The 360-Page Proof That 1+1=2 

This is what happens when the child keeps asking 'why' and the parent only breaks the discussion at 'because existence is assumed to be possible'

I can't believe you left out the best part! Accompanying the proof is the statement that 'the above [i.e. 1+1=2] is occasionally useful'

I have a degree in Mathematics. When he showed that first snippet of the proof I questioned my whole existence before he pointed out half of it was just old fashioned theorem references.

In my second-year real analysis class, we used "1 + 1" as our definition of 2. "Define 1" and "Define +" were two of those "laugh politely and stop talking to you forever" questions. It looks like the authors of this book maybe had "Define 1" and "Define 2" among their "laugh politely and stop talking to you forever" questions, and a very long-winded answer for "Define +".

As far as I know 360 pages is where they got the basics needed to prove 1+1=2. The full rigorous proof itself took more than 300 pages on top of that

Can't believe you didn't mention the fact that right after this proof, the authors write "The above proposition is occasionally useful"

As a mathematician, I have *never* liked proofs that used symbols like this. Some symbols *greatly* simplify things, but there's a certain line between making things easier to work with, and getting a headache trying to remember the heiroglyphics. Projects like this crossed that line a *long* time ago!

Friend: What's 1 + 1? Me: 2 Friend: No, it's 11! Me: *Pulls out Prinicipia Mathematica*

As someone currently studying maths and physics, I think this video does a pretty good job by showing how complicated mathematical proofs can be. I hated them for my entire first semester because proofing theorems is not something you can learn in a day. It is a long time learning process and I am hoping to improve over time.

As the saying goes, "to make an apple pie, one must first create the universe" - the universe here being the basic tenets of mathematics that had to be rigorously, logically defined before even being able to parse the concept of addition

Maybe I’m biased given my math degree but the proof description here is much more satisfying than the “this is so simple lol” jokes. In math, we can prove so much with so little. Most people accept 1+1=2 as a concept without much question but for those who question it, it can be proven. Most other fields can’t prove their widely accepted core concepts like this and most who can are based in math.

"If two things exist, then one of them exists, and the other one exists." This is the single thing that kept me from my PhD in Mathematics. It's called the Axiom of Choice, or as I called it, "Duh."

In Bertrand Russell's biography he is described in his later years recounting a nightmare he once had: "Russell was in the top floor of the University Library, about A.D. 2100. A library assistant was going round the shelves carrying an enormous bucket, taking down books, glancing at them, restoring them to the shelves or dumping them into the bucket. At last he came to three large volumes which Russell could recognize as the last surviving copy of Principia Mathematica. He took down one of the volumes, turned over a few pages, seemed puzzled for a moment by the curious symbolism, closed the volume, balanced it in his hand and hesitated…."

I remember my math teacher (i was about 13-16 at the time) telling the class about writing an essay that 1+1=2. I never believed that people would go ridiculous extents for such a simple problem. I guess I was wrong.

What the teacher expects you to do when they say "Show your solution"

Ah, this brings me back to takeing a crash course in logic a few years back. Loved it, understood nothing :)

Taking 300+ pages to prove 1+1=2, with lines like "if two things exist, they each exist" just sounds like the greatest work of procrastination in human history. And you know what? I respect it.

4:17 that's actually funny 😂

Wasn't expecting this from this channel, but you actually did a really good job of explaining this proof! Probably the most accessible explanation out there for this one page.

Principia Mathematica was very useful, even if it relies on principles which cannot be proven (axioms). It is basically the foundation of modern mathematic. Then Gödel came along and showed it was fine if you relied on principles which couldn't be proven

I remember in my advanced Mathematics class back in college, the professor said he made a joke in another class, and as extra credit on an exam, he put what is 1 + 1. The students were caught off guard (since they've been studying really advanced math), that they got confused and weren't sure how to solve it. One even tried to write a proof why 1 + 1 isn't one, thinking it was a trick. 🤣

Dude, I read this super old book on discrete mathematics and then tried to use it in class to prove something and no one knew what I was talking about. Took a second to realize the symbols were antiquated.

