January 22, 2024 • 38 mins
Creativity and maths have never been two words that I have put together… until now. In this episode of Creativity: Uncovered, I chatted with data scientist Noah Healy, who is a recreational mathematician in his spare time.
He shared with me how maths is one of the most creative acts out there, how it helps us understand the world around us and how it requires a curious, imaginative mind.
This episode will encourage you to rethink what you know about maths and creativity, and hopefully broaden your view on both.
Happy listening!
xo Abi
P.S. For more information about this episode and our guest, head to: http://www.crispcomms.co/podcast-episodes/recreational-maths-is-fun-and-creative
Creativity: Uncovered is lovingly edited by the team at Crisp Communications.
Creativity: Uncovered is a registered Australian Trade Mark.
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Transcript
Episode Transcript
Available transcripts are automatically generated. Complete accuracy is not guaranteed.
Hello and welcome to Creativity (00:00):
Uncovered. My name is Abi Gatling and I am on a journey
(00:12):
to uncover how everyday people find inspiration, get inventive and open their imagination.
Basically I want to know how people find creative solutions at home, work, play and everything
in between. And my goal for this podcast is that by the end of it you'll be armed with
a whole suite of tried and tested ways to some creativity whenever you need it.
(00:42):
Now today I'm speaking with Noah Healy who is a professional data scientist and a recreationally
use a mathematician just for fun. And now if you're anything like me, you probably never
thought of maths as a creative field. So I'm definitely keen to find out what it means
to be a mathematician just for fun and discover where the creativity lies in that field.
(01:08):
So welcome Noah.
Thanks for having me here Abi.
So before we start, I kind of want to get a gauge from you because on this podcast we've
had so many different people on here and it's meant that we've had dozens of different definitions
of what creativity is. So before we jump into our conversation I'd love to know what is
(01:31):
creativity to you and what does it mean to be creative?
Well for me creativity is the production of something not seen before. And I consider
that at a categorical level. So there's a certain creativity in craft. Every time you
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say throw a cup on a potter's wheel that specific cup hasn't been produced before.
But you can shuffle a deck of cards and produce a sequence of cards that probably will never
be repeated again in our universe. So that's not creative but not actual creativity from
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my lights. But the bringing forth of an entirely new idea even where it has to build on things
that's creativity to me. And in terms of what it means I think it's sort of a miracle in
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the palm of your hand. We live surrounded by things that were brought forth from just
ideas and imagination. And we can add more to that list and it's amazing.
Wow. Do you think that creativity is expensive?
(02:56):
I think yeah actually. There's certain aspects. The specific thing that I like to pursue is
inexpensive economically but it takes a lot of time and effort for me. I'm not as smart
as the smartest people who have ever lived. And societally there is a cost. You have to
(03:28):
step back from just staying sort of at the coal face or plowing the furrow and taking
the time and taking the leap of faith that is imagination. And that's a cost. And I'm
very happy to live in a time when many, many more people are available to have that cost.
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And quite specifically I am capable of generating enough free revenue that I've been able to
carve that time out without starving myself to death in the process.
Well that's good. No one wants you to be starving to death. But tell me, what is, why is creativity
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important to you?
I think it's sort of foundational. It's a big universe. We have a small piece of it.
It's a big planet. We have a small piece of that. There's so much that's possible and
available. And while amazing things can happen at our level and at levels that we've been
(04:42):
at before, each of those things is leaving aside things that we know for a fact could
have been created that would have made them so much more impressive. One of the ones that
I typically trot out is Block and Tackle, which was invented by Archimedes. That means
(05:06):
that the great wonders of the ancient world from the pyramids which still stand to not
one of the official ones, but things like the Great Wall of China, were built by people
who didn't actually understand what Block and Tackle was, which is one of the most powerful
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simple machines that human beings ever created. And for over half of recorded history, people
didn't know how to do that. But another fun one I thought of recently is hot air balloons.
The enabling technologies to have allowed people to have constructed hot air balloons
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probably predate writing by two to four thousand years. We don't have precise timing on when
high fat linen was generated. But hot air balloons weren't actually created until the
Montgauflier brothers, I can't remember which one of the two of them was, watched clothes
(06:10):
drying above a fire and noticed that they billowed upwards in the smoke and decided
to build a very large drying dress more or less and demonstrate that you could float above
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the ground. And this is a technology that Ramses or Tutt or early Chinese emperors could
have had, but certainly didn't.
Wow. Okay. So it's like creativity is kind of helping people make sense of the world,
but also make leaps and bounds forward.
(06:53):
Yeah. Yeah. There's a very real sense in which the chaos of the universe contains vastly more
than we can be aware of. And by finding information that makes sense that we can interact with,
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are leveling ourselves up in a way that's very deep and difficult to fully comprehend
all the aspects of. And that gets into sort of my passion around computational mathematics
because computational mathematics is essentially about the treatment of information as a, what's
(07:42):
known as a first class object. So information about how information can be processed, which
you can then use on the information that's doing the processing and create these things
that are sort of at the same time daydreams, but also machines that can do things. And that
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is, that is effectively what the computer is a the computer, the computers that we surround
ourselves with are small manifestations of the partial concept of this general or universal
computing machine, a imaginative system that is capable of encapsulating all of the possibilities
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of imagination. And which, like a gin from, you know, ancient myth will do anything that
you tell it to do, but only specifically and exactly what you tell it to do.
