"An equilibrium is defined to be stable if all sufficiently small disturbances away from it damp out in time. Thus stable equilibria are represented geometrically by stable fixed points. Conversely, unstable equilibria, in which disturbances grow in time, are represented by unstable fixed points." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)
"[…] chaos and fractals are part of an even grander subject
known as dynamics. This is the subject that deals with change, with systems
that evolve in time. Whether the system in question settles down to equilibrium,
keeps repeating in cycles, or does something more complicated, it is dynamics
that we use to analyze the behavior." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)
"The qualitative structure of the flow can change as parameters
are varied. In particular, fixed points can be created or destroyed, or their
stability can change. These qualitative changes in the dynamics are called
bifurcations , and the parameter values at which they occur are called
bifurcation points." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)
"Why are nonlinear systems so much harder to analyze than linear ones? The essential difference is that linear systems can be broken down into parts. Then each part can be solved separately and finally recombined to get the answer. This idea allows a fantastic simplification of complex problems, and underlies such methods as normal modes, Laplace transforms, superposition arguments, and Fourier analysis. In this sense, a linear system is precisely equal to the sum of its parts." (Steven H Strogatz, "Non-Linear Dynamics and Chaos, 1994)
"A depressing corollary of the butterfly effect (or so it was
widely believed) was that two chaotic systems could never synchronize with each
other. Even if you took great pains to start them the same way, there would
always be some infinitesimal difference in their initial states. Normally that
small discrepancy would remain small for a long time, but in a chaotic system,
the error cascades and feeds on itself so swiftly that the systems diverge
almost immediately, destroying the synchronization. Unfortunately, it seemed,
two of the most vibrant branches of nonlinear science - chaos and sync - could
never be married. They were fundamentally incompatible." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"[…] all human beings - professional mathematicians
included - are easily muddled when it comes to estimating the probabilities of
rare events. Even figuring out the right question to ask can be confusing." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"Although the shape of chaos is nightmarish, its voice is
oddly soothing. When played through a loudspeaker, chaos sounds like white
noise, like the soft static that helps insomniacs fall asleep." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"At an anatomical level - the level of pure, abstract
connectivity - we seem to have stumbled upon a universal pattern of complexity.
Disparate networks show the same three tendencies: short chains, high
clustering, and scale-free link distributions. The coincidences are eerie, and
baffling to interpret." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"Average path length reflects the global structure; it
depends on the way the entire network is connected, and cannot be inferred from
any local measurement. Clustering reflects the local structure; it depends only
on the interconnectedness of a typical neighborhood, the inbreeding among nodes
tied to a common center. Roughly speaking, path length measures how big the
network is. Clustering measures how incestuous it is." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"But linearity is often an approximation to a more
complicated reality. Most systems behave linearly only when they are close to
equilibrium, and only when we don't push them too hard." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"By its very nature, the mathematical study of networks
transcends the usual boundaries between disciplines. Network theory is
concerned with the relationships between individuals, the patterns of interactions.
The precise nature of the individuals is downplayed, or even suppressed, in
hopes of uncovering deeper laws. A network theorist will look at any system of
interlinked components and see an abstract pattern of dots connected by lines.
It's the pattern that matters, the architecture of relationships, not the
identities of the dots themselves. Viewed from these lofty heights, many
networks, seemingly unrelated, begin to look the same." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"Chaos theory revealed that simple nonlinear systems could
behave in extremely complicated ways, and showed us how to understand them with
pictures instead of equations. Complexity theory taught us that many simple
units interacting according to simple rules could generate unexpected order.
