Why it took 379 pages to prove 1+1=2
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Russell's Paradox video:
• Russell's Paradox - A ...
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
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At 14:31 I meant to say "complete" rather than "consistent". Thanks for pointing it out!
That's a HUGE difference. Personally, I never understood why people were so reluctant to surrender completeness. You can want a pony too, but sometimes you can't have one.
The good news is that Godel showed us that if we DO surrender completeness, we can have our perfect logical structure, just as Russell and Whitehead intended. So maybe they didn't fail after all.
Wow a linguistics rabbit hole ha-thanks wow wild stuff indeed!♾♾♾♾♾☮️💟🌈🤯🤩😍😘🥰😻🗽🗽🗽🎬🎬🎬…
@@sindyr ll
I got an advertisement before I could watch your video! the doctor says he is having a mental crisis, he actually said we,, I'm serious talking about mentally ill psychotic doctors and psychiatrists LOL,,(they are having the mental crisis, people are finding out), definitely 1 + 1 = 2,,, but if you're an identical twin? you could be in more than one place at the same time,,, that means ones plus one only equals one,,,😂
what the heck is she talking about? I'm the greatest mathematician! and I can't even add one plus one,, there's always three,, if they were able to write a book on that, there would be a thousand pages spend all night reading it!
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”.
He did? Wow. I don't. 1 = 1 does not include time. So, one apple = one apple is not true unless you say 1 apple equals itself and only as long as you don't say when (leave time out of it)! Math is only a mechanism to solving a problem in the physical universe. In such instances, there are assumptions that are made and made with all equations. It is interesting to talk number theory but 1 + 1 = 2 does not need to be proven. It is an assumption right from the get go! If you don't agree with it, the proof will not be valid. If you do, the proof is valid. I find that VERY interesting!
@albert lipschutz that's why axioms exist
How does a wrong assumption lead to correct results
@@Krrish006 I assume you have a human great-grandchild, so you should be human too. While I doubt you have a great-grandchild, you're probably still human.
@@felipedamascenosilva3011 so how does this answer my question
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."
So easy. I took 11 chem courses and 39 non-chem courses. I needed courses in at least 2 other languages, and that was just a state college.
Assuming there's no significantly shorter formal language to prove that 1 + 1 = 2, a concept probably embedded into even relatively simple organisms like ants, raises the philosophical question, where this incredible expansion respectively compression comes from, going from a few bits to about 200 KByte of text.
Sure, the 200 KByte is the proof, while on the other hand "1 +1 = 2" is the fact, behaviour, instanciated rule, algorithm, automat, mechanic, universal invariant, empirical experience or how one wishes to call it. However, the latter must always "observe" the former, follow it at all times, be always governed by it - there must be a permanent link - in thought, information and physics.
pure mathematics is a hell of a thing.
@@user-hm3ni1wd3f Do you work in the field of pure mathematics? Now that I read my comment again, "empirical" and "experience" forming a pleonasm wasn't intended, makes it appear silly or unreflected, haha.
@@DarkSkay i do not work in the field of pure mathematics, at the moment.
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!!!
There is no Nobel Prize for mathematics
@@eljanrimsa5843 Could have won the Fields Medal, though.... awarded for- "Outstanding contributions in mathematics attributed to young scientists"
Considered to be the "Nobel Prize" of the mathematical world.
en.wikipedia.org/wiki/Fields_Medal
In 9th grade I was led to believe there was no proof that a tangent to a circle was perpendicular to the radius line touching it. So I came up with my own proof! Excite.
Next year's math teacher told me it was one of Euclid's basic proofs. Though apparently my proof was actually somewhat novel. Instead of Euclid's proof I proved you could construct a square bounding a circle from any tangent line in (Euclidean) space.
@@eljanrimsa5843 I'm aware of that now, but didn't know it in 1985.
logic is math for words. it's very important to acknowledge different systems of logic though. aristotilian logic is useful but it's not the only way to think about logic. it seems to be so widely held as the standard though due to how simplified it is.
but a simple set of rules to analyze something very complicated is not always going to work, even it if appears to.
Some Indigenous cultures formed their language around logical systems that were able to approach these more complex ideas that aristotilian logic has trouble with.
Some of these kinds of logical systems make sense to describe quantum mechanics or the concepts around multiple dimensions.
5:29 math explained so well, even a cat will show up and understand it
The way you simplify and explain the matter is really fantastic! Thanks for the vid.
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.
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 !
"and then headache ensues" sums it very well lol
I made a long comment above. Most of this math was thought up using classical mechanics as the valid universe. Since we all know classical mechanics is wrong , most of the math is just wrong. You can have things that are both false and true at the same time. This is one of the basic tenants of Quantum Mechanics.