Wow. I feel like maths is kind of the ultimate problem solver for you. If I'm reading between
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the lines there, it helps you make sense of that. But then apply it, it only does what
you tell it to do. How do you know what to tell it to do? Is this where this comes in
when you're noodling around with your maths? You're like, how about this and just applying
different strategies? That's, that's the basic challenge. So to find ways to hook these
(09:25):
things into our actual experience, we have to have actual experiences. So I read and
have been reading extensively for quite some time. And actually, you know, have jobs here
and there throughout my life. And have done some traveling, not much. I'm kind of a home
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body for the most part. I still live in the town I was raised in. But once again, it comes
back to sort of that, that there's a big universe. So you can, you can concentrate on basically
any, any piece of it and get a an incoming stream of information that's that's bigger
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and deeper and wider than you can possibly hope to contain. And so you can take these
sense impressions and inspirations back into your problem space and then see, see what
happens. So, for example, my major project is CDM, this system for doing price discovery.
(10:36):
I was initially thinking about sort of consensus or fairness on networks. So people who have
children might be familiar with the cake sharing problem. If you've got two kids, you get one
of them to cut the last piece of cake and you get the other one to pick and they both
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then have to agree that the outcome is fair. Because if it's not fair, it's their own
fault.
Very practical. Yes. Yes. As it happens, there are fair division algorithms for dividable
objects that work for any number of people. Unfortunately, those algorithms are what is
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known as exponential time. So if you have three kids, it's a lot more complicated. If
you have four kids, it's much, much, much more complicated than the three kids example.
If you have five kids, they're gonna, they're gonna be in college before they figure out
how to divide the cake. Like, it gets bad fast. And so this is, this is obviously an
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important general problem, particularly as we're cranking up the amount of things that
we plug into the internet. And so I thought it'd be an interesting thing to think about
ways that you could negotiate a common interest in, in these sorts of fair outcomes. And I
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came up with an approach and then I was asked a question and it occurred to me that I could
add recursion to the approach. And recursion is when a system can refer back to its own
behavior as part of its operation. And it, it makes systems much harder to think about
and much more powerful, which is like catnip to me, basically, like that's, that's exactly
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what I love thinking about. So I was like, well, that's a fascinating problem. Let's,
let's dive into that and see what happens. And what happens, it turns out when you think
about that specific set of things is a brand new kind of marketplace falls out of the,
the system at you. And that marketplace is more efficient than the ones we actually use.
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And that's, that's something that's essentially been sitting there for people to figure out
and communicate to each other forever. You know, this is this as far as I can tell math
is a is sort of a feature of the universe. So human beings innovating fire talking around
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the camp could have come up with this idea, but there's no record of it anywhere. So
that's, that's creativity.
So are you saying that there's limitless amounts of maths problems out there that we can find
and solve and work on?
(13:45):
Oh, beyond, beyond your imagination, quite literally, in fact, there's, there's math
problems about that specific issue.
Very meta.
Yes, about the scope of imagination. There's, it's, there's something based around computational
mathematics called the mathematicians infinite employment theorem. And there's a number
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of infinite employment theorems, actually. And basically what this theorem says is that
there's no system that we could conceivably deploy that would be able to get to all the
interesting math problems. And so anyone that wants to sort of break loose and throw their
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hat into the ring, there's, there's low hanging fruit somewhere and you might be the person
that picks it up.
And do you get to name it if you come up with a new maths problem?
In general, yes. If you come up with something that's really awesome, then, then usually
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the community will wind up naming the conjecture or the proof after you. In some cases, this
has led to a certain amount of confusion, because certain superstar mathematicians like
Leonard Oiler, who's basically famous for having a name that is pronounced differently
(15:17):
from the way it's spelled, created so much mathematics that there's so many things that
are named after him that many people actually get them confused with one another. So that
becomes, that becomes quite the challenge.
(15:40):
Surely there is a mathematical formula that could help that situation in the naming of
new maths formulas.
Yeah, yeah. Well, that's, that's not something that's currently excited the community, but
maybe someday.
I've got an opportunity here. I can just see it. So tell me, let's jump back a tiny
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little bit. So this is, so doing this type of maths, this is not your regular nine to
five job, is it? You just, you have an interest in this outside of work.
Yeah, yeah. So typically, when I'm actually employed, it's doing data analysis or programming.
(16:26):
And while programming has some relationship to computational mathematics, many actual
valuable programming involves more or less plumbing, as it were, which again, is is
a craft and respectable thing, but not not a deep creative object. So the if you have
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incoming information, say web requests that need to be served at some particular rate,
then you, you have a basic engineering problem, you need to be able to pull in these things,
process them at speed, you know, carve off the information that's that's appropriate.
It involves a certain amount of measuring what's going on and sort of figuring out what
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parts are slow, what parts are fast, and then figuring out how to use well worn libraries
that exist to hook in to get them to operate the way it's supposed to be.
Even there, there's a certain amount of design that exists that that overlaps pretty heavily
with the kinds of the kinds of things that I find actually interesting. In terms of data
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analysis, again, there's so many different areas of mathematics. These days, the AI approaches
are largely dominating data analysis, but they usually require fairly large scale sets.
And so many people are operating small or medium sized companies and don't really generate
(18:02):
millions or billions of records on a daily basis. And if you're generating tens or hundreds
of records, then techniques that go back hundreds of years are actually not merely sufficient,
but actually better on average, because they're easier and cheaper to deploy.
