There is a hole in the bottom of Math

Everyone, Leland Bartlett from the reporting training consultancy, this video played a very interesting video that I found not only for its content, but also for its deep questions and answers, but it starts this video with these statements. There is a

hole

at the

bottom

of

math

ematics. All of this means that we will never know everything for sure; there will always be true statements that cannot be proven. No one knows what exactly those claims are. I discovered that Dr. Derek Miller from the very in-depth Veritasium YouTube channel shows us three questions about

math

ematics. Is complete? consistent and decidable progress with logic to show us that, in fact, mathematics is not complete consistency is questionable and not decidable below is the full video I hope you watch it it is very interesting I think the conclusions can shake the foundations of our beliefs because we put a lot of emphasis on science, but the basis of science is mathematics and those who are best at it the state is not bulletproof, I hope you enjoy it, leave me a comment what you thought, there is a

hole

in the

bottom

of mathematics. hole that means we will never know everything for sure there will always be true statements that cannot be proven now no one knows what exactly those statements are but they could be something like the twin prime conjecture twin primes are prime numbers that are separated by only one numbers like 11 and 13 or 17 and 19. and as you go down the number line, primes occur less frequently and twin primes are even rarer, but the twin prime conjecture is that there are infinitely many twin primes than ever You're exhausted from now on, no. one has proven that this conjecture is true or false, but the crazy thing is that we may never know because what has been proven is that in any system of mathematics where you can do basic arithmetic there will always be true statements that are impossible to prove. . specifically, this is the game of life created in 1970 by mathematician John Conway, sadly he passed away in 2020 from Covet 19.

Conway's game of life is played on an infinite grid of square cells, each of which She is alive or dead and there are only two. rules one, any dead cell with exactly three neighbors comes to life and two, any live cell with less than two or more than three neighbors dies once you have set up the initial arrangement of cells, the two rules are applied to create the next generation and then the next and the next and so on is completely automatic. Conway called it a zero-player game, but although the rules are simple, the game itself can generate a wide variety of behaviors.

Some patterns are stable once they emerge, others never change. they oscillate back and forth in a loop some can travel through the network forever like this glider here many patterns just disappear but some keep growing forever keep generating new cells now you would think that given the simple rules of the game they could Just look at any pattern and determine what will happen to it, whether it will eventually reach a stable state or continue to grow without limit, but it turns out that this question is impossible to answer. The ultimate fate of a patron in Conway's game of life is undecidable.

There is no possible algorithm that guarantees answering the question in a finite period of time. You can always try running the pattern and see what happens. I mean, the rules of the game are some kind of algorithm after all, but that doesn't guarantee that it will give it to you. one answer because even if you run it for a million generations you won't be able to tell if it will last forever or just 2 million generations or a billion or a googleplex. Is there something special about the game of life that does it? undecidable, no, there are actually a lot of systems that are undecidable, like Wang Tiles' quantum physics airline ticketing systems and even the magical meeting to understand how undecidability appears in all these places.

We have to go back 150 years to a full-blown revolt. In mathematics in 1874, Georg Cantor, a German mathematician, published a paper that launched a new branch of mathematics called set theory. A set is simply a well-defined collection of things, so the two shoes you are wearing are a set. , like all the planetariums in the world. In the world there is a set with nothing, the empty set and a set with everything. Now Cantor was thinking about sets of numbers like natural numbers, positive integers like 1, 2, 3, 4, etc., and real numbers that include fractions like one-third and five halves. and also irrational numbers like pi e and the square root of two basically any number that can be represented as an infinite decimal it was asked if there are more natural numbers or more real numbers between 0 and 1 the answer may seem obvious there is an infinite number of each, so both sets must be the same size, but to verify this logic, I imagined writing an infinite list that pairs each natural number on one side with a real number between zero and one on the other, now that each real number is a decimal infinite there. is not the first, so we can write them in any random order.

The key is to make sure you get them all without duplicates and align them one by one with an integer. If we can do it without any left, then we know that the set of natural numbers and the set of real numbers between 0 and 1 are the same size, so suppose we have made it so that we have a complete infinite list with each integer acting as a index number, a unique identifier for each real number in the list now says cantor start writing a new real number and the way we're going to do that is by taking the first digit of the first number and adding one, then take the second digit of the second number and again add one, take the third. third number digit add one and continue doing this throughout the list, if the digit is nine just convert it back to eight and at the end of this process you will have a real number between zero and one but here is what this number does not will appear nowhere in our list is different from the first number in the first decimal different from the second number in the second decimal and so on down the line it has to be different from each number in the list by at least one digit the number on the diagonal , that's why it is called Cantor's diagonalization test.

