Four Ways of Thinking: Statistical, Interactive, Chaotic and Complex - David Sumpter
Summary
TLDRIn this enlightening talk, the speaker, an applied mathematician, delves into the power and limitations of mathematical thinking in understanding the world. He explores four distinct modes of thinking: statistical, interactive, chaotic, and complex. Through engaging stories from football, science, and everyday life, he illustrates how these approaches can provide insight but also emphasizes the importance of recognizing when statistical significance does not equate to practical importance. The speaker highlights the beauty of chaos theory and the concept of complexity, as defined by Kolmogorov, suggesting that true understanding comes from finding succinct descriptions of complex phenomena.
Takeaways
- 📚 The speaker is an applied mathematician whose motivation stems from a desire to understand the world around them, rather than a love for calculations themselves.
- 🔢 The talk emphasizes four different ways of thinking: statistical, interactive, chaotic, and complex, using stories from football, science, and personal life to illustrate each.
- 👨💼 The speaker discusses the importance of experimental design and the contributions of Ronald Fisher, highlighting the power of statistics in understanding phenomena but also its limitations.
- 🏈 Football is used as a medium to demonstrate how statistics can measure aspects of a player's performance and attitude, challenging the notion that some elements are immeasurable.
- 🧬 The speaker introduces Alfred J. Lotka and his work on differential equations to model ecological interactions, such as predator-prey dynamics, showcasing interactive thinking.
- 🦋 The concept of chaos theory is introduced through the story of Margaret Hamilton and Edward Lorenz, illustrating how small differences can lead to vastly different outcomes, as in weather prediction.
- 🤖 An experiment is conducted during the talk to demonstrate the divergence of outcomes from small initial differences, reflecting the butterfly effect in chaos theory.
- 👨🚀 Margaret Hamilton's story is highlighted to show how a deep understanding of chaos led to meticulous control in critical situations, such as the Apollo moon mission.
- 🌐 The speaker discusses the balance between order (Yang) and chaos (Yin), suggesting that while we can control some aspects of life, we must accept the randomness in others.
- 🌟 The final point touches on complexity theory, suggesting that finding simple rules that capture the essence of complex systems can lead to a deeper understanding of the world.
Q & A
What is the main theme of the speaker's talk?
-The main theme of the speaker's talk is to provide insight into their thinking process in four different stages: statistical, interactive, chaotic, and complex thinking, using examples from football, science, and personal life.
Why did the speaker choose to discuss Ronald Fisher in their talk?
-The speaker chose to discuss Ronald Fisher because he was a significant figure in the development of applied mathematics and experimental design, and his work exemplifies the application of statistical thinking.
What is the significance of the milk-first or tea-first experiment mentioned in the talk?
-The milk-first or tea-first experiment is significant because it demonstrates the application of combinatorics and experimental design to solve a seemingly trivial problem, highlighting the power of mathematical thinking in everyday situations.
How does the speaker use football as an example to illustrate statistical thinking?
-The speaker uses football to illustrate statistical thinking by analyzing player performance metrics, such as how a player's performance changes when their team concedes a goal, and comparing these metrics to evaluate player attitudes and impact on team dynamics.
What is the 'Gary Neville statistic' referred to in the talk?
-The 'Gary Neville statistic' is a measurable statistic developed by the speaker to quantify a player's performance change after their team concedes a goal, which was initially thought to be immeasurable by Gary Neville.
What is the limitation of using statistics to measure concepts like 'grit'?
-The limitation of using statistics to measure concepts like 'grit' is that while it can provide some predictive power, it often only explains a small percentage of the variance in outcomes, indicating that many other factors contribute to success.
What is the role of Alfred J. Lotka in the development of interactive thinking?
-Alfred J. Lotka played a crucial role in the development of interactive thinking by introducing the concept of unbalanced chemical equations to model ecological interactions, such as predator-prey dynamics, which laid the foundation for understanding complex systems.
How does the speaker use the applause experiment to illustrate interactive thinking?
-The speaker uses the applause experiment to illustrate interactive thinking by showing how social behaviors, like clapping, can be modeled using the same principles as ecological interactions, demonstrating the spread of a 'social epidemic'.
What is the concept of chaos theory and how does it relate to the speaker's talk?
-Chaos theory is the concept that small changes in initial conditions can lead to vastly different outcomes in complex systems, making long-term prediction impossible. It relates to the speaker's talk by illustrating the limitations of control and predictability, even with advanced mathematical models.
How does the speaker's personal life example of moving a sofa with friends relate to interactive thinking?
-The speaker's personal life example of moving a sofa relates to interactive thinking by demonstrating how social interactions and cooperation can lead to a desired outcome, which is a simple model of how collective behavior can be understood and predicted.
What is the significance of the experiment involving doubling numbers and subtracting from 100 in the context of chaos theory?
-The significance of the experiment involving doubling numbers and subtracting from 100 is to demonstrate how small differences in initial conditions can quickly lead to divergent outcomes, illustrating the concept of sensitive dependence on initial conditions in chaos theory.
What is the 'butterfly effect' in chaos theory and how was it discovered?
-The 'butterfly effect' in chaos theory refers to the idea that the flap of a butterfly's wings in Brazil could set off a chain of events leading to a tornado in Texas. It was discovered by Edward Lorenz in the context of weather prediction models, where small differences in initial data led to vastly different forecasts.
Who is Margaret Hamilton and what is her contribution to the field of mathematics?
-Margaret Hamilton was a mathematician who worked on the Apollo moon mission, creating the software that helped navigate the spacecraft and control the thrusters. She is known for her attention to detail and her work in ensuring error-free computation in critical systems.
What is the fourth way of thinking introduced by the speaker and how does it relate to complexity?
-The fourth way of thinking introduced by the speaker is complex thinking, which is related to the concept of complexity as defined by Kolmogorov. It involves finding the shortest description that can produce a pattern, capturing the essence of complexity without losing detail or nuance.
What is the significance of cellular automata models in understanding complexity?
