Russell's Paradox - A Ripple in the Foundations of Mathematics
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- Опубликовано: 26 сен 2024
- Bertrand Russell's set theory paradox on the foundations of mathematics, axiomatic set theory and the laws of logic. A celebration of Gottlob Frege.
Thank you to Professor Joel David Hamkins for your help with this video.
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What is a number? (no using the word number)
Up and Atom
The existence of an element.
whether it exists or not or repeated
An object that characterizes (represents) an equivalence class of sets that are in a one to one correspondence
Note: cheated, I already did some set theory
A element that represents position on a particular manifold, such as the space of real numbers.
I'll bite: The quality of a category that describes the observed occurrence of discrete instances within that category.
"Instances" is kind of number related, but strictly speaking the concept only requires understanding of the idea of individual things existing, distinct from others. It's enabled by counting, but it doesn't depend on it.
You want to know what a number is? Here you go: ruclips.net/video/UmWmfTt4VBQ/видео.html It's the first of a 8-part series of 4-min videos. Next video has a tag (G1). Here are all the videos: ruclips.net/channel/UCRUATK39-y_dSwmN59_-aNQvideos
What I learned from this is if you get a letter from Bertrand Russell, don't read it.
Honestly, if you get a letter from Bertrand Russell today, bring that shizz right to James Randi and get yourself a million bucks.
@@DampeS8N
There's nothing supernatural in that. They call it 'snail mail' for a reason.
Joe, letter from Bertrand Russell: can you go fetch my teapot, please? (Read as: maybe make a video about interesting thought experiments?)
Russell: Dear Joe,
I wholeheartedly enjoy your channel and would not change anything.... Except this small thing that will ruin your confidence for eternity.
Joe: Wanna collab?
but it could have money inside!
The barber was pulling his hair out trying to solve this problem, which ironically did solve the problem.
until it grew back
@@golfgod1017 when the hair grew back, the problem also came back. So, he found himself once more pulling his hair out again trying to solve it.
@@post1305 unless he learned from the experience and chose a different approach.
@@golfgod1017 There is no evidence to suggest that happened.
@@post1305 or you just didn't see the evidence
I enjoyed this video so much. The animation and the explanation are so good!
WoW! , youre here... yay!
Hey dude!!!
You chad LEGEND
@@dakshdheer1681 whaaaaa?
super cool dude? what the hell!
The clarity you bring to these difficult to articulate and comprehend topics is exceptional.
This was fantastic. Please don't worry about being overly-nuanced or complex--there is already plenty of dumbed-down content available elsewhere, and you have a skill at presenting complex concepts in a straightforward, understandable manner. Thanks.
I second this message. This is why this channel is great.
I on the contrary think that this video did not explain anything substantial or get into sufficient detail. The talk was mostly about history, and introducing various concepts. For me it only managed to define two paradoxes.
Ivko Stanilov agreed- I absolutely enjoyed the video but was waiting to jump into the abstractions and into the weeds a bit after the intro warning. Hope for a part two going deeper!
Brilliant point! I think if the delivery was simplified, a different audience would be attracted. But I'm grateful that these videos were made in a way that appeals to me.
Well said, sir. I completely agree.
"So Baldrick, if I have some beans and add one more bean, what does that make?"
"A very small casserole m'lord."
"Three beans and that one."
Great video, but my favorite moment is when you said "Tifa is a dog" and she looks at you as if saying "Wait?? I'm a DOG???"
xacharon insulted that we assumed she couldnt smell cause she’s old. She’s a dog, not a smoker, dammit!
@@alex0589 old people lose the acuity of their senses as they become old, even when they really, really take care of them selves... I think it's just how genetics works, man. i suppose we are probably eventually going to figure out how to prolong this, but i doubt out doggie friend has been genetically modified.
This does bring up a shortcoming of building things out of simple logic: given “dogs have a good sense of smell,” if Tifa does not have a good sense of smell then “Tifa is not a dog” is a logical conclusion, but we can all see she a good gurl
xacharon ohhhhhh shit
This is brilliant. I was trained as a physicist and last night - over a bottle of wine - tried to explain the Russel paradox to my baffled adolescent daughters 😃. I now sent them this link 😂
yeah i try to explain too ,but most of them have no idea what am i talking
You just put the barber into a superposition with himself. You do this by putting him in an isolation box as in Schrodinger's box and you use an electron gun pointed at a spin detector. The detector will reveal if the electron is spin up or spin down. Tell the barber to shave himself if the spin is up and not to shave himself if the spin is down. Then start the gun up and close the box. Inside the box the Barber will be in a state of having shaved himself and not having shaved himself at the same time. You can also solve Russell's paradox using this method and any self referral paradox. You have to use the real world which is quantum mechanics and stop living in Newtons Classical world. Let's face it zero and infinity can't exist. Mathematicians completely ignore the uncertainty principle when they do their thought process to develop math. You can't create math that is impossible. That is what they have done.
For Russell's paradox just create two sets R1 is the set of all sets that don't contain themselves and include R1 in the set. R2 is the set of all sets that don't contain themselves and exclude R2 from the set. Put these two sets in writing on two papers in a box and have a random quantum event burn one of the papers. Close the box and inside the box will be a superposition of R1 and R2. The superimposed set is labelled R3 and it contains itself and doesn't contain itself at the same time.
This was great. I have heard Russell's Paradox before and my response was usually, "Ok, but so what?" What you did here was put a seemingly uninteresting paradox into both the historical and mathematical context to help me see _why_ this paradox is so important and interesting. Thank you.
Thank you
I mean the paradox is another way of saying that an axiom cannot prove itself. That happens in logic therefore in math. If you want to go more in depth you can check out the incompleteness theorem of goedel.
@@lucashuerga1368 A paradox means using classical logic that there is a basic mistake in the foundations of math. Quantum math would fix this but no one seems to be working on this. You can have things that are true and false at the same time if you create a superposition. So put the barber in Schrodinger's box and have a random quantum event like measuring whether an electron is spin up or down. Close the box and if the spin is up , then the barber shaves himself and if it is spin down he doesn't. So you have a box and in the box is the barber in a state of having shaved himself and not having shaved himself at the same time. This cures Russell's paradox. You can do the same set up with the set of all sets that don't contain themselves.
@@jeffbguarinoRussel's paradox has been resolved for over a hundred years. It is a problem of naive set theory, but modern set theory gets around it by simply restricting what is considered a set.
Can you please eloborate what "quantum" math is? I doubt such a thing would be very meaningful as quantum physics is already easily described by classical mathematics.
What you are describing sounds very similar to so called "fuzzy" sets. These can easily be modeled in set theory like this:
If we have a superset S of all things we want consider, a universe of discussion so to say, all relevant sets are subsets of S which can be uniquely identified with a function f : S -> {0, 1},
where every element of S is assigned the value 1 if it is considered to be in the subset.
A fuzzy set is defined in a similar way as f : S -> [0, 1], where [0, 1] is the interval of real numbers from 0 to 1. So every element is assigned a probability of being in a particular subset.
The point is that these kinds of set are easily modeled in classical mathematics.
Consider a sets of all sets that have never been considered. Oh wait, they’re all gone now, never mind.