It shows that there must be more real numbers between 0 and 1 than there are natural numbers that extend to infinity, so not all infinities are the same size. Cantor calls them countable and uncountable infinities respectively and in fact there are many more uncountable infinities that are even larger now Cantor's work was just the latest blow to mathematics for 2000 years Euclid's elements were considered the basis of the discipline , but in the early 19th century Lobazewski and Gauss discovered non-Euclidean geometries and this prompted mathematicians to take a closer look at the foundations of their field and they didn't like what they saw.

The idea of ​​a limit at the heart of calculus turned out to be ill-defined and now Cantor was proving that infinity. In itself, it was much more complex than anyone had imagined. In all this turmoil, mathematics fractured and a great debate broke out among mathematicians at the end of the 19th century. On the one hand, there were intuitionists who thought that Cantor's work was a nonsense, they were convinced that mathematics was a pure creation of the human mind and that infinities like gallops were not real auri puenkere said that later generations will consider set theory as a disease from which one has recovered leopold kronecker He called Cantor a scientific charlatan and a corrupter of youth and worked to prevent Cantor from getting the job he wanted.

On the other side were the formalists who thought that mathematics could be based on absolutely secure logical foundations through Cantor's set theory. informal leader of the formalists was the german mathematician david hilbert hilbert was a living legend an enormously influential mathematician who had worked in almost all areas of mathematics almost beat einstein in general relativity developed completely new mathematical concepts that were crucial for the quantum mechanics and knew that Kanter's work was brilliant Hilbert was convinced that a more formal and rigorous mathematical proof system based on set theory could solve all the problems that had arisen in mathematics over the last century and most of the other mathematicians agreed with him: "No one will expel us from the paradise that Cantor has created," declared Hilbert, but in 1901, Bertrand Russell pointed out. solved a serious problem in Cantor's set theory, Russell knew that if sets can contain anything, they can contain other sets or even themselves, for example, the set of all sets must contain itself, Just like the set of sets with more than five elements, we could even talk about the set of all sets that contain themselves, but this leads directly to a problem.

What happens to r the set of all sets that do not contain themselves? If r does not contain itself well, then it must contain itself, but if r contains itself. then by definition it must not contain itself, so r contains itself if and only if it does not. Russell had found another paradox of self-reference and then explained his paradox using a hairy analogy: Let's say there is a village populated entirely by adult men with a strange law against beards, specifically the law states that the village barber must shave everyone and only the men in the town who don't shave, but the barber also lives in the town, of course, and he's a man, so who shaves him if not? he doesn't shave, so the barber has to shave him, but the barber can't shave himself because the barber doesn't shave anyone who shaves, so the barber must shave only if and only if he doesn't shave, it's a contradiction.

Intuitionists rejoiced at Russell's paradox, thinking that it had shown that set theory was hopelessly flawed, but Zermelo and other mathematicians of the Hilbert school solved the problem by restricting the concept of set so that the collection of all sets, For example, it is no longer a set, any more than it is the collection of all sets that do not contain themselves. This eliminated the paradoxes that come with self-reference. Hilbert and the Formalists lived to fight another day, but self-reference refused to die so easily. Fast forward to the 1960s and mathematician Hal Wang was looking at the square. tiles with different colors on each side like these, the rules were that the touching edges must be the same color and you can't rotate or mirror the tiles, just slide them around the question was if you are given an arbitrary set of these tiles, what? can you know?

If they will tile the plane, that is, they will connect without spaces to infinity, it turns out that one cannot know for an arbitrary set of tiles whether they will tile the plane or not, the problem is undecidable just like the fate of a pattern in conway's game of life is in fact exactly the same problem and that problem ultimately comes from self-reference as Hilbert and the formalists were about to discover Hilbert wanted to secure the foundations of mathematics by developing a new system for proofs math. Testing systems were an ancient idea dating back to the ancient Greeks.

