Chapter 4: The Constants of Change by Ian Stewart

Summary

In 'Nature's Numbers' by Ian Stewart, the video explores the dichotomy between viewing the universe as governed by fixed laws versus a fluid, ever-changing reality. It delves into the historical significance of Newton's laws and calculus in describing nature's changes through differential equations. The narrative progresses to the challenges of solving complex systems, like the three-body problem, and the emergence of chaos theory, illustrating the evolution of 'solving' from exact formulas to understanding patterns. The video concludes by emphasizing the importance of qualitative understanding in addition to quantitative analysis, showcasing how constants generate change.

Takeaways

• 📚 Ian Stewart's book 'Nature's Numbers' explores the mathematical perspective on the natural world.
• 🌌 Two contrasting views of the universe are discussed: one where the universe obeys predictable laws, and another where objective reality is fluid and ever-changing.
• 🔍 The rise of science has been largely governed by the first viewpoint, emphasizing objective reality and predictable laws.
• 📈 Stewart discusses the role of calculus in describing change in nature, particularly through the use of differential equations.
• 🍎 Sir Isaac Newton's laws of physics, including gravity and his development of calculus, are highlighted as foundational to understanding change.
• 🌊 Differential equations are used to model various phenomena, such as wave behavior, exponential growth, and the spread of diseases.
• 🌐 Newton's law of gravitation is noted for its ability to describe the universe in terms of differential equations, influencing our understanding of celestial mechanics.
• 🧮 The script touches on the historical struggle to solve equations for systems with three or more bodies, leading to the development of approximation methods.
• 🔄 The concept of chaos theory is introduced, with examples like the three-body problem and the double pendulum, showing that solutions can be qualitative rather than exact.
• 🔑 The meaning of 'solving' a problem has evolved from finding exact formulas to understanding patterns and behaviors, reflecting a shift in scientific thinking.

Q & A

What is the main theme of Ian Stewart's book 'Nature's Numbers'?

The main theme of 'Nature's Numbers' is how mathematics is used to understand and describe the natural world, with a focus on the constants and change in nature.

What are the two contrasting views of nature mentioned in the script?

The two contrasting views of nature are: one that believes the universe obeys predictable, immutable laws and everything exists in a well-defined, objective reality; and the other that believes there is no objective reality, and everything is subject to flux and change.

Who is credited with the discovery of gravity and the invention of calculus, as mentioned in the script?

Sir Isaac Newton is credited with the discovery of gravity and the invention of calculus.

What is the significance of differential equations in describing change in nature?

Differential equations are significant in describing change in nature because they can model rates of change, such as the wave equation, which describes the rate of change of the height of a wave.

What is the wave equation, and how does it relate to calculus?

The wave equation is a differential equation that describes how the height of a wave changes over time. It relates to calculus because it involves rates of change, which are calculated using calculus techniques like integration and differentiation.

How does the script connect Newton's laws of physics to the concept of change?

The script connects Newton's laws of physics to the concept of change by explaining that Newton's laws can be used to describe changes in nature mathematically, particularly through the use of differential equations.

What is the significance of the three-body problem in the context of the script?

The three-body problem is significant because it demonstrates the limitations of finding exact solutions in complex systems. It led to the discovery of chaos theory and the understanding that some problems may not have exact solutions but can still be analyzed through approximate methods.

What does the script imply about the evolution of the concept of 'solving' in mathematics?

The script implies that the concept of 'solving' in mathematics has evolved from finding exact formulas to finding approximate numbers and, more recently, to describing the behavior and patterns of solutions.

How does the script relate the discovery of chaos theory to the understanding of change in nature?

The script relates the discovery of chaos theory to the understanding of change in nature by showing that seemingly random behaviors in systems can be analyzed and understood through the study of chaotic dynamics, which can provide insights into the patterns of change.

What is the role of qualitative understanding in the study of nature as per the script?

The role of qualitative understanding in the study of nature, as per the script, is to provide an alternative or complementary approach to mathematical processes, allowing for an understanding of nature's patterns in its own terms.

How does the script suggest that the meaning of constants in nature has changed over time?

The script suggests that the meaning of constants in nature has changed over time with the introduction of new scientific discoveries and theories, such as chaos theory, which have expanded our understanding of how constants generate change.

Outlines

00:00

📚 The Interplay of Mathematics and Nature

The video script begins by discussing Ian Stewart's book 'Nature's Numbers,' which explores the perspective of a mathematician on the natural world. It contrasts two historical views of nature: one that believes in a predictable, immutable universe, and another that sees reality as ever-changing and subjective. The script then transitions into a discussion of how science has been largely influenced by the first viewpoint, but societal shifts suggest a growing acceptance of the second. The focus is on the constants of reality and their role in generating change, with a particular emphasis on differential equations and their applications in describing natural phenomena. The video introduces the concept of 'constants of change' and how they have evolved over time, starting with the Renaissance and Sir Isaac Newton's discoveries, including his laws of physics and calculus, which laid the groundwork for understanding change through mathematical processes.

