Mathematics and Continuing Creation
Is mathematics the “language of God?” Or rather, as we will ask in this Book, “Is mathematics the language of G>O>D>?” We will discuss that at length, but one thing is certain: knowing just a few things about mathematics can greatly augment our appreciation of G>O>D>: The Growing>Organizing>Direction of Continuing Creation. How does mathematics augment our appreciation of Continuing Creation? By showing us the logical steps that create the Patterns we describe in our sister-Essay, Patterns in Continuing Creation. Those logical steps are described in the language of mathematics. Mathematics can also describe the processes by which forces move and assemble matter. (We discuss those processes in our Essay, Physics and Continuing Creation.) And most importantly, mathematics has a profound beauty which we can all appreciate with just a little introduction.
Note: Readers may be interested in watching some of the fine television programs and motion pictures that deal with mathematics and mathematicians:
- Documentary programs on the PBS series, NOVA, including “The Great Math Mystery,” Cosmic Code Breakers,” and “Prediction by the Numbers.”
- Major motion pictures: Stand and Deliver, A Beautiful Mind, Good Will Hunting, Pi, and
Introduction to Mathematics and Continuing Creation
Our purpose in this Essay is not to teach you mathematics, but to demonstrate that its power and beauty are fundamental features of Continuing Creation.
While mathematics cannot portray the wonderful fragrance of a rose, it can show how Nature uses a simple progression of numbers to create the Spiral Pattern of every rose blossom. For many of us, this kind of understanding produces a “wow factor” just as intense as our sense of smell does.
As we go along in this Essay, we will see how the descriptive power and the beauty of mathematics result from its ability to idealize forms, explain symmetry, achieve economy, and link seemingly remote phenomena. We will show how mathematics can make processes more efficient and predictable.
In sum, the real world is beautiful to our senses, mathematics is beautiful to our thinking, and each type of apprehension enhances the beauty of the other – the whole is greater than the sum of its parts.
Don’t worry — the aspects of mathematics we present can be understood by nearly everyone who reads this Essay. And if some readers find this Essay too challenging, they can set it aside without doing much damage to one’s understanding of The Path of G>O>D>.
There is something fundamental, something “not physically real” about mathematics. There is no perfect circle in nature, yet we can perfectly define a “circle” using words: “On a flat plane, a circle is the set of all points that are equidistant from one single point on that plane.” So, which came first – the ideal circle, or all the real approximate circles in Nature? Is mathematics invented by the human brain, or discovered in Nature by the human brain? Is mathematics the language of G>O>D>? Is mathematics more fundamental than physical reality? At the end of his Essay, will we be better equipped to discuss these philosophical questions.
The Pentagram, Golden Ratio, Fibonacci Numbers, and Spirals
In this Essay we also learn that same series of numbers creates hundreds of other spirals found in the Cosmos and in Nature – including the spirals of galaxies, pinecones, snail shells, and staircases.
The homepage of our website, www.ContinuingCreation.org, begins with photographs of spiral of galaxies, pinecones, snail shells, and staircases. Here they are again:
As we shall see, the Spiral Pattern is related to a special number called the Golden Ratio; to pentagons and pentagrams, and to a unique string of numbers called the Fibonacci Sequence.
The Golden Ratio, (or Golden Mean, or Golden Proportion)
The Golden Ratio is the number 1.6180339887498948482… (The use of three dots conventionally indicates that the series goes on forever.). Mathematicians named this number after the Greek letter “Phi,” (pronounced “Fie”) which the ancient Greeks wrote as a small oval with a vertical line drawn through it — ϕ.
The simplest way to calculate the Golden Ratio is to take any straight line and divide it into two segments in a certain way. Say our line is 34 inches long. We want to cut it so we have a longer segment on the left, and a shorter segment on the right. We also want the proportion of the original 34-inch line to the big segment, to be the same as the proportion of the big segment to the little segment.
By trial-and-error measurement and estimating, we would find that we should we cut the 34-inch line into a big segment that’s about 21 inches long and the little segment that’s about 13 inches long. We can check it like this:
34 divided by 21 = 1.619…, and …
21 divided by 13 = 1.615…
Since 1.619… is pretty close to 1.615…, we are pretty close to making the cut in the right place.
Here’s a drawing of this “line-dividing method,” plus a restatement of the line concept using a bit of simple algebra: we define the golden ratio as the number we get when we divide a line into two parts so that:
It can be also proven in mathematics that Phi more generally equals (1+√5) / 2. But we are not going to present that proof here. Do note the formula’s reference to the number 5 as you read the following section.
Pentagons and Pentagrams
If we approach the Golden Ratio (also known as the Golden Mean) historically, the story starts with the Ancient Greek mathematician Pythagoras.
Pythagoras and his followers thought that numbers, mathematics, and geometric shapes were the fundamental reality underlying the universe. They thought the number 5 was sacred because it included the 4 “basic substances” of water, fire, air, and earth. The creative union of those four substances into the whole of Creation is represented by the number 5. For the Pythagoreans, the 5-sided pentagon, along with the 5-pointed pentagram based on it, symbolized what we today might call the Processes of Creation. 1
Golden Ratios in the Pentagon
Phi shows up in several places within the regular five-sided pentagon and in the 5-pointed pentagram. In fact, modern mathematicians have continued to find new incidences of Phi in these geometric shapes.
The figure below shows a small inner pentagon and a larger pentagram made by extending its sides out to make five points. Then, those 5 points are used to define the 5 sides of a larger outer pentagon:
Where are the Golden Ratios? Where are the Phi’s?
To point out a few, consider the length of the line between point B and point E (called line BE); vs. the length between points A and E (called line AE), etc. Then Compute AE / BE. The answer is Phi.
Then do the same for all the following pairs of lines:
DE/EX = EX/XY = UV/XY = EY/EX = BE/AE = all equal Phi.
All five of the above line-length ratios = Phi =1.6180339887498948482…
The Golden Ratio — An Ideal of Beautiful Proportion
For centuries, artists and architects have used the Golden Ratio segments to define the proportions of their work because they seem to be aesthetically pleasing to the human eye and mind. This use has been called the “divine proportion.”
A rectangle with its sides in this ‘golden’ ratio is thought of as aesthetically pleasing. So, a painting 34 inches wide would often be made 20 7/8 inches high, because 34 / 21 = 1.619, which is about equal to Phi.
The architecture of the hundreds of thousands of buildings, old and new, including the Parthenon in Athens, Greece, incorporate the Golden Ratio. In the Parthenon, the height of the lower columned façade is 1.618 times the height of the face (the “pediment”) above the columns. 2
So, if you wanted to build a post office like the Parthenon, and if you wanted a total height of 70 feet, you would make the lower (columned) section of the façade about 43 ¼ feet height, and the upper (pediment) section of the façade about 26 3.4 feet high.
For readers who would like to know how we figure that out, we would use a little algebra:
Let a = height of the lower and taller “columns” part
Let b = height of the upper and shorter part (the “pediment”) on top of the columns
We require: b + a = 70 feet = total height.
