Calculus is cool. Spiritual, even. Infinity rocks.

Astoundingly, I'm enjoying a book about calculus, Infinite Powers by Steven Strogatz. The subtitle is the reason: How Calculus Reveals the Secrets of the Universe.

Hey, I'm all in on having the secrets of the universe revealed to me. Especially when the cost was a mere $16.52 to have Amazon deliver the book to my doorstep. 

I started this post with the word "astoundingly" because I was forced to take a calculus class in the first year of my Systems Science Ph.D. program studies way back when. (I completed the course work, but then gave up on being called Dr. Hines, as appealing as that prospect was.)

All I remember about the class was that I didn't like it. I passed, but I can't recall a single deep philosophical statement about the universe during that enforced dive into the equations of calculus, introductory variety.

I've only read 25 pages of Infinite Powers, yet already I'm fascinated by calculus -- because Strogatz, a Professor of Applied Mathematics at Cornell, is both an excellent writer and a talented expositor of this difficult subject. 

Here's a sampling in Strogatz' own words of what's intrigued me in the few pages I've perused so far. (The headlines are mine.)

Calculus is key to modern civilization.
Without calculus, we wouldn't have cell phones, computers, or microwave ovens. We wouldn't have radio. Or television. Or ultrasound for expectant mothers, or GPS for lost travelers. We wouldn't have split the atom, unraveled the human genome, or put astronauts on the moon. We might not even have the Declaration of Independence.

The universe is mathematical. 
For reasons nobody understands, the universe is deeply mathematical. Maybe God made it that way. Or maybe it's the only way a universe with us in it could be, because nonmathematical universes can't harbor life intelligent enough for us to ask the question.

In any case, it's a mysterious and marvelous fact that our universe obeys laws of nature that always turn out to be expressible in the language of calculus as sentences called differential equations. Such equations describe the difference between something right now and the same thing an instant later or between something right here and the same thing infinitesimally close by.

Calculus expresses the universe's operating system.
To put this awesome assertion another way, there seems to be something like a code to the universe, an operating system that animates everything from moment to moment and place to place. Calculus taps into this order and expresses it.

Infinity is what gives calculus its power.
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 means of calculus is that it takes this divide-and-conquer strategy to its utmost extreme -- all the way out to infinity.

Instead of cutting a big problem into a handful of bite-size pieces, it keeps cutting and cutting relentlessly until the problem has been chopped and pulverized into its tiniest conceivable parts, leaving infinitely many of them. Once that's done, it solves the original problem for all the tiny parts, which is usually a much easier task than solving the initial giant problem.

The remaining challenge at that point is to put all the tiny answers back together again. That tends to be a much harder step, but at least it's not as difficult as the original problem was.

Even the quantum world bows down to calculus. 
Admittedly, some aspects of our ever-changing world lie beyond the approximations and wishful thinking inherent in the Infinity Principle. In the subatomic realm, for example, physicists can no longer think of an electron as a classical particle following a smooth path in the same way that a planet or a cannonball does.

According to quantum mechanics, trajectories become jittery, blurry, and poorly defined at the microscopic scale, so we need to describe the behavior of electrons as probability waves instead of Newtonian trajectories. As soon as we do that, however, calculus returns triumphantly. It governs the evolution of probability waves through something called the Schrodinger equation.

It's incredible but true: Even in the subatomic realm where Newtonian physics breaks down, Newtonian calculus still works. In fact, it works spectacularly well.

Completed infinity is the original sin of calculus. 
Should we take the plunge and say that a circle truly is a polygon with infinitely many infinitesimal sides? No. We mustn't do that, mustn't yield to that temptation. Doing so would be to commit the sin of completed infinity. It would condemn us to logical hell.

...Like the biblical original sin, the original sin of calculus -- the temptation to treat a circle as an infinite polygon with infinitesimally short sides -- is very hard to resist, and for the same reason. It tempts us with the prospect of forbidden knowledge, with insights unavailable by ordinary means.

The sin of dividing by zero.
All across the world, students are taught that division by zero is forbidden... The root of the problem is infinity. Dividing by zero summons infinity in much the same way that a Ouija board supposedly summons spirits from another realm. It's risky. Don't go there.

...The transgression that dragged us into this mess was pretending that we could actually reach the limit, that we could treat infinity like an attainable number. 

...In the context of chopping a line into pieces, potential infinity would mean that the line could be cut into more and more pieces, as many as desired but still always a finite number and all of nonzero length. That's perfectly permissible and leads to no logical difficulties.

