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.
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!
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!
@@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.
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."
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.
@@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.
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!!!
@@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.
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.
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 !
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.
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.
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
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!
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.
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.
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
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.
Explaining not just PM, but also its inherent shortcomings, within 17 minutes is a marvellous achievement. Great video, and very clear, thank you, Jade!
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.
@@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.
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. 🙂
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.
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.
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
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.
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.
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.
the quantity property of each apple is the same, and addition deals with the quantity property of the apples, and thus empirically, 1 apple + 1 apple = 2 apples every single time.
@@miff227 :: The point of Russel's thinking is that numbers are human inventions, except perhaps natural numbers - but the number abstraction is extremely useful.
I can demonstrate that 1+1 does not always equal 2 using only a few paragraphs. To start, if we add 1 atom plus 1 universe... what does that equal? The problem here, you might argue, is that I'm adding apples and oranges (so to speak). But the problem with that suggestion is that if we change how we define those objects we can easily conclude that we have 2 fruit. More over, if I'm adding two actual apples, there is a very real sense in which [1 apple] + [1 apple] does not equal 2. Because the apples themselves are different sizes. Thus, the only reason they equal 2 is because we are turning two things (which are clearly different) into a concept that we can call the same. In pure mathematics, the unit is "theoretically" identical. 1 serves as the unit of measure, which defines the nature of all the other numbers. But this only works because we are now dealing (once again) with a pure concept. 1 raindrop + 1 raindrop = 1 raindrop 1 man + 1 woman = 1 couple. 1 heap + 1 heap = 1 heap 1 infinity + 1 infinity = 1 infinity All of these cases are true because of the peculiar nature of the concepts that we are adding. The man plus woman equals couple works because of the definition of a couple. Raindrops work because raindrops have no distinct size. In other words, a distinct magnitude is not part of the definition of a raindrop. The same is true for a heap and for infinity. All that infinity means is that something is endless. And, in my opinion infinities of different sizes can be added together to produce other infinities, in exactly the same way that heaps (and raindrops) of different sizes can be added together to create heaps and raindrops... which (conceptually speaking) are identical... just as two apples (of different sizes) are still conceptually identical, at least in terms of being apples.
For that matter, how do you define an object? You have a rock (1 rock) and you drop it and break it in two. Now you have two rocks. And so on (though at some point we cease to call it a rock and call it dust but that is a viewpoint of scale...). So objects are two or more particles connected together in some fashion. Now imagine I had a molecule in one hand and it was connected to another molecule which was 30,000 light years away. By definition, I would have an object as they are connected. I keep looking for the latter... It would seem that the important part of the definition is the connecting line between the two objects, or, as I call it, the "third" pole the objects are poles and the line that connects them must exist or we do not have an object. But if we were to ask what this connecting line is the best we can come up with is "space". OMG! What is definition of space? Could it be that it is that which keeps poles apart? All I know is no physics text defines "space." Uses the words a lot and we have experience with it but no definition. But lookee here! If we take the space away, the particles collapse into one another. This mimics the behaviour of a number of phenomenon we observe like magnets. Could it be that a magnet is creating and destroying space simultaneously? Food for thought certainly.
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....
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: {{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{},{{},{{}}}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{{},{{}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{},{{},{{}}}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{{},{{}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}}}
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
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.
Isn't that so because of definitions? 1 plus 1 equals 2 because the defined it to be true? If a meaning of 1 or 2 or plus or equal were to be different, then so would the equation
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 🙂
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.
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.
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...
@@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.
@@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.)
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.
@theshadeow5103 true, but perhaps we might add, that the alien will only be able to grasp what 1 means if they already possess the concept of "one-ness" in their mind (from birth); because if they didn't have that concept in an innate way, no amount of life-experience (showing it 1 apple versus 2 apples; 1 duck versus 3 ducks) could teach it that concept. It seem strange to suggest this, after all humans and a lot of mammals (if not non-mammals) seem to have an intuitive grasp of this simple concept, but who knows how our alien in this imaginary scenario "evolved" and what concepts its mind has or lacks
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.
