The 360-Page Proof That 1+1=2
Vložit
- čas přidán 26. 04. 2022
- Sign up for the free Morning Brew newsletter: morningbrewdaily.com/halfasin...
Get a Half as Interesting t-shirt: standard.tv/collections/half-...
Suggest a video and get a free t-shirt if we use it: halfasinteresting.com/suggest
Follow Sam from Half as Interesting on Instagram: / sam.from.wendover
Follow Half as Interesting on Twitter: / halfinteresting
Discuss this video on Reddit: / halfasinteresting
Video written by Ben Doyle
Check out my other channel: / wendoverproductions
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'
But why is the existence assumed to be possible?
@@valentinmitterbauer4196 Because there is a possibility that the reality we live in exists or that it doesn't. And we rest upon the assumption that it exists mainly due to the fact that it is the easier possibility to comprehend or to make sense of
@@AltimeFAILS why is it considered easier?
@Just some guy who cares about privacy Why shouldn't we understand it?
@@b3nl555 Because there is a possibility that the reality we live in exists or that it doesn't. And we rest upon the assumption that it exists mainly due to the fact that it is the easier possibility to comprehend or to make sense of
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'
@@dannypipewrench533 It's a bot. A very clever bot.
Lol…my favorite line: “From this proposition it will follow, when arithmetical addition has been
defined, that 1 + 1 = 2.”
@@GunboyzElite "Most people" being about 57 people, ever! :)
@@GunboyzElite I am here to tell you MOST people never open Volume I either! Only mathematicians would even CONSIDER doing such a thing.
@@drewmortenson Danny could be a bot himself. Bots replying to each other is a thing.
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.
My degree is ling, but when I first started reading this book that was my reaction lmao. Thank goodness for Standford's Bernard Linsky who took the time to explain it on the plato resources or I wouldn't have ever managed to even start.
im really good at math but dont got a degree im hoping for coding
@@Qiibli haha for real
@@Qiibli coding is trivial to a seasoned mathematician
@@snared_ -Said someone who isn't proficient in either.
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 +".
It is not necessary to prove because it is the definition of the decimal system. Now if we are talking about binary math. Then 1 + 1 is 10.
Actually, defining 1 is one of the first things done in college math, it is defined as a neutral element for multiplication (more simply, a number which does not change the number multiplied by it)
@@warmike But then you didn't define what is an "element"
The important thing to realize is that numbers greater than 1 are basically just shorthand. If you want to be fundamental, then 0 and 1 are basically the only fundamental numbers (arguably you could also say -1 fits here too). 2 is the number after 1, 3 is the number after 2, etc. And addition is merely a means of moving on the number line. But the symbols you use on the number line can literally be anything. C is 100 in roman numerals, and that's just as valid as any choice. In any case both are just shorthanded for a chain of 1 followed by +1 99 times.
"there exists a number 1 such that 1≠0 and 1*n=n"
Can't believe you didn't mention the fact that right after this proof, the authors write "The above proposition is occasionally useful"
That's after *110.643 (i.e. the actual proof that 1+1=2) not after *54.43, which is what he's talking about here.
It was useful for the author to get published in the first place
What a meme lord
Well, really the only use for the proof is for people to go, "huh, there's a 300-page proof that one plus one is two. that's funny."
I was waiting for “the proof of this proposition is left as an exercise for the reader”
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
Veritasium does a video on the incompleteness of math, (also a great vid) I believe he said it took over 700 pages. In that video they cover the basics of why it takes so many pages.
@@Iamthelolrus yeah you’re right
You literally saw the full rigorous proof in this video. The goal of the book was not to prove 1 + 1 = 2. Literally only a few lines are dedicated to doing so.
@@Iamthelolrus And then because it relies on certain assumptions, the conclusion is that "1+1=2 *can* be true, but doesn't have to be" (assuming you use a different set of assumptions)
The "360 page proof" is a bit of a stretch, to be honest. Russell and Whitehead spent 360 pages developing a rigorous, axiomatic background to set theory, and then on page 360, they used their previous results to prove (in a few pages) that 1+1=2. You could argue that, because their proof used lemmas established earlier in the book, that it would require "360 pages of reading to fully understand the proof," but then by that logic, nearly every proof in advanced mathematics could be considered several hundred pages long.
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!
