The following is a conversation with Donald Knuth,
one of the greatest and most impactful computer scientists
and mathematicians ever.
He's the recipient of the 1974 Turing Award,
considered the Nobel Prize of Computing.
He's the author of the multi-volume work,
The Magnum Opus, The Art of Computer Programming.
He made several key contributions
to the rigorous analysis of computational complexity
of algorithms, including the popularization
of asymptotic notation that we all affectionately
know as the big O notation.
He also created the tech typesetting system, which
most computer scientists, physicists, mathematicians,
and scientists and engineers in general
use to write technical papers and make them look beautiful.
I can imagine no better guest to end 2019 with than Don,
one of the kindest, most brilliant people in our field.
This podcast was recorded many months ago.
It's one I avoided because, perhaps counterintuitively,
the conversation meant so much to me.
If you can believe it, I knew even less
about recording back then, so the camera angle is a bit off.
I hope that's OK with you.
The office space was a big cram for filming,
but it was a magical space, where Don does most of his work.
It meant a lot to me that he would welcome me into his home.
It was quite a journey to get there, as many people know.
He doesn't check email, so I had to get creative.
The effort was worth it.
I've been doing this podcast on the side for just over a year.
Sometimes I had to sacrifice a bit of sleep,
but I was happy to do it, and to be part of an amazing community
of curious minds.
Thank you for your kind words of support
and for the interesting discussions,
and I look forward to many more of those in 2020.
This is the Artificial Intelligence Podcast.
If you enjoy it, subscribe on YouTube,
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Alex Friedman, spelled F-R-I-D-M-A-N.
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And now here's my conversation with Donald Knuth.
In 1957, at Case Tech,
you were once allowed to spend several evenings
with an IBM 650 computer,
as you've talked about in the past,
and you fell in love with computing then.
Can you take me back to that moment with the IBM 650?
What was it that grabbed you about that computer?
So the IBM 650 was this machine
that, well, it didn't fill a room,
but it was big and noisy.
But when I first saw it, it was through a window
and there were just a lot of lights flashing on it.
And I was a freshman.
I had a job with the statistics group,
and I was supposed to punch cards for data
and then sort them on another machine,
but then they got this new computer, came in
and it had interesting lights, okay.
So, well, but I had a key to the building,
so I could get in and look at it and got a manual for it.
And my first experience was based on the fact
that I could punch cards, basically,
which is a big thing for the,
but the IBM 650 was big in size,
but incredibly small in power, in memory.
In memory, it had 2,000 words of memory
and a word of memory was 10 decimal digits plus a sign.
And it would do, add two numbers together,
you could probably expect that would take,
I'll say three milliseconds.
So...
It's still pretty fast.
It's the memory's the constraint,
the memory's the problem.
That was why it took three milliseconds
because it took five milliseconds for the drum to go around.
And you had to wait, I don't know, five cycle times.
If you have an instruction in one position on the drum,
then it would be ready to read the data
for the instruction and you know, go three notches.
The drum is 50 cycles around
and you go three cycles and you can get the data
and then you can go another three cycles
and get to your next instruction.
If the instruction is there,
otherwise you spin until you get to the right place.
And we had no random access memory whatsoever
until my senior year.
In senior year, we got 50 words of random access memory
which were priceless.
And we would move stuff up to the random access memory
in 60 word chunks and then we would start again.
So subroutine went to go up there and...
Could you have predicted the future 60 years later
of computing from then?
No, in fact, the hardest question I was asking
was ever asked was, what could I have predicted?
In other words, the interviewer asked me,
she said, you know, what about computing has surprised you?
You know, and immediately I ran,
I rattled off a couple of dozen things
and then she said, okay, so what didn't surprise?
And I tried for five minutes to think of something
that I would have predicted and I couldn't.
But let me say that this machine,
I didn't know, well, there wasn't much else
in the world at that time.
The 650 was the first machine that was,
that there were more than 1,000 of ever.
Before that, there were, you know,
there was each machine, there might be a half a dozen
examples, maybe, maybe...
This is the first mass market mass produced.
The first one, yeah, done in quantity.
And IBM didn't sell them, they rented them,
they rented them, but they rented them
to universities at great, you know, had a great deal.
And so that's why a lot of students learned
about computers at that time.
So you refer to people, including yourself,
who gravitate toward a kind of computational thinking
as geeks, at least I've heard you use that terminology.
It's true that I think there's something
that happened to me as I was growing up
that made my brain structured in a certain way
that resonates with computers.
So there's this space of people,
it's 2% of the population you empirically estimate.
That's been fairly constant over most of my career.
However, it might be different now
because kids have different experiences when they're young.
So what does the world look like to a geek?
What is this aspect of thinking that is unique to...
That makes, yeah.
That makes a geek.
This is a hugely important question.
In the 50s, IBM noticed that there were geeks
and non-geeks, and so they tried to hire geeks
and they put out ads with papers saying,
if you play chess, come to Madison Avenue
and for an interview or something like this,
they were trying for some things.
So what is it that I find easy
and other people tend to find harder?
And I think there's two main things.
One is the ability to jump levels of abstraction.
So you see something in the large
and you see something in the small
and you pass between those unconsciously.
So you know that in order to solve some big problem,
what you need to do is add one to a certain register
and it gets you to another step and below the...
Yeah, I mean, I don't go down to the electron level,
but I knew what those milliseconds were,
what the drum was like on the 650.
I knew how I was gonna factor a number
or find a root of an equation or something
because of what was doing.
And as I'm debugging, I'm going through,
did I make a key punch error?
Did I write the wrong instruction?
Do I have the wrong thing in a register?
And each level, each level is different.
And this idea of being able to see something at lots of levels
and fluently go between them seems to me to be more pronounced,
much more pronounced in the people that resonate with computers.
So in my books, I also don't stick just to the high level,
but I mix low-level stuff from the high level.
And this means that some people think that I should write better books,
and it's probably true.
But other people say, well, if you think like that,
then that's the way to train yourself,
keep mixing the levels and learn more and more how to jump between.
So that's the one thing.
The other thing is that it's more of a talent,
to be able to deal with non-uniformity,
where there's case one, case two, case three,
instead of having one or two rules that govern everything.
So it doesn't bother me if I need, like an algorithm has 10 steps to it.
Each step does something else that doesn't bother me.
But a lot of pure mathematics is based on one or two rules,
which are universal.
And so this means that people like me sometimes work with systems
that are more complicated than necessary because it doesn't bother us
that we didn't figure out the simple rule.
And you mentioned that while Jacobi, Buol, Abol,
and all the mathematicians in the 19th century may have had symptoms of geek.
The first 100% legit geek was Turing, Alan Turing.
I think he had a lot more of this quality than anyone,
just from reading the kind of stuff he did.
So how does Turing, what influence has Turing had on you?
Well, I didn't know that aspect of him until after I graduated somewhere.
As an undergraduate, we had a class that talked about computerability theory
and Turing machines, and it sounded like a very specific kind of purely theoretical approach to stuff.
So how old was he when I learned that he had a design machine
and that he wrote a wonderful manual for Manchester machines
and he invented subroutines and he was a real hacker that he had his hands dirty.
And I thought for many years that he had only done purely formal work.
As I started reading his own publications, I could feel this kinship.
And of course, he had a lot of peculiarities.
He wrote numbers backwards because, I mean, left to right instead of right to left
because it was easier for computers to process them that way.
What do you mean left to right?