05:01

🌌 Chaos and the Evolution of Problem-Solving

The second paragraph delves into the concept of chaos and its implications for problem-solving in mathematics and physics. It mentions Jihong Sha's proof in 1994 that a system of three bodies is not integrable, leading to the discovery of Arnold diffusion, a phenomenon that causes a slow, random drift in orbital positions. This behavior, now recognized as chaos, is seen in other systems like the Lorenz attractor and the double pendulum. The script highlights the evolution of the meaning of 'solving' a problem, from finding exact formulas to approximating numbers and, more recently, to describing the patterns of solutions. It emphasizes that while some problems, like the three-body problem, may not have exact solutions, there are always ways to address them. The video concludes by connecting these mathematical concepts to a broader understanding of nature's patterns, suggesting that a qualitative approach can complement mathematical processes in understanding the natural world.

Keywords

💡Nature's Numbers

Nature's Numbers refers to Ian Stewart's book that explores the mathematical perspectives on the natural world. The book provides insights into how mathematicians view and analyze nature through mathematical models and equations. In the video, this concept is central as it sets the stage for discussing how constants and change are interwoven in natural phenomena.

💡Objective Reality

Objective reality is the idea that the universe follows predictable, immutable laws, and everything exists within a well-defined framework. This concept is contrasted with the idea of subjective reality in the script, highlighting the historical shift in scientific thinking. The video discusses how the rise of science has been largely governed by this viewpoint, emphasizing the deterministic nature of natural laws as described by classical physics.

💡Differential Equations

Differential equations are mathematical equations that describe the rate of change of a quantity over time or space. In the video, they are presented as a tool to describe change in nature, such as the wave equation, which describes the rate of change of the height of a wave. They are integral to understanding how calculus is applied to model real-world phenomena, including the growth of cancer or the spread of diseases.

💡Calculus

Calculus is a branch of mathematics that deals with the study of change and motion. It is mentioned in the script in relation to Newton's development of calculus alongside his laws of motion. Calculus provides the mathematical framework necessary for solving differential equations, which are crucial for understanding and predicting changes in various systems.

💡Isaac Newton

Sir Isaac Newton is a pivotal figure in the history of science, known for his laws of motion and universal gravitation. The script mentions Newton's contributions to calculus and his role in providing a mathematical framework to describe the universe. His work is foundational to the understanding of how the constants of nature generate change.

💡Constants of Change

Constants of change refers to the underlying mathematical principles and laws that govern the transformations and dynamics in the natural world. The video uses this term to discuss how these constants, through differential equations and calculus, help us understand and predict changes in various natural phenomena, such as the motion of celestial bodies or the spread of diseases.

💡Chaos Theory

Chaos theory is a field of study in mathematics that deals with systems that are highly sensitive to initial conditions, exhibiting unpredictable and seemingly random behavior. The script mentions Arnold diffusion and the three-body problem as examples of chaotic systems, where traditional solutions are not possible, and the focus shifts to understanding the qualitative behavior of the system.

💡Approximation

Approximation in the context of the video refers to the process of finding close estimates or numerical solutions for complex problems where exact solutions are not feasible. This concept is highlighted in the discussion of the three-body problem, where exact solutions could not be found, and mathematicians turned to approximations to understand the system's behavior.

💡Qualitative Understanding

Qualitative understanding is the process of comprehending the nature of things in terms of their properties and how they change, rather than relying solely on quantitative measures. The video suggests that a qualitative understanding of nature, which includes recognizing patterns and behaviors, is essential for a deeper comprehension of the natural world beyond just mathematical processes.

💡Three-Body Problem

The three-body problem is a classic problem in physics and mathematics that involves predicting the motion of three bodies under the influence of gravity. The script discusses how this problem led to the discovery of chaos theory and the realization that not all systems can be solved with exact formulas, shifting the focus to understanding the qualitative aspects of the system's behavior.

Highlights

Ian Stewart's book 'Nature's Numbers' explores the mathematician's view of the natural world.

Two contrasting views of nature have been formed throughout human history: one of predictable, immutable laws, and another of subjective reality and constant change.

The rise of science has largely been governed by the first viewpoint, emphasizing objective reality.

There are signs that the prevailing cultural background is shifting towards the second viewpoint, acknowledging the subjective and changing nature of reality.

The discussion will cover the constants of reality and how they generate change, altering the findings of the class.

Sir Isaac Newton's discoveries, including the law of universal gravitation and calculus, are pivotal in describing change in nature.

Newton's laws of physics allow the change in nature to be described using mathematical processes, such as differential equations.

Differential equations are essential in modeling real-life phenomena like wave behavior, exponential growth, and the spread of diseases.

