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

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Principia Mathematica
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Russell's Paradox video:
• Russell's Paradox - A ...
0:00 Intro
0:52 All was well in the land of math
1:39 Oh no! Trouble is brewing
3:47 The heroes of the story
5:06 Principia Mathematica
5:49 Logic
7:42 Formal Systems
9:52 Struggles :(
11:49 Ideas in 1+1=2
14:26 Failure
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Комментарии : 3,1 тыс.

@upandatom Год назад

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

@sindyr Год назад

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.

@pingnick Год назад

Wow a linguistics rabbit hole ha-thanks wow wild stuff indeed!♾♾♾♾♾☮️💟🌈🤯🤩😍😘🥰😻🗽🗽🗽🎬🎬🎬…

@ishajangir7664 Год назад

@@sindyr ll

@barneyrubble1431 Год назад

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,,,😂

@barneyrubble1431 Год назад

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!

@bramverhees755 Год назад

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”.

@albertlipschutz Год назад

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!

@Diamond-ji2gv Год назад

@albert lipschutz that's why axioms exist

@Krrish006 Год назад

How does a wrong assumption lead to correct results

@felipedamascenosilva3011 Год назад

@@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.

@Krrish006 Год назад

@@felipedamascenosilva3011 so how does this answer my question

@ptorq Год назад

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."

@mikemondano3624 Год назад

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.

@DarkSkay Год назад

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 Год назад

pure mathematics is a hell of a thing.

@DarkSkay Год назад

@@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.

@user-hm3ni1wd3f Год назад

@@DarkSkay i do not work in the field of pure mathematics, at the moment.

@JohnKarro Год назад

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.

@exdejesus Год назад

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.

@michaelmcchesney6645 Год назад

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 Год назад

There is no Nobel Prize for mathematics

@timbeaton5045 Год назад

@@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

@ariphaos Год назад

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.

@michaelmcchesney6645 Год назад

@@eljanrimsa5843 I'm aware of that now, but didn't know it in 1985.

@deltahat880 Год назад

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.

@formerunsecretarygeneralba9536 Год назад

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

@carinatus1758 Год назад

It's evolving just backwards

@jasonp7091 Год назад

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.

@user-ch2px4jy4b 9 месяцев назад

If you can't prove it, you have to assume it as an axiom. And that has consequences.

@gdmathguy 9 месяцев назад

@@carinatus1758But then going from backwards and ending up with something way better than the original

@bingusiswatching6335 9 месяцев назад

idk why you're acting like that's a bad thing, it's basic because it used to be unproven, circular argument idiot

@civotamuaz5781 5 месяцев назад

Wait until they hear about 1+2

@georgevladimirovich7190 Год назад

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.

@BallotBoxer Год назад

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

@NathanFarb Год назад

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!

@Supremax67 Год назад

Did you see the video where it took 758 pages to prove 2 + 2 = 4 ?

@andy764 Год назад

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

@FunWithBits 8 месяцев назад

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.

@syjwg 3 месяца назад

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.

@CHOCOLATIONZ Год назад

0:12 or can we? *VSauce theme plays*

@ivanscottw Год назад

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 !

@josefanon8504 9 месяцев назад

"and then headache ensues" sums it very well lol

@jeffbguarino 6 месяцев назад

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.

@franchise8633 5 месяцев назад

@@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.

@jeffbguarino 5 месяцев назад

@@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 ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-R3OkCxhjDmQ.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.

@jeffbguarino 5 месяцев назад

@@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 ru-vid.com/video/%D0%B2%D0%B8%D0%B4%D0%B5%D0%BE-xauCQpnbNAM.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.

@louisgiokas2206 Год назад

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 Год назад

Route I went...

@louisgiokas2206 Год назад

@@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?

@albertlipschutz Год назад

@@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.

@louisgiokas2206 Год назад

@@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.

@albertlipschutz 11 месяцев назад

@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.

@donaldaxel Год назад

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.

@parabellum4622 8 месяцев назад

Kitty Pride

@ColorwaveCraftsCo Год назад

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

@DarkSkay Год назад

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.

@marioluigi9599 Год назад

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

@jeenee_ Год назад

@@marioluigi9599i thought i was the only one who felt the same

@theshadeow5103 Год назад

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.

@17utk 14 дней назад

@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

@miramosa7768 Год назад

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

@Nick-lm9hg Год назад

The problem is it always leads to a contradiction

@miramosa7768 Год назад

@@Nick-lm9hg Yeah, she... Says so in the video?

@nonavad Год назад

@@Nick-lm9hg prove it! what contradiction is present in the law of identity? The unfalsifiability of the unfalsifiable?

