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Stream: theory: philosophy

Topic: Teaching CT before university

Morgan Rogers (he/him) (Nov 22 2024 at 12:20):

I moved some messages about the attitude of 'the generic mathematician' to category theory from the tail end of this discussion to here.

Madeleine Birchfield (Nov 22 2024 at 16:07):

Morgan Rogers (he/him) said:

I disagree. Understanding how a number can be decomposed with respect to different bases is almost as important as understanding how a number decomposes into prime factors, and having access to these different ways of thinking about numbers is really essential to understanding (rather than just memorizing) operations we perform on numbers.

I don't find alternate bases to be all that important when it comes to understanding operations we perform on natural numbers. There are a number of things that I find to be much more important for that cause, such as

• teaching students about set theory and the relation between set operations (disjoint unions, cartesian products, function sets) on finite sets and the arithmetic operations on natural numbers (addition, multiplication, exponentiation)
• teaching students early the concept of a function between natural numbers so they can understand that arithmetic operations are functions on the natural numbers
• teaching students the induction and recursion principles of the natural numbers, since the usual definitions of multiplication as repeated addition and exponentiation as repeated multiplication are a direct result of the induction and recursion principles of the natural numbers

Teachers only have a limited amount of time to teach mathematical concepts in their classes and I would rather them spend their time teaching the three concepts above which will have a lot of use in their future life and in university level mathematics, rather than teach something like alternate bases, which in all reality will not have much use in life or in university mathematics - unless they end up working with binary and hexadecimal in computers.

As for alternate base positional notation representations of the real numbers, if those are to be in the curriculum, those really belong in the part of the curriculum which talks about infinite sequences and limits.

John Baez (Nov 22 2024 at 17:51):

Morgan Rogers (he/him) said:

Understanding how a number can be decomposed with respect to different bases is almost as important as understanding how a number decomposes into prime factors, and having access to these different ways of thinking about numbers is really essential to understanding (rather than just memorizing) operations we perform on numbers.

To me, learning about different bases as a kid helped free me from thinking that numbers are the numerals we use to denote them. One can argue that this realization is mainly important for mathematicians, but one can also argue that getting a grip on the idea that we have choices in how we describe the world, and separating out the significant from the purely conventional, is crucial for human freedom.

John Baez (Nov 22 2024 at 17:56):

But I don't think teachers should spend a lot of time teaching students to get good at arithmetic operations in base 7, or something like that. I agree with Madeleine that's there's a huge amount of important material to cover. It's sort of amazing how we - that is, we here - have learned all that stuff.

Alex Kreitzberg (Nov 22 2024 at 19:25):

I guess I want to make explicit a certain assumption that's important to how I think about all this.

If you believe "Theory is how we give our collective experience to the next generation, so that science can progress", then in my view good theory is dual to good education. And my political assumption is, as far as it's practical, good education should be public.

So now, if we have a question like "can category theory be learned before learning university mathematics?" I immediately think "Is pre university theory sufficient context to engage with categorical questions?" And therefore dually wonder "Is pre university education a good context from which to make sense of category theory education?"

I've worked with many brilliant kids, and have seen a couple prodigies. I fully believe these kids can learn essentially whatever they want with the proper mentorship. So "Can someone learn category theory before learning university mathematics?" To me is obviously true, but has little relevance as to whether it's good theory, or good education, to think about category theory with only experience in pre university mathematics.

Zulip has advantages over mentorship, because multiple people can help a learner, and the work can be recorded for others to read later. But if some people can learn category theory on Zulip with only a high school education, it doesn't immediately follow to me that "Category theory can be learned before university mathematics" in the generic sense I explained above.

You're free to disagree with this, so I felt it was polite for me to make my implicit assumptions here very clear. Sorry if folks think this point of view derailed the original topic, that wasn't my intent :sweat_smile:.

Madeleine Birchfield (Nov 22 2024 at 20:49):

The original question was

Jean-Baptiste Vienney said:

• Can we start learning category theory before learning any university level math?

Perhaps we should have begun by asking who the "we" in the question is.

Are we referring to the generic person, or if there exists a person, or for all persons?

Madeleine Birchfield (Nov 22 2024 at 21:04):

This can also be interpreted personally: could you personally have started learning category theory before learning any university level maths?

Madeleine Birchfield (Nov 22 2024 at 21:26):

Alex Kreitzberg said:

So now, if we have a question like "can category theory be learned before learning university mathematics?" I immediately think "Is pre university theory sufficient context to engage with categorical questions?" And therefore dually wonder "Is pre university education a good context from which to make sense of category theory education?"