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So is that why is that why you've sort of strayed into maths outside of ours? Because
your regular work is kind of logical and procedural and is very set as to what you do. Is maths
to a charge and a let loose? Not really. So math is something that has always come relatively
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easy to me compared to most people. And so I learned quite a lot of it in school. But
I would have more fun reading books or, or, you know, taking walks through the woods or
the mountains and getting some daydreaming in that way. But once I became a professional
(19:13):
programmer, my sort of engineering training kicked in. And we'd had some ethics classes
on the concept of what a professional is. And a professional is someone that has mastered
a body of knowledge that allows them to make decisions on behalf of others for others. And
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so you have to master a body of knowledge. And so now that I had this sort of engineering
ish job, I was like, well, I need to master this body of knowledge so that I actually
am capable of doing it. And in doing that mathematical study, that's where I ran across
this, this genie in a box, you know, characteristic of computational mathematics. And that's where
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the this sort of, you know, having fun daydreaming type of type of pastime linked in hard to
this, this actual engineering result where your daydreams could be turned into reality
by the machines. And the machines would show you other things that were the results of
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the daydreams you were having. And, and this kind of entirely new world opened up. So it
was, it was something that I kind of fell into, rather than, than being my release.
Wow. So you're having, you're having all these thoughts about various things and then maths
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is your way of being able to actually enact that.
Yeah, yeah, it's, it's a language that allows you to express ideas in much more effective
ways. And so without that language, you would never really even be able to think these ideas.
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One of the classic examples is how sort of bad Rome was at math, because Roman numerals
basically are terrible. Adding Roman numerals isn't so bad, multiplying Roman numerals is
an enormous pain in the neck. Dividing Roman numerals is basically an object of torture.
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Whereas our positional notation that we use these days is much more natural for those
types of operations. And there's other ways of thinking about and representing numbers
that are even more effective at doing those things. So there's, there's a few different
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sort of angles going on where you're attempting to develop language to allow you to think
about things, but that language development is itself embedded in this language of helping
you to think about things. So as you get better at developing these, these new language ideas,
you get better at being better at developing these new language ideas. And you find yourself
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deep, deep into the weeds very, very rapidly.
Yes. How do you, how do you do that? You're getting deeper and deeper and deeper. It's
a, it's also the cool thing. How do you know when to just stop?
Well, sleep, food, you know, there's, the body will stop you at some point. And, and
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that's usually what, what happens.
So you say you like, you have loved maths, like your entire life. Did you, did you always
think it was a creative thing or is it the daydreaming that has made you realise that?
I really didn't. I have, have quite the bone to pick pedagogically. I was extremely fortunate
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in, in sort of my schooling, I was exposed to enormous amounts of math, but I was still
exposed to math as sort of a series of tricks, if you will. So, you know, this is, this is
how you count. This is the order that numbers are in. And oh, by the way, now that you know
that numbers are in this order, here's these shortcuts, like addition and multiplication.
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And then there's these, you know, division and fractions and, and you keep building out
and out from there. And each of these things, whether set theory or geometry or probability
were sort of presented as mechanical objects. It was just like, this is, this is how the
world exists. These are how these things are related. So it was like chemistry or physics.
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It's just like, you know, if you, if you stick some of this and some of the blue powder and
some of the purple powder, and then you heat it up, then, then this happens. And so that's
like how you add. And that was, that was cool. I mean, I was, I was very good at it. It was
very easy for me to do these things. But the idea that there would be like different kinds
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of addition. I actually recall, I believe I was actually exposed to negative numbers,
but in, in first grade, we were given an exercise set that involved subtracting. I believe it
was seven from five. And I wrote down negative two and they marked it wrong, because they
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were teaching first graders. And so they weren't going to talk about negative numbers. And,
and so there was a bit of a disconnect where, where, you know, the boundaries were very
much in place. And it was all presented as a set of boundaries that these are, these
are exactly what you can do. And that's precisely what, what comes out. And that's fascinating
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in its own right that there is this exactness that's possible. Certainly children don't
have a lot of experience with exactness and precision in their ordinary tainted day lives.
And I certainly didn't. I'm still a bit of a slob to this day. But, but that, that creative
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aspect didn't really sort of exist for me. And, and that broke a little bit when we got
into geometric proofs. It broke a little bit more when I started studying computer programming.
But it didn't really shatter until I sort of wrapped my head around exploratory mathematics.
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And that would have been by the time I was in college.
Yeah, wow. Yeah, because I mean, I suppose my schooling was similar, not the set of five
minus seven, which was what was the answer if it wasn't minus two, by the way?
The answer was there's no answer.
Oh, oh, there you go. Now, if I feel like certainly the schooling that I had in a few different
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countries, I actually went to school in three different countries. It was always at maths
is logical is definitive. There's one right answer and there's one right way of doing
things. And that's the opposite of creativity. It never sort of crossed my brain that it
could be a creative pursuit. But I feel as though if you do consider it to be problem
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solving, problem solving is a very creative pursuit. And that's essentially what you're
saying there, isn't it?
Well, problem solving, yes, is essentially a very creative pursuit. And although, again,
for my standards of creativity, problem solving doesn't necessarily hit the high notes because
the problem would have to exist. And so to some extent, the thing that you're dealing
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with is is still sort of just existing within its scope.
So you create the problem and solve the problem.
Yeah, yeah, yeah, that might be more where the creativity lies. But within mathematics,
there are these these facts about ideas that in particular, in the last two centuries, as
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the tools have become more and more sophisticated and some of the things that have been thinking
about it more and more abstract that are shocking and surprising. Things things like the the
Quintic. So many people are familiar with the quadratic formula. If you have an algebra
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problem that with secondary equations, there's a straightforward way you can plug in the
coefficients of the equation and pop out what the the roots, the zeros of the equation are.