He would write pi as, you know, nine, five, one, four, point three, I mean, okay.
Right, got it.
Four, one, point three.
On the blackboard, I mean, he had trained himself to do that
because the computers he was working with worked that way inside.
He trained himself to think like a computer.
Well, there you go. That's geek thinking.
You've practiced some of the most elegant formalism in computer science
and yet you're the creator of a concept like literate programming
which seems to move closer to natural language type of description of programming.
Yeah, absolutely.
Do you see those two as conflicting as the formalism of theory and the idea of literate programming?
So there we are in a non-uniform system where I don't think one size fits all
and I don't think all truth lies in one kind of expertise.
And so somehow, in a way, you'd say my life is a convex combination of English and mathematics.
You're okay with that.
And not only that, I think.
Thrive in it.
I wish, you know, I want my kids to be that way.
I want, et cetera, you know.
Use left brain, right brain at the same time.
You got a lot more done.
That was part of the bargain.
And I've heard that you didn't really read for pleasure until into your 30s in literature.
That's true.
You know more about me than I do, but I'll try to be consistent with what you read.
Yeah, no, just believe me.
I just go with whatever story I tell you.
It'll be easier that way.
The conversation works.
Right, yeah.
No, that's true.
So I've heard mention of Philip Roth's American Pastoral, which I love as a book.
I don't know if it was mentioned as something, I think, that was meaningful to you as well.
In either case, what literary books had a lasting impact on you?
What literature?
Yeah, okay.
Good question.
So I met Roth.
Oh, really?
Well, we both got doctors from Harvard on the same day.
So we had lunch together and stuff like that, but he knew that computer books would never
sell.
Well, all right, so you say you were a teenager when you left Russia.
So I have to say that Tolstoy was one of the big influences on me.
I especially like Anna Karnina, not because of a particular plot of the story where, but
because there's this character who, you know, the philosophical discussions, the whole way
of life is worked out there among the characters.
And so that I thought was especially beautiful.
On the other hand, Dostoevsky, I didn't like it all because I felt that his genius was mostly
because he kept forgetting what he had started out to do and he was just sloppy.
I didn't think that he polished his stuff at all.
And I tend to admire somebody who dots the eyes and crosses the T's.
So the music of the prose is what you admire more than...
I certainly do admire the music of the language, which I couldn't appreciate in the Russian
original, but I can in Victor Hugo because French is closer.
But Tolstoy, I like, the same reason I like Herman Wolk as a novelist.
I think like his book, Marjorie Morningstar has a similar character in Hugo who developed
his own personal philosophy and it goes in the...
Was consistent.
Yeah.
Right.
Yeah.
And it's worth pondering.
So yeah.
So you don't like Nietzsche and...
Like what?
You don't like Friedrich Nietzsche or...
Nietzsche.
Yeah.
No, no.
I keep seeing quotations from Nietzsche and they never tempt me to read any further.
Well, he's full of contradictions, so you will certainly not appreciate him.
But Schiller, I'm trying to get across what I appreciate in literature and part of it
is, as you say, the music of the language, the way it flows and take Raymond Chandler
versus Dashel Hammett, Dashel Hammett's sentences are awful and Raymond Chandler's are beautiful.
They just flow.
So I don't read literature because it's supposed to be good for me or because somebody said
it's great, but I find things that I like.
I mean, you mentioned you were dressed like James Bond, so I love Ian Fleming.
I think he's got it.
He had a really great gift for...
If he has a golf game or a game of bridge or something and this comes into his story,
it'll be the most exciting golf game or the absolute best possible hands of bridge that
exists and he exploits it and tells it beautifully.
So in connecting some things here, looking at literate programming and being able to
convey encode algorithms to a computer in a way that mimics how humans speak, what do
you think about natural language in general and the messiness of our human world, about
trying to express difficult things?
So the idea of literate programming is really to try to understand something better by seeing
it from at least two perspectives, the formal and the informal, if we're trying to understand
a complicated thing, if we can look at it in different ways.
And so this is, in fact, the key to technical writing, a good technical writer trying not
to be obvious about it but says everything twice, formally and informally, or maybe three
times, but you try to give the reader a way to put the concept into his own brain or her
own brain.
Is that better for the writer or the reader or both?
Well, the writer just tries to understand the reader.
Just the goal of a writer is to have a good mental image of the reader and to say what
the reader expects next and to impress the reader with what has impressed the writer
why something is interesting.
So when you have a computer program, we try to, instead of looking at it as something
that we're just trying to give an instruction to the computer, what we really want to be
is giving insight to the person who's going to be maintaining this program or to the programmer
himself when he's debugging it as to why this stuff is being done.
And so all the techniques of exposition that a teacher uses or a book writer uses make
you a better programmer if your program is going to be not just a one-shot deal.
So how difficult is that?
Do you see hope for the combination of informal and formal for the programming task?
Yeah, I'm the wrong person to ask, I guess, because I'm a geek, but I think for a geek,
it's easy.
I don't know.
Some people have difficulty writing and that might be because there's something in their
brain structure that makes it hard for them to write or it might be something just that
they haven't had enough practice.
I'm not the right one to judge, but I don't think you can teach any person any particular
skill.
I do think that writing is half of my life, and so I put it together in a letter program.
Even when I'm writing a one-shot program, I write it in a literate way because I get
it right faster that way.
Now does it get compiled automatically or?
So I guess on the technical side, my question was how difficult is a design, a system where
much of the programming is done informally?
Informally.
Yeah, informally.
I think whatever works to make it understandable is good, but then you have to also understand
how informal it is.
You have to know the limitations.
So by putting the formal and informal together, this is where it gets locked into your brain.
You can say informally, well, I'm working on a problem right now, so...
Let's go there.
Can you give me an example of connecting the informal and the formal?
Well, it's a little too complicated an example.
There's a puzzle that's self-referential, it's called a Japanese arrow puzzle, and you're
given a bunch of boxes, each one points northeast, south or west, and at the end, you're supposed
to fill in each box with the number of distinct numbers that it points to.
So if I put a three in a box, that means that, and it's pointing to five other boxes, that
means that there's going to be three different numbers in those five boxes.
And those boxes are pointing, one of them might be pointing to me, one of them might
be pointing the other way, but anyway, I'm supposed to find a set of numbers that obeys
this complicated condition that each number counts how many distinct numbers it points
to.
And so a guy sent me his solution to this problem where he presents formal statements
that say either this is true or this is true or this is true, and so I try to render that
formal statement informally, and I try to say I contain a three, and the guys I'm pointing
to contain the numbers one, two, and six.
So by putting it informally and also I convert it into a dialogue statement, that helps me
understand the logical statement that he's written down as a string of numbers in terms
of some abstract variables that he had.
That's really interesting.
So maybe an extension of that.
There has been a resurgence in computer science and machine learning and neural networks.
So using data to construct algorithms.
So it's another way to construct algorithms really.
Yes, exactly.
If you can think of it that way.
So as opposed to natural language to construct algorithms, use data to construct algorithms.
So what's your view of this branch of computer science where data is almost more important
than the mechanism of the algorithm?
It seems to be suited to a certain kind of non-geek, which is probably why it's taken
off that it has its own community that really resonates with that.
But it's hard to trust something like that because nobody, even the people who work with
it, they have no idea what has been learned.
That's a really interesting thought that it makes algorithms more accessible to a different
community, a different type of brain.
Yep.