You make a living writing in greek but would like quick clarity in a 360 page extremely technical work on the bleeding edge of an obscure and abandoned philosophical project, thats interesting. Tell me professor, how much research in mathmatics is legible to ordinary people? Like everyone else you see symbols you know well as useful shorthand and those you dont as needless tedium. How much effort would it take to write "change in x" instead of "dx" (just for publications)? Very little, but they arent written for lay people they're written for mathematicians, theres no reason to waste ink when your readers immediately recognizes dx. This was similarly written to experts in Russell's field.
To be fair* at least in principal thats who it was written to, im really not confident any notable number of people actually read all of this shit
True. I remember being absolutely lost when covering the formal definition of a limit in AP calculus which is very mild compared to whatever this proof is.
They were basically trying to invent LEAN but on paper.
The intention of the book is not to actually be a reasonably readable proof by anyone. They basically wanted to show "is this possible?" and then they tried their best.
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.
I studied computer science, which is only tangentially related to mathematics so I was spared most of it. But I still have Vietnam flashbacks whenever I remember the mind melting hell that is proof by contradiction and similar bullshit
Was doing cp geometry and I absolutely hated proofs, glad that’s over
If you hate proving theorems then I have bad news for you because literally 99% of math comes down to proving theorems.
@@LOLquendoTV To be honest, I always felt proof by contradiction is actually the easiest type of proof. You basically just have to find some kind of loophole in the equation and that's it. The real issue is if you cannot proof something by contradiction, because now you need to make sure that there are no loopholes in your proof that somebody else (aka the professor that studied that shit way longer than you) can find.
@@LOLquendoTV Wut? Proof by contradiction is mind melting?
That stuff is straightforward as shit. And it's immensely useful outside of mathematics as well. It's basically the backbone of every argument I ever won.
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
Well, the saying is "To make an apple pie _from scratch,_ one must first create the universe."
There's also a joke along similar lines about a scientist who had told God that mankind's understanding had grown to the point that we were essentially on his level ourselves. Our science was so advanced, we could even craft a living person out of dirt the same way God made Adam. So God says "Alright, show me." The scientist gets a shovel and starts digging, but God stops him and says, "Woah, hang on... make your own dirt."
Which mathematical universe though?
@@omargoodman2999 A correction to your correction; the actual quote is "If you wish to make an apple pie from scratch, you must first invent the universe."
Carl Sagan.
But can you prove the existence of apple pie?
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.
There are a lot of proofs like this.
Just casually make an impossible operation when nobody noticea and boom, youre a mathemagician
Ah, this brings me back to takeing a crash course in logic a few years back. Loved it, understood nothing :)
could you share the video you watched? I've been looking for a good one on logic
@@vcuberx I don't have a video for you, but as a computer science student, I suggest you look up discrete mathematics, especially propositional logic and rules of inference. They're simple yet useful in forming your foundations of logical thinking.
Seems about inline with most people. A is A.
taking*
Discrete math is simultaneously fun and traumatizing
"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."
Can you please elaborate? I rarely heard a story of how someone chose his PhD topic that's not half as interesting.
Yours sound three quarter interesting. I'm intrigued
Technically the axiom of choice only refers to the (countably?) infinite case. For the finite case, it’s either an elementary axiom, or a result of one or more elementary axioms, of basic ZF, no ZFC needed.
@@Pablo360able Basic ZF... if I didn't know what that short meant, I'd be so confused right now. Zermelo-Fraenkel set theory for those who aren't into maths.
You're just confusing the viewers by writing it in shorthand.
@@Pablo360able No the axiom of choice works on all sets. The difference is that choice is not needed for finite sets, but it is useful.
@@livedandletdie I don’t think “Zermelo-Fraenkel set theory” is any less opaque for people who don’t know Zermelo-Fraenkel set theory. Also, we live in an era where the Internet exists (obviously), anyone who doesn’t know what something means in a comment has the choice to either immediately learn what it means or remain ignorant by their own volition.
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.
ok but i know another way of proving it take one object than take another object and than count bove objects
Those other fields don’t have mathematical proofs, but they still absolutely prove things in a way appropriate for the subject. I mean like give me an example, biologists definitely aren’t blindly assuming that plants are different than animals they’ve proven it
they cant prove some of the math they use because it is not their job, its for the mathematicians. a lot of ppl think that if physics is mostly math why both of them dont combine into one, because they arent the same, you cant use pure math logic to explain physics and you cant prove physic laws without math
question is what is 0+0 equal to?
