Thank you for watching! I recently hit 10K subscribers and planning a Q&A video. Head over to the Another Roof subreddit to ask your questions. If I get enough questions, I'll make the video -- should be a fun, less scripted one. www.reddit.com/r/anotherroof/comments/wj8hhn/10k_subscriber_qa/ If common questions arise related to this video, I'll respond here!
After all we built the natural numbers starting from 0 and so starting induction at 1 would mean that we would have to prove all those properties separately for 0. Also starting induction at 0 makes the computations less messy
I'm having trouble understanding how once could construct commutativity from 0. Specifically I'm stuck at showing how the general from a+(k+1)=(k+1)+a without having to specifically prove the case for 1. Am I stupid and you have to prove that 0 and 1 commute with every number?
@@ratatouille5172 It seams to me, yes, we do need to prove a+1=1+a for all a but we can still start the induction on a from zero and ,yes, we will need to prove a+0=0+a separately
I love how sequential these are. At first we learned numbers, then we learned counting, and then we learn arithmetic. Can't wait to do insanely difficult integrals on empty sets!
Eventually you "forget" the specific constructions of natural numbers, integers, rationals, reals, etc. And just work with their properties (like we do in school). So "underneath" the integral notation you could think of countless nested empty sets, but of course that would be super impractical.
Doing any integral on an empty set is easy, because Integrals are measures and one of the axioms of a measure m is that m({})=0. Without that axiom, you'd get inconsistent measures, because the measure of a countable union of disjunct sets is the sum of their individual measures. So you could just note that for an arbitrary number n, m({})=n*m({}) and that is unique if and only if m({})=0.
While probably making it harder for new viewers, I like how the videos don't stand on their own, but slowly evolve into a mathematics cinematic universe
Just wait for my video set in the mathematical cinematic universe on the dispute over the validity of Cantor's Diagonal Argument, or as I call it, the Infinity War.
@@AnotherRoof If you ever you do a game theory set please use chess examples with video titles such as "The Two Towers", "The Return of The King" and I'm sure there's a maths problem called "The Fellowship of the Ring" - A fellow maths teacher.
@@julianbushelli1331 It was a joke. However it is true that, at the time, Cantor's proofs and results were considered incorrect and laughable by many of his contemporaries. He became depressed as a result of the hostility and by the time he published his diagonal argument (which most mathematicians seemed to realise was an exceptional insight) his reputation was still shaky with people like Kronecker. Giving a very brief overview here but it's worth reading about!
Before watching this series, I had known that numbers could be represented as sets, but I'd never really understood how. I'm super excited to find out how things like negatives and fractions are encoded!
Negatives are super easy with Church encodings. A negative is represented as a partial application of subtraction on a positive. So a negative integer is a function with one free variable. In this system not only is a number a function, but a negative can be both a number and a function that subtracts whatever you feed it.
There are a lot of parallels here with array indexing in computer science. An array of size 5 would be indexed using the integers 0 through 4. In a way, we're doing the same thing here, but we're referring to each array by its size and assuming a standard set contained within.
I'm loving this series. It's way better taught than any of my university maths lectures were. I would like to suggest that you put this series in a playlist, though.
Can't wait to see the video. Really love you way of presenting information. This channel, though small, is already a part of mathematical RU-vid for me, alongside Numberphile and Mathologer.
44:14 "I hate to end on a NEGATIVE, but that will have to wait until next time. Don't give me that look. Be RATIONAL about it. I wanna keep things REAL in this video and not make it too COMPLEX." Oh man, I love that sentence containing all the sets of numbers. N, Z, Q, R, C
When you were showing your thought process for induction, you wrote down the goal you were aiming for (at 29:09). I think it might be really useful if you did that every time you used induction, since knowing what you're aiming for makes it a lot easier to follow how the last step actually proves the thing you're trying to prove.
It would be nice if as you went on with this series, on the closing cards, we have a picture of all the bricks, ideally (if possible) laid out in such a way that one brick is positioned above other bricks if it is derived from those bricks (if it is a theorem), whereby all the axioms would be the literal foundation of the brick wall, that is, on the very bottom. Kind of funny that I am suggesting a channel named Another Roof lay literal bricks to build a wall. I just realized that there would probably be vastly more theorems than axioms, so I'm not sure how such a wall would hold physically, under the influence of a gravitational force. Answer: the mathematical force of logical proof.
I love the purity of the recursive definition of addition. From now on, in my code I will be implementing addition of two numbers as a loop finding the successors of successors.
I genuinely love watching these videos. You caught my attention with "what is a number." I always liked math because I understood how to do what the teacher told me to do. But watching the most basic principles be broken down in ways I didn't know existed is honestly fascinating.
