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 i mean the rules of the game are a kind of 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 actually 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 basically 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 the formalists lived to fight another day but self-reference refused to die quite that easily

fast forward to the 1960s and mathematician Hao Wang was looking at square tiles with different colors on each side like these

the rules were that touching edges must be the same color and you can't rotate or reflect tiles only slide them around

the question was if you're given an arbitrary set of these tiles can you tell if they will tile the plane that is will they connect up with no gaps all the way out to infinity

it turns out you can't tell 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

in fact it's 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 mathematical proofs

systems of proof were an old idea going back to the ancient greeks

a system of proof starts with axioms basic statements that are assumed to be true like a straight line can be drawn between any two points

proofs are then constructed from those axioms using rules of inference methods for using existing statements to derive new statements

and these are chosen to preserve truth the existing statements are true then so are the new ones

Hilbert wanted a formal system of proof a symbolic logical 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 would be a then b and no human is immortal would be expressed like this

Hilbert and the formalists wanted to express the axioms of mathematics as symbolic statements in a formal system and set up the rules of inference as the system's rules for symbol manipulation

so Russell along with Alfred North Whitehead developed a formal system like this in their three-volume Principia Mathematica published in 1913.

Principia Mathematica is vast a total of nearly 2 000 pages of dense mathematical notation

it takes 762 pages just to get to a complete proof that one plus one equals two

at which point Russell and Whitehead dryly note the above proposition is occasionally useful

the authors had originally planned a fourth volume but unsurprisingly they were too worn out to complete it

so yes the notation is dense and exhausting but it is also exact unlike ordinary languages it leaves no room for errors or fuzzy logic to creep in

and most importantly it allows you to prove properties of the formal system itself

there were three big questions that Hilbert wanted answered about mathematics

number one is math complete meaning is there a way to prove every true statement does every true statement have a proof

number two is mathematics consistent meaning is it free of contradictions

i mean if you can simultaneously prove a and not a then that's a real problem because you can prove anything at all

and number three is math 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 of these questions was yes

at a major conference in 1930 Hilbert gave a fiery speech about these questions

he ended it with a line that summed up his formalist dream in opposition to the foolish ignorabimus which means we will not know our slogan shall be we must know we will know

these words are literally on his grave

but by the time Hilbert gave this speech his dream was already crumbling

just the day before at a small meeting at the same conference a 24 year old logician named Kurt Gödel explained that he had found the answer to the first of Hilbert's three big questions about completeness 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 who pulled Gödel aside to ask a few questions

but the next year Gödel published a proof of his incompleteness theorem and this time everyone including Hilbert took notice

this is how Gödel's proof works

Gödel wanted to use logic and mathematics to answer questions about the very system of logic and mathematics

and so he took all these basic symbols of a mathematical system and then he gave each one a number

this is known as the symbol's Gödel number

so the symbol for not gets the number one or has the Gödel number two if then it's Gödel number three

now if you're expressing all of these symbols with numbers then what do you do about the numbers themselves

well zero gets its own Gödel number six and if you want to write a one you just put this successor symbol next to it

the immediate successor of zero is 1. and if you want to write 2 then you have to write ss0 and that represents 2 and so on

so you could represent any positive integer this way granted it is cumbersome but it works and that is the point of this system

so now that we have Gödel numbers for all of the basic symbols and all of the numbers we might want to use we can start to write equations like we could write zero equals zero

so these symbols have the Gödel numbers six five six and we can actually create a new card that represents this equation zero equals zero

and the way we do it is we take the prime numbers starting at 2 and we raise each one to the power of the Gödel number of the symbol in our equation

so we have 2 to the power of 6 times 3 to the power of 5 times 5 to the power of 6 equals 243 million

so 243 million is the Gödel number of the whole equation zero equals zero

we can write down Gödel numbers for any set of symbols that you can imagine

so really this is an infinite deck of cards where any set of symbols that you want to write down in a sequence you can represent with a unique Gödel number

and the beauty of this Gödel number is you can do a prime factorization of it and you can work out exactly what symbols made up this card

so in this whole pack of cards there are going to be true statements and there are going to be false statements

