Four Ways of Thinking: Statistical, Interactive, Chaotic and Complex - David Sumpter

Oxford Mathematics
4 Oct 202356:07

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

00:00

📚 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.

05:00

🎲 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.

10:01

📊 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.

15:01

🔍 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.

20:01

🧬 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.

25:01

🤖 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.

30:03

🦊 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.

35:03

🐟 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.

40:04

🌀 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.

45:05

🔄 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.

50:08

🌐 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.

55:09

🛰️ 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

Applied Mathematics refers to the use of mathematical methods to solve practical problems in science, engineering, and other disciplines. In the video, the speaker, an applied mathematician, emphasizes its utility as a 'toolkit' for understanding the world around us, rather than focusing solely on calculations and theoretical proofs. The concept is central to the video's theme, as it underpins the speaker's approach to various problems, from football statistics to social interactions.

💡Statistical Thinking

Statistical Thinking involves the application of statistical methods and theories to analyze data and draw inferences. The speaker discusses this concept as one of the four stages of his thinking process. It is exemplified in the script through historical references to Ronald Fisher, who contributed significantly to statistical methods, and through modern applications like analyzing football player performance and TED Talk claims, demonstrating its relevance to both historical and contemporary contexts.

💡Effect Size

Effect Size is a measure of the magnitude of a phenomenon or an outcome in statistical analysis. It is mentioned in the video to illustrate the limitation of statistical significance, where a result may be statistically significant but have a small effect size, meaning it does not have a large practical impact. The speaker uses the example of 'grit' and its impact on success to clarify this concept, noting that grit only explains four percent of the variance, a small effect size in the context of predicting success.

💡Chaos Theory

Chaos Theory is a branch of mathematics that deals with systems that are highly sensitive to initial conditions, meaning small changes can lead to significantly different outcomes. The speaker introduces this concept by discussing the work of Edward Lorenz and Margaret Hamilton, highlighting the 'butterfly effect' through a simple doubling experiment. The concept is integral to understanding the limits of predictability and control in various systems, including weather prediction and football match outcomes.

💡Cellular Automata

Cellular Automata are mathematical models used to simulate complex systems based on simple rules and local interactions. The speaker refers to a project by a master's student, Michael Hansen, who created a cellular automata model that generates complex patterns from simple rules. This concept is used to illustrate the emergence of complexity from simplicity, a key idea in understanding complex systems without needing to control every variable.

💡Ecological Interactions

Ecological Interactions refer to the relationships and influences between different organisms and their environment. The speaker uses the Lotka-Volterra equations, which model the predator-prey dynamics between foxes and rabbits, to explain how ecological interactions can be mathematically represented. This concept is crucial for understanding the balance and changes in ecosystems, as well as the broader theme of interactive thinking.

💡Correlation vs. Causation

Correlation vs. Causation is a fundamental statistical concept that distinguishes between two variables being associated with each other (correlation) and one variable causing changes in the other (causation). The speaker warns against confusing these two, using the example of a player's activity when their team goes a goal down in a football match. The concept is essential for understanding the limitations of statistical analysis and the need for careful interpretation of data.

💡Ronald Fisher

Ronald Fisher is a key figure in the development of modern statistical methods, known for his work in experimental design and statistical inference. The speaker discusses Fisher's contributions to statistics, including his innovative approach to experimental design and his development of various statistical tests. Fisher's work is foundational to the speaker's discussion of statistical thinking and its applications and limitations.

💡Interactive Thinking

Interactive Thinking is an approach that involves building models to understand the interactions within a system. The speaker describes this as a progression from statistical thinking, where one uses mathematical models to explore how different elements within a system influence each other. Examples from the script include modeling the behavior of football players and fish, illustrating how this approach can lead to a deeper understanding of complex systems.

💡Complexity

Complexity, in the context of the video, refers to the intricate patterns and behaviors that emerge from simple rules or interactions. The speaker discusses the concept of complexity in relation to cellular automata and the work of Kolmogorov, emphasizing that a pattern's complexity is tied to the shortest description that can reproduce it. This concept is central to the video's exploration of how simple interactions can lead to complex outcomes, which is a key theme in understanding various systems in the world.

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

play00:02

[Music]

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thank you

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thank you very much for the lovely

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introduction and and being allowed to

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come here for the third time it's a real

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privilege to come and and talk to you

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here I worked earlier in Oxford

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University I think I left here about

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2005 so it's very nice for to walk

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around and all these old memories come

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back so it's lovely to be here so what

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am I going to talk about today well for

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me

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and this this was a quote that Daryl

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took and used actually in advertising

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this talk and I hadn't thought so much

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about it but I wanted to to lift up

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again because for me as an applied

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mathematician the calculations are

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secondary now I'm quite good at maths

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I'm quite good at calculating things and

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doing the manipulations required and I

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suppose I teach it so I have to be

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reasonably good at it but that's never

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been the thing that motivates me first

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I'm not one of these people who likes to

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sit and do long calculations or well

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maybe I like it a little bit but not a

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great deal not as much as many of my

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pure mathematics colleagues the reason I

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got interested in mathematics and this

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really came from an early age the reason

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I got interested was I wanted to

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understand the world around me and I

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felt that mathematics was the toolkit

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which I could use to get that

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understanding

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and there that's what we're going to

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look at today that's the story I'm going

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to tell I'm going to try and give you an

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insight

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into how my own thinking Works in four

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different stages looking at statistical

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interactive chaotic and complex thinking

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and I'm going to illustrate it with

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stories some of them are going to come

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from football because Daryl has told me

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that uh I'm very popular in my football

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talks I thought I'd throw in a little

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bit of football for you

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some of it is going to come from other

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parts of science so I'm an active

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scientist working on a large range of

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different areas

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and some of it also comes from my

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personal life so how I use mathematics

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to think about the types of social

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problems that I encounter every day how

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how I interact with people when things

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go wrong how I can find a solution for

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them so I'm going to take all of those

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three different branches Science

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Football and our personal lives and use

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them to illustrate these four different

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ways of thinking

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so the first way of thinking

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we have is statistical

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and when I did this I also went back in

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time and really really Applied

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Mathematics is just a little bit more

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than 100 years old really the the things

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that we use today so I went back in time

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and I started with various

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historical figures who built these

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things

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and this is Ronald Fisher this is him in

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1912 when he was a student at Cambridge

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University

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and Fisher

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was an incredibly arrogant young man he

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believed that he was smarter than

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everybody around him and at school he

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was smarter than everybody around him