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.
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 🌹🌹
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.
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.
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.
At 5:34 the cat sits down, looks at the book and is like "Jade, you've got my book." That cat is brilliant; I hear he's working on a proof that 2+2 = 4. So far, the proof is up to 157 pages. Smartest cat ever!
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 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.
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)
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.
(Holds up one finger, looks at it for a second, holds up second finger...thinks for a second, looks at super super long paper, looks back at fingers...looks puzzled 🧐)
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.
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
Good. You can prove in 379 pages that one plus one equals two. Two plus two still equals five though. - BROTHER, Big (1984). Jokes aside: after watching the video, it basically explains the 2+2=5 from Orwell's book too (wich I've always had a hard time trying to explain to people). I loved it! Anxious for the volume 2 (of the video, not the Principia Mathematica =S ). Cheers!
One is an abstract idea. If I add an abstract 1 to another 1 those two ones aren’t the same. If they were the same you would have to clone the first 1. But then the second 1 would just be similar to the first one. In fact it would be another one and we couldn’t call it 1 because 1 already described the first 1. We simply accept that misconception because it is functional. Here we have an apple and then we have another one so we have two apples. But apple is an abstract definition of similarities of actual objects. There can never be 2 of the exact same apple. They would not only have each atom in the same relative position, also the absolute position would need to be the same. One also has no size. Even if you add up the size of every number the total would be 0. Your cognition wants to tell you that 1 starts at 0.0 or even smaller numbers and goes until 1. But that’s wrong. That would just be summing up a lot of numbers which each don’t have any size. Math is just a functional illusion that we can use to describe the world in a meaningful way.
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?
The book was based on logic, but then, who decided that logic (true/false) is the most pure form that can cause no ambiguity or paradoxes? I mean, even in nature, a quantum state or qbit can have a state inbetween true and false. By saying that true = not false feels like introducing a new axiom. We also thought once that all real numbers were "complete" until we discovered complex numbers. Anyway, the book also goes far above my understanding so thank you Jade for this video! ❤
A quantum state is OUR knowledge about the system. Not the system itself. Stop being brainwashed by TH-cam videos into thinking that a) you can learn Quantum Mechanics from TH-cam and b) Quantum Mechanics is somehow magical or paradoxical. Quantum Mechanics is still work in progress, and there are many interpretations of it, we just don't know which one is the correct one. If we start arguing if logic is or isn't, the most pure form that can cause no ambiguity or paradoxes. Then we have to spend 1000 years talking about what is "is" what is "or" what is "isn't" what is "," what is "the" what is "most" what is "pure" what is "form" what is "that" what is "can" what is "cause" what is "no" what is "ambiguity" what is "paradoxes" and what is ".". And then what is ... what is "is" what is what is "or" wh... etc.
Why is quantum mechanics the go-to response by cranks to dismiss anything logical anyone says? Like it's some sort of magic? I'm tired of seeing comments like this. A qbit isn't "between true and false"; it's a linear combination of two states. If you put a tomato and potato in your shopping cart, the contents of your cart isn't "something between a tomato and a potato", the contents of your cart is a linear combination of a tomato and a potato: 1 tomato + 1 potato. It's the two things TOGETHER. It's not magical nor illogical. And any "number system" (i.e. any ring) can be made larger, not just the reals, and even the complex numbers. I could right now in this comment make something bigger by defining C[x]/(x^3+4x-9). The thing that makes the complex numbers more "special" than the reals is that they're algebraically closed. And I could ALSO make a bigger "number system" than it that is also algebraically closed. You should either get an accredited degree in math and/or physics, or stop talking about things you haven't studied at all. Because as it stands, all of this is "far above your understanding" as well. Delete your comment and stop making more like it.
en.wikipedia.org/wiki/Logic#Systems_of_logic A quick visit to Wikipedia shows other systems of logic that may or may not have the "true = not false" properties.