They haven't been considered, just the set now no longer contains itself :)
I see the joke you did there
so underrated lol
Nice one. I think paradoxes should be hunted and taken as gateways toward unpacking primitives and axioms.
@@muhaimin244 lookup Vsauce
Gottlob Frege: * makes a definition of number*
Bertrand Russell : I'm about to end this man's whole career
Foundation is not the right word. These sciences existed for thousands of years before their 'foundations' were even known to exist.
@@Jordan-Ramses It doesn't matter if the word came before the sciences, it can still be considered a foundation. Just as foundations for houses were foundations long before the name "foundation" was invented. This is actually the case with most things. Think of it as "common source" or "common basis".
lol
@@Jordan-Ramses Right, And the same thing is true for mathematics. Science, and math can begin anywhere you like. Whatever you happen to discover, observe, experiment with first. And then it can grow from there in any direction. I think a problem arises when we try to put knowledge into a "tree" format. We make an unfounded assumption that there is a "base" or "foundation" or "root" of the tree, and the rest of science/math grows upward from that. Logic does not have to be at the very "bottom" of the math "tree". Arithmetic works correctly and consistently anyway. We can start with that, and then explore "upwards" or "downwards" as far as we like.
@@fredrikekholm3718 - No, it's not a foundation because they aren't actually built on them. Math and science existed for thousands of years before someone decided to try to come up with a 'foundation'.
Math is not based on the definition of a number.
This is the best and clearest explanation of Russell's paradox that I've ever heard/seen. Thank you so much. I think I actually get it now :)
Yeah..
In the UK, we use the Brexit method to solve the Barber Paradox: the barber keeps saying he's going to shave himself, but he never does :-)
usvalve
If you can only define what you don’t want but not what the heck you do want instead, that’s what you end up with.
Just like in mathematics: it’s much easier to debunk than to confirm something.
he can Vax himself and shave others 😂😂
Please don't refer to the Europeans as armpit hair, they are sensitive about that
You mean, it's much easier to bunk than to debunk? I'd say that's true... @@yaff1851
Barberxit... Barbrexit... Barbarella?
"my nose will now grow" said pinnochio
Pinnochio's nose would disappear.
Ah...I see
lol
In this case, isn't Pinnochio making a promise and not lying?
Pinnochio's nose doesn't grow when he breaks a promise.
@@argumengenichyperloquaciou4115 Let us put it in this way, "My nose is growing now"
Just discovered this channel and spent most of the day just watching a bunch of your videos. Seriously some of the best and most accessible, entertaining science content I've ever come across.
Frege: here's the neat systematic set theory I made.
Russell: *I'm about to end this man's whole career*
And hospitalize him
I love how Immanuel Kant "soon came along" after Aristotle. I once had to teach a Phil 101 course, and our textbook jumped from Aristotle to (I think) Descartes. In the final exam one of my students wrote, "Descartes was a student of Plato, but you'd never know it from the things he wrote."
mathematics students
mmanuel Kant was a real pissant
Who was very rarely stable
Heidegger, Heidegger was a boozy beggar
Who could think you under the table
David Hume could out-consume
Wilhelm Freidrich Hegel
And Wittgenstein was a beery swine
Who was just as schloshed as Schlegel
There's nothing Nietzsche couldn't teach ya
'bout the raising of the wrist
Socrates, himself, was permanently pissed
John Stuart Mill, of his own free will
On half a pint of shandy was particularly ill
Plato, they say, could stick it away
Half a crate of whiskey every day
Aristotle, Aristotle was a bugger for the bottle
Hobbes was fond of his dram
And Rene Descartes was a drunken fart
"I drink, therefore I am."
Is that because the middle ages philosophy only had to do with religion and Plato and Aristotle's Organon, until renaissance humanism came along?
@@captainzork6109 Not exactly. Even the so-called "churchmen" looked at what we would call philosophical questions about epistemology and ontology and the philosophy of language. There were also philosophers in the Caliphate that I know very little about. More modern thinkers have created theories that people nowadays take more seriously than the medieval ideas, so the medieval philosophers tend to get overlooked and forgotten. It is, however, a deep vein, and I think philosophy is as much about the thought processes as about the end result. Journey vs. destination.
@@sourisvoleur4854 I'm a psychology graduate, and although my Master is in Theory and History of Psychology, it has only been since a year or so I've started learning philosophy and history more generally. But thus far it seems like their epistemological questions have been very broad: What is the world, and how can we know of it? And, as Nietzsche pointed out, even until Schopenhauer the hinterwelt had always been part of the most prominent thinker's philosophies. That is to say, scholars in the past put so much emphasis on some 'more perfect world', getting lost in a convoluted mythos of heaven and hell, that they failed to make any sense of the here and now. As far as Francis Bacon was concerned, those scholars were all just armchair scientists, who come to the wildest conclusions based on singular experiments
Except, of course, when it came to practical things, such as geometry and algebra, which presumably was also helpful for engineering
This is all to say: People's worldview used to be wild and stupid, and we are much more sensible nowadays
But despite the sources I've come across, I can't help but wonder if it's really all that true there really weren't any unsung heroes from those middle ages. After all, the ancient Greeks had people like Ptolemy, Socrates, Plato, Aristotle, and Galen, and though they believed in the gods, they still came to great thoughts and discoveries
I wish there'd be such nice examples of the medieval times, who were influential, but were just overlooked by those in the 14-15th century, who called themselves renaissance humanists
And then Gödel wrote a letter to Russel.
I was going to say: Why are people still looking for a foundation post-Godel?
@@GrantDexter Exactly. Every meta-system that could provide a foundation is itself subject to incompleteness, infinite regress.
@@GrantDexter Because they are no longer looking for a complete coherent foundational framework. They are just looking for a list coherent list of axioms that lines up with what we commonly picture as a set. Not complete, just coherent and with the least possible amount of vagueness.
Then Schrödinger asked him if the barber was observed, since he obviously was both shaved and unshaved
@@crackedcandy7958 That's interesting. If we consider a foundational theory such as ZFS to be equivalent to quantum states in physics, is it possible for a theory to be superpositional? If so, Russel's paradox becomes a superposition, not a contradiction.
5:05 ‘Another philosopher, Immanuel Kant, soon came along’ Soon, as in 2000 years!
hey, like jesus!
relative to the history of people, 2000 years is soon. relative to how long this problem has been around, probably not so soon.
Soon is a relative term
Brilliant. There’s nothing else like this. I’ve been struggling with this for too long to mention and this graphic presentation is the clearest I’ve encountered.
I'm Russell and I approve this paradox.
If you're Russell you cannot approve your own paradox, if you approve your paradox you're not Russell. :-)
@@sumeshrajurkar5922 First of all, I'm not going to publish something that I don't approve of. Secondly, I'm the other Russell.
@@sumeshrajurkar5922 IT WAS A JOKE YOU ABSOLUTE DUMBASS. HOW DID YOU NOT GET THAT?
Plus one.
@@derylpetersonnnnnnnnn Yes, how dare he reply to a joke! Please be more mean to him.
13:23 "Apparently he didn´t know about the breakdown." 😂😂😂 I think this says something about us all; happiness lies in not trying to belong to the set of all sets because this action alone just excludes ourselves 😉.