So the statement she makes about eating cheese is wrong. The correct statement is this " I will not eat the cheese or I will eat the cheese or I will be in a superposition of doing both"
If you apply this to Godel and Turning and other infinites and paradoxes they all go away.
An electron shot at a double slit goes through the left slit or the right slit ......or it goes through both. That is the real world. Electrons have a long wavelength so encounter these situations all the time. People and the classical mathematical ideas have a very very short wavelengths that none of the mathematicians incorporate into their mathematics or even acknowledge or attempt to develop this math. The wavelengths are so short that they are never noticed. No one even knew about these wavelengths until the 1920's.
Quantum mechanics has a way of getting around what at first might seem impossible. So it might just be possible to have a math theory that can completely explain itself , as in pull itself up by it's own bootstraps.
@@jeffbguarinoreality and formal systems are inherently in a classical mechanics. I guess it depends on the interpretation of QM you use but the existence of axioms validates godels theorem.
@@franchise8633 I don't know where exactly but most of these theorems like Godel's and Turing machines stopping are leaving out QM in their logical presentation.
I don't know where it has gone wrong but something is wrong.
The law of noncontradiction for one. "The Law of Non-Contradiction
The Law of Non-Contradiction is almost the opposite of the Law of Identity and states that if something is true it cannot NOT be true at the same time."
Obviously this law is wrong. In the double slit experiment it can be true and false at the same time that an electron goes through the left slit , as long as you see an interference pattern. At 1:40
th-cam.com/video/R3OkCxhjDmQ/w-d-xo.html He demonstrates the example of Russel's teapot and states the fact that the teapot in orbit cannot be entirely made of steel and entirely made of china at the same time. But this is not true. You just need to launch two teapots into orbit , one made of steel and one made of china in a box and a quantum electron is produced by an apparatus in the box , if the spin is up then the steel teapot is destroyed and it the spin is down then the china teapot is destroyed. After the destruction there is only one teapot and it is in a superposition of being all steel or all china at the same time. If you open the box then it will jump into being one of the two teapots but if you never open the box then it will forever be both at the same time.
I haven't figured out yet how to get the barber to shave himself without shaving himself. I think you would have to put all the men including the barber into a superposition, so that we can't know if the barber actually shaved himself or not.
@@franchise8633 R is the set of all sets that don't contain themselves. So if R a member of itself ? Russel wrote Frege and asked him about this set. Frege had a mental breakdown and landed in the hospital. 9:40 th-cam.com/video/xauCQpnbNAM/w-d-xo.html
You just need to write the these two sets on a two pieces of paper. S1 is the set of all sets that don't contain themselves not including S1 itself and S2 is the set of all sets that don't contain themselves with S2 included. Put the papers in box and have an electron produced and if it is spin up then the first paper is burned and if it is spin down the second paper is burned. So therefore the two sets S1 and S2 are in a superposition and the resulting set contains itself without containing itself at the same time. So there is no contradiction.
Math started becoming so complicated that mathematicians even question something basic such as 1+1 = 2.
It's evolving just backwards
Yes, that is what this video is about. There's really no reason that math works so well. Why does 1 + 1 always equal 2? Why doesn't it sometimes equal 3? Or blue?
We spent 2000 years just assuming things and nobody bothered to check those assumptions. These guys checked it, thoroughly.
If you can't prove it, you have to assume it as an axiom. And that has consequences.
@@carinatus1758But then going from backwards and ending up with something way better than the original
idk why you're acting like that's a bad thing, it's basic because it used to be unproven, circular argument idiot
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.
We still lack this sort of proof in computer science.
Someone saying "This happened because of that" is hard to prove, but it's easy to say.
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!
Did you see the video where it took 758 pages to prove 2 + 2 = 4 ?
Your animations add so much to the storytelling, one of the many things I love about your channel
Gödel blowing up the whole house with explosives makes him appear quite evil. He was a good friend of Einstein. And in a certain sense he could be seen as the "ultimate constructivist": trying to prove the existence of God.
Is it just me or did she not answer the question of why it took 379 pages. Yeah, sure you have to define what 1 is and what + is and =... but why does it take that long
@@marioluigi9599i thought i was the only one who felt the same
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.
Route I went...
@@albertlipschutz My older son did as well, and with almost the same timing that I did. Interestingly, my younger son finished his degree in normal time. He majored in CS with a minor (or perhaps double major) in math. It just so happened that my younger brother also finished his degree in normal time. He majored in architecture.
I went back to school when I was working full time at an aerospace firm. It was fully paid for. What about you?