And that's really interesting because just like literate programming perhaps could make
programming more accessible to a certain kind of brain.
There are people who think it's just a matter of education and anybody can learn to be a
great programmer, anybody can learn to be a great skier.
I wish that were true, but I know that there's a lot of things that I've tried to do and
I was well-motivated and I kept trying to build myself up and I never got past a certain
level.
I can't view, for example, I can't view three-dimensional objects in my head.
I have to make a model and look at it and study it from all points of view and then
I start to get some idea of what other people are good at four dimensions.
Physicists.
Yeah.
So let's go to the art of computer programming.
In 1962, you set the table of contents for this magnum opus, right?
Yep.
It was supposed to be a single book with 12 chapters.
Now today, what is it, 57 years later, you're in the middle of volume four of seven.
You're in the middle of volume four B more precisely.
Can I ask you for an impossible task, which is try to summarize the book so far, maybe
by giving a little examples.
So from the sorting and the search and the combinatorial algorithms, if you were to give
a summary, a quick elevator summary, depending how many floors there are in the building.
The first volume called fundamental algorithms talks about something that you can't do without.
You have to know the basic concepts of what is a program, what is an algorithm.
And it also talks about a low-level machine so you can have some kind of an idea of what's
going on and it has basic concepts of input and output and subroutines.
Induction.
Induction, right, mathematical preliminary.
So the thing that makes my book different from a lot of others is that I try to not
only present the algorithm, but I try to analyze them, which means that quantitatively I say
not only does it work, but it works this fast and so I need math for that.
And then there's a standard way to structure data inside and represent information in the
computer.
So that's all volume one.
Volume two talks, it's called semi-numerical algorithms and here we're writing programs,
but we're also dealing with numbers.
Algorithms deal with any kinds of objects, but when there's objects or numbers, well
then we have certain special paradigms that apply to things that have a lot of numbers
and so there's arithmetic on numbers and there's matrices full of numbers, there's random numbers
and there's power series full of numbers, there's different algebraic concepts that
have numbers in structured ways.
And arithmetic in the way a computer would think about arithmetic, so floating point.
Floating point arithmetic, high precision arithmetic, not only addition, subtraction,
multiplication, but also comparison of numbers.
So then volume three talks about sorting and search.
I like that one, sorting and search.
I love sorting.
Right.
So here we're not dealing necessarily with numbers because you sort letters and other
objects and searching we're doing all the time with Google nowadays, but I mean we have
to find stuff.
So again, algorithms that underlie all kinds of applications, none of these volumes is
about a particular application, but the applications are examples of why people want to know about
sorting, why people want to know about random numbers.
So then volume four goes into combinatorial algorithm.
This is where we have billions of things to deal with and here we keep finding cases where
one good idea can make something go more than a million times faster and we're dealing with
problems that are probably never going to be solved efficiently, but that doesn't mean
we give up on them and we have this chance to have good ideas and go much, much faster
on them.
So that's combinatorial algorithms and those are the ones that are, you said sorting is
most fun for you.
Well, it's true, it's fun, but combinatorial algorithms are the ones that I always enjoyed
the most because that's when my skillet programming had the most payoff and the difference between
an obvious algorithm that you think of first thing and an interesting subtle algorithm
that's not so obvious but circles around the other one.
That's where computer science really comes in and a lot of these combinatorial methods
were found first in applications to artificial intelligence or cryptography.
In my case, I just liked them and it was associated more with puzzles.
Do you like the most in the domain of graphs and graph theory?
Graphs are great because they're terrific models of so many things in the real world
and if you throw in numbers on a graph, you got a network and so there you have many more
things.
But combinatorial in general is any arrangement of objects that has some kind of a higher
structure, non-random structure and is it possible to put something together satisfying
all these conditions like I mentioned arrows a minute ago, is there a way to put these
numbers on a bunch of boxes that are pointing to each other, is that going to be possible
at all?
That's volume four.
That's volume four.
What is the future hold?
Volume four A was part one and what happened was in 1962 when I started writing down a
table of contents, it wasn't going to be a book about computer programming in general,
it was going to be a book about how to write compilers and I was asked to write a book
explaining how to write a compiler and at that time there were only a few dozen people
in the world who had written compilers and I happen to be one of them and I also had
some experience writing for like the campus newspaper and things like that.
So I thought, okay, great, I'm the only person I know who's written a compiler but hasn't
invented any new techniques for writing compilers and all the other people I knew had super
ideas but I couldn't see that they would be able to write a book that would describe
anybody else's ideas with their own.
So I could be the journalist and I could explain what all these cool ideas about compiler writing
were and then I started putting down, well, yeah, you need to have a chapter about data
structures, you need to have some introductory material, you want to talk about searching
because a compiler writer has to look up the variables in a symbol table and find out which
when you write the name of a variable in one place it's supposed to be the same as the
one you put somewhere else so you need all these basic techniques and I had to know some
arithmetic and stuff so I threw in these chapters and I threw in a chapter on combinatorics
because that was what I really enjoyed programming the most but there weren't many algorithms
in known about combinatorial methods in 1962.
So that was kind of a short chapter but it was sort of thrown in just for fun and chapter
12 was going to be actual compilers applying all the stuff in chapters one to eleven to
make compilers.
Well, okay, so that was my table of contents from 1962 and during the 70s the whole field
of combinatorics went through a huge explosion.
People talk about a combinatorial explosion and they usually mean by that the number of
cases goes up and n plus one and all of a sudden your problem has gotten more than ten
times harder but there was an explosion of ideas about combinatorics in the 70s to the
point that it takes 1975 I bet more than half of all the journals of computer science were
about combinatorial methods.
What kind of problems were occupying people's minds?
What kind of problems in combinatorics?
Was it satisfiability, graph theory?
Yeah, graph theory was quite dominant.
But all of the NP hard problems that you have like Hamiltonian path going beyond graphs
you had operation research whenever there was a small class of problems that had efficient
solutions and they were usually associated with matrilinear special mathematical construction.
But once we went to things that involved three things at a time instead of two all of a sudden
things got harder.
So we had satisfiability problems where if you have clauses every clause has two logical
elements in it then we can satisfy it in linear time.
We can test for satisfiability in linear time but if you allow yourself three variables
in a clause then nobody knows how to do it.
So these articles were about trying to find better ways to solve cryptography problems
and graph theory problems where we have lots of data but we didn't know how to find the
best subsets of the data like with sorting we could get the answer didn't take long.
So how did it continue to change from the 70s to today?
Yeah, so now there may be half a dozen conferences whose topic is combinatorics, different kind
but fortunately I don't have to rewrite my book every month like I had to in the 70s.
But still there's huge amount of work being done and people getting better ideas on these
problems that don't seem to have really efficient solutions but we still get into a lot more
with them and so this book that I'm finishing now is I've got a whole bunch of brand new
methods that as far as I know there's no other book that covers this particular approach
and so I'm trying to do my best of exploring the tip of the iceberg and I try out lots
of things and keep rewriting as I find better methods.
So what's your writing process like?
What's your thinking and writing process like every day?
So what's your routine even?
Yeah, I guess it's actually the best question because I spend seven days a week doing it.
The most prepared to answer it.
Okay, so the chair I'm sitting in is where I do...
That's where the magic happens.
Well, reading and writing.
The chair is usually sitting over there where I have some reference book but I found this
chair which was designed by a Swedish guy anyway.
It turns out this is the only chair I can really sit in for hours and hours and not
know that I'm in a chair.