I have a question observation.
We routinely define math such that we exclude certain conditions because there isn’t a clean definition. We cannot divide by zero. We used to not be able to take the square root of negative numbers. And we used to insist on only rational numbers. We have determined a means to work around these issues, except we still say that dividing by zero is undefined. The other place I think we see the rules change is when we talk about sets of infinite size. We have limitations on what we can compare with these sets. Hence we exclude properties because of the paradoxes that arise.
The Russel paradox looks like the divide by zero concern. He’s just pointing out that there are these cases that tend to act like dividing by zero. These cases are self referral cases. Any set that refers to itself can create this paradox. In fact, all of the paradoxes I’ve seen here have this same property that the rule because it applies to itself changes the state of the object and so self referral creates the same type of condition as dividing by zero. Hence, for the same reasons we exclude divide by zero; can’t we also just exclude cases of self referral that create the paradox? If it works for dividing by zero, it appears that it works here as well?
That's basically what happened in the future. Some dude's (Zermelo and Fränkel) developed a new axiomatic set theory (Zermelo-Fränkel set theory) specifically to exclude paradoxes like this.
@@epicmarschmallow5049 Thx!
I'm three years late to the party, but I really enjoyed this video and wanted to offer an answer to the important question you asked, "What Is A Number?" The most perfect definition of what a number is that I've ever come across was over 25 years ago when I first read a book called "Mister God This Is Anna." Anna was a truly remarkable 5 year old girl who asked the same question and shared her incredible answer.
Anna knew that 1 planet and 1 ant were in no way equal, but wanted to find how and why the number 1 made them equally countable as "1" mathematically. She discovered her answer through a light and shadow experiment. She had an adult set up an overhead projector so a blank square of light shined on a wall. She then placed an apple on the overhead projector screen which made a 2D shadow of the apple on the wall. She then taped a piece of paper on the wall, traced the outline of the apple's shadow and cut it with scissors. She then placed the paper cutout of the apple's shadow in front of the projector holding it at a 90 degree angle, which created the 1D shadow of a line on the wall. She put another piece of paper on the wall, traced the line and cut it out. Then she took the paper cutout of the line and held it over the projector at a 90 degree angle...and was left with a zero dimensional dot on the wall. Then she pointed in excitement and said. "That's what a number is!"
No matter what the size, weight or shape of the object was that she conducted this experiment with, she was always left with the exact same dot. She then realized that if there was a projector and a wall big enough, her experiment would get the same dot putting a planet in front of it as an ant. And so Anna concluded that in our three dimensional universe, a number is light's shadow of a shadow of a shadow. I've never found a more beautiful or perfect definition that doesn't use the word "number" and is fully supported by experiment with completely repeatable results.
Underrated
Thanks for this!
@@carlosraventosprieto2065 You're most welcome!
Bravo!!
Dammit! I just waisted 5 minutes of my life because I read really slow.
A great little video so well scripted and cut and a testament to the ability of its creator. I got to the end without needing to rewind but I can call on a degree in Philosophy to help me. I have never seen set theory explained so well. Thanks and well done.
Frege: I'm finally done with my work!
Russel: I'm about to end this whole man career.
Lmao love this meme
Story of PhDs
Frege's work got known thanks to Russel though.
And although the problem he found was at the base of the theory, most of the work still was very important for the future develpment of formal logic.
@@Deguiko So he destroyed his mind in order to build him back up? I've heard of tough love, but savage love? Damn.
Frege's breakdown almost made me cry. I can't even imagine how it must have felt to have his life's work be disproved by a single sentence. Great video, you've earned a subscriber!
I literally laughed out loud when she said that.... Time to find the Ted Talk about why we laugh at other people's pain...
@@broffutt Well it is funny from a dark comedy point of view and also we are all different I guess. So I think there is nothing wrong with you 😄
He wouldn't be the only one of these clowns with a screw loose. Pondering different flavors of infinity defies all intuition, until you're deep enough into it to develop a new kind of intuition.
Same...
There is no evidence Frege had a breakdown due to Russel's letter.
nice video, but OMG i feel so bad for frege. imagine being so determined that you would solve all of math and then your years of hard work is just crushed. i understand math is like that because theres paradoxes and all, but i feel like me and lots of other people can relate to the poor man mentally
"Apparently he didn't know about the breakdown?!" 😂
People: Imagine if everything was absurd!?
Quantum Mechanics: Well hello there :)
Deep philosophical question that comes to my mind from watching this video:
Who shaves the turtle?
Achilles
Turtles all the way down to the Mock.
"Who shaves the turtle?" ==> Mitch McConnel's wife.
One thing I am sure of, turtles don’t get electrolysis. That would leave them shell shocked.
@@bobbimke82 yeah, he definitely is the turtle.
The claim that Frege had a breakdown due to Russel's letter is a fiction added for dramatic purposes. Frege was going through a combination of poor health, the early loss of his wife in 1904, and disappointment over the continued poor reception of his work. There is no evidence that frustration with his failure to find an adequate solution to Russell’s paradox was the primary reason for his hospitalization.
The Buttersotch Paradox - It tastes neither like butter or scotch.
This Butterscotch Ripple is more upsetting to the foundation of life than Russell’s Paradox ever could be.
J J if you throw butterscotch hard enough, it tears space-time so you can step out of this reality and can taste thoughts and concepts instead. Try it.
@@alex0589 we've got a synesthete!
My butterscotch paradox is that my butter is usually messy while my scotch is always neat
I refuse to join any club that would have me as a member - Groucho Marx
And this leads us to a different paradox. Imagine a town where every possible set of citizens forms a club. Would it be possible to name all clubs after a citizen, in such a way that every club is named and no two clubs have the same name? Of course, this can't be done with a finite town; it would have more clubs for than citizens. (For example, with just 10 people there would be 2^10=1024 clubs.) But could it be done in an infinite town?
Turns out, the answer is: no, it can't be done either. Take any naming scheme (where no two clubs have the same name), and ask: does it cover all clubs? If every set is a club, then so is the set of all citizens who are not a member of their own clubs. But this club can't be named; otherwise, can the citizen who the club is named after be its member? It can be seen that he's a member of our club if and only if he isn't a member. This is an impossibility, so the club can't have a name.
@@MikeRosoftJH do these clubs come with membership benefits? otherwise I must decline your offer.
@@MikeRosoftJH Anyways, you seem to be running out of letter combinations, so here I propose an infinite alphabet to go along with the naming.
@@MikeRosoftJH The number of clubs is summation(n choose r) for 0
I thought that was Woody Allen
12:50 WOW, did Russel clearly know his way about elegantly rubbing salt in the wound ^^!!
Russell s question was plain stupid n irrelevant. Idiotic man, overrated as f
Quick point: There isn't actually any paradox with Frege's theory of concepts and extensions at all (as it was presented in this video at least); that idea is used in ZFC set theory all the time (every well-defined property φ induces a class of sets satisfying φ). The reason this isn't a contradiction is that there is no notion of a class containing another class - so you can't have a class of classes that do not contain themself.
The contradiction seems to just be in the way he defined "set". If you swap it for the modern idea of a set, then you get a perfectly good model for set theory.