-Cellular automata models, like the Game of Life, are significant in understanding complexity because they demonstrate how simple local interaction rules can lead to the emergence of complex patterns and behaviors, showcasing the self-organizing nature of complex systems.
Outlines
📚 A Mathematician's Journey and the Power of Statistics
The speaker, an applied mathematician, expresses gratitude for the invitation to speak and reflects on their previous work at Oxford University. They introduce the theme of the talk, which is to explore their thought process through four stages: statistical, interactive, chaotic, and complex thinking. The speaker emphasizes the importance of mathematics as a tool for understanding the world, rather than merely for calculations. They plan to illustrate their points with stories from football, science, and personal life, highlighting the multidisciplinary nature of their work.
🎲 The Birth of Applied Mathematics and Ronald Fisher's Legacy
This paragraph delves into the history of applied mathematics, highlighting the contributions of Ronald Fisher. Fisher, known for his arrogance and brilliance, sought to understand the real-world applications of mathematical theories. His work in experimental design and statistics, including the famous tea-tasting experiment with Dr. Muriel Bristol, demonstrated the power of statistical methods in practical scenarios. Fisher's innovative approaches to experimental design and his book on the subject have had a lasting impact, shaping the field of applied mathematics.
📊 Statistics in Football: Measuring Intangibles
The speaker discusses the application of statistical thinking in football, challenging the notion that certain aspects of the game, like a player's attitude, cannot be measured. They recount their experience with former footballer Gary Neville and present an analysis that quantifies the impact of a team conceding a goal on individual player performance. The 'Gary Neville statistic' is introduced as an example of how statistics can reveal the tangible effects of intangible aspects of the game.
🔍 The Limits of Statistics and the Dangers of Misinterpretation
The speaker acknowledges the limitations of statistical analysis, using the example of Angela Duckworth's 'grit' concept to illustrate how statistics can be misinterpreted. They explain that while 'grit' may predict some variance in success, it is only a small fraction of the whole picture. The speaker warns against confusing statistical significance with practical significance and emphasizes the importance of context and the difference between correlation and causation.
🧬 The Misuse of Statistics: Eugenics and Smoking
This paragraph explores the darker side of statistics, where they have been misused to support false theories, such as eugenics and the denial of the link between smoking and cancer. The speaker uses the example of Ronald Fisher, who, despite his significant contributions to statistics, used his skills to argue for scientifically unfounded and morally repugnant ideas, demonstrating the need for ethical considerations in the application of statistical methods.
🤖 Interactive Thinking and the Work of Alfred J. Lotka
The speaker introduces interactive thinking through the story of Alfred J. Lotka, who applied mathematical models to understand complex biological and ecological systems. Lotka's work on unbalanced chemical equations, such as the rabbit and fox population model, demonstrated how simple mathematical models could capture the dynamics of real-world interactions, leading to the development of new fields of study like mathematical biology.
🦊 The Spread of Social Phenomena: Modeling with Lotka's Equations
The speaker discusses the application of Lotka's equations to model social phenomena, such as the spread of applause among a group of people. They highlight the importance of understanding the cues and social dynamics that lead to collective behavior. The speaker also touches on the personal aspects of social interactions and the value of reflecting on these dynamics in everyday life.
🐟 Mathematical Modeling in Biology and Football
The speaker describes the process of creating mathematical models to understand complex systems, using the behavior of fish and football players as examples. They explain how simple rules of interaction can lead to complex collective behavior and how these models can be used to predict and analyze movements, escapes, and strategic decisions in both biology and sports.
🌀 The Emergence of Chaos Theory and its Implications
The speaker introduces chaos theory through the story of Margaret Hamilton and Edward Lorenz, who discovered the concept of the 'butterfly effect' while working on weather prediction models. They illustrate how small changes in initial conditions can lead to vastly different outcomes, emphasizing the inherent unpredictability in certain systems and the limitations of control and prediction.
🔄 The Experiment of Chaos: Diverging Numbers and Randomness
The speaker engages the audience in a simple numerical experiment to demonstrate the concept of chaos theory. By doubling numbers below 50 or subtracting from 100 and doubling for numbers above 50, the audience experiences how quickly numbers diverge from their initial values. This activity highlights the unpredictable nature of chaotic systems and the rapid spread of outcomes from similar starting points.
🌐 The Yin and Yang of Chaos and Order in Life
The speaker reflects on the personal implications of chaos theory, drawing a parallel between the concepts of Yin and Yang to represent order and disorder, respectively. They discuss the importance of finding a balance between controlling aspects of life that are important and embracing chaos for less critical matters. The speaker shares personal anecdotes to illustrate how understanding chaos theory has influenced their approach to life and relationships.
🛰️ Embracing Complexity and the Work of Kolmogorov
In the final paragraph, the speaker introduces the concept of complexity through the work of Kolmogorov, who defined a pattern's complexity as the length of the shortest description needed to produce it. The speaker suggests that finding concise yet detailed descriptions of complex phenomena is a key goal in science and understanding the world. They encourage younger researchers to seek these descriptions as a way to capture and make sense of complexity.
Mindmap
Keywords
💡Applied Mathematics
💡Statistical Thinking
💡Effect Size
💡Chaos Theory
💡Cellular Automata
💡Ecological Interactions
💡Correlation vs. Causation
💡Ronald Fisher
💡Interactive Thinking
💡Complexity
Highlights
The speaker, an applied mathematician, emphasizes the secondary role of calculations in understanding the world through mathematical tools.
The speaker's interest in mathematics stems from a desire to understand the world around them, rather than a passion for calculations.
The talk is structured around four different ways of thinking: statistical, interactive, chaotic, and complex.
Ronald Fisher's work in experimental design and combinatorics is highlighted, showing the power of using mathematics to solve real-world problems.
The story of Fisher's tea party experiment demonstrates the application of combinatorics in a light-hearted scenario.
The speaker discusses the limits of statistics, using the example of 'grit' and its minimal explanatory power in predicting success.