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and he went to Cambridge University

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which was this is going to sound

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controversial but at the time if you

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wanted to study mathematics Cambridge

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University was the best place to study

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mathematics in the world and he went

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there and he found that he was pretty

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much smarter than all the other students

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and he also thought that he was smarter

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than pretty much all of the professors

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and so you can imagine Fisher he was

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sitting in his room a few weeks before

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the tripos exams which of course he aced

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he was sitting in the room not studying

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for the exams but he was trying to work

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out how the mathematics he was using was

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coupled to reality he felt that when the

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the people who taught in mathematics the

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professors who taught him when they used

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it they didn't see the coupling between

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what they were proving and the results

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they had and how that actually could be

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used and that's what he wanted to find

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he was sitting looking for that solution

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and it didn't go well for him right he

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wrote an article nobody read it nobody

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was interested in his ideas and he ended

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up pretty much in the wilderness he

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wanted to fight in World War One in

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1914. he couldn't because he was too

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short-sighted and he ended up buying a

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farm and trying to run this Farm which

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he was absolutely terrible at the

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terrible at he just couldn't manage to

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he wasn't very good at working hard he

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was good at having theoretical ideas but

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not getting anything anything done but

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he was rescued and he was rescued by a

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guy called Sir John Russell and Sir John

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Russell

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he ran rothemsted experimental station

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and he actually said he was looking for

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an oddball mathematician to look at all

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their experimental results and so he

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recruited Fisher and here is Fisher

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pictured on the left-hand side this is

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how he could be typically be seen at

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rothenstead he would be sort of puffing

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smoke and explaining ideas to people

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and in this picture he's at a tea party

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now Sir John Russell

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um he instigated the tea party because

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at rothamstead in 1919

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um they started to have women working

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there and he felt that if they had lady

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employees he didn't really know what to

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do with them but he knew one thing they

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needed to drink tea and so he made he

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made tea for these uh so they started

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having a tea party for these um uh for

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the woman ostentation essentially but he

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went on oh well everybody had the tea in

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the end and they became a very central

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part of what was done at rothenstead

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now at one of these tea parties

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um though there was a a Dr Muriel

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Bristol who was one of the

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experimentalists and one of the people

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who did the studies on crops and Fisher

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was about to serve her tea and she said

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stop

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I need to have my milk first and Fisher

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as usual in his arrogant way he just he

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said nonsense this can't be true it

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doesn't matter if you have your milk

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first in your tea you can have your milk

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afterwards you know it all mixes up

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together there's no difference if you

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have your milk first or if you have your

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milk afterwards

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and so and he wasn't he wouldn't just

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stop there he wasn't happy until he'd

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done an experiment and tested if Dr

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Muriel Bristol could tell the difference

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between uh if she had her milk first or

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her tea first in her tea

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and so he set up an experiment to do the

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test or he got his some of his

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colleagues they suggested various

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methods how they might do that

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and

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I'm now going to ask you actually so

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we want to test if Dr Muriel Bristol can

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tell the difference between if the milk

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goes into her tea first or if the tea is

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put in before the milk

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and I'm going to allow you to consider

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three different ways or two different

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ways plus another alternative for doing

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this test so the first is to offer her a

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pairwise challenge we offer her a milk

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cup and a non-milk cup

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and then we maybe randomize these in

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different ways and we have four tests

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for her

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the other one is we present her with a

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tray with milk and non-milk tea on the

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tray and we ask her to identify the ones

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that will milk first and the ones that

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were not milk first okay so hands up who

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thinks that the pairwise test is the

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best way to do this

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we've got a few people at the back hands

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up who thinks that the milk first tea

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first tray is the best

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you can do better than this okay now

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we've got a few more hands up so I'm

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taking the rest of you uh going for

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option C and you're also thinking that

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there's no difference between these two

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methods that they're both the same

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okay so I'm not sure if that's okay

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hands up if you do think there's no

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difference between the two methods

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I think we've got a little bit of a

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majority there for for option b okay so

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let's have a look at this and this is

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the working that um

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that fisher did

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so if you've got a pairwise challenge

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option A

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when you're setting up the experiment

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and this is key to how Fischer was

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thinking when you're setting up the

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experiment

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you have 16 different ways you can

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organize the cups so the black one the

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black circle there is is T first the

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white circle is milk first and you have

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four different pairs and there's 16

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different ways of arranging those pairs

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so that's two to the power of 4 is 16

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and the probability and this is the the

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key here the probability that Muriel

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Bristol gets this right is 1 in 16. she

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has to in order to get them all right

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well if she if she can't tell the

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difference the probability she gets them

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all right is 1 in 16.

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now if we do hit option b

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the one that you liked and this is going

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to prove to be the better choice

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there are eight places to put out the

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first cup seven places for the second

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cup these are the milk cups

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um six places for the third cup and five

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places for the fourth cup and you've put

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them out at random and then you fill up

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with the these would be the the milk

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ones then you fill up with the non-milt

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ones and then you can also think of the

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ordering of the cups there's four times

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three times two times one ways to do it

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and using combinatorics you can find out

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well there's 70 ways of placing the cups

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and if you don't believe the math you

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can sit and write them all out and I got

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I got a little yeah I told you I don't

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like calculating so I got a little bit

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bored before I wrote all of the

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different ways you can arrange the cups

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out but you can arrange them in these

play10:17

different ways and so if Muriel Bristol

play10:20

can't tell the difference the

play10:22

probability of getting all four right is

play10:25

now only one in in seventy so the the

play10:27

this test is the better way to do it and

play10:31

this I think is a perfect example of

play10:34

using a nice piece of mathematics in the

play10:36

form of combinatorics and that's what

play10:38

Fisher did he used different parts of

play10:40

combinatorics to solve a problem in

play10:44

experimental design

play10:46

and he went on to write a book

play10:48

which became is still a sort of handbook

play10:51

used today of how you design different

play10:54

experiments this is a slightly this is

play10:56

his Latin Square design which is a

play10:58

different design than the randomized

play10:59

design we've just talked about but he

play11:01

could then from there start to spread

play11:04

his statistical ideas

play11:07

and there's just an incredible power I

play11:10

think in being able to think about the

play11:12

correct way to do an experiment or the

play11:14

correct way to analyze data

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now I like to so I've as Daryl said I I

play11:21

have worked on football and I've worked

play11:24

in mathematics and football and this has

play11:25

taken me on some amazing Journeys and

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it's very nice to for example I was I

