Originally from Math4Wisdom public listserv, hosted by Andrius Kulikauskas, PhD. https://www.freelists.org/list/math4wisdom From: kirby urner <kirby.urner@gmail.com> Date: Wed, Apr 5, 2023 at 8:16 AM Subject: Re: [math4wisdom] Re: What is algebra? How do we describe adding, subtracting, multiplying, dividing? To: M4W clique Glad to see David chiming in. I'm continuing my conversation with Jon re the Bucky stuff on another listserv (also public archives, like M4W) [1] [1] https://groups.io/g/synergeo/message/2260 <https://groups.io/g/synergeo/message/2235> On Tue, Apr 4, 2023 at 2:24 PM David Pinto <david@ecosquared.co.uk> wrote:
Thanks Andrius et al
Not philosophy. Just what is going on when we count, do addition, subtraction, multiplication, addition.
What is Algebra? I really like M4W's focus on Pascal's Triangle (binomial theorem) as a doorway into (a) combinatorics and (b) polynomials more generally, leading (eventually) to the classification of polynomials (their taxonomy) into such as Bernoulli, Hermite, Sheffer, Chebyshev. Pascal's Triangle is what I call a "grand central station" on our planet (aka "Math World" if we wanna call it that, or, more parochially, for me, "Python Planet"). As a curriculum developer for the Silicon Forest (Oregon Curriculum Network is my public vehicle, backed by my proprietary 4D Solutions (4dsolutions.net)), I've been injecting more Abstract Algebra into Algebra (the pre-college academic subject) by means of Python. As many of you know, I'm one of those math teachers who thinks we should ditch graphing calculators for big screen computers, even though this disrupts the factory-based "every kid has a locker, moves from room to room" picture of what "school" is supposed to be. Python can do anything a graphing calculator does, including plots (xy-curves, histogram, pie chart... i.e. it's great for statistics, i.e. data science, as well as for generic geometry). Students might have to take a van or bus to specific buildings to access their workspaces though, as these take up too much room per the standard model classroom. https://i.pinimg.com/originals/7f/4d/99/7f4d99ec33a9736f3fb890b6127894eb.jpg https://thetechnodiary.com/wp-content/uploads/2021/07/Cooler-Master-ORB-X-Ga... Safe to say: ergonomics is tier one, in terms of priority topics. Cramped desk or game-ready pod? For some, it's a no brainer. Python allows operator overloading i.e. we can define new classes of object that control what the operators mean to them. Not every computer language (e.g. Java) allows this. Lesson Plan: A typical Silicon Forest lesson plan features the function type object, which originally, out of the box, comes unequipped with any meaning for multiplication. We then create a new class (new type) based on the function, its instances callable with the same arguments, giving the same outputs, that makes "multiply" mean "compose", the archetypal thing we do with functions. Instead of: f ( g (x) ) i.e. f after g of x we now get to write: (f * g)(x) or even: h = f * g and later: h(x) Example (screen shots) https://www.flickr.com/photos/kirbyurner/52795727030/sizes/c/ https://flic.kr/s/aHsknZHG75 (album) https://flic.kr/s/aHBqjzJdXN (a school the let me prototype; Saturday Academy another) Part of abstract algebra is elementary group theory and learning that subtraction and division are "syntactic sugar" for "adding the additive inverse of" and "multiplying by the multiplicative inverse of" respectively. Then we dive into all that totative / totient stuff, i.e. groups of totatives of N mod N, aiming towards Fermat's Little, and Euler's Theorems used to implement the RSA algorithm (public key crypto). This is already an established east coast college prep pathway at Phillips Academy / Andover [1]. The Silicon Forest is playing catch up in some respects. I'm a big fan of the Abstract Algebra lectures on the YouTube channel Socratica, also good on Python. [2] [1] https://dl.acm.org/doi/book/10.5555/1855295 (authors were faculty at the time) [2] https://youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6 === Philosophy of Mathematics I mentioned to Andrius my interest in Ludwig Wittgenstein's later philosophy, which later philo LW encouraged us to take in by using his earlier philo (Tractatus) for contrast, and for clues as to what he was aiming to show and/or induce. His Philosophical Investigations (posthumous) dovetails with his Remarks on the Foundations of Mathematics. I consider him another philosopher of mathematics, albeit under-appreciated (somewhat like RBF in that regard -- I merge the two, featuring Bucky-style language games under a philosophers' microscope (LW's technique)). EJA said he enjoyed my RBF/LW confluence. Two points on LW's contribution: (1) he had a strong interest in 'aspect shifts' i.e. the schooling of our perceptions, such that immersion in a discipline changes how we see the world. Example: it's all but impossible to "unsee" a language we know well i.e. to have the symbols revert to meaningless squiggles, as when staring at a language we do not know (e.g. Japanese for me -- I can't unsee English to make it seem that opaque). Many math languages certainly have that power: to induce new ways of seeing, perhaps irrevocably (and perhaps not always for the better, let's accept). (2) he took the meanings of such words as "addition", "computation", "understanding", "thinking" to be intrinsically public i.e. private sensations, observations of mental processes, internal states would not be essential to our investigations. Introspective forays in search of "what X really means" in the sense of "what X points to internally" were in his estimation misguided (in the sense of superstitious). Symbols do not gain their meanings from pointing, but from patterns of use. Think not of pointers, but of tools (e.g. a screwdriver), or of pieces on a game board (e.g. a pawn in chess). The upshot was: most readers, including other philosophers, could not shift their own perceptions sufficiently, per (1) above, to make sense of what he was getting at per (2). Here's a mental exercise (a thought experiment): imagine a non-experiential non-sentient AI bot that uses human voice boxes as peripherals, and also propagates in writing, eventually becoming so ubiquitous that we internalize it, imagine it speaking. Spend a few minutes to hours someday having this be your new gestalt. Do any aspect shifts occur? As I wrote to an old friend on Telegram (we were discussing AI): kirby urner, [Mar 26, 2023 at 8:40:25 AM]: Probably my main influence is Wittgenstein, who famously, according to said book, threatened Popper with a fire poker, although in my camp's telling, he's doing another Zen koan type gesture, saying language is a lot more than we're able to say. As a thought experiment in my 1980 About Wittgenstein thesis, I had a non-sentient AI GPT computer named Adam speaking through all human beings, saying "I" this and "I" that, but not really having subjectivity. Adam = Language. Of course back then I didn't say GPT or AI, so I'm updating my terms. As GPT would. As I would, as Adam. Kirby