@@louisgiokas2206 HI! I was in astrophysics (of all things) but had been flying since I was 14 and had licenses as well. I turned to aviation as a career before I was out of university but on the way found I had a penchant for programming. Back then it was FORTRAN and I had used it to solve a number of questions posed in classes. In those days (early 1970s) computer printouts were not accepted by professors and I had to demonstrate the solutions by hand! I laugh at this now, but it simply was the way back then. Made me a much better programmer. I had a career in aerospace (even have the slide rule I used back then) in which I got my own desktop with, can you believe it, an 8" floppy disk!!! Ta about privilege! I programmed using a text editor called SPF which I would write and if others needed the program, got put on the company's mainframe. Later I freelanced my talents to other companies. I'm retired now but I still code and still take jobs when it suits me.
@@albertlipschutz Sounds like we had very similar experiences. I started with FORTRAN as well. SPF rings a bell. I also worked in aerospace and defense. Mostly satellites. I worked on at least ten. The first ground control systems I worked on were actually programmed in assembler on a mainframe. I mean the whole thing was one program taking the whole mainframe. It was wild. Debugging using panel lights and switches for input. I am working on a couple of startups. I like to keep busy.
@@FredCarpenter-pm8bfHate to tell you this Fred, but Pavlov's experiments unequivocally DO NOT WORK. They were political propaganda insisted upon by Stalin (which Pavlov willingly supplied to curry favour) so he could "prove" that life could be programmed and all men were animals. I tried it. The dogs hated the bells. They got mad at me. I've never seen anyone salivate over money, only euphemistically or comedically. Not one salivated on a bell ring though I probably did not have too big a cross section of dogs (they were ours and our friends pets) and I'm sure the percentage of people who do salivate over money is incredibly small. Suggest you "give it a ring" and verify for yourself. Amazingly, these "results" have permeated Western thought. Shows you what governments want of their people. It's enough for me that this disproved the "theory" of Pavlov Skinner and those who blindly follow this stuff. Most likely people are "baffled by the b___s__t and give up trying to understand it and give in. This is why you should always dig into a concept to a) determine EXACTLY what the speaker is saying and b) realizing that often, people are promoting self ideas, not knowledge. Meaning THEY don't understand it either or want a pre-ordained outcome. Whole subjects can go by the wayside if you use this approach.
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
The problem is it always leads to a contradiction
@@Nick-lm9hg Yeah, she... Says so in the video?
@@Nick-lm9hg prove it! what contradiction is present in the law of identity? The unfalsifiability of the unfalsifiable?
Forget it
Beautiful channel, well researched and adorable animations. You deserve a mill + subs soon!
0:04 "Hey guys! Spirit Of The Law, here."
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.
That's a surprising and interesting association, particularly given that Church and Turing each had their own closely related (equivalent?) theorems, and they went to influence computer science greatly with the tools they developed for those theorems.
Unless you work with a Harvard architecture where instructions and data are separate
The halting problem and the incompleteness theorem feel very similar. Years ago I did some digging trying to justify this feeling and learned of a couple obscure but amazing ideas:
1) Programs are proofs -- Namely constructive proofs from one type to another type.
2) Curry Howard Correspondence -- Every logic has an associated computational model / programming language.
3) Computational Trintitarianism -- And both have a corresponding category.
Basically, (almost) any concept in one domain is translatable (or has a dual) in the other two domains. So its no surprise a similar proof works in both domains, the theroems could be duals of each other under a certain model/logic/category triple.
So the foundation of mathematics is set theory? Or not?
@@EM-qr4kz No. Gödel proved that the work demonstrating such a foundation was incorrect.
0:12 or can we?
*VSauce theme plays*
This video took me back to my freshman year at uni, when I was attending Discrete Mathematics course. I can confirm that we went thru all of that. Defining what is a number, what is equality and all basic mathematical functions such sum and subtraction.
A Summary of the Formal Logic and Semantics course I took in the University, but fun to watch. Awesome video!
Explaining not just PM, but also its inherent shortcomings, within 17 minutes is a marvellous achievement. Great video, and very clear, thank you, Jade!
agreed! 👏
Yep - this is the clearest short explanation of this topic that I've ever heard. Nice one Jade!
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. 🙂
thanks! i just bought this after reading your comment. it looks wonderful 😊
Thank you for this, I had never heard of this book before, just checked it out and now I’m definitely going to buy it!!
So that makes the reader a bird brain? Argue if this is mocking or a logical conclusion to the question given your statements lol
Another great book on the topic is Goedel Escher Bach by Douglas Hofstadter! I'm working my way through it now but it can be a tough read at times. I'll have to check out your recommendation!
@@monkeygame7 Definitely!