Qbits are kindof irrelevant, but if you're looking for different ideas about logic, consider looking up intuitionistic/constructive logic systems. In these types of systems, a proposition is true if and only if there's a proof. A "false" judgement for a proposition is just "sugar," e.g. saying "A is false" is actually saying that "the existence of a proof of A implies a contradiction". One difference from classical logic is that you no longer have the law of the excluded middle (or equivalently, double-negation elimination), so you can't just do truth-table proofs where (as you point out) true = not false. Another difference is that proofs in intuitionistic logic correspond to algorithms, so a proof of an existential statement actually gives a way (not always the best way) to produce a the thing you're asserting exists.
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.
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!
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.
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
Cool video, I like the philisophical angle. Non-euclidean geometries would be an interesting thing to hear more about if you were looking for future topics.
I’d be interested in covering it further but there are already a lot of great videos about it with awesome simulations and visuals, which are not my strength unfortunately. But who knows maybe I’ll go crazy and learn to code one day :)
2 is a symbol. It needs to be defined somehow for us to know what it is. 1 + 1 = 2 can be taken as a definition of 2 (no proof required), depending on where you start the formalization of arithmetic. You will only have to prove that 1 + 1 = 2 if 2 is defined differently. For instance, I like the way natural numbers can be built up from definitions over set operations. Define: 0 := {} 1 := {0} X + 1 := X U {X} Then also define: 2 := 1 + 1 = 1 U {1} = {0, 1} 3 := 2 + 1 = 2 U {2} = {0, 1, 2} 4 := 3 + 1 = 3 U {3} = {0, 1, 2, 3} 5 := 4 + 1 = 4 U {4} = {0, 1, 2, 3, 4} ... ℕ = {0, 1, 2, 3, 4, 5, ... } = ∞
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)
3:06 could have swore i invented more of that by accident in my geometry class (Once made a triangle with 4 sides, and the third one was imaginary side) God. What did i do?
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!
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!
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.
there's a general problem i have with proofs: how can we prove that there's no error in the proof? in practice, "to prove" really only means "to convince"
In practice it is common to do two things: 1. Agree on the “rules” 2. Follow the “rules” The typical “rules” include but are not limited to: writing line-by-line, using commonly used notation when possible, using the “normal” logic, providing proofs that don’t rely on intuition (but can be motivated by one). Agreeing on the rules is done collectively by the people. For example, we use the symbol “2” for the number two, because everyone else understands it. Another example is from logic. “If we have some starting assumptions S and we now assume A, such that now B logically follows; then we can conclude that given S we have proven A->B.“ It is common practice to list all “unusual” assumptions before you give the proof. This way the readers can combine “normal” assumptions with your assumptions and follow the “normal” rules to assure the validity of your proof.
@@ribosomerocker but how can you prove to me that those theorem provers are perfect? i claim you can't. even if they get it right 99.999...% of the time, there will be an error rate > 0. the more complex a proof gets the higher the chance that an error was made and overlooked at a certain step. if you claim they are 100% correct, then you need to demonstrate the mechanism that guarantees this - but if this mechanism is "human brain", then we can never reach 100%. because if brains were perfect, we wouldn't need theorem checkers in the first place.
14:55 Lesson learned: If you fail in logic, become a politician. I guess we have many failures in logic becoming politicians, and why most politicians fail at making sense.
When I first learned that proof in college, I was a bit stunned at how much it took to prove it. Another fun one to prove is 2n + 1 is odd for all integers.
It doesn't take 379 pages to prove 1+1=2 unless you choose a formalism with an extremely esoteric agenda, such as proving Gödel's theorem later on. I suppose in a year or two I will ask ChatGPT 5.0 to furnish a formal system capable of proving its own incompleteness, where the proof of 1+1=2 is also concise. And it will attempt to clarify: Do you wish to stick to set theory as the underlying framework, or am I free to propose something else?
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.
Most of this is just why I simply disagree with modern math. Logic and reasoning shouldn't be valued more than what's observable. What's observable should take the main priority. Modern math is just like a game where a few people invent the rules then force others to follow the rules of their game, trying to make their math real rather than trying to understand real math. Modern, advanced math is just a pyramid scheme sold to kids to learn so that they're only useful teaching it to other kids so they can teach it. It's not helping understand or describe the world - it's creating their own world. So many college-level math courses rely on axioms and inferences that are arbitrary and only exist because without them, all the other math they invented doesn't work. (3+3)^3 is both (9+9) AND 9^3 with not a single one being correct until you make up some random rule as to why you're right instead of showing that you're right with observable proof instead of fancy words and symbols
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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.
