The recursive, binary structures at the root of math

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The recusive, binary structures at the root of modern math. Visualized to show their unity-in-diversity.

This builds on this recent, unexpectedly viral diagram. I'm adding to the parallelism the structures of set theory, combinatorics & order theory: the Venn diagram, Pascal's triangle & Hasse diagram. Also being more explicit about the duality in computation between code & data. twitter.com/elzr/status/1622022572530606083

For friends who'd like to see me 🫳🤌🫴🙌 emoting & gesturing excitedly about abstractions, I recorded this 12min screencast unfurling this new diagram. cc @jamescham, @dela3499, @maxkriegers loom.com/share/afca55c3ce9845c9b4796fdce906dc5e

These "recursive, binary structures" — are they in the room with us right now? (They've surely been with me this past week! 🤣) twitter.com/stevekrouse/status/1444281751682142214

I integrated Pascal's triangle thanks to @prathyvsh's many kind prompts 😅 (combinatorics is still very foreign to me): twitter.com/prathyvsh/status/1622027820431183872

Instead of the more common sums in Pascal's triangle, I preserve the binary "vertices" to make it match with the other diagrams. Their stacking makes very clear the connection with the Galton board —how combinatorics gives rise to probability! Left diagram by @AlanZucconi.

Seeing the pebbles falling down the Galton board makes me remember this visualization of luck that's been a sort of lucky charm itself since it periodically resurfaces: twitter.com/elzr/status/1381105950602461189

My visual investigation into logical fundamentals can be traced back to this 2019 tweet, where I come up with a tetragram to represent logic gates visually (iconically, as Peirce would say), then build up additive circuits! twitter.com/elzr/status/1169409893197111296

Back then, deep in the comments (with @prathyvsh, in an exchange that would start our friendship) we ended up with Venn-like diagrams too: twitter.com/elzr/status/1171641749057495041

I was also led to Euler diagrams and how they match with trees & Hasse diagrams in this beautiful & clear thread on lattices by @prathyush (often it's only until he makes a graphic thread that I can finally grok his references ;) Going Venn instead of Euler was a hard choice! 💔 twitter.com/prathyvsh/status/1476974452705480712

Key inspirations for including Venn diagrams were the layouts by Russell Johnston (1987) & Stephen H. Cullinae (2007) finitegeometry.org/sc/16/venn.html as well as the many ones found across logical articles in Wikipedia, like Greg Bard's (2007) en.wikipedia.org/wiki/Logical_connective

Back to the head diagram. Notice the 2-arity of all the structures. Binary seems the simplest, but nature may speak more natively in higher-arities, as @stephen_wolfram & @getjonwithit point in @wolframphysics. We need @TheTedNelson in our math to hyper-ify all the things! twitter.com/elzr/status/1622022587277799424

Would also like to recommend here the fantastic work Mark Jeffery is doing at the @lasttheory_ project (videos, podcast, newsletter). He has several great tutorials on arity & this explanation of why hypergraphs by @getjonwithit recently blew me away. 🤯 twitter.com/lasttheory_/status/1606055527141646339

Spivak's Calculus has an appendix on how you can build up the ordered pair from sets. "Abstract nonsense," said my teacher when I mentioned it excitedly. Spivak too frames it as if pesky construction-ism is just to assuage nerves about abstraction. But it's also just beautiful!

And having many alternative constructions of abstractions can be of world-historic importance! It wasn't until logic gates & boolean operators that we really figured out how to instantiate abstractions physically, dynamically, as computers. twitter.com/elzr/status/1615057520585842688

Anyway, what I'm really trying here is to "sensualize abstractions" (to mangle Hamilton's 1858 phrase), to share with you the aesthetic chills I feel at the convergence we find as we discover new bedrock for the formal science of math.

It's like the Swift quote Spivak has as epigraph to an epilogue constructing the ℝeals. Below modern math's rarefied heights, we're also building & discovering its foundation —and always will, for it's 🐢foundations all the way down, as @DavidDeutschOxf coulda but never said ;)

Let's end with a visual joke: Here's another kind of binary tree, in the red/black styling I'm using ( which @gwern calls rubrication: gwern.net/red ). PS: There are non-binary Easter eggs in the original comic! (2 & 4 but seemingly no 3). twitter.com/turnoff_us/status/781662589965172740