What's verboten is to imagine going all the way to a completed infinity of pieces of zero length.

That, Aristotle felt, would lead to nonsense -- as it does here, in revealing that zero times infinity can give any answer. And so he forbade the use of completed infinity in mathematics and philosophy. His edict was upheld by mathematicians for the next twenty-two hundred years.

Infinity is a dangerous temptress. 
Somewhere in the dark recesses of prehistory, somebody realized that numbers never end. And with that thought, infinity was born. It's the numerical equivalent 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 and science and mathematics, infinity has befuddled the world's finest minds for thousands of years.

Discrete and continuous become one. 
Consider how an old-fashioned analog clock differs from a modern-day digital/mechanical monstrosity. On the analog clock, the second hand sweeps around in a beautifully uniform motion. It depicts time as flowing. Whereas on the digital clock, the second hand jerks forward in discrete steps, thwack, thwack, thwack. It depicts time as jumping. 

Infinity can build a bridge between these two very different conceptions of time. Imagine a digital clock that advances through trillions of little clicks per second instead of one loud thwack. We would no longer be able to tell the difference between that kind of digital clock and a true analog clock. 

Likewise with movies and videos: as long as the frames flash by fast enough, say at thirty frames a second, they give the impression of seamless flow. And if there were infinitely many frames per second, the flow truly would be seamless. 

...So in everyday life, the gulf between the discrete and the continuous can often be bridged, at least to a good approximation. For many practical purposes, the discrete can stand in for the continuous, as long as we slice things thinly enough. In the ideal world of calculus, we can go one better.

Anything that's continuous can be sliced exactly (not just approximately) into infinitely many infinitesimal pieces. That's the Infinity Principle. With limits and infinity, the discrete and the continuous become one. 

Get real. Accept limits. Embrace continuous. 
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.

If all we ever used are decimals with only sixty digits of precision, the number line would be pockmarked and cratered. There would be holes where pi, the square root of two, and any other numbers that need infinitely many digits after the decimal point would exist. 

Even a simple fraction such as 1/3 would be missing, because it too requires an infinite number of digits (o.333...) to pinpoint its location on the number line. If we want to think of the totality of all numbers as forming a continuous line, those numbers have to be real numbers.

They may be an approximation to reality, but they work amazingly well. Reality is too hard to model any other way. With infinite decimals, was with the rest of calculus, infinity makes everything simpler.

Comments

[Vinny]

There is a joker, who says English, Maths, Physics and Gurudom; they are all one/nothingness. From nothingness/zero is coming infinite atomic energy. That joker might now say Calculus and Gurudom are one.

[Fairy]

The most beautiful equation or formula in the world: Euler's Equation!
"Euler's identity is beautiful because it combines five of the most important constants ( numbers ) in mathematics into a single equation. These are: 1 – the basis of all other numbers and the multiplicative identity. 0 – the concept of nothingness and the additive
identity.Jun 20, 2017"

N+

[Fairy]

2003
AN ESOTERIC THEORY ON TIME
If it takes light from Alpha Centaurus 4.5 years to reach Earth then does it take Earth's light the same time to reach
AC? I guess not, as the position of the sun is different for a viewing reflective perspective and also the energy of
both masses would be different. So what they radiate would be different.
Now, they both sit in the same time – or do they? But from my perspective they both exist now. But at the same time
they exist in different times, as one is deemed (probably) older than the other. Who really knows? A lot is based on
theory and known physics. What about unknown physics?
*Take two human beings in a room for example, separated by space and time. They occupy different space in the
same broader space (room) and are functioning during the same present time, which immediately becomes the past,
while still having a future, which is different for both entities supposedly living in the same time.
a. There must only be a present given the above. –0+ (ppf or past/present/future) = trinity =
negative/positive=neutral.
b. One entity’s energy light must take a certain amount of time to reach the other entity’s sensor field, and vice
versa. Does this energy source travel at the same speed for each? I would say no, as each energy source would
be different, as different entities and the sensory abilities of each entity to pick up/translate/assimilate would be
different for each.
c. They would not be the same age. One would have manifested before the other in its life span. (I am talking
about this physical region of humans).
d. Because one is older than the other, (assuming this is so, even if they are the same age, they would have come
in at a different space) then all of its past is its present, and that which has not come into being, is also its
present.
Logically, if there is infinite space and time and it is the energy that puts it all into motion, then is it that without
energy there is only eternal emptiness?
No motion without energy
No time without motion
No space without time.
They all need one another to exist; to manifest or is energy the supreme manifestation? Because it activates all
states of being. Is energy the logos? Where did it originate from? What is the vital force? Does it stand alone
without time and space? Does it need time and space or do time and space need it?
Is this the same argument as the sum having knowledge of the parts that make it up or can the sum or total stand-
alone? Zero needs nothing to be recognised or does it become zero or emptiness only because it is not negative –1
to n- or positive +1n+?? Does it matter? What is matter? All the above or none of the above????