My review, I guess: GEB is a must read for people who are interested at all in the philosophy of mathematics and our logical systems’ simultaneous simplicity and chaos. It flips between easily understandable examples, to dense portions (such as walking you through symbolic logical proofs such as those in Principia Mathematica). Took ages to get through, but I think that flip flopping was a brilliant device to keep me reading. In essence it’s sort of just a collection of interesting features of logic and math, but Hofstadter has a magical way of connecting it all together.
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.
Kitty Pride
So happy to hear you are planning to do a video on Gödel!
5:40 cat is inarguably the best part of the video. Its self evident
I love watching your videos. They are simple enough to understand yet open up vast areas to keep researching.
Brillant storytelling! Wonder why this channel was just now brought to my attention. And i'm in love.
I love the way you cut off the clip of you about to eat the cheese at 6:58. It reminds me of Inception's ending, haha-we will never know if you ate the cheese or not (unless you were to reply?). I haven't seen anyone else mention this and I thought it was quite clever.
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....
ah yes a 50+ year old watching youtube
which college has Principa Mathematica in their library..?
@@szamurainagy7644It's great to see old people in here
Every single one?
Because gravity, like relativity, is fake... it's a subjective definition, not a law.
I recommend checking out the graphic novel Logicomix. It’s a historical fiction about Russell’s quest to formalize mathematics, and it’s one of my favorite books 🙂
When your teacher tells you to show your work
Love that video you did a great job explaining 😊
I delved a lot into math history when I was much younger and videos like this want me to read up on it again. There is so much I forgot. Thanks for the vid. Looking forward to the one on Godel.
This is my new favourite video from this channel! Jade is such a great storyteller and she picks great topics
Thanks! I worked really hard on this video so I appreciate that :)
@@upandatom , This seems like a very important subject; so thanks for doing it.
@@upandatom ur awesome
She kinda bad too NGL.
I never thought I would question what the number one even is, absolutely mind blowing video!
I know! I guess I’m too literal. I can see 1. One apple. One chair.
Well, anyone (hopefully) can count [something] or [some other thing], but what is "1" of that thing, in no uncertain words? The issue is defining the number 1, at the most basic level, purely by logic and without the definition being circular (because "this is one apple, because there's a single apple here" is not informative at all). Hell to be quite honest you probably have to start defining the idea of countability and sets before you get to numbers...and it takes a few hundred more pages at least to lay out what addition is.
Besides, it's kind of arbitrary when you think about it. Is one apple still one apple with a few atoms shaved off? What if you stab it? Slice it to pieces? Grow it into a tree that bears more apples? How far do you go before it stops being one apple?
So I was trying to teach abstract algebra to my daughter, and thought I would pick one of the old school ones that is a bit more accessible - van der waarden. Basically I just wanted to intro group theory, ring theory field theory, show some polynomial calculations like gcd, resultant, and see if I could jump over to Galois theory; at least prove abel's theorem.
Well, the first chapter was number system. Integers using Peano axioms. It was fun, (to do the exercises), but much longer than I anticipated.
If I remember correctly, in this approach, 1 + 1 = 2 by definition, but the harder work is to prove 3 = 2 + 1 = 1 + 2
In currently taking a directed studies course in Zermelo Frankes Set Theory, you’re explanations here are spot on!
Your stuff is informative and interesting, but maybe even more importantly, you're a great storyteller which makes it easier to learn. Keep up the good work
As always, a fantastic presentation in a very dry subject, Jade. Years later, you channel is still such a joy to watch!
Simple yet damn elegant way to explain... 🌟
Loved the whole video...
Great video. Especially kitty cameo at 5:29
I'll never forget the one lecture, dealing with examples of arithmetic as a result of ZFC axioms...
My prof wrote down an example of representations of two numbers.
(5 and 10... Yes. He was that patient and pedantic to do all the curly brackets, an yes he ran out of space, after using the full width of the board!)
He then proceeded to go through the algorithmic process of using the 'set theoretic' definition of the symbol '+'.
After fully enforcing and explaining all the axioms/lemmas/theorems, he looked at the board and goes: "huh! Looks like I've proved '10 + 5 = 15'... If any of you want a quick PHD, copy down this."
Best lecture ever! Being pedantic on lower order logical systemae is tedious, yes, but also insightful.
I really doubt he could have written the full expansion of 10 on the blackboard:
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And 15 is much worse, having 81919 characters (braces and commas), there is no way he could have written that by hand
I always thought that 0 = O where O is the empty set and then 1 = {O}, 2 = {{O}, O}, and 3 = {{{O}, O}, {O}, O} and so on in Von Neumann ordinals. How would this be that difficult to right out? Or is there another representation of the naturals that I don’t know?