And we require: a = 1.618034 x b
Substituting, we get b + 1.618034b = 70 feet
So, 2.618034b = 70 feet
So, b = 70 / 2.618034
So b = 26.74 feet. = the height of the columnar base
So a = 1.618034 x 26.74 = 43.26 feet = the height of the upper pediment
And a + b = 43.26 + 26.74 = 70 feet total height
Note: For more information about the intersection of art and mathematics, including beautiful illustrations, search online for the Wikipedia article, Mathematics and Art
The Golden Ratio and the Fibonacci Sequence
We’re almost ready to show how the Golden Ratio is used to create the spirals we see in the Cosmos and Nature.
In 1202, an Italian mathematician named Fibonacci (short for “Son of Bonacci”) wrote the Liber Abaci (Book of Calculation) which introduced the Hindu-Arabic number system to the West. In this book, Fibonacci also described the idealized pattern of population growth using a sequence of numbers we now call the Fibonacci Sequence.
In the Fibonacci Sequence, each number is the sum of the previous two numbers. Here is the sequence:
1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610… and so on.
We see that in this sequence, 8 + 13 = the next Fibonacci number after 13, which is 21. Then, 13 + 21 yields the next number in the Fibonacci sequence, which is 34. And so on.
Centuries later, in the late 1700’s, Johannes Kepler discovered that when you take any two adjacent numbers in the Fibonacci Sequence, and divide the bigger one by the smaller, you get an approximation of the Golden Ratio! And if you do this using neighbor-numbers further and further out on the Sequence, the approximation of Phi becomes more and more exact. 3
Thus, using Fibonacci numbers 13 / 8 = 1.625000.
Using more distant Fibonacci numbers, 610 / 377 = 1.618037 (awfully close to Phi, which is 1.618034).
Therefore, the Golden Mean is connected to the Fibonacci Number Sequence. This is one of the many relationships in mathematics that are amazing and almost magical when they are first discovered.
Now, let’s stop right there. When your author, J.X. Mason, took math in high school, the teacher would point to a mathematical connection like this and say… “Now, isn’t that interesting?” He or she would say the same thing when pointing out some of the Golden Ratios present in a pentagon.
Well, yes, these connections are interesting, but The Way of Continuing Creation says they are also wondrous, powerful, and beautiful. Mathematics provides the compact language and the logical tools by which we can see these connections.
(In J.X. Mason’s high school, we never dwelled too long even on the word “interesting,” because in those days, before computers, teaching mathematics was about learning to do accurate manipulations and calculations. The school system was trying to produce accurate and efficient future engineers, not appreciative spiritual beings. )
In fact, if you take the Fibonacci Sequence out to infinity, i.e. to its limit, a mathematician named Jacques Binet proved in 1843 that the Golden Ratio is exactly described by a short and formula (now called the Binet’s Formula):
Phi = (1 + the square root of 5) / 2 = 1.6180339887….
Again, we’re not going to show that proof here; just note that it exists. Also, note the appearance of the Pythagoreans’ favorite number, 5, in this formula.
Another “interesting” (amazing) thing about Phi is that (Phi-times-Phi) equals Phi + 1. Or, using mathematical notation, we write, Phi x Phi = Phi+1. Or we can write Phi2 = Phi+1
Try it and see: 1.6180339887499 x 1.6180339887499 = 2.6180339887499 = 1.6180339887499 + 1.
Spiral Patterns Come from The Golden Ratio (and from Fibonacci Numbers)
We can use the Fibonacci numbers to draw spirals if we use each number in the sequence to make a square. So, the “size-21-square” would be 21 inches (or centimeters, etc.) on a side. Next, we set these squares next to each other in a pattern like this:
We add new squares by moving in a counter-clock-wise direction. We place the squares so that each square touches the smaller square we drew in just before it. The figure above shows how to do it.
Then, if we draw an arc through the opposite corners of the 3-square, and then through the opposite corners of the 5-square, the 8-square, the 13-square, and so on – we will have drawn a spiral. Specifically, a Golden Ratio or Fibonacci Spiral.
Now, “Isn’t that interesting?”
What’s even more interesting about the Fibonacci Spiral is it appears over and over in the natural world – ranging from impossibly massive spiral galaxies to the small spirals of rose blossoms, pinecones, and snail shells.
In fact, in pinecones, sunflowers, and other plants, there are two Fibonacci Spirals – one clockwise and one counter-clockwise — interweaving with each other. This pattern is shown in the photograph of sunflower, below:
“Flower petals mostly occur in counts equal to Fibonacci numbers…There are daisy varieties with 8, 13, 21, and even 55 (all Fib-numbers) petals. Why? Because there is a “little machine” in the genetic code of the daisy that tells the emerging flower to generate flower petals using a Fibonacci-like counter” 4
Note: The Wikipedia article on Spirals explains that Fibonacci (Golden Ratio) spirals are just one of several classes of spirals. The classes can be collectively described using trigonometry to express the angle of the spiral’s opening. So, some spirals are tight, and some open up rapidly. But most of the spirals found in Nature are Fibonacci spirals.5
Clearly, there is something about the forces, materials, and/or processes of the Growing> Organizing> Direction> that incorporates this spiral pattern. It even shows up in the crystals of certain aluminum alloys. 6 Likely gravity and centrifugal force generates the spiral galaxy and tornadoes. Likely the shape of certain molecules creates the spiral arrangement of rose petals and aluminum alloy crystals.
Could we go so far as to say that the spiral mathematics causes (i.e. precedes, or is foundational to) those physical spirals? Are the roses using the spiral pattern, or is the spiral pattern using the rose? Is mathematics somehow behind Continuing Creation? We’ll talk more about these philosophical questions at the end of this Essay.
Branching Patterns Come From The Golden Ratio
In addition to being central to spirals, the Golden Ratio often plays a role in the branching patterns of plants. For example, trees usually have their longest branches nearest the ground. When they reach a certain length, they fork into two or more branches. These grow to a shorter length than their predecessors, and then they branch again — into twigs, which are even shorter. In trees that have the Golden Ratio branching pattern, the longest (bottom) branches are 1.618 times (“phi times”) as long as the mid-sized branches; and the mid-sized branches are 1.618 times as long as the twigs.
“Adolf Zeising (1810-1876), whose main interests were mathematics and philosophy, found the golden ratio expressed in the arrangement of branches along the stems of plants and of veins in leaves. He extended his research to the skeletons of animals and the branchings of their veins and nerves.” 7
Branching patterns permit efficient transportation and communication in both downward and upward directions. Branching is also a chain of command communications system – information flowing up, and then directives coming down. But branching does not transport or communicate things between “peers.” Transport among peers requires a web pattern, as we have in the connections between the brain’s neurons.
The Weave of G>O>D> says — The entire process of evolution, represented by the Tree of Life, is also is a branching pattern.
Other Patterns Found in Nature
The Fibonacci spiral and branching patterns are just two of many patterns that occur in Nature. Most of these patterns can be described, or even predicted, by mathematics.
Patterns occur everywhere in Nature, including cracks, spots, stripes, tiles, bubbles, waves, meanders, foams, arrays, cracks, spirals, fractals, tessellations, branchings, webs, knots, symmetries, and even in the Thomson Structures of nickel-iron crystals found in certain meteorites.
We also touch on these Patterns in our Essay, Patterns in Continuing Creation, although that Essay is more about the theory of patterns. See also our Essay, Nature and the Path of G>O>D>.
At the time of this writing, there is an excellent article in Wikipedia online called Patterns in Nature which explains how natural patterns arise and provides photographs of them. (https://en.wikipedia.org/wiki/Patterns_in_nature).