Wait until they hear about 1+2
Wait can it be proved using mathematical induction? 😢
Wait can it be proved via mathematical induction? 😢😂
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.
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.
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.
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
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 ?
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.
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
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
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.
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!
5:29 math explained so well, even a cat will show up and understand it
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.
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.
0:12 or can we?
*VSauce theme plays*
*Theme playing*
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
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.
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
the quantity property of each apple is the same, and addition deals with the quantity property of the apples, and thus empirically, 1 apple + 1 apple = 2 apples every single time.
@@miff227 :: The point of Russel's thinking is that numbers are human inventions, except perhaps natural numbers - but the number abstraction is extremely useful.
5:40 cat is inarguably the best part of the video. Its self evident
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 can demonstrate that 1+1 does not always equal 2 using only a few paragraphs. To start, if we add 1 atom plus 1 universe... what does that equal? The problem here, you might argue, is that I'm adding apples and oranges (so to speak). But the problem with that suggestion is that if we change how we define those objects we can easily conclude that we have 2 fruit. More over, if I'm adding two actual apples, there is a very real sense in which [1 apple] + [1 apple] does not equal 2. Because the apples themselves are different sizes. Thus, the only reason they equal 2 is because we are turning two things (which are clearly different) into a concept that we can call the same. In pure mathematics, the unit is "theoretically" identical. 1 serves as the unit of measure, which defines the nature of all the other numbers. But this only works because we are now dealing (once again) with a pure concept.
1 raindrop + 1 raindrop = 1 raindrop
1 man + 1 woman = 1 couple.
1 heap + 1 heap = 1 heap
1 infinity + 1 infinity = 1 infinity
All of these cases are true because of the peculiar nature of the concepts that we are adding.
The man plus woman equals couple works because of the definition of a couple. Raindrops work because raindrops have no distinct size. In other words, a distinct magnitude is not part of the definition of a raindrop. The same is true for a heap and for infinity. All that infinity means is that something is endless. And, in my opinion infinities of different sizes can be added together to produce other infinities, in exactly the same way that heaps (and raindrops) of different sizes can be added together to create heaps and raindrops... which (conceptually speaking) are identical... just as two apples (of different sizes) are still conceptually identical, at least in terms of being apples.
For that matter, how do you define an object? You have a rock (1 rock) and you drop it and break it in two. Now you have two rocks. And so on (though at some point we cease to call it a rock and call it dust but that is a viewpoint of scale...). So objects are two or more particles connected together in some fashion.
Now imagine I had a molecule in one hand and it was connected to another molecule which was 30,000 light years away. By definition, I would have an object as they are connected.
I keep looking for the latter...
It would seem that the important part of the definition is the connecting line between the two objects, or, as I call it, the "third" pole the objects are poles and the line that connects them must exist or we do not have an object.
But if we were to ask what this connecting line is the best we can come up with is "space".
OMG!
What is definition of space?
Could it be that it is that which keeps poles apart?
All I know is no physics text defines "space."
Uses the words a lot and we have experience with it but no definition.
But lookee here!
If we take the space away, the particles collapse into one another.
This mimics the behaviour of a number of phenomenon we observe like magnets. Could it be that a magnet is creating and destroying space simultaneously?
Food for thought certainly.
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'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.
0:04 "Hey guys! Spirit Of The Law, here."
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?
...{{{{{{{...}}}}}}}...
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.
Isn't that so because of definitions? 1 plus 1 equals 2 because the defined it to be true? If a meaning of 1 or 2 or plus or equal were to be different, then so would the equation
And at 5:34 the start of the show arrives!
YAY U POSTED!!
Me, a physicist: Well, if you have one of something, and you add another one, then you have two. 🤷🏻♀️😂
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 🙂
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.
