Main Greg Channel!

Main Greg Channel!

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Main Greg Channel! episodes

  • Slim Lim: "Concrete syntax matters, actually"

    Topos Institute Colloquium, 4th of December 2025.

    ———
    Too many programming languages researchers dismiss concrete syntax as an afterthought: arbitrary, superficial, or distracting from matters of “actual” (semantic) importance. This received view ignores a critical factor: the human at the computer. Concrete syntax defines the principal interface through which programmers interact with the vast majority of programming languages. Moreover, this interface is hardly decoupled from semantics; even trivial-seeming differences in keywords, sigils, and indentation can affect how programmers utilize and reason about language behavior. Using examples from asynchronous control flow, gradual subtyping, first-class functions, and more, I will make a case for the importance of concrete syntax, why language designers often overlook it, and what this implies for those of us who care about the usability of abstractions. Finally, I will describe some preliminary work evaluating the role of lexical ambiguity in programmer comprehension of type system features.

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  • What Is the Best Bible Translation?

    Register for our online Koine Greek class here!: https://religiondepartment.com/

    Join our Patreon community! https://www.patreon.com/religionforbreakfast

    Thumbnail art by @ECHenry

    Select imagery/video supplied by Getty Images

    00:00 Setting the parameters of the question

    3:01 Which languages are being translated?
    3:58 How manuscript traditions affect translation
    13:20 Translation philosophy
    21:31 Case Study: NRSV
    29:12 Case Study: ESV
    33:54 Case Study: NIV
    38:15 Case Study: NAB
    42:29 Case Study: JPS Tanakh
    48:20 Case Study: KJV
    51:48 Conclusion

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  • Solving polynomials is HARD. Here's why.

    Use my code, 2SWAPYT20, for a 20% discount on your first purchase of a Lovable Pro plan before December 26!

    Click here to automatically apply the discount at checkout: https://lovable.link/2swapyt
    Check out the interactive polynomial solutions demo here: https://2swap.github.io/LittlewoodFractal/

    If you liked this, please support me on Patreon! https://www.patreon.com/2swap

    You can support the musician, 6884, on Kofi: https://ko-fi.com/iam6884
    Check out his music here! https://6884.bandcamp.com/
    This video was animated using SwapTube! https://github.com/2swap/swaptube
    Join our discord server here, where we talk about math, CS, puzzles, etc! https://discord.gg/XDb7x2BZAE

    === Nerd Notes ===

    There are a number of comments asking "But what if I use [something] in my quintic formula?" One such [something] being the Bring radical, which I will take as an example here. The Bring radical is defined as the solution of a particular quintic, which can then be used to solve quintics in the general case. (Aside: I'm not sure this can be called a formula because even by invoking the Bring radical in the general case requires casework, per all of the sources I have found. If you know of one which does not, leave a comment.)

    But as far as I am concerned, this yields a larger question of what can be called a formula in the first place. If I say that an oracle which yields the solutions of the quintic is a valid symbol in my "formula", then the formula is merely the identity statement that the solution to the quintic is the solution to the quintic. I think this sidesteps the question- and furthermore, in such a world, the halting problem would even be solvable, by merely invoking such an oracle. The idea of a formula, if you ask me, is that we represent a complex behavior with simpler constituent symbols which we accept as atomic computations. Where the line is drawn is, of course, arbitrary, (although there is certainly interesting things to be learned by accepting alternate symbols,) but in this video we consider the arithmetic operators and traditional radicals.

    === Errata ===

    Found by @samuelwaid: at 12:26 I said "we define i as being the [square root of -1]" but the actual convention is to define i as being an element such that i^2 = -1. As mentioned in the video, there are various valid definitions of the square root function, and for some of them, the "definition" of i mentioned in the video would not be respected.

    Found by @fumeal: at 11:37 I said that the set of operators on screen were continuous and single valued. However the division operator is not continuous at z=0.
    Found by myself: in the conclusion I said that there is no solution writable in terms of algebraic functions. This isn't at all correct, since the solution itself is an algebraic function! So, one may just say that the solution of the quintic is the solution of the quintic, and be done with it. For some reason I thought that algebraic functions were the functions definable with radicals and traditional operators, which is not the case (and obviously so, by juxtaposition with the definition of the algebraic numbers.) Sigh. If you replace "algebraic" with "using radicals and arithmetic operators", this would be true.
    Found by many: the fundamental theorem of Algebra does not comment on constant polynomials, which I didn't mention in the video.