The importance of context in statistics is underscored, noting that correlation does not imply causation.
Alfred J. Lotka's introduction of interactive thinking through his work on ecological interactions and differential equations is presented.
The concept of chaos theory is introduced through the work of Margaret Hamilton and Edward Lorenz, emphasizing the unpredictability in systems.
The 'butterfly effect' in chaos theory is illustrated through a simple mathematical process showing how small differences can lead to vastly different outcomes.
The speaker uses football as an analogy to explain the balance between control (order) and randomness (chaos) in various aspects of life.
The importance of understanding the difference between statistical significance and practical significance is discussed.
The speaker's personal journey in mathematics and science, including working on a wide range of different areas, is shared.
The application of mathematical models in understanding social behaviors, such as applause in a group, is demonstrated.
The limitations of using mathematics to explain everything are discussed, using Lotka's unsuccessful attempt to create a 'Grand Theory of Everything' as an example.
The final way of thinking, complexity, is introduced with a cellular automata model showcasing how simple rules can create complex patterns.
The concept of complexity is related to the shortest description that can produce a pattern, as defined by Kolmogorov.
Transcripts
[Music]
thank you
thank you very much for the lovely
introduction and and being allowed to
come here for the third time it's a real
privilege to come and and talk to you
here I worked earlier in Oxford
University I think I left here about
2005 so it's very nice for to walk
around and all these old memories come
back so it's lovely to be here so what
am I going to talk about today well for
me
and this this was a quote that Daryl
took and used actually in advertising
this talk and I hadn't thought so much
about it but I wanted to to lift up
again because for me as an applied
mathematician the calculations are
secondary now I'm quite good at maths
I'm quite good at calculating things and
doing the manipulations required and I
suppose I teach it so I have to be
reasonably good at it but that's never
been the thing that motivates me first
I'm not one of these people who likes to
sit and do long calculations or well
maybe I like it a little bit but not a
great deal not as much as many of my
pure mathematics colleagues the reason I
got interested in mathematics and this
really came from an early age the reason
I got interested was I wanted to
understand the world around me and I
felt that mathematics was the toolkit
which I could use to get that
understanding
and there that's what we're going to
look at today that's the story I'm going
to tell I'm going to try and give you an
insight
into how my own thinking Works in four
different stages looking at statistical
interactive chaotic and complex thinking
and I'm going to illustrate it with
stories some of them are going to come
from football because Daryl has told me
that uh I'm very popular in my football
talks I thought I'd throw in a little
bit of football for you
some of it is going to come from other
parts of science so I'm an active
scientist working on a large range of
different areas
and some of it also comes from my
personal life so how I use mathematics
to think about the types of social
problems that I encounter every day how
how I interact with people when things
go wrong how I can find a solution for
them so I'm going to take all of those
three different branches Science
Football and our personal lives and use
them to illustrate these four different
ways of thinking
so the first way of thinking
we have is statistical
and when I did this I also went back in
time and really really Applied
Mathematics is just a little bit more
than 100 years old really the the things
that we use today so I went back in time
and I started with various
historical figures who built these
things
and this is Ronald Fisher this is him in
1912 when he was a student at Cambridge
University
and Fisher
was an incredibly arrogant young man he
believed that he was smarter than
everybody around him and at school he
was smarter than everybody around him
and he went to Cambridge University
which was this is going to sound
controversial but at the time if you
wanted to study mathematics Cambridge
University was the best place to study
mathematics in the world and he went
there and he found that he was pretty
much smarter than all the other students
and he also thought that he was smarter
than pretty much all of the professors
and so you can imagine Fisher he was
sitting in his room a few weeks before
the tripos exams which of course he aced
he was sitting in the room not studying
for the exams but he was trying to work
out how the mathematics he was using was
coupled to reality he felt that when the
the people who taught in mathematics the
professors who taught him when they used
it they didn't see the coupling between
what they were proving and the results
they had and how that actually could be
used and that's what he wanted to find
he was sitting looking for that solution
and it didn't go well for him right he
wrote an article nobody read it nobody
was interested in his ideas and he ended
up pretty much in the wilderness he
wanted to fight in World War One in
1914. he couldn't because he was too
short-sighted and he ended up buying a
farm and trying to run this Farm which
he was absolutely terrible at the
terrible at he just couldn't manage to
he wasn't very good at working hard he
was good at having theoretical ideas but
not getting anything anything done but
he was rescued and he was rescued by a
guy called Sir John Russell and Sir John
Russell
he ran rothemsted experimental station
and he actually said he was looking for
an oddball mathematician to look at all
their experimental results and so he
recruited Fisher and here is Fisher
pictured on the left-hand side this is
how he could be typically be seen at
rothenstead he would be sort of puffing
smoke and explaining ideas to people
and in this picture he's at a tea party
now Sir John Russell
um he instigated the tea party because
at rothamstead in 1919
um they started to have women working
there and he felt that if they had lady
employees he didn't really know what to
do with them but he knew one thing they
needed to drink tea and so he made he
made tea for these uh so they started
having a tea party for these um uh for
the woman ostentation essentially but he
went on oh well everybody had the tea in
the end and they became a very central
part of what was done at rothenstead
now at one of these tea parties
um though there was a a Dr Muriel
Bristol who was one of the
experimentalists and one of the people
who did the studies on crops and Fisher
was about to serve her tea and she said
stop
I need to have my milk first and Fisher
as usual in his arrogant way he just he
said nonsense this can't be true it
doesn't matter if you have your milk
first in your tea you can have your milk
afterwards you know it all mixes up
together there's no difference if you
have your milk first or if you have your
milk afterwards
and so and he wasn't he wouldn't just
stop there he wasn't happy until he'd
done an experiment and tested if Dr
Muriel Bristol could tell the difference
between uh if she had her milk first or
her tea first in her tea
and so he set up an experiment to do the
test or he got his some of his
colleagues they suggested various
methods how they might do that
and
I'm now going to ask you actually so
we want to test if Dr Muriel Bristol can
tell the difference between if the milk
goes into her tea first or if the tea is
put in before the milk
and I'm going to allow you to consider
three different ways or two different
ways plus another alternative for doing
this test so the first is to offer her a
pairwise challenge we offer her a milk
cup and a non-milk cup
and then we maybe randomize these in
different ways and we have four tests
for her
the other one is we present her with a
tray with milk and non-milk tea on the
tray and we ask her to identify the ones
that will milk first and the ones that
were not milk first okay so hands up who
thinks that the pairwise test is the
best way to do this
we've got a few people at the back hands
up who thinks that the milk first tea
first tray is the best
you can do better than this okay now
we've got a few more hands up so I'm
taking the rest of you uh going for
option C and you're also thinking that
there's no difference between these two
methods that they're both the same
okay so I'm not sure if that's okay
hands up if you do think there's no
difference between the two methods
I think we've got a little bit of a
majority there for for option b okay so
let's have a look at this and this is
the working that um
that fisher did
so if you've got a pairwise challenge
option A
when you're setting up the experiment
and this is key to how Fischer was
thinking when you're setting up the
experiment
you have 16 different ways you can
organize the cups so the black one the
black circle there is is T first the
white circle is milk first and you have
four different pairs and there's 16
different ways of arranging those pairs
so that's two to the power of 4 is 16
and the probability and this is the the
key here the probability that Muriel
Bristol gets this right is 1 in 16. she
has to in order to get them all right
well if she if she can't tell the
difference the probability she gets them
all right is 1 in 16.