@@ryanlangman4266 Well, if you look at that order you just wrote down, the length goes from 3, to 7, to 15. I didn't see this at first, until I calculated all the numbers up to 15, and noticed this extended pattern: 3, 7, 15, 31, 63, 127, 255, 511, 1023, 2047, it's always 1 short of 2^(x+1), or:
Len([Von Nueman Ordinal].x) = -1 + 2^(x + 1)
This "only" comes out to 65535, tho, not almost 90k, so I'm not sure what Federico is going on about, exactly.
EDIT: For the curious, just start with ="{0}" in cell D4, and make cell D5 =LEFT(D4,LEN(D4)-1)&","&D4&RIGHT(D4,1). Drag down D5 for as many digits as excel can handle, which in this case is only actually 14 digits before the maximum cell length is reach. The last bit of that function, RIGHT(D4,1), is really just "}".
@@kindlin Oh, of course. That makes sense. I don’t know how I didn’t think of it being exponential growth. Thanks.
I think Frederico may have double counted the O or perhaps used a different representation that had an extra character. At each step >0 you will have 2^(n-1) empty elements, so if you double count that and add it to your calculation for the number of characters, 2^(n-1) + 2^(n+1) - 1 = 5 * 2^(n-1) - 1 = 81919 characters for n = 15. There are many other ways that you could get this number as well, but I think this is the simplest.
I actually prefer the method of not counting the O or the , elements, since neither of them are technically elements of any of the sets, and aren’t technically needed if you want to write quickly. (The empty set is not an element and neither are the commas) so if you only count {} then for all n>0 you end up with 2^n bracket characters. Which is a much cleaner formula.
This is also the fastest possible method you could use to write these ordinals. So, if we assume that their prof. was using this method and could write 6 brackets per second at a constant rate (which is very fast to keep up for very long). They could write the number 15 in approx 2^15 / (6*60) = 91 minutes. Which would make for an extremely long lecture of just watching someone write brackets. But perhaps they simply misremembered, and it was really something like 5+5=10 which could be written in about 3 minutes if you can write 6 brackets per second. Exponential growth is crazy!
Btw, I’m just curious, but why are you using excel notation? (If that’s what it is) That seems much more likely to confuse than simply using mathematical formulae.
11:54
I have always thought maths in this way, that if we have to explain mathematics to some extraterrestrial being then how it is going to be done. And it's really a tough and important task to do at the same time
This idea needs to be explored more. I would love to listen more about that from you.
And as always great video 👍 ma'am
Keep explaining, keep growing
Luckily, this alien understood (non-math-related) English.
This is stupid. You should focus on the culture which makes you have these silly ideas.
Why ho to aliens? Do you have cats in your home, would you ever teach it math. If you tried to your mom would call you mad.
Hello, Thamks for valuable videos related to science. What programmes do you use while creating that kind of amazing videos?
Thanks
I've watched this video 3 times. Your program brings a Joy to my day
One is the loneliest set of all sets containing a number of elements equal to the number one. 🎵
Beautiful! What about {} containing {} containing {} containing {} ...and so on. Lonely? Empty? Zero or infinity?
...{{{{{{{...}}}}}}}...
Great! Always constant high quality video here! On proofwiki you can find the 1+1=2 with Peano axioms, it's a good exercice and humanly manageable (it's a good way to train going back to axioms). I am wondering if there is not a proof done by someone for fun in ZFC, possibly shorter than the russel proof because if I remember well the natural way to map natural numbers to set with zfc is to take 0 = empty set and each integer being the set of the parts of the set we use for the previous integer.
thank you. This really helped me to understand what this book was about.
Hey, I love the video! I was wondering if you read or heard of Hofstadter’s Gödel, Escher, Bach?
Wonderful as always Jade!
When I was a kid I hated mathematics. They were so apathic explaining. Later on HS I started to enjoy this subject.
I wish more teachers could see your videos and find your way of explaining as a model to follow 🌹🌹
maybe you got good teachers or profs then at hs
Really clear, fun storytelling. I am terrible at math but I enjoyed listening, as I do to all your videos. Nice work.
Godel showing up with his incompleteness theorem made me laugh much harder than it should have
Awesome video... Anybody who attended KNUST in Ghana, and studied under Dr. F. T. Oduro will love this and miss the good old professor.
YAY U POSTED!!
I want to bring attention to your wonderful chapter titles, and the fact that they coincide perfectly with the narrative!
Hehe thanks :)
Such an impressive and useful video. Subscribed.
Love your assistant at 5:45
You have an amazing unique quality of being so clear and easy with complex/difficult topics.
My daughter spent a whole damn semester in non-euclidean (mostly hyperbolic) geometry (she was a math major). All too familiar with the Parallel Postulate (I reviewed her papers). They should include this video in the syllabus. Funny though that they bring in Linear Algebra mapping concepts.
All of geometry can be described using various bits of algebra
I love that around 5:35 your cat comes to sit next to you!