@bargainwallart653 Год назад

Forget it

@notgad3130 Год назад

5:40 cat is inarguably the best part of the video. Its self evident

@natepeace1737 Год назад

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

@sahilsardeshpande1889 4 месяца назад

Or can we , ( Vsauce)

@archivist17 Год назад

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

@oliver_siegel Год назад

agreed! 👏

@argh01hass Год назад

Yep - this is the clearest short explanation of this topic that I've ever heard. Nice one Jade!

@masonwheeler6536 Год назад

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.

@ccgarciab Год назад

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.

@dwightk.schrute8696 Год назад

Unless you work with a Harvard architecture where instructions and data are separate

@ttthttpd Год назад

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.

@EM-qr4kz Год назад

So the foundation of mathematics is set theory? Or not?

@masonwheeler6536 Год назад

@@EM-qr4kz No. Gödel proved that the work demonstrating such a foundation was incorrect.

@ericask4666 Год назад

So happy to hear you are planning to do a video on Gödel!

@yddemper5517 9 месяцев назад

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.

@Bodyknock Год назад

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. 🙂

@jamieg2427 Год назад

thanks! i just bought this after reading your comment. it looks wonderful 😊

@bxnny0374 Год назад

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!!

@terryarmbruster9719 Год назад

So that makes the reader a bird brain? Argue if this is mocking or a logical conclusion to the question given your statements lol

@monkeygame7 Год назад

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!

@Michael-kp4bd Год назад

@@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.

@theprinceofinadequatelighting Год назад

One is the loneliest set of all sets containing a number of elements equal to the number one. 🎵

@DarkSkay Год назад

Beautiful! What about {} containing {} containing {} containing {} ...and so on. Lonely? Empty? Zero or infinity? ...{{{{{{{...}}}}}}}...

4 месяца назад

Hello, Thamks for valuable videos related to science. What programmes do you use while creating that kind of amazing videos? Thanks

@rachelblenkin437 Год назад

Me, a physicist: Well, if you have one of something, and you add another one, then you have two. 🤷🏻‍♀️😂

@yossiea Год назад

I love watching your videos. They are simple enough to understand yet open up vast areas to keep researching.

@alexbennie Год назад

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.

@FedericoStra Год назад

I really doubt he could have written the full expansion of 10 on the blackboard: {{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{},{{},{{}}}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{{},{{}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{},{{},{{}}}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{{},{{}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{},{{{},{{}},{{},{{}}}},{{{},{{}},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}},{{}},{{},{{},{{}},{{},{{}}}},{{}},{{},{{}}}},{},{{},{{}}}}}

@FedericoStra Год назад

And 15 is much worse, having 81919 characters (braces and commas), there is no way he could have written that by hand

@ryanlangman4266 Год назад

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?

@kindlin Год назад

@@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 "}".

@ryanlangman4266 Год назад

@@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.

@Lamalas 6 месяцев назад

I love that around 5:35 your cat comes to sit next to you!

@MisaelCastilloBrenes 5 месяцев назад

A Summary of the Formal Logic and Semantics course I took in the University, but fun to watch. Awesome video!

@Artaxo Год назад

This is my new favourite video from this channel! Jade is such a great storyteller and she picks great topics

@upandatom Год назад

Thanks! I worked really hard on this video so I appreciate that :)

@NickRoman Год назад

@@upandatom , This seems like a very important subject; so thanks for doing it.

@notyourbusiness1773 Год назад

@@upandatom ur awesome

@deadspline3252 Год назад

She kinda bad too NGL.

@AltecE Год назад

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 🙂

@HazFrostYT Год назад

I never thought I would question what the number one even is, absolutely mind blowing video!

@Kajisdaddy Год назад

I know! I guess I’m too literal. I can see 1. One apple. One chair.

@natchu96 Год назад

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?

@archismanrudra9336 Год назад

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

@mad_samster Год назад

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.

@johnmccarthy2594 Год назад

And at 5:34 the start of the show arrives!

@ericgenaroflores7069 Год назад

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.

@Roxve 4 месяца назад

very great story...

@ericgenaroflores7069 4 месяца назад

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

@livelifejolly8608 Год назад

Love that video you did a great job explaining 😊

@oakleypankratz8547 Год назад

In currently taking a directed studies course in Zermelo Frankes Set Theory, you’re explanations here are spot on!

@mskellyrlv Год назад

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....

@szamurainagy7644 Год назад

ah yes a 50+ year old watching youtube

@fishy1TTV Год назад

which college has Principa Mathematica in their library..?

@sumsarsiranen Год назад

@@szamurainagy7644It's great to see old people in here

@mrkeki1 6 месяцев назад

Every single one?