now if we do hit option b
the one that you liked and this is going
to prove to be the better choice
there are eight places to put out the
first cup seven places for the second
cup these are the milk cups
um six places for the third cup and five
places for the fourth cup and you've put
them out at random and then you fill up
with the these would be the the milk
ones then you fill up with the non-milt
ones and then you can also think of the
ordering of the cups there's four times
three times two times one ways to do it
and using combinatorics you can find out
well there's 70 ways of placing the cups
and if you don't believe the math you
can sit and write them all out and I got
I got a little yeah I told you I don't
like calculating so I got a little bit
bored before I wrote all of the
different ways you can arrange the cups
out but you can arrange them in these
different ways and so if Muriel Bristol
can't tell the difference the
probability of getting all four right is
now only one in in seventy so the the
this test is the better way to do it and
this I think is a perfect example of
using a nice piece of mathematics in the
form of combinatorics and that's what
Fisher did he used different parts of
combinatorics to solve a problem in
experimental design
and he went on to write a book
which became is still a sort of handbook
used today of how you design different
experiments this is a slightly this is
his Latin Square design which is a
different design than the randomized
design we've just talked about but he
could then from there start to spread
his statistical ideas
and there's just an incredible power I
think in being able to think about the
correct way to do an experiment or the
correct way to analyze data
now I like to so I've as Daryl said I I
have worked on football and I've worked
in mathematics and football and this has
taken me on some amazing Journeys and
it's very nice to for example I was I
had a um thing with Gary Neville so I
like to show off now about my sort of
football and contacts I think Gary
Neville is the most famous uh footballer
I've I've met on my journey so I just
wanted to sort of throw that in there
um and what's what's invariably happens
when I talk to footballers or former
footballers or coaches about about
mathematics and football
is that they have this thing where they
say oh well you know numbers can tell us
something numbers can tell us a few
things but you can't measure a player's
attitude you can't measure and so Gary
said that when we did this thing
together he said oh you can't measure if
a team goes a goal down
you can't measure the player who really
gets everyone going and really rallies
the team and gets them going again
and I was sitting there thinking yeah
actually that's exactly what you can
measure with Statistics and so a few
days after
um Gary Gary said this I sent him an
analysis where we analyzed exactly that
we look to see what happens when a team
um so when a team can see the goal so
this is Trent Alexander Arnold in a game
in the 21-22 season and the line the
central line here the central dotted
line
which is highlighted
is when they conceded a second goal
against United they lost the game 2-1 in
the end and just after half time they
conceded a second goal
and the squiggly line that's going up
and down that's Trent Alexander Arnold's
performance on the ball
and
as time goes on you can see that he's
getting better and better he's he's
actually producing lots of good passes
for his teammates
and then the goal goes in dotted line
and then you see his performance drops
back down again and so in this
particular case if we do the Gary so we
ended up calling this the Gary Neville
statistic so if we do the Gary Neville
statistic on this his performance goes
down after they concede a goal compared
to his performance in the 15 minutes
before they concede a goal so it's a
measurable statistic and in this way you
can for example we looked at the top
Strikers in 2122 and it was quite
interesting because you might call
um Jamie vardy a player who's got a lot
of attitude or character or something
like that and then it turns out that's
what we got when we measured it when
when
um when Jamie vardy's team went down
they went down a little bit more often
than some of the other teams but he got
better 29 of the times worse 13 of the
times and his performance was the same
about five of the times and you can see
yeah there's a ranking of the top five
players for That season in these
different situations
and it's very nice I've put in
parenthesis here because we used
precisely one of Ronald Fisher's tests
Fisher's exact test in order to test if
these players were statistically better
when they're when their team went down
or not
and so there's all types of ways in
which we can use statistics another
um example
and now we're kind of moving over to the
edges of the limits of statistics
and I think that this is a very
important point because
while while I think Gary is wrong in
that you can't measure attitude at all
he's right in another way because you
can't measure everything you can measure
certain aspects of how a player gets
better it's one piece of information you
have but you can't measure everything
and this is from a study uh from a TED
Talk and during the writing of the book
I watched the 25 most popular TED Talks
because I was very interested how they
use statistics in order to assess the
validity of claims in TED Talks and this
was a talk by Angela Duckworth and she
said that Grit and grit is the idea of
determination how determined you are to
succeed the grit is the strongest
predictor of success and it comes from a
study that she conducted
um together with some colleagues and
what they did is they looked at Ivy
League undergraduates I think at Yale
they looked at U.S military cadets and
they looked at people who were competing
in a spelling bee and before they
started doing these activities they
asked them questions about how if you
start something do you always see it
through it those types of questions a
series of 12 questions about if they
were determined gritty types of people
and they found that grit this the
answers that people gave in those
questions were some of the biggest
predictors of success and that's what
she said in the in the talk and that
sounds very impressive a bit like me
trying to persuade Gary Gary Neville
that we can we can measure
um attitude in football players
but
if you look a little bit more closely at
this the actual study as opposed to the
yes we shouldn't measure we shouldn't
measure success on how many times people
have watched the YouTube video because
this YouTube video has been watched 25
million times
and it's it's not Angela Duckworth who
wrote the headline on it but
her paper
reveals quite clearly and as she doesn't
try to hide this in any way at all the
grit just explains four percent of the
variance between people
now four percent of the variance how
much is that it isn't
it doesn't mean that only four percent
of people are explained by this I'm
going to try and show you what four
percent of the variants look like four
percent of the variance looks like this
so
if you measure grit on the scale one to
five on the bottom here and you look at
grade point averages for example now
this isn't real data this is data I've
just made up in order to illustrate what
four percent of the variance looks like
four percent of the variants would have