play11:31

had a um thing with Gary Neville so I

play11:34

like to show off now about my sort of

play11:36

football and contacts I think Gary

play11:38

Neville is the most famous uh footballer

play11:40

I've I've met on my journey so I just

play11:42

wanted to sort of throw that in there

play11:45

um and what's what's invariably happens

play11:49

when I talk to footballers or former

play11:52

footballers or coaches about about

play11:54

mathematics and football

play11:56

is that they have this thing where they

play11:58

say oh well you know numbers can tell us

play12:00

something numbers can tell us a few

play12:01

things but you can't measure a player's

play12:03

attitude you can't measure and so Gary

play12:06

said that when we did this thing

play12:08

together he said oh you can't measure if

play12:10

a team goes a goal down

play12:12

you can't measure the player who really

play12:14

gets everyone going and really rallies

play12:16

the team and gets them going again

play12:18

and I was sitting there thinking yeah

play12:21

actually that's exactly what you can

play12:23

measure with Statistics and so a few

play12:25

days after

play12:26

um Gary Gary said this I sent him an

play12:29

analysis where we analyzed exactly that

play12:31

we look to see what happens when a team

play12:37

um so when a team can see the goal so

play12:40

this is Trent Alexander Arnold in a game

play12:43

in the 21-22 season and the line the

play12:47

central line here the central dotted

play12:49

line

play12:50

which is highlighted

play12:52

is when they conceded a second goal

play12:55

against United they lost the game 2-1 in

play12:57

the end and just after half time they

play12:59

conceded a second goal

play13:01

and the squiggly line that's going up

play13:04

and down that's Trent Alexander Arnold's

play13:07

performance on the ball

play13:09

and

play13:10

as time goes on you can see that he's

play13:12

getting better and better he's he's

play13:14

actually producing lots of good passes

play13:16

for his teammates

play13:18

and then the goal goes in dotted line

play13:20

and then you see his performance drops

play13:23

back down again and so in this

play13:25

particular case if we do the Gary so we

play13:27

ended up calling this the Gary Neville

play13:29

statistic so if we do the Gary Neville

play13:31

statistic on this his performance goes

play13:34

down after they concede a goal compared

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to his performance in the 15 minutes

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before they concede a goal so it's a

play13:41

measurable statistic and in this way you

play13:44

can for example we looked at the top

play13:46

Strikers in 2122 and it was quite

play13:50

interesting because you might call

play13:54

um Jamie vardy a player who's got a lot

play13:56

of attitude or character or something

play13:58

like that and then it turns out that's

play14:00

what we got when we measured it when

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when

play14:02

um when Jamie vardy's team went down

play14:04

they went down a little bit more often

play14:07

than some of the other teams but he got

play14:08

better 29 of the times worse 13 of the

play14:12

times and his performance was the same

play14:15

about five of the times and you can see

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yeah there's a ranking of the top five

play14:20

players for That season in these

play14:22

different situations

play14:24

and it's very nice I've put in

play14:25

parenthesis here because we used

play14:27

precisely one of Ronald Fisher's tests

play14:29

Fisher's exact test in order to test if

play14:32

these players were statistically better

play14:33

when they're when their team went down

play14:36

or not

play14:37

and so there's all types of ways in

play14:39

which we can use statistics another

play14:41

um example

play14:43

and now we're kind of moving over to the

play14:46

edges of the limits of statistics

play14:49

and I think that this is a very

play14:51

important point because

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while while I think Gary is wrong in

play14:56

that you can't measure attitude at all

play14:58

he's right in another way because you

play15:00

can't measure everything you can measure

play15:03

certain aspects of how a player gets

play15:05

better it's one piece of information you

play15:07

have but you can't measure everything

play15:10

and this is from a study uh from a TED

play15:12

Talk and during the writing of the book

play15:14

I watched the 25 most popular TED Talks

play15:18

because I was very interested how they

play15:19

use statistics in order to assess the

play15:22

validity of claims in TED Talks and this

play15:26

was a talk by Angela Duckworth and she

play15:29

said that Grit and grit is the idea of

play15:32

determination how determined you are to

play15:35

succeed the grit is the strongest

play15:37

predictor of success and it comes from a

play15:39

study that she conducted

play15:42

um together with some colleagues and

play15:43

what they did is they looked at Ivy

play15:45

League undergraduates I think at Yale

play15:48

they looked at U.S military cadets and

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they looked at people who were competing