But then I have the stand-up desk right next to us and so after I write something with
pencil and eraser, I get up and I type it and revise and rewrite, I'm standing up.
The kernel of the idea is first put on paper.
Yeah.
That's where...
Right.
And I'll write maybe five programs a week, of course, literate programming and these
are...before I describe something in my book, I always program it to see how it's working
and I try it a lot.
So for example, I learned at the end of January, I learned of a breakthrough by four Japanese
people who had extended one of my methods in a new direction.
And so I spent the next five days writing a program to implement what they did and then
I...they had only generalized part of what I had done so then I had to see if I could
generalize more parts of it and then I had to take their approach and I had to try it
out on a couple of dozen of the other problems I had already worked out with my old methods.
And so that took another couple of weeks and then I started to see the light and I started
writing the final draft and then I would type it up and involve some new mathematical questions
and so I wrote to my friends who might be good at solving those problems and they solved
some of them and so I put that in as exercises.
And so a month later, I had absorbed one new idea that I learned and I'm glad I heard
about it in time.
Otherwise, I wouldn't put my book out before I'd heard about the idea.
On the other hand, this book was supposed to come in at 300 pages and I'm up to 350
now.
That added 10 pages to the book.
But if I learn about another one, my publisher is going to shoot me.
Well, so in that process, in that one month process, are some days harder than others?
Are some days harder than others?
Yeah, my work is fun, but I also work hard and every big job has parts that are a lot
more fun than others.
And so many days, I'll say, why do I have to have such high standards?
Why couldn't I just be sloppy and not try this out and just report the answer?
But I know that people are calling me to do this and so okay, so okay, Don, I'll grit
my teeth and do it.
And then the joy comes out when I see that actually, you know, I'm getting good results
and I get even more when I see that somebody has actually read and understood what I wrote
and told me how to make it even better.
I did want to mention something about the method.
So I got this tablet here where I do the first, you know, the first writing of concepts, okay.
So, so...
And what language is that, Don?
Right, so take a look at it, but you know, here, random say, explain how to draw such
skewed pixel diagrams, okay.
I got this paper about 40 years ago when I was visiting my sister in Canada and they
make tablets of paper with this nice, large size and just the right size.
And a very small space between like two minus five.
Yeah, yeah, particularly.
Maybe I'll also just show it.
Yeah.
Yeah.
Wow.
You know, I've got these manuscripts going back to the 60s.
And those are when I get my ideas on paper, okay.
But I'm a good typist.
In fact, I went to typing school when I was in high school and so I can type faster than
I think.
So then when I do the editing, you know, stand up and type, then I revise this and it comes
out a lot different than what, you know, for style and rhythm and things like that come
out at the typing stage.
And you type in tech.
And I type in tech.
And can you, can you think in tech?
No.
So to a certain extent, I have only a small number of idioms that I use, like, you know,
I'm beginning with theorem, I do something for, just by the equation, I do something
and so on.
But I have to see it and.
In the way that it's on paper here.
Yeah, right.
So for example, Turing wrote what, the other direction, you don't write macros.
You don't think in macros.
Not particularly, but when I need a macro, I'll go ahead and do it.
But the thing is, I also write to fit.
I mean, I'll change something if I can save a line.
You know, it's like haiku.
I'll figure out a way to rewrite the sentence so that it'll look better on the page.
And I shouldn't be wasting my time on that, but I can't resist because I know it's only
another 3% of the time or something like that.
And it could also be argued that that is what life is about.
Ah, yes.
In fact, that's true.
Like I worked in the garden one day a week and that's kind of a description of my life
is getting rid of weeds, you know, removing bugs for programs.
So you know, a lot of writers talk about, you know, basically suffering, the writing
process as having, you know, it's extremely difficult.
And I think of programming, especially the technical writing that you're doing can be
like that.
Do you find yourself methodologically?
How do you every day sit down to do the work?
Is it a challenge?
You kind of say it's, you know, it's fun.
But it would be interesting to hear if there are non-fun parts that you really struggle
with.
Yeah.
So the fun comes when I'm able to put together ideas of two people who didn't know about
each other.
And so I might be the first person that saw both of their ideas.
And so then, you know, then I get to make the synthesis and that gives me a chance to
be creative.
But the dredge work is where I've got to chase everything down to its root.
This leads me into really interesting stuff.
I mean, I learn about Sanskrit and I, you know, I try to give credit to all the authors.
And so I write to people who know the people, authors if they're dead, or I communicate
this way.
And I got to get the math right.
And then I got to tackle all my programs, try to find holes in them.
And I rewrite the programs over after I get a better idea.
Is there ever dead ends?
Dead ends.
Oh yeah, I throw stuff out.
Yeah.
One of the things that I, I spent a lot of time preparing a major example based on the
game of baseball.
And I know a lot of people who, for whom baseball is the most important thing in the world,
you know.
And I also know a lot of people, for whom cricket is the most important in the world
or soccer or something, you know.
And I realized that if I had a big example, I mean, it was going to have a fold out illustration
and everything.
And I was saying, well, what am I really teaching about algorithms here where I had this baseball
example?
And if I was a person who knew only cricket, wouldn't they, what would they think about
this?
And so I've ripped the whole thing out.
And I, you know, I had something that would have really appealed to people who grew up
with baseball as a major theme in their life.
Which is a lot of people.
But just, yeah.
But still a minority.
Small minority.
I took out bowling too.
Even a smaller minority.
What's the art in the art of programming?
Why is there, of the few words in the title, why is art one of them?
Yeah.
Well, that's what I wrote my touring lecture about.
And so when people talk about art, it really, I mean, what the word means is something that's
not in nature.
So when you have artificial intelligence, the art comes from the same root, saying that
this is something that was created by human beings.
And then it's gotten a further meaning off in a fine art, which has this beauty to the
mix.
And so as you know, we have things that are artistically done, and this means not only
done by humans, but also done in a way that's elegant and brings joy.
And has, I guess, Tolstoy versus Dostoevsky.
But anyway, it's that part that says that it's done well, as well as not only different
from nature.
In general, then, art is what human beings are specifically good at.
And when they say artificial intelligence, well, they're trying to mimic human beings.
But there's an element of fine art and beauty.
You are one.
That's what I try to also say, that you can write a program and make a work of art.
So now, in terms of surprising, what ideas in writing from in search to the combinatorial
algorithms, what ideas have you come across that were particularly surprising to you,
that changed the way you see a space of problems?
I get a surprise every time I have a bug in my program, obviously, but that isn't really
what you're at.
More transformational than surprising.
For example, in volume 4A, I was especially surprised when I learned about data structure
called BDD, Boolean Decision Diagram, because I sort of had the feeling that as an old-timer,
you know, I'd been programming since the 50s, and BDDs weren't invented until 1986.
And here comes a brand new idea that revolutionizes the way to represent a Boolean function.
And Boolean functions are so basic to all kinds of things in, I mean, logic is underlyzic.
And we can describe all of what we know in terms of logic somehow and propositional logic.
I thought that was cut and dried and everything was known, but here comes Randy Bryant and
discovers that BDDs are incredibly powerful.
So that means I have a whole new section to the book that I never would have thought of
until 1986, not until 1990s, when people started to use it for a billion dollar of applications.
And it was the standard way to design computers for a long time until Satsalvers came along
in the year 2000, so that's another great big surprise.
So a lot of these things have totally changed the structure of my book.