But where complexity theory has largely failed is in explaining where the order
comes from, in a deep mathematical sense, and in tying the theory to real phenomena
in a convincing way. For these reasons, it has had little impact on the thinking
of most mathematicians and scientists." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"[…] equilibrium means nothing changes; stability means
slight disturbances die out." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"From a purely mathematical perspective, a power law
signifies nothing in particular - it's just one of many possible kinds of
algebraic relationship. But when a physicist sees a power law, his eyes light
up. For power laws hint that a system may be organizing itself. They arise at
phase transitions, when a system is poised at the brink, teetering between
order and chaos. They arise in fractals, when an arbitrarily small piece of a
complex shape is a microcosm of the whole. They arise in the statistics of
natural hazards - avalanches and earthquakes, floods and forest fires - whose sizes
fluctuate so erratically from one event to the next that the average cannot
adequately stand in for the distribution as a whole." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"In colloquial usage, chaos means a state of total disorder.
In its technical sense, however, chaos refers to a state that only appears
random, but is actually generated by nonrandom laws. As such, it occupies an
unfamiliar middle ground between order and disorder. It looks erratic
superficially, yet it contains cryptic patterns and is governed by rigid rules.
It's predictable in the short run but unpredictable in the long run. And it
never repeats itself: Its behavior is nonperiodic." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"Just as a circle is the shape of periodicity, a strange
attractor is the shape of chaos. It lives in an abstract mathematical space
called state space, whose axes represent all the different variables in a
physical system." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"Like regular networks, random ones are seductive
idealizations. Theorists find them beguiling, not because of their
verisimilitude, but because they're the easiest ones to analyze. [...] Random networks are small and poorly clustered; regular ones
are big and highly clustered." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"One of the most wonderful things about curiosity-driven
research - aside from the pleasure it brings - is that it often has unexpected
spin-offs." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"Scientists have long been baffled by the existence of
spontaneous order in the universe. The laws of thermodynamics seem to dictate
the opposite, that nature should inexorably degenerate to - ward a state of
greater disorder, greater entropy." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"Structure always affects function. The structure of social
networks affects the spread of information and disease; the structure of the
power grid affects the stability of power transmission. The same must be true
for species in an ecosystem, companies in the global marketplace, cascades of
enzyme reactions in living cells. The layout of the web must profoundly shape
its dynamics." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"The best case that can be made for human sync to the
environment (outside of circadian entrainment) has to do with the possibility
that electrical rhythms in our brains can be influenced by external signals." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"The butterfly effect came to be the most familiar icon of
the new science, and appropriately so, for it is the signature of chaos. […] The
idea is that in a chaotic system, small disturbances grow exponentially fast,
rendering long-term prediction impossible." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"The nonlinear dynamics of systems with that many variables
is still beyond us. Even with the help of supercomputers, the collective
behavior of gigantic systems of oscillators remains a forbidding terra
incognita." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"[...] the transition to a small world is essentially undetectable at a local level. If you were living through the morph, nothing about your immediate neighborhood would tell you that the world had become small." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"The uncertainty principle expresses a seesaw relationship between
the fluctuations of certain pairs of variables, such as an electron's position
and its speed. Anything that lowers the uncertainty of one must necessarily
raise the uncertainty of the other; you can't push both down at the same time.