AMAZING VIDEO!!!! from the beginning to the end... i wish there was a super like for videos like these
congratulations :)
I would love it if you cover the candidates for modern foundations of mathematics like type theory etc. Also if you can start a series focused on logicians themselves like Wittgenstein,Saul Kripke, Godel etc.
Type Theory is so important to serious computing and programming, it can't be overstated.
One of the weird things about Bertrand Russel, 3rd Earl Russel, is that he was mainly brought up by his grandfather, the 1st Earl Russel, who was twice Prime Mister of the UK the mid-1800's. The first Earl also was sent to meet with Napoleon Bonaparte as an emissary.
Considering that Bertrand Russel was mainly a figure of the twentieth century, passing away in 1970, it always catches me off guard to think that he was brought up by someone who met with Napoleon.
and someone who helped Lincoln by keeping GB out of the civil war. RIP both the first and three earls.
Russell was also Nobel prized in literature and a founder of the Analythical philosophy, and Wittgenstein doctoral advisor. He was a beast.
Aristotle invented analytical philosophy, not Russel
@@keylanoslokj1806 A founder, not the very first founder or pioneer. And the modern one I meant.
I am not sure that I would call their work a "failure" since even though we cannot have consistency or completness at least we know that math can be formalized.
And thanks to this it is now possible for a computer to check if a given proof is correct (coq software) and we are also trying to make it possible for computers to find proofs.
And at 5:34 the start of the show arrives!
In the first lecture at university, our teacher said that we (students) thought that natural numbers were natural, and that he would show that they weren't. We spent the next two months building toward natural numbers. One day, we also reached things like 1+1=2. Then went on gradually to metric spaces, Hilbert space, integrals etc. All with the formalism shown in this video. Doing exams with him was an otherworldly experience. On one occasion, my exam took close to 9 hours. He had three students in the room, and alternated between us all day, to give us a mark in the evening. He had the philosophy that if a student could present and prove everything on the given topic, then the student reached the equivalent of a D score. At that point, the "discussions" started...
That sounds marvelous! Can you recall the name of the course or any textbooks used? I'd love to learn more.
Is this... The true hell?
@@BJ52091 I don't think it will be useful to you, as it was in Hungarian. It was a 5 semester "Introduction into the foundation of calculus" course at university, by János Kristóf. A slightly abridged version of the pdf is available online from his uni page, if you want to have a look at the mathematical formalism.
@@zoltanposfai3451 Fascinating! Was this an undergraduate, or graduate course (towards Master's or PhD)? How many total students were in your class?
@@bennettjoseph9970 Undergrad. First five semesters in the physicist faculty. We had a class of about 40. This was one of the subjects where the university made sure that no matter how many students started, by second year, the classes were trimmed down to around 40. (The uni got the money based on the numbers admitted, and not not students attending. So, they were incentivised to bring down the admittence criteria unreasonably low, but then get rid of most students to keep the good international stats and standard for those who made it.)
Hi Jade, thank you very much for your extremely interesting episode. The only thing I would suggest is the pronunciation of the "Principia" that would be probably similar to "principle" not "prinkiple".
Latin "c" is frequently pronounced as "s" (in fact as "ts") but sometimes as "k", like in caecum/cecum ("tse-kuhm" or in English form of pronunciation
"seek-uhm"). I suspect however that Italians would probably like to pronounce this as "preen-chip-yah" to add to the mess with the different understanding of Latin pronunciation :-)
Lrn2Latin pls
That is incorrect. The latin "c" is pronounced as "k" in this case. "Prinkipia" is the Latin pronunciation from the Classical period, "Princhipia" is Church Latin and Italian, and "Prinsipia" is the modern English accent.
The Greek zeimbekiko song in the Euclides section was a nice touch
Loved the bouzouki music while talking about the Greeks. I'm Greek and I'm pretty sure we didn't have bouzouki back then. We did have were pipes and weird scales called tonoi, akin to modern modes (Dorian, Frygian, Mixolydian, those things)
You are clearly a very ancient Greek
Maybe in this context, it is also interesting to mention the work of Nicolas Bourbaki, not a real person, but a collective of French mathematicians whose goal it was to document mathematics in a formally consistent way.
Good video, BUT I found your blanket statement of non-Euclidean geometry, like the 270 degree triangle being unobservable things. It is very easy to draw triangles onto a ball or any other spherical object to observe this geometry. Non-Euclidean simply means that the background e.g. the paper on which lines are draw on isn't flat.
Thank you for placing non-adsense at front/back/both of video rather than interrupting somewhere between, so I can justify liking the video and listening to offer.
Aside that, your method and presentation ever delight and inform :>
Suddenly I'm thinking about what it has been like to help my son with his math homework. It takes half a page to do the simplest problems. Learning the method of the solution appears to be more important than getting to a correct conclusion.