play15:53

in a spelling bee and before they

play15:55

started doing these activities they

play15:57

asked them questions about how if you

play16:00

start something do you always see it

play16:02

through it those types of questions a

play16:04

series of 12 questions about if they

play16:06

were determined gritty types of people

play16:09

and they found that grit this the

play16:11

answers that people gave in those

play16:13

questions were some of the biggest

play16:15

predictors of success and that's what

play16:18

she said in the in the talk and that

play16:21

sounds very impressive a bit like me

play16:23

trying to persuade Gary Gary Neville

play16:25

that we can we can measure

play16:27

um attitude in football players

play16:29

but

play16:31

if you look a little bit more closely at

play16:33

this the actual study as opposed to the

play16:36

yes we shouldn't measure we shouldn't

play16:38

measure success on how many times people

play16:40

have watched the YouTube video because

play16:41

this YouTube video has been watched 25

play16:43

million times

play16:45

and it's it's not Angela Duckworth who

play16:48

wrote the headline on it but

play16:51

her paper

play16:53

reveals quite clearly and as she doesn't

play16:56

try to hide this in any way at all the

play16:58

grit just explains four percent of the

play17:00

variance between people

play17:02

now four percent of the variance how

play17:04

much is that it isn't

play17:06

it doesn't mean that only four percent

play17:08

of people are explained by this I'm

play17:10

going to try and show you what four

play17:12

percent of the variants look like four

play17:13

percent of the variance looks like this

play17:16

so

play17:18

if you measure grit on the scale one to

play17:22

five on the bottom here and you look at

play17:24

grade point averages for example now

play17:26

this isn't real data this is data I've

play17:28

just made up in order to illustrate what

play17:30

four percent of the variance looks like

play17:32

four percent of the variants would have

play17:34

some type of relationship a bit like

play17:36

that

play17:37

and you can see if you squint carefully

play17:40

I can't see it from this angle but I

play17:42

think maybe you can squint and see this

play17:44

there is a kind of increasing Trend

play17:46

there between grit and grade point

play17:49

average but you also see that some

play17:51

people who yeah and there's there's lots

play17:54

of people who are very gritty and are

play17:55

successful and there's people who aren't

play17:57

gritty and unsuccessful there's also

play17:59

lots of people who are very gritty and

play18:01

don't get a high grade and there's lots

play18:03

of people who aren't very gritty and do

play18:05

get a high grade and so when you're

play18:08

interpreting this you shouldn't confuse

play18:10

so I often say that you shouldn't

play18:12

confuse the forest for the tree you're a

play18:15

tree right every person in this in this

play18:18

room is a tree so if we tested all of

play18:20

your grittiness and your success in life

play18:22

we'd find some kind of relationship like

play18:24

this but it wouldn't mean that you

play18:26

necessarily as an individual

play18:29

um had this relationship between grit so

play18:32

if you're not a gritty person if you

play18:33

never see through any projects you start

play18:35

you don't need to worry at all you're

play18:37

going to be absolutely fine there's lots

play18:39

and lots of ways in which your life can

play18:41

succeed

play18:43

and this what I've tried to do here and

play18:46

I'm not sure if I've got the art quite

play18:48

right here but the the arc I want to

play18:50

describe here is I I want to start by

play18:53

saying statistics is very powerful I can

play18:55

show off to Gary Neville about it but

play18:58

then at some point statistics doesn't

play19:00

actually give you all of the answers

play19:02

right and that's very well Illustrated

play19:07

if we actually go back to Ronald Fisher

play19:09

this rather arrogant young undergraduate

play19:12

student because Ronald Fisher also has

play19:15

another scientist story

play19:18

he was from a very early age and this

play19:21

picture was taken in 1912

play19:24

very interested in Eugenics and he had

play19:27

this idea that we needed to breed people

play19:29

to be more like him like to be more

play19:33

clever and smart and good at maths and

play19:35

so on and he campaigned all the way up

play19:39

to the war I think after the war he kept

play19:42

more quiet about this

play19:43

but all the way up to the war he

play19:46

campaigned for for example sterilizing

play19:48

people who were considered feeble-minded

play19:51

and this is this is of course horrible

play19:54

and I mean it's a horrible thing to

play19:56

think about but not only is it like

play19:58

morally repugnant it's also

play20:00

scientifically wrong they couldn't find

play20:03

any kind of Gene for single uh for

play20:05

feeble-mindedness there is no

play20:07

correlation between or there's a very

play20:09

weak correlation or no correlation at

play20:11

all between feeble-mindedness in one

play20:14

generation in mothers and in their

play20:16

daughters so this was a relationship

play20:18

that was scientifically dubious that he

play20:21

continued to press forward and the

play20:23

reason he was successful is because or a

play20:26

reason he was successful in pushing us

play20:27

forward is is thing forward was that he

play20:30

would use statistics in order to sort of

play20:33

attack his opponents he would call them

play20:35

all stupid they didn't understand

play20:36

statistics so he would actually use

play20:38

statistics to undermine other people's

play20:41

arguments in a really counterproductive

play20:43

way he took a fake Theory and then used

play20:46

statistics to defend it and he didn't

play20:48

just do this once after the war when

play20:50

he'd given up on on Eugenics or at least

play20:54

stopped talking about it

play20:55

he then did the same thing on smoking so

play20:59

as we saw in the first picture he was a

play21:01

very keen smoker

play21:02

and for him there was just no

play21:04

possibility that smoking could cause

play21:06

cancer

play21:07

and so he spent a lot of time

play21:09

investigating very narrow areas of

play21:12

science doing his statistics on that and

play21:15

trying to convince people that smoking

play21:16

didn't cause cancer and I don't know

play21:19

the effect that this research had but

play21:21

certainly you've got one of the leading

play21:22

statisticians in the world who's

play21:24

defending this position for a long long

play21:26

time

play21:28

and that

play21:29

illustrates a lot of why statistics is a

play21:33

limited approach so I've written down a

play21:37

few a few sort of bullet points here so

play21:39

statistical thinking doesn't provide all

play21:41

the answers

play21:43

um

play21:44

one problem is and I I didn't really get

play21:47

into this but I mean what a dick he is

play21:50

right

play21:52

I mean why do you need to test if she

play21:55

you know can tell the difference between

play21:57

the milk first everyone was very happy

play21:58

they were all just enjoying their tea

play22:00

party and suddenly he's doing a

play22:02

statistical day I mean you know he's an

play22:03

idiot so so that's one thing and and I

play22:06

really think you know we joke about that

play22:09