so how do you prove that something is true well you need to go to the axioms

the axioms also have their own Gödel numbers which are formed in the same way

so here's an axiom which says not the successor of any number x equals zero which makes sense because in this system there are no negative numbers and so the successor of any number cannot be zero

so we can put down this axiom and then we can substitute in zero for x saying that one does not equal zero and we've created a proof

this is the simplest proof i can think of that shows that one does not equal zero

now this card for the proof of one does not equal zero gets its own Gödel number

and the way we calculate this Gödel number is just as before we take the prime numbers and we raise 2 to the power of the axiom times 3 to the power of 1 does not equal 0 and we get a tremendously large number

it would have 73 million digits if you calculated it all out so i've just left it here in exponential notation

as you can see these numbers get very large and so you might want to start just calling them by letters so we could say this one is Gödel number a this is Gödel number b Gödel number c and so on

Gödel goes to all this trouble to find this card which says there is no proof for the statement with Gödel number g

the trick is that the Gödel number of this card is g

so what this statement is really saying is this card is unprovable

there is no proof anywhere in our infinite deck for this card

now think about it if it's false and there is a proof then what you have just proven is there is no proof so you're stuck with a contradiction

that would mean the mathematical system is inconsistent

the other alternative is that this card is true there is no proof for the statement with Gödel number g

but that means this mathematical system has true statements in it that have no proof so your mathematical system is incomplete

and that is Gödel's incompleteness theorem

this is how he shows that any basic mathematical system that can do fundamental arithmetic will always have statements within it which are true but which have no proof

there's a line in the tv show the office that echoes the self-referential paradox of Gödel's proof

jim is my enemy but it turns out the jim is also his own worst enemy and the enemy of my enemy is my friend so jim is actually my friend

but because he is his own worst enemy the enemy of my friend is my enemy so actually jim is my enemy but

Gödel's incompleteness theorem means that truth and provability are not at all the same thing

Hilbert was wrong there will always be true statements about mathematics that just cannot be proven

now Hilbert could console himself with the hope that at least we could still prove maths consistent that is free of contradictions

but then Gödel published his second incompleteness theorem in which he showed that any consistent formal system of math cannot prove its own consistency

so taken together Gödel's two incompleteness theorems say that the best you can hope for is a consistent yet incomplete system of math

but a system like that cannot prove its own consistency so some contradiction could always crop up in the future revealing that the system you'd been working with had been inconsistent the whole time

and that leaves only the third and final big question from Hilbert is mathematics decidable that is is there an algorithm that can always determine whether a statement follows from the axioms

and in 1936 Alan Turing found a way to settle this question but to do it he had to invent the modern computer

in his time computers weren't machines they were people often women who carried out long and tedious calculations

Turing imagined an entirely mechanical computer

he wanted it to be powerful enough to perform any computation imaginable but simple enough that you could reason through its operation

so he came up with a machine that takes as input an infinitely long tape of square cells each containing a zero or a one

the machine has a read write head that can read one digit at a time then it can perform one of only a few tasks overwrite a new value move left move right or simply halt

halting means the program has run to completion

the program consists of a set of internal instructions you can think of it like a flow chart 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 perform in exactly the same way as the first

although this sounds simple a Turing machine's arbitrarily large memory and program mean it can execute any computable algorithm if given enough time

from addition and subtraction to the entire youtube algorithm it can do anything a modern computer can

that's what made the Turing machine so useful in answering Hilbert's question on decidability

when a touring machine halts the program has finished running and the tape is the output

but sometimes a touring machine never halts maybe it gets stuck in an infinite loop

is it possible to tell beforehand if a program will halt or not on a particular input