As it happens, there's a cubic formula. It's a lot more complicated. There's a quartic
formula for power. It's a lot more complicated than cubic formula. And for a while, people
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were looking for the next the fifth power one, they assumed it would be very, very complex,
but they were checking it out. It turns out that there is no formula in fifth for in fifth
powers. You have to use other much more powerful techniques. And for five and all higher powers,
because of because of the symmetries of the patterns of the roots that are possible in
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the complex plane, it's not possible for there to be enough information in the in a set of
six numbers to to narrow that down to five answers. And and that's true at every level.
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So there's for 100 degree polynomial, you'd have 101 numbers of information. That's that
wouldn't contain enough information to tell you what the outcome is.
Wow, okay, this is getting so complex. And I feel as though, you know, is it important
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for you to for people to feel like maths is created?
I I think I think that would have a lot of value because of the technology and the tools
that exist today, that because we have computers, and because we use them so poorly, or in some
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cases, not really at all. The I it looks to me like there's a lot of this low hanging
fruit lying around, there's there's got to be a lot of Montgoughly brothers out there
that could just, you know, notice, oh, you know, when I do this ordinary thing with with
(30:15):
computational technology, obviously, it would be possible to no longer have to some very
important part of human society that takes up a lot of time and effort and whatever, because
it's just like we could just we can just slip the bonds of earth and float in the sky now.
And I don't know what those things are, but I can't imagine that they're not there, because
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nobody knew any of this stuff, you know, three weeks ago, basically, like this is all very,
very new. And so, having a lot more human minds, looking at and getting curious and throwing
themselves into this particular deep end strikes me as a very sensible way to let our imaginations
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loose. Yeah, I mean, because all of this is so complex, it's hard to even comprehend what
you're talking about half the time, because it's so theoretical, right? And you can't
put your hands on it necessarily, it's not tangible to me anyway. That perhaps is like
a PR approach to rebrand maths as a creative pursuit, if it will engage more people in
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it and thus we'd be able to solve more problems in the world faster.
I would hope so. And in particular, the problem that might be most important to solve is to
deal with the language and work on some of these issues. So communication is hard in
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its own right. But we now have the mathematics of information, which is one of the key parts
of computational mathematics, to talk about language. And there are some very powerful
ideas there that are not part of the general discussion about how language and communication
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is structured. Sadly, most of that has been dominated by the marketing department, which
is mostly did it sell product? Good. Did it not sell product bad? Which isn't entirely
unimportant. But there are there are other things that you can do with it. So that's
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something that I think is part of that fertile ground is finding ways to think about how
to visualize or conceptualize and in most important fact, communicate a common understanding
(33:00):
of abstract ideas to get people on the same page. Because there have been a few, at least
half a dozen that I'm aware of projects that where Internet scale interested parties got
involved in a mathematical proof. And and the progress is is pretty considerable. One
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of the more famous ones, there's something called the twin primes conjecture. So going
back to Euclid, we've had a well known proof that there's an infinite number of prime numbers
prime numbers have the property that the only whole number that divides them wholly is themselves
and one of course, one divides all the numbers. So two, three, five, seven, so on. Some primes
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and the early ones are sort of close together enough that we don't really count them. But
primes like 11 and 13 primes like 17 and 19 primes like 41 and 43 are only two apart
from each other. And so they're known as twin primes. And we don't know whether or not there's
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an infinite number of twin primes. We know that they keep showing up as far as we can
check. But you know, obviously, we can't check very far from an infinite standpoint. And it
turns out that it's a very hard problem to think about. Essentially, zero progress was
made up until the like 2010s, or maybe maybe he was back in the aughts. So this this math
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professor, whose wife had a job out of state. And so he said he had a lot of time to think
for himself, came up with a proof that there were in fact an infinite number of primes
that were separated by no more than it was something like a few million, which might not
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sound like much, but nobody had ever made any progress on this problem at all in the
thousands of years that had been proposed. And he had sort of the very first stake in
the ground. Yeah, well, an internet collaboration of thousands of people took his approach and
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and did the work of chasing down all the, you know, screw tight nings and doing all the
computational pieces and got that number down into the thousands about a month after he
published his his paper. Wow. And so there's there's a lot of prospects for wide scale
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collaboration, if people can have a common understanding and interest in some of these
abstract problems. Yeah, well, many hands may like word, right? And I guess it's just
so hard for people to understand because the numbers you're talking about there are so
big that you almost you can't even count them. And that's why you couldn't understand these
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things, right? And so people have no frame of reference for most people would have no
frame of reference for the scale of these numbers and the complexity of all these theorems
and formulas and whatnot. So there are there are people who work in areas. This isn't sort
of my thing, but there are people who work in areas of figuring out numbers that are
(36:32):
so big that they have no reference except themselves.
See, that that is it is like, what does that mean? What does that mean? But no, I agree.
I agree. I feel like it is something that we should talk about more because obviously
it maps have so so many applications across all of our universe and understanding and taking
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us to the next level that why not have more people involved, you know, and well, maybe
this podcast will be that step is just to lifting the veil and maths a little bit more.
Hopefully so.
No, I want to thank you so much for joining me today. I really enjoyed our complex chat.
(37:21):
Thank you. Yeah, this was a lot of fun for me too.
And I also want to say thank you to everyone who's tuned in to Creativity (37:26):
Uncovered today.
I hope this this episode has inspired you to pick up a little maths and that it helps
you summon creativity the next time you need it.
. If you've made it this far, thank you. I'll see you next time. Bye.
(38:09):
If you've made it this far, a huge thank you for your support and tuning into today's
episode. Creativity (38:15):
Uncovered has been lovingly recorded on the land of the Kabi Kabi people
and we pay our respects to elders past, present and emerging. This podcast has been produced
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(38:40):
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You
Uncovered. My name is Abi Gatling and I am on a journey
(00:12):
to uncover how everyday people find inspiration, get inventive and open their imagination.