but we see it at work all the time you

play22:11

know that's always being told that we

play22:13

should be there should be statistical

play22:15

tests and metrics made about us and

play22:18

things like that and we don't need as

play22:21

much quantification as we have

play22:23

then there's the effect size thing and

play22:25

we sometimes talk about statistical

play22:27

significance you can have statistical

play22:29

significance but still have a very small

play22:31

effect size and that's the confusing the

play22:34

tree in the forest many non-gritty

play22:36

individuals are successful in life

play22:38

context is always important so just

play22:41

because a player is more active when

play22:43

their team goes a goal down does not

play22:45

imply the team plays better just because

play22:47

Ronaldo demands that everyone gives him

play22:49

the ball when they go a goal down or

play22:52

Jamie vardy demands that he gets the

play22:54

ball that doesn't mean that the team

play22:55

actually does better as a result of that

play22:58

so it's very important to think about

play22:59

the context of these types of things and

play23:02

then as we all know correlation and

play23:04

causation they're not the same thing

play23:06

but that leads us to the to the next

play23:09

step we need to think about ways to

play23:12

tease out causation we want to be able

play23:15

to tease out our understanding of the

play23:17

world one thing causes another

play23:19

and it brings us very nicely on to

play23:22

interactive thinking and that's the next

play23:25

step on from statistical thinking

play23:28

and I have another hero I can reassure

play23:31

you that this hero is not going to turn

play23:32

out to be a raving racist who bullies

play23:35

all of his co-workers so um there are a

play23:38

few mathematicians who haven't done that

play23:41

in their lives just a few of them but

play23:43

they're around

play23:44

and this is Alfred J Locker

play23:47

um and he was originally a chemist

play23:51

and he started his uh he's he was

play23:55

originally from Poland but he did his

play23:56

undergraduate degree in Birmingham

play23:59

and

play24:00

I I really like his story because

play24:03

he started working in this chemistry lab

play24:07

and he was kind of disappointed with

play24:09

what he saw when he was doing his

play24:11

experiments I mean it's a long time

play24:13

since I did chemistry at school but it

play24:16

can be a little bit disappointed you

play24:17

know you get the acid and The Alkali you

play24:20

mix them together and there's some salt

play24:21

and water it's not always the most

play24:23

exciting thing you've ever seen and so

play24:26

he'd see these stable stable reactions

play24:29

just come to equilibrium

play24:31

but in the evening he was reading all of

play24:33

these books like Charles Darwin's book

play24:36

and he was thinking about biology and

play24:38

just all of the exciting patterns we see

play24:41

all the motion and movement of animals

play24:44

or the firing of our brains everything

play24:46

that happens in society and he was

play24:48

thinking why can't chemistry produce

play24:51

anything like that I mean we know that

play24:53

chemistry must be the underlying

play24:55

building block of it but it's not

play24:57

something we see we can't make that

play24:58

relationship together

play25:00

and the way he solved the problem was he

play25:04

basically cheated and he he did the

play25:06

following thing so if

play25:08

we've all done this in school

play25:10

we've balanced equations right through a

play25:13

balanced reactions and if you've got uh

play25:15

two two h two well you've got four

play25:17

hydrogen atoms two oxygen atoms and they

play25:20

react to get to make two water molecules

play25:22

so that's a a standard chemical reaction

play25:25

and the important Point here is that

play25:26

this is balanced so there's four

play25:28

hydrogens and two oxygens on the left

play25:30

and there's four hydrogens and two

play25:33

oxygens on the right

play25:35

but what loter said is well I'll just

play25:39

forget about that balance thing even

play25:41

though I can't find a chemical reaction

play25:43

that is unbalanced I'll just think in my

play25:46

head I'll do a thought experiment and

play25:48

this is where it's lovely with

play25:49

mathematics I'll do a thought experiment

play25:51

where I ignore the fact that I can't

play25:53

balance my reactions and so he take he

play25:55

wrote down these equations he said that

play25:57

imagine an r that becomes two r's and

play26:00

imagine an R plus an F which becomes two

play26:03

F's and you can see that these aren't

play26:04

balanced there's one r on the left hand

play26:06

side of the first one two r's on the

play26:07

right yeah there's you can see that

play26:09

they're just not balanced and I've

play26:12

written down here below the way you can

play26:13

think about these things are rabbits and

play26:15

foxes so

play26:17

and and they're not it's not a realistic

play26:19

model of rabbits and foxes you've got to

play26:21

think of the idea of one little rabbit

play26:23

hopping around and suddenly there's two

play26:24

little rabbits hopping around we know

play26:26

it's a little bit more complicated than

play26:27

that but we'll start with that idea and

play26:29

then a fox comes and eats a rabbit and

play26:31

then it makes another Fox that's that's

play26:32

what the model says it gives a a rough

play26:35

idea of how ecological interactions work

play26:40

and he took that and he wrote down

play26:43

differential equations

play26:45

I wanted to put in a few of these

play26:47

equations here to give you a feeling for

play26:49

how they work

play26:50

the the equations on the left here one

play26:53

of them I I'm not going to get you to

play26:55

understand every detail of the equation

play26:56

but what I want to give you a feeling

play26:58

for is on the left is the rate of change

play27:01

of the rabbits and the foxes so Dr by DT

play27:04

is a rate of change of the rabbits DF by

play27:07

DT is the rate of change of the foxes

play27:09

and on the right are the things which

play27:11

cause that change so I mentioned here

play27:14

that we want to get causation into our

play27:15

equations so on the right of the things

play27:17

which cause that change and rabbits

play27:20

increase when there's lots of other

play27:21

rabbits they have lots of little bunny

play27:22

rabbits and then they are eaten by the

play27:27

foxes so the more F they are if we look

play27:29

at this brf term that's the rate at

play27:32

which the

play27:33

um the rabbits are eaten by the foxes

play27:35

and then when we come down that we have

play27:37

the opposite relationship for for The

play27:38

Foxes the foxes grow when there's more

play27:41

rabbits and then they eventually die off

play27:44

um of old age there's nothing which

play27:46

hunts the foxes in in this in this

play27:48

scenario

play27:50

now again I'm not going to solve all of

play27:52

these equations but I did want to

play27:54

mention a little bit about how you can

play27:56

think about them and understand them

play27:58

and on the right here actually learned

play28:01

this from Philip Maney when I was here

play28:04

in in Oxford about these types of method

play28:06

but he had a very lovely method

play28:09

a professor of mathematical biology here

play28:12

for solving these equations without

play28:15

