Chapter 4: The Constants of Change by Ian Stewart
Summary
TLDRIn '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
📚 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.
🌌 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.
Mindmap
Keywords
💡Nature's Numbers
💡Objective Reality
💡Differential Equations
💡Calculus
💡Isaac Newton
💡Constants of Change
💡Chaos Theory
💡Approximation
💡Qualitative Understanding
💡Three-Body Problem
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
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