Because you didn't ask me to subscribe and hit the notification bell, I did. How's that for a paradox?
Bruno Bronosky that is not a paradox. It was an exploitation of your nature. The fact that it
simultaneously is and was, that’s a paradox.
@@rolyf100 is and is not*
Try subscribing to the channels of all RUclipsrs who don't subscribe to their own channels.
"You're all individuals."
"I'm not."
Tom van Leeuwen but you are in the set of all individuals
No, no. It goes:
"You're all individuals."
"Yes, we're all individuals!"
"You're all different."
"Yes, we're all different!"
"I'm not..."
@@AvanToor I know, just tried to simplify so that it was easily read. :-)
The ones who know the scene understand.
@@fakkmorradi I am the set of all individuals.
@@AvanToor The very best movie quote of all time.
I have subscribed. This video was as clear and concise of a description of Russell's paradox as I have seen. It was enjoyable. Good work, Jade.
I've seen the Jeffrey Kaplan tribute to the Russel's Paradox, but, I got Way More value from your approach. Those adorable animated Thingies remind me a lot of those Delightful Saturday Morning Educational Interstitials. "Interplanet Janet", "Conjunction Junction ", "Bill ", Twelve Toes", "Hero Zero"... There Were So Many!🙀💕
You'd have Loved My Century!😻
THANK YOU FOR WORKING SO HARD ON THIS!💖
Tifa might not be able to smell things as well as when she was younger, but, we could Use Logicism to infer, correctly that "All Dogs Smell!" And, This is accurate to At Least, ONE POV concerning Dogs.
How to describe a Number w/o using the Word "Number" Might be expressed as
:the value and/or *"Quantity" (I know 🤪) of a something denotes how it may be combined with another Value and/or Quantity.
Ie: I have a quantity of Shoes, but, only enough for One person at a time to Wear."
I think that THIS is How Most animals determine if everyone in the family is present! NOT with artificial inventions, but, by direct Observation and memory.
The Linnaean system of classification operates similarly by determining What items Belong together based on intensity and plurality of similarities. The Cow clearly doesn't Belong with Humans and Apes. But, Could belong with Cats and Dogs, but, moreso with Sheep and Goats.
Phylogenetic classification deals with ancestral relationships between organisms.
1+1=3! Is True because there are Three value symbols shown. But, we know that This is Incorrect.
The Quantity of All symbols in the equation would Be 5! Which is how an animal would perceive this equation's Value.
1+1=2 is logically correct to the purpose of combining integers.
My Dyscalclia operates like this giving me an instinctive value of the symbols First and then resulting in everything getting all mixed up in my mind disrupting computation.
*But, like you said: Words Such as "Quantity" are related to the Word "Numbers"! But, what if I use a Made Up term such as "Accumulation", but, here, again, we've got a number related word. Humans Are the ONLY Indigenous Species Currently Extant on Earth to Utilize Complex Structured Utterances to convey information. It's completely artificial and unnatural. That's Why It has to be taught and learned!
But, a mother Duck determining if everyone in Her brood Are present, is Not Likely accomplished with artificiality.
Even words Such as Concepts and Sets are, precisely The same sort of thing As Quantity and Value! And, again, we're back to the Linnaean system!
Try conveying Any Information w/o employing Artificiality!
Like That Party Game "Taboo"!
And, that's At the Root of this type of Paradox: Just like with The Raven Paradox!
Although, you might be able to Break it down with Sets, Subsets, Ifrasets, Quantum Sets. You can say "All Ravens in Subset A are black. But, NOT All Ravens in Set 1 Are Black. And, only some of the Ravens in Collection 3 10:06 are Black!"
Er .........🤪😵💫
Does the Concept of "Nothing" have an Extension?!🙀😱💫
Getting Off Tangent... Sorry 😹
I am currently reading Logicomix and this video really helped me understand the novel. Thanks!!👍
I bought it thanks to you…
Frege: I have the most fundamental theory about maths
Russell: I'm about to end this man's whole career
“Life” is immeasurably and incomprehensibly complex. Words are at best rough approximations of anything resembling “life” or “reality”.
omg i've seen videos on this paradox so often but this is the first time i actually got it!! Thank you sm🙏🏼
11:27 -- I love that Jade's gives the camera that same look you'd give anybody when you're pretty sure you've said something that's gone over their head.
I'm so glad Up and Atom is a channel on youtube. Keep up the good work. I can't wait to binge on all your videos.
As for the barber paradox, a similar solution (to my solution for Russell's paradox) can be applied. Once again, the trick is to divide what the barber is into two reciprocal aspects. Instead of sets and elements, we must divide the barber into that part of himself that is an actual barber and that which is just an ordinary person. Now, the definition of a barber is someone who shaves or cuts the hair of another person for money. Now, these two aspects of the barber must be kept separate because (like sets and elements) they have certain characteristics that are incompatible with each other. For instance, the [person aspect] is a necessary characteristic while the [barber aspect] is optional since he could choose to be something other than a barber in a way that he cannot choose to be a different person.
Now the barber aspect is the aspect that shaves people. This is true whether he's shaving other people or himself. Thus, if his barber aspect shaves his person aspect, then the person aspect is NOT shaving himself. Now, there are two possibilities. Since the barber aspect isn't charging his self aspect any money to shave himself, then the barber isn't functioning as a barber, since that requires the acceptance of money. Thus, if the person aspect shaves himself, the barber aspect is not involved in the shaving. And the situation is not paradoxical. On the other hand, if the person did not shave himself, he would have to pay someone else to do it, and thus, he is receiving value (the absence of need to pay someone else) by shaving himself... but if we acknowledge that value, then we must also admit that the barber aspect kicks in and it is the barber who is shaving his person aspect, not the person, and so once again the barber is shaving an aspect that is not shaving itself. Either way, there is no paradoxical confusion.
Excellent description of the paradox. Any similar insights on the essence of the Russell paradox or Godel incompleteness
@@mikedougherty1011 Thanks for asking and yes, I do, although a detailed look at Godel is probably beyond the capabilities of this format.
Russell's Paradox... can be resolved (I think) by distinguishing between the nature of an element and that of a set. A set is that which contains elements, an element is that which is contained by a set. It's like the relation between a father and a son. The same person can be both a father and a son, but he can't be both of these things to the same person. Similarly, a set is like a [container], while an element is like [that which is contained]. You can but a small box (that contains something into a larger box) but the relation between the boxes is such that only one contains the other. Thus, since R is the set of all sets that do not contain themselves, R is necessarily the [set of all set], since no set contains itself. The opposite of R is the empty set. We can think of this distinction as the [name of the set] vs [what the set actually is]. Like the single word "English" versus the set that contains all the English words, [English].
The set [English] contains a the name of itself, which is that single word "English" but it does not "contain" the set [English] it simply is that set. In the same way, we can create a set R that contains all the sets that do not contain their own name. But since a [name] is not the same thing as the [thing named], there is no paradox.