solving them

play28:16

so you can split up the plane of foxes

play28:19

and rabbits and you can identify a point

play28:21

on that plane and look to see do the

play28:24

rabbits increase or do the fox's

play28:26

increase so in the bottom left-hand

play28:28

Corner the foxes go down because there's

play28:31

not enough rabbits to eat but the

play28:33

rabbits go up because there's not enough

play28:35

foxes eating them and that goes on until

play28:38

there's a sufficient number of rabbits

play28:41

and then the foxes start increasing so

play28:44

in the bottom right hand corner this

play28:46

Arrow points up and if that Arrow points

play28:48

up well then the foxes increase

play28:51

and when the foxes increase they start

play28:54

to eat the rabbits and the rabbits go

play28:55

down and you start to get this circling

play28:58

round and round of foxes and rabbits and

play29:00

you could do basically show this without

play29:04

explicitly solving the equations that

play29:07

there's going to be Cycles round and

play29:10

round of these foxes and rabbits we're

play29:13

going to get this interesting

play29:14

interaction and if we look at it over

play29:16

time

play29:18

um we have these periodic oscillations

play29:21

of the foxes and the rabbits

play29:23

now this whole way of thinking

play29:26

which latke introduced turned out to be

play29:29

useful in all sorts of situations now

play29:32

the one thing we should we try not to

play29:34

mention but can't be unmentioned in any

play29:37

mathematical modeling thing we try not

play29:38

to mention this but but it's also used

play29:40

in pandemic modeling and you've all

play29:43

heard about these epidemic curves and

play29:45

our values and so on

play29:48

um but that's the same thing that a

play29:50

susceptible plus an infective becomes

play29:52

too infectives and that's an example of

play29:54

one of lotka's unbalanced equations

play29:57

which allows us to describe how an

play30:00

epidemic reaction epidemic will spread

play30:02

through a group of people

play30:05

now

play30:07

I'm not going to go into as I said I

play30:09

don't want to talk too much about

play30:10

epidemics but this example I really love

play30:12

so I'm going to talk about this example

play30:13

this is a study we did I did together

play30:16

with some colleagues and this is a very

play30:19

cruel experience it's not it's not it's

play30:20

not that cruel but we have we had a

play30:21

group of undergraduate students we got a

play30:24

third a third year undergraduate student

play30:26

to give a seminar to first-year

play30:28

undergraduate students and we told the

play30:30

first year undergraduate students you

play30:32

know remember to give a round of

play30:34

applause after the seminar just to show

play30:38

your appreciation and then what we were

play30:41

really interested in was how people

play30:43

applaud it what are the cues that make

play30:46

people applaud and we could see that

play30:48

people start applauding when other

play30:50

people around them start applauding and

play30:53

you basically have an epidemic of

play30:55

Applause and that's what the first green

play30:57

curve shows that's the number of people

play30:58

clapping that's the spread of the

play31:01

Applause virus going through the group

play31:04

but then

play31:06

and this doesn't happen in real diseases

play31:08

you also have a social recovery

play31:11

so when people stop clapping they look

play31:15

around and they hear the other people

play31:16

have stopped clapping and it was

play31:18

actually a little bit like when Daryl

play31:20

left the room just now there was a sort

play31:22

of start signal there that there might

play31:24

be something going on that we might be

play31:26

about to start and you all started to go

play31:28

down in volume and suddenly everyone was

play31:30

quiet

play31:31

and that's the type of social effect

play31:33

we're very aware of all kinds of small

play31:37

social details and these spread through

play31:40

us in a group and the the conclusion

play31:43

that what I love about the recovered

play31:44

thing is because we have that social

play31:46

recovery so

play31:48

and I try to always remember this that

play31:50

if at the end of a talk you've given or

play31:53

a presentation if the Clapping goes on

play31:55

for a long time that's not because you

play31:57

gave a good talk it's just because your

play31:59

audience aren't particularly coordinated

play32:01

so they couldn't they could they

play32:03

couldn't manage to to stop together

play32:06

and I think that's what that's what I

play32:09

would encourage you to think about and I

play32:10

said that I also wondering about the

play32:13

personal aspects of this

play32:15

it's nice just to sit sometimes and

play32:18

think about the social reactions that

play32:20

you have in your life and how they work

play32:22

and I've written down a few of them I

play32:23

haven't told you what they are yet so

play32:25

I'll tell you what what they are the top

play32:27

the top one

play32:28

that I was imagining this is a person p

play32:31

and then it's plus an O this is a sofa

play32:35

that the person has outside the house

play32:37

and so we have P plus o goes to P plus o

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if you're just one person and you've got

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a sofa outside the house you're going to

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still be one person and you can't get

play32:46

the sofa into the house so what you have

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to do and this is the bottom equation is

play32:50

you have to get a friend and so this

play32:52

bottom one is 2p 2 people plus a sofa

play32:56

that's outside the house is still two

play32:58

people you've still got your people

play33:00

afterwards but you've moved your sofa

play33:02

into the living room and you can write

play33:04

those social interactions for every type

play33:06

of activity the one on the right here

play33:08

the ones on the right here I was

play33:10

thinking about smiling so if you're a

play33:12

smiley person why and you meet a

play33:16

non-smiley person whose ex then if you

play33:19

smile then hopefully they become a

play33:21

smiley person too but that's not always

play33:23

the case sometimes you know you don't

play33:26

always start smiling because somebody

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else is smiling they might just be an

play33:29

idiot who's just smiling for no reason

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at all what happens most often actually

play33:34

in human social interactions is they as

play33:36

you believe the following Equator the

play33:38

one the equation at the bottom this is

play33:40

the most common equation I think that

play33:41

describes human behavior and that's that

play33:44

a non-smiling person plus two smiley

play33:47

people will become three Smiley people

play33:49

because then they're convinced that

play33:50

actually must be something to smile at

play33:53

and I use that a lot in my thinking if

play33:56

if I'm thinking about how to in the book

play33:59

I take an example of if I'm trying to

play34:00

get a group if a group of friends are

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trying to get going with some kind of

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healthy activity maybe they spend a lot