Mathematics Can Describe Patterns In Nature
Mathematics is often able to describe the patterns found in nature, and also the mechanisms that give rise to those patterns. The processes of creating shapes and patterns in Nature is called morphogenesis. Here are three important examples:
Spirals, Stripes, and Spots
In the 20th century, British mathematician and WWII code-breaker Alan Turing predicted oscillating chemical reactions, in particular the Belousov–Zhabotinsky reaction. These activator-inhibitor mechanisms, Turing suggested, can contribute to the spiral patterns seen in plants and also generate patterns of stripes and spots in animals. 8
Fractal Patterns in Nature
In 1968, Hungarian theoretical biologist Aristid Lindenmayer (1925–1989) developed the L-system, a formal mathematical grammar which can be used to model plant growth patterns in the style of fractals. 9
A fractal is a pattern that repeats itself over different size scales. These are also called “self-similar” patterns. For example, the irregular, in-and-out shape of a coastline looks the same from a mile above, from 10 feet above, and (under magnification) from a millimeter above. The ins-and-outs are so numerous, that the true measured length of a coastline is practically infinite.
Fractals are created by repeating a simple process over and over in an ongoing feedback loop. Driven feedback loops, fractals are images of dynamic, recursive systems. A perfect example of a fractal pattern in nature is a floret of broccoli:
This photograph of broccoli shows a series of large spiraled-cones. Each of those cones has a surface composed of smaller spiraled-cones. And each of those cones has a surface composed of still smaller- spiraled cones. Thus, we say that the broccoli is similar to itself (“self-similar”) when we look at it on different size scales.
Coastlines and the Mandelbrot Set
After centuries of slow development of the mathematics of patterns, Benoît Mandelbrot wrote a famous paper in 1975 called, How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension, which synthesized many decades of mathematics and created the concept of the fractal. 10
The image below is a computer simulation of the most complex fractal ever modeled, a Mandelbrot Set, in “full bloom.” (Benoît B. Mandelbrot, The Fractal Geometry of Nature, 1983, Macmillan. ISBN 978-0-7167-1186-5)
The Circle and the Number Pi
We have looked at the pentagon, its relationship to the number Phi, and the wonderful spiral and branching patterns that grow out of them. Next, let’s talk about the circle and its relationship to the number Pi (or π). (Remember – Phi and Pi are not the same number!)
There are no perfect circles (nor squares or straight lines, for that matter) in Nature or the cosmos. So, are the mathematically-defined perfect circle and all other geometric figures part of nature or are they mathematical concepts that exist only in the human mind? This is one of those philosophical questions we will take up at the end of the Essay. However, let us say a few things right here.
The Book of G>O>D> holds that the abstract mathematical concept of a circle is actually part of the Process of Continuing Creation, in the same way a blueprint is part of the process of constructing a house. Here are some supporting reasons:
- Because it’s what you would get if a plant tries to colonize outward equally in all directions at the same time.
- Because a circle mathematically contains the largest area for a given perimeter. Therefore, the circle is the “cheapest” way to enclose an area of territory. The circle also minimizes the average distance of the perimeter from its center, which can make it easier to defend, and quicker to communicate across.
- In three dimensions, a circle becomes a sphere, and the above arguments apply to spheres as well as circles. And we also know that a “boundary,” or membrane is one of the three basic requirements for every living organism, all of which are of course three-dimensional. (The other two requirements for life are a metabolism and the ability to reproduce.) 11
- Also, spheres are what you get if when the force of gravity pulls hydrogen, dust and rock together to make suns, planets, and moons.
Cycles Are “Circles in Motion”
Cycles are dynamic analogues to circles. A circle extends across a plane of space; a sphere extends across three-dimensional space; and a cycle extends across time. Our Earthly seasons are a perfect example of a cycle. The seasonal cycle runs from spring, to summer, to fall, to winter…. and then back to spring again. Similarly, a human life extends, according to the Bible, “from dust to dust.” Hinduism describes the goddess Shiva’s endless “cycle of creation and destruction.”
We all recognize circles and spheres. Because they return to their beginning, they seem to be the perfect two-dimensional and three-dimensional objects. The Circle can stand for unity and can symbolize cycles, which also return to their beginning. Here is a poem written by Black Elk that clearly shows the power of the Circle and the Cycle as metaphors for Nature and life:
Everything the Power of the World does
Is done in a circle. The sky is round,
And I have heard that the earth is round
Like a ball, and so are all the stars.
The wind, in its greatest power, whirls.
Birds make their nest in circles,
For theirs is the same religion as ours.
The sun comes forth and goes down again
In a circle. The moon does the same
And both are round.
Even the seasons
Form a great circle in their changing,
And always come back again to where they were.
The life of a man is a circle from childhood to childhood,
And so it is in everything where power moves.
— Black Elk, Holy Man of the Oglala Sioux.
The Book of G>O>D> says — If we think about it, the spiral is a circle that progresses. A spiral both circles and cycles, and it grows larger with each turn. Since repeating processes of growth are the central theme of The Way of G>O>D>, the spiral is a potent symbol for our Spiritual Path of G>O>D>.
Defining the Number Pi (π)
A circle, as we’ve said, is the set of points on a flat plane that are equidistant from a given central point.
In school, most of us leaned that:
- The Circumference (C) is the distance around the outside of a circle
- The Diameter (D) of a circle is the distance across the widest part of the circle, and
- The Radius (r) is the distance from the center point to the outer ring of a circle.
The number Pi is defined as the Circumference of any Circle divided by its Diameter:
Circumference / Diameter = C/D = 3.1415926535… This is true regardless of the size of the circle.
In school, we also learned:
Diameter, D, = 2r.
Circumference, C, = 2(Pi)r
Area, A, = Pi(r2)
Let’s take a closer look at the number Pi:
Like phi, pi is a number that goes on forever. π = 3.1415926535…
“Moreover, the pattern of pi’s digits never repeats. The digits are “seemingly random – except they can’t be random because they embody the order inherent in all perfect circles.” 13
Pi Links Distant Areas of Mathematics and Science
Here is perhaps the most amazing thing about pi: Pi ( π ) appears in areas of mathematics and the sciences having little to do with the geometry of circles, such as number theory and statistics. It is also found in cosmology, thermodynamics, mechanics, and electromagnetism.
(Note: This section quotes extensively from the beautifully written article, “Why Pi Matters,” by Cornell University Mathematics Professor Steven Strogatz, which appeared in The New Yorker, 3-13-15.)
For example, “There are astonishing formulas in which an endless procession of smaller and smaller numbers adds up to pi. One of the earliest such infinite series to be discovered says that:
“Pi equals four times the sum 1 – (1 ⁄ 3) + (1 ⁄ 5) – (1 ⁄ 7) + (1 ⁄ 9) – (1 ⁄ 11) + (1 / 13)⋯ and so on, forever.”
“This formula connects all odd numbers to pi, thereby also linking number theory to circles and geometry. In this way, pi joins two seemingly separate mathematical universes, ‘like a cosmic wormhole.’ ” 14 (Alternatively, we might call this link an “unexpected connection,” a “hidden passage” a “back-channel” or simply a “bridge.”)