A proof system begins with axioms. Basic statements that are assumed to be true, as if a straight line could be drawn between any two points. Proofs are then constructed from those axioms using inference rules. methods of using existing statements to derive new statements and these are chosen to preserve the truth the existing statements are true so so are the new ones hilbert wanted a formal proof system a symbolic logic language with a rigid set of manipulation rules for those symbols logical and mathematical statements could then be translated into this system. If you drop a book, then it will fall.

It would be a, then b, and no human being is immortal, it would be expressed like this. Hilbert and the formalists wanted to express the axioms of mathematics as symbolic statements in a formal format. system and established the rules of inference as system rules for symbol manipulation, so Russell together with Alfred North Whitehead developed a formal system like this in his three-volume Principia Mathematica published in 1913. Principia Mathematica is enormous, with a total of almost 2,000 pages. of dense mathematical notation it takes 762 pages to arrive at a complete proof that one plus one equals two, at which point russell and whitehead dryly note that the above proposition is occasionally useful the authors had originally planned a fourth volume but, unsurprisingly they were too worn out to complete it, yes the notation is dense andexhausting, but it is also exact, unlike ordinary languages, it leaves no room for errors or confusing logic, and, most importantly, it allows testing the properties of the formal system itself.

There were three big questions. that Hilbert wanted an answer about mathematics number one is mathematics complete meaning is there a way to prove each statement true? every true statement has a proof number two is consistent mathematics meaning is it free of contradictions? I mean, if you can simultaneously prove one and not one then that is a real problem because you can prove anything and the number three is a mathematically decidable meaning. Is there an algorithm that can always determine whether a statement follows from the axioms? Now Hilbert was convinced that the answers to all three questions were yes.

At an important conference in 1930, Hilbert gave an impassioned speech on these questions and ended it with a phrase that summed up his formalist dream as opposed to the foolish ignorabus, meaning that we will not know what our motto will be, we must know that we will know that these words are. literally in his grave, but when Hilbert gave this speech, his dream was already crumbling just the day before, in a small meeting at the same conference, a 24-year-old logician named Kurt Gerdel explained that he had found the answer to the first of Hilbert's three questions. Big questions about integrity and the answer was no.

A complete formal system of mathematics was impossible. The only person who paid much attention was John von Neumann. One of Hilbert's former students took Goodl aside to ask him some questions, but the following year Goodell published a proof of his incompleteness theorem and this time everyone, including Hilbert, they realized that this is how the Goodell test works. Goodell wanted to use logic and mathematics to answer questions about the same system of logic and mathematics, so he took all of these basic symbols of a mathematical system and then gave each one a number, this is known as the good symbol number. , so the zero symbol gets the number one or has the good number. two if then it's good, the whole number three now, if you're expressing all these symbols with numbers, then what do you do with the numbers themselves?

Well, zero has its own good number, six, and if you want to write one, just put this successor symbol below. the immediate successor of zero is one and if you want to write two then you have to write ss0 and that represents 2 and so on so you can represent any positive integer this way, it's cumbersome but it works and that's the point of this system, now that we have all the numbers for all the basic symbols and all the numbers that we would want to use, we can start writing equations as if we could write zero equals zero, so that these symbols have the numbers good six five 6 and In fact, we can create a new card that represents this equation: zero equals zero and the way we do this is by taking the prime numbers starting at two and raising each one to the power of the Goodell number of the symbol in our equation so that we have 2. to the power of 6 times 3 to the power of 5 times 5 to the power of 6 is equal to 243 million so 243 million is the good number of the entire equation zero is equal to zero we can write small numbers for any set of symbols that You can imagine that this is really an infinite deck of cards where any set of symbols that you want to write in a sequence you can represent it with a single Goodal number and the beauty of this Goodal number is that you can do a prime factorization of it and I can find out exactly what symbols make up this card, so throughout this card pack there will be true statements and there will be false statements, so how do you prove something to be true?

Well, you must go to the axioms. The axioms also have their own good numbers that are formed in the same way, so here is an axiom that says that the successor of any number x is not equal to zero, which makes sense because in this system there are no negative numbers and , therefore the successor of any number cannot be zero so we can write this axiom and then we can replace x with zero saying that one is not equal to zero and we have created a proof. This is the simplest test I can think of that shows that one is not equal to zero.