@kwimms 4 месяца назад

Because gravity, like relativity, is fake... it's a subjective definition, not a law.

@gabeteuton 10 месяцев назад

Brillant storytelling! Wonder why this channel was just now brought to my attention. And i'm in love.

@bishopwulf Год назад

Great video. Especially kitty cameo at 5:29

@stormlord1984 Год назад

As always, a fantastic presentation in a very dry subject, Jade. Years later, you channel is still such a joy to watch!

@PasseScience Год назад

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.

@kolty99 11 месяцев назад

thank you. This really helped me to understand what this book was about.

@WhiteChocolate74 Год назад

Your stuff is informative and interesting, but maybe even more importantly, you're a great storyteller which makes it easier to learn. Keep up the good work

@Purified-Bananas 17 дней назад

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, ... } = ∞

@zoltanposfai3451 Год назад

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 Год назад

That sounds marvelous! Can you recall the name of the course or any textbooks used? I'd love to learn more.

@nickeni3050 Год назад

Is this... The true hell?

@zoltanposfai3451 Год назад

@@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 Год назад

@@zoltanposfai3451 Fascinating! Was this an undergraduate, or graduate course (towards Master's or PhD)? How many total students were in your class?

@zoltanposfai3451 Год назад

@@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.)

@nekoverse6611 Год назад

YAY U POSTED!!

@davidwood2387 8 месяцев назад

1 rain drop and one rain drop equals 2 rain drops. Until they go together and become one .

@clubx1000 Год назад

I've watched this video 3 times. Your program brings a Joy to my day

@eckligt Год назад

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.

@rlkinnard 7 месяцев назад

and someone who helped Lincoln by keeping GB out of the civil war. RIP both the first and three earls.

@NovaWarrior77 Год назад

I want to bring attention to your wonderful chapter titles, and the fact that they coincide perfectly with the narrative!

@upandatom Год назад

Hehe thanks :)

@ManNamedBrUce 11 месяцев назад

Hey, I love the video! I was wondering if you read or heard of Hofstadter’s Gödel, Escher, Bach?

@ericsperling9724 Год назад

a equals b: a = b multiply both sides by a: a^2 = ab subtract both sides by b^2: a^2 - b^2 = ab - b^2 refactor: (a+b)(a-b) = b(a-b) divide both sides by (a-b): a+b=b??? algebraically correct but last step is division by zero

@yushKumarBaranwal-01 Год назад

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

@SgtSupaman Год назад

Luckily, this alien understood (non-math-related) English.

@amourdesoipittie2621 Год назад

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.

@Mel-95 Год назад

Really clear, fun storytelling. I am terrible at math but I enjoyed listening, as I do to all your videos. Nice work.

@kito1san Год назад

This just shows how something we take for granted that requires so much to arrive.

@koskarvounis Год назад

The Greek zeimbekiko song in the Euclides section was a nice touch

@Phoebus82 Год назад

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 🌹🌹

@punguru1360 11 месяцев назад

maybe you got good teachers or profs then at hs

@JimAllen-Persona Год назад

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.

@epicmarschmallow5049 Год назад

All of geometry can be described using various bits of algebra

@tylerdurden4006 Год назад

(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 🧐)

@shortnotes-bds2621 Год назад

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.

@monad_tcp Год назад

Type Theory is so important to serious computing and programming, it can't be overstated.

@MarcelCox1 Год назад

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.

@user-tq1hi8pn2h 3 месяца назад

your videos are really fun and informative

@rohanbiswas3388 Год назад

Simple yet damn elegant way to explain... 🌟 Loved the whole video...

@gregnixon1296 Год назад

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.

@bestaround3323 Год назад

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.

@timothyhendricks3004 10 месяцев назад

Learning the methodology of math is more important. Really learning math is really learning an extremely logical way to think.

@grapy83 Год назад

You have an amazing unique quality of being so clear and easy with complex/difficult topics.

@osikani Год назад

Awesome video... Anybody who attended KNUST in Ghana, and studied under Dr. F. T. Oduro will love this and miss the good old professor.

@grproteus Год назад

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)

@zetristan4525 Год назад

You are clearly a very ancient Greek

@paulschumacher1263 Год назад

Your cat seems to get interested at about 5:46.

@anomalous3039 Год назад

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.

@Spoohkybooh666 Год назад

@Humdebel Год назад

1:35 that turn... was epic. Instant like!

@justinbohemier118 Год назад

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.

@mrsnidesmin Год назад

Completely agree and commented a similar remark. ☺️

@Lexivor Год назад

One of my favorite books.