some type of relationship a bit like
that
and you can see if you squint carefully
I can't see it from this angle but I
think maybe you can squint and see this
there is a kind of increasing Trend
there between grit and grade point
average but you also see that some
people who yeah and there's there's lots
of people who are very gritty and are
successful and there's people who aren't
gritty and unsuccessful there's also
lots of people who are very gritty and
don't get a high grade and there's lots
of people who aren't very gritty and do
get a high grade and so when you're
interpreting this you shouldn't confuse
so I often say that you shouldn't
confuse the forest for the tree you're a
tree right every person in this in this
room is a tree so if we tested all of
your grittiness and your success in life
we'd find some kind of relationship like
this but it wouldn't mean that you
necessarily as an individual
um had this relationship between grit so
if you're not a gritty person if you
never see through any projects you start
you don't need to worry at all you're
going to be absolutely fine there's lots
and lots of ways in which your life can
succeed
and this what I've tried to do here and
I'm not sure if I've got the art quite
right here but the the arc I want to
describe here is I I want to start by
saying statistics is very powerful I can
show off to Gary Neville about it but
then at some point statistics doesn't
actually give you all of the answers
right and that's very well Illustrated
if we actually go back to Ronald Fisher
this rather arrogant young undergraduate
student because Ronald Fisher also has
another scientist story
he was from a very early age and this
picture was taken in 1912
very interested in Eugenics and he had
this idea that we needed to breed people
to be more like him like to be more
clever and smart and good at maths and
so on and he campaigned all the way up
to the war I think after the war he kept
more quiet about this
but all the way up to the war he
campaigned for for example sterilizing
people who were considered feeble-minded
and this is this is of course horrible
and I mean it's a horrible thing to
think about but not only is it like
morally repugnant it's also
scientifically wrong they couldn't find
any kind of Gene for single uh for
feeble-mindedness there is no
correlation between or there's a very
weak correlation or no correlation at
all between feeble-mindedness in one
generation in mothers and in their
daughters so this was a relationship
that was scientifically dubious that he
continued to press forward and the
reason he was successful is because or a
reason he was successful in pushing us
forward is is thing forward was that he
would use statistics in order to sort of
attack his opponents he would call them
all stupid they didn't understand
statistics so he would actually use
statistics to undermine other people's
arguments in a really counterproductive
way he took a fake Theory and then used
statistics to defend it and he didn't
just do this once after the war when
he'd given up on on Eugenics or at least
stopped talking about it
he then did the same thing on smoking so
as we saw in the first picture he was a
very keen smoker
and for him there was just no
possibility that smoking could cause
cancer
and so he spent a lot of time
investigating very narrow areas of
science doing his statistics on that and
trying to convince people that smoking
didn't cause cancer and I don't know
the effect that this research had but
certainly you've got one of the leading
statisticians in the world who's
defending this position for a long long
time
and that
illustrates a lot of why statistics is a
limited approach so I've written down a
few a few sort of bullet points here so
statistical thinking doesn't provide all
the answers
um
one problem is and I I didn't really get
into this but I mean what a dick he is
right
I mean why do you need to test if she
you know can tell the difference between
the milk first everyone was very happy
they were all just enjoying their tea
party and suddenly he's doing a
statistical day I mean you know he's an
idiot so so that's one thing and and I
really think you know we joke about that
but we see it at work all the time you
know that's always being told that we
should be there should be statistical
tests and metrics made about us and
things like that and we don't need as
much quantification as we have
then there's the effect size thing and
we sometimes talk about statistical
significance you can have statistical
significance but still have a very small
effect size and that's the confusing the
tree in the forest many non-gritty
individuals are successful in life
context is always important so just
because a player is more active when
their team goes a goal down does not
imply the team plays better just because
Ronaldo demands that everyone gives him
the ball when they go a goal down or
Jamie vardy demands that he gets the
ball that doesn't mean that the team
actually does better as a result of that
so it's very important to think about
the context of these types of things and
then as we all know correlation and
causation they're not the same thing
but that leads us to the to the next
step we need to think about ways to
tease out causation we want to be able
to tease out our understanding of the
world one thing causes another
and it brings us very nicely on to
interactive thinking and that's the next
step on from statistical thinking
and I have another hero I can reassure
you that this hero is not going to turn
out to be a raving racist who bullies
all of his co-workers so um there are a
few mathematicians who haven't done that
in their lives just a few of them but
they're around
and this is Alfred J Locker
um and he was originally a chemist
and he started his uh he's he was
originally from Poland but he did his
undergraduate degree in Birmingham
and
I I really like his story because
he started working in this chemistry lab
and he was kind of disappointed with
what he saw when he was doing his
experiments I mean it's a long time
since I did chemistry at school but it
can be a little bit disappointed you
know you get the acid and The Alkali you
mix them together and there's some salt
and water it's not always the most
exciting thing you've ever seen and so
he'd see these stable stable reactions
just come to equilibrium
but in the evening he was reading all of
these books like Charles Darwin's book
and he was thinking about biology and
just all of the exciting patterns we see
all the motion and movement of animals
or the firing of our brains everything
that happens in society and he was
thinking why can't chemistry produce
anything like that I mean we know that
chemistry must be the underlying
building block of it but it's not
something we see we can't make that
relationship together
and the way he solved the problem was he
basically cheated and he he did the
following thing so if
we've all done this in school
we've balanced equations right through a
balanced reactions and if you've got uh
two two h two well you've got four
hydrogen atoms two oxygen atoms and they
react to get to make two water molecules
so that's a a standard chemical reaction
and the important Point here is that
this is balanced so there's four
hydrogens and two oxygens on the left
and there's four hydrogens and two
oxygens on the right
but what loter said is well I'll just
forget about that balance thing even