Now this. The card for the proof that one is not equal to zero gets its own Goodal number and the way we calculate this Goodal number is the same as before: we take the prime numbers and raise 2 to the power of the axiom multiplied by 3 a the power of 1 is not equal to 0 and we get a tremendously large number. It would be 73 million digits if you calculated it all out, so I left it here in exponential notation, since you can see these numbers get very big, so you might want to just start. calling them by letters so we can say this is the goodall number a this is the goodall number b the goodall number c and so on goodall goes to all the trouble to find this card which says there is no proof for the statement with the goodall number g The trick is that the good number for this card is g, so what this statement really says is that this card cannot be proven, there is no proof anywhere in our infinite deck for this card.

Now think about it, if it is false and there is a proof, then I just proved that there is no proof, so you are stuck in a contradiction which would mean that the mathematical system is inconsistent. The other alternative is that this card is real. There is no proof of the claim with the number g, but that means this mathematical system. you have true statements that have no proof, so your mathematical system is incomplete and that is Goodall's incompleteness theorem. This is how it shows that any basic mathematical system that can do fundamental arithmetic will always have statements that are true but have no proof.

There is a line in the television show The Office that echoes the self-referential paradox of the Goodell test. Jim is my enemy, but it turns out that Jim is also his worst enemy and the enemy of my enemy is my friend, so Jim is actually my friend. but because he is his worst enemy my friend's enemy is my enemy so actually jim is my enemy but goodall's incompleteness theorem means that truth and provability are not the same thing at all hilbert was wrong there will always be true statements about mathematics that simply cannot be proven now Hilbert could console himself with the hope that at least we could still prove that mathematics is consistent and free of contradictions, but then Goodell published his second incompleteness theorem in which he showed that any consistent formal system mathematics cannot prove its own consistency.

So, taken together, Goodell's two incompleteness theorems say that the best that can be expected is a consistent but incomplete mathematical system, but such a system cannot prove its own consistency, so some contradiction could always arise in the future that reveals that the system you would be working on had been inconsistent all along and that leaves only Hilbert's third and final big question: is mathematics decidable, that is, is there an algorithm that can always determine whether a statement is decidable? does it follow from the axioms? And in 1936, Alan Turing found a way to solve this question, but to do so he had to invent the modern computer.

In his time, computers were not machines, they were people, often women, who carried out long and tedious calculations. Turing envisioned a completely mechanical computer and wanted it to be powerful enough to perform any calculation imaginable but simple enough that you could reason through its operation, so he came up with a machine that takes a tape as input infinitely long square cells, each of which contains a zero or a one. The machine has a read and write head that can read one digit at a time. a while and then you can perform one of the few tasks overwrite a new value move left move right or just stop stop means that the program has run to completion the program consists of a set of internal instructions that you can think of as a flowchart that tells the machine what to do based on the digit it reads and its internal state.

You could imagine exporting these instructions to any other Turing machine which would then work in exactly the same way as the first, although this sounds simple. The arbitrarily large memory of a Turing machine and the program means that it can run any computable algorithm if given enough time from addition and subtraction to the entire YouTube algorithm. It can do anything a modern computer can do. That's what made the Turing machine so useful in answering Hilbert's question about decidability when a spinning machine stops the program. has finished running and the tape is the output, but sometimes a Turing machine never stops, maybe it gets stuck in an infinite loop.

Is it possible to know in advance whether or not a program will stop at a particular input? Turing realized that this stopping problem was very similar to the decidability problem. If he could find a way to determine whether a Turing machine would stop, then it would also be possible to decide whether a statement followed by the axioms, for example, could solve the twin prime conjecture by writing a Turing machine program that starts with the axioms and constructs all the theorems that can be produced in one step using the rules of inference, then constructs all the theorems that can be produced in one step by Starting from those and so on, every time it generates a new theorem, it checks if it is the twin prime. conjecture and if it is it stops, if not it never stops so if you could solve the stopping problem you could solve the twin prime conjecture and all sorts of other unsolved questions so said turing suppose that we can make an h machine that can determine if there is any The processing machine will stop or not on a particular input, you insert the program and its input and h simulates what will happen, the printing stops or never stops for now, We don't worry about how h works, we just know that it always works, always. gives you the correct answer now we can modify the machine h by adding additional components one if it receives the output it stops immediately it goes into an infinite loop another if it receives it never stops then it stops immediately we can call this new complete machine h plus and now h is simulating what h plus would do given its own input, essentially h has to determine the behavior of a machine which itself is part of this exact circumstance if h concludes that h plus never stops well, this causes h plus to stop immediately if h believes that h plus will stop fine, then that necessarily forces h plus to loop whatever output the stopping machine h provides. be wrong there is a contradiction the only explanation can be that a machine like h cannot exist there is no way to know in general whether a spinning machine will stop or not at a given input and this means that the mathematics is undecidable there is no algorithm We can always determine whether a statement is derivable from the axioms, so something like the twin prime conjecture might be unsolvable.