Basically I want to know how people find creative solutions at home, work, play and everything
in between. And my goal for this podcast is that by the end of it you'll be armed with
a whole suite of tried and tested ways to some creativity whenever you need it.
(00:42):
Now today I'm speaking with Noah Healy who is a professional data scientist and a recreationally
use a mathematician just for fun. And now if you're anything like me, you probably never
thought of maths as a creative field. So I'm definitely keen to find out what it means
to be a mathematician just for fun and discover where the creativity lies in that field.
(01:08):
So welcome Noah.
Thanks for having me here Abi.
So before we start, I kind of want to get a gauge from you because on this podcast we've
had so many different people on here and it's meant that we've had dozens of different definitions
of what creativity is. So before we jump into our conversation I'd love to know what is
(01:31):
creativity to you and what does it mean to be creative?
Well for me creativity is the production of something not seen before. And I consider
that at a categorical level. So there's a certain creativity in craft. Every time you
(01:53):
say throw a cup on a potter's wheel that specific cup hasn't been produced before.
But you can shuffle a deck of cards and produce a sequence of cards that probably will never
be repeated again in our universe. So that's not creative but not actual creativity from
(02:13):
my lights. But the bringing forth of an entirely new idea even where it has to build on things
that's creativity to me. And in terms of what it means I think it's sort of a miracle in
(02:34):
the palm of your hand. We live surrounded by things that were brought forth from just
ideas and imagination. And we can add more to that list and it's amazing.
Wow. Do you think that creativity is expensive?
(02:56):
I think yeah actually. There's certain aspects. The specific thing that I like to pursue is
inexpensive economically but it takes a lot of time and effort for me. I'm not as smart
as the smartest people who have ever lived. And societally there is a cost. You have to
(03:28):
step back from just staying sort of at the coal face or plowing the furrow and taking
the time and taking the leap of faith that is imagination. And that's a cost. And I'm
very happy to live in a time when many, many more people are available to have that cost.
(03:53):
And quite specifically I am capable of generating enough free revenue that I've been able to
carve that time out without starving myself to death in the process.
Well that's good. No one wants you to be starving to death. But tell me, what is, why is creativity
(04:16):
important to you?
I think it's sort of foundational. It's a big universe. We have a small piece of it.
It's a big planet. We have a small piece of that. There's so much that's possible and
available. And while amazing things can happen at our level and at levels that we've been
(04:42):
at before, each of those things is leaving aside things that we know for a fact could
have been created that would have made them so much more impressive. One of the ones that
I typically trot out is Block and Tackle, which was invented by Archimedes. That means
(05:06):
that the great wonders of the ancient world from the pyramids which still stand to not
one of the official ones, but things like the Great Wall of China, were built by people
who didn't actually understand what Block and Tackle was, which is one of the most powerful
(05:28):
simple machines that human beings ever created. And for over half of recorded history, people
didn't know how to do that. But another fun one I thought of recently is hot air balloons.
The enabling technologies to have allowed people to have constructed hot air balloons
(05:49):
probably predate writing by two to four thousand years. We don't have precise timing on when
high fat linen was generated. But hot air balloons weren't actually created until the
Montgauflier brothers, I can't remember which one of the two of them was, watched clothes
(06:10):
drying above a fire and noticed that they billowed upwards in the smoke and decided
to build a very large drying dress more or less and demonstrate that you could float above
(06:33):
the ground. And this is a technology that Ramses or Tutt or early Chinese emperors could
have had, but certainly didn't.
Wow. Okay. So it's like creativity is kind of helping people make sense of the world,
but also make leaps and bounds forward.
(06:53):
Yeah. Yeah. There's a very real sense in which the chaos of the universe contains vastly more
than we can be aware of. And by finding information that makes sense that we can interact with,
(07:13):
are leveling ourselves up in a way that's very deep and difficult to fully comprehend
all the aspects of. And that gets into sort of my passion around computational mathematics
because computational mathematics is essentially about the treatment of information as a, what's
(07:42):
known as a first class object. So information about how information can be processed, which
you can then use on the information that's doing the processing and create these things
that are sort of at the same time daydreams, but also machines that can do things. And that
(08:05):
is, that is effectively what the computer is a the computer, the computers that we surround
ourselves with are small manifestations of the partial concept of this general or universal
computing machine, a imaginative system that is capable of encapsulating all of the possibilities
(08:34):
of imagination. And which, like a gin from, you know, ancient myth will do anything that
you tell it to do, but only specifically and exactly what you tell it to do.
Wow. I feel like maths is kind of the ultimate problem solver for you. If I'm reading between
(08:59):
the lines there, it helps you make sense of that. But then apply it, it only does what
you tell it to do. How do you know what to tell it to do? Is this where this comes in
when you're noodling around with your maths? You're like, how about this and just applying
different strategies? That's, that's the basic challenge. So to find ways to hook these
(09:25):
things into our actual experience, we have to have actual experiences. So I read and
have been reading extensively for quite some time. And actually, you know, have jobs here
and there throughout my life. And have done some traveling, not much. I'm kind of a home
(09:47):
body for the most part. I still live in the town I was raised in. But once again, it comes
back to sort of that, that there's a big universe. So you can, you can concentrate on basically
any, any piece of it and get a an incoming stream of information that's that's bigger
(10:09):
and deeper and wider than you can possibly hope to contain. And so you can take these
sense impressions and inspirations back into your problem space and then see, see what
happens. So, for example, my major project is CDM, this system for doing price discovery.
