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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