Pi is an irrational number, meaning that it cannot be computed by dividing one integer by another. Pi is one of several important and famous irrational numbers, like the square root of 2 and the number e (epsilon in the Greek alphabet) which is the base of natural logarithms. Those two important irrational numbers also “bridge different arenas of mathematics, and they, too, have never-ending, apparently random sequences of digits. 15
“What distinguishes pi from all other numbers is its connection to cycles. For those of us interested in the applications of mathematics to the real world, this makes pi indispensable. Whenever we think about rhythms—processes that repeat periodically, with a fixed tempo, like a pulsing heart or a planet orbiting the sun—we inevitably encounter pi.” 16
For example, Pi emerges when we study winding rivers. A river’s “windiness” is expressed by its “meandering ratio,” or the ratio of the river’s actual length to the *distance from its source to its mouth as the crow flies. Rivers that flow fairly straight have small meandering ratios, while rivers with many extreme bends have high ratios. Statistically, the average meandering ratio of rivers approaches pi.
Why? Because the length of a near-circular riverbend is like the circumference of a circle, while the straight-line distance from one bend to the next is diameter-like. The ratio of these lengths approximates pi. 17
We could say that a bending river cycles from left to right. It turns out that any process that periodically repeats (cycles) can be represented by the formula for a Fourier Series. This formula uses Pi along with the sine and cosine concepts from trigonometry to mathematically describe cycles. These cycles include such things as “the gentle breathing of a baby and the circadian rhythms of sleep and wakefulness that govern our bodies. [And] when structural engineers need to design buildings to withstand earthquakes, pi always shows up in their calculations. Pi is inescapable because cycles are the temporal cousins of circles; they are to time as circles are to space. Pi is at the heart of both.” For this reason, Pi is intimately associated with waves, from the ebb and flow of the ocean’s tides to the electromagnetic waves that let us communicate wirelessly.” 18
“At a deeper level, Pi appears in both the statement of Heisenberg’s uncertainty principle and the Schrödinger wave equation, which capture the fundamental behavior of atoms and subatomic particles. In short, pi is woven into our descriptions of the innermost workings of the universe.”19
Pi Is Mathematically Related to Phi !
When we discussed phi, we talked about pentagons and pentagrams. When we talk about pi, we started by talking about circles. However, we can easily see that every pentagon (and every pentagram) fits perfectly inside a circle that touches its outer five points. Therefore, it should not surprise us that phi and pi are mathematically related!
Divide a 360° circle into 5 sections of 72° each and you get the five points of a pentagon, whose dimensions are all based on phi relationships.
It turns out that phi, pi and the number 5 (a Fibonacci number) can be related through trigonometry by the following equation. (Note: we’re not going to explain this equation; we just want the reader to note that it exists.)
Or, we can express the same thing with this simpler equation: Pi = 5 arccos (0.5 Phi)
…..where “arccos” means “Inverse-cosine.”
(Gary B. Meisner, Golden Number.net, https://www.goldennumber.net/pi-phi-fibonacci/)
Pi Is Also Related to the Prime Numbers… and More
We’re still not done cataloging Pi’s rich network of relationships. Amazingly, Pi is also related to the Prime Numbers.
First, we have to introduce the Prime Numbers. (References for this section are largely from the documentary, Cosmic Code Breakers, WHJJ (PBS), Dec, 2012)
The Prime Numbers
The prime numbers are the integers (the numbers we use for counting) that cannot be divided by anything except themselves and 1.
The primes are 2, 3, 5, 7, 11, 13,17, 19,23, 29, 31, 37, 39, 41, 43, 47, 53, 59, 61, 71, 73, 79, …and so on; and they go on to infinity.
All the non-prime numbers are the product of multiplying two or more prime numbers. So, why isn’t “9” a prime? Because it can be produced by multiplying 3 x 3 = 9. Why isn’t “45” a Prime? Because it can be produced by multiplying 5 x 9 = 45.
Prime numbers seem to be fundamental building blocks in mathematics. “Like the chemical elements, they combine to form a universe. To a chemist, water is two atoms of hydrogen and one of oxygen. Similarly, in mathematics, the number 12 is composed of three prime numbers (“atoms”): two ‘‘atoms’’of the prime number 2, and one ‘‘atom’’ of the prime number 3. Because 12 = 2 x 2 x 3.” 20
Another thing about primes is that every positive integer has a unique representation as a product of primes. For example, 30 is 2 * 3 * 5, a product of primes, and no other product of primes yields 30.
However, the intervals between the primes appear to be random! Sometimes there is just 1 or two steps between successive primes; other times the interval is long… one very long gap is 72 steps long. To us humans, whose brains are evolved to seek and expect to find patterns, this randomness seems very mysterious.
Pi Is Related to the Prime Numbers by the Zeta Function
Now we can address how Pi relates to the prime numbers – namely by the Zeta function, uncovered by the mathematicians Leonhard Euler (working in 1774) and Bernhard Riemann (working in 1859).
Note: General readers don’t need to fully follow this; getting a feel for it will suffice.
(Note: We define “functions” a bit later. For now, just think of it as the “Zeta Mathematical Relationship.”)
Using the Prime numbers 2, 3, 5, 7, 11, 13, 17…. put them into the following equation:
(2squared) / (2squared -1) x (3squared) / (3squared -1) x (5squared) / (5squared -1)…etc.
The answer turns out to be = pi-squared / 6.
So: 4/3 x 9/8 x 25/24 x 49/48 x 121/120 x 169/168 x 289/288.…
= 1.33333 x 1.12500 x 1.04167 x 1.02083 x 1.00832 x 1.000347… = 1.60888.
If we use more of these chained fractions (and show more decimal places for each fraction), our result steadily increases from 1.60888 above to approach the final answer 1.644934, which = pi-squared / 6.
To check, calculate Pi-squared as follows: 3.14159265 x 3.14159265 = 9.869604.
Then divide by 9.869604 by 6 = 1.644934…, which is itself Pi.
Clearly there is something important and fundamental going on between the prime numbers and pi. And since pi is the ratio of circumference to diameter of ALL circles, clearly there is also something going on between the prime numbers and circles.
Another “hidden relationship” appears when this Zeta function is plotted on a 3-dimensional graph (which we will not try to show here). The graph shows that the zero points for the x-axis of the graph lie along a straight line. (While is very difficult for non-mathematicians to grasp this surprising regularity, see: http://mathworld.wolfram.com/RiemannZetaFunction.html.)
Later, mathematician Hugh Lowell Montgomery found that the gaps between the zero points on the graph were a function of this equation: (sin pi x u) / (pi x u)2 We simply remark that this simple equation exists, without going into the meaning of the symbol “u.”
This hidden regularity – that all the Zeta zero-points lie along one straight line — is amazing, because, as we mentioned earlier, the gaps between the primes appear to be random and highly variable.