@@unknowngod8221 0, because 0 is
well
it's 0
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.
The point wasn't really to prove 1+1=2. The point of the book was to set a foundation for the entirety of mathematics, to unify analysis, algebra, geometry etc.
It tried to provide a system that could rigorously be applied in any branch. Proof for 1+1=2 itself is quite short
Well in this book there is proof that 1 is greater than 0 at the beginning so at this low level 1+1=2 sound not so obvious
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
We don’t have to prove everything. If there is a wide consensus that something is true, then we can assume it is true.
Proof is needed when someone questions this consensus.
For example there is no need for proof that stars exist on the sky, we all see them.
@@juzoli You're confusing the concept of proof in mathematics, and the concept of proof in science or day to day life. They have the same name, but they are not quite the same thing.
@@juzoli Yes but mathematics doesn't objectively exist. It is a logical framework where people use basic facts together to gain new insight. In this field, it doesn't matter if everyone agrees that something is true. If there is no direct reasoning that can show certain existing facts can ONLY mean a new fact is true or false, then it cannot be considered a proven part of mathematics.
Conjectures are a great example of this: We have tons of different ideas people have put forth about new facts in math, but we haven't figured out any logical path that shows that these facts have to be true or false. So despite being intuitive, probable, and sometimes even assumed true, they aren't proven parts of math.
@Ben 🅥 Your ticket out of the comments section for life
It's here FINALLY!
I'm pretty sure no one uses Principia Mathematica?
At least not that I've heard. I'm pretty sure everyone just says 1 plus 1 is 2 and moves on.
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…."
That is kind of terrifying, kind of like having a nightmare where someone is silently fidgeting with a matchbook in the library of Alexandria
Damn that is terrifying. Seeing your life's work be dismissed as nothing more than a useless stack of paper
It's like the end of Inception. We don't get to know if the book was actually kept or thrown away. 😅
What the teacher expects you to do when they say "Show your solution"
teacher: why didnt you use my strategy?
her strategy:
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.
to me (a programmer) that line sounds more like a (part of) definition of how the "and" and "exists" operator(s) work and interact:
[a exists] and [b exists] == [a and b] exists
which in turn, basically defines how merging sets works.
which... seems useful.
This is why you have the expression:
If and only if
@@pedrofilardo might be mixing 'languages' here, "if" all on it's own includes the only if part. though I suppose it could be expanded with an if not that includes all other cases, but an else is more or less the same thing. If case a is true do a thing, however case a is defined already includes only if case a is true.
Of course this can go to hell pretty easy when you use an xor (exclusive or), as even if case a is, in fact true, if case b is also true, then the value od 'if a xor b' is false.
@@johngaltline9933 if and only if you can use logical language. I only know English and Portuguese
@@johngaltline9933 "A if B" is the same as "if B, then A."
"A only if B" is the same as "if A, then B."
"A if and only if B" is the same as "(if A, then B) and (if B, then A)."
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. 🤣
ah yes, "confusion", the greatest weapon of all
why don't just answer with: "1 + 1" is an addition of two number. then provide the definition of addition and number.
@@muhammadqatrunnadaahnaf9453actually, proving 1+1=2 straight from Peano's axioms is much easier than providing general definition for addition
@@Noname-67 What do you mean? Peano's axioms also has a "general" definition for addition, including its properties such as commutativity and associativity. No proof system can prove something without stating the operator's general definition and its properties.
@@muhammadqatrunnadaahnaf9453 I was wrong about that, for some reason I thought that it was possible to prove without using all the axioms of addition. It's like product with 0, you don't need the definition, as long as there is an axiom state that the product of any number with 0 is 0, you don't need to bother the other part.
I want to point out commutativity and associativity are not in the axioms, at least in the most commonly used, they are the consequences.
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.
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.)
Mahlo cardinal supremacy smh smh
Don't overestimate what math PhD people are. They'll waste time watching these on the toilet just like me.
Why Tf I was watching it on toilet seat too? 😭
0:36 did I just got no bitched by a math video?
Yes bro 💀😭
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
You were thinking about mathematical proof’s in elementary school? They don’t even teach that in elementary, they’re still trying to teach you that 1+1=2 in the first place. Then like 8 years later they make you prove why, and it sucks lol.