By Fairy Gyani

25 June 2014
I would suggest that it is gravity that puts the energy of the bosons into effect; the “Higgs Boson” must be the
infinite dark matter “emptiness”. We will never know what quantum gravity is as at present it is just too far beyond
our research. Then, in the distant future, once discovered, (or should I say realised) there will be something else that
even puts that into effect in a progression of matter/anti-matter interdependence for existence or non-existence.
Therefore all matter and non-matter must be interdependent or dependent for arising or non-arising. Meanwhile we
mere and meek humans continue our destructive paths of killing one another in the name of a non-existent God and
man-made religions with unbelievably (in this day and age) silly beliefs. Philosophical religions (Buddhism) for one
advocate and explain quantum physics. What a clever quantum physicist Buddha was!

[Fairy]

A Fairy's Theories on Thought Speed and Mass or Lack Of and on What is a Neutrino
13 December 2015

c= speed of light constant
m= mass
0=No mass (thought)
t=no time all dimensions (no limitations)
v=velocity
n=infinity
v(e)=neutrino

The Buddhist theory of "Emptiness" and dependent origination

0=c(n-)+m(n-)=v(n+)=v(e) or 0
v
0

Therefore, I conclude that if a neutrino has zero rest matter, then its true independent state must be in emptiness, and its opposite must be in immeasurable speed. However, the paradox is that once it gains momentum, does it pass through its "emptiness" to create gravity in order to gain mass and create cause and effect states that are dependent on light and vibration to manifest all matter of matter?

I am just an average person of meagre intelligence, but I am so very interested in SETI@home and the activities at CERN and the LHC.

I never studied physics but I am so very interested in quantum and theoretical physics - not that I understand fully.

Is the above too far off the mark?

"Discovering the Higgs boson has allowed us to understand how particles acquire mass. Now with more data and with LHC collisions at higher energies, we hope to understand even more puzzling phenomena like, what is dark matter? And why is gravity so much weaker than the other forces? These are exciting times in particle physics." ( I must have found this quote from the Cern website).

By Fairy Gyani

P.S. Clearly I am no mathematician!

[Spence Tepper]

The creation, to a mystic, is entirely mathematical...

1. Divide by zero, you get infinity. Zero is the denominator of the creation. When we become zero, we become all.

2. Time doesn't exist. It is an infinite summation of an infinite number of discrete static points. Newton identified digital aspect right from the start. And time and all detection of time arises from taking static snapshots. We can't actually measure movement, only the elapsed infinity of static points between two static measures. Time is all illusion.

[anami]

Speaking of illusion...

"Did we evolve to see reality is it exists?"
https://bigthink.com/surprising-science/does-reality-exist

[Georgy Porgy]

Calculus is many things but ‘cool’ it is not.

Spence strikes me as a calculus Jedi.

[Dungeness]

Time is all illusion

Other mystics say "time is just an ordinate of space"... that
"elapsed infinity of static points". Yet, time gave birth to
space and zero, the magician. laughs "You belong to me."

[j]

It's really pretty simple. Just substitute x=cos\alp,\alp∈[0:;π]

And then \; |x+\sqrt{1-x^2}|=\sqrt{2}(2x^2-1)\Leftright |cos\alp +sin\alp |=\sqrt{2}(2cos^2\alp -1)Then∣x+ 1−x 2∣= 2 (2x 2 −1)\Leftright∣cos\alp+sin\alp∣= 2 (2cos 2\alp−1)
\displaystyle |\N {\sqrt{2}}cos(\alp-\frac{\pi}{4})|=\N {\sqrt{2}}cos(2\alp )\Right \alp\in[0\: ;\: \frac{\pi}{4}]\cup [\frac{3\pi}{4}\: ;\: \pi]∣N 2 cos(\alp− 4 π )∣=N 2 cos(2\alp)\Right\alp∈[0; 4 π ]∪[ 43π ;π]
1) \displaystyle \alp \in [0\: ;\: \frac{\pi}{4}]\alp∈[0; 4π = Sach Khand

You're welcome
​

[Jesse]

math is great i assume. but nothing about mathematics solves a single existential question which is all people seem to care about.

thinking and calculating are preferred distractions for modern man, but we'd likely be better off innumerate, illiterate and immersed in enthogenic trances.