    === Sources and References ===

    This video was primarily inspired by a paper by Leo Goldmakher. It covers the main proof of this video in a more academic fashion. https://web.williams.edu/Mathematics/lg5/394/ArnoldQuintic.pdf

    The proof itself is by Vladimir Arnold: https://en.wikipedia.org/wiki/Vladimir_Arnold

    While I was interested in Leo's paper, I saw this post about fractals of polynomial solutions. The images were somewhat low-resolution, which inspired me to make my own in conjunction with the topic of the video. https://les-mathematiques.net/vanilla/discussion/2335252/images-des-entiers-algebriques / https://web.archive.org/web/20240530132413/https://les-mathematiques.net/vanilla/discussion/2335252/images-des-entiers-algebriques

    This excellent talk by Andrej Bauer was recommended to me while making my video. It goes much deeper into detail about the fractals of roots seen here. His talk prompted the comment about the dragon curve in this video. https://www.youtube.com/watch?v=8wrYPqEU1x0

    Here are some other resources about the Abel-Ruffini theorem which I found along the way that helped with my own understanding:

    This tool helped me enormously in animating the paths of my roots and coefficients through complex space. Note the box that shows the roots of the determinant- compare to the "poles" in my animation: https://duetosymmetry.com/tool/polynomial-roots-toy/
    Another easy-to-read paper covering this proof: https://arxiv.org/abs/2011.05162
    Two more youtube videos covering this proof: https://www.youtube.com/watch?v=BSHv9Elk1MU https://www.youtube.com/watch?v=RhpVSV6iCko

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  • Richard Feldman, "New Ways to Roc!"

    In the past year, Roc has gotten a new syntax, a new foundational abstraction, and a new compiler. A lot has changed! This talk is a reintroduction to the language, including how it now compares to other functional languages as well as to popular imperative languages – which it now resembles more, despite still being fully functional!

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  • Is This the End of Handwritten Math? Introducing Lean

    A first guide to the Lean 4 Proof Assistant.

    To learn for free on Brilliant, go to https://brilliant.org/AnkYog . You’ll also get 20% off an annual premium subscription.

    You can try out Lean at Lean 4 Web: https://live.lean-lang.org/ (no installation required)

    The code for the video is given at https://github.com/AnkYog/LeanTutorial1
    You can copy this code into Lean 4 Web to follow along the video.

    Learning Lean 4: https://leanprover-community.github.io/learn.html

    Proof assistants are insane -

    Errors that might take months to find can be done in minutes.
    Math problems become open-source projects as anyone can contribute verified math code.
    AIs now have an environment to learn and generate verified math proofs.

    This video is the ultimate tutorial on the Lean 4 proof assistant.

    By the end, we’ll be able to formalize these three big proofs:

    1. An irrational number raised to the power of an irrational number can be rational (using the law of the excluded middle)
    2. The existence of infinite primes (using a proof by contradiction)
    3. Every onto function has a right inverse (using the axiom of choice)

      4. This video was sponsored by Brilliant.

      Twitter: https://twitter.com/ank_yog

      00:00 Intro

      00:46 Basic Syntax
      01:41 Prop-as-Types
      03:28 Tactic Mode
      04:09 Mathlib
      07:00 Intro and Elim Rules
      08:52 Proof: 10 is even
      09:43 Proof: If a number is even, 2 divides it
      12:53 Define PrimeNum
      13:19 Proof: 1 is not prime
      14:18 Proof: 9 is not prime
      16:01 Proof: 5 is prime
      19:37 Final 3 Proofs
      20:01 Irrational to the power Irrational can be Rational
      24:44 There exist infinite primes
      31:22 Every onto (surjective) function has a right inverse
      34:21 End

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    4. うずまき

      Provided to YouTube by Symphonic Distribution

      うずまき · Machine Girl

      …Because I’m Young Arrogant and Hate Everything You Stand For

      ℗ 2017 Machine Girl

      Released on: 2017-09-22

      Writer: Matt Stephenson

      Auto-generated by YouTube.

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    5. Building a container from scratch in Go - Liz Rice (Microscaling Systems)

      Everyone has heard of Docker, but what is a container? Is it really "a lightweight VM"? In this talk we'll dispel the magic by writing a container in about 100 lines of Go.

      Liz Rice has a wealth of software development, team, and product management experience from her years working on network protocols and distributed systems as well as in digital technology sectors including VOD, music, and VoIP. When not building startups and writing code, Liz loves riding bikes in places with better weather than her native London.

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