.) or transfinite
| ordinal (like oo, o+1o+1, etc.). The way the network expands
| and evolves could follow the ordering and progression
| inherent in the Von Neumann ordinal system.
|
| This could lead to an architecture where early layers (low
| ordinals) represent simpler, more basic computations (e.g.,
| feature extraction or basic transformations). Later layers
| (higher ordinals) could correspond to more complex, abstract
| processing or deeper, more abstract representations.
|
| But I'm afraid there is no hardware substrate upon which to
| build such a thing.
| nthingtohide wrote:
| I 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
|
| https://youtu.be/H5SocLNkT9M?si=Fk2Hmpw3yOtDW7GS
| bubblyworld wrote:
| Thanks, that was an interesting rabbit hole. Although I can't
| help but feel there's a philosophical map/territory confusion
| here. Like, sure, numbers can't possibly just "be" sets because
| there are many different models of the naturals in (ZF) set
| theory, even in higher order logics. But I feel like a
| Platonist would just counter that of course this is the case -
| we are simply _modelling_ the properties of the "true"
| naturals with these sets, in the same way that differential
| equations can model the behaviour of fluids without "being"
| water. Nobody writes down Navier-Stokes and expects to get wet!
| itishappy wrote:
| Because 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.
| diyseguy wrote:
| Perhaps 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.
| epgui wrote:
| Setting aside the rest of the article, there is one thing I've
| never really understood the motivation for, and I think this
| article really highlights it well.
|
| > "Well congratulations this works! We can represent numbers
| using singleton sets (sets with one element). However, it would
| be nice if our sets had some more structure. Specifically we
| would like the set corresponding to the number n to have n
| elements."
|
| Why? What's the motivation here?
|
| It seems to me like `next(x) = {x}` is simpler than `next(x) = x
| [?] {x}`, and I'm not totally clear on what the extra complexity
| buys us.
|
| I am of course familiar with the structure `next(x) = x [?] {x}`,
| having seen it in textbooks and in a set theory / mathematical
| foundations class, but I feel like I've never really understood
| what insight this structure captured. It seems like it's always
| presented matter-of-factly.
|
| Anyone?
| tromp wrote:
| Representing the natural number n by a set of size n turns out
| to be rather useful. And the obvious choice for a set of n
| elements is the representations of the n natural numbers
| 0..n-1.
| epgui wrote:
| I think I can see why... But the obvious question to me
| becomes:
|
| - What do the operations of addition and multiplication look
| like with this structure? Doesn't this seem a bit complex?
| bpoudev wrote:
| I think it's (like so many things) a question of tradeoffs.
| Programmers often think of complexity (and hence
| performance) of operations, but that is not important to a
| mathematician.
|
| The fundamental operation, the successor function, does not
| look much different, S(n) = n [?] {n} vs S(n) = {n}.
| Mathematics usually defines addition in terms of this
| function, so that n + m = 1 + (n-1) + m and 0 + m = m. This
| can be done via induction and works equally well,
| regardless of which "implementation" we choose. Similarly,
| multiplication is repeated addition. Seen in this way, both
| "implementations" of natural numbers leads to horribly
| inefficient, but ultimately very similar, addition and
| multiplication operations.
|
| However, the representation S(n) = n [?] {n} leads to a
| very simple definition of "a finite set of size n". It is
| simply a set, which has a bijection between it and n. This,
| in turn, leads to a much easier arithmetic. Instead of
| manipulating a specific set representing a given number, we
| can say that any set of size n can represent the number n.
| Then addition simply becomes disjoint union, and
| multiplication becomes Cartesian product, from which things
| like associativity and commutativity can be proven much
| easier than in the inductive definition.
| xanderlewis wrote:
| Look up 'transitive sets' if you haven't heard of them already.
|
| The primary answer to your question is that in the von Neumann
| definition the ordinals are transitive and well ordered by the
| epsilon (membership) relation, which is a pair of stipulations
| that can then be used as a _definition_ of the ordinals if you
| like. This in turn is nice for many other reasons!
|
| Also, you can make convenient definitions like defining the
| supremum of a set of ordinals to be the union of that set. The
| union of two of your ordinals usually won't be an ordinal.
|
| In general, it's nice to have an ordinal simply be _the set of
| its predecessors_ , which is something that this definition
| implies.
| epgui wrote:
| Thanks for the pointers!
| noam_k wrote:
| Not a professional mathematician, but you have the benefit of
| set operations mapping to functions you're familiar with.
|
| For example, set union becomes the max function.
| epgui wrote:
| That one is neat, but how do you define addition and
| multiplication on these structures?
| xanderlewis wrote:
| You use (transfinite, if your ordinals are large enough)
| recursion! Just define a + 1 to be the successor of a --
| succ(a) -- and then, assuming we've defined a + b, define
|
| a + (b + 1) := (a + b) + 1 = succ(a + b)
|
| (it's only slightly more complicated for infinite ordinals)
|
| You can do a similar thing for multiplication, and
| exponents, and so on.
|
| Technically, you have to use induction to prove that this
| definition indeed works to define the operations for all
| ordinals.
| Chinjut wrote:
| Encoding n as the set of all m < n is called "von Neumann
| ordinals". Encoding (positive) n as just the singleton set
| containing n's immediate predecessor is called "Zermelo
| ordinals". The main advantage of using the former
| representation rather than the latter is that it allows
| uniformly encoding not just finite ordinals, but also
| transfinite ordinals, many of which do not have an immediate
| predecessor. E.g., in the von Neumann ordinal system, the
| infinite set of all finite ordinals may itself be interpreted
| as an ordinal value larger than every finite one. (And then the
| set of finite ordinals [?] {the set of finite ordinals} becomes
| yet a larger transfinite ordinal still, and so on...)