The passage from the first question to the second question would have made more sense before the internet connected everybody together and allowed teenagers to talk to experts online to learn whatever topics, bypassing the education system entirely.

But for all practical purposes, the average person who ends up browsing the internet and entering mathematics forums to learn more mathematics is probably going to end up encountering and learning some other university mathematics before discovering the existence of category theory.

Jean-Baptiste Vienney (Nov 22 2024 at 21:31):

I think it could make sense pedagogically to learn category theory before any other university level math in the future. If all the main branches of mathematics, computer science, engineering etc… get a widely used categorical foundation, then it will make sense. But today that’s not yet where we are.

Alex Kreitzberg (Nov 22 2024 at 21:33):

Jean-Baptiste Vienney said:

I think it could make sense pedagogically to learn category theory before any other university level math in the future. If all the main branches of mathematics, computer science, engineering etc… get a widely used categorical foundation, then it will make sense. But today that’s not yet where we are.

Right, even if it doesn't make sense today to literally implement something like that, I still think understanding how that could work might usefully inform the theory.

That's what I mean, education can be used as a heurstic to understand the theory. That you feel category theory could work in K-12 someday is notable, I think it's worth exploring why from that lens, and that this is a valid way to engage with the original prompt.

Alex Kreitzberg (Nov 22 2024 at 21:35):

Madeleine Birchfield said:

The passage from the first question to the second question would have made more sense before the internet connected everybody together and allowed teenagers to talk to experts online to learn whatever topics, bypassing the education system entirely.

While true, after a bit of time thinking about this before this conversation, I still think public school is much better than an internet connection, and maybe I should've explicated that as well. I can see now, how I explored the premise was probably a major failure in communication on my part, but it's difficult for me to lay out all of my assumptions like this.

Madeleine Birchfield (Nov 22 2024 at 21:38):

Jean-Baptiste Vienney said:

I think it could make sense pedagogically to learn category theory before any other university level math in the future. If all the main branches of mathematics, computer science, engineering etc… get a widely used categorical foundation, then it will make sense. But today that’s not yet where we are.

It is going to be very difficult to teach category theory before university level maths in some cases. If I remember correctly, some schools in the United States teach differential and integral calculus, vector calculus, statistics, and even linear algebra at the high school level, even though these are traditionally considered university level mathematics. Then you're back to the question of curriculum debates at the secondary school level.

Alex Kreitzberg (Nov 22 2024 at 21:49):

I want to add somewhere in this conversation, I think Eugenia Cheng's book "Joy of Abstraction" is brilliant, and I bet there's a good chance somebody will decide to be a math major after reading it. I think "Conceptual Mathematics" is also very good, but doesn't succeed at its goal as well as Cheng succeeds at hers.

Actually, what kills me is everybody who I've recommended Cheng's book to has told me they already know everything in it, which I know for a fact they don't. So incredibly, the only criticism I've heard of it so far is that she made the concepts too relatable!

Peva Blanchard (Nov 22 2024 at 21:52):

The way you recommend Eugenia Cheng's book just made me add it to my bibtower.

Peva Blanchard (Nov 22 2024 at 21:56):

I have another question related to the topic (but feel free not to respond).

Since CT deals a lot about "building language", can it be used to teach better what is currently being taught before university? i.e., it would not be about the content, but more about the presentation.

Madeleine Birchfield (Nov 22 2024 at 22:06):

Probably the best bet would be in set theory, combinatorics (aka finite sets), and Euclidean geometry.

I have a really hard time seeing categorical language being applied to school elementary number theory and real analysis / school algebra, since those subjects aren't talking about morphisms between mathematical structures.

Peva Blanchard (Nov 22 2024 at 22:35):

I was thinking of using CT when preparing the content, not really put CT in the content itself.

Often, to learn something, it is useful to multiply the perspectives on that thing. Hence, as a teacher if I know, thanks to some experience in CT, that some concept has many instances that are all mutually isomorphic, or equivalent, or etc. then maybe it can help design multiple exercises.

For instance, at some point children learn about remarkable identities like $(a + b)^2 = a^2 + 2ab + b^2$. The underlying notion is distributivity, which is one of the defining properties/data of rigs/rig categories. The natural numbers are one instance, and the groupoid of finite sets with bijections is another (of a different kind). I find that the groupoid perspective suggests teaching distributivity with "arrangements of marbles". Of course, there is no need to mention what a groupoid is with the children.

I don't find my example very convincing, because I made it up to support my case. But I find the idea of having multiple perspectives, yet rigorously related, useful to design a learning process.