@@monhi64 It was a very crude idea of proofs. It boiled down to something like "what if this super-complicated thing existed just to show 1+1=2".
I had no idea what that super-complicated thing was at the time. I just imagined whatever it was took up an entire page of work.
@@monhi64 have you never asked yourself 'dumb' questions, especially about math? Like the water is wet because you can always verify by jumping into it. But 1+1? Why isn't 1+1 idk, equal to 3 or something? That's the kind of question i'm referring to. Maybe not a proof as you know it now but something similar in the spirit.
@@itsfreakinharry7370 Yeah I also had some kind of idea of proofs before even knowing they were an actual thing in mathematics.
@@monhi64 when did they make us prove 1+1=2
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.
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
haha so true, pure logic is for nerds, we use math to make stuff works, not to answer why and start doing pure logic brainfuckery...
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!
Meanwhile logicians quabble quabble quabble about GCH
@@vrowniediamond6202 mostly we quabble about the axiom of choice, the C in ZFC. So we’re not that different after all
Greetings, college! May I ask your field of study? Mine is in hyperbolic geometry and dynamical systems.
I hold 1+1=3 to be true
@@vrowniediamond6202 that discussion is not over yet, it never will be I think
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").
Ahh cardinality. That's the good stuff
Objection: This is not Legal Eagle, so your comment does not need to be in the form of an objection.
You can write 0 as {}, 1 as { {} } and 2 as { {}, { {} } }. Those sets have the cardinality 0, 1 and 2 and contain all natural numbers (including zero) smaller than themselves.
@@pi_xi That's the von neumann definition
Admittedly though it's much easier with the von neumann defintion, the peano axioms (which can be proven if you assume ZF/ZFC which of course we're doing here) and the definition of + as
a + 0 = 0
a + S(b) = S(a + b) which is just a set-theoretical function that gives you something that represents a + 1
@@lox7182 I guess, you mean a + 0 = a, as 0 is the neutral element of addition.
4:17 that's actually funny 😂
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.
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.
iirc (and it was a long time ago) I think they ran into a problem with set theory that whilst it can be used to define 1+1=2 the same rules can also be used to show 1+1=1 which is why it didn't really catch on with all the cool kids.
@@davidb9036 nah, that only works when you divide by 0 which is very iffy mathwise and so isn't legit
"...basically all of mathematics"
Gödel: hold my beer..
@@azursmile smiled :)
This is so sad and pathetic.
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.
it’s not so much why as much as it is how in mathematics. that’s the whole point-HOW can i prove this. not why. we don’t really care why, just that we can.
@@feline.equation Perfectly said. It's so annoying when people say, "what's the point of learning this?"
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.
What the teacher expects when she says show your work:
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.
Hi I'm blind and for some reason I don't think that method would work
@pyropulse It would be, if you could successfully describe blue to a blind person. The most incredible thing about the proof that 1+1=2 isn't that it's hundreds of pages long. It's that the proof exists and is finite.
Theoretically, there are ways to describe blue to a blind person, even if the method ends up being reconstructing the person's eye, optic nerve, and visual cortex, and then showing them blue. In that way, it is a bit like proving 1+1=2.
@@jetison333 take it from a blind person but showing is on no way describing
@@jetison333 You're reversing the blindness. I guess that's one way to do it, even though it breaks the simile. The point is that it's impossible to describe blue to a blind person because they are, well, blind, and lack the necessary frame of reference. You can make analogies ("red feels hot, blue feels cold"), but that's not what colour is. The idea is connected to the "qualia problem". Quite a rabbit hole...
this is a certified hood classic
Everything on this channel is
This video slaps
This is a certified bot comment
I love how with every new HAI video sam's humor gets even better 😂
This feels like making a computing system from scratch. But even more abstract
One could literally write any equation & a mathematician will work the hell out of him to prove it
GAGAGAGAGAGA! I will now count to 3 and then I am still the unprettiest CZcamsr of all time. 1...2...3. GAGAGAGAGAGA!!! Thank you for your attention, dear uz
Except... You don't write an equation out of nowhere and expect people to prove it.
@@segmentsAndCurves Obviously it has to make sense.
@@IamUzairSajid "can you spell that more rigorously?"
@@segmentsAndCurves No offence to anyone. I'm just a random person on the internet.