A Quick look at Godel's Incompleteness. Without getting into the weeds, G can be essentially understood as a set that makes a self-reference to itself, as follows:
(G) [G is false]
Again, the error is to assume that (G) and [G is false] are the same thing and that they are interchangeable. In reality, the G in [G is false] is only a name. It is not the whole set [G is false]. We could try to substitute the whole set for the name, in order to get rid of the name aspect, but this only produces
[G is false is false]
We can substitute as many times as we want, but it will never get rid of the name aspect. And this creates a necessary vicious circle that is identical to the way two mirrors partially reflecting each other create an "infinite" series of mirrors in mirrors. We see the same thing with a camera records it's own monitor. We see an infinite series of smaller monitors. Again, with sound feedback, etc. Every time we encounter this same structure, we always see an infinite regression. Godel's error was to treat the [name] and the [thing named] as if they are the same thing, when clearly they cannot possibly be the same. His trick of using astronomically large numbers to represent the name and the thing named, however, makes it very difficult to see what is actually happening, since it is literally impossible to actualize either the [name] or the [thing being named] in his proof. This makes it very easy to ignore the infinite regression that must occur. However, since the infinite regression is unavoidable, the construction of the proof is invalid and thus it does not show what it claims to show.
If you're interested in a more detailed analysis, still using layman's language, but definitely much more precise and closer to Godel's original language, let me know your email, or some other place where we can discuss more and I'll be happy to expand.
Your take on the paradox is intriguing, you have divided the barber into two personalities, one who is a barber and one who is just another random guy who doesn't shave himself. You are suggesting that the barber shave his non barber self from what i understand. However that does not actually solve the problem, in fact the problem remains. You are just proposing he has schizophrenia, which might solve the problem from his point of view, but what if we change the frame of reference and set it as an observer? The paradox would be deemed solved only if everybody agrees. To other people he is still the barber who shaves himself.
My solution to the barber paradox is that the barber is a woman. Easy.
@@abigailcooling6604 She has legs. The concept is that a set cannot contain itself or can it. The description of it being like a mirror is intriguing. I think the solution is that a set cannot contain itself. Just as much as you cannot divide by 0 and get anything that exists. If you divide something into nothing then you get undefined due to the limit of 1/.X as n approaches 0. It leads to infinity. A set that contains itself would divide by nothing because it technically wouldn't contain itself. Therefore, it would create an infinite loop like dividing 0 does. It is the same. If you divide a set that contains nothing but itself, it would be 0. Dividing by 0 leads to an infinite loop which creates the paradox. I hope that makes sense as to why the paradox exists and why a set cannot contain itself if it is the only thing within the set. It is because there is nothing but the description of the set which means that the set contains nothing but itself so that it is 0 and you cannot divide by 0 as the limit leads to infinity. I have to go now.
11:00 "Most sets are not members of themselves"
I disagree.
An object, any object, always is a collection of one element, namely itself. There is no way to avoid this, because there is no property that might distinguish the said object from the collection it constitutes.
Let us take a set. Any set, be it the empty set, a set with one element, with a finite number of members, or with infinite elements. Now, let us consider the set of such a set. This superset has only one member, the above-mentioned set, and therefore, as per the reasoning in the above paragraph, is indistinguishable from its contained set. Therefore it contains itself. Any set contains itself. Every set contains itself. If the definition of the set indicates that it does not contain itself, it is a self-contradictory definition and defines no set; maybe it defines something else, but not a set. Bertrand Russell´s paradox, solved.
Of course, this means that a set, always being a member of itself, indistinguishable from itself, always has a cardinality of one. This is in addition to the cardinality of the set as measured by the number of members that it has. A set with n members has a cardinality of n and also a cardinality of one. How is this possible? Answer: the nature of a thing is not necessarily singular, it may be several things at the same time. For instance, I am a vertebrate, am a mammal, am a person . . . . am a collection of ten trillion biological cells, ten percent of which are human cells.
My concept of set matches the concept of collection of objects very tightly. It is not that of Zermelo-Fraenkel sets, which cannot contain themselves (and are, in my set concept, not sets but something else: a Zermelo-Fraenkel-qualified collection of objects; if mathematicians want to call them sets, who am I to stop them? But they do no favour to people seeking clarity).
A set containing another set is not the same as the set itself. A wrapper is not the same thing as a container, even if a wrapper contains only one container. A larger box shipped by UPS containing a smaller box like the larger box is not the same thing as the smaller box inside it.
@@Syndiate__ It depends on your notion of what a set should be and what it should not be. I've introduced a concept of set that's grounded not on a set being a container but on it being the collection of objects, nothing added, nothing taken. Just the members.
@@wafikiri_ So by your logic, a set is not a container.... but it contains objects?
@@Syndiate__ No, a set is not a container. A set is a collection of objects.
Up and Atom Kurt Gödel shed a lot of light on self referential statements with his work. You should consider a follow up video covering his work on meta-mathematics and consistency vs completeness. I really enjoyed this video btw! 😁
She's done Gödel I think, and the related Halting Problem. But yes, a follow-up video on how one led to the other would be well warranted.
Psychologist: hey mr. Frege, Gödel doesn't exists, it can't hurt you...
Intuitively, numbers are variables:
“One” : l
“Two” : ll
“Three” : lll
…
Those variables point to a collection of objects (not quite…)
1) A number can be considered a scalar, is a tensor, is a sequence, is a class, is an object. Given a sequence, we can apply the system from Tensor Theory, and define numbers like that…
2) A number can be considered 1 more than the previous number, where 0 represents nothing.
3) A number can be considered as the *magma* of generic collections mapping to the explicitly defined property: “quantity”.
Barber paradox solution: The Barber doesn't need to shave, because she is a woman.
@E Mathematics has nothing to do with it. Gender bias to assume the barber must be a man.
I had a grandmother who could have used a barber...
@@isadoreladuca1112 When you know her. Women can get more hirsute after menopause.
Other options, barber is a kid, so he doesn't have a beard and adults are not allowed to be barbers. Or barbers are required to have bears by law.
According to Russell - not a paradox because the barber simply does not exist. This does not work with class theory however, as he points out.
Really interesting, what I like about this paradox is that in a way it’s the same as the problem with quantizing gravity.
One of the problems with quantizing gravity is that it’s not a quantum field on top of space time, it is space time, I see here similarity to this paradox, the set of the sets that are not members of themselves is sort of different than other sets in the same way that gravity is from the other fundamental forces.
Yes! I was thinking the same thing
@7:50 oh, gotta try defining number, because I love trying to do this kind of stuff.
So, I thought through stuff like she said, quantity, amount. Then I went on to stuff like sets of things (the things you would count). Then I thought of a series, the series of numbers, and a correspondence between each number and the item in the set being counted.
So, removing the word "number" and such to make the definition non-circular:
A label selected from a series of unique labels that are in a fixed order, with a beginning. For a given set of items, each item receives a unique label given out in the same order as the list of labels is defined, beginning with the beginning label. A set itself can receive a label that is the same as the final label applied to the items in the set.
Then you just have to invent names for those labels. The beginning we call "one", and so on.
And the above also shows how "quantity" and stuff comes from this.
Anyways, rough sketch, now on to watch the rest of the video!
@9:10 Huh, looks like Gottlob Frege got extensions and concepts reversed when he talks about numbers...
Clearly, 4 is the concept, and the extension is every set of objects that have that amount.