I love that around 5:35 your cat comes to sit next to you!
In currently taking a directed studies course in Zermelo Frankes Set Theory, you’re explanations here are spot on!
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.
She has been calmly waiting for terrance to appear.
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.
@@TheZoltan-42 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.)
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.
@theshadeow5103 true, but perhaps we might add, that the alien will only be able to grasp what 1 means if they already possess the concept of "one-ness" in their mind (from birth); because if they didn't have that concept in an innate way, no amount of life-experience (showing it 1 apple versus 2 apples; 1 duck versus 3 ducks) could teach it that concept. It seem strange to suggest this, after all humans and a lot of mammals (if not non-mammals) seem to have an intuitive grasp of this simple concept, but who knows how our alien in this imaginary scenario "evolved" and what concepts its mind has or lacks
1 rain drop and one rain drop equals 2 rain drops. Until they go together and become one .
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
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.
1 + 1 can equal 3, if you've been to room 101 at the Ministry of Love.
According to non Euclidean neo Orwellian geometry
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
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.
The way you simplify and explain the matter is really fantastic! Thanks for the vid.
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.
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.
At 5:34 the cat sits down, looks at the book and is like "Jade, you've got my book." That cat is brilliant; I hear he's working on a proof that 2+2 = 4. So far, the proof is up to 157 pages. Smartest cat ever!
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.
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.
I think Schrödinger’s cat escaped from the box during this video.
I love watching your videos. They are simple enough to understand yet open up vast areas to keep researching.
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
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.
Your cat seems to get interested at about 5:46.
I want to bring attention to your wonderful chapter titles, and the fact that they coincide perfectly with the narrative!
Hehe thanks :)
(Holds up one finger, looks at it for a second, holds up second finger...thinks for a second, looks at super super long paper, looks back at fingers...looks puzzled 🧐)
As always, a fantastic presentation in a very dry subject, Jade. Years later, you channel is still such a joy to watch!
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.
"Bertrand and Russell weren't trying to prove 1+1=2, they were trying to prove..." KITTY!
Sorry, you were saying?
Really clear, fun storytelling. I am terrible at math but I enjoyed listening, as I do to all your videos. Nice work.
You have an amazing unique quality of being so clear and easy with complex/difficult topics.
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
Or can we , ( Vsauce)
Good. You can prove in 379 pages that one plus one equals two. Two plus two still equals five though. - BROTHER, Big (1984).
Jokes aside: after watching the video, it basically explains the 2+2=5 from Orwell's book too (wich I've always had a hard time trying to explain to people). I loved it!
Anxious for the volume 2 (of the video, not the Principia Mathematica =S ). Cheers!
One is an abstract idea. If I add an abstract 1 to another 1 those two ones aren’t the same. If they were the same you would have to clone the first 1. But then the second 1 would just be similar to the first one. In fact it would be another one and we couldn’t call it 1 because 1 already described the first 1.
We simply accept that misconception because it is functional. Here we have an apple and then we have another one so we have two apples. But apple is an abstract definition of similarities of actual objects. There can never be 2 of the exact same apple. They would not only have each atom in the same relative position, also the absolute position would need to be the same.
One also has no size. Even if you add up the size of every number the total would be 0. Your cognition wants to tell you that 1 starts at 0.0 or even smaller numbers and goes until 1. But that’s wrong. That would just be summing up a lot of numbers which each don’t have any size.
Math is just a functional illusion that we can use to describe the world in a meaningful way.
The hell is an “abstract 1”?
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?
The book was based on logic, but then, who decided that logic (true/false) is the most pure form that can cause no ambiguity or paradoxes? I mean, even in nature, a quantum state or qbit can have a state inbetween true and false. By saying that true = not false feels like introducing a new axiom. We also thought once that all real numbers were "complete" until we discovered complex numbers. Anyway, the book also goes far above my understanding so thank you Jade for this video! ❤
A quantum state is OUR knowledge about the system. Not the system itself.
Stop being brainwashed by TH-cam videos into thinking that a) you can learn Quantum Mechanics from TH-cam and b) Quantum Mechanics is somehow magical or paradoxical.