You're the solution for the problem, that we've never had time in school for the ground of mathematics. The "What are we even doing here". Love it. Thanks for not leaving out any detail. 💚💚
as a kid i always thought of how could civilization advance from nothing to computers, and how could computers advance from 0 and 1 to all of the complicated computation My answer to that question was just like this videos, you start defining little "bricks" of proved truth and use it to define a new "brick"; you could spend so much time proving a brick, but once you prove it, you can now freely use it to build very complicated stuff in these videos you illustrate very well this concept and i just wanted to say thank you for making this so well and rigorous
This is actually a lesson in number theory, but more fun. My teacher made a really good work explaining this, but I like how you did it. Thanks for the video.
I find interesting that in terms of elements of a set the order doesn't matter and we need to use some other sets to encode a situation in which order matters, but in the case of addition and multiplication, we go on our way to prove we can change the order, instead of assuming it just can. It does makes sense overall
What really brings it home is when you start trying to mess around with subtraction the same way: (5-3)-2 is different both from (3-5)-2 and from 5-(3-2). Or there are systems where the equivalent of multiplication in that system doesn't commute - where the order matters.
@@rmsgrey well that wouldn't be an equivalent anymore, would it? I think we already have this in the form of function composition. f(x) = 3x+1 g(x) = 2x+2 f(g(x)) = f(2x+2) = 3(2x+2)+1 = 6x+6+1 = 6x+7 g(f(x)) = g(3x+1) = 2(3x+1)+2 = 6x+2+2 = 6x+4 6x+7 ≠ 6x+4 f(g(x)) ≠ g(f(x))
Yeah, function composition is an example of a thing that resembles multiplication, but doesn't commute. Other common examples include the vector cross product (which anti-commutes: ab=-ba and isn't even associative) and matrix multiplication.
Mate, these are incredible. You must be putting so much effort into creating this series that it boggles my mind. I did physics in uni first (highly recommended if your favourite sensations in life involve headaches), but moving to IT after a long break. We didn't really learn why and how the numbers worked in the three semesters I did, we just worked with them. I love catching up on the things I missed to get back into thinking logically to start into a new attempt at a degree.
we've spent so much time, over 2 hours (summing the time from the previous videos in this set) to learn how to add 1 + 1, im crying here, on to the next one
Looking forward to this. The presentation of counting in Godel Escher Bach was my first introduction into this really primitive type of math, but I'm liking your videos better.
I like that you actually construct math rather than introducing everything axiomatically. But let’s face it: the reals as the equivalence classes of rational Cauchy sequences or Dedekind cuts will be a mess a few episodes in the future, and proving the properties you need to do anything interesting will be … long
@@AnotherRoof As someone who took real analysis and could never really wrap my head around either of these constructions, this is something I would love to see! The clarity of your explanations and the way you build up all definitions from intuition is just fantastic. No doubt you'll be able to make a great video on the topic.
Yeah, the jump from rationals to reals is a pretty big one - you're suddenly introducing uncountably many new numbers, most of which no-one will ever hear of... And no-one ever bothers with the algebraics...
@@rmsgrey The problem is that when you are working with real numbers or the algebraic numbers, it becomes impractical and conceptually useless to work with explicit constructions and encodings. To properly give a formal introduction to the algebraic numbers, you need ring theory, and to properly give a formal introduction to the real numbers, you need lattice theory on top of ring theory. Set-theoretic constructions are not appropriate when dealing with these higher-level mathematical objects. For instance, axiomatically, it is very easy to write a list of simple axioms that uniquely define what the real numbers are. Talking about the algebraic numbers is even easier: the field of algebraic numbers is the algebraic closure of the field of rational numbers. However, while it is very easy to understand the axioms, actually constructing these objects using nothing but sets is complicated, and to be honest, a waste of time. That is not to say that it cannot be done, but rather, that it should not be done.
thank you so much for this series! it reminds me a lot of "Good Math" by Mark C Chu-Carroll, but the visuals make it easier to understand. This is a great series for adults wanting to understand fundamentals! There are very few channels where I don't mind watching 45min videos (15min is usually my max), but I find myself engrossed and looking forward to the time well spent!
These videos can branch into WAY more abstract areas of mathematics like Group Theory. Group theory starts easy enough and only uses a small number of the blocks we have so far.
My PhD was in group theory so I'm all about it -- I want to wait until I've made a lot of videos before I venture into this topic because I want much more experience in order to ensure I do justice to the topic I love the most!
I have a graduate degree in "math", so I completely follow this and it's utterly familiar. I'm curious if anyone with no more than high school algebra is following this with understanding. I think it's amazing that Alex is taking on such a formal and structured approach to "This is maths." But there's also a reason it hasn't really been done (this well) on RU-vid to date. Hoping this breaks through.