So, mathematician here. I was actually going in to this expecting I'd feel compelled to write a long comment explaining in detail everything Sam got wrong. But this is actually very good. I do have one specific quibble: The system in Principia Mathematica does in fact do what it sets out to do in the sense of making a system which can work as a general foundation. The part about any system having "holes" is roughly true, and refers to Godel's incompleteness theorem, which says (roughly speaking) that any sufficiently powerful axiomatic system must either be inconsistent (that is, it contradicts itself) or must be incomplete in the sense that there are statements in the system which can't be proven or disproven within the system itself. So the system of PM is incomplete, but it is usable as a foundation. Modern math doesn't use PM as a foundation, not because it has "holes" but primarily because it has some additional philosophical baggage and because we have a system, ZFC en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory , which for most purposes works pretty well as a foundation, is more intuitive, and is not nearly as complicated for many purposes. (There's some issues here which I'm shoving under the rug here involving what are called "large cardinals" where you sometimes throw in another axiom that says that some very mindbogglingly large set exists.)

When I was in High School, I saw a proof that 1+1=3. It depended on an implied division by zero. That is, by a term, which would evaluate to zero.

0:36 did I just got no bitched by a math video?

In elementary school, I always thought to myself “I wonder if there’s a page-long proof that 1+1=2” I’m happy to report to my younger self that I got my wish 360 times over

I think it's so fascinating that it doesn't matter what words we use for numbers, and yet whatever words we end up choosing, someone can write a mathematical statement that proves the consistency of whatever value "word" we choose to define a given value by, and it's relation to other numbers. In this case 1 and 2. It doesn't matter that they are called 1 and 2, what matters is that they are different in value, and that there is a specific difference in those values. In other words 1 and 2 are different and they differ by 1. And all that need be done is for any mathematical statement using 1 or 2 or any number, the usage of those values must remain consistent among all statements and their relationship to other values. And it's just so fascinating that we can prove that those values are consistent aside from obviously using the same word to talk about the same number consistently. It's almost like synonyms in language. Some words are spelt different but mean the same thing. This is like a proof that proves there is no synonym for the number 1 or 2 or any other number. They are each individual distinct values with no synonyms. 1 is 1, it is consistently that value, and there is no other value that is "kinda" like 1.

This feels like making a computing system from scratch. But even more abstract

Think of it this way, rather than them trying to say "this is why 1+1=2" they've essentially used new and existing "math tools" that are not only is extremely useful in their own right, but they've shown they can use them to define 1+1=2 , and so anything that is derived from that can also be defined in terms of their systems. Which is basically all of mathematics.

Objection: *54.43 is not a proof that 1+1=2. It's a proof that two sets which both have cardinality 1 are disjoint if and only if their union has cardinality 2. That 1+1=2 is an easy consequence of this once you've defined what "+" means, but it takes them another 300 pages for them to do that, finally proving that 1+1=2 at *110.643 (after which they remark that "the above proposition is occasionally useful").

After going through engineering, I've just resigned myself to the camp of "if it works, I don't care about why" for math stuff

I love how with every new HAI video sam's humor gets even better 😂

This is what happens when you have that kid that keeps saying "Why?" nonstop, and someone decided to write a whole book to shut him up long enough for the kid to grow up and get a PhD in philosophy.

I like this because it shows what mathematicians actually do. I feel like most people don't know. We try to prove things! Generally more interesting statements than what 1+1 is!

teacher: why didnt you use my strategy? her strategy:

1:22, IT'S SPELLED "than" cause "then" refers to afterwards or like "back then". And "than" means "I should probably be doing my homework rather THAN watching these videos"

this is a certified hood classic

god, I love this so much. It's like if 2 guys were bantering and pooped out a method to describe blue to a blind person.

Makes me think of my discrete math course. Once my prof asked why x is less than x + 1 after we used it to explain a problem and we all just stared at him blankly.

I love details like this, because when I look at these there seems to be so many assumptions baked and they even have names for my distinctions

One could literally write any equation & a mathematician will work the hell out of him to prove it

2 + 2 is four, minus one that's three, quick maths But 1 + 1 is 2 is long maths

When I was teaching myself math (didnt care in hs and went into liberal arts anyways so even logic, let alone math, wasnt always needed lol), I started with Serge Lang's books on basics then abstract algebra then this book. Reading it was wild. Learning the notation used was almost more effort than the actual book because the notation can differ widely from modern logic/set notation. It was however a book I loved reading through because it bent, melted, and reshaped my brain in a lot of great ways for understanding proofs, not considering arbitrary things useless, and manipulation. A lot of other things too, but there is even more to it than just the one volume, but the first was great.