Turing realized this halting problem was very similar to the decidability problem

if he could find a way to figure out if a Turing machine would halt then it would also be possible to decide if a statement followed from the axioms

for example you could solve the twin prime conjecture by writing a Turing machine program that starts with the axioms and constructs all theorems that can be produced in one step using the rules of inference

then it constructs all theorems that can be produced in one step from those and so on

each time it generates a new theorem it checks if it's the twin prime conjecture and if it is it halts if not it never halts

so if you could solve the halting problem you could solve the twin prime conjecture and all sorts of other unsolved questions

so Turing said let's assume we can make a machine h that can determine whether any Turing machine will halt or not on a particular input

you insert the program and its input and h simulates what will happen printing out either halts or never halts

for now we don't worry about how h works we just know that it always works it always gives you the right answer

now we can modify the h machine by adding additional components one if it receives the output halts immediately goes into an infinite loop another if it receives never holds then it immediately halts

we can call this entire new machine h plus and we can export the program for that entire machine

now what happens if we give this machine its own code both as program and input

well now h is simulating what h plus would do given its own input essentially h has to determine the behavior of a machine that it itself is a part of in this exact circumstance

if h concludes that h plus never halts well this makes h plus immediately halt

if h thinks h plus will halt well then that necessarily forces h plus to loop

whatever output the halting machine h gives it turns out to be wrong there's a contradiction

the only explanation can be that a machine like h can't exist there's no way to tell in general if a touring machine will halt or not on a given input and this means mathematics is undecidable

there is no algorithm that can always determine whether a statement is derivable from the axioms

so something like the twin prime conjecture might be unsolvable we might never know whether there are infinite twin primes or not

the problem of undecidability even crops up in physical systems in quantum mechanics

one of the most important properties of a many-body system is the difference in energy between its ground state and its first excited state

this is known as the spectral gap

some systems have a significant spectral gap whereas others are gapless there are a continuum of energy levels all the way down to the ground state

and this is important because at low temperatures gapless quantum systems can undergo phase transitions while gapped systems cannot they don't have the energy needed to overcome the spectral gap

but figuring out if a system is gapped or gapless has long been known to be a very difficult problem

only recently in 2015 did mathematicians prove that in general the spectral gap question is undecidable

to quote the authors even a perfect complete description of the microscopic interactions between a material's particles is not always enough to deduce its macroscopic properties

now remember that Turing designed his machines to be as powerful computers as it is possible to make

to this day the best computational systems are those that can do everything a touring machine can

this is called touring completeness and it turns out there are many such during complete systems

although they are powerful every touring complete system comes with a catch its own analog of the halting problem some undecidable property of the system

Wang tiles are touring complete their halting problem is whether or not they'll tile the plane

complex quantum systems are Turing complete and their halting problem is the spectral gap question

and the game of life is Turing complete its halting problem is literally whether or not it halts

there are many more examples of this like airline ticketing systems magic the gathering powerpoint slides and excel spreadsheets

nearly every programming language in existence is designed to be Turing complete

but in theory we only need one programming language because well you can program anything at all using any Turing complete system

so here's a touring machine inside the game of life and since the game of life is itself touring complete it should be capable of simulating itself and indeed it is

this is the game of life running on the game of life

the real legacy of David Hilbert's dream is all of our modern computational devices

Kurt Gödel suffered bouts of mental instability later in life convinced that people were trying to poison him

he refused to eat eventually starving himself to death

Hilbert died in 1943 his epitaph was his slogan from 1930. we must know we will know

the truth is we don't know sometimes we can't know but in trying to find out we can discover new things things that change the world

Alan Turing put his ideas about computing to practical use in world war ii leading the team at Bletchley Park that built real calculating machines to crack nazi codes for the allies

by one estimate the intelligence Turing and his colleagues gathered from decrypted messages shortened the war by two to four years

after the war Turing and John von Neumann designed the first true programmable electronic computer eniac based on touring's designs

but Turing didn't live to see his ideas develop much further

in 1952 the british government convicted him of gross indecency upon learning he was gay he was stripped of his security clearance and forced to take hormones

in 1954 he committed suicide

touring changed the world he is widely considered to be the most important founding figure in computer science

all modern computers are descended from his designs

but touring's ideas about computability came from his concept of the Turing machine which came from thinking about Hilbert's question is math decidable

so touring's code breaking machines and indeed all modern computers stem from the weird paradoxes that arise from self-reference

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

and you might think this realization would have driven mathematicians mad and led to the disintegration of the entire mathematical enterprise

but instead thinking about this problem transformed the concept of infinity changed the course of a world war and led directly to the invention of the device you're watching this on right now

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