So, while we may not yet understand the full connection between circles and primes, between Pi and primes, our understanding has clearly improved; at least for the mathematicians who fully comprehend these equations. 21
Mathematics Can Unexpectedly Illuminate Areas of Physics
Then, in 1992, a chance conversation between physicist Freeman Dyson and mathematician Hugh Montgomery (whom we mentioned just above) led them to realize that Montgomery’s equation was the same as the equation Dyson had for the distribution of the energy levels in the nucleus of atoms, namely:
(sin pi x r) / (pi r)2
Their 1992 meeting led to a conference in 1996 attended by 200 professors from these two and other disciplines. At this conference, mathematician Alain Connes said that these two distributions fit very well into his new Non-commutative Geometry, which says that all space is discontinuous – that space has seams in it; all through it. Some theorists believe that this new geometry could lead to a final theory of everything – the blueprint for the universe.22
The Weave of Continuing Creation remarks that these historical connections and cross-applications between Pi and the Primes show the power and beauty of mathematics in the workings of the Growing Organizing> Direction> of the Cosmos.
However, there are some statements about prime numbers which appear to be true, but which generations of mathematicians have not yet proved to be true. A mathematical statement which appears to be true; but has so far not been proven true, is called a conjecture. 23
A famous example is Goldbach’s Conjecture, which simply proposes that “Every even integer (whole number) greater than 2 can be expressed as the sum of at least one pair of odd prime numbers.”
We can try out the Goldbach Conjecture for some even numbers picked at random:
4 = 1+3;
8 = 5+3;
26 = 3+23 or 7+19;
60 = 7+53, or 13+47, or 17+43, or 19+41, or 23+37
Alas, “trying out,” Goldbach’s Conjecture, no matter how many times we do it, is not the same as proving it. Proving Goldbach’s Conjecture would be showing, with full force of logic, that there is no, and can be no even number which is not the sum of two primes. We will talk more about mathematical proofs later in this Essay. (For more on Goldbach’s Conjecture, see https://www.youtube.com/watch?v=O4ndIDcDSGc.)
The Union of Algebra and Geometry – Graphs & Equations
Before we leave our discussion of the Circle, we want to use it to illustrate another deep connection. This deep connection is between two of the most useful (and most basic) areas in all mathematics: algebra and geometry.
Translating from geometry to algebra, and back again, involves graphs. The French mathematician Rene Descartes (1596-1650) was inspired to invent graphs by looking at architectural drawings and maps. Maps were marked off with crisscrossing lines, making a reference grid for distances. (After an accurate way to determine longitude at sea was invented in the 18th century, every point on the globe could be designated by a unique pair of numbers – one for latitude and one for longitude.)
Descartes reasoned that the same thing could be done for geometric figures drawn on a flat piece of paper. 24
Get a piece of graph paper; or make one by ruling blank paper off with horizontal and vertical lines making equal-sized squares. Designate a point near the middle of the paper as the place where your horizontal and vertical “baselines” cross, making it your “home base” or “starting point.” Descartes called the horizontal baseline line the x-axis and the vertical baseline the y-axis. The point where they cross, i.e. where x=0 and y=0, is called the “zero-point, or “origin,” (denoted as “0,0”). The location of every possible point anywhere on this graph paper can now be designated by a unique set of Cartesian coordinates (x,y) (“Cartesian” meaning “in the fashion of Descartes.”)
So, we say that the Cartesian coordinates, (x,y) of the Origin are (0,0). Place your pencil at (0,0) and march out to the right 5 squares on the x-axis to coordinates (5,0) and make a little mark. Then, place your pencil at (0,0) and march up 5 squares on the y-axis to point (0,5) and make another little mark.
Take a drawing-compass (or use a piece of string held down on the graph paper at one end) to draw a circle having 5-units’ radius all the way around the Origin. You will pass through your marked points (5,0) and (0,5).
(On the other side of the circle, you also pass through -5,0 and -5,-5; but we don’t need to deal with those areas of negative numbers to say what we want to say in this Essay.) We’ll focus on the upper-right quadrant of the circle, where all the coordinates are positive numbers.
We always knew that a 5-unit circle was the “set of all points that are 5 units from the center point.” But now, thanks to Cartesian coordinates, we could actually name every one of those points if we wanted to by labeling each one with its own unique set of coordinates. (Of course, it would take an infinity of time and an infinity of coordinate pairs, and an infinite number of pencils to do so.)
Descartes’ next stroke of genius was to show that the geometric figure of a circle could also be described by a simple equation in algebra:
x2 + y2 = r2 is the algebraic equation for a circle centered on the point (0,0)…
…where x is the distance on the x-axis, y is the distance on the y-axis, and r is the radius of the circle.
Let’s see if the algebraic formula works:
We know the point 5,0 is on the circle.
Test: 5-squared + 0-squared = 5-squared.
So, 25 + 0 = 25. It works for that point.
And we know the point 0,5 is on the circle.
Test: 0-squared + 5-squared = 5-squared.
So, 0 + 25 = 25. It works for that point as well.
Now, look again at the grid. It appears that points 4,3 and 3,4 are also on the circle!
Test: At the circle-point 3,4 we have: 3×3 + 4×4 = 25. So, r = the square-root of 25 = 5
Test: At the circle-point 4,3 we have: 4×4 + 3×3 = 25. So, r = the square-root of 25 = 5.
Correct…but many readers will remark that algebraic equation, x2 + y2 = r2 , looks familiar; like something we saw in high school algebra. We will explain the familiarity of this equation later in the Essay.
Then, Descartes had a third stroke of genius. The intersection of geometry and algebra led him to the concept of mathematical functions.
Equations are static; they describe a state of being in the moment, without any change. For example, 22 =4. We can use an equation to see change change only when we think of the equation as a function.
In mathematics, a function is a relationship between a set of inputs and a set of resulting outputs, where each set of inputs is related to exactly one output. A function has 3 elements: the inputs, the operation (action, computation, conversion) and the output. There are two kinds of inputs: variables (which can be changed) and constants (which cannot be changed). Functions can usually be plotted on graphs. 25
An example is the function that relates each real number to its square, as expressed by the equation y = x2. When we input different values for x, we get different values for y. If x = 2, then y = 4. If x = 3, then y = 9. In other words, we “run” the equation as if it were a little machine, and we generate the resulting outputs of y. So, when we input 1, 2, 3, and 4 as our values of “x”, we output 1, 4, 9, and 16 for our values of “y.”
In addition to being dynamic (i.e., dealing with change), functions are recursive (i.e., they repeat a sequence of steps over and over, making small changes in just one or a few variables with each iteration of the sequence.)
For a second example, input distance and also input time, use a “calculating machine” called division, and we get the output speed: Speed = distance / time.
Functions are Dynamic – They Describe Processes
We can use our circle equation, x2 + y2 = r2, as a function: If we make r a constant by setting it equal to fixed number, say 5, and then we input various numbers for x (a variable), and we generate (solve for) the output values of y, thereby drawing a circle with a radius of 5.
Note: We could re-arrange the algebra in the form often adopted for functions:
f (y2) = 25 – x2
…where “f” stands for “function,” “y2” is named as our output, and “x2” is our input.
Or, we could say that the f(y) = the square root of ( 25 – x2 ).
The drawn circle is a continuously changing string of Cartesian Coordinates plotted on graph paper. The circle-equation-function describes a dynamic process: If we sequentially feed in larger and larger values of x, and plot each point as we go, the function gives us the coordinates for all the points on the circle, thus describing the process of circle-drawing. In fact, written as a computer program, the function could direct a machine to actually draw the circle.
Note: If we also vary the value for “r,” we can get an endless number of concentric circles, all of the with a center-point at the origin: (0,0).