though I can't find a chemical reaction
that is unbalanced I'll just think in my
head I'll do a thought experiment and
this is where it's lovely with
mathematics I'll do a thought experiment
where I ignore the fact that I can't
balance my reactions and so he take he
wrote down these equations he said that
imagine an r that becomes two r's and
imagine an R plus an F which becomes two
F's and you can see that these aren't
balanced there's one r on the left hand
side of the first one two r's on the
right yeah there's you can see that
they're just not balanced and I've
written down here below the way you can
think about these things are rabbits and
foxes so
and and they're not it's not a realistic
model of rabbits and foxes you've got to
think of the idea of one little rabbit
hopping around and suddenly there's two
little rabbits hopping around we know
it's a little bit more complicated than
that but we'll start with that idea and
then a fox comes and eats a rabbit and
then it makes another Fox that's that's
what the model says it gives a a rough
idea of how ecological interactions work
and he took that and he wrote down
differential equations
I wanted to put in a few of these
equations here to give you a feeling for
how they work
the the equations on the left here one
of them I I'm not going to get you to
understand every detail of the equation
but what I want to give you a feeling
for is on the left is the rate of change
of the rabbits and the foxes so Dr by DT
is a rate of change of the rabbits DF by
DT is the rate of change of the foxes
and on the right are the things which
cause that change so I mentioned here
that we want to get causation into our
equations so on the right of the things
which cause that change and rabbits
increase when there's lots of other
rabbits they have lots of little bunny
rabbits and then they are eaten by the
foxes so the more F they are if we look
at this brf term that's the rate at
which the
um the rabbits are eaten by the foxes
and then when we come down that we have
the opposite relationship for for The
Foxes the foxes grow when there's more
rabbits and then they eventually die off
um of old age there's nothing which
hunts the foxes in in this in this
scenario
now again I'm not going to solve all of
these equations but I did want to
mention a little bit about how you can
think about them and understand them
and on the right here actually learned
this from Philip Maney when I was here
in in Oxford about these types of method
but he had a very lovely method
a professor of mathematical biology here
for solving these equations without
solving them
so you can split up the plane of foxes
and rabbits and you can identify a point
on that plane and look to see do the
rabbits increase or do the fox's
increase so in the bottom left-hand
Corner the foxes go down because there's
not enough rabbits to eat but the
rabbits go up because there's not enough
foxes eating them and that goes on until
there's a sufficient number of rabbits
and then the foxes start increasing so
in the bottom right hand corner this
Arrow points up and if that Arrow points
up well then the foxes increase
and when the foxes increase they start
to eat the rabbits and the rabbits go
down and you start to get this circling
round and round of foxes and rabbits and
you could do basically show this without
explicitly solving the equations that
there's going to be Cycles round and
round of these foxes and rabbits we're
going to get this interesting
interaction and if we look at it over
time
um we have these periodic oscillations
of the foxes and the rabbits
now this whole way of thinking
which latke introduced turned out to be
useful in all sorts of situations now
the one thing we should we try not to
mention but can't be unmentioned in any
mathematical modeling thing we try not
to mention this but but it's also used
in pandemic modeling and you've all
heard about these epidemic curves and
our values and so on
um but that's the same thing that a
susceptible plus an infective becomes
too infectives and that's an example of
one of lotka's unbalanced equations
which allows us to describe how an
epidemic reaction epidemic will spread
through a group of people
now
I'm not going to go into as I said I
don't want to talk too much about
epidemics but this example I really love
so I'm going to talk about this example
this is a study we did I did together
with some colleagues and this is a very
cruel experience it's not it's not it's
not that cruel but we have we had a
group of undergraduate students we got a
third a third year undergraduate student
to give a seminar to first-year
undergraduate students and we told the
first year undergraduate students you
know remember to give a round of
applause after the seminar just to show
your appreciation and then what we were
really interested in was how people
applaud it what are the cues that make
people applaud and we could see that
people start applauding when other
people around them start applauding and
you basically have an epidemic of
Applause and that's what the first green
curve shows that's the number of people
clapping that's the spread of the
Applause virus going through the group
but then
and this doesn't happen in real diseases
you also have a social recovery
so when people stop clapping they look
around and they hear the other people
have stopped clapping and it was
actually a little bit like when Daryl
left the room just now there was a sort
of start signal there that there might
be something going on that we might be
about to start and you all started to go
down in volume and suddenly everyone was
quiet
and that's the type of social effect
we're very aware of all kinds of small
social details and these spread through
us in a group and the the conclusion
that what I love about the recovered
thing is because we have that social
recovery so
and I try to always remember this that
if at the end of a talk you've given or
a presentation if the Clapping goes on
for a long time that's not because you
gave a good talk it's just because your
audience aren't particularly coordinated
so they couldn't they could they
couldn't manage to to stop together
and I think that's what that's what I
would encourage you to think about and I
said that I also wondering about the
personal aspects of this
it's nice just to sit sometimes and
think about the social reactions that
you have in your life and how they work
and I've written down a few of them I
haven't told you what they are yet so
I'll tell you what what they are the top
the top one
that I was imagining this is a person p
and then it's plus an O this is a sofa
that the person has outside the house
and so we have P plus o goes to P plus o
if you're just one person and you've got
a sofa outside the house you're going to
still be one person and you can't get
the sofa into the house so what you have
to do and this is the bottom equation is
you have to get a friend and so this
bottom one is 2p 2 people plus a sofa
that's outside the house is still two
people you've still got your people
afterwards but you've moved your sofa
into the living room and you can write
those social interactions for every type
of activity the one on the right here
the ones on the right here I was
thinking about smiling so if you're a
smiley person why and you meet a
non-smiley person whose ex then if you
smile then hopefully they become a
smiley person too but that's not always
