The most misunderstood equation in math (associative property)

Associativity is one of the first properties taught to students. Let's explore this property together.

For the curious among you, the mathematical theorem at the heart of this video is Cayley’s theorem for semigroups.

00:00 The equation
02:20 Exploration
13:36 What is associativity? Part 1
20:46 Application: group theory
25:20 What is associativity? Part 2
27:34 Reflections

Function composition interpretation
Function as object: A machine taking input and transforming it into output
Function as action: modify any starting machine by connecting the output of another machine to its input
Object-object: A machine for which the output of a squaring machine is connected to the input of a square rooting machine is equivalent to an absolute value machine
Object-action: starting with a square rooting machine and then attaching a squaring machine to its input produces an absolute value machine
Action-action: attaching a square rooting machine to the input of any starting machine, then attaching a squaring machine to the input of the result is equivalent to attaching an absolute value machine to the input of the starting machine.

This was my submission to #3B1B #SoME1 competition

[MaximQuantum]

William Shakespear: "To be, or not to be"
Lingua Mathematics: "To be, and then to see."
14:55

[spacelightning]

One of the proofs at the start of my first abstract algebra course showed that the "basic" associativity (ab)c = a(bc) implied general associativity for finite expressions via structural induction. In other words, an associative operation is one in which parentheses are no longer important, extending a binary operation into an n-ary operation as long as the sequence of object symbols is constant. So with that in mind, let's take a swing at 2:16

Suppose you have a string A1☆A2☆...☆An and a closed computation of strings of the form X☆Y. Let P be an algorithm which

1) Chooses an arbitrary substring of A1☆A2☆...☆An of the form X☆Y,

2) Computes the substring in accordance with its binary definition,

3) Substitutes the computation for X☆Y,

4a) Loops if the new string contains any ☆ symbols, or

4b) Terminates otherwise

The operation ☆ is associative if and only if for every natural n>1 the string A1☆A2☆...☆An reduces to the same result for every such algorithm P meeting the above 4 properties.

In fewer words, "☆ is associative if and only if well-formed formulas of n-ary instances of ☆ are well-defined under arbitrary recursive parsing algorithms which evaluate well-formed sub-formulas of 2-ary instances of ☆."

[jks]

This is an example of a very simple idea completely losing meaning because it became a rule and people used it only for its usefulness instead of its meaning.

Thank you for recovering why associativity is used in the first place.

[SoundsOfTheWildYT]

This comes across as more of a philosophical look at associativity than a mathematical one. I think it would be a struggle to define objects and actions in a meaningful way that provides any mathematical context, as mathematical constructs are by their very nature able to represent a great many things, far more than just the “dual” in duality. Objects and actions as described in this video are far more tangible than abstract, and lean more toward philosophical constructs than mathematical ones,

[cmilkau]

In a sense, concatenation is "the" associative operation. Every equation satisfied by concatenation is satisfied by every associative operation. This is easy to understand keeping in mind that associative operations allow you to drop parentheses, so you can write terms just by writing sequences of arguments to the operation.

[dennissweeney]

My intuition is that an associative operation is an operation that is well-defined on finite strings of objects

[OrangeC]

That was a really cool video. I'll be honest, at the beginning I thought it was all pretty obvious, but by the part of the video when you related all of it to group theory I realized just how important it was for you to really concretely define what "and then" really is. You explained everything very clearly and the visuals made it very easy to understand. I really think I understand this topic much more clearly than I did before, even though I thought that I already understood it well.

[Qhartb]

Seems odd to forbid the word "order" when challenging us to explain associativity, then you proceed to use "and then" and timelines, which are very "ordery" things. I stayed more object- than action-oriented and considered the natural input type of the variadic extension of an operation. A general operation takes a binary tree, an associative one takes a finite sequence, and an associative & commutative one takes a finite set [edit: multiset]. But I had the same criticism of this -- a sequence is an "ordery" thing.

[otesunki]

EDIT: i now realize i made a fool of myself. and this is commutativity.

multiplicative: if you take a -cube- rectangular prizm with side lengths a, b, and c, no matter how you rotate it, it has the same volume

additive: say you have a shelf that is packed full of books, each with different thickness [a, b, c, d, e, ...], and you take one out randomly, then need to put it back in
you don't have to remember _where_ you put it, you can just make a space and jam it in there, and it is guaranteed to fit.

