Math's Fundamental Flaw

There is a hole at the bottom of math a hole that means we will never know everything with certainty There will always be true statements that cannot be proven. Now no one knows what those statements are exactly but they could be something like the Twin Prime Conjecture. Twin primes are prime numbers that are separated by just one number like 11 and 13, or 17 and 19. And as you go up the number line primes occur less frequently and twin primes are rarer still.

But the Twin Prime Conjecture is that there are infinitely many twin primes. You never run out them. As of right now no one has proven this conjecture true or false. But the crazy thing is this: 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. That is life. 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 is either live or dead and there are only two rules one any dead cell with exactly three neighbors comes to life and two any living cell with less than two or more than three neighbors dies once you've set up the initial arrangement of cells the two rules are applied to create the next generation and then the one after that

and the one after that and so on it's totally automatic Conway called it a zero player game but even though the rules are simple the game itself can generate a wide variety of behavior some patterns are stable once they arise they never change others oscillate back and forth in a loop a few can travel across the grid forever like this glider here many patterns just fizzle out

but a few keep growing forever they keep generating new cells now you would think that given the simple rules of the game you could just look at any pattern and determine what will happen to it will it eventually reach a steady state or will it keep growing without limit but it turns out this question is impossible to answer the ultimate fate of a pattern in Conway's game of life is undecidable meaning there is no possible algorithm that is guaranteed to answer the question in a finite amount of time you could always just try running the pattern and see what happens the rules of the game are a algorithm after all but that's not guaranteed to give you an answer either because

even if you run it for a million generations you won't be able to say whether it'll last forever or just 2 million generations or a billion or a googleplex is there something special about the game of life that makes it undecidable nope there are a huge number of systems that are undecidable like Wang tiles quantum physics airline ticketing systems and even magic the gathering to understand how undecidability shows up in all of 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 just a well-defined collection of things so the two shoes on your feet are a set as are all the planetariums in the world there's a set with nothing in it the empty set and a set with everything in it now Cantor was thinking about sets of numbers like natural numbers positive integers like 1 2 3 4 and so on and real numbers

which include fractions like a third five halves and also irrational numbers like pi e and the square root of two any number that can be represented as an infinite decimal he wondered are there more natural numbers or more real numbers between zero and one the answer might seem obvious there are an infinite number of each so both sets should be the same size but to check this logic canter imagined writing down an infinite list matching up each natural number on one side with a real number between zero and one on the other now since each real number is an infinite decimal there is no first one so we can just write them

down in any random order the key is to make sure we get them all with no duplicates and line them up one to one with an integer if we can do that with none left over well then we know that the set of natural numbers and the set of real numbers between 0 and 1 are the same size so assume we've done that we have a complete infinite list with each integer acting like an index number a unique identifier for each real number on the list now Cantor says start writing down a new real number

and the way we're going to do it 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 digit of the third number add one and keep doing this all the way down the list if the digit is a nine just roll it back to an eight and by the end of this process you'll have a real number between zero and one but here's the thing this number won't appear anywhere on our list it's different from the first number

in the first decimal place different from the second number in the second decimal place and so on down the line it has to be different from every number on the list by at least one digit the number on the diagonal that's why this is called Cantor's diagonalization proof it shows there must be more real numbers between 0 and 1 then there are natural numbers extending out to infinity so not all infinities are the same size Cantor call these countable and uncountable

infinities respectively and in fact there are many more uncountable infinities which are even larger now Cantor's work was just the latest blow to mathematics for 2000 years Euclid's elements were considered the bedrock of the discipline but at the turn of the 19th century Lobashevsky and gauss discovered non-Euclidean geometries and this prompted mathematicians to examine more closely the foundations of their field and they did not like what they saw the idea of a limit

at the heart of calculus turned out to be poorly defined and now Cantor was showing that infinity itself was much more complex than anyone had imagined in all this upheaval mathematics fractured and a huge debate broke out among mathematicians at the end of the 1800s on the one side were the intuitionists who thought that Cantor's work was nonsense they were convinced that math was a pure creation of the human mind and that infinities like Cantors weren't real

Henri Poincaré said that later generations will regard set theory as a disease from which one has recovered Leopold Kronecker called Cantor a scientific charlatan and a corrupter of the youth and he worked to keep Cantor from getting a job he wanted on the other side were the formalists they thought that math could be put on absolutely secure logical foundations through Cantor's set theory the informal leader of the formalists was the german mathematician David Hilbert

Hilbert was a living legend a hugely influential mathematician who had worked in nearly every area of mathematics he almost beat Einstein to the punch on general relativity he developed entirely new mathematical concepts that were crucial for quantum mechanics and he knew that Cantor's work was brilliant Hilbert was convinced that a more formal and rigorous system of mathematical proof based on set theory could solve all the issues that had cropped up in math over the last century

and most other mathematicians agreed with him no one shall expel us from the paradise that Cantor has created Hilbert declared but in 1901 Bertrand Russell pointed out 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 as does the set of sets with more than five elements in them you could even talk about the set of all sets that contain themselves

but this leads straight to a problem what about r the set of all sets that don't contain themselves if r doesn't contain itself well then it must contain itself but if r does contain itself then by definition it must not contain itself so r contains itself if and only if it doesn't Russell had found another paradox of self-reference and he later explained his paradox using a hairy analogy let's say there's a village populated entirely by grown men with a strange law against

beards specifically the law states that the village barber must shave all and only those men of the village who do not shave themselves but the barber himself lives in the village too of course and he's a man so who shaves him if he doesn't shave himself then the barber has to shave him but the barber can't shave himself because the barber doesn't shave anyone who shaves themselves so the barber must shave himself if and only if he doesn't shave himself it's a contradiction

the intuitionists rejoiced at Russell's paradox thinking it had proven set theory hopelessly flawed but zermelo and other mathematicians from Hilbert school solved the problem by restricting the concept of a set so the collection of all sets for example is not a set anymore and neither is the collection of all sets which don't contain themselves this eliminated the paradoxes that come with self-reference Hilbert and