the case sometimes you know you don't
always start smiling because somebody
else is smiling they might just be an
idiot who's just smiling for no reason
at all what happens most often actually
in human social interactions is they as
you believe the following Equator the
one the equation at the bottom this is
the most common equation I think that
describes human behavior and that's that
a non-smiling person plus two smiley
people will become three Smiley people
because then they're convinced that
actually must be something to smile at
and I use that a lot in my thinking if
if I'm thinking about how to in the book
I take an example of if I'm trying to
get a group if a group of friends are
trying to get going with some kind of
healthy activity maybe they spend a lot
of time sitting in the pub together
don't really go out and do any exercise
together it's not enough for one of them
to become a why to to try and get them
going that you have to have two of them
and they have to have a really sustained
effort and over time then you get this
Tipping Point effect where everybody
starts to move over and starts it starts
to engage in the healthy activity so
those are the types of things you have
to think about what type of chemical
reaction what type of social reaction am
I involved in
and that's been a lot of to be honest
this has taken up a lot of my adult life
is studying these types of things and to
give you a little bit of a flavor of the
sorts of things we do
this is just to give a sort of overall
representation but
um when we modeled fish for example we
would create models which described
their social interactions described how
one fish turned left if another fish
turned left if another fish turned right
and so on then we were building the top
there it's a mathematical model we've
built a fish movement and so on so we'd
show that these simple rules of
interaction would produce their
Collective Behavior then we'd study also
the the movement of individual fish
that's the colored idea at the the
bottom then we'd actually frighten all
of the fish and we'd look at how they
made an escape wave we we'd measure that
escape wave and then we'd use models to
understand that escape wave and it's a
very powerful way of thinking throughout
science that you can build up these
models of interaction you compare them
to reality and build a better and better
understanding of fish Behavior and we do
a lot of similar things in football so
this is an example of an attacking run
by Marcus rashford and the model that we
build for these types of situations
this red area here shows the territory
that he controls
and this is a physics-based model where
we say how far how fast can he run where
can he get to and we can actually
describe what area he he occupies and
also the value of that area so How
likely is he is getting a pass at that
particular Point going to lead to a goal
and that allows us to actually Scout
players based on their runs and it even
allows us to scout runs where they don't
get the ball so in this example
where
we're interested in Luke Shaw here and
he's doing a run here on the left
and he doesn't get the pass he'd love to
have this pass but he doesn't get it but
we can still measure the value that that
pass created so you can look at these
counterfactual situations for
um for for football players
and this is a very powerful method the
interactive way of thinking allows us to
build up our understanding of systems it
doesn't have the same kind of
I I suppose the statistics has a sort of
more of a grounding feeling to it this
we use our imagination much more we try
to use our imagination to increase our
understanding and then build
mathematical models to test that
understanding
now I wanted to go back to lotka because
um
there are also limits to this way of
thinking and of course I wouldn't have
four if we'd if we'd solved it all now
so there's there's limits to this and
there's limits were limits that lotker
himself hit he wrote a book called
elements of physical biology and
he's one of these these mathematicians
who and this happens a lot to us
is we sort of just get carried away and
we believe that we can just explain
everything with mathematics that there's
nothing that we can't explain and so he
built models he built models of
Consciousness he built models of of our
whole society and he believed that all
of them could be understood using his
reaction Dynamics and it really
yeah he didn't I mean and this was I
suppose it was a very Valiant effort
this is in 1922 he finished his his
Magnus Opus so he didn't even have a
computer or anything to simulate these
types of models on but he never really
succeeded in pinning down one essential
way in which you should approach all
sorts of problems he he ended up kind of
split between lots and lots of different
small things and that I can personally I
can relate to that very well because
that tends to be how I work with lots of
problems there's lots of different
methods and you're doing lots and lots
of small different things in order to
get your solution the day of day of an
applied mathematician isn't it's not
like these theoretical physicists you
know they have like this beautiful
Theory of Everything and they can come
here and just say oh it's all this and
wow but no it's not like that it's more
that you're sort of tinkering around
with small different problems in lots
and lots of different ways so lotska
never found his Grand Theory of
Everything using interactive thinking
and one of the reason
one of the reasons he never found is
Grand theory was because he didn't know
about chaos
which is the third way of thinking
now to introduce chaos
I'm going to go to another mathematical
hero
this is Margaret Hamilton and
she was also like the other two we've
met prodigious at school
very talented undergraduate student she
wanted to go on and do a PhD in pure
mathematics
but her husband also wanted to do a PhD
and this is now in the 1960s and she
ended up moving to Boston and she also
had to get a job she had a daughter to
support a husband to support and so she
had to get a job to support them but the
job that she got was programming this
machine the lgp 30 and she fell
immediately in love with this Computing
machine because
she hated making mistakes she hated
errors whenever she calculated anything
she calculated it perfectly and now she
found that she could actually program
this first computer to do the same
calculations and she got access to this
computer because she was working in the
lab of a person called Edward Lorenz who
was a professor of meteorology but also
with a mathematical background there are
a lot of mathematicians in this talk so
um and he
he wanted to predict the weather he
wanted to predict the future weather
based on temperature pressure and so on
in different areas could he predict the
weather into the future and she started
writing a computer code to do this and
this involved writing and doing Punch
Cards at the time and she'd run her
computer code
and they did this one thing is that they
simulated they simulated the weather one
day and the next day they decided to
check their results by simulating making
the exact same simulation on the
computer to check that everything worked
but they found on the second day they
got a different result than on the first
day
and Margaret was distraught because she
didn't like making mistakes she didn't
want to think there was a mistake in her