I just graduated high school and am following nicely, but I have taken more than algebra. I have taken calculus and statistics, and I have explored many math RU-vid videos.
I know you've said you're curious to see what will you do on this channel later, but I'm really enjoying this structured series of fundamental concepts and definitions and I wish it keeps going. I'm hooked for the negatives now.
Thanks for your comment :) My plan is for video 4 to be the fourth (and last, for now) in this series of building mathematics from the ground up. From there, I have many ideas, but as a purist you can be sure I'll use a similar approach to whatever subject I tackle.
I've always viewed adding as an extension of counting. if 8 and 17 represent groups of items, then whatever path you walk to count the total of both groups added together in terms of ordering, will always lead to the same result. Multiplication is a series of self-additions, so this video makes a ton of sense to me.
Gotta love those paper dolls :D Even all theree video topics are somewhat familiar, I really enjoy these videos. They are very nicely thought trough and conducted. Very clear road forward. Thanks and keep posting.
Excellent tutorial! I would point out that the "Axiom of Induction", which plays a crucial role in these proofs, is, I recall, not allowed in "First Order Logic"; thus these proofs make use of "Second Order Logic". This is significant because "First Order Logic" is well understood and has very nice properties. "Second Order Logic" is a bit "wilder", and Logicians don't have as clear an understanding of higher order Logic. Most practical mathematics requires "Second Order Logic" for stating and proving theorems. There may be ways of proving these basic Arithmetic theorems using only "First Order Logic", without the "Axiom of Induction", but this might require the introduction of additional axioms. Actually, there is the famous "Skolem Lowenheim Paradox" that shows that "First Order Logic" can not uniquely describe infinite number systems like the integers.
This leaves so many questions, like how do you define adding irrational numbers? What is their set definition, and how does their successor work? Really hyped for next video :D
spoiler alert: all the different types of numbers are actually different. The above-mentioned definitions of addition and multiplication only work for natural numbers. To get a new number system, you'll have to define the new number system (usually in terms of different number system), define new arithmetic, and prove all the properties all over again. Actually defining what "real numbers" are is a very controversial topic. They require defining results of infinite chains of operations. Which, according to some, constitutes a proof by contradiction that they don't exist. They certainly don't exist, as far as computation and arithmetic is concerned.
if he can explain such easy looking concepts in such an intuitive and comprehensive way, I can inly imagine what will his Real numbers and Complex numbers video will gonna be! never stop uploading
I've been doing some looking into category theory, and think it's neat that you had to define the identity functions of addition and multiplication so it has certain properties. (It's A+0=A and A*1=A on the bricks. Basically you need some way to do "nothing" with a function or it doesn't necessarily have certain useful properties)
To anyone trying to do the proof of associativity and commutativity of multiplication, it’s best to start with commutativity! Start with associativity and you’ll run into the problem of needing left distributivity.
Infinity is greater than any number you could arbitrarily choose to compare to it. So in this way, it's more of a function than a number. A function that gives you the successor of its input is one example of infinity. So asking "how big is infinity?" is like asking "how big is addition?" The questions don't apply to non-numbers. Similarly, asking which of two infinities is larger is like asking "Is multiplication bigger than addition?" The answer is undefined until you choose inputs to those functions and compare the results. Luckily, any application of the concept of infinity only needs to use a finite amount of precision to be a useful tool. For example, we only know or use so many digits of pi. Yet pi is still extremely useful.
Wait, you're just tricking me into taking Real Analysis again with this series of videos! I just know this'll eventually land on sequences and lebesgue integration
Ok, I think I figured out the homework he gave us. If you’re good at math please grade me: If a x k = k x a Then a x (k + 1) a x S(k) Addition def (a x k) + a Multiplication def (k x a) + a Given (k x a) + (1 x a) Multiplication def (k + 1) x a Right Distributivity And with that proof (hopefully) under my belt: If (a x b) x k = a x (b x k) Then (a x b) x (k + 1) (a x b) x S(k) Addition def ((a x b) x k) + (a x b) Multiplication def (a x (b x k)) + (a x b) Given ((b x k) x a) + (b x a) Commutativity a x ((b x k) + b) Right Distributivity a x (b x S(k)) Multiplication def a x (b x (k + 1)) Addition def P.S. The IPhone keyboard wasn’t designed for this.
I mean, this is really wonderful stuff. I'm so glad the almighty algorithm chose to show me the beginning of this series a month ago. I'd love to eventually see you do the Banach-Tarski Paradox. Or Gödel's incompleteness theorems! Or Cantor's diagonal argument!! Even though a lot of this stuff is familiar, it's like hearing an old familiar tale told by a master storyteller - you realize there's a lot more to the story than you first thought.