[Meditator]

Fantastic and interesting and entertaining (as by my Master) piece, at least defining the extant of available tools as Maths as also Science about the world as 'live fiction' the Observers ( us) observe with obvious emotions and not as machines (the task could be simple if that be so and that's what The Saints expects us to be in real time if not have the capacity or flavour to lift veils , go inside to 'observe with inner eyes' the subtle reality not obvious to feeble mind power.

In fact we have been going past discrete frames with our pursuit to better , comfortable frames in the making prospectively for us all through the analysis and simulations by maths and science as carriers.

Beyond a range of thinking and extrapolative analysis the mind may become helpless even in best of the 'minds' as the desired parameters to observe may not be observable in real-time making minds to hang repetitively if subjected to more thinking and analysis using intricate mathematics and physics, thus, limiting the scope of the intents to unravel the Reality of a phenomenon to a considerable accuracy.

We begin as infinite (obvious egos) when born and grow observing and analysing infinite World around us and become 'zero', meaningless at death with all the observations and symmetries and postulates of Life and Universe and God useless for ourselves.

Then Maths defines anything at the best between say, limits of Zero and Infinity in terms of quantity ( and not as opposite or in minus terms) but what if Life and Universe even exist at lesser than zero and more than infinity state of existence and precisely definable there.

I think we are just helpless creatures subjected to intense mental pressures (those who think about it) of Life and beyond(Life) at the same time finding ourselves in each next frames of existence, may be in our beyond deaths. Rewarding ourselves an escape from these frames has been the target set for ourselves by the Mystics of yore and the Present

Of late we are in a catch-22 situation as we find it hard to submit ourselves to largely exceptionally money driven Mahatmas or Perfect Masters of present who assert and certify perfect answers to a Perfect World or Universe - not answerable perfectly in Maths and Science.

[OshoRobbins]

@J
I calculated that equation and guess what

And then \; |x+\sqrt{1-x^2}|=\sqrt{2}(2x^2-1)\Leftright |cos\alp +sin\alp |=\sqrt{2}(2cos^2\alp -1)Then∣x+ 1−x 2∣= 2 (2x 2 −1)\Leftright∣cos\alp+sin\alp∣= 2 (2cos 2\alp−1)
\displaystyle |\N {\sqrt{2}}cos(\alp-\frac{\pi}{4})|=\N {\sqrt{2}}cos(2\alp )\Right \alp\in[0\: ;\: \frac{\pi}{4}]\cup [\frac{3\pi}{4}\: ;\: \pi]∣N 2 cos(\alp− 4 π )∣=N 2 cos(2\alp)\Right\alp∈[0; 4 π ]∪[ 43π ;π]
1) \displaystyle \alp \in [0\: ;\: \frac{\pi}{4}]\alp∈[0; 4π = Sach Khand

It’s wrong
It equates to
Sach Khand +3

And now I am wondering if the +3
Is Anami, Agam, Alakh

[Spence Tepper]

When you go within, there is a stage where florescent geometric shapes emerge from the dark like living fractals, forming and expanding into a moving florescent mesh that appears alive, almost like the skin of an animal.

Where mathematics confirms spirituality is in the capacity for abstract concepts to form into hard laws by which the entire creation runs. Do you really think logic is the source of creative insight? That rational thought without creative insight has the capacity to piece it together? Hardly.

We experience in parts and conjecture the rest. And all error resides within the conjecture.

But for the scientist whose mind is disciplined with absolute focus on their work, insights arise that often prove 100% accurate. All proven mathematics and scientific principles are evidence of this mysterious process. The proof can take lifetimes to develop, after the insight. And then the rest of us learn to trust that insight. We are all aware of how errors are found and eradicated, but generally unaware of how the creative genius of insight works.

And until we can duplicate that in a machine, until a computer generates reliably its own insights that prove to be true, insight after insight, genius remains a mystery, a sacred trust to humanity from this even more mysterious creation.