code but she started going through the
code and there was no errors in the code
and what they found was that the output
of the simulation was in six decimal
places
well the input they put into it was in
three decimal places
so there was an error in the input in
the fourth decimal place and this meant
that the weather simulation made
completely different predictions in the
future going like a few 10 days into the
future in the simulated World it made
completely different predictions in the
future and I didn't mention that this
was a system of 14 differential
equations that she solved we've moved on
from lockter and Voltaren too so she
solved these 14 differential equations
and they make just this small error in
the value you put in makes a massive
difference and that is the first
indication of the butterfly of chaos
which many of you will be familiar with
and Lorenz went on he worked with
um I say Lorenz went on Margaret
Hamilton we're going to find also went
on to do some very impressive things but
Lorenz went on with the help of Ellen
Fetter who replaced Margaret Hamilton as
his programmer to produce what we now
know as the we often think of this
picture I think or I think of it as
being the butterfly of chaos and what it
illustrates is if you do start with two
points very close to each other we've
moved now down from 14 Dimensions to
three dimensions again if you start with
two points very close together and they
start to diverge
they'll move around on the same
attractor on this shape that we have
here but they'll never come close or
they might come close to each other for
a short amount of time but they'll then
live their own life and so when we move
from two Dimensions up to three we have
this chaotic movement where things never
come back to the same place again
I think I think I think we're going to
do my experiment okay so I think I think
we're going to do the experiment and
then I'll I'll boot out something else
because you've listened to me patiently
for 50 minutes so you have to get to do
the experiment okay so here we're going
to we're going to do this I want you to
work in pairs
I want you one of you should think of it
so it's a look at the person next to you
and you might be a new friend that
you've got today
um or it might be somebody that you came
with
and then I think I want one of you to
think of a number between 1 and 99
then you tell that number to the other
person
and the other person follows the follow
the the following rules so if a number
is less than 50 double it and this is
the new number I chose 42 because you
can never have a math talk without 42 in
it so 42 times 2 is 84 and so that's all
you do you just double the number now if
the number is greater than 50 take it
away from 100 and then double it to get
the new number so if I have 84 then I
have 100 minus 84 is 16 times 2 is 32.
now say the new number to your partner
and they repeat step one and two so
we'll do this for um do this with either
with the person you came with or
somebody who's nearby to you we'll do
this for about two minutes and then
we'll see where we get to
I think you've done it very nicely done
I can see the I can see I hear the
murmur of numbers everywhere very very
lovely
um
I'm I'm not going to get you all to come
up here and present your results I just
wanted to give you get get you to get a
feeling of this type of process
um you're not generating purely chaotic
numbers when you do this if you'd
started with 20 for example you would
have found yourself cycling around quite
quickly but if you started with a number
that's not divisible by five you would
have probably been on quite a long
trajectory through different numbers
and the point I want to make about this
process is the following is that close
together numbers very quickly diverge so
if one group over there had started with
13 and another group over here had
started with 14
by the end of just this short period
where you got to say the numbers to each
other you would have been on very
different numbers so you have 13 26 52
96 8 16 32 64. 14 28 28 is not so far
from 26 56 52 they're still together
88.96 are starting to get away from each
other but the big jump is now one of
them sort of goes over the threshold and
one of them doesn't so you've got 8 and
24.
1648 and then you've got 32 and 96 and
64 and 8. So within a few steps these
numbers have diverged quite far from
each other I don't know if any any of
you took decimal numbers
um you didn't think of that but if you
do take decimal numbers then you get
true chaos from this thing for almost
any real number you choose you will get
if you take plus 0.1 in this case only
so this is 14.1 compared to 14.2 you
start to they're together for a few
steps but after about seven eight nine
ten they go apart they come a little bit
together again for a while but then they
diverge and you've got very different
paths for those two numbers
and we often illustrate this
um using something called a cobweb
diagram so the idea here is you take the
number from One Step and the previous
number might be around 20 for example it
will jump up to be around 40 then it
will go to 80 then it will crash down to
around 20 again and then it will start
to move around everywhere on this
and one of the reasons I wanted you to
do this experiment is what was being
what I could hear from your perspective
was a Mumble of uniform distributed
random numbers you were essentially
going through a lot of integers and
everywhere in the room there was a
different point in this distribution you
basically had this uniform distribution
of numbers that were sort of kind of
coming up to me and I think it's really
lovely to think of that that you're all
doing the same process you're all doing
exactly the same thing yet you kind of
have this hum this distribution this
background of very different numbers
and
that is the butterfly of chaos and and
for me it illustrates
there's an important Point here
I I think chaos is wonderful
Margaret Hamilton she hated chaos right
and she she left Lorenzo's lab and she'd
learned a valuable lesson from working
on these weather simulations and it was
that she doubled down and made even
fewer errors and she wanted to work in
the most extreme
conditions possible where you couldn't
make errors and so she got a job for
NASA
and she became the head of the software
engineering which created the software
that sent that was on the Apollo moon
mission and so she was she created the
software that the astronauts used to
tell them how to to do navigational
decisions to control the thrusters to um
uh to update to know where the position
of the ship was
and she was in the control room when
they when they made the actual landing
on the moon and so I see this as a
situation where you sort of have to
choose right in if you're if you're
going to control something because of
chaos if there's something you really
care about or there's something that's
really important then you have to treat
it like Margaret Hamilton does you had
to treat it like the moon landing
there's no error there's no room for any
type of error
but you can't have control over
everything so I often think about this
in football because
there's always going to be butterflies
in other situations so here this isn't
the uniform distribution as you
generated but it's the poisson
distribution
there is lots of other situations
football being one of them where we just