Madeleine Birchfield (Nov 22 2024 at 22:42):

So that's a question of putting category theory in the mathematics education curriculum in university.

Sascha Haupt (Nov 22 2024 at 23:32):

I am no education expert whatsoever, but I think there is an option to use "categorical thinking" in high school teaching without explicitly mentioning category theory. It could be applied to different topics in math, physics, and possibly other subjects (computer science? biology? ...) which might be more suitable for high school education than introducing category as such, as an abstract mathematical theory.

Actually, there was an educational experiment conducted in the UK where high school students where taught quantum mechanics using the [[ZX-calculus]] without category theory being mentioned (explicitly). Basically, the students where learning quantum mechanics and the basics of quantum computing using string diagrams instead of lengthy expressions involving matrices or bra-ket notation. You can learn and apply the rules for string diagram rewriting without knowing that the diagrams you are dealing with are actually morphisms in a monoidal category.
Here's the paper and a talk about the experiment.
I believe there are thoughts about pushing the age limit for such an approach to even younger students (middle school or younger).

Another topic where I could imagine a similar approach would be the graphical approach to linear algebra.

Jorge Soto-Andrade (Nov 23 2024 at 01:30):

Peva Blanchard said:

I was thinking of using CT when preparing the content, not really put CT in the content itself.

Often, to learn something, it is useful to multiply the perspectives on that thing. Hence, as a teacher if I know, thanks to some experience in CT, that some concept has many instances that are all mutually isomorphic, or equivalent, or etc. then maybe it can help design multiple exercises.

For instance, at some point children learn about remarkable identities like $(a + b)^2 = a^2 + 2ab + b^2$. The underlying notion is distributivity, which is one of the defining properties/data of rigs/rig categories. The natural numbers are one instance, and the groupoid of finite sets with bijections is another (of a different kind). I find that the groupoid perspective suggests teaching distributivity with "arrangements of marbles". Of course, there is no need to mention what a groupoid is with the children.

I don't find my example very convincing, because I made it up to support my case. But I find the idea of having multiple perspectives, yet rigorously related, useful to design a learning process.

@Peva Blanchard in fact you can approach an identity like $(a + b)^2 = a^2 + 2ab + b^2$ (and n-th powers too) diagrammatically, as descending paths in a grid (not a tree), which suggests a categorical rendering...

Jean-Baptiste Vienney (Nov 23 2024 at 01:55):

@Jorge Soto-Andrade Could you explain this a bit more? How do you approach this identity with paths in a grid?

Jean-Baptiste Vienney (Nov 23 2024 at 02:02):

Are you talking about a geometric Greek antiquity-style proof, where you draw a square of side $a+b$ and divide it into a square of side $a$ plus two rectangles of sides $a,b$ and a square of side $b$?

Damiano Mazza (Nov 23 2024 at 07:20):

I think he is talking about a 2-dimensional grid where, say, $a$ means "move right" and $b$ means "move down". The identity

$$(a+b)^n = \sum_{k=0}^n \binom n k a^k b^{n-k}$$

may be interpreted as saying that, if you start from any point on the grid, there are $n+1$ different points you may reach in exactly $n$ moves of type $a$ or $b$, and the $k$-th point has $\binom{n}{k}$ different paths leading to it.

Peva Blanchard (Nov 23 2024 at 08:46):

Oh nice! The grid perspective highlights commutativity. This could be used to learn Pascal's triangle too. On a side note, this perspective is also used in concurrency theory to detect deadlocks (grids with holes, directed homotopy).

Jean-Baptiste Vienney (Nov 23 2024 at 13:46):

Thank you!

Morgan Rogers (he/him) (Nov 23 2024 at 15:18):

Madeleine Birchfield said:

Teachers only have a limited amount of time to teach mathematical concepts in their classes and I would rather them spend their time teaching the three concepts above which will have a lot of use in their future life and in university level mathematics, rather than teach something like alternate bases, which in all reality will not have much use in life or in university mathematics - unless they end up working with binary and hexadecimal in computers.

Short intervals of time are recorded in base 60, so bases other than decimal aren't all that exotic. I would also argue that knowing about binary representation of numbers rather than people feeling alienated by it would be good for society. People encounter different representations of numbers far more than they encounter concepts from set theory.

Morgan Rogers (he/him) (Nov 23 2024 at 15:31):

Sascha Haupt said:

I am no education expert whatsoever, but I think there is an option to use "categorical thinking" in high school teaching without explicitly mentioning category theory.

Suggestions like this bemuse me. You should definitely explicitly mention the domain that an idea or approach comes from, even if you won't have time to lay the groundwork for students to learn about the domain (as in Leinster's recent ETCS course).