For example, the more tightly you confine an electron, the more wildly it
thrashes. By lowering the position end of the seesaw, you force the velocity
end to lift up. On the other hand, if you try to constrain the electron's velocity
instead, its position becomes fuzzier and fuzzier; the electron can turn up almost
anywhere." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"These, then, are the defining features of chaos: erratic,
seemingly random behavior in an otherwise deterministic system; predictability
in the short run, because of the deterministic laws; and unpredictability in
the long run, because of the butterfly effect." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"This synergistic character of nonlinear systems is precisely
what makes them so difficult to analyze. They can't be taken apart. The whole
system has to be examined all at once, as a coherent entity. As we've seen
earlier, this necessity for global thinking is the greatest challenge in
understanding how large systems of oscillators can spontaneously synchronize
themselves. More generally, all problems about self-organization are
fundamentally nonlinear. So the study of sync has always been entwined with the
study of nonlinearity." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"[…] topology, the study of continuous shape, a kind of generalized
geometry where rigidity is replaced by elasticity. It's as if everything is
made of rubber. Shapes can be continuously deformed, bent, or twisted, but not
cut - that's never allowed. A square is topologically equivalent to a circle,
because you can round off the corners. On the other hand, a circle is different
from a figure eight, because there's no way to get rid of the crossing point
without resorting to scissors. In that sense, topology is ideal for sorting
shapes into broad classes, based on their pure connectivity." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"Unanticipated forms of collective behavior emerge that are
not obvious from the properties of the individuals themselves. All the models
are extremely simplified, of course, but that's the point. If even their idealized
behavior can surprise us, we may find clues about what to expect in the real
thing. […] the collective dynamics of a crowd can be exquisitely sensitive to
its composition, which may be one reason why mobs are so unpredictable, which may be one reason why mobs are so unpredictable." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"We’re accustomed to in terms of centralized control, clear chains of command, the straightforward logic of cause and effect. But in huge, interconnected systems, where every player ultimately affects every other, our standard ways of thinking fall apart. Simple pictures and verbal arguments are too feeble, too myopic." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"When you’re trying to prove something, it helps to know it’s true. That gives you the confidence you need to keep searching for a rigorous proof." (Steven Strogatz, "Sync: The Emerging Science of Spontaneous Order", 2003)
"Change is most sluggish at the extremes precisely because the derivative is zero there." (Steven Strogatz, "The Joy of X: A Guided Tour of Mathematics, from One to Infinity", 2012)
"In mathematics, our freedom lies in the questions we ask - and in how we pursue them - but not in the answers awaiting us." (Steven Strogatz, "The Joy of X: A Guided Tour of Mathematics, from One to Infinity", 2012)
"Proofs can cause dizziness or excessive drowsiness. Side effects of prolonged exposure may include night sweats, panic attacks, and, in rare cases, euphoria. Ask your doctor if proofs are right for you." (Steven Strogatz, "The Joy of X: A Guided Tour of Mathematics, from One to Infinity", 2012)
"[...] things that seem hopelessly random and unpredictable when viewed in isolation often turn out to be lawful and predictable when viewed in aggregate." (Steven Strogatz, "The Joy of X: A Guided Tour of Mathematics, from One to Infinity", 2012)
"A limit cycle is an isolated closed trajectory. Isolated
means that neighboring trajectories are not closed; they spiral either toward
or away from the limit cycle. If all neighboring trajectories approach the
limit cycle, we say the limit cycle is stable or attracting. Otherwise the
limit cycle is unstable, or in exceptional cases, half-stable. Stable limit
cycles are very important scientifically - they model systems that exhibit
self-sustained oscillations. In other words, these systems oscillate even in the
absence of external periodic forcing." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)
"An equilibrium is defined to be stable if all sufficiently
small disturbances away from it damp out in time. Thus stable equilibria are
represented geometrically by stable fixed points. Conversely, unstable equilibria,
in which disturbances grow in time, are represented by unstable fixed points." (Steven
H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics,
Biology, Chemistry, and Engineering", 2015)
"[…] chaos and fractals are part of an even grander subject
known as dynamics. This is the subject that deals with change, with systems
that evolve in time. Whether the system in question settles down to equilibrium,