Actually, if we consider more recent ZFC model as a fundation of mathematics, we can induce the Peano axioms in a few pages, thus we DEFINE 2 as S(1), the successor of one 1 (and 1 as S(0), the successor of 0, 0 being an element defined by the axioms). Then we define the addition as such -n+0=n, for all integer n (prop *1) -n+S(m)=S(n+m), for all integers n and m (prop *2) With this definition, we have : 1+1=1+S(0), by definition of 1 1+S(0)=S(1+0), by prop *2 S(1+0)=S(1) , by prop *1 S(1)=2 by definition of 2 And there you go, by transitivity of the equality, 1+1=2

0:03 It's actually not obvious that A comes before B, because the order of the alphabet is rather arbitrary. It's probably based on some mnemonic in a language that nobody speaks anymore.

You know, as an engineer that had a lot of calculus and algebra and geometry, I can tell you that 1+1 is not always 2. Sometimes, it's 0.

I heard of this proof years ago and it was just a fun fact to me. Now I'm studying mathematics with philosophy as a side-subject, I accidentally took this book from the university library, actually being curious about several things in there, and I feel like I might actually read and understand this one day.

I study Pure Maths and when I started reading Principia Mathematica, I was like so amazed by how Russell was executing this demonstration, I remember that once an algebra teacher said “we all as mathematicians aspire to have at least one demonstration such like this”

"if you have a PhD in mathematics, you probably have better things to be doing than watching this video" I mean, that's true, but I'm still here aren't I? (Foundations-of-math and type theory stuff makes my head hurt though. My degree is in algebraic combinatorics.) I wouldn't say, btw, that Godel makes the Principia obsolete. Just because no system can prove its own consistency doesn't mean that having a very solid and rigorous foundation is a bad thing. (even if most working mathematicians just use ZFC)

Teacher: "Don't forget to show your work" Student: *hands over the above book* Teacher: *instant regret*

So basicly the proof is as simple as: "If you have an apple and i give you one other apple, you have two apples. Hence 1+1=2"

Don't overestimate what math PhD people are. They'll waste time watching these on the toilet just like me.

I read the title too quick and thought it would be about the mathematical “proof” that Terrance Howard (the actor that played Rhody in iron man 1 then got replaced) wrote because he thinks 1 x 1 equals 2

Bertrand Russell was such a bro!! Tons of philosophers are pompous assholes but he has so many great quotes about being a good person, and about how you should never be too assured of something and always be willing to second guess when you have new information. Absolutely humble guy and smart as hell too.

As someone who is currently taking a class that is all about proving stuff like this, this video pretty much sums this up.

1:50 "it didn't actually work, it turned out it's actually impossible to do that" - The system of Principia is perfectly fine. Also, it wasn't based on Logic alone, it was a version of Type Theory. The guy who wanted to base all math on Logic alone was Gottlob Frege and, yes, he failed and quite spectacularly. But yea, as you say, the fact that _complete_ systems in the same vein as the Principia could not exist turned out to be the case (by Gœdel's incompleteness theorems).

There's a pretty good reason for this actually. People have generally just accepted the notion that "everyone agrees on the basic assumptions of reality". Nowadays however, that notion is no longer valid. If it were, then proof of something would prove it, but think of how many things there are that people believe despite there being proof to the contrary, just because you can't show proof of the negative.

I always thought proofs were the hardest in math, arithmetic, algebra, calculus, way easier. I can't recall how to do an easy proof like proving the sum of two odd numbers is an even number.

0:38 WHY I WAS LITTERALLY WATCHING THIS VIDEO TO DISTRACT MYSELF FROM THAT

What the teacher expects when she says show your work:

somebody took "show your working" a bit too seriously

4:29 lol

0:33 He really said "why aren't you getting bitches?" 💀

so many high quality stock videos I can't believe it.

Now we need a proof that 1+2=3

It's amazing that when he goes really philosophical around 3:40, he just comes back saying "It's something like that"

I'm an idiot talking to an idiot that makes us 2 idiots

3:32 The incomprehensibility of "absence of light" is actually called Olbers' Paradox. This is a super important question in the cosmology and one of the key observations that led to the big bang theory.

2:11 "ow oof my normal brain hurts!"