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.
I know, right? I thought that was a bad example, lol.
You are right, the order of the letters is totally arbitrary. But the point is that if you get to the question, “Why does A come before B?” The answer is ultimately that it just does. That’s the rule and everyone agrees that A comes before B. Reminds me of flat earthers thinking they can disprove gravity…
@@derekeastman7771 because alpha was before beta, and because aleph came before bet. But in fact from one of the oldest known books we know classification happened due to how tongue is placed in the mouth on those letters.
0:33
He really said "why aren't you getting bitches?" 💀
2:17. Guarding the door is an important job...
2 + 2 is four, minus one that's three, quick maths
But 1 + 1 is 2 is long maths
Yooo dank meme
"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)
That's cool. I agree about Godel not making it obsolete.
algebraic combinatorics? so big numbers, then?
@@MABfan11 Not quite. Basically my research concerned geometric objects in a huge number of dimensions. As part of my research I discovered a new object in 13,056-dimensional space with certain special properties that hadn't been found before.
@@Tesseract_King Wow. What's the new property?
Trans pfp
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.
"But if you have a PhD in mathematics, you'd probably be doing something more important than watching this video."
You overestimate my power.
Teacher: "Don't forget to show your work"
Student: *hands over the above book*
Teacher: *instant regret*
That student may regret it, too.
If I was the teacher, I'd ask the student to explain it. If they can't then the student gets an F for cheating. If the student can then neither the student nor I should have any regrets. In fact, I'd probably ask the student to see me after the lesson, so we can discuss ways to get them into an advanced math course.
sHoW yOuR WoRK got so annoying holy shit. There were so many times where i just didn't have a method it was just basic logic to figure it out
@@comet.x My entire study of mathematics was dedicated to figuring out what the steps were because I had exactly no idea how to make things easier for my teachers. That's why I read this book and I can safely say I know how to show my work now, and I teach others how to do it too. It's literally my entire personality.
@@abebuckingham8198 if I have children i'm gifting them this insanity just so they can 'show their work' on stupid questions
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.
He said “There was a footpath leading across fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics.” and that has kept me alive more times than I care to recount.
He might not have been pompous as his contemporaries but he was still probably a complete brazen idiot
No, he's the same as any other philosopher. In philosophical circles, he's famous for dismissing half of all philosophical research that was being done, and lost a debate with Frederick Copleston, author of the authoritative series of the history of philosophy - the same topic that Russell half-assed his way through in his own book.
@@marchdarkenotp3346 What debate did he lose against Copleston? In what way?
@@splatted6201 nonsense by the OP, he has a debate with Copleston which he hardly lost. But of course that’s what theists would like to believe.
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.
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
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.
Except for the times when a lightswitch, with two positions, is switched from initial position to the second position, then back, but it results in a third state for the light the switch controls.
Then 1 + 1 = 3.
Clearly.
@@tyelerhiggins300 hi-Z / high impedance is what I live for.
Well, to be safe, lets make it 3.
@@fltchr4449 nah fam, i use safety factor of 2, so it should be 4
@@fltchr4449 nah, it should be sqrt(g)
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
@Ben 🅥 No
@@survivinggamer2598 Don`t reply to the bots, just silently report them. They're trying to churn up false engagement and every reply encourages it.
@@dustinbrueggemann1875 I know thanks, but I was just referencing the Talking Ben meme.
i thought so too, kinda hoping for a video on that now….
"The 360-Page Proof That 1+1=2"
Me with two apples: "You underestimate my power."
I laughed for like half a minute straight at the delivery of "ow! oof! my normal brain HURTS!"
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.
Literally all religious people ever
the problem is that people who don't agree on the basic assumptions of reality are also the people who don't give a flying fuck about proofs, even IF they were ever able to understand them, which they are certainly not, since they all studied history of queer african dance theory instead of something useful.
You can prove negatives just fine. Proof by contradiction, etc.
@@revan552 "what's inherently wrong with studying the 'history of queer African dance?' "
the fact that it's a useless, made up subject created as a front for indoctrination into the "woke" cult.
(the subject doesn't exist (yet) as far as i know, but many others that are similarly absurd and useless do. i was just trying to bring a bit of humor into my comment by inventing a specific thing instead of saying "useless subjects that only exist to indoctrinate people into woke leftist cult")
@@rpavlik1 My bad, I'm pretty sure it was you can't disprove a negative.
somebody took "show your working" a bit too seriously
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"
Then is also used to form a consequence. I read your comment, then I replied to it.