keeps repeating in cycles, or does something more complicated, it is dynamics
that we use to analyze the behavior." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)
"The qualitative structure of the flow can change as parameters
are varied. In particular, fixed points can be created or destroyed, or their
stability can change. These qualitative changes in the dynamics are called bifurcations,
and the parameter values at which they occur are called bifurcation points. Bifurcations
are important scientifically - they provide models of transitions and instabilities
as some control parameter is varied." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)
"[…] what exactly do we mean by a bifurcation? The usual definition
involves the concept of 'topological equivalence': if the phase portrait
changes its topological structure as a parameter is varied, we say that a
bifurcation has occurred. Examples include changes in the number or stability
of fixed points, closed orbits, or saddle connections as a parameter is varied." (Steven H Strogatz, "Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering", 2015)
"Why do mathematicians care so much about π? Is it some kind of weird circle fixation? Hardly. The beauty of π, in part, is that it puts infinity within reach. Even young children get this. The digits of π never end and never show a pattern. They go on forever, seemingly at random - except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of π." (Steven Strogatz, "Why π Matters" 2015)
"Although base e is uniquely distinguished, other exponential
functions obey a similar principle of growth. The only difference is that the
rate of exponential growth is proportional to the function’s current level, not
strictly equal to it. Still, that proportionality is sufficient to generate the
explosiveness we associate with exponential growth." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"An infinitesimal is a hazy thing. It is supposed to be the
tiniest number you can possibly imagine that isn’t actually zero. More
succinctly, an infinitesimal is smaller than everything but greater than
nothing. Even more paradoxically, infinitesimals come in different sizes. An infinitesimal
part of an infinitesimal is incomparably smaller still. We could call it a
second-order infinitesimal." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Because of its intimate connection to the backward problem,
the area problem is not just about area. It’s not just about shape or the relationship
between distance and speed or anything that narrow. It’s completely general.
From a modern perspective, the area problem is about predicting the
relationship between anything that changes at a changing rate and how much that
thing builds up over time." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Because of the geometry of a circle, there’s always a
quarter-cycle off set between any sine wave and the wave derived from it as its
derivative, its rate of change. In this analogy, the point’s direction of
travel is like its rate of change. It determines where the point will go next
and hence how it changes its location. Moreover, this compass heading of the
arrow itself rotates in a circular fashion at a constant speed as the point
goes around the circle, so the compass heading of the arrow follows a
sine-wave pattern in time. And since the compass heading is like the rate of
change, voilà! The rate of change follows a sine-wave pattern too." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Calculus succeeds by breaking complicated problems down into simpler parts. That strategy, of course, is not unique to calculus. All good problem-solvers know that hard problems become easier when they’re split into chunks. The truly radical and distinctive move of calculus is that it takes this divide-and-conquer strategy to its utmost extreme - all the way out to infinity." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Chaotic systems are finicky. A little change in how they’re
started can make a big difference in where they end up. That’s because small changes
in their initial conditions get magnified exponentially fast. Any tiny error or
disturbance snowballs so rapidly that in the long term, the system becomes
unpredictable. Chaotic systems are not random - they’re deterministic and hence
predictable in the short run - but in the long run, they’re so sensitive to
tiny disturbances that they look effectively random in many respects." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Generally speaking, things can change in one of three ways: they can go up, they can go down, or they can go up and down. In other words, they can grow, decay, or fluctuate. Different functions are suitable for different occasions." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"In analysis, one solves a problem by starting at the end, as
if the answer had already been obtained, and then works back wishfully toward
the beginning, hoping to find a path to the given assumptions. [….] Synthesis
goes in the other direction. It starts with the givens, and then, by stabbing
in the dark, trying things, you are somehow supposed to move forward to a
solution, step by logical step, and eventually arrive at the desired result.