During my Ph. D studies, I took advanced math. My Canadian class mate and I tried to prove that 1 plus one was equal to 2. We brainstormed to solve the equation for almost a week. One evening, I resolved it and I started jumping up and down yelling Eureka, Eureka.. My Canadien friend gently reminded me to put my pants on before I rushed into the street. Your video reminded me of my graduate studies. 🤣

Grab a potato with you left hand and put it in an empty balcony. Now grab another potato with your right hand and put it in the same balcony. Now count how many potatos are in the balcony

POV: when the statement is not left as an excercise to the reader

Why is there stock footage of "women in suits crossing their arms underwater"?

Mitochondria is the power house of the cell.

Btw. When Principia Mathematica published, the set theory was in a big controversy.

These days, it takes far less pages to prove this statement whether you're using peano arithmetic or something like ZFC

shared this to my maths teacher, suddenly my grade changed from A to F, can someone tell why?

"I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain"

the ability to withstand high stimulation without confusion, yet also the low stimulation levels to reason profoundly is essential to getting out of and into boxes at will. if you've heard him talk, you know he was a whip of a wit; yet he could steady his mind enough to do THAT. I hope for the chance to go bungee-jumping with him in the Afterlife. PS: thanks for the copy of Principia you left for me in the dumpster; I'd been wanting one of those

I had to do a bunch of math courses during my undergraduate chemistry program, including linear algebra. There was a proof on each assignment and on each exam. I'm fairly certain that I completed that course without ever getting a proof correct.

This video is the epitome of Half as Interesting.

An explanation of this, in simple terms, now that I finally understand enough to get what's written on those pages: a small foreword first: sets contain things and we tell them apart by the things they contain. it doesn't matter what number of the same thing a set contains, nor the order, so {a, a, a,} = {a} and {a, b, c} = {b, c, a} sets can contain anything* two sets are different when they contain different things we say numbers are the set of all sets with that number of elements, so the number 1 is the set of all sets with one element. (negative numbers are defined as the additive inverse for these, and fractions as the multiplicative) when we have one thing - say one pen - then we would say we have a set with one element in it (the pen), which is an element of the number 1 (as a set of all elements with a single thing in them), so we can then say we have 1 pen finally, if we have two sets which have one element, and we combine them, if those sets are not the same set, then the resulting set will have two elements, and will be an element of the number 2, so we can say 1+1=2 rest assured that every little thing said with words here must and has been explicitly defined. rest assured that these definitions are very paradoxical and the person who invented this entire framework had a panic attack so bad he was hospitalized upon reading a letter which proved it so :)

I appreciate how you were describing aleph null without just saying it.

If it took 360 pages to prove that 1+1=2, imagine how thicc that book would have to be to prove Einstein’s theory of relativity

1+1=2 took 360 pages to proves 3x4(1+6): Finally a wortht challenger, OUR BATTLE WILL BE LEGENDRY

"But if you have a PhD in mathematics, you'd probably be doing something more important than watching this video." You overestimate my power.

0:33 It takes 361 pages to do that so we are watching what's best for us now

"Then you'd probably be doing something more important than watching this video" *sweats in doing a PhD in AI and still watches every HAI video*

I had a teacher for my Math Analysis (pre-calc) class in 11th grade who was a Ph.D. in math. This was the first assignment we were given. Those who completed it got it wrong, because you can't prove 1 plus 1 = 2 until you prove that 1=1 and they hadn't done that. I hated that class. I had straight A's in math my whole life up until that point and loved it, but he ruined math for me.

Ok I think I’ll read this book before my test sheets

I laughed for like half a minute straight at the delivery of "ow! oof! my normal brain HURTS!"

ah yes finally a question i never knew existed yet in the same time i been longing someone to answer

Some matmaticians: we made a complete consistent axiomatic system without any contradictions Kurt Gödel: you missed something

We did this in a maths theory class at my local community college and it involves defining successor.

0:28 every grad student asks this question at least a few times a day

Your videos are underrated my guy. If their is a team working with you to put these videos out y'all are killing it! Love the little jokes throughout just just sneak up on the viewer as we learn useless information. I'm here for it!

I love how the writer almost managed to get a month of paid vacation

The transition to the ads is really smooth. 👍

This video is evidence you can expand your essay to make it way longer. You could say 1+1=2 or write out that whole entire book. Just remember, whatever you write can always be expanded to hit that word quota.

77+33=100