If a = b (statement), then b = a (consequence)
@@gito4066 You're right. I was wrong. Learned something new : )
Teacher: Ferb, I know what we're doing today. 💀
It's amazing that when he goes really philosophical around 3:40, he just comes back saying "It's something like that"
2:11 "ow oof my normal brain hurts!"
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.
"I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain"
Why is there stock footage of "women in suits crossing their arms underwater"?
The next time Sam is sponsored by his stock footage company, he definitely needs to bring this up, to prove their versatility.
Rule 34, that's why.
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. 🤣
Kindergarten teacher : you have an apple, your friends have an apple. if the apples were combined, then how much apple now
Whitehead & Russel: "1 + 1 + 2"
Banach & Tarski: "Yeah about that..."
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.
Any odd number can be represented as 2n + 1 where n is an integer
let a = 2p + 1 and b = 2q + 1, where both p and q are integers
a + b = 2p + 2q + 1 + 1 = 2p + 2q + 2
since all terms of 2p + 2q + 2 are multiples of 2, a + b must also be divisible by 2, thus concludes the proof that the sum of two odd numbers is even
Let me take a stab at it :D
Consider two odd numbers, A and B. A and B are odd implies they can be expressed in the form 2q+1, where q is an arbitrary integer. Then, without loss of generality, A + B = 2q + 1 + 2q + 1 = 2q + 2q + 2 = 2(q + q + 1). Then, since integer addition results in an integer, q+q+1 = an integer, c. Thus, for odd A and B, A+B = 2c, which implies the sum is even.
Of course, the fun part about this proof is realizing how many assumptions are already made, like the rules of addition, multiplication, integers etc.
Proofs basically are math. I don't know what else in math you're referring to.
@@BreezyInterwebs Actually what you proved is that an odd number added to itself is even. Not that any two odd number added together are even.
€ is like the symbol "belongs to":
let n € Z
even number are defined as:
2n
odd numbers are:
2n + 1
so:
(2n + 1) + (2n + 1) = 4n + 2
4n + 2 = 2*(2n + 1)
now, from the principles of whole numbers, (2n + 1) is just another whole number, so we can replace it with n:
2n
as you can see, this is the same as the even numbers, which proves your statement
4:29 lol
"if i have one apple, and then i have another apple, and i put them into a box together, how many apples does that box have?"
"11"
"correct"
You should've make it clear that Russell didn't write that 360 pages of Principia because he felt a need to prove 1+1=2. His job was to lay a logical foundation for all branches of mathematics, and proving 1+1=2 is just a relatively minor byproduct of his work on logic. No mathematician would remember him as "the guy who proved 1+1=2", because it would be as ridiculous/superficial as remembering Newton for his observation that apples fall onto the ground. Russell's work laid the foundation for the more fancy things people love to talk about in pop science like the Goedel's Incompleteness Theorem. Without the logical language he helped to create, we wouldn't have fancy mathematical theories about infinity and, more importantly for most people, the foundation of computer science.
I just hate to see some people in the comment dissing mathematicians for supposedly doing useless pretentious over-complication when they have no clue about what those works are meant for in the big picture.
Now we need a proof that 1+2=3
Yep also 1+3=4
Don’t forget about 1+5=6
and also 1×1=1
@@HufflepuffBaseball42313 wait a minute ... we forgot 2+1 !!!
@@dihaozhan shit…which one is that again? Need a 1700 page proof for that one
"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*
Why did you do your PhD in As Interesting?
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
"Why are you watching something like this instead of kissing a beautiful woman?" *vine boom*
These days, it takes far less pages to prove this statement whether you're using peano arithmetic or something like ZFC
Using the Peano axioms is cheating. :)
@pyropulse
Never read principia mathematica, but I can't see how they could possibly reduce mathematics to formal logic (i.e. prove mathematical theorems without adding any axioms on top of logical ones), and I even think Gödel's first incompleteness theorem prohibits that (since if math were Reducible to logic, the fol is incomplete by the first and completeness theorem which contradicts the completeness theorem).