Mike Shulman (Nov 23 2024 at 15:59):

Short intervals of time are recorded in base 60

I would say they use a hybrid of base 10 and base 6. There is a units digit, a 10s digit, a 60s digit, a 600s digit, a 3600s digit, and a 36000s digit. (Is there a general name for hybrid base systems like this?)

Morgan Rogers (he/him) (Nov 23 2024 at 16:11):

Mixed-radix, apparently!

Mike Shulman (Nov 23 2024 at 16:17):

Thanks! Looks like Wikipedia agrees with you about base 60, but it seems to me that when I write 12:34 there are four digits, not two...

Peva Blanchard (Nov 23 2024 at 16:19):

base 2 occurs as well with am/pm notation

Mike Shulman (Nov 23 2024 at 16:46):

Haha, that's right. But either 12 or 24 hours do only work if you consider the hours to be one "digit", at least with the simple notion of mixed-radix.

John Baez (Nov 23 2024 at 23:00):

Morgan Rogers (he/him) said:

Sascha Haupt said:

I am no education expert whatsoever, but I think there is an option to use "categorical thinking" in high school teaching without explicitly mentioning category theory.

Suggestions like this bemuse me. You should definitely explicitly mention the domain that an idea or approach comes from, even if you won't have time to lay the groundwork for students to learn about the domain (as in Leinster's recent ETCS course).

I don't agree in high school or earlier. For example I think teaching kids in elementary school about the natural numbers and their addition and multiplication should definitely be informed by a perspective based on the category FinSet, coproducts and products, and treating $\mathbb{N}$ as a decategorification of $\mathsf{FinSet}$ - but I don't think the kids or even their teachers need to hear about the technical names of these concepts, or even the phrase "category theory". I just mean that they should learn 2+3 = 5 summarizes a lot of facts like

"You can take these rocks

o o o o o

and move them to look like this

o o o o o "

and so on.

Alex Kreitzberg (Nov 24 2024 at 04:42):

Could you talk a bit about why facts like that are categorical? Or have you written about that somewhere?

I think I know what you mean, but I'd love an explanation that really solidified the visual intuition of those rocks into category theory statements.

John Baez (Nov 24 2024 at 04:45):

When we say

"You can take these rocks

o o o o o

and move them to look like this

o o o o o "

we categorifying $2 + 3 = 5$, which is an equation between elements in $\mathbb{N}$, by treating it as an isomorphism between objects in $\mathsf{FinSet}$, namely "the coproduct of a set of 3 rocks and a set of 2 rocks" and "a set of 5 rocks". The process of moving the two sets of rocks together is the isomorphism.

John Baez (Nov 24 2024 at 04:47):

Didn't we already spend a long time talking about how $\text{Decat}$ maps $\mathsf{FinSet}$ with its product and coproduct to $\mathbb{N}$ with the usual $\times$ and $+$?

Yeah, we talked about it starting here. I'm just referring to that sort of thing.

Alex Kreitzberg (Nov 24 2024 at 05:27):

Yes we talked quite a bit about $Decat$, thank you for the extra bit of visual context!

John Baez (Nov 24 2024 at 06:17):

Yes, it's good to illustrate all the features of $\times$ and $+$ in $\mathsf{FinSet}$ using sets of pebbles. For example $X \times Y$ is a rectangle of pebbles:

o o o o o o o o o o o o

John Baez (Nov 24 2024 at 06:18):

The isomorphism $X \times (Y + Z) \cong X \times Y \, + \, X \times Z$ is....

Alex Kreitzberg (Nov 24 2024 at 07:21):

o o o o o o o o o   o o o o    o o o o o
o o o o o o o o o ≅ o o o o    o o o o o
o o o o o o o o o   o o o o    o o o o o

Right?

Morgan Rogers (he/him) (Nov 24 2024 at 11:32):

John Baez said:

Morgan Rogers (he/him) said:

Sascha Haupt said:

I am no education expert whatsoever, but I think there is an option to use "categorical thinking" in high school teaching without explicitly mentioning category theory.

Suggestions like this bemuse me. You should definitely explicitly mention the domain that an idea or approach comes from, even if you won't have time to lay the groundwork for students to learn about the domain (as in Leinster's recent ETCS course).

I don't agree in high school or earlier. For example I think teaching kids in elementary school about the natural numbers and their addition and multiplication should definitely be informed by a perspective based on the category FinSet, coproducts and products, and treating $\mathbb{N}$ as a decategorification of $\mathsf{FinSet}$ - but I don't think the kids or even their teachers need to hear about the technical names of these concepts, or even the phrase "category theory".

