Episode 147: Gabriella Gonzalez discusses the intersection of algebra and programming
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What is abstract algebra and how does it differ from high‑school algebra?
Yeah. Mm.
Hello and welcome to Elucidations, an unexpected philosophy podcast. I'm Matt Tikman. And with me today is Gabriela Gonzalez, creator of the Dahl Programmable Configuration Language, longtime evangelist for the Haskell Programming Language, author of the Haskell for All blog, and engineering manager at Arista Networks. And she is here to discuss the intersection of algebra and programming. Gabriella Gonzalez, welcome. Thank you very much for having me on your podcast. Absolute pleasure to have you. So I think that when a lot of people Hear the word algebra. The thing they think of is like high school algebra. Like somebody gives you the math teacher gives you a problem, two times x equals sixteen. Now solve for x.
Whereas people who've gone off to study a little bit more math in college often come across this thing called abstract algebra, which is like somehow related to the high school algebra, but it's a little more general. So what exactly is
abstract algebra? So typically when we think about algebra, we think numbers, you know, I I maybe in simple arithmetic. The idea behind abstract algebra is that you take these algebraic operations, but they work on things that are not numbers. So for example, we're gonna be talking about how you can add things like code, for example, or programs, not just simple numbers. That's what makes it abstract. The fact that it's no longer on numbers anymore. Abstract algebra is a bit more general than that. That's kind of a first order approximation of what's going on. But I feel like that's a useful way to get started in in thinking about it.
So it's like you can add and multiply Other stuff besides just numbers. In in some sense that we can actually make precise.
Mm-hmm.
Yep. Well that's crazy. So w how w like what would I dunno, I have two cups of water and I pour one into the other. Is that like adding I'm like yeah, what would be an example of adding something that's not two numbers?
So one example I like to give is a recipe. So for example, you can imagine that let's take plus and multiplication. So in a recipe, imagine that times is that you specify that you want to use more than one ingredient. In s typically when you read a recipe you'll say like okay, the ingredients are A B and C So you can imagine that you could represent that algebraically as a Times B. Time C And then often in recipes you can substitute one ingredient for another ingredient. For example Maybe you substitute I don't know. I'm bad at this like honey with sugar or I'm making that up.
Yeah,
or
agave or some other sweetener.
Exactly. Yeah. And so the way you can represent that is using plus. So the idea is that A plus B means that you can use either ingredient A Or you can use ingredient B. And if you think about it, many of our algebraic intuitions are correct if we get define recipes using that convention. So let me give you an example. So one of the arithmetic rules that you've probably learned in school is the concept of distributivity, where you say that if I take A and I multiply it by B plus C. That's the same thing as A times B plus A times C plus. And if you think about that from like a recipe standpoint, it's saying, Okay, if I need ingredient A And I need ingredient B or C That's the same thing as saying I need either ingredient A and ingredient B
Or I need ingredient A and ingredient C. So that algebraic intuition still holds when talking about recipes, even though we're not really talking about numbers anymore.
Okay, right.
How can algebraic operations like “plus” and “times” model recipes and cooking?
So at first I was a little bit thrown. By the idea that times means and and plus means or Because I don't know, I'm just used to thinking of times as the number thing. But it seems like what we're getting at here is Maybe like part of the essence of being times and part of the essence of being plus is for this distributivity law to r apply. Um that that you know, y if you try to strip What does it mean for something to be a times operation versus and what does it mean for something to be a plus operation down to the bare essentials? One of the things maybe that you'd get is this distributivity property.
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Chapters
8 chapters
1
What is abstract algebra and how does it differ from high‑school algebra?
0:15–4:02
2
How can algebraic operations like “plus” and “times” model recipes and cooking?
4:02–9:17
3
Why do regular expressions form an algebraic structure and what are their zero and one elements?
9:17–14:18
4
What are semigroups, monoids, and semi‑rings and how do they relate to code reuse?
14:18–19:37
5
How does functional programming expose algebraic concepts such as product, sum, and exponentiation types?
19:37–24:30
6
Why does equational reasoning make reasoning about programs easier and more scalable?
24:30–28:36
7
How does abstract algebra provide a timeless, reusable foundation compared to programming fads?
28:36–34:04
8
What are the key takeaways on applying algebra‑driven design to real‑world software?
34:04–39:43
Speakers
1 identifiedMore from Elucidations
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