Why it took 379 pages to prove 1+1=2

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

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”.

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."

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.

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.

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!!!

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

I am not stranger to mathematics and these presentations do affirm one belief; At the base of absolutely everything, from science, to arts, to biology and philosophy, there is always a math concept. A pleasure to watch you Jade.

The way you simplify and explain the matter is really fantastic! Thanks for the vid.

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

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

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.

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.

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 !

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

11:49 As soon as the questions was asked, I came up with the solution, but I instead came up with a comparison of 1 apple and 2 apples. Comparing items in a set is great and all, but because you don’t compare the sets against each other the alien could come up with “they are all made of matter” in all instances. If you compare them to each other, the difference can be spotted right away.

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!

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

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.

Beautiful channel, well researched and adorable animations. You deserve a mill + subs soon!