If you can get the correct conclusion, but do not understand how you got there, then it is far more difficult to build off it.
Learning the methodology of math is more important. Really learning math is really learning an extremely logical way to think.
Please more logic videos! This is a great niche you're serving that other math channels don't hit hard enough. And so often when I do see it hit, it's not modern logic, or it's not formal logic. Thank you so much!
Also, how far did you get? I think I read the first half of volume 1 before Principia started collecting dust for me.
Thank you! Ok I'll consider it, I didn't know it was a niche people were interested in!
And wow half way through is excellent, really hats off to you. I read the first chapter and gave up.
@@upandatom What's your academic background?
Ha! I just pulled out the book. My notes suddenly stop at page 180. Significantly less than half-way. But I wouldn't wish more than that on anyone else. Also don't know how big the logic audience is. Probably something wrong with me!
isn't this a mathematics video? or are you talking about mathematical logic?
Yes, mathematical logic, but if you're seeing it as distinct from non-mathematical logic, then you more or less may be a victim of what I'm seeing in many presentations of logic. Formal truth/false based logic with logical operations (and/or/not, etc.), mixed with set-theory ('for all x in such-and-such', 'for some x in such-and-such') should be stressed to the public as the first presentation of the field. Instead, I'm seeing little verbal riddles, Socrates, and Aristotle: presented as though they were state-of-the-art. I'm seeing logical fallacies stressed (e.g. ad-hominem, straw-man). These are legit to discuss, but often presented as though a listing of these gives you a good idea of the field. I'm seeing applications in debates and arguments, to knock down an opponent, as opposed to it being a tool to seek out deeper truths in a more positive sense. And of course I'm often not seeing it being presented together with set-theory, the latter making it powerful enough to allow it to become the foundation to build the vast majority of mathematics, which is the story of Principia Mathematica. And I'm seeing kids and a society uncomfortable with proofs. When a good education in logic and set-theory may make this more natural, and have us all be much better thinkers in a way that won't be compensated or made obsolete by a calculator.
Thank you for your explanation. I am not going to say I understood all, but I did get some idea of why 1+1 = 2 is not as easy as one thinks. There is also the fact that people will insists on 1+1 = 4 just to contradict for contradicting and fobbing one off.
a equals b: a = b
multiply both sides by a: a^2 = ab
subtract both sides by b^2: a^2 - b^2 = ab - b^2
refactor: (a+b)(a-b) = b(a-b)
divide both sides by (a-b): a+b=b???
algebraically correct but last step is division by zero
Your cat seems to get interested at about 5:46.
Always great to see a new video from you. I've heard this story before, however, never in such a easy to understand way. Excellent job!
To be quite precise it should be written as 1+1=2 (mod n) n>2. because if n=2 then 1+1=0(mod2). Your expression could mean 1+1=2 (mod 10) but then 12=2.
Thanks for video on principia mathematica...I could understand how set theory was used to explain large statements in shirt scientific symbols
In graduate school more than 50 years ago, I took a course on PM *1-*56 (1962 Cambridge paperback edition) in a philosophy department and then took a follow-up course on Godel's impossibility (incompleteness) theorem the following semester. That is what happens when you attempt to construct a formal system that is complete and consistent. C'est la vie! Also, the theory of types I found to be contrived. I was neither a graduate student in mathematics nor philosophy, but I thoroughly enjoyed this extracurricular activity.
most of Bertrand Russell's ideas are contrived imo. his famous paradox is literally because of making abstract objects into predicates....which is a huge no
If this video interests you, I recommend reading Gödel, Escher, Bach: an Eternal Golden Braid, by Douglas Hofstadter. It won the Pulitzer prize and is a must read. A masterpiece in literature.
Completely agree and commented a similar remark. ☺️
One of my favorite books.
I've read this book several times. It is an absolute masterpiece.
your videos are really fun and informative
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.
very great story...
Footnote: This is from steiner and for the very longest time caused me great anxiety showing that mathematics is divorced from reality...however he was using this example as a mode to get persons to think,feel,and will critically....there are ways to cogitate over mathematics that shows that causal active power is available. if you look at rudolf steiners other works on mathematics he provides other counter examples implying that there are no limits to knowledge and that the only factor needed is will through and through
Set theory and Foundations have always interested me. I enjoyed your video on Russell's Paradox and others on issues with infinity like Gabriel's Horn. It would be cool to see your opinion on different types of set theory (ZF, ZF+ Choice, NBG) and how Russell's Paradox relates. Plus your take on Gödel. Maybe even Inaccessible and Surreal Numbers (ok, I'm getting greedy now)
I've been thinking about a video on surreal numbers actually...