Also, I slightly cheated because I know Richard Carrier said that set theory is the foundation of mathematics or something, so I knew "sets" had to be important, that helped me! Yes, Richard Carrier is the cheat-sheet for philosophy.
And, to handle two dimensional numbers (so-called "complex numbers") can generalize from a list in one dimension to naming locations on another dimension as well.
When I was a little lad of five I started school, the teacher started talking about the ' numbers ' one, two, three, four ..... We were encouraged to count these on our fingers.
It was several decades later that for me the big intellectual leap was made that from ' one ' to ' two ' is to accept the fiction that two objects are the same, so the concept of ' two ' is a DOUBLING of the original object. It is a matter of convenience [ a first level of abstraction ] that the second one is the same in the present concept. So that a right shoe is for the moment treated as no different from a left shoe, or a red sock is equivalent to a blue sock to serve as working assumptions for some a priori result. The question of whether or not such a priori results are useful in some real-world application seems to be an important consideration for most of us. So if you are selling oranges it's an important abstraction that each individual orange has the same properties as every other orange on the stall. This is an essential abstraction necessary to facilitate the sale of oranges.
My real analysis professor gave us a definition of a set as "any mathematical object" which in retrospect punts the definition down the road, as it leaves open the question of what is "mathematical". He also raised the question of when a set can itself be a member of the set and the example he gave was the "set of ideas".
I would claim sets are groups, specifically groups of images.
But first I will show that numbers are built from images
Example , 4 always represents 4 images, like 4 squares for instance.
To be specific numbers are "labels" for groups of images
1. The main idea here is that maths is built from images
(a) example , geometry is clearly made of images
b) example 2, We claim numbers are built from images too, as say 4 , always represents 4 images, like 4 squares for instance.
C) imaginary numbers are connected to images too , which is why they have applications in physics
D) In general any mathematical symbol that comes to mind is connected to images too.
Q- What is mathematics?
A- Something cool
Q- Show me a proof
A- This video
"Show me proof" isnt a question
@@marcperez2598 Q is not a question. Q is just a name of someone.
@@lewis3774 absolutely
That bottom quark will haunt my dreams...
Oh no XD
I too, have nightmares about Michael Jackson.
Frege didn't coin the term "General Comprehension Principle". It was simply his Basic Law #5 (a property of concepts). W. V. Quine pointed out in his 1955 paper that Frege's Basic Law 5 was implied by the Principle of Abstraction/Comprehension, but Frege wasn't directly concerned about the properties of sets per se.
I’d argue the most shocking and interesting part of this story comes after, when Gödel throws a spanner in the works
"Immanuel Kant soon came along." Well that took some 'human' time but on the scale of the universe I'll let that one stand :)
I mean, what does soon even mean?
This thread xD
A "number" is a collective grouping noun. It can also be used as a verb to assign titles to objects.
Bertrand Russell has been one of my heroes since I first heard about mathematical philosophy
It was actually first discovered by German mathematicians before him, but he was the first to publish it.
"Tifa is a dog" Tifa's like, "I am?!" lol
"Always knew I liked balls way too much."
It's worth mentioning that Russell's aim also was destroyed by the great Kurt Gödel, guess karma is a thing?
haha i know. poor logicians... such great men too
But hey, we got computer science out of all of it.
@@JoshuaHillerup I'm waiting for the Wolfram Alpha idea (A new kind of Mathematics) that Computing is the basis of Mathematics. I'm all for HoTT.
Gödel, like others working on the foundation of logic, ended up mentally unstable. Guess this stuff is very hard.
@@Krmpfpks That raises an interesting question, because it means that people will lose their senses when trying to make sense of logic, which in itself should be logical, but apparently we as humans can't deal with this logic?
Jade, you have a gift in presenting complex concepts. The only thing more fascinating is you!
So, I walk into the barber shop and I saw the barber all lathered up with a razor ready to go. He says,
"I'm not feeling myself today."
Number - A manmade reference, used for logical analysis of objects and the interactions with their environments.
Are you implying that logic is man made? We cannot use a man made concept to conclude something absolutely given in a logical proof...
Yes, you can use manmade objects as tools to simplify and organize your thoughts in the prime concept of logic. That’s all numbers are are placeholders for our minds so we can keep track of what we know and draw implications from what we know, especially when we can’t focus on too much knowledge at once using our brains at their current evolutionary state.
10 month later ....
This definition contains many things that aren't numbers. like words, language, philosophy and logic itself.
@@AriaNight What if you add "to a quantity" after "A manmade reference" and keep the rest the same? Yes, it uses the word "quantity" which is a word that's a direct reference to the word "number," but it seems like a pretty good definition at least.
@@elliem7339 you answered yourself, it's recursive definition, ofcourse sounds good. But it's not gonna be useful
”Plato, Aristotle..”
”Kant soon came along!”
That's the paradox of time.
I love this. I would say, not having watched further in the video yet then the preposition to define a number without using number, that a number is: a mathematical object defined by a relation to the empty set and non-empty sets using logical operators.
While the paradox holds a folklorish status, there's some less reported work too.
Frege talked of sense and reference, not just extension. Russell worked on theory of description, not just theory of types. Wittgenstein came up with ideas that led to Russell's logical atomism.
Before you proceed to Godel, a shoutout to these shall be appreciated much 😊. All in the spirit of giving a 360° overview!
Your graphics are so cute that I sometimes have to rewind and focus my attention on what you are saying. Totally worth it, though, nice explanation.
Thank the animators lol
It's one of those divide by zeros situations where you just have to axiomatically declare the set of all things which are not sets an undefined set.
It doesn't have an answer. There's no answer to some number divided by zero. There's no answer to the whether the barber shaves himself as presented, so you add an axiom that the barber shaves people who don't shave themselves *and aren't barbers* and himself.
@@tthung8668 i do believe this human nature at fault, because we try categories nature from sentience. Nature true particle physics does not bind to mathematics, albeit a great measuring stick does not give you the full scope of what is at play.
@@tthung8668 Yes, it's a very different situation.
The so-called "imaginary" numbers (as derisively named by Descartes) are no more imaginary than zero or negative numbers, both of which were in the past equally derided. It's just that in our current state of mathematical sophistication many non-mathematicians have a comfortable intuition for zero and negative numbers that was not previously widespread. In the same way that if you're comfortable with extending a number line infinitely in both directions, giving you negative numbers, you can get comfortable with having two perpendicular number lines extending infinitely in both directions, giving you a plane on which each complex number is a point, at which point you have an easy way of developing an intuition about how _i_ and the like work.
The issue with division by zero is that (in a _very_ arm-wavy sense here) division itself tends to be intuited as an inverse of multiplication in the same way that subtraction is an inverse of addition. Thus, "if we can subtract any numbers we can add, we should be able to divide any numbers we can multiply." But division is not that at all, which is why it doesn't work as I just described, and there _are_ numbers that can be multiplied but not divided.
@@Curt_Sampson friendly reminder that RUclips use Markdown not LaTex.
* for *bold* and _ for italics
@@therealjezzyc6209 Actually, my attempt (which I should have checked after posting) was HTML markup, not LaTeX. But thanks for the reminder; I can never keep straight which commenting systems use which markup, but I've made a note to myself about what RUclips uses and fixed the comment.
