Russell's Paradox - A Ripple in the Foundations of Mathematics

Hello everyone, Jade, here today's video is about one of the most famous

paradox

es in the history of

mathematics

. It is called Russell's

paradox

and is named after Bertrand Russell. Cool guy, a very influential philosopher, mathematician, logician, social activist and Nobel Prize winner, so yeah, we'll meet. all about it, but this is probably one of my favorite paradoxes and it's really one of those that makes you question everything you know, question everything you know, so yeah, I'm really excited for this video right before we start. I mean this is probably the newest study I've ever done, there's a lot of subtlety and a lot of philosophy, so if you notice that you're getting a little lost or need to rewind or watch the video again, that's totally normal and in fact, It would be pretty surprising if you got it all the first time, so yeah, let's get right to it, so first I'll present you with a somewhat simpler version of the paradox just to soften some of the concepts that we're going to talk about later, like this This is called the barber's paradox and it goes like this: there is a town with a barber and his task is to shave everyone and only those who do not shave themselves, now the question is, does the barber shave?

Think about it if the barber doesn't shave then he should and if he shaves then he shouldn't. So keep this little riddle in the back of your mind for the rest of the video, but now let's get straight to the point, so Russell's paradox is ultimately about the

foundations

of

mathematics

, but it's possible that Some of you may not be familiar with that concept. What is a foundation of mathematics? Well, let's make an analogy with the other sciences. Sciences are like a tree. Biology is. the study of living organisms that are the result of millions and millions of chemical interactions, so one could conclude that all branches of biology come from chemistry or that chemistry is somehow fundamental to biology;

Likewise, chemical reactions are ultimately physical interactions, so one might conclude that The old branches of chemistry arise from physics. If we continue descending this way, we will eventually reach the elementary particles that make up our universe. Electrons and quarks. The laws that govern these particles can be considered the

foundations

of science in the sense that the rest of the sciences are derived from them, if we discovered something completely new about them, it would somehow affect our understanding of the other sciences, even if only outside on a theoretical level, in the 19th century each branch of mathematics was quite disconnected from the other branches and there was no unifying mathematician at that time who wanted to unify the branches of mathematics and therefore a fundamental theory was needed, but it What made it complicated is that the approach to discovering the foundations of mathematics is very different from the other sciences that we learn about mathematics. other sciences by looking at the outside world, whether it's looking at cells under a microscope, measuring orbits through a telescope, or smashing particles at incredibly high speeds, but we can't really learn more about mathematics by looking at the outside world.

Can we imagine a universe where the fundamental components are different or where time goes backwards or where everything is made of antimatter, but we can't really imagine a universe where two plus two equals five? The math is different in the sense that it appears to be true. by definition, and although it is very good at describing our universe, it doesn't seem to be about what the universe is like, why that is the question at the center of this paradox. Now this is where we move away from the more difficult sciences and move more towards study. of what reality really is philosophy we come to the first character in our story the question why mathematics is the way we capture one of the greatest philosophical minds of all time Plato Plato thought that mathematical objects, numbers, forms and the relationships between them were objective truths that is, objects that existed independently of us and our worlds, he thought that they existed in their own world which he called the world of forms, but one of Plato's students, Aristotle, had his own views, unlike Plato, he thought that numbers themselves were not objects but properties of objects, for example, if there were four cows in a field, it was not that we cows were an object and four was also a object, but the number four was a property of the collection of cows, he also thought that they did not exist in their own world of forms independent of our world, but they described characteristics of our world and therefore belonged to our world.

Soon another colorful philosopher, Immanuel Kant, appeared and he did not like the view that mathematical objects were objective truths and he believed that to understand basic concepts. principles of mathematics you needed some kind of intuition, but yes, mathematics described our world, but we also brought it into the world from our experience. Now this is where our silent hero of the story, Gottlob Frager, appears. Yes, interestingly, Bertrand Russell is not the main actor. Russell's paradox, but a German mathematician called Frager, Freya's story is somewhat tragic as he was largely ignored during his lifetime and twenty years of his life's work were unfairly washed down the drain, but here he will be remembered as a God among us.

Frager disagreed. Aristotle's view that numbers were properties of objects is due to this argument, if numbers are properties of objects then only one number should belong to any object and it should not be influenced by matters of opinion, but then imagine a pair of shoes, is it a pair of shoes? shoes or it is shoes depending on how we conceptualize an object the number that belongs to it changes again imagine a deck of cards depending on how we conceptualize it it could be a deck of cards or 52 cards so Frago concluded that numbers do not apply to objects , but he took concepts, he also did not agree with Kent that to understand mathematics intuition and experience were needed, stating that the laws of arithmetic can be known only from reason.