@@upandatom yay!!
Even though I personally don't like math, you did a pretty good job explaining what it's all about. Truthfully I've had this thought of math having a formal system cross my mind more than a few times but no matter how hard I thought about it, nothing really seemed to make sense objectively so, just like Russell and co I also gave up on trying to understand it, instead deciding to settle on the fact that math is nothing but a fancy name for a game born purely from abstraction or abstract thinking. Don't get me wrong! I'm no mathematician, however regardless i find it universally agreeable that the inconsistencies which exist in this field are way too conspicuous to miss.
Even though you don't like math , it is still your boss! Like it or not it controls you and like your boss will fire your ass if you disobey it.
What a tiresome, bullying response, No.6. Very revealing.
Of course, you could've just meant it to be lighthearted.... In that case, I'm sorry. But an emoji may have helped 🙂
Keep up I’m learning so much ❤
5:52 I've never heard it said so plainly, how (theoretical) mathematicians killed physics:
observations unnecessary, the logical consistency of a model is all that's needed.
this works great for purely mathematical landscapes but applying mathematics to the physical universe requires closing the loop on the process...actually testing the models with physical observations AND abandoning them if their output isn't in agreement with/can't predict observable phenomena.
too many otherwise intelligent ppl can't step back from their pet theory/model when it doesn't produce accurate predictions.
what does an infinity in a model's output mean? that density is infinite inside a black hole or that the theory/model is mistaken...the former is physically impossible yet we hear the concept talked about as if its veracity is self evident & that it has physical meaning 😮
thx for the insightful video ma'am!
I dont understand why this channel is so underrated. it should have millions of views per video, Jade and her team do an excellent job. Another great video, thanks guys.
Thank you so much!
It is because Math is racists ahahahahahahahahahahaha.
This was wonderful. I remember first hearing about this book, a several hundred page proof that 1+1=2, from my electricity and magnetism professor in college. I remember being both amused and completely fascinated.
1 + 1 = 2 in geometry means vertex + vertex = vertex . Because vertex means digital root = 1 , edge means "digital root = 2" , so 3,5,6 are edges but 2^i are vertices of simplex.
So digital root is dimension of number. ( d-vertex simplex and d-digit binary number system of course are the same thing).
How about this statement: "The empty set {} contains itself infinitely many times."
@@DarkSkay lets d=3, then: 000 - is externity of triangle; 111 is internity of triangle; 001, 010, 100 (digital root=1) are vertices of triangle; 110,101,011(digital root=2) are edges of triangle. So any binary number system(geometrically simplex) consist of zero (geometrically externity), infinity (geometrically internity) and numbers (geometrically faces). "Go to infinity" ( your "infinitely") is impossible for fixed "d" because numbers lies directly between zero and infinity and can go only around them.
@@DarkSkay If you try to do something infinitely times you "open" infinity and break the law: "Infinity is closed but zero is open".
Euclid's Fifth Axiom (I.e. Parallel Postulate) actually reads: If two straight lines in a plane are met by another line, and if the sum of the internal angles on one side is less than two right angles (i.e. less than 180 degrees), then the straight lines will meet if extended sufficiently on the side on which the sum of the angles is less than two right angles. If you think about this, this should actually be called the Triangular Postulate. It essentially observes that if the lines are not parallel, they form two sides of a triangle, and the transversal (the line that crosses them) form a triangle.
"Bertrand and Russell weren't trying to prove 1+1=2, they were trying to prove..." KITTY!
Sorry, you were saying?
You are really doing a great job. I'm a physics undergraduate but I also love math and your videos really gets me more interested in fundamental math and logic. We are currently doing real analysis and complex analysis in college, just started with real math and loving it so far 😀 Keep making these videos, you're an inspiration to us!
Fascinating stuff if you can get the hang of it! Thank you for watching and good luck in your degree :) I did a physics degree too but these days seem more absorbed in abstract math!
@@upandatom That's relatable, and thank you so much!!
You should see if your university has a class in the Philosophy of Mathematics that you could pick up. I did when I was in uni. It had no pre-reqs, but it was a 4th year class.
@@Crushnaut thanks for the nice suggestion, I don't think my college has a philosophy of math class but I'll try to learn something online, it really interests me!
14:04 P. 379 Wtf! Was that written in Alien Language?
ikr 😞
Written in axiom that no one can understand ( except a hard-core mathematician )
I had heard a bit about Russel and Whitehead's Principia and the ridiculous effort they went through to arrive at 1+1=2 and I had heard about Godel from a book I bought called Godel Escher and Bach The Eternal Golden Braid or something like that but I did not know that poor Godel was their party pooper! 😁😁
I think Schrödinger’s cat escaped from the box during this video.