@simongross3122 Год назад

I've read this book several times. It is an absolute masterpiece.

@Fudandori Год назад

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.

@upandatom Год назад

Thank you so much!

@pulsar22 Год назад

It is because Math is racists ahahahahahahahahahahaha.

@morantNO1 6 месяцев назад

Man, that was sad. I am really sorry for these guys.

@FIREBRAND38 Год назад

Such an impressive and useful video. Subscribed.

@theta4625 Год назад

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.

@upandatom Год назад

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.

@scholarlyanalyst7700 Год назад

@@upandatom What's your academic background?

@theta4625 Год назад

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!

@user-hm3ni1wd3f Год назад

isn't this a mathematics video? or are you talking about mathematical logic?

@theta4625 Год назад

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.

@AwestrikeFearofGods 10 месяцев назад

0:04 "Hey guys! Spirit Of The Law, here."

@parsimoniousdialog 7 дней назад

Google relates to googol, a quantity or number. Obviously googol is not prime because googol is an even number greater than 2. In reality, the author of the prime number system was discussing a proof relating to quantities and terminology (i.e. the theory of types). Realistically, 1 and itself are the quantities 1 and 1. These quantities are prime in a sense that defies the fundamental theorem of arithmetic.

@Dr.TJ1 Год назад

I think Schrödinger’s cat escaped from the box during this video.

@ndubuisimessiah3011 Год назад

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.

@Number6_ Год назад

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.

@ralphmccawley1554 9 месяцев назад

What a tiresome, bullying response, No.6. Very revealing.

@ralphmccawley1554 9 месяцев назад

Of course, you could've just meant it to be lighthearted.... In that case, I'm sorry. But an emoji may have helped 🙂

@kentcurrie7326 Год назад

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.

@jacobpeters5458 Год назад

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

@shawntco 9 месяцев назад

Godel showing up with his incompleteness theorem made me laugh much harder than it should have

@ryvyr Год назад

Thank you for placing non-adsense at front/back/both of video rather than interrupting somewhere between, so I can justify liking the video and listening to offer. Aside that, your method and presentation ever delight and inform :>

@soesan801 Год назад

I never expected that 1+1=2 is almost unprovable.

@luudest Год назад

14:04 P. 379 Wtf! Was that written in Alien Language?

@upandatom Год назад

ikr 😞

@GOOD_FARMER Год назад

Written in axiom that no one can understand ( except a hard-core mathematician )

@antoneogzewalla2040 6 месяцев назад

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.

@superivan300100 Год назад

I am not sure that I would call their work a "failure" since even though we cannot have consistency or completness at least we know that math can be formalized. And thanks to this it is now possible for a computer to check if a given proof is correct (coq software) and we are also trying to make it possible for computers to find proofs.

@JasonAStillman Год назад

This was wonderful. I remember first hearing about this book, a several hundred page proof that 1+1=2, from my electricity and magnetism professor in college. I remember being both amused and completely fascinated.

@user-ys3ev5sh3w Год назад

1 + 1 = 2 in geometry means vertex + vertex = vertex . Because vertex means digital root = 1 , edge means "digital root = 2" , so 3,5,6 are edges but 2^i are vertices of simplex. So digital root is dimension of number. ( d-vertex simplex and d-digit binary number system of course are the same thing).

@DarkSkay Год назад

How about this statement: "The empty set {} contains itself infinitely many times."

@user-ys3ev5sh3w Год назад

@@DarkSkay lets d=3, then: 000 - is externity of triangle; 111 is internity of triangle; 001, 010, 100 (digital root=1) are vertices of triangle; 110,101,011(digital root=2) are edges of triangle. So any binary number system(geometrically simplex) consist of zero (geometrically externity), infinity (geometrically internity) and numbers (geometrically faces). "Go to infinity" ( your "infinitely") is impossible for fixed "d" because numbers lies directly between zero and infinity and can go only around them.

@user-ys3ev5sh3w Год назад

@@DarkSkay If you try to do something infinitely times you "open" infinity and break the law: "Infinity is closed but zero is open".

@carmabound 9 месяцев назад

I love the way you cut off the clip of you about to eat the cheese at 6:58. It reminds me of Inception's ending, haha-we will never know if you ate the cheese or not (unless you were to reply?). I haven't seen anyone else mention this and I thought it was quite clever.

@geektoys370 Год назад

Keep up I’m learning so much ❤

@ArpanD Год назад

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!

@upandatom Год назад

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!

@ArpanD Год назад

@@upandatom That's relatable, and thank you so much!!

@Crushnaut Год назад

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.

@ArpanD Год назад

@@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!