(10:36):
I was initially thinking about sort of consensus or fairness on networks. So people who have
children might be familiar with the cake sharing problem. If you've got two kids, you get one
of them to cut the last piece of cake and you get the other one to pick and they both
(10:58):
then have to agree that the outcome is fair. Because if it's not fair, it's their own
fault.
Very practical. Yes. Yes. As it happens, there are fair division algorithms for dividable
objects that work for any number of people. Unfortunately, those algorithms are what is
(11:21):
known as exponential time. So if you have three kids, it's a lot more complicated. If
you have four kids, it's much, much, much more complicated than the three kids example.
If you have five kids, they're gonna, they're gonna be in college before they figure out
how to divide the cake. Like, it gets bad fast. And so this is, this is obviously an
(11:50):
important general problem, particularly as we're cranking up the amount of things that
we plug into the internet. And so I thought it'd be an interesting thing to think about
ways that you could negotiate a common interest in, in these sorts of fair outcomes. And I
(12:11):
came up with an approach and then I was asked a question and it occurred to me that I could
add recursion to the approach. And recursion is when a system can refer back to its own
behavior as part of its operation. And it, it makes systems much harder to think about
and much more powerful, which is like catnip to me, basically, like that's, that's exactly
(12:34):
what I love thinking about. So I was like, well, that's a fascinating problem. Let's,
let's dive into that and see what happens. And what happens, it turns out when you think
about that specific set of things is a brand new kind of marketplace falls out of the,
the system at you. And that marketplace is more efficient than the ones we actually use.
(12:58):
And that's, that's something that's essentially been sitting there for people to figure out
and communicate to each other forever. You know, this is this as far as I can tell math
is a is sort of a feature of the universe. So human beings innovating fire talking around
(13:23):
the camp could have come up with this idea, but there's no record of it anywhere. So
that's, that's creativity.
So are you saying that there's limitless amounts of maths problems out there that we can find
and solve and work on?
(13:45):
Oh, beyond, beyond your imagination, quite literally, in fact, there's, there's math
problems about that specific issue.
Very meta.
Yes, about the scope of imagination. There's, it's, there's something based around computational
mathematics called the mathematicians infinite employment theorem. And there's a number
(14:12):
of infinite employment theorems, actually. And basically what this theorem says is that
there's no system that we could conceivably deploy that would be able to get to all the
interesting math problems. And so anyone that wants to sort of break loose and throw their
(14:34):
hat into the ring, there's, there's low hanging fruit somewhere and you might be the person
that picks it up.
And do you get to name it if you come up with a new maths problem?
In general, yes. If you come up with something that's really awesome, then, then usually
(14:57):
the community will wind up naming the conjecture or the proof after you. In some cases, this
has led to a certain amount of confusion, because certain superstar mathematicians like
Leonard Oiler, who's basically famous for having a name that is pronounced differently
(15:17):
from the way it's spelled, created so much mathematics that there's so many things that
are named after him that many people actually get them confused with one another. So that
becomes, that becomes quite the challenge.
(15:40):
Surely there is a mathematical formula that could help that situation in the naming of
new maths formulas.
Yeah, yeah. Well, that's, that's not something that's currently excited the community, but
maybe someday.
I've got an opportunity here. I can just see it. So tell me, let's jump back a tiny
(16:06):
little bit. So this is, so doing this type of maths, this is not your regular nine to
five job, is it? You just, you have an interest in this outside of work.
Yeah, yeah. So typically, when I'm actually employed, it's doing data analysis or programming.
(16:26):
And while programming has some relationship to computational mathematics, many actual
valuable programming involves more or less plumbing, as it were, which again, is is
a craft and respectable thing, but not not a deep creative object. So the if you have
(16:52):
incoming information, say web requests that need to be served at some particular rate,
then you, you have a basic engineering problem, you need to be able to pull in these things,
process them at speed, you know, carve off the information that's that's appropriate.
It involves a certain amount of measuring what's going on and sort of figuring out what
(17:14):
parts are slow, what parts are fast, and then figuring out how to use well worn libraries
that exist to hook in to get them to operate the way it's supposed to be.
Even there, there's a certain amount of design that exists that that overlaps pretty heavily
with the kinds of the kinds of things that I find actually interesting. In terms of data
(17:38):
analysis, again, there's so many different areas of mathematics. These days, the AI approaches
are largely dominating data analysis, but they usually require fairly large scale sets.
And so many people are operating small or medium sized companies and don't really generate
(18:02):
millions or billions of records on a daily basis. And if you're generating tens or hundreds
of records, then techniques that go back hundreds of years are actually not merely sufficient,
but actually better on average, because they're easier and cheaper to deploy.
(18:26):
So is that why is that why you've sort of strayed into maths outside of ours? Because
your regular work is kind of logical and procedural and is very set as to what you do. Is maths
to a charge and a let loose? Not really. So math is something that has always come relatively
(18:50):
easy to me compared to most people. And so I learned quite a lot of it in school. But
I would have more fun reading books or, or, you know, taking walks through the woods or
the mountains and getting some daydreaming in that way. But once I became a professional
(19:13):
programmer, my sort of engineering training kicked in. And we'd had some ethics classes
on the concept of what a professional is. And a professional is someone that has mastered
a body of knowledge that allows them to make decisions on behalf of others for others. And
(19:33):
so you have to master a body of knowledge. And so now that I had this sort of engineering
ish job, I was like, well, I need to master this body of knowledge so that I actually
am capable of doing it. And in doing that mathematical study, that's where I ran across
this, this genie in a box, you know, characteristic of computational mathematics. And that's where
(20:03):
the this sort of, you know, having fun daydreaming type of type of pastime linked in hard to
this, this actual engineering result where your daydreams could be turned into reality
by the machines. And the machines would show you other things that were the results of
(20:24):
the daydreams you were having. And, and this kind of entirely new world opened up. So it
was, it was something that I kind of fell into, rather than, than being my release.