Quantum Mechanics is still work in progress, and there are many interpretations of it, we just don't know which one is the correct one.
If we start arguing if logic is or isn't, the most pure form that can cause no ambiguity or paradoxes. Then we have to spend 1000 years talking about what is "is" what is "or" what is "isn't" what is "," what is "the" what is "most" what is "pure" what is "form" what is "that" what is "can" what is "cause" what is "no" what is "ambiguity" what is "paradoxes" and what is ".". And then what is ... what is "is" what is what is "or" wh... etc.
Why is quantum mechanics the go-to response by cranks to dismiss anything logical anyone says? Like it's some sort of magic? I'm tired of seeing comments like this.
A qbit isn't "between true and false"; it's a linear combination of two states. If you put a tomato and potato in your shopping cart, the contents of your cart isn't "something between a tomato and a potato", the contents of your cart is a linear combination of a tomato and a potato: 1 tomato + 1 potato. It's the two things TOGETHER. It's not magical nor illogical.
And any "number system" (i.e. any ring) can be made larger, not just the reals, and even the complex numbers. I could right now in this comment make something bigger by defining C[x]/(x^3+4x-9). The thing that makes the complex numbers more "special" than the reals is that they're algebraically closed. And I could ALSO make a bigger "number system" than it that is also algebraically closed.
You should either get an accredited degree in math and/or physics, or stop talking about things you haven't studied at all. Because as it stands, all of this is "far above your understanding" as well.
Delete your comment and stop making more like it.
en.wikipedia.org/wiki/Logic#Systems_of_logic
A quick visit to Wikipedia shows other systems of logic that may or may not have the "true = not false" properties.
Qbits are kindof irrelevant, but if you're looking for different ideas about logic, consider looking up intuitionistic/constructive logic systems. In these types of systems, a proposition is true if and only if there's a proof. A "false" judgement for a proposition is just "sugar," e.g. saying "A is false" is actually saying that "the existence of a proof of A implies a contradiction".
One difference from classical logic is that you no longer have the law of the excluded middle (or equivalently, double-negation elimination), so you can't just do truth-table proofs where (as you point out) true = not false. Another difference is that proofs in intuitionistic logic correspond to algorithms, so a proof of an existential statement actually gives a way (not always the best way) to produce a the thing you're asserting exists.
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.
6:30 No. It hinges on your definition of eat and whether eating part of it and stopping constitutes not eating it or eating it. It is not so clear.
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
Cool video, I like the philisophical angle. Non-euclidean geometries would be an interesting thing to hear more about if you were looking for future topics.
I’d be interested in covering it further but there are already a lot of great videos about it with awesome simulations and visuals, which are not my strength unfortunately. But who knows maybe I’ll go crazy and learn to code one day :)
@@upandatom maybe do a collaboration with CodeParade 🤭
2 is a symbol. It needs to be defined somehow for us to know what it is.
1 + 1 = 2 can be taken as a definition of 2 (no proof required), depending on where you start the formalization of arithmetic.
You will only have to prove that 1 + 1 = 2 if 2 is defined differently.
For instance, I like the way natural numbers can be built up from definitions over set operations.
Define:
0 := {}
1 := {0}
X + 1 := X U {X}
Then also define:
2 := 1 + 1 = 1 U {1} = {0, 1}
3 := 2 + 1 = 2 U {2} = {0, 1, 2}
4 := 3 + 1 = 3 U {3} = {0, 1, 2, 3}
5 := 4 + 1 = 4 U {4} = {0, 1, 2, 3, 4}
...
ℕ = {0, 1, 2, 3, 4, 5, ... } = ∞
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!!
Love to see your meow meow make an appearance, you should have that happen more often. Every video is better with cats!
3:06 could have swore i invented more of that by accident in my geometry class
(Once made a triangle with 4 sides, and the third one was imaginary side)
God. What did i do?
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!