Functions allow mathematics to be applied to cause-and-effect situations – situations involving change, energy, force, and even evolution. Today, most of mathematics deals with functions. Mathematical functions are perfect for science and engineering. They can describe many, many real-world processes, such as stress on a bridge, electric currents, radio waves, and metabolisms. Note: Wikipedia’s current entry (April, 2018) is well worth reading.
Algorithms: “Functions” For Solving Problems
Like functions, algorithms are also dynamic, and also describe processes. But while functions describe the steps of generating an output, of building something, algorithms describe the steps used to solve a problem. Algorithms can do calculations, data processing, and automated reasoning. A computer program is an algorithm. 26
Connecting the Circle and the Pythagorean Theorem
Now, let’s return to the hunch we had earlier that the formula “x2 + y2 = r2” looks familiar. Why does it look familiar?
Here’s a hint: Every time we find a new point on the circle, we essentially do it by drawing a right triangle (i.e. a triangle with one 90-degree angle) inside our circle. For example, the three points of the last triangle we drew were (0,0); (4,0); and (0,3). We drew it by starting at the origin, moving right 4 squares to the second point, and then up 3 squares to the third point.
If we connect those three coordinates with straight lines, we draw a right-triangle inside our circle. A right-triangle is defined as a triangle where one of the three angles is a right-angle, i.e. a 90-degree angle. The longest side of a right triangle is called the hypotenuse.
The three sides of our right-triangle are:
The x-axis side of the right triangle = 4.
The y-axis side of the right triangle = 3.
The radius (hypotenuse) side of the right triangle = 5.
Now we can make the connection: our formula x2 + y2 = r2 for the circle is also the formula for the Pythagorean Theorem!
The Pythagorean Theorem states that “For every right triangle, the sum of the squares on the two smaller sides equals the square on the largest side (called the hypotenuse).”
For our example triangle, this theorem says that 4-squared plus 3-squared = 5-squared. And it does, because 16 + 9 = 25.
In fact, every point on our circle has a right-triangle associated with it, and all of those triangles have hypotenuses of 5. So, there is yet another mathematical bridge (passage, connection, “wormhole”) — namely this connection between circles and right-triangles.
The Map of Mathematics
We’ve pointed out important bridges between several fields of mathematics – geometry, algebra, and number theory (when we talked about primes). We’ve also discussed the connections between algebraic equations and functions. Let’s pause here to talk more generally about the connections between the various fields or branches of mathematics.
The Map of Mathematics, recently drawn by physicist and science writer, Dominic Walliman, is a map in gentle cartoon style that names all the main areas of mathematics and positions them in relation to each other. We have a picture of the Map on this page, but it is too small for readers to make out the label names. However, you can see a much larger Map by googling for it online (or see the links just below).
Also, Dr. Walliman presents a wonderful narrated and animated mini-lecture about the Map. Readers who would like to take a “break” at this point in our Essay are encouraged to check his presentation http://www.openculture.com/2017/02/the-map-of-mathematics.html. You can also see it on YouTube: https://www.youtube.com/watch?v=OmJ-4B-mS-Y.
The Map of Mathematics and its accompanying mini-lecture tell us that there are specialized branches of mathematics for knots, crystals, sets, groups, fractals, vectors, matrices, statistics, probabilities, waves, chaotic systems, partitions, codes, games, computations, and others. You can usually find a clear basic description of what each of these fields of mathematics are about by googling their names.
While Walliman divides mathematics into two main realms of Pure and Applied, we instead might divide mathematics into these two categories:
- Numbers and the counting of objects
- Patterns and the positioning of objects
Number one leads to arithmetic, algebra, and number theory. Number two leads to the various geometries, topology, and knot theory. Both #1 and #2 grow out of the existence of difference – the difference between object and field — which we discussed in our Essay, Patterns in Continuing Creation.
Dr. Walliman is also the author of Maps about other areas of knowledge, of four highly-rated books on research methods, and children’s books about science, featuring “Professor Astro Cat.” (See www.dominicwalliman.com.)
The Power & Beauty of Mathematical Proofs
Now let’s talk about mathematical proofs. Why? Because they demonstrate the power and beauty of mathematics within the Continuing Creation of G>O>D>.
To begin, we must first describe formal mathematical systems. Each system of mathematics (e.g., algebra, geometry, and calculus) has a formal logical structure. It begins with a relatively small number of statements that are so obvious to us, so self-evident, that we can accept them as foundation blocks on which to erect the rest of the mathematical structure. These foundation blocks are called axioms or postulates.
A mathematical proof uses the axioms plus a series of small, usually obvious logical steps to show that a more complicated mathematical statement (called a theorem) is-and-must-be true. In a mathematical proof, each little step is logical and unassailable. But the end conclusion looks wondrous to someone standing at the very beginning.
For example, Classical geometry (called Euclidean Geometry after its ancient Greek founder, Euclid of Alexandria, 323-283 BCE) had just 5 axioms:
- A straight-line segment may be drawn from any given point to any other.
- A straight line may be extended to any finite length.
- A circle may be described with any given point as its center and any distance as its radius.
- All right angles are congruent (are identical).
- Parallel lines never meet.
Using just these 5 axioms, it can be proven that the area of a rectangle equals its length times its width. (We’re not going to show that proof here, but we can reach that conclusion ourselves simply by placing any rectangle on a sheet of graph paper and counting the number of grid squares inside it.)
In Appendix A, we illustrate what we mean by a formal mathematical proof by presenting a proof of the Pythagorean Theorem, which says that for any right triangle, a square drawn on the longest side of the triangle is equal to the sum of the two squares drawn on the other two sides.
There are hundreds of different proofs of the Pythagorean Theorem. The proof we show in Appendix A is thought to be the proof that Pythagoras did himself. This proof is particularly interesting because it is logically sound and complete without using any algebra. The theorem is proved simply by making paper cutouts of a few geometric figures and moving them around on a table top. Mathematicians would say that this particular proof has elegance.
A proof of a mathematical theorem exhibits mathematical elegance if it is surprisingly simple yet effective and constructive. Similarly, a computer program or algorithm is elegant if it uses a small amount of programming code to great effect. 27
The Book of G>O>D> says — The ability of “obvious” steps of reasoning to reach conclusions that are not obvious shows the amazing power and beauty of mathematics. It is similar to using the small steps of tiny chemical reactions to assemble a complex protein molecule or using small steps of evolution to tell the tale of how the human eye came to be. In all of these processes the whole is greater than the sum of its parts. All these processes are prime examples of Continuing Creation: The Growing> Organizing> Direction of the Cosmos.
Conjectures Not Yet Proven
Unlike the Pythagorean Theorem, there are many theorems in mathematics that have only one proof. Some of these proofs are elegant, and some are lengthy, sophisticated, and cumbersome.
Often, there are patterns in mathematics which seem to be always present, or seemingly logical statements in mathematics which mathematicians suspect might be true, but which they have not yet been able to prove.
Unproven theorems are called conjectures. Some conjectures hang around for hundreds of years until someone finally proves them. Often, the proof is quite elaborate and uses very advanced mathematics.
An example would be Catalan’s Conjecture about prime numbers; put forth in 1844 and finally proved in 2002, becoming Mihailescu’s Theorem 28
The motion picture Proof, starring Gwyneth Paltrow and Anthony Hopkins, is a wonderful dramatization about two present-day mathematicians – a daughter and her father – whose struggles to complete and publish proofs are complicated by their family relationship and the father’s declining health.