| d-lisp wrote:
| It always felt arbitrary to me : next(x)={x}
| would give 1={0} 2={{0}}
| 3={{{0}}} Kuratowski's encoding gives :
| 0=O=() 1={O}=(0) 2={O,{O}}=(0,1)
| 3={O,{O},{O,{O}}}=(0,1,2) The cardinal of N is
| n and every element in N are the predecessors of n.
| Von Neuman's encoding gives : 0=O
| 1=0U{0}={O,{O}} 2=1U{1}={O,{O},{O,{O}}}
| Now the cardinal of N is n+1, and n is the maximum of
| the set N defining n. Both Von Neuman's and
| Kuratowski's encoding allows us to define ordered tuples, but I
| cannot understand how to write the tuples for Von Neuman's in
| the context of natural numbers.
|
| 2 is {O,{O},{O,{O}}} with Von Neuman's we can recognize 0 and 1
| as the first and second element of the tuple : what is the
| third one ?
| openquery wrote:
| 2 = {0, 1} = {O,{O}} by the Von Neumann ordinal definition.
| epgui wrote:
| For Von Neuman: 1 = O U {O} = {O}
| 2 = 1 U {1} = {O,{O}}
| d-lisp wrote:
| True, then it doesn't differ from Kuratowski's encoding or
| am I missing something ?
| sagebird wrote:
| A more compact and beautiful relation exists between integers and
| finite rooted trees exist, imo.
|
| David W. Matula found a correspondence between trees and integers
| using prime factorization, and reported it in 1968 in SIAM: "A
| Natural Rooted Tree Enumeration by Prime Factorization", SIAM
| Rev. 10, 1968, p.273 [1]
|
| Others have commented on it before, search the web for Matula
| Numbers
|
| I independently found this relation when working on a bar code
| system that was topologically robust to deformation. I wrote a
| document that explained this relation here[2].
|
| I created an interactive javascript notebook that draws related
| topological diagrams for numbers. [3]
|
| [1] http://williamsharkey.com/matulaSIAM.png
|
| [2] https://williamsharkey.com/integer-tree-isomorphism.pdf
|
| [3]
| https://williamsharkey.com/MatulaExplorer/MatulaExplorer.htm...
| sagebird wrote:
| Sorry - I believe I am off topic as this is not relevant given:
|
| "This indirectly enforces the idea that sets cannot have
| duplicate elements, as set membership is defined purely by the
| presence or absence of elements. For example:"
|
| So there is a constraint on what sort of trees are allowed in
| this -forrest- which would preclude most finite rooted trees.
| photonthug wrote:
| A related rabbit hole that you can jump down in FoM is that
| Zermelo's ordinals and von Neumann ordinals cannot both be true
| at the same time. Wikipedia's intro to the topic [0] is a
| starting place, see also [1] which might be more in depth.
|
| [0]
| https://en.m.wikipedia.org/wiki/Benacerraf%27s_identificatio...
| [1] https://plato.stanford.edu/entries/philosophy-
| mathematics/#W...
|
| > If you don't know why set theory is important, it is because
| set theory is the foundation of all of mathematics.
|
| Nitpick maybe but the types and categories people might prefer
| sets as "a" foundation instead of "the"? These 3 things are the
| most useful, get the most attention, and have benefited from the
| most serious efforts. But IMHO one of the cool things about math
| is that if you're willing to squint and work at it, then many
| alternative foundations are possible. For example Conway's
| surreals[2] hint that you can get numbers/sets by starting with
| even _games_ as a primitive. I can 't quickly find refs, but the
| visualizations here hint that starting with graph theoretic
| axioms can lead to sets instead of vice-versa and I think people
| have worked on that too. Who knows whether alien math builds
| everything else up starting from geometry or probability, etc.
|
| [2] https://en.wikipedia.org/wiki/Surreal_number
| sproutini wrote:
| > If you don't know why set theory is important, it is because
| set theory is the foundation of all of mathematics.
|
| Sorry to burst your bubble, but as far as we know, that isn't
| true in the slightest. It's a logical positivist view abandoned
| after Goedel and Turing.
|
| At best: What we hope is true is that there often is some
| axiomatic system where a specific mathematical lemma makes sense
| when redefined into something similar but not the same.
| afpx wrote:
| You may be interested in percolation theory
|
| https://en.m.wikipedia.org/wiki/Percolation_theory
| some-unique-x wrote:
| Despite several negative comments, I thought the author did a
| great job of explaining and playing with set theory (which, as
| can be seen by the response, is good fun.)
|
| I do take some issue with intepreting set theory's membership
| relation in terms of the tree child relation, though.
|
| First, the child relation is presumably transitive, whilst set
| membership is not. (The _subset_ relation is transitive.
| Presumably it is 'direct child' relation we have in mind here.)
|
| Second, as seen in the third diagram, nodes don't map well to set
| entities, because the same entity can be a member of distinct
| sets, but these would count as distinct nodes on some trees.
| E.g., in the diagram both leaf nodes are distinct, but they both
| represent the empty set, and hence should be identical. So the
| identity of sets is not preserved in the tree encoding.
|
| But this is picky -- a lovely read. Thanks author!
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