Wow. So you're having, you're having all these thoughts about various things and then maths
(20:46):
is your way of being able to actually enact that.
Yeah, yeah, it's, it's a language that allows you to express ideas in much more effective
ways. And so without that language, you would never really even be able to think these ideas.
(21:08):
One of the classic examples is how sort of bad Rome was at math, because Roman numerals
basically are terrible. Adding Roman numerals isn't so bad, multiplying Roman numerals is
an enormous pain in the neck. Dividing Roman numerals is basically an object of torture.
(21:34):
Whereas our positional notation that we use these days is much more natural for those
types of operations. And there's other ways of thinking about and representing numbers
that are even more effective at doing those things. So there's, there's a few different
(21:55):
sort of angles going on where you're attempting to develop language to allow you to think
about things, but that language development is itself embedded in this language of helping
you to think about things. So as you get better at developing these, these new language ideas,
you get better at being better at developing these new language ideas. And you find yourself
(22:22):
deep, deep into the weeds very, very rapidly.
Yes. How do you, how do you do that? You're getting deeper and deeper and deeper. It's
a, it's also the cool thing. How do you know when to just stop?
Well, sleep, food, you know, there's, the body will stop you at some point. And, and
(22:48):
that's usually what, what happens.
So you say you like, you have loved maths, like your entire life. Did you, did you always
think it was a creative thing or is it the daydreaming that has made you realise that?
I really didn't. I have, have quite the bone to pick pedagogically. I was extremely fortunate
(23:14):
in, in sort of my schooling, I was exposed to enormous amounts of math, but I was still
exposed to math as sort of a series of tricks, if you will. So, you know, this is, this is
how you count. This is the order that numbers are in. And oh, by the way, now that you know
that numbers are in this order, here's these shortcuts, like addition and multiplication.
(23:37):
And then there's these, you know, division and fractions and, and you keep building out
and out from there. And each of these things, whether set theory or geometry or probability
were sort of presented as mechanical objects. It was just like, this is, this is how the
world exists. These are how these things are related. So it was like chemistry or physics.
(24:01):
It's just like, you know, if you, if you stick some of this and some of the blue powder and
some of the purple powder, and then you heat it up, then, then this happens. And so that's
like how you add. And that was, that was cool. I mean, I was, I was very good at it. It was
very easy for me to do these things. But the idea that there would be like different kinds
(24:25):
of addition. I actually recall, I believe I was actually exposed to negative numbers,
but in, in first grade, we were given an exercise set that involved subtracting. I believe it
was seven from five. And I wrote down negative two and they marked it wrong, because they
(24:46):
were teaching first graders. And so they weren't going to talk about negative numbers. And,
and so there was a bit of a disconnect where, where, you know, the boundaries were very
much in place. And it was all presented as a set of boundaries that these are, these
are exactly what you can do. And that's precisely what, what comes out. And that's fascinating
(25:09):
in its own right that there is this exactness that's possible. Certainly children don't
have a lot of experience with exactness and precision in their ordinary tainted day lives.
And I certainly didn't. I'm still a bit of a slob to this day. But, but that, that creative
(25:30):
aspect didn't really sort of exist for me. And, and that broke a little bit when we got
into geometric proofs. It broke a little bit more when I started studying computer programming.
But it didn't really shatter until I sort of wrapped my head around exploratory mathematics.
(26:00):
And that would have been by the time I was in college.
Yeah, wow. Yeah, because I mean, I suppose my schooling was similar, not the set of five
minus seven, which was what was the answer if it wasn't minus two, by the way?
The answer was there's no answer.
Oh, oh, there you go. Now, if I feel like certainly the schooling that I had in a few different
(26:25):
countries, I actually went to school in three different countries. It was always at maths
is logical is definitive. There's one right answer and there's one right way of doing
things. And that's the opposite of creativity. It never sort of crossed my brain that it
could be a creative pursuit. But I feel as though if you do consider it to be problem
(26:50):
solving, problem solving is a very creative pursuit. And that's essentially what you're
saying there, isn't it?
Well, problem solving, yes, is essentially a very creative pursuit. And although, again,
for my standards of creativity, problem solving doesn't necessarily hit the high notes because
the problem would have to exist. And so to some extent, the thing that you're dealing
(27:15):
with is is still sort of just existing within its scope.
So you create the problem and solve the problem.
Yeah, yeah, yeah, that might be more where the creativity lies. But within mathematics,
there are these these facts about ideas that in particular, in the last two centuries, as
(27:42):
the tools have become more and more sophisticated and some of the things that have been thinking
about it more and more abstract that are shocking and surprising. Things things like the the
Quintic. So many people are familiar with the quadratic formula. If you have an algebra
(28:02):
problem that with secondary equations, there's a straightforward way you can plug in the
coefficients of the equation and pop out what the the roots, the zeros of the equation are.
As it happens, there's a cubic formula. It's a lot more complicated. There's a quartic
formula for power. It's a lot more complicated than cubic formula. And for a while, people
(28:25):
were looking for the next the fifth power one, they assumed it would be very, very complex,
but they were checking it out. It turns out that there is no formula in fifth for in fifth
powers. You have to use other much more powerful techniques. And for five and all higher powers,
because of because of the symmetries of the patterns of the roots that are possible in
(28:52):
the complex plane, it's not possible for there to be enough information in the in a set of
six numbers to to narrow that down to five answers. And and that's true at every level.