There are also conjectures which have never been proved… and perhaps never can be proved. A famous example is the Goldbach Conjecture about prime numbers, which we discussed earlier.
An Unproved Statement in Geometry Leads to New Fields of Mathematics
In the classical geometry we have been discussing (the geometry we studied in high school), there are 5 axioms (or postulates), which we have already listed earlier in the Essay. (In 1889, mathematician David Hilbert made Euclid’s axioms more air-tight by expanding them to a total of twenty. But Euclid’s original five still cover the waterfront for us general readers.)
Here, we want to further discuss the fifth axiom:
- “Parallel lines never meet.” (More formally: “If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.”)
The fifth axiom is known as the parallel axiom or parallel postulate. Across the centuries, many mathematicians have felt that this fifth statement was not a true axiom. It seemed like a theorem to them, a theorem which they ought to be able to derive from the first four of Euclid’s axioms. However, while many purported proofs of the parallel axiom have been published, none of them has ever stood up to scrutiny.
Nevertheless, it just didn’t seem useful to deny the Parallel Postulate, because it makes so much “common sense” in everyday life; so it was left in the standard list of 5 axioms.
The Book of Continuing Creation says — The unprovability of the Parallel Axiom as a theorem is a kind of “gap” or hole in Euclidean geometry. But, here’s the wondrous thing: a logical gap in a mathematical system can actually be a hidden passage to entirely new and powerful branches of mathematics. (And if the mathematics is being used to model physics, a door can open onto entirely new theories, and/or discoveries, in the science of physics.)
This is exactly what happened to Euclidean geometry. In 1823, Janos Bolyai and Nikolai Lobachevsky, and then Bernhard Riemann, realized that entirely self-consistent “non-Euclidean geometries” could be created by assuming that the parallel axiom (or, parallel postulate) does not hold.
In fact, seafaring navigators were already quite familiar with a geometry where parallel lines do meet. On a globe of the Earth, longitudinal (vertical) lines, which are always parallel to each other at the equator, do in fact meet at the North Pole, and again at the South pole. Also, it was clear that while the 3 angles of flat-surface triangles always add up to 180 degrees, the angles of triangles drawn on a globe-surface add up to more than 180 degrees. Today, this geometry is called elliptical geometry (which includes the spherical geometry used to navigate and map the Earth.
Thus, when pilots fly from Washington D.C. to Tokyo, the shortest distance is not the straight line we could draw on a flat map, but a “great circle route,” i.e. a curved line that arcs over the surface of the Earth. For example, the great circle route from Washington D.C. to Tokyo can be seen by tightly stretching string between those two cities on a globe.
A second new geometry was also developed for saddle-shaped surfaces (and saddle-shaped three dimensional spaces), where lines that start out being parallel move away from each other over distance. In this hyperbolic geometry, the angles of a triangle add up to less than 180 degrees. 29
Interesting, we might say, but so what? What are the new geometries “good for”? Well, astrophysicists realized that the three geometries – flat (Euclidean), elliptical, and hyperbolic — were ideally suited to describing three different theories about the shape of space-time itself. These shapes are dubbed the Flat, Closed, and Open universes. (However, we’re not going to explain them here in this Essay.)
Today, there are many formal, logical geometries in mathematics, including Elliptical, Hyperbolic, Projective, and Thurston Geometries. Still, for most of our uses here on Earth, it safe to assume that two parallel lines never meet…except when navigating across the Earth.
No Mathematical System Can Be Fully Consistent
So, there is a logical “trapdoor” in Euclidean geometry that was really a “hidden passage.”
Actually, it turns out that no mathematical system can be fully consistent! There are always true mathematical statements that lie outside a particular system’s set of axioms and theorems. Mathematician Kurt Goedel actually proved this in 1931. However, we need to defer discussion of this important phenomenon until later in this Essay. For now, let us just say this:
The Weave of Continuing Creation says — The unprovability of the parallel postulate turned out not to be a weakness of flat geometry, but a door to other regions of mathematics. This “hidden passage” phenomenon happens over and over in mathematics. Apparent “holes” in some mathematical system turn out to be opportunities to discover (or create) new realms of mathematical beauty and power. It reminds us of the poetic metaphor in the Bible — “In my Father’s House are many mansions.” (John 14:2.)
Does Mathematics Have Other “Weaknesses”?
Okay, so mathematical systems have logical inconsistencies, which can be both a weakness and a strength. Does math have other weaknesses? Yes.
First, the rigor and precise notation of mathematics can be very off-putting to people who have not been trained to understand it. Most people prefer to appreciate the interconnections of Creation through their five senses, not through their logical intelligence.
Second, mathematics can be very descriptive of how the four fundamental forces work, but not very descriptive of what the four forces are. Math has no verbal connotations, no metaphorical power, no poetic power; at least for people without mathematical training.
This is also true in science. When physicists talk about “fields,” Humans get some “meaning” of the field concept by analogizing the concept of scientific fields to athletic fields. Similarly, most non-physicist humans rightly balk when physicists call a tiny sub-atomic energy-ball or field-fluctuation a “virtual particle.” Our confusion only increases when physicists “explain” that a virtual particle is “like a particle of matter, but not the same as a particle of matter.”
The Mathematics of Uncertainty — Statistics and Probability Theory
So far in this Essay, we have described the mathematics for several patterns: the pentagon, spiral, branching and circles. And we talked about the math for two kinds of dynamic patterns: “patterns through time” – functions and algorithms.
The mathematics of physical things like gravity, or the spreading and attenuation of sound, are described by functions having a concrete outcome for each input. But iron-clad predictions do not apply to many things in the real world, where the best we can do is see tendencies and likelihoods. In daily life, we constantly run into patterns which are often true, but not always true. People who over-eat are often overweight, but not always overweight. All around us, we see mostly imperfection, not perfection. As we discussed earlier in this Essay, there are no perfect circles in the physical world.
In recent decades, discoveries in physics show that even the behavior of sub-atomic particles is governed by Quantum Mechanics, which also follows the laws of probability. In quantum mechanics, “it is impossible to predict the outcome of every experiment, even in principle. Rather, the theory can only predict the probabilities for different results.” 30 For more on this, see our Essay, Physics and Continuing Creation.
The Path of G>O>D> says — Processes that produce tendencies and likelihoods are as much a part of Continuing Creation as are processes that have certain outcomes. Happily, there are branches of mathematics that describe and measure these phenomena. The mathematics of tendencies include statistics, probability theory, game theory, and complexity theory.
(The use of the word “theory” is unfortunate, because for most of us “theory” means something that may or may not be true. Similarly, in science we have unfortunate names like the “theory of gravity” and the “theory of evolution.” We will try to avoid this confusing use of the word “theory,” and speak instead of probability-mathematics, game-mathematics, and complexity-science.)
Everywhere we look in real life (especially in our human lives of economic and social relationships), we see tendencies instead of certain outcomes. So, now we turn to statistical patterns:
When cities grow, electricity use and water use increase in direct (linear) proportion to population. If population rises 20%, so do the use of both electricity and water use; and this happen regardless of the cities’ underlying cultures or forms of government. 31 When relationships like these are plotted on a graph, they form a slanting straight line. These relationships (also called functions or scalings) are called direct, proportional, or straight-line relationships.