I tried to make it plausible that arithmetic is a branch of logic and not necessarily so. For me, borrow any motive of proof, whatever experience or intuition, Frege's main goal was to reduce mathematics to logic and show that logic is, in fact, the basis of mathematics. This idea is known as logis ISM. Now the notion of logic that Frager had in mind is quite similar to how we would use it in everyday language, but to be super clear, logic is a tool for reasoning about how different statements affect each other through nothing more. than two ductions and inferences, for example, if we take as a fact that all dogs have a good sense of smell and Tifa is a dog.

Deductive reasoning will tell you that Tiffer has a good sense of smell. I don't know if this is the best example, because she is quite old. I don't know if she can smell. You get the idea that Freya began her quest to reduce mathematics to logic by finding a definition of what a number was. Now this might seem like a somewhat trivial question to some of you: what is a number? But remember, at the heart of Frege's question. The paradox was the idea that you don't need any kind of intuition or experience to understand mathematics, and when you think about it, have you ever tried to describe what a number is without using the word number or words derived from the word number?

I spent some time trying while writing this video and found it. I always ended up using the word number again or a word that was directly related to a number like amount or amount. Try it right now and let me know what you come up with. In the comments, Frager defined numbers using this idea of ​​concepts and extensions, a concept is pretty much any idea you can think of, the color red, the shape of the beak, and the works of Bob Shakespeare, earless hipster elephants, literally any idea. and an extension is the set. of all the things that fall under that concept for the concept the color red its extension would be the set of all past, present and future red things the concept bill and Bob would be made of the extension of the bill and Bob that you are referring to if they have a concept like square circles that has no meaning its extension would simply be the empty set as a set with nothing in it the numbers declared by Frege are extensions of concepts, for example, the number four is the extension of the concept of all things composed of one collection of so many objects the number seven is the extension of the concept of all things composed of a collection of so many objects Frager built his foundations of mathematics from the axiom that all the concepts that we can think of have a corresponding extension and therefore There are as many extensions as there are concepts.

He called it the principle of general understanding and it sounds reasonable. Can you think of a concept that does not have a corresponding set? I don't think I could, but Bertrand Russell could. Frago was about to print his work when he received a letter from Bertrand Russell in June 1901 that said something like Dear Colleague. I completely agree with you in all essentials regarding many particular issues that I find in his work. discussion of distinctions and definitions that one seeks in vain in the works of other logicians there is only one point where I have found a difficulty in considering the set of all sets that are not members of themselves is that the set is a member of itself This simple question itself shattered the Freitas Foundation and caused him a mental crisis so serious that he ended up in the hospital and then made him write.

My efforts to shed light on the question surrounding the word number seem to have ended in complete failure, let's analyze that first. What about the sets? Sets are a collection of things that you can even have sets within sets, take for example the set of all sets of bird species, there is the set of all penguins, the set of all seagulls, the set of all pigeons and the set of all other bird species, and they themselves form the set of all sets of bird species, most sets are not members of themselves, that is, they do not include each other, e.g. , the set of all teapots is not a teapot, the set of all turtles is not a turtle, but what about the set of all things that are not Turtles because it is a set and a turtle is not a member of the set of all things that are not Turtles therefore the set of all things that are not Turtles is a member of itself many sets of members of themselves and there is nothing extraordinary about that, but the question Russell asked was: the set of all sets that are not members of themselves, a member of itself, think about it, if it is a member of itself then it is not and if it is not it is a member of itself, sounds familiar, I'll say it again if it is a member of itself, then it is not and if it is not a member of itself, now the reason why this was so catastrophic is because the last thing you want in the verbose systematized acts that the EMA links, the description of any thing is a contradiction.

Imagine if one of the fundamental laws of physics predicted that gravity always pulls and always pushes, it would be a pretty useless law when your system is able to derive two opposing theorems from this. The entire theory was questioned and numerous attempts were made to correct this problem, such as Russell's type theory, which attempted to put sets in some kind of hierarchy, but many thought this solution was too artificial and Frager's ad-hoc eventually came to fruition. He felt forced to abandon many of his views on logic and mathematics, yet as Russell Franco points out, he received the news of the paradox with remarkable strength.

As I think about acts of integrity and grace, I realized that there is nothing in my knowledge that is can compare with Frege's dedication to truth, his life's work was nearing completion, much of his work had been ignored in favor of men infinitely less capable, his second volume was about to be published, and upon discovering that his fundamental assumption was a mistake, he responded with intellectual pleasure, clearly submerging any feeling of personal responsibility. The disappointment was almost superhuman and a telling indication of what men are capable of if their dedication is to creative work and knowledge rather than cruder efforts to dominate and be known.

Apparently he didn't know about the collapse anyway after a series of fortunate events. It is now known that the foundations of mathematics are a system called Zermelo Fraenkel's set theory which has many of Frege's original ideas incorporated, but although this is currently the most accepted theory, the search is not yet over with candidates such as set theory. categories and type theoryof homotopy are becoming serious contenders for that highly desired position at the bottom of the mathematical tree, but that's for another video.