(29:13):
So there's for 100 degree polynomial, you'd have 101 numbers of information. That's that
wouldn't contain enough information to tell you what the outcome is.
Wow, okay, this is getting so complex. And I feel as though, you know, is it important
(29:34):
for you to for people to feel like maths is created?
I I think I think that would have a lot of value because of the technology and the tools
that exist today, that because we have computers, and because we use them so poorly, or in some
(29:54):
cases, not really at all. The I it looks to me like there's a lot of this low hanging
fruit lying around, there's there's got to be a lot of Montgoughly brothers out there
that could just, you know, notice, oh, you know, when I do this ordinary thing with with
(30:15):
computational technology, obviously, it would be possible to no longer have to some very
important part of human society that takes up a lot of time and effort and whatever, because
it's just like we could just we can just slip the bonds of earth and float in the sky now.
And I don't know what those things are, but I can't imagine that they're not there, because
(30:40):
nobody knew any of this stuff, you know, three weeks ago, basically, like this is all very,
very new. And so, having a lot more human minds, looking at and getting curious and throwing
themselves into this particular deep end strikes me as a very sensible way to let our imaginations
(31:06):
loose. Yeah, I mean, because all of this is so complex, it's hard to even comprehend what
you're talking about half the time, because it's so theoretical, right? And you can't
put your hands on it necessarily, it's not tangible to me anyway. That perhaps is like
a PR approach to rebrand maths as a creative pursuit, if it will engage more people in
(31:31):
it and thus we'd be able to solve more problems in the world faster.
I would hope so. And in particular, the problem that might be most important to solve is to
deal with the language and work on some of these issues. So communication is hard in
(31:55):
its own right. But we now have the mathematics of information, which is one of the key parts
of computational mathematics, to talk about language. And there are some very powerful
ideas there that are not part of the general discussion about how language and communication
(32:16):
is structured. Sadly, most of that has been dominated by the marketing department, which
is mostly did it sell product? Good. Did it not sell product bad? Which isn't entirely
unimportant. But there are there are other things that you can do with it. So that's
(32:39):
something that I think is part of that fertile ground is finding ways to think about how
to visualize or conceptualize and in most important fact, communicate a common understanding
(33:00):
of abstract ideas to get people on the same page. Because there have been a few, at least
half a dozen that I'm aware of projects that where Internet scale interested parties got
involved in a mathematical proof. And and the progress is is pretty considerable. One
(33:27):
of the more famous ones, there's something called the twin primes conjecture. So going
back to Euclid, we've had a well known proof that there's an infinite number of prime numbers
prime numbers have the property that the only whole number that divides them wholly is themselves
and one of course, one divides all the numbers. So two, three, five, seven, so on. Some primes
(33:53):
and the early ones are sort of close together enough that we don't really count them. But
primes like 11 and 13 primes like 17 and 19 primes like 41 and 43 are only two apart
from each other. And so they're known as twin primes. And we don't know whether or not there's
(34:13):
an infinite number of twin primes. We know that they keep showing up as far as we can
check. But you know, obviously, we can't check very far from an infinite standpoint. And it
turns out that it's a very hard problem to think about. Essentially, zero progress was
made up until the like 2010s, or maybe maybe he was back in the aughts. So this this math
(34:40):
professor, whose wife had a job out of state. And so he said he had a lot of time to think
for himself, came up with a proof that there were in fact an infinite number of primes
that were separated by no more than it was something like a few million, which might not
(35:02):
sound like much, but nobody had ever made any progress on this problem at all in the
thousands of years that had been proposed. And he had sort of the very first stake in
the ground. Yeah, well, an internet collaboration of thousands of people took his approach and
(35:24):
and did the work of chasing down all the, you know, screw tight nings and doing all the
computational pieces and got that number down into the thousands about a month after he
published his his paper. Wow. And so there's there's a lot of prospects for wide scale
(35:47):
collaboration, if people can have a common understanding and interest in some of these
abstract problems. Yeah, well, many hands may like word, right? And I guess it's just
so hard for people to understand because the numbers you're talking about there are so
big that you almost you can't even count them. And that's why you couldn't understand these
(36:10):
things, right? And so people have no frame of reference for most people would have no
frame of reference for the scale of these numbers and the complexity of all these theorems
and formulas and whatnot. So there are there are people who work in areas. This isn't sort
of my thing, but there are people who work in areas of figuring out numbers that are
(36:32):
so big that they have no reference except themselves.
See, that that is it is like, what does that mean? What does that mean? But no, I agree.
I agree. I feel like it is something that we should talk about more because obviously
it maps have so so many applications across all of our universe and understanding and taking
(36:57):
us to the next level that why not have more people involved, you know, and well, maybe
this podcast will be that step is just to lifting the veil and maths a little bit more.
Hopefully so.
No, I want to thank you so much for joining me today. I really enjoyed our complex chat.
(37:21):
Thank you. Yeah, this was a lot of fun for me too.
And I also want to say thank you to everyone who's tuned in to Creativity (37:26):
Uncovered today.
I hope this this episode has inspired you to pick up a little maths and that it helps
you summon creativity the next time you need it.
. If you've made it this far, thank you. I'll see you next time. Bye.
(38:09):
If you've made it this far, a huge thank you for your support and tuning into today's
episode. Creativity (38:15):
Uncovered has been lovingly recorded on the land of the Kabi Kabi people
and we pay our respects to elders past, present and emerging. This podcast has been produced
by my amazing team here at Crisp Communications and the music you just heard was composed
by James Gatling. If you liked this episode, please do share it around and help us on
(38:40):
our mission to unlock more creativity in this world. You can also hit subscribe so you don't
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