And the middle third of Volume 4B is about Satsalvers, and that's 300-plus pages, which
is all about material, mostly about material that was discovered in this century.
And I had to start from scratch and meet all the people in the field and write 15 different
Satsalvers that I wrote while preparing that, and seven of them are described in the book,
others were from my own experience.
So newly invented data structures or ways to represent?
A whole new class of algorithm.
A whole new class of algorithm.
Yeah.
And the interesting thing about the BDDs was that the theoreticians started looking at
it and started to describe all the things you couldn't do with BDDs.
So they were getting a bad name because they were useful, but they didn't solve everything.
I'm sure that the theoreticians are in the next 10 years are going to show why machine
learning doesn't solve everything.
But I'm not only worried about the worst case, I get a huge delight when I can actually
solve a problem that I couldn't solve before.
Even though I can't solve the problem that it suggests is a further problem, I know that
I'm way better than I was before.
And so I found out that BDDs could do all kinds of miraculous things.
And so I had to spend quite a few years learning about that territory.
So in general, what brings you more pleasure at proving or showing a worst case analysis
of an algorithm or showing a good average case or just showing a good case that something
good pragmatically can be done with this algorithm?
Yeah, I like a good case that is maybe only a million times faster than I was able to
do before and not worry about the fact that it's still going to take too long if I double
the size of the problem.
So that said, do you popularize the asymptotic notation for describing running time?
Obviously, in the analysis of algorithms, worst case is such an important part.
Do you see any aspects of that kind of analysis as lacking and notation too?
Well, the main purpose is to have notations that help us for the problems we want to solve.
And so they match our intuitions.
And people who worked in number theory had used asymptotic notation in a certain way,
but it was only known to a small group of people.
And I realized that, in fact, it was very useful to be able to have a notation for something
that we don't know exactly what it is, but we only know partial about it and so on.
So for example, instead of big O notation, let's just take a much simpler notation where
I'd say zero or one or zero, one or two.
And suppose that when I had been in high school, we would be allowed to put in the middle
of our formula, x plus zero, one or two equals y, okay, and then we would learn how to multiply
two such expressions together and deal with them.
Well, the same thing, big O notation says, here's something that's, I'm not sure what
it is, but I know it's not too big.
I know it's not bigger than some constant times n squared or something like that.
And so I write big O of n squared.
Now I learn how to add big O of n squared to big O of n cubed and I know how to add
big O of n squared to plus one and square that and how to take logarithms and exponentials
to have big O's in the middle of them.
And that turned out to be hugely valuable in all of the work that I was trying to do
is I'm trying to figure out how good an algorithm is.
So have there been algorithms in your journey that perform very differently in practice
than they do in theory?
Well, the worst case of a combinatorial algorithm is almost always horrible.
But we have set solvers that are solving one of the last exercises in that part of my
book was to figure out a problem that has 100 variables that's difficult for a set solver.
But you would think that a problem with 100 Boolean variables requires you to do two to
the 100th operations because that's the number of possibilities when you have 100 Boolean
variables and two to the 100th is way bigger than we can handle.
But 10 to the 17th is a lot.
You've mentioned over the past few years that you believe P may be equal to NP.
But that it's not really, you know, if somebody does prove that P equals NP, it will not directly
lead to an actual algorithm to solve difficult problems.
Can you explain your intuition here?
Has it been changed?
And in general, on the difference between easy and difficult problems of P and NP and
so on?
So, the popular idea is if an algorithm exists, then somebody will find it.
And it's just a matter of writing it down.
But many more algorithms exist than anybody can understand or ever make use of.
Or discover, yeah.
Because they're just way beyond human comprehension, the total number of algorithms is more than
mind boggling.
So we have situations now where we know that algorithm exists, but we don't know the farthest
idea what the algorithms are.
There are simple examples based on game playing where you have, where you say, well, there
must be an algorithm that exists to win in the game of Hex, because for the first player
to win in the game of Hex, because Hex is always either a win for the first player or
the second player.
But what's the game of Hex?
There's a game of Hex which is based on putting pebbles onto a hexagonal board.
And the white player tries to get a white path from left to right, and the black player
tries to get a black path from bottom to top.
And how does capture occur?
And there's no capture, you just put pebbles down one at a time.
But there's no draws, because after all the white and black are played, there's either
going to be a white path across from each to west or a black path from bottom to top.
So there's always, it's a perfect information game, and people take turns like tic-tac-toe.
And the hex board can be different sizes, but there's no possibility of a draw.
And players move one at a time.
And so it's got to be either a first player win or a second player win.
Mathematically, you follow out all the trees, and either there's always a win for the first
player, second player, okay.
And it's finite.
The game is finite.
So there's an algorithm that will decide, you can show it has to be one or the other.
Because the second player could mimic the first player with kind of a pairing strategy.
And so you can show that it has to be one or the other.
But we don't know any algorithm anyway.
We don't know if there's another way.
There are cases where you can prove the existence of a solution, but nobody knows anyway how
to find it.
More like the algorithm question, there's a very powerful theorem in graph theory by
Robinson and Seymour that says that every class of graphs that is closed undertaking
minors has a polynomial time algorithm to determine whether it's in this class or not.
Now, a class of graphs, for example, planar graphs, these are graphs that you can draw
in a plane without crossing lines.
And the planar graph, taking minors means that you can shrink an edge into a point or
you can delete an edge.
And so you start with a planar graph and shrink any edge to a point, it's still planar.
Deleting an edge is still planar, okay.
But there are millions of different ways to describe a family of graphs that still remains
the same undertaking minors.
And Robinson and Seymour proved that any such family of graphs, there's a finite number
of minimum graphs that are obstructions, so that if it's not in the family, then it has
to contain, then there has to be a way to shrink it down until you get one of these
bad minimum graphs that's not in the family.
In the case of a planar graph, the minimum graph is a five-pointed star where everything
points to another and the minimum graph consisting of trying to connect three utilities to three
houses without crossing lines.
And so there are two bad graphs that are not planar, and every non-planar graph contains
one of these two bad graphs by shrinking and removing edges.
Sorry, can you say that again so he proved that there's a finite number of these bad
graphs?
There's always a finite number.
So somebody says, here's a family of...
It's hard to believe.
And they proved the next sequence of 20 papers, I mean, and it's deep work, but it's...
Because that's for any arbitrary class.
So it's for any class.
Any arbitrary class that's closed undertaking minors.
That's closed under...
Maybe I'm not understanding because it seems like a lot of them are closed undertaking
minors.
Almost all the important classes of graphs are.
There are tons of such graphs, but also hundreds of them that arise in applications.
I have a book over here called Classes of Graphs.
And it's amazing how many different classes of graphs that people have looked at.
So why do you bring up this theorem, lower this proof?
So now there's lots of algorithms that are known for special class of graphs.
For example, if I have a chordal graph, then I can color it efficiently.
If I have some kinds of graphs, it'll make a great network of various things.
So you'd like to test it.
Somebody gives you a graph that's always in this family of graphs.
If so, then I can go to the library and find an algorithm that's going to solve my problem
on that graph.
Okay.
So we want to have an algorithm that says, you give me a graph, I'll tell you whether
it's in this family or not.
And so all I have to do is test whether or not that does this given graph have a minor
that's one of the bad ones.
A minor is everything you can get by shrinking and removing it.
And given any minor, there's a polynomial time algorithm saying I can tell whether this
is a minor of you.
And there's a finite number of bad cases.