So, the barber was given an impossible task, and you have to make an exeption for him to complete his task, makes sense.
However, if you are trying to create a theory that is supposed to be a completely logical foundations of all of maths, creating an arbitrary exeption like that would completely defeat the purpose of what you are doing
My definition of number: An individual member of a fully abstract series.
For example: If I create a uni-dimensional series, I get all real numbers. If I create a uni-dimentional space made out of individual member with a set distance between them, I get natural numbers.
The names "one", "two", "1/3", etc. are just that, names.
Sets like even numbers are only defined based on another set. If you take a fully abstract series that is only described with the most basic concepts like dimension or space you get the numbers.
I love your videos :) great explanation for complicated topics, and the animation is amazing and creative, thank you jade
Um the question I'm struggling to get a grip on is whether mathematics just exists and we "unearth"/"discover" it or if we "invent" it. If we do just discover mathematics using our intellectual capabilities as they develop, then is it even right to look for a "foundation" of mathematics in the first place? If we're really just discovering numbers woven into the fabric of our reality then where does the logic idea fit in?
They’re not just embedded in our reality, they’re embedded outside our reality too! They are embedded in truth itself.
The idea of using different foundations these days is less about finding one that is “correct” (internally consistent), since there can be more than one correct foundation, but to find one that is both the simplest and most descriptive. All a foundation is is something that other math concepts can be described in terms of, so for example if the idea of a triangle can be described as either a set of points or a set of lines, and in both cases you would still be able to prove all of the properties of a triangle. Math concepts don’t require a foundation to exist, foundations are just useful for describing them in simpler and more unifying terms
@@Adraria8 So it's mostly just us trying to find the simplest set of axioms to construct the rest of mathematics with? And it doesn't clash with the idea of us discovering mathematics rather than inventing it?
Surya Shivaprasad Exactly. And no it doesn’t because you could think of looking for foundations as either discovering them or inventing them.
Personally I don't think math is a fundamental property of the universe. Instead it is a fundamental property of the human mind.
A number is an item or items with a label depicting how many items are existing, being referred to, or are non-existent. This works for zero and integers. Negatives, decimals, etc. are expressions of those labeled items.
Note to self, next time make sure you are sufficiently caffeinated before stumbling round RUclips and clicking on a random mathematics paradox video. I liked and subscribed after my coffee.
Thank you.
Coffee and math
2 thumbs way up! Lol
Thank you, great explanation! ALso loved the graphics, especially the birds :)
Please include more of Tifa! Dogs are so lovely and older dogs deserve love just as much as puppies!
8:00 “Numbers are something we can hold onto in face of the infinite”
More poetic then useful but this is one definition for “numbers” I came up with.
Kant 'soon came along'? I guess time is relative.
Jade, you're like a scientist disney princess, love you!!
mathematician
Thanks for this really superbly explained and illustrated video. I have no idea how to define a number; but I think it has to encompass the concept you described whereby different numbers can apply depending on context (one pair of shoes vs two shoes).
The context can go all the way up to everything: One universe vs however many points in space; even one multiverse vs however many universes it contains. All the way down to one proton vs three quarks.
As I see it, for there to be any numbers other than 1 or 0, there has to be some mechanism of differentiation or segmentation. Consider an infinite, completely empty space. Number would have no meaning as far as I can tell. Even if one proposes an imaginary grid or coordinate system, one is imposing differentiation or segmentation onto that empty space.
I sometimes have pondered that I think we can generate our whole number system from just the digit 1 and a set of operations (which I think may implicitly assume the existence of differentiation or segmentation): 1 - 1 = 0, 1 + 1 = 2, 1 + 1 + 1 = 3, .... So I suspect the essence of numbers somehow has to do with 1 (unity/emptiness/etc) and ways of breaking up that unity. Anyway, I will stop my rambling now lol.
A brilliant video, thank you! I have sipped it in small doses, and then tried to explain it to a fictitious friend. Now, I am eager to learn about Zermelo-Fraenkel :-).
Very interesting video. Now my interpretation of them might be wrong but I believe Gödel's theorems prove that there can't be a "fundation of maths".
Since no system of axioms can be both coherent and complete, and we can't even use a given system of axiom to determine whether it is incoherent or incomplete.
No system of axioms advanced enough for arithemtic, i.e. second order logic. First order logic still can be both coherent and complete.
@@redvel5042 You're right! That's still a problem for any candidate to the title of "fundation of maths"
As long as it's first order logic, which I'd say sounds fairly fitting for the *foundation* of mathematics, there isn't much problem. Of course, you'd still have to get a good candidate and prove its coherence and completeness, and indeed, it won't be capable of even arithmetic, but that may still be sufficient for the foundation of mathematics.
But if you want the foundation to be second order logic [or above, I guess], then yes, there's the probelm of the necessity of either incoherence or incompleteness. Thankfully, though, not all of them would be on the same level, so I'd say you could perhaps scream pragmatism once you're tired and pick the one with least problems and most benefits, lol.
@@redvel5042 We may not be using "fundation" in the same way here. I think about it like I think about the laws of physics in relation to chemistry and biology. You only have to accept those axioms and the rest will sort of appear.
If a system of axiom does not allow for arithmetic I don't understand how it could be considered fundamental. Perhaps you could explain?
Oh, on, it's not like the system of axioms doesn't allow for arithmetic. Instead, it simply doesn't go that far. Since you brought up physics as an example for a foundation for chemistry and biology, I guess it'd be kinda like how physics isn't exactly chemistry. It's got the building blocks for chemistry, but it's not so much concerned with chemical reactions and as far as I know [not a physicist, so feel free to correct me if I'm wrong], doesn't actually deal with such reactions or even say anything about it in particular.
So while first order logic is too simple for arithmetic, it's not like it outright doesn't allow it. You have to get to second order logic to get arithmetic. The good thing about first order logic is that it can be both coherent and complete. Only because it doesn't go as far as arithmetic, which does indeed render it mostly useless, trivial, but that's as far as you can go with coherent and complete systems of axioms.
I'm not a mathematician - I'm a programmer. The common thread from this and other sources I have read, is that a system cannot describe itself. It can only be described from or by another independent system.
If a system attempts to describe itself, paradoxes seem follow. This video only seems to reinforce this point of view, which is a little unfortunate as we tend to use mathematics to model things. The things we tend to model are in this universe. Along with us and the mathematics we use to describe those models. Presumably, as all of these things are in the same system - the universe - it would be impossible to describe said universe with said mathematical models without introducing paradoxes.
What got me into this, is that once upon a time I used to be assured in the truthfulness of logic until someone told me about the chap living on the island of Crete.... That one broke my logic box, much like your examples in this video.
For those that don't know the Cretan's story.... Basically, this chap (or chapess) from Crete, proudly stood up and proclaimed that all Cretans are liars. That one statement floored me, and I wrestled with it and its implications for a long time. How could logic break so easily?
Of course, one could resolve the above paradox by having somebody else, not from Crete, and outside of the system describe that system (Crete). For example, someone from mainland Greece could say 'All Cretans are liars'. Then you have a perfectly reasonable statement that in no way contradicts itself.