play34:07

of time sitting in the pub together

play34:08

don't really go out and do any exercise

play34:11

together it's not enough for one of them

play34:14

to become a why to to try and get them

play34:17

going that you have to have two of them

play34:19

and they have to have a really sustained

play34:21

effort and over time then you get this

play34:24

Tipping Point effect where everybody

play34:26

starts to move over and starts it starts

play34:28

to engage in the healthy activity so

play34:30

those are the types of things you have

play34:32

to think about what type of chemical

play34:33

reaction what type of social reaction am

play34:36

I involved in

play34:38

and that's been a lot of to be honest

play34:40

this has taken up a lot of my adult life

play34:43

is studying these types of things and to

play34:45

give you a little bit of a flavor of the

play34:47

sorts of things we do

play34:49

this is just to give a sort of overall

play34:51

representation but

play34:54

um when we modeled fish for example we

play34:56

would create models which described

play34:58

their social interactions described how

play35:00

one fish turned left if another fish

play35:02

turned left if another fish turned right

play35:04

and so on then we were building the top

play35:06

there it's a mathematical model we've

play35:08

built a fish movement and so on so we'd

play35:11

show that these simple rules of

play35:13

interaction would produce their

play35:15

Collective Behavior then we'd study also

play35:19

the the movement of individual fish

play35:21

that's the colored idea at the the

play35:23

bottom then we'd actually frighten all

play35:25

of the fish and we'd look at how they

play35:27

made an escape wave we we'd measure that

play35:30

escape wave and then we'd use models to

play35:32

understand that escape wave and it's a

play35:35

very powerful way of thinking throughout

play35:37

science that you can build up these

play35:39

models of interaction you compare them

play35:41

to reality and build a better and better

play35:43

understanding of fish Behavior and we do

play35:46

a lot of similar things in football so

play35:49

this is an example of an attacking run

play35:51

by Marcus rashford and the model that we

play35:55

build for these types of situations

play35:59

this red area here shows the territory

play36:01

that he controls

play36:04

and this is a physics-based model where

play36:06

we say how far how fast can he run where

play36:08

can he get to and we can actually

play36:10

describe what area he he occupies and

play36:14

also the value of that area so How

play36:16

likely is he is getting a pass at that

play36:19

particular Point going to lead to a goal

play36:22

and that allows us to actually Scout

play36:25

players based on their runs and it even

play36:29

allows us to scout runs where they don't

play36:31

get the ball so in this example

play36:34

where

play36:36

we're interested in Luke Shaw here and

play36:38

he's doing a run here on the left

play36:40

and he doesn't get the pass he'd love to

play36:42

have this pass but he doesn't get it but

play36:44

we can still measure the value that that

play36:47

pass created so you can look at these

play36:49

counterfactual situations for

play36:52

um for for football players

play36:55

and this is a very powerful method the

play37:00

interactive way of thinking allows us to

play37:03

build up our understanding of systems it

play37:06

doesn't have the same kind of

play37:08

I I suppose the statistics has a sort of

play37:11

more of a grounding feeling to it this

play37:15

we use our imagination much more we try

play37:18

to use our imagination to increase our

play37:19

understanding and then build

play37:21

mathematical models to test that

play37:23

understanding

play37:24

now I wanted to go back to lotka because

play37:28

um

play37:29

there are also limits to this way of

play37:31

thinking and of course I wouldn't have

play37:33

four if we'd if we'd solved it all now

play37:35

so there's there's limits to this and

play37:37

there's limits were limits that lotker

play37:40

himself hit he wrote a book called

play37:42

elements of physical biology and

play37:46

he's one of these these mathematicians

play37:49

who and this happens a lot to us

play37:52

is we sort of just get carried away and

play37:55

we believe that we can just explain

play37:57

everything with mathematics that there's

play37:59

nothing that we can't explain and so he

play38:02

built models he built models of

play38:03

Consciousness he built models of of our

play38:06

whole society and he believed that all

play38:09

of them could be understood using his

play38:11

reaction Dynamics and it really

play38:15

yeah he didn't I mean and this was I

play38:17

suppose it was a very Valiant effort

play38:19

this is in 1922 he finished his his

play38:21

Magnus Opus so he didn't even have a

play38:23

computer or anything to simulate these

play38:25

types of models on but he never really

play38:27

succeeded in pinning down one essential

play38:30

way in which you should approach all

play38:32

sorts of problems he he ended up kind of

play38:35

split between lots and lots of different

play38:37

small things and that I can personally I

play38:39

can relate to that very well because

play38:41

that tends to be how I work with lots of

play38:44

problems there's lots of different

play38:45

methods and you're doing lots and lots

play38:47

of small different things in order to

play38:49

get your solution the day of day of an

play38:51

applied mathematician isn't it's not

play38:53

like these theoretical physicists you

play38:55

know they have like this beautiful

play38:56

Theory of Everything and they can come

play38:58

here and just say oh it's all this and

play39:00

wow but no it's not like that it's more

play39:03

that you're sort of tinkering around

play39:04

with small different problems in lots

play39:06

and lots of different ways so lotska

play39:08

never found his Grand Theory of

play39:10

Everything using interactive thinking

play39:12

and one of the reason

play39:14

one of the reasons he never found is

play39:16

Grand theory was because he didn't know

play39:18

about chaos

play39:20

which is the third way of thinking

play39:23

now to introduce chaos

play39:25

I'm going to go to another mathematical

play39:28

hero

play39:29

this is Margaret Hamilton and

play39:32

she was also like the other two we've

play39:35

met prodigious at school

play39:37

very talented undergraduate student she

play39:40

wanted to go on and do a PhD in pure

play39:42

mathematics

play39:44

but her husband also wanted to do a PhD

play39:47

and this is now in the 1960s and she

play39:50

ended up moving to Boston and she also

play39:53

had to get a job she had a daughter to

play39:55

support a husband to support and so she

play39:57

had to get a job to support them but the

play39:59

job that she got was programming this

play40:03

machine the lgp 30 and she fell

play40:06

immediately in love with this Computing

play40:09

machine because

play40:10

she hated making mistakes she hated

play40:13

errors whenever she calculated anything

play40:16

she calculated it perfectly and now she

play40:19

found that she could actually program

play40:20

this first computer to do the same

play40:23

calculations and she got access to this

play40:25

computer because she was working in the

play40:27

lab of a person called Edward Lorenz who

play40:31

was a professor of meteorology but also

play40:33

with a mathematical background there are

play40:35

a lot of mathematicians in this talk so

play40:38

um and he

play40:39

he wanted to predict the weather he

play40:41

wanted to predict the future weather

play40:43

based on temperature pressure and so on

play40:46

in different areas could he predict the

play40:48

weather into the future and she started

play40:50

writing a computer code to do this and

play40:53

this involved writing and doing Punch

play40:55

Cards at the time and she'd run her

play40:56

computer code

play40:58

and they did this one thing is that they

play41:00

simulated they simulated the weather one

play41:01

day and the next day they decided to

play41:04

check their results by simulating making

play41:06

the exact same simulation on the

play41:08

computer to check that everything worked

play41:10

but they found on the second day they

play41:13

got a different result than on the first

play41:15

day

play41:16

and Margaret was distraught because she

play41:20

didn't like making mistakes she didn't

play41:21

want to think there was a mistake in her

play41:23

code but she started going through the

play41:24

code and there was no errors in the code

play41:26

and what they found was that the output

play41:29

of the simulation was in six decimal

play41:32

places

play41:33

well the input they put into it was in

play41:36

three decimal places

play41:37

so there was an error in the input in

play41:40

the fourth decimal place and this meant

play41:43

that the weather simulation made

play41:45

completely different predictions in the