We may never know if there are infinitely many twin primes or not. The problem of undecidability even arises in physical systems. In quantum mechanics, one of the most common problems. An important property of a many-body system is the energy difference between its ground state and its first excited state. This is known as the spectral gap. Some systems have a significant spectral gap while others have no gap. There is a continuum of energy levels at all times. to the ground state and this is important because at low temperatures quantum systems without gaps can undergo phase transitions, while systems with gaps cannot.

They don't have the energy to bridge the spectral gap, but it has long been difficult to determine whether a system has gaps or not. It is known to be a very difficult problem, only recently, in 2015, did mathematicians show that, in general, the question of the spectral gap is undecidable, citing the authors, including a complete and perfect description of the microscopic interactions between particles of a material is not always sufficient to deduce its macroscopic properties. Remember that Turing designed his machines to be the most powerful computers possible. To this day, the best computer systems are those that can do everything a touring machine can do.

This is called tour integrity and it turns out there are a lot of these complete systems. Although powerful, each complete tour system has a drawback: its own analogue of the stopping problem, some undecidable property of thesystem. The wang mosaics are on tour complete, their stopping problem is whether or not they are going to tile the plane. Complex quantum systems are being completed and their stopping problem is the question of the spectral gap and the game of life is being completed and their stopping problem is literally whether to stop or not. There are many more examples of this, such as airline ticketing systems, the magic of PowerPoint slides, and Excel spreadsheets that accrue to almost every programming language.

Existence is designed to be complete, but in theory we only need one programming language because well, you can program anything using any complete tour system, so here is a tour machine inside the game of life and since the game of life itself is being completed. it should be able to simulate itself and in fact it is this is the game of life running in the game of life the true legacy of david hilbert's dream are all our modern computing devices kurt godel later suffered attacks of mental instability in life convinced that people were trying to poison him he refused to eat and finally starved to death hilbert died in 1943 his epitaph was his slogan from 1930. we should know, we will know, the truth is that we don't know, sometimes we don't we may know, but by trying to find we can discover new things things that change the world alan turing put his ideas about computing into practice in the second world war leading the team at bletchley park that built real calculating machines to crack nazi codes for the allies according to an intelligence estimate turing and his colleagues gathered from decrypted messages shortened the war between two and four years after the war turing and john von neumann designed the first eniac programmable electronic computer based on turing's designs, but turing did not He lived to see his ideas develop much further in 1952.

The British government convicted him of gross indecency upon learning that he was gay. They took away his security clearance and forced him to take hormones in 1954. He committed suicide. Touring changed the world. He is widely considered the most important founding figure in computing. All modern computers. descended from his designs, but Turing's ideas about computability come from his concept of the Turing machine, which arose from thinking that Hilbert's question is mathematically decidable, so Touring's code-breaking machines and, In fact, all modern computers arise from the strange paradoxes that arise from self-reference there. There is a hole at the bottom of mathematics.

A hole that means we will never know everything for sure. There will always be true statements that cannot be proven. You would think that this discovery would have maddened mathematicians and led to the disintegration of the entire mathematical enterprise, but instead of thinking about this problem, it transformed the concept of infinity, changed the course of a world war, and led directly to the invention of the device you're viewing this on now. Hey, this video was sponsored by a brilliant website full of interactive courses and quizzes that go deeper into topics like maths, physics and computer science.

If you've made it this far in this video, you're probably someone who would love it. Great, I've been following your logic course and it really made me think that the problems start out easy but get more and more challenging as you develop your understanding. and that's what I love about the site. I love how he guides you, not telling you. exactly, but by exposing you to increasingly sophisticated puzzles and in the end I feel like I've solved it for myself, which I essentially have done through its carefully curated selection of problems and with helpful hints and explanations for when I get stuck now There were a lot of things I wanted to include in this video but I couldn't because it's already over half an hour long, so if you want to explore these ideas further, I highly recommend his courses on number theory and computer science fundamentals for viewers of this channel. bright is offering 20 off an annual subscription to the first 200 people who sign up.

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