Newton's law of gravitation was based on solving differential equations to describe the universe.

Attempts to solve equations for systems of three or more bodies led to the development of approximation methods.

Chaotic behavior, such as Arnold diffusion, challenges the traditional notion of solving equations by introducing unpredictability.

The meaning of 'solving' has evolved from finding exact formulas to approximating numbers and now to describing the nature of solutions.

The three-body problem illustrates the limitations of finding exact solutions and the shift towards understanding the qualitative nature of solutions.

Ian Stewart's work emphasizes the importance of understanding nature's patterns in its own terms, not just through mathematical processes.

The book 'Nature's Numbers' provides new perspectives on the application of mathematics in understanding the natural world.

The concept of constants of change, as explained through differential equations, shows how mathematical constants can lead to dynamic changes.

The meaning of 'solved' has changed over time with the introduction of new constants, reflecting a deeper understanding of nature's complexity.

Transcripts

in 1997 ian stewart

published his book called nature's

numbers

and in that book we got a glimpse of how

a mathematician

views our natural world but one chapter

really caught my eye

throughout human history two views have

been formed about how we view

nature one of you believes the universe

obeys predicts

immutable loss and everything is in a

well-defined

objective reality while the other

believes that there is no such thing as

objective reality

that all is flocks in all this change

the rise of science is largely been

governed by the first viewpoints

but as we advance together as a society

there have been increasing signs that

prevailing cultural background

is starting to switch to the second way

of thinking

in a span of seven minutes we're going

to talk about the constants of this

reality

and how they generate change and alter

the findings of the class

i am hella nicolas aragosa and today we

are going to dive deep

into the constants of change

and joining me today are aisling fate

strabusa

irene lava and michael raymond zamora

now we begin as we take a step back into

the renaissance

with the discoveries of sir isaac newton

when we hear the word newton what is the

first

thing that comes into mind

is it gravity the apple

falling from a tree both gases are

correct

however did you know that together with

madness

newton invented calculus

with their discovery they provided the

techniques of

integration and differentiation

both techniques work side by side or in

one

and thus the other

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between them they tell you that if you

know any of the functions

position velocity or

acceleration at every instant

then you can work out the other two

due to newton's law of physics the

change in nature can be described

using mathematical processes

for example wave equation

wave equation describes the rate of

change

of the height of the wave a rate of

change

is about the difference between some

quantity now and its value

and instant into the future equations of

this

kind are called differential equations

and with that we get a call back to

calculus

other examples where differential

equations are applied in real life

include

explaining the exponential growth and

the composition

the modification of return on investment

over time and the modeling of cancer

growth

or the spread of abcs to learn more

about newton's discoveries

let us take a trip to outer space

isn't it fascinating how the earth is

just floating magically in the darkness

of space

who am i kidding it's not magic it's the

sun's gravity that keeps us in place

a discovery made over 300 years ago

before we even set foot on the moon

newton's discovery of love gravitation

rested upon solution of describing the

universe

in terms of differential equations and

then solving it

he assumed that the same attractive

force must exist

between any two bodies in the universe

in those days solved meant finding a

mathematical formula for the emotion

other examples that rested upon a

solution of this kind

are paul's law laws of friction and

joule's law

when utahn and his successors tried

solving the equations for a system of

three or more bodies

they failed to find exact solutions

instead

they tried to find ways to calculate

approximate numbers

for example around 1860 charles eugene

de lunay

filled an entire book with a single

approximation to the motion of the

movement

other problems that have approximate

algorithms

are the bill packing problem the vertex

copper

and the shortest suppression perhaps you

might think we have hit a dead end

but as it was said before times are

changing

and so are the ways we think in 1994

jihongsha proved that a system of three

bodies

is not integrable since it demonstrates

arnold diffusion

which was discovered with vladimir arnin

this phenomena produces an extremely

slow

random drift in the relative orbital

positions

however this drift is not truly random

this behavior is now known as chaos

other examples of chaotic behavior

include the lord's attractor

double pendulum and the bonimovic

stadium it is worth noting that this

again

changes the meaning of solve it has

transitioned from finding a formula

to finding approximate numbers and now

it has become telling how resolutions

look like

it is wrong to see this development as a

repeat for what this change of meaning

has taught us

is that for questions like the dream

body problem for instance

no formulas can exist but there's always

a way to solve it

overall ian stewart's book nature's

numbers

is filled with useful information as

well as new perspectives of the use

of mathematics in nature seward

expounded

the concepts of constants of change

through differential equations

and its relation to change furthermore

he advanced it by illustrating how the

meaning was solved

changed over time with the introduction

of new constants

through the passing of you ions to work

provides an understanding of nature

that does not only rely on mathematical

processes

but an understanding of nature's

patterns using its own terms

by qualitative and with that

we can truly say that constant generate

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change

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you