play41:48

future going like a few 10 days into the

play41:50

future in the simulated World it made

play41:52

completely different predictions in the

play41:55

future and I didn't mention that this

play41:57

was a system of 14 differential

play41:58

equations that she solved we've moved on

play42:00

from lockter and Voltaren too so she

play42:02

solved these 14 differential equations

play42:04

and they make just this small error in

play42:08

the value you put in makes a massive

play42:10

difference and that is the first

play42:12

indication of the butterfly of chaos

play42:15

which many of you will be familiar with

play42:17

and Lorenz went on he worked with

play42:20

um I say Lorenz went on Margaret

play42:23

Hamilton we're going to find also went

play42:24

on to do some very impressive things but

play42:26

Lorenz went on with the help of Ellen

play42:29

Fetter who replaced Margaret Hamilton as

play42:32

his programmer to produce what we now

play42:34

know as the we often think of this

play42:37

picture I think or I think of it as

play42:39

being the butterfly of chaos and what it

play42:41

illustrates is if you do start with two

play42:43

points very close to each other we've

play42:45

moved now down from 14 Dimensions to

play42:47

three dimensions again if you start with

play42:50

two points very close together and they

play42:52

start to diverge

play42:54

they'll move around on the same

play42:57

attractor on this shape that we have

play43:00

here but they'll never come close or

play43:01

they might come close to each other for

play43:03

a short amount of time but they'll then

play43:05

live their own life and so when we move

play43:07

from two Dimensions up to three we have

play43:10

this chaotic movement where things never

play43:12

come back to the same place again

play43:17

I think I think I think we're going to

play43:19

do my experiment okay so I think I think

play43:21

we're going to do the experiment and

play43:22

then I'll I'll boot out something else

play43:24

because you've listened to me patiently

play43:25

for 50 minutes so you have to get to do

play43:27

the experiment okay so here we're going

play43:29

to we're going to do this I want you to

play43:30

work in pairs

play43:32

I want you one of you should think of it

play43:34

so it's a look at the person next to you

play43:36

and you might be a new friend that

play43:37

you've got today

play43:39

um or it might be somebody that you came

play43:41

with

play43:42

and then I think I want one of you to

play43:44

think of a number between 1 and 99

play43:48

then you tell that number to the other

play43:50

person

play43:51

and the other person follows the follow

play43:53

the the following rules so if a number

play43:55

is less than 50 double it and this is

play43:58

the new number I chose 42 because you

play44:00

can never have a math talk without 42 in

play44:02

it so 42 times 2 is 84 and so that's all

play44:05

you do you just double the number now if

play44:08

the number is greater than 50 take it

play44:10

away from 100 and then double it to get

play44:12

the new number so if I have 84 then I

play44:15

have 100 minus 84 is 16 times 2 is 32.

play44:19

now say the new number to your partner

play44:21

and they repeat step one and two so

play44:24

we'll do this for um do this with either

play44:26

with the person you came with or

play44:28

somebody who's nearby to you we'll do

play44:30

this for about two minutes and then

play44:31

we'll see where we get to

play44:41

I think you've done it very nicely done

play44:43

I can see the I can see I hear the

play44:45

murmur of numbers everywhere very very

play44:47

lovely

play44:49

um

play44:50

I'm I'm not going to get you all to come

play44:52

up here and present your results I just

play44:54

wanted to give you get get you to get a

play44:56

feeling of this type of process

play44:59

um you're not generating purely chaotic

play45:01

numbers when you do this if you'd

play45:03

started with 20 for example you would

play45:05

have found yourself cycling around quite

play45:06

quickly but if you started with a number

play45:09

that's not divisible by five you would

play45:11

have probably been on quite a long

play45:13

trajectory through different numbers

play45:16

and the point I want to make about this

play45:18

process is the following is that close

play45:21

together numbers very quickly diverge so

play45:25

if one group over there had started with

play45:27

13 and another group over here had

play45:29

started with 14

play45:31

by the end of just this short period

play45:33

where you got to say the numbers to each

play45:34

other you would have been on very

play45:36

different numbers so you have 13 26 52

play45:39

96 8 16 32 64. 14 28 28 is not so far

play45:45

from 26 56 52 they're still together

play45:48

88.96 are starting to get away from each

play45:50

other but the big jump is now one of

play45:53

them sort of goes over the threshold and

play45:54

one of them doesn't so you've got 8 and

play45:56

24.

play45:57

1648 and then you've got 32 and 96 and

play46:01

64 and 8. So within a few steps these

play46:05

numbers have diverged quite far from

play46:09

each other I don't know if any any of

play46:11

you took decimal numbers

play46:13

um you didn't think of that but if you

play46:15

do take decimal numbers then you get

play46:17

true chaos from this thing for almost

play46:20

any real number you choose you will get

play46:24

if you take plus 0.1 in this case only

play46:27

so this is 14.1 compared to 14.2 you

play46:31

start to they're together for a few

play46:34

steps but after about seven eight nine

play46:37

ten they go apart they come a little bit

play46:39

together again for a while but then they

play46:41

diverge and you've got very different

play46:43

paths for those two numbers

play46:46

and we often illustrate this

play46:49

um using something called a cobweb

play46:51

diagram so the idea here is you take the

play46:54

number from One Step and the previous

play46:56

number might be around 20 for example it

play46:58

will jump up to be around 40 then it

play47:01

will go to 80 then it will crash down to

play47:03

around 20 again and then it will start

play47:06

to move around everywhere on this

play47:10

and one of the reasons I wanted you to

play47:12

do this experiment is what was being

play47:14

what I could hear from your perspective

play47:17

was a Mumble of uniform distributed

play47:21

random numbers you were essentially

play47:23

going through a lot of integers and

play47:27

everywhere in the room there was a

play47:28

different point in this distribution you

play47:31

basically had this uniform distribution

play47:33

of numbers that were sort of kind of

play47:35

coming up to me and I think it's really

play47:37

lovely to think of that that you're all

play47:39

doing the same process you're all doing

play47:42

exactly the same thing yet you kind of

play47:44

have this hum this distribution this

play47:46

background of very different numbers

play47:50

and

play47:51

that is the butterfly of chaos and and

play47:54

for me it illustrates

play47:56

there's an important Point here

play47:59

I I think chaos is wonderful

play48:02

Margaret Hamilton she hated chaos right

play48:06

and she she left Lorenzo's lab and she'd

play48:09

learned a valuable lesson from working

play48:11

on these weather simulations and it was

play48:14

that she doubled down and made even

play48:17

fewer errors and she wanted to work in

play48:20

the most extreme

play48:22

conditions possible where you couldn't

play48:24

make errors and so she got a job for

play48:26

NASA

play48:27

and she became the head of the software

play48:30

engineering which created the software

play48:34

that sent that was on the Apollo moon

play48:37

mission and so she was she created the

play48:39

software that the astronauts used to

play48:42

tell them how to to do navigational

play48:44

decisions to control the thrusters to um

play48:49

uh to update to know where the position

play48:52

of the ship was

play48:54

and she was in the control room when

play48:56

they when they made the actual landing

play48:58

on the moon and so I see this as a

play49:00

situation where you sort of have to

play49:02

choose right in if you're if you're

play49:04

going to control something because of

play49:06

chaos if there's something you really

play49:08

care about or there's something that's

play49:10

really important then you have to treat

play49:12

it like Margaret Hamilton does you had

play49:14

to treat it like the moon landing

play49:15

there's no error there's no room for any

play49:18

type of error

play49:20

but you can't have control over

play49:21

everything so I often think about this

play49:23

in football because

play49:26

there's always going to be butterflies

play49:27

in other situations so here this isn't

play49:31

the uniform distribution as you

play49:32

generated but it's the poisson

play49:33

distribution

play49:35

there is lots of other situations

play49:37

football being one of them where we just