Summer of Math Exposition

I like the storyline, as well as the smaller images. The artificial reader becomes distracting afte a while and takes up quite a bit of space. The background music gets a bit loud too... The various ways different number concepts float about is very nice and keeps one considering alternatives. Your idea of ending the video at some point made sense - the raminders seems like something you might like to workt out in more detail in a separate video.

The video has a unique style that captures the attention of the viewer. The introduction explains well what the problem that the author is trying to solve is. I especially liked the distinction between construction and definition, the example of houses, and the comment on having multiple possible constructions that satisfy the same properties. However I find that this video is lacking in many aspects: - the introduction seems too long, and at times the viewer gets the impression that the author is going around in circles without ever getting to the point. It is not even clear where the introduction ends and the main content of the video begins. - In certain points it seems as though the author is going to start talking about a certain topic, but then ends up not talking about that. For example the Church encoding is introduced and placed on the screen, the author says something about it, but then 'dismisses' it. - It is not made clear that the Peano axioms for natural numbers are proposed in the video as a candidate solution of the problem. It should have first been stated that what we are looking for is properties that characterise what we call 'numbers'. Then the Peano axioms should have been presented as the solution given by Peano to this problem, in which Peano specifically states what properties characterise the counting numbers according to him. Perhaps it should have also been commented that the numbers constructed in the video by "AnotherRoof" are a way of constructing a Peano system using only the axioms of set theory (this would also give an example of the distinction between definition and construction) - In my opinion the Peano axioms are poorly presented: the author states the axioms too quickly and doesn't explain them enough, and, most importantly, doesn't take enough time to explain why each axiom is needed, and what the purpose of each axiom is. The problem of having loops in the number system without certain axioms is not really explained, and is only commented on quickly; the diagrams on screen are really helpful, but the author should acknowledge them and explain what they represent, and use them to explain which behaviour each axiom prevents. The most important axiom of all (the induction axiom) is given way way to speedily. Actually I do not understand what the author is doing in this portion of the video: the author says "this is one of the things I'm going to have to lie about. For every property [...]" with no pause between the two sentences. And what does the author lie about? Does the author want the viewer to understand the idea of the induction axiom without having to talk about the technicalities (logic, axiom schema, first order vs second order...)? Or does the author want the viewer to know there is another axiom, but too complicated to talk about in the video? In either case I think the author fails in their presentation here: the axiom gets thrown at the viewer quickly and with no explanations, leaving more doubt and questions, and hindering comprehension. (the induction axiom should have either been talked about in a way to make it understandable (with a comment about the technicalities of logic), or should have been omitted entirely, with just a comment about its existence). - The use of the "square" and "triangle" as variables can be confusing, especially because "S square" sounds like "S squared" and just creates confusion. (Saying "S of square" would already have been better). - The various proposals of definitions of numbers are weak and do not give a satisfying answer. For example the author talks about commutativity of multiplication: this is done in a not well-thought out way, and the other properties of operations are not mentioned (for instance, associativity seems more important). Order relations are only mentioned in the comparison table, but deserve their own section of the video. - Talking about transfinite ordinals as extension of the counting numbers but not talking about transfinite cardinals was, in my opinion, not a good thing to do: the cardinals are probably a better extension for what we want to do. - At the end of the video I feel like the video never got to an actual point, and no satisfying definition of number was given, but also the problem of this difficulty in giving a definition was not talked about enough. I feel like the video keeps giving anticipation that we are about to give some solid candidate definitions of what to consider a number, but this never happens. There is a comparison of properties of different types of numbers, but the thing is just put there, and the viewer is left with not much added value. (At 15:39 the author also acknowledges that up to this point they are as ignorant about a definition of numbers as at the beginning of the video. (And the remaining couple of minutes of the video do not change this)). In any case I think the author is great at presenting maths topics, and can easily improve on these points.