Interestingly, while the diagonalizaion is relatively simple, incompleteness is quite a bit more advanced - requiring an introduction to mathematical logic and proof systems (and very technical). And compared to the other two Banach-Tarski is extremely advanced (I think it requires at the very least a lengthy introduction about groups and a lengthy introduction about geometry, which in itself contains multiple topics to cover) so presenting even a sketch of the proof is probably unfeasible (though I would be happy to be proven wrong :))
Thanks for your comments! As I say at the end of the video, I only want to cover topics that I don't feel have been covered in the RU-vid maths community, or, topics in which I think I can add a new perspective. Ori is right about Banach-Tarski, I think vsauce's video is excellent in terms of balancing depth and accessibility. I do have some things I'd love to talk about, quite advanced stuff which would require a multi-video approach, but we'll see if my channel grows big enough that I can justify doing those!
@@AnotherRoof Yeah this is why I should watch the whole video before commenting... I had to pause and go to sleep but I had to share my excitement. Anyway, wherever you go next is great with me! I'm just excited for whatever that happens to be.
End on a negative… Be rational… Keep it real… Don’t get too complex… I think I know where you’re going with this next time! Great video again; the explanations and humor are top notch!
Aw, I could have sworn I was already subscribed to your channel months ago... Sad that I missed this 5 months ago but happy to be seeing it now! I love the feeling of gradually grokking more and more as the video continues! Wonderful work!
"...this is something we can prove using our concrete deffenition of addition..." he says while pointing to a concrete brick with the definition of addition written on it XD
Little late to influence the algorithm, but commenting anyway. Love your stuff, man! Keep up the great work! p.s. Absolutely love the use of stone blocks for axioms/important theorems. They're a great practical reminder as you explain things and they're metaphorically evocative. Just something very simple and genius that work perfectly for your videos.
Would love to see your response to Nigel Cheese proving 1+1=1…most just make fun of him, but would be interesting to see precisely where he goes wrong in definitions. He uses magnets but could work with any combinable object like a water droplet, a breeze, etc. Love your stuff!
I really think it would be amazing if this series continued. I was very much looking forward to hopefully covering basic division defined in such a low level way. And then showing how more advanced operations are just implementations of core 4 (+-*/)
I love how you boiled down a book the length of 300 pages into 3 interesting (albeit long) videos. I wish you'd keep this series going even beyond 4 eps.
14:50 I was thinking we could just do n+(k+1) = n + S(k)? And then we'd do m = k+1 so n+m = n + S(m-1). But then I realized that's really just using the precessor function again, in the form of subtraction, and the whole point was to avoid that, since we haven't rigorously defined/constructed it yet. These videos are making me really think about just how mathematical/logical structure underlies even the most basic math that we teach kids, let alone the more advanced math used in science, engineering, and other STEM fields. As an engineer and data science student (electrical engineering major, data science minor) who uses advanced math regularly in my coursework, this is making me really appreciate all the work mathematicians have done throughout history to build up all this structure that the rest of us can stand on without even thinking about it most of the time. It also amazes me that our _brains/minds_ are able to abstract away all this structure, and to recognize the patterns and properties we want certain objects to have so we can build up the structure in the first place. We do really do "stand on the shoulders of giants"! On a related note, I took a course in digital logic design a couple years ago and we did a similar process of starting from a few simple mathematical objects (the logic functions/operators AND, OR, and NOT, which themselves can be constructed using set theory axioms and the rules of logic) and we built up the complex mathematical structure used to design and model logic circuits, which are themselves the basis of computer architecture. To be able to build such a _versatile_ system out of a mere three simple objects (well, OK, technically 5, [or 4, depending on how we count] since we also need either {1,0} or {true, false} as the domain and range of the logic functions) just amazed me and made me really appreciate the process of abstraction. The saying "don't miss the forest for the trees" kept coming to mind as we kept going up levels of abstraction, ignoring the details of lower level objects we'd previously previously explicitly constructed, to focus on the bigger picture details of the higher level of abstraction.
Hey, thanks for your videos. Coincidently I am in exam this coming Sunday on relations and functions and yoga videos have helped me understand 80% of the chapter in less than 10% of the time. And I understood it very well as compare to the class. I will be eagerly waiting for your next video to come out. Wish me the best of luck for the exam. :-)
You can continue to apply the same concept - that turned succession into addition and addition into multiplication - To define higher order versions (i.e. knuth up-arrow notation), exponentiation, tetration, etc. You should make a video on that