Sure, you don't need to give a full account in elementary school, but by the end of high school young people are mature enough to appreciate where ideas originate from and benefit from being told as much. As a 13/14-year-old, I appreciated my science teachers pointing out where explanations represented simplifications of scientific understanding (eg. the electron shell model of atoms) even if there wasn't room to explain deeper nuances, because it revealed ways in which further study of these subjects would be rewarding. At that stage I didn't necessarily need to know domain-specific words because the curriculum is already organized so that I would learn more by simply continuing to pursue those subjects. But at the end of high school, young people start to specialize (this is more the case in the UK & Europe than the US, but there are still choices to make). If they want to learn more about something specific, they need to know the associated vocabulary to be able to identify whether the opportunities they are choosing from will enable them to do so. If they specialize away from some of the things that interest them, then without key words/terminology to orient them, they will lose contact with it completely.

Posina Venkata Rayudu (Nov 24 2024 at 23:03):

Dear All, if I may, I'd like to address some of the issues raised above; I hope it's OK to post them all as one message, instead of answering individually. Thanking you, yours truly, poison

Category theory is not removed from our everyday experience, which is also categorical: objects populating my perceptual experience are objects of one or another category (e.g., when I see a cat, I see it as an exemplar of a category of cats; see Professor Tom Albright's chapter on categorical perception in Kandel et al., Principles of Neural Science; our neural correlates of categorical perception might be of some interest to some of you).

Professor F. William Lawvere, with his answer: 'a space is an object of a category of spaces' to Professor Stephen Schanuel's question: 'What is a space?', appears to be drawing attention to the fact that by virtue of being an object of a category of objects, all of which partake in the essence (which can be thought of as: objects have souls ;) characterizing the category, morphisms of objects necessarily preserve the corresponding categorical essence / theory. For example, a loop to which an arrow can be mapped is quantitatively different (1 vs. 2 dots); so am I: child --> geriatric, but, in spite of all the quantitative changes, not unlike graph morphisms, aging preserved my poisoness, my soul, so to speak.

Since we already learned / teach all about numbers in place-value notation, it's not beyond the grasp of students to construe numbers, say 2024, as functions from a domain set of places {10^0 10^1 ... } to a codomain set of values {0 1 2 ... 9}. Equivalence relation is not that far from here: both places of tens (10^1) and thousands (10^3) are in the same basket (not of deplorables, but of 2 ;)

Speaking of place-value notation, can we construe it as a presentation (cf. generators, relations; since presentations are discussed in Conceptual Mathematics in terms of dynamical systems and graphs, we can think of functions as a subcategory of graphs, i.e., loops, where target = source; here the basic shapes are a dot D and a loop L, which along with a structural inclusion map: D --> L, constitute the soul of the category of functions, pp. 114 - 119)?

Composition (of functions) is also all too familiar in our everyday experience: putting together those that fit together, with the "fit" being something in common to the two, thanks to which the given two becomes one.

Yes yes, d = vt is amazing in protecting us from having to face the harsh reality (reminds me of Buddha's dad ;) out there (Perugia Notes, p. 128).

Reciting universal mapping property definitions of, say SUM might be an easy way to shoo away people and enjoy my solitude, but it's just saying that: A sum is a whole that is completely determined by its parts (the exactly one map from the sum to a test object, say, T = {u, v} is about the aforementioned 'determined by', i.e., every f: A --> T is determined by i: A --> A + B (sum), with the "exactly one map" (p: A + B --> T), a fixture in universal mapping property definitions, being the proof of the determination.

There's a contrasting notion that might help see the sensibility of the definition of SUM (actually a shrill cry for help much louder than Obama screaming: change), which is the Gestalt maxim: whole is different from the sum of its parts, which spawned all sorts of whatever like holism. I, for one, can't figure out what's so shocking about a commonplace fact that once in a while we might miss a part or two or there might be interactions between parts, which can be taken care of with, say, colimits, while saving the unsuspecting students from having to endure, as if it's a divine condemnation, the trauma of having to make sense of His Eminence talking in tongues ;)

I am not sure what's so natural about natural numbers (if anything Eulers numbers, by virtue of counting extended bodies, are more natural than those counting those way too abstract lauter Einsen / all ones ;)

John Baez (Nov 25 2024 at 01:17):

Alex Kreitzberg said:

o o o o o o o o o   o o o o    o o o o o
o o o o o o o o o ≅ o o o o    o o o o o
o o o o o o o o o   o o o o    o o o o o

Right?

Right! I like these almost wordless discussions!