[justinzamora]

Thank you for this great video. I think the idea of object-action duality exposes something fundamental that is missing in elementary math education. As some of your early examples show, we commonly switch between object-object interaction and object-action interaction without ever acknowledging the distinction. Sometimes this leads to confusion for students, especially those without a good sense of math intuition.

My daughter struggles with math (she almost certainly has dyscalculia, a math version of dyslexia), and, after watching this video, it seems clear that part of her struggles stem from the way we casually interchange object-object and object-action models. A good example is negative numbers: we treat -1 as both an object and an action without ever acknowledging that is what we are doing. This actually became a stumbling block for my daughter in an Algebra 1 problem. She had to simplify something like 5 - (x + 3). Distributing the subtraction, turns the action of subtraction, into negative objects. She had trouble with the this (partly because of the implicit 1 as a coefficient), and I was at a loss to explain it to her, and I now I see that it is exactly because of the action-object switch. I'm still not sure how to explain object-action in a way that would make sense to young students, but it seems important.

Another great example is fractions. Is a fraction a number or an operation? We use it as both interchangeably and almost never acknowledge the shift. I wonder if this is part of the reason so many kids have trouble with fractions. I would be very interested to read a math educator's/researchers view on this.

[bcthoburn]

Formalization: every semigroup (i.e. set with an associative, binary operation) is isomorphic to some set of functions under the operation of function composition.

[Vova__]

I find it to be more a matter of semantics rather than anything else. The "and then" is pretty much precisely how one would understand what the order of operations is, thus it seems wrong to claim, that the description of associativity as the ability to change the order of the operations is faulty. I guess, sure, if this helps someone to understand the concept, which seems to be the case scrolling through the comments, it certainly is useful, but I would not claim it is anything more than either a more in-depth explaination or a slightly alternative approach to look at associativity.

[conoroneill]

This is probably the clearest explanation of an idea from category that I've seen. Well done.

[EricKolotyluk]

Normally I like to praise people for their stories, but as you finally noted, you shared your journey with me, where a story is merely a memory of a journey. Well done!

As a computer programmer with over 50 years of experience, this was a refreshing journey that helps me better appreciate my own journey. While learning functional programming back in the 70s, it was not until 2005 and later with Scala did I start practicing functional programming. In particular, I have seen dozens of explanations of Monoid and Monad, and they rarely helped, they usually hurt. This journey has really helped me understand monoids and monads better... thank you. I hope to see some of your presentations on Functional Programming, in particular grounding like this.

[maxkolbl]

At 20:11 when the dark background becomes white is probably the most epic thing I've ever seen! No better allegory for enlightenment than literal enlightening.
But apart from that, this video is a bloody masterpiece! The first time I really started thinking about what it actually means was when I learned that the operations of a Rubik's cube have the structure of a group and I wanted to explain to someone why the associative operation actually holds, but I've never thought about it that deeply before

[kodirovsshik]

This is one of the moments when I remember what I love math for. Thank you for this amazing video!

[austin]

Excellent presentation and unique interpretation. Just one suggestion : Counter examples like Sedenions and the interpretation of this difference would have been appreciated.

[madhadamard]

Nice work. You explain it so much better than any YouTube video I have ever seen. It’s also explained so dryly by most teachers as I think even they don’t have an intuitive grasp of what the associative property really means. Keep it up man. 👍

[captainsnake]

This was an amazing video, however, I think there was one change you should have considered making. As you mentioned in the description, this video was essentially an intuitive proof of Cayley’s theorem for semigroups, a theorem I was not familiar with before watching this video. After googling it, I found the formal statement of the theorem made the video make a lot more sense to me. Maybe it would have been a good idea to throw in this formal statement of the theorem at the end?

I hope you don’t see this as me saying “this video needs more rigor! Get rid of the intuition.” Because personally, I don’t think I would have understood how meaningful cayley’s theorem for semigroups was without the intuitive arguments given in the video. I just think maybe 1 minute at the end of the video that explained how all of this translated to rigorous mathematics would have been nice.

All in all, this is definitely one of my favorite SoME videos. Will def be subscribing (and checking out that poll you mentioned at the end of the video.)

[hhbxxeo]

Associativity is also a coherence requirement.
Its meaning is that the result of calculation doesn't depend on the path taken for calculating. Or better said all paths are equivalent.

As an example of a non associative process, take the transformation of velocity through frames in special relativity. With the same starting frame and velocity and the same ending frame, you would get different results depending on the intermediate frame used.