Synthesis tends to be much harder than analysis because you don’t ever know how
you’re going to get to the solution until you do." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"In mathematical modeling, as in all of science, we always
have to make choices about what to stress and what to ignore. The art of
abstraction lies in knowing what is essential and what is minutia, what is
signal and what is noise, what is trend and what is wiggle. It’s an art because
such choices always involve an element of danger; they come close to wishful
thinking and intellectual dishonesty." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"In mathematics, pendulums stimulated the development of
calculus through the riddles they posed. In physics and engineering, pendulums
became paradigms of oscillation. […] In some cases, the connections between
pendulums and other phenomena are so exact that the same equations can be
recycled without change. Only the symbols need to be reinterpreted; the syntax
stays the same. It’s as if nature keeps returning to the same motif again and
again, a pendular repetition of a pendular theme. For example, the equations
for the swinging of a pendulum carry over without change to those for the
spinning of generators that produce alternating current and send it to our
homes and offices. In honor of that pedigree, electrical engineers refer to
their generator equations as swing equations." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"If real numbers are not real, why do mathematicians love
them so much? And why are schoolchildren forced to learn about them? Because
calculus needs them. From the beginning, calculus has stubbornly insisted that
everything - space and time, matter and energy, all objects that ever have been
or will be - should be regarded as continuous. Accordingly, everything can and
should be quantified by real numbers. In this idealized, imaginary world, we pretend
that everything can be split finer and finer without end. The whole theory of
calculus is built on that assumption. Without it, we couldn’t compute limits, and
without limits, calculus would come to a clanking halt." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Mathematically, circles embody change without change. A point
moving around the circumference of a circle changes direction without ever
changing its distance from a center. It’s a minimal form of change, a way to
change and curve in the slightest way possible. And, of course, circles are
symmetrical. If you rotate a circle about its center, it looks unchanged. That
rotational symmetry may be why circles are so ubiquitous. Whenever some aspect
of nature doesn’t care about direction, circles are bound to appear. Consider what
happens when a raindrop hits a puddle: tiny ripples expand outward from the
point of impact. Because they spread equally fast in all directions and because
they started at a single point, the ripples have to be circles. Symmetry
demands it." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Mathematicians don’t come up with the proofs first. First
comes intuition. Rigor comes later. This essential role of in- tuition and
imagination is often left out of high-school geometry courses, but it is
essential to all creative mathematics." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Nonlinearity is responsible for the richness in the world,
for its beauty and complexity and, often, its inscrutability. […] When a system
is nonlinear, its behavior can be impossible to forecast with formulas, even
though that behavior is completely determined. In other words, determinism does
not imply predictability. […] Chaotic systems can be predicted perfectly well
up to a time known as the predictability horizon. Before that, the determinism of
the system makes it predictable." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"On a linear system like a scale, the whole is equal to the sum of the parts. That’s the first key property of linearity. The second is that causes are proportional to effects. […] These two properties - the proportionality between cause and effect, and the equality of the whole to the sum of the parts - are the essence of what it means to be linear. […] The great advantage of linearity is that it allows for reductionist thinking. To solve a linear problem, we can break it down to its simplest parts, solve each part separately, and put the parts back together to get the answer." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Pi is fundamentally a child of calculus. It is defined as the unattainable limit of a never-ending process. But unlike a sequence of polygons steadfastly approaching a circle or a hapless walker stepping halfway to a wall, there is no end in sight for pi, no limit we can ever know. And yet pi exists. There it is, defined so crisply as the ratio of two lengths we can see right before us, the circumference of a circle and its diameter. That ratio defines pi, pinpoints it as clearly as can be, and yet the number itself slips through our fingers." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"So there is a lot to be said for following one’s curiosity
in mathematics. It often has scientific and practical payoff s that can’t be
foreseen. It also gives mathematicians great pleasure for its own sake and reveals
hidden connections between different parts of mathematics." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Somewhere in the dark recesses of prehistory, somebody
realized that numbers never end. And with that thought, infinity was born. It’s
the numerical counterpart of something deep in our psyches, in our nightmares
of bottomless pits, and in our hopes for eternal life. Infinity lies at the
heart of so many of our dreams and fears and unanswerable questions: How big is
the universe? How long is forever? How powerful is God? In every branch of
human thought, from religion and philosophy to science and mathematics, infinity
has befuddled the world’s finest minds for thousands of years."(Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"[…] the derivative of a sine wave is another sine wave, shifted by a quarter cycle. That’s a remarkable property. It’s not true of other kinds of waves. Typically, when we take the derivative of a curve of any kind, that curve will become distorted by being differentiated. It won’t have the same shape before and after. Being differentiated is a traumatic experience for most curves. But not for a sine wave. After its derivative is taken, it dusts itself of f and appears unfazed, as sinusoidal as ever. The only injury it suffers - and it isn’t even an injury, really - is that the sine wave shifts in time. It peaks a quarter of a cycle earlier than it used to." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"The great advantage of infinitesimals in general and differentials
in particular is that they make calculations easier. They provide shortcuts.