So I just try, you know, does it have this bad case, polynomial time, I got the answer.
Does it have this bad case, polynomial time, I got the answer, total polynomial time.
And so I've solved the problem.
However, all we know is that the number of minors is finite.
We don't know what, we might only know one or two of those minors, but we don't know
that if we've got 20 of them, we don't know there might be 21, 25, all we know is that
it's finite.
So here we have a polynomial time algorithm that we don't know.
That's a really great example of what you worry about or why you think p equals NP won't
be useful, but still, why do you hold the intuition that p equals NP?
Because you have to rule out so many possible algorithms as being not working.
You can take the graph and you can represent it in terms of certain prime numbers and then
you can multiply those together and then you can take the bitwise and construct some certain
constant and polynomial time and that's a perfectly valid algorithm.
And there are so many algorithms of that kind, a lot of times we see random take data and
we get coincidences that some fairly random looking number actually is useful because it
happens to solve a problem just because there's so many hairs on your head, but it seems unlikely
that two people are going to have the same number of hairs on their head.
But you can count how many people there are and how many hairs on their head.
So there must be people walking around in the country that have the same number of hairs
on their head.
Well, that's a kind of a coincidence that you might say also this particular combination
of operations just happens to prove that a graph has a Hamiltonian path.
I see lots of cases where unexpected things happen when you have enough possibilities.
So because the space of possibility is so huge, you have to rule them all out and so
that's the reason for my intuition is by no means a proof.
Some people say, well, P can't equal NP because you've had all these smart people, the smartest
designers of algorithms have been racking their brains for years and years and there's
a million dollar prizes out there and none of them, nobody has thought of the algorithm.
So it must be no such thing.
On the other hand, I can use exactly the same logic and I can say, well, P must be equal
to NP because there's so many smart people out here trying to prove it unequal to NP
and they've all failed.
This kind of reminds me of the discussion about the search for aliens.
They've been trying to look for them and we haven't found them yet therefore they don't
exist but you can show that there's so many planets out there that they very possibly
could exist.
And then there's also the possibility that they exist but they all discovered machine
learning or something and then blew each other up.
On that small quick tangent, let me ask, do you think there's intelligent life out there
in the universe?
I have no idea.
Do you hope so?
Do you think about it?
I don't spend my time thinking about things that I could never know really.
And yet you do enjoy the fact that there's many things you don't know.
You do enjoy the mystery of things.
I enjoy the fact that I have limits but I don't take time to answer unsolvable questions.
Got it.
Because you've taken on some tough questions that may seem unsolvable.
You have taken on some tough questions that may seem unsolvable.
It gives me a thrill when I can get further than I ever thought I could.
But I don't.
But much like with religion, I'm glad that there's no proof that God exists or not.
I think...
It would spoil the mystery.
It would be too dull, yeah.
So to quickly talk about the other art of artificial intelligence, what's your view?
Artificial intelligence community has developed as part of computer science and in parallel
with computer science since the 60s.
What's your view of the AI community from the 60s to now?
So all the way through, it was the people who were inspired by trying to mimic intelligence
or to do things that were somehow the greatest achievements of intelligence that had been
inspiration to people who have pushed the envelope of computer science maybe more than
any other group of people.
So all the way through, it's been a great source of good problems to sink teeth into
and getting partial answers and then more and more successful answers over the year.
So this has been the inspiration for lots of the great discoveries of computer science.
Are you yourself captivated by the possibility of creating, of algorithms having echoes of
intelligence in them?
Not as much as most of the people in the field, I guess I would say, but that's not to say
that they're wrong or that it's just you asked about my own personal preferences.
But the thing that I worry about is when people start believing that they've actually succeeded
and because there seems to be this huge gap between really understanding something and
being able to pretend to understand something and give the illusion of understanding something.
Everything is possible to create without understanding.
I do that all the time, too, I mean, that's why I use random numbers, but there's still
this great gap.
I don't assert that it's impossible, but I don't see anything coming any closer to
really the kind of stuff that I would consider intelligence.
So you've mentioned something that on that line of thinking, which I very much agree
with.
So the art of computer programming as the book is focused on single processor algorithms
and for the most part, you mentioned...
It's only because I set the table of contents in 1962, you have to remember.
For sure.
There's no...
I'm glad I didn't wait until 1965 or something.
One book, maybe we'll touch on the Bible, but one book can't always cover the entirety
of everything.
So I'm glad the table of contents for the art of computer programming is what it is.
But you did mention that you thought that an understanding of the way ant colonies
are able to perform incredibly organized tasks might well be the key to understanding human
cognition.
So these fundamentally distributed systems.
So what do you think is the difference between the way Don Knuth would sort a list and an
ant colony would sort a list or perform an algorithm?
Sorting a list isn't the same as cognition, though.
But I know what you're getting at is, well, the advantage of ant colony, at least we can
see what they're doing.
We know which ant has talked to which other ant and it's much harder with the brains to
know to what extent neurons are passing signals.
So I'm just saying that ant colony might be...
If they have the secret of cognition, think of an ant colony as a cognitive single being
rather than as a colony of lots of different ants, I mean, just like the cells of our brain
are and the microbiome and all that is interacting entities, but somehow I consider myself to
be a single person.
Well, you know, an ant colony, you can say, might be cognitive and somehow and it's...
Yeah, I mean, you know, okay, I smash a certain ant and the organism thinks, hmm, that's
stung.
What was that?
But if we're going to crack the secret of cognition, it might be that we could do so
by psyching out how ants do it because we have a better chance to measure, they're communicating
by pheromones and by touching each other and psych, but not by much more subtle phenomenon
like current's going through.
But even a simpler version of that, what are your thoughts of maybe Conway's game of life?
Okay, so Conway's game of life is able to simulate any computable process and any deterministic
process is...
I like how you went there.
I mean, that's not its most powerful thing, I would say, I mean, they can simulate it,
but the magic is that the individual units are distributed and extremely simple.
Yes, we understand exactly what the primitives are.
The primitives, just like with the ant colony, even simpler though.
But still, it doesn't say that I understand life, I mean, it gives me a better insight
into what does it mean to have a deterministic universe?
What does it mean to have free choice, for example?
So, do you think God plays dice?
Yes.
I don't see any reason why God should be forbidden from using the most efficient ways to, I mean,
we know that dice are extremely important in efficient algorithms.
There are things that couldn't be done well without randomness, and so I don't see any
reason why God should be prohibited from...
When the algorithm requires it, you don't see why the physics should constrain it.
So in 2001, you gave a series of lectures at MIT about religion and science.
No, that was 1999.
You published...
Sorry.
The book came out in 2001.
So in 1999, you spent a little bit of time in Boston enough to give those lectures.
And I read in the 2001 version, most of it.
It's quite fascinating to read.
I recommend people, it's transcription review lectures.
So what did you learn about how ideas get started and grow from studying the history
of the Bible?
So you've rigorously studied a very particular part of the Bible.
What did you learn from this process about the way us human beings as a society develop
and grow ideas, share ideas, and define by those ideas?
Well, it's hard to summarize that.
I wouldn't say that I learned a great deal of really definite things, like where I could
make conclusions, but I learned more about what I don't know.
You have a complex subject, which is really beyond human understanding.
So we give up on saying, I'm ever going to get to the end of the road and I'm ever going
to understand it, but you say, but maybe it might be good for me to get closer and closer
and learn more and more about something.
And so how can I do that efficiently?
And the answer is, well, use randomness.