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 🙂
Well, that just completely blows away the conjecture that we used up 379 pages to prove that 1+1=2 because we really just needed to use up more trees.
there's a general problem i have with proofs: how can we prove that there's no error in the proof?
in practice, "to prove" really only means "to convince"
by rigorously checking the logic. theres a reason theorem provers exist, and utilise type checking that's rigorously checked as well.
In practice it is common to do two things:
1. Agree on the “rules”
2. Follow the “rules”
The typical “rules” include but are not limited to: writing line-by-line, using commonly used notation when possible, using the “normal” logic, providing proofs that don’t rely on intuition (but can be motivated by one).
Agreeing on the rules is done collectively by the people. For example, we use the symbol “2” for the number two, because everyone else understands it.
Another example is from logic.
“If we have some starting assumptions S and we now assume A, such that now B logically follows; then we can conclude that given S we have proven A->B.“
It is common practice to list all “unusual” assumptions before you give the proof. This way the readers can combine “normal” assumptions with your assumptions and follow the “normal” rules to assure the validity of your proof.
@@ribosomerocker but how can you prove to me that those theorem provers are perfect? i claim you can't. even if they get it right 99.999...% of the time, there will be an error rate > 0. the more complex a proof gets the higher the chance that an error was made and overlooked at a certain step. if you claim they are 100% correct, then you need to demonstrate the mechanism that guarantees this - but if this mechanism is "human brain", then we can never reach 100%. because if brains were perfect, we wouldn't need theorem checkers in the first place.
@@HoD999x Mathematics. That was easy.
@@ribosomerocker mathematics is not a physically available device. how do you actually *apply* it *reliably*?
5:10 "Principia" is pronounced with a soft "s" not a hard "k". It's "prinSipia" not "prinKipia".
6:23 ~ melt the cheese, drink it! 🧠
Don’t be fooled. This is anunnaki propaganda to distract from the true geometries of Terryology.
The video is excellent, but given the conclusions a more appropriate title would rather be "Why 379 pages were NOT enough to prove that 1+1=2"
5:28 Even the cat is interested.
1:36 :"pure and perfect"
Gus:👀
2:28 that is not the 5th axiom
14:55 Lesson learned: If you fail in logic, become a politician.
I guess we have many failures in logic becoming politicians,
and why most politicians fail at making sense.
Eureka ! I found a way to make sense of politicians.
Politician = nonsense.
Politician + politician = 2 x nonsense.
Where the number of politicians increase so does the nonsense.
Please carry some clothes along. Junk need not to be Jeopardized.
When I first learned that proof in college, I was a bit stunned at how much it took to prove it.
Another fun one to prove is 2n + 1 is odd for all integers.
It doesn't take 379 pages to prove 1+1=2 unless you choose a formalism with an extremely esoteric agenda, such as proving Gödel's theorem later on. I suppose in a year or two I will ask ChatGPT 5.0 to furnish a formal system capable of proving its own incompleteness, where the proof of 1+1=2 is also concise. And it will attempt to clarify: Do you wish to stick to set theory as the underlying framework, or am I free to propose something else?
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.
"I have 1 candy you have 1 candy if you take mine how many candy will you have" "you will have 2 you doofus" meanwhile Gödel:
Most of this is just why I simply disagree with modern math. Logic and reasoning shouldn't be valued more than what's observable. What's observable should take the main priority. Modern math is just like a game where a few people invent the rules then force others to follow the rules of their game, trying to make their math real rather than trying to understand real math. Modern, advanced math is just a pyramid scheme sold to kids to learn so that they're only useful teaching it to other kids so they can teach it. It's not helping understand or describe the world - it's creating their own world. So many college-level math courses rely on axioms and inferences that are arbitrary and only exist because without them, all the other math they invented doesn't work. (3+3)^3 is both (9+9) AND 9^3 with not a single one being correct until you make up some random rule as to why you're right instead of showing that you're right with observable proof instead of fancy words and symbols
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.
When you really REALLY need that doctarite but have no idea what to write for your thesis.
2:10 why calculus is so cracked? XD hahaha XD
Hello, Thamks for valuable videos related to science. What programmes do you use while creating that kind of amazing videos?
Thanks