They free the mind for more imaginative thought, just as algebra did for
geometry in an earlier era. […] The only thing wrong with infinitesimals is
that they don’t exist, at least not within the system of real numbers." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"The important point about e is that an exponential function with this base grows at a rate precisely equal to the function itself. Let me say that again. The rate of growth of ex is ex itself. This marvelous property simplifies all calculations about exponential functions when they are expressed in base e. No other base enjoys this simplicity. Whether we are working with derivatives, integrals, differential equations, or any of the other tools of calculus, exponential functions expressed in base e are always the cleanest, most elegant, and most beautiful." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"The reason why integration is so much harder than differentiation
has to do with the distinction between local and global. Local problems are
easy. Global problems are hard. Differentiation is a local operation. [...] when we are calculating a derivative, it’s like we’re looking under a
microscope. We zoom in on a curve or a function, repeatedly magnifying the field
of view. As we zoom in on that little local patch, the curve appears to become
less and less curved. […] Integration is a global operation. Instead of a
microscope, we are now using a telescope. We are trying to peer far of f into
the distance - or far ahead into the future, although in that case we need a
crystal ball. Naturally, these problems are a lot harder. All the intervening
events matter and cannot be discarded. Or so it would seem." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"There’s something so paradoxical about pi. On the one hand, it
represents order, as embodied by the shape of a circle, long held to be a
symbol of perfection and eternity. On the other hand, pi is unruly, disheveled
in appearance, its digits obeying no obvious rule, or at least none that we can
perceive. Pi is elusive and mysterious, forever beyond reach. Its mix of order
and disorder is what makes it so bewitching." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"Thus, calculus proceeds in two phases: cutting and rebuilding. In mathematical terms, the cutting process always involves infinitely fine subtraction, which is used to quantify the differences between the parts. Accordingly, this half of the subject is called differential calculus. The reassembly process always involves infinite addition, which integrates the parts back into the original whole. This half of the subject is called integral calculus." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"To shed light on any continuous shape, object, motion, process, or phenomenon - no matter how wild and complicated it may appear - reimagine it as an infinite series of simpler parts, analyze those, and then add the results back together to make sense of the original whole." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"We feel we are discovering mathematics. The results are
there, waiting for us. They have been inherent in the figures all along. We are
not inventing them. […] we are discovering facts that already exist, that are
inherent in the objects we study. Although we have creative freedom to invent
the objects themselves - to create idealizations like perfect spheres and
circles and cylinders - once we do, they take on lives of their own." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"[…] when a curve does look increasingly straight when we zoom in on it sufficiently at any point, that curve is said to be smooth. […] In modern calculus, however, we have learned how to cope with curves that are not smooth. The inconveniences and pathologies of non-smooth curves sometimes arise in applications due to sudden jumps or other discontinuities in the behavior of a physical system." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"With a linear growth of errors, improving the measurements
could always keep pace with the desire for longer prediction. But when errors
grow exponentially fast, a system is said to have sensitive dependence on its
initial conditions. Then long-term prediction becomes impossible. This is the
philosophically disturbing message of chaos." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
"With its yin and yang binaries, pi is like all of calculus in miniature. Pi is a portal between the round and the straight, a single number yet infinitely complex, a balance of order and chaos. Calculus, for its part, uses the infinite to study the finite, the unlimited to study the limited, and the straight to study the curved. The Infinity Principle is the key to unlocking the mystery of curves, and it arose here first, in the mystery of pi." (Steven H Strogatz, "Infinite Powers: The Story of Calculus - The Most Important Discovery in Mathematics", 2019)