@@seneca983
s0 + s0 = s(s0 + 0) = ss0 go brrrrrrrr
@@mohammedbelgoumri Well Godel’s theorem was created by Godel specifically to prove that the stated goal of the Principia (to create a system by which all of mathematics was based on a foundation that was wholly logical and complete in nature) was flawed so yes it does contradict it.
@@mohammedbelgoumri the foundations of principia mathematica aren’t quite fol, but rather type theory. in a way, whereas set theory postulates a universe of sets on top of an existing logic, type theory bakes a universe of types more directly into the logic.
This video is the epitome of Half as Interesting.
The old man's face at 3:47 perfectly describes my feeling about this video.
Oh look a HAI video on Buffalo Buffalo Buffalo *click*
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).
shared this to my maths teacher, suddenly my grade changed from A to F, can someone tell why?
Probably because it's an HAI video you're sending.
You had topped out at A and, like Ghandi, it rolled over to "nuke everyone".
prob cuz ur lying
Mitochondria is the power house of the cell.
Sun rises in the east.
POV: when the statement is not left as an excercise to the reader
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 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.
Some matmaticians: we made a complete consistent axiomatic system without any contradictions
Kurt Gödel: you missed something
"Oh, wanted complete and consistent? I'm sorry that's not how the menu works." - Kurt Godel, probably.
When I was like 8 or 9, I wrote a strange, obstinate little essay, called "Logic," trying to prove that hydrogen and oxygen combined don't make water. I argued that since hydrogen added to more hydrogen doesn't change anything, why would adding a different kind of gas?
Once, during a trumpet lesson where my teacher said something I already knew and I thought he was being condescending, I mentioned this essay, TOTALLY out of the blue. I mumbled, "I wrote this book called Logic..." and tried to explain, even though there was NO relevance to playing trumpet. He gruffly said something like "Well, I'd like to see what you've written," and went on to say that he couldn't teach me if I wouldn't cooperate. I respected my teachers after that.
Me: mixes 1 cup of water and 1 cup of alcohol to NOT get 2 cups of liquid*
Mathematicians: "impossible"
ah yes finally a question i never knew existed yet in the same time i been longing someone to answer
Kinda like wondering why 37 potatoes? I didn't know I needed to know the answer to that but now I need to know. And are those russets or yukon gold. Normal grocery store or Costco size? Urgh... What have you done to me?
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
This looks like my presentation when I still have 2 minutes left on the clock XD
When the teacher tells u to write out ur solution
I love how the writer almost managed to get a month of paid vacation
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.
Sounds like he just wanted to feel like he was good at math by comparing himself to kids. What a jerk. Even if you hand someone something more obvious, like something basic on the peano axioms, you still need to walk them through it for a day or two before they get a feel for it. I love proofs, but my first couple days were awful. Sorry you had him.
but he's correct. you should first define what "=" means and then provide the proof of its property; it is also needed for "+". and only then you can proof 1 + 1 = 2.
that sounds like a dumb assignment and I hope that guy doesn't get to teach HS children again.
If you aren't willing to sit down and try to solve and unsolvable problem for a couple of decades in a row math is probably not a good fit for you anyway. I feel like it's more about frustration tolerance than talent.
@@abebuckingham8198 To be ruthlessly honest, this is a bad take. There are far more mathematical problems out there than the ones that have been stalling for decades. Beyond that, there’s plenty of math to be done in figuring out new facets of things that we already know. Plenty of skilled mathematicians like Freeman Dyson never dedicated themselves to the same problem for very long. You’re only obligated to if you’re going for your PhD, some random award, or if a certain problem’s really caught your eye.
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”
Not even a minute into the video and he’s calling me maidenless. 😭
1+1=2 took 360 pages to proves
3x4(1+6): Finally a wortht challenger, OUR BATTLE WILL BE LEGENDRY
Sorry to be a boomer, the answer is 84
@@outsideconfidence12 prove it
@@b4594 hold on gimme 3 years I'll write a 2000 page book.
Order of operations: brackets first so 1+6=7, next just multiply everything 3x4x7 = 84. Sorry I'm a maths geek 🤓
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!
As someone who is currently taking a class that is all about proving stuff like this, this video pretty much sums this up.
Teacher gives a question:Prove 1+1=2
Students:Take 2apples on one side and 1 apple on the other side add 1 to the other side and we have 2apples on both side L.H.s=R.H.S.
Teacher: