The 379 page proof that 1+1=2

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Principia Mathematica

Russell's Paradox video:

0:00 Intro
0:52 All was well in the land of math
1:39 Oh no! Trouble is brewing
3:47 The heroes of the story
5:06 Principia Mathematica
5:49 Logic
7:42 Formal Systems
9:52 Struggles :(
11:49 Ideas in 1+1=2
14:26 Failure
15:04 Sponsor

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Creator - Jade Tan-Holmes
Animations - Tom Groenestyn
Music - epidemic sound

[upandatom]

At 14:31 I meant to say "complete" rather than "consistent". Thanks for pointing it out!

[astralshore]

In an exam, I once incorrectly used Gauss’s theorem to end up with the equation 1=1. The professor wrote down: “thanks, but we knew that already”.

[ptorq]

I was a chemistry major in college, and one of the requirements for chemistry majors was "take at least two classes from this list of about six non-chemistry classes." One of these was called "Math Foundations", and a couple of friends of mine decided to take that, assuming that with a name like that it ought to be easy. They came up to the lounge one day with extremely dazed looks. I asked them what was wrong and they said "We just spent an entire class talking about 1 + 1 = 2." I said "You said you wanted an easy class, and that sounds pretty simple, " and they said "No, no, you don't get it. First you have to show that numbers are even a thing, and then we have to show that there's something called addition that you can do to them. The professor says because we'll be glossing over a lot of the finer details we ought to be able to prove that 1 + 1 = 2 sometime next week."

[michaelmcchesney]

When I was a junior in high school, almost 40 years ago, I had to write a term paper about a math topic. I had really enjoyed geometry. Euclid's Parallel Postulate or given a line and a point not on that line only 1 line could be drawn through the point that is parallel to that line, had always seemed like it should be provable. I realized that since many much smarter people than I had been unable to do so for 2, 000 years, it was unlikely I could do it. I had to return my geometry textbook at the end of 10th grade. But my father had bought a geometry textbook at a garage sale. I have no idea why he bought it, but it meant I had it as a reference source. After a few hours I had figured out a proof that used only postulates. I checked it over very careful and could not find my mistake. I was pretty sure I must have made one. Instead of a Nobel Prize for my proof, I got a B on my paper. My mistake? Trusting a textbook my Dad had bought at a garage sale for $1. It turned out that one of the postulates given in the book was actually a theorem that was proved using Euclid's Parallel Postulate. Apparently the textbook author didn't feel like including the proof of the theorem I used and just listed it as a postulate. Their laziness cost me a Nobel Prize!!!

[ColorwaveCraftsCo]

Your animations add so much to the storytelling, one of the many things I love about your channel

[JohnKarro]

This is arguably your best video -- really nicely done in tone, production, visuals and (most importantly) content. i'd had not gotten round to watching it for a while, thinking I already knew the material. Very glad I did take the time; well worth it.

[ivanscottw]

Philosophically, I always thought that Gödel's incompleteness theorem was both depressing (in a (non trivial) defined system, there are always problems that we cannot solve) and infinitely fascinating - we can always build (one, multiple, an infinite number of) more complex system(s) over the previous one where the problem can be solved - but yes - then it becomes recursive - and then headache ensues !

[miramosa]

The attempt at formalism to define all maths is such a fascinating project. I've known about it before, but thanks for putting out a video about it! It's always good to hear about it again, especially in such a concise and easy-to-understand way <3

[exdejesus]

I'm impressed that you were able to explain this so well and so simply. I was a math major in college, and took many courses on logic and set theory. And I've read some of Principia Mathematica. Your explanation is amazing.

[Bodyknock]

One of my favorite books on logic is To Mock a Mockingbird by Raymond Smullyan which essentially walks the reader through a predicate logic course in the form of logic puzzles involving birds as the basic symbols. In fact working through the entire book does get you from start to finish through proving Goedel's Incompleteness theorem and also how numbers and arithmetic are derived from fundamental set theory and logic. 🙂

[masonwheeler]

Gödel's Incompleteness Theorem is a very interesting thing, because the system of "Gödel numbers" he came up with to describe the problem is immediately recognizable if you work in software. There are some significant differences in the implementation, but it maps quite well to the numerical "instruction set" concept that lies at the core of the CPUs that power all of modern computing.

[louisgiokas]

Computer languages are strictly formal systems. That is what drew me to the field. I was good with languages and math. I was studying physics and was actually doing a bit better in my math classes at university. I was also working as a programmer (we were all self-taught at that time) and High Energy Physics, where I worked, used a lot of computers. One of the co-heads of the department had a joint appointment with the then new computer science program (which was only a graduate program). I thought about changing to mathematics, so I asked my professor what a theoretical mathematician did. His response was that he thought up theorems and proved them. I found that unsatisfying. Of course, that leaves out all of applied mathematics and statistics. The other reason for leaving physics was that there were few opportunities to do physics academically. Many physicists became programmers.

[NathanFarb]

Amazing to see how much more sophisticated your videos are becoming without feeling like the content is changing or being lost. Multiple locations, animations... every video is more interesting to watch than the last!

[FunWithBits]

Great videos as always, Jade! In college, I was a math major, and I always joke around (but I also feel it is true) that the "1+1=2" topic in my first week in proofs class is what made me lose my joy for math and switch to computer science. I still enjoy math 20 years later though as a side hobby.

[archivist]

Explaining not just PM, but also its inherent shortcomings, within 17 minutes is a marvellous achievement. Great video, and very clear, thank you, Jade!

[formerunsecretarygeneralba]

Math started becoming so complicated that mathematicians even question something basic such as 1+1 = 2.

[donaldaxel]

Our Physics teacher mentioned Russell and Principia, briefly: You need to define numbers - two objects are never the same, but a sequence converging is a good representation of what we mean when two objects are the same.
Emphasize that two objects can never occupy same space and time - or in other words, not any two apples are the same.

[BallotBoxer]

5:29 math explained so well, even a cat will show up and understand it

[ericgenaroflores]

There was once a small boy in a village who was sent regularly by his parents to fetch bread. He used always to have ten kreuzer, and bring back in exchange six rolls. If you bought one such roll it cost two kreuzer, but he always brought back six rolls for his ten kreuzer. The boy was not particularly good at arithmetic and never troubled himself as to how it worked out that he always took with him ten kreuzer, that a roll cost two and yet he brought home six rolls in return for his ten. One day a boy was brought into the family from another part and he became for our small boy a kind of foster-brother. They were of about the same age, but the foster-brother was a good arithmetician. And he saw how his companion went to the baker's, taking with him ten kreuzer, and he knew that a roll cost two. So he said to him, “You must bring home five rolls.” He was a very good arithmetician and his reasoning was perfectly accurate. One roll costs two kreuzer (so he reasoned), he takes with him ten, he will obviously bring home five rolls. But behold, he brought back six. Then said our good arithmetician: “But that is quite wrong! One roll costs two kreuzer, and you took ten, and two into ten goes five times; you can't possibly bring back six rolls. You must have made a mistake or else you have pinched one ...” But now, lo and behold, on the next day, too, the boy brought home six rolls. It was, you see, a custom in those parts that when you bought five you received an extra one in addition, so that in fact when you paid for five rolls you received six. It was a custom that was very agreeable for anyone who needed five rolls for his household.
The good arithmetician had reasoned, quite correctly, there was no fault in his thinking; but this correct thinking did not accord with reality. We are obliged to admit the correct thinking did not arrive at the reality, for reality does not order itself in accordance with correct thinking. You may see very clearly in this case how with the most conscientious, the most clever logical thinking that can possibly be spun out, you may arrive at a correct conclusion and yet, measured by reality your conclusion may be utterly and completely false. That can always happen. Consequently a proof that is acquired purely through thought can never be a criterion for reality — never.

[mskellyrlv]

Great video. I tried reading Principia Mathematica 44 years ago, when I was in college. I didn't know at the time that I was both severely ADHD and dyslexic (not knowing even of the existence of either of these things), which made getting very far virtually impossible. I was lucky to get my BS and MS in Mechanical Engineering (which involved liberal application of my own non-dimensional number, the Kelly Number - "the right answer divided by the answer I got", which, multiplied by the answer I got, yielded the right answer. It could take on any real or complex - or alphanumeric - value, though ideally its value would be 1 but I digress). I don't know if you've tried delving into Newton's Principia Mathematica, but it is just as formidable. The first 19 pages took me two months to read, and contains the entire set of concepts of engineering statics I was ever taught. I still have neve finished it. But then, when I found out that Richard Feynman had been unable to duplicate Newton's derivation of universal gravitation, I didn't feel so badly....