And so try a random subset that is within my grasp and study that in detail, instead
of just studying parts that somebody tells me to study, or instead of studying nothing
because it's too hard.
So I decided for my own amusement at once that I would take a subset of the verses of
the Bible and I would try to find out what the best thinkers have said about that small
subset.
And I had about, let's say, 60 verses out of 3,000, I think is one out of 500 or something
like this.
And so then I went to the libraries, which are well-indexed.
I spent, for example, at Boston Public Library, I would go once a week for a year.
And I went to have done time-stuck and over Harvard Library to look at this book that
weren't in the Boston Public, where scholars had looked and you can go down the shelves
and you can look at the index and say, oh, is this verse mentioned anywhere in this book?
If so, look at page 105.
So in other words, I could learn not only about the Bible, but about the secondary literature
about the Bible, the things that scholars have written about it.
And so that gave me a way to zoom in on parts of the thing so that I could get more insight.
And so I look at it as a way of giving me some firm pegs, which I could hang pieces
of information, but not as things where I would say, and therefore, this is true.
In this random approach of sampling the Bible, what did you learn about the most, you know,
central, one of the biggest accumulation of ideas in our...
It seemed to me that the main thrust was not the one that most people think of as saying,
oh, don't have sex or something like this, but that the main thrust was to try to figure
out how to live in harmony with God's wishes.
I'm assuming that God exists, and as I say, I'm glad that there's no way to prove this,
because I would run through the proof once and then I'd forget it, and I would never
speculate about spiritual things and mysteries otherwise.
And I think my life would be very incomplete.
So I'm assuming that God exists, but a lot of the people say God doesn't exist, but that's
still important to them.
And so in a way that might still be whether God is there or not, in some sense, God is
important to them.
One of the verses I studied, you can interpret it as saying it's much better to be an atheist
than not to care at all.
So I would say it's similar to the P equals NP discussion.
You mentioned a mental exercise that I'd love it if you could partake in yourself, a mental
exercise of being God.
So if you were God, Dr. Nooth, how would you present yourself to the people of Earth?
You mentioned your love of literature, and there's this book that really I can recommend
to you.
Yeah, the title I think is Blast for Me.
It talks about God revealing himself through a computer in Los Alamos.
It's the only book that I've ever read where the punchline was really the very last word
of the book and explained the whole idea of the book.
And so I don't want to give that away, but it's really very much about this question
that you raised.
But suppose God said, okay, my previous means of communication with the world are not the
best for the 21st century, so what should I do now?
And it's conceivable that God would choose the way that's described in this book.
Another way to look at this exercise is looking at the human mind, looking at the human spirit,
the human life in a systematic way.
I think it mostly, you want to learn humility, you want to realize that once we solve one
problem, that doesn't mean that all of a sudden other problems are going to drop out.
And we have to realize that there are things beyond our ability.
I see hubris all around.
Yeah, well said.
If you were to run program analysis on your own life, how did you do in terms of correctness,
spending time, resource use, asymptotically speaking, of course.
Okay, yeah.
Well, I would say that question has not been asked me before, and I started out with library
subroutines and learning how to be a automaton that was obedient, and I had the great advantage
that I didn't have anybody to blame for my failures.
If I started getting, not understanding something, I knew that I should stop playing ping-pong,
and it was my fault that I wasn't studying hard enough or something, rather than that
somebody was discriminating against me in some way, and I don't know how to avoid the existence
of biases in the world, but I know that that's an extra burden that I didn't have to suffer
from.
And then I found the, from parents, I learned the idea of service to other people as being
more important than what I get out of stuff myself.
I know that I need to be happy enough in order to be able to be of service, but I came to
a philosophy, finally, that I phrase as 0.8 is enough.
There was a TV show once called Eight is Enough, which was about, somebody had eight kids,
but I say 0.8 is enough, which means if I can have a way of rating happiness, I think
it's good design to have an organism that's happy about 80% of the time.
And if it was 100% of the time, it would be like everybody's on drugs, and everything
collapses, nothing works, because everybody's just too happy.
Do you think you've achieved that 0.8 optimal balance?
There are times when I'm down, and I know that I'm chemically, I know that I've actually
been programmed to be depressed a certain amount of time, and if that gets out of kilter
and I'm more depressed than usual, sometimes I find myself trying to think, now, who should
I be mad at today?
There must be a reason why I'm, but then I realize, it's just my chemistry telling
me that I'm supposed to be mad at somebody, and so I truck it up, say, okay, go to sleep
and get better, but if I'm not 100% happy, that doesn't mean that I should find somebody
that's screaming and try to silence them, but I'm saying, okay, I'm not 100% happy,
but I'm happy enough to be a part of a sustainable situation.
So that's kind of the numerical analysis I do.
You've converged towards the optimal, which for human life is a 0.8.
I hope it's okay to talk about, as you talked about previously, in 2006, you were diagnosed
with prostate cancer.
Has that encounter with mortality changed you in some way or the way you see the world?
Well, yeah, it did.
The first encounter with mortality when my dad died, and I went through a month when
I sort of came to be comfortable with the fact that I was going to die someday, and
during that month, I felt okay, but I couldn't sing, and I couldn't do original research
either.
I sort of remember after three or four weeks, the first time I started having a technical
thought that made sense and was maybe slightly creative, I could sort of feel that something
was starting to move again, but that was, so I felt very empty until I came to grips with
the, I learned that this is sort of a standard grief process that people go through.
So then now I'm at a point in my life, even more so than in 2006, where all of my goals
have been fulfilled except for finishing the art of computer programming.
I had one major unfulfilled goal that I'd wanted all my life to write a piece of music
that, and I had an idea for a certain kind of music that I thought ought to be written,
at least somebody ought to try to do it, and I felt that it wasn't going to be easy, but
I wanted a proof of concept, I wanted to know if it was going to work or not, and so I spent
a lot of time, and finally I finished that piece, and we had the world premiere last
year on my 80th birthday, and we had another premiere in Canada, and there's talk of concerts
in Europe and various things, so that, but that's done.
It's part of the world's music now, and it's either good or bad, but I did what I was hoping
to do.
So the only thing that I have on my agenda is to try to do as well as I can with the
art of computer programming until I go see now.
Do you think there's an element of 0.8 that might apply there?
0.8?
Yeah.
Well, I look at it more that I got actually to 1.0 when that concert was over with.
So in 2006 I was at 0.8, so when I was diagnosed with prostate cancer, then I said, okay, well
maybe this is, you know, I've had all kinds of good luck all my life, and I have nothing
to complain about, so I might die now, and we'll see what happens.
And so it's quite seriously, I had no expectation that I deserved better, I didn't make any
plans for the future.
I had my surgery, I came out of the surgery and spent some time learning how to walk again
and so on, it was painful for a while.
But I got home and I realized I hadn't really thought about what to do next.
I hadn't any expectations, I said, okay, hey, I'm still alive, okay, now I can write
some more books, but I didn't come with the attitude that, you know, this was terribly
unfair and I just said, okay, I was accepting whatever turned out, you know, I had gotten
more than my share already, so why should I, why should I, and I didn't, and I really
thought, when I got home, I realized that I had really not thought about the next step,
what I would do after I would be able to work again, I'd sort of thought of it as if this
might, I was comfortable with the fact that it was at the end, but I was hoping that I
would still, you know, be able to learn about satisfiability and also someday even write
music.
I didn't start seriously on the music project until 2012.