It's pretty much like your hairdresser's example. Have a car mechanic make the statement about that hairdresser and all the paradoxes disappear. Having the hairdresser make exactly the same statement results in the whole world falling apart!
As my job entails coding logic into software, I was naturally quite alarmed by these discoveries! :)
PS: the answer to the number question posed above is: It's a symbol that describes a quantity!
And soon after, Kurt Godëvil stroke, with his infernal indecidilities: math would never be the same... Poor Hilbert could not even dream for long of his perfect formalistic math-topia, ruined forever...
What is a number without using the word number: "a descriptor of the size of a set"
This works for natural numbers, but how does this describe decimal or even irrational numbers?
number n. 1. An indication of precise magnitude, degree, or extent.
I said, a unit of value.
@@RoiEXLab we can just go Euclidian and say that numbers are a symbol system that corresponds to a specific line segment.
A number is just a symbol. A symbol to interpret what we sense of a thing or things. As red would be a symbol of a certain wavelength of energy.
Infinity is a symbol used in math, but is it a number? Is zero a number or nothing? All of this pondering seems to show we have too many brain cells, and can postulate possibly impossible challenges.
If only more people could acknowledge such ideas. They still get called names. Poetic or emotional whereas as these are facts and is the missing trunk for botj math, science as well as spirituality.
"Common denominator approach": Patterns (including patterns of influence). Concepts are patterns of neural activity, and application of patterns influences other patterns. A number seems to be a mental pattern where rule patterns are set up, including associations to other patterns. Anyway, you know where I'm driving in this...this may help understand where some patterns cannot have influence over targeted patterns (the rule patterns impede association or influence).
I wonder if the barber also had a breakdown..
"To shave or not to shave?"
He later became Sweeney Todd.
The demon barber of fleet street
fifteen yard penalty for "alternative" use of a razor.
and another thirty yard penalty for undermining the innocent trust a Briton deserves to keep, concerning meat pies and sausages.
and yes, I'm using American football terms. Britons also deserve a better sport than continental football.
This is a demonstration that paradox seem to be inherently semantic.
But you need semantics to define stuff, right? Otherwise, there's nothing to study...
@@irrelevant_noob Yes, but definitions are descriptive not prescriptive.
Edmond Dantez Not all definitions though... Defining the series of Fibonacci numbers using recursion is quite prescriptive IMO. ^^
@@irrelevant_noob Um... no. It's still merely descriptive. You're not really "defining" the Fibonacci series using recursion, you're just applying an already established description (definition) to the case of solving the larger problem, now based on the original result. That essentially IS the definition (description) of a Fibonacci series.
Edmond Dantez uhh, wth are you on about?! What "larger problem"? What "original result"?!
And if not-even-that is an example of a prescriptive definition, then what was your point again? Why would it matter that "definitions are descriptive not prescriptive"? o.O
A set of sets is like a search query of prior search queries.
A search for prior searches will never show up in its own set of results. But it will show up in all such searches in the future.
Thus this paradox is easily resolved by requiring its formal resolution at a precise time. I think most most paradoxes might be resolved by requiring all infinite concepts or parameters be defined or constrained.
Which leads us to Godel, and incompleteness. Your videos are brilliant;)
I thought this as well
I've always struggled to express an idea or I guess a question I have about mathematics. So, mathematics is GREAT at modeling things in the real world, but I would like to emphasize the word MODELING because we all know that systems in nature are not actually performing complex mathematical operations in their regular functioning. I know that probably sounds dumb, I'm not an expert so I'm just trying to get some help from someone who might be able to point out the flaw in my thinking about this concept. I do really well in math classes so I'm not a complete dolt lol, currently taking Calc 2 & it's not as challenging as all the horror stories I've heard but I do think those stories are truly from people who never developed a solid foundation while they were in Calc 1 but somehow managed to pass & now they are having a hard time.
Anyway, we know that when a kid is riding a bike, he is not performing complicated math operations, the brain is using an entirely different processing system in order to maintain balance, speed & all that. I can't remember who said it but the famous quote of "the unreasonable effectiveness of mathematics" really resonated with me when I first read it because of how true it is, it's remarkable how USEFUL it continues to be & how POWERFUL it is at accurately describing phenomena but it's NOT even how reality operates!
Yup, math is a descriptive language of things. It isn't the things themselves.
The reason it's effective at describing stuff is because we made the language good at describing stuff! (Including describing how reality operates). Not so unreasonable, actually.
The brain is not carrying out those calculations, but they are in a sense baked into the structure and function of the brain. This happened through evolution. So in a sense it IS carrying out those calculations, but int a shorthand, compressed, and optimized way that makes many assumptions, etc. That's why it's very good at certain things like pattern recognition, but can also be tricked with optical illusions. The calculations being carried out are not actually doing all the math to model the entire world.
As to why math is good at modelling the world, it's because math is just a subset of logic, and the universe works according to logical principles.
I don't know I kind of feel that the brain IS making those complex calculations. The brain is taking in a serious amount of information through your senses and then issuing commands to your body. For example, if I throw a ball to someone I am taking into account it's are, the wind, the air resistance, gravity, spin, and all that. I am making the calculation in real time, I am just not doing it in a mathematical language. I am doing it through my experiential knowledge of the world both encoded in my DNA and learned through experience. It took me years and years of effort as a child to learn fine motor control and how to throw a ball well. If you added all the time I literally spent learning to control my body well and compared that to the amount of time it would take to obtain a PhD in mathematics or physics. Well you would quickly see that it takes significantly (orders of magnitude) less time to get that PhD than it did to master fine motor control and athleticism.
The actions and calculations your brain make are equivalent to doing the math, but just done in a much more efficient and natural way. Math like all languages is a poor representation of reality. None the less every atom or electron in your body conforms to quantum mechanics which is essentially the love child of statistics and calculus on steroids. And those bastard little particles break reality and can only be truly destined by mathematics.....though that right there might be why physics really hasn't advanced much since the1920's and has only become more and more fractured. How many different unprovable string theories do we have now? 50? 500? We kind of hit a dead end in physics 80 years ago and have been doing little other than fleshing out or confirming those theories, or coming up with wild unprovable predictions and theories.
Me: Can Immanuel do it, Jade?
Jade: 5:07
😂😂😂😂😂🤣
Immanuel Kant, but Genghis Khan!
There are many of us who are nuanced beyond adequate descriptions. I enjoyed your presentation. This comment is what came to mind at the problem description. I thought that according to Kant, the physical world does not imply the existence of mathematics. Therefore, mathematics is a synthetic construct that may or may not have descriptive use. This can only be determined posteriori or after the fact through analysis.
Language is the real issue. There are no paradoxes, there are only illogical circular statements.
Precisely
our definitions define our state of being, and our state of being influences our definitions.
:(
@C c The whole question is an illogical circular statement, but I'll play your silly game. He can only shave at home when he's off the clock.
@C c Only in your eyes and with people of your ilk. Doesn't an intellect of your stature have better things to do?
@C c Projecting are you?
The barber is a woman....................................
So she doesn't shave herself therefore she should shave herself but she can't because she does shave herself but then again she shouldn't....
This is a gender-neutral paradox dude
@@hicri9739then yeah its unsolvable but it aint gender neutral then this is a solution
Or a Sikh