So I'm going to be in huge trouble if I don't talk to you about this.
In the 70s, you've created the tech typesetting system together with MetaFont language for
font description and computer modern family of typefaces.
That has basically defined the methodology in the aesthetic of the countless research
fields, right?
Math, physics, well, beyond computer science, so on, okay, well, first of all, thank you.
I think I speak for a lot of people in saying that, but a question in terms of beauty, there's
a beauty to typography that you've created, and yet beauty is hard to quantify.
How does one create beautiful letters and beautiful equations?
Like what, so, I mean, perhaps there's no words to be describing the process, but.
So the great Harvard mathematician, George Deberg, wrote a book in the 30s called Aesthetic
Measure where he would have pictures of vases and underneath would be a number, and this
was how beautiful the vase was, and he had a formula for this.
And he actually also wrote about music, and so he could, you know, so I thought maybe
I would, for part of my musical composition, I would try to program his algorithms and,
you know, so that I would write something that had the highest number by his score.
Well, it wasn't quite rigorous enough for a computer to do, but anyway, people have
tried to put numerical value on beauty, but, and he did probably the most serious attempt
and, and George Grishwin's teacher also wrote two vines where he talked about his method
of composing music, but you're talking about another kind of beauty and beauty and letters
and let.
Elegance and whatever that curvature is.
Right.
So, and so that's the beholder, as they say, but kind of striving for excellence and whatever
definition you want to give to beauty, then you try to get as close to that as you can
somehow with it.
I guess, I guess I'm trying to ask, and there may not be a good answer, what loose definitions
were you operating under with the community of people that you were working on?
Oh, the loose definition, I wanted, I wanted it to appeal to me, to me.
To you personally.
Yeah.
That's a good start.
Yeah.
I just pulled that test when, when I got volume two came out with this, with the new printing
and I was expecting to be the happiest day of my life and, and I felt like a burning,
like how angry I was that I opened the book and it was in the same beige covers and, and
but, but it didn't look right on the page.
The number two was particularly ugly.
I couldn't stand any page or had a two in his page number.
And I was expecting that it was, you know, I spent all this time making measurements
and I had, had, had looked at stuff in different, different ways and I, I had, I had great
technology, but, but it didn't, you know, but I, but I wasn't done.
I had, I had to retune the whole thing after 1961.
Has it ever made you happy finally?
Oh.
Oh, yes.
Or is it a point eight?
No, no.
And so many books have come out that would never have been written without this.
I just, it just, it's just, it's a joy, but I can, but now I, I mean, all these pages
that are sitting up there, I, I, I, I don't have a, if I didn't like them, I would change
them.
That's my, nobody else has this ability.
They have to stick with what I gave them.
Yeah.
And in terms of the other side of it, there's the typography, so the look of the type and
the curves and the lines.
What about the spacing?
What, what about the spacing between the white space?
Yeah.
It seems like you could be a little bit more systematic about the layout or technical.
Oh yeah.
You can always go further.
I, I, I, I, I didn't, I didn't stop at point eight.
But I stopped, but I stopped at about point nine eight.
It seems like you're not following your own rule for happiness, or is, no, no, no, I,
there's, okay.
Of course, there's this, what is the Japanese word, wabi-sabi or something where, where
the most beautiful works of art are those that have flaws because then the person who,
who perceives them as their own appreciation and that give, if you were more satisfaction
or so on, but, but I, but no, no, with typography, I wanted it to look as good as I could in,
in the vast majority of cases.
And then when it doesn't, then I, I say, okay, that's 2% more work for the, for the author.
But I didn't want to, I didn't want to say that my job was to get 200% with and take
all the work away from the author.
That's what I meant by that.
So if you were to venture a guess, how much of the nature of reality do you think we humans
understand?
So you mentioned, you appreciate mystery.
How much of the world about us is shrouded in mystery?
Are we, are we, if you were to put a number on it, what, what percent of it all do we
understand?
Are we totally-
How many leading zeros do you point, 0.00, I don't know, no, I think it's infinitesimal.
How do we think about that?
And what do we do about that?
Do we continue one step at a time?
Yeah, we muddle through, I mean, we, we, we do our best.
We realize that none, that nobody's perfect and we, and we try to keep advancing, but
we don't spend time saying we're not there, we're not all the way to the end.
Some mathematicians that, that would be in the office next to me when I was in the math
department, they would never think about anything smaller than countable infinity.
And I never, you know, we intersect that countable infinity because I rarely got up to countable
infinity.
I was always talking about finite stuff.
But even, even limiting to finite stuff, which is, which the universe might be, there's
no, no way to really know whether the universe isn't, isn't just made out of capital N, or
whatever you want to call them, quarks or whatever, where capital N is some finite number.
All of the numbers that are comprehensible are still way smaller than most, almost all
finite numbers.
I, I, I got this one paper called supernatural numbers where I, what, what, I guess you probably
ran into something called Knuth arrow notation.
Did you run into that where, or anyway, so you take the number, I think it's like, I,
and I called it super K, I named it after myself, but it's, it's, but in arrow notation
it's something like 10 and then four arrows and a three or something like that.
Okay.
Now, the arrow notation, if you have, if you have no arrows, that means multiplication.
X Y means X times X times X times X Y times.
If you have one arrow, that means exponentiation.
So X one arrow Y means X to the X to the X to the X to the X Y times.
So I found out, by the way, that this, this notation was invented by a guy in 1830 and,
and he was, he was a, a, a, one of the English nobility who, who spent his time thinking
about stuff like this.
And it was exactly the same concept that I, that I, that I used arrows and he used a slightly
different notation.
But anyway, this, and then this Ackerman's function is, is based on the same kind of
ideas, but Ackerman was 1920s.
But anyway, we got this number 10 quadruple arrow three.
So, so that, that says, well, we take, you know, we take 10 to the 10 to the, 10 to the
10 to the 10 to the 10.
How many times do we do that?
Oh, 10 double arrow two times or something.
I mean, how tall is that stack?
But, but then we do that again, because that was only 10 triple or quadruple arrow two.
We take quadruple arrow three means we have to be a pretty large number.
It gets way beyond comprehension.
Okay.
And, and, and so, but it's so small compared to what finite numbers really are because
I want to using four arrows and, you know, and 10 and a three, I mean, let's have that,
let's have that many number of arrows.
I mean, the, the boundary between infinite and finite is incomprehensible for us humans.
Anyway,
Infinity is a good, is a useful way for us to think about extremely large, extremely
large things.
And, and, and we can, we can manipulate it, but, but we can never know that the universe
is actually, and we're near that.
So, it just, so I realize how little we know, but, but we, we found an awful lot of things
that are too hard for any one person to know even with, even in our small universe.
Yeah.
And we did pretty good.
So, when you go up to heaven and meet God and get to ask one question that would get
answered, what question would you ask?
What kind of browser do you have up here?
No, no, I actually, I don't think it's meaningful to ask this question, but, but I certainly
hope we had good internet.
Okay.
On that note, that's, that's beautiful, actually.
Don, thank you so much.
It was a huge honor to talk to you.
I really appreciate it.
Okay.
Well, thanks for the gamut of questions.
Yeah, it was fun.
Thanks for listening to this conversation with Donald Knuth and thank you to our presenting
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And now, let me leave you with some words of wisdom from Donald Knuth.
We should continually be striving to transform every art into a science.
And in the process, we advance the art.
Thank you for listening and hope to see you next time.