However, the data points often do not fall exactly on the straight line. They cluster around a slanting straight line. For example, older children are only generally taller than young children. If we drew a graph with age (2 year to 9 years old) on the x-axis and height on the y-axis, and then plotted the data of 100 kids by making a dot (data point) for each one, the dots would lay roughly along a straight line from the lower left to the upper right.
The points would not be exactly on this line because, as we all know, some kids will become tall adults and some will become short adults due to their genetics and nutrition; and also because children grow at different rates.
If the dots are mostly close to the line, we say that height and age are highly correlated; and if the dispersion from the line is wide, we say the correlation is weak. The branch of applied mathematics called statistics give us a way to calculate the degree of closeness and present it as a single number, called the coefficient of correlation.
In our example, increasing age actually, but imperfectly, causes increased height. However, “correlation does not always imply causation.” For example, smoking is correlated with alcoholism, but smoking doesn’t cause alcoholism. Instead, alcohol addiction and nicotine addiction are both caused by (a) a genetic propensity for addiction, and (b) cultural influences on people, especially when they are young. 32
If we plot the height of people a different way, we get another very important statistical phenomenon: The Normal Distribution, also called the Bell Curve. These curves are everywhere in the real world.
We’ll stick with subject of people’s height; but ask a different question about it. We ask: “For fully-grown American adults, “What’s the average height,” and “How different are the individuals from that average?” This question is no longer concerned about height versus weight; we just want to know about the number of people having certain heights.
So, we’ll place our data points on a graph where the x-axis is height, and the y-axis is the number of people having any given height. We’ll get a curve like this:
The average height is at the highest point on the curve: 5’ 9” for if we plot only American men, and 5’ 4’’ if we just plot American women. 33
Depending on the phenomenon we plot, the Bell Curve can be tall and narrow, meaning that people (or cities, or frogs, etc.) are tightly concentrated around the average, or wide and flat, meaning that they are loosely concentrated around the average. Statistics has developed mathematical way to compute the degree of concentration, and compute other characteristics of these data “distributions.”
Power Law Distributions
However, straight-line relationships and Normal Distributions are hardly the only statistical relationships (patterns) present in nature, culture, and technology.
For example, some things increase much more rapidly than a straight-line (linear) relationship. If the increase in something is exponential, its growth accelerates over time. Things that grow exponentially are called Power-law distributions or super-linear scaling.
Let’s take the growth of cities as our example.
While water use increases directly (linearly) with the growth of cities, for other activities the increase is exponential, and different activities have different exponents in their equations. For example, Research and Development employment increases with an exponent of 1.34. New patents rise with an exponent of 1.27. Gross domestic product increases at between 1.13 and 1.26.
The equation for new patents would be: New Patents = (city-size)1.27 So, if city-size increases by 2, New Patents would increase by 2.411615… (Turn to Google to learn how this is computed.)
“Exponential growth [happens] because the increasing cooperation and specialization in the larger city increases the productivity (on average) of each person in the city. Physical proximity promotes collaboration and innovation. Basically, super-linear scaling is why cities are formed in the first place, and why they tend to get bigger over time. Unfortunately, new AIDS cases also increase at 1.23, and serious crime at 1.16. And some things increase at exponents of less than one, e.g. the number of gas stations (0.77) and the total surface area of roads (.08), and the total length of wiring in the electrical grid (0.87).” 34
(Note: When something changes by an exponent less than one, it is actually increasing more slowly than the city population is increasing; i.e. by less-than-linear growth.)
When graphed, inverse power law distributions form hyperbolic curves – one that starts low at the left and then swoops upward to the high right. The curve below shows that as storms (or earthquakes, or salespersons’ performance) are graphed according to their increasing power, the number of them rapidly declines.
A prevalent type of Power Law Distribution, observable in fields like geography, botany, and economics, is Zipf’s Law.
- Most nations have a largest city that’s about twice the size of the next-largest income, and about three times the size of the next-next income, and so on.
- The largest tree in a forest will often be 2x the size of the next largest tree, 3x the times of the third-largest tree, etc.
- The tenth-smallest lake in a large nation will be about one-tenth as big as the largest lake.
- The 15th most frequent word in a book occurs about 1/15th as often as the most frequent word.
These are called Power Law Distributions when we focus our attention on the larger items, and they are called Inverse Power Law Distributions when we are more interested in the smaller items.
When graphed, power-law distributions form hyperbolic curves. The inverse distribution’s curve starts high at the upper-left and then swoops downward to the lower-right, hugging (but not touching) the bottom axis (x-axis) of the graph. 35
One important Inverse Power Law relationship says that “The second richest person in a free-market society owns about half as much as the richest, and the 50th richest person owns about 1/50th as much as the richest person.” In a large company, the CEO might earn $2,000,000 a year, the 50th-ranked worker will only earn about 1/50th of $2,000,000 = $40,000 per year. 36 Actual statistics for the U.S. show that the top 1% of households received approximately 20% of the pre-tax income in 2013, versus approximately 10% from 1950 to 1980. 37
Why is it that the items at the top do so very much better than the items near the bottom? So far, no one knows. Inverse-Power Distributions are a fundamental feature of reality. They are “as inevitable as turbulence, entropy, or the law of gravity.” 38
The Pareto Principle (the “80 /20 Rule”)
Similar to the Power Law is the Pareto principle (also known as the 80/20 Rule, or the Law of the Vital Few. This rule states that for many events, roughly 80% of the effects come from 20% of the causes.
In business, it very often true that “80% of our business comes from 20% of our clients.” Similarly, we often hear that “20% of the people do 80% of the work.”
Vilfredo Pareto developed his 80/20 Rule in 1896. It accurately describes social, scientific, geophysical, actuarial, and many other types of observable phenomena.
Probability Mathematics (Probability “Theory”) deals with games of chance. It is used to calculate the odds of something happening after a sequence of game events. To save time, we are not going to discuss Probabilities here, except to say this:
When you flip a coin once, the probability of getting “tails” is ½, or 50%.
When you flip a coin twice, the probability of getting “tails” twice in a row is ½ x ½ = ¼, or 25%.
For more about Probability Theory, see Wikipedia’s article on it.
Game-mathematics (Game “Theory”) uses mathematics to describe conflict and cooperation between intelligent rational decision-makers. Game-mathematics is an umbrella term for the science of logical decision making by humans, animals, and computers.
Game Theory is a great example of a branch of mathematics created to formally describe interactions in society instead of in physics and chemistry
The film, A Beautiful Mind, tells the real-life story of Dr. John Forbes Nash, a mathematician who was one of the founders of Game Theory and a Nobel Laureate in Economics. A Beautiful Mind won the Academy Award for Best Picture in 2002.
In this film, Nash has a flash of insight about game theory while he is at a tavern near the university along with four of his friends. All five of them are fairly attractive young men. Then, five young women enter the tavern. Four of the women are fairly attractive, and one woman is super-attractive (we’ll call her “Jenn Ten.”)
Dr. Nash watches as the men begin to converse with and pursue the women. While he expects to see all of his pals chat up Jenn Ten, he is surprised to see that none of his friends elect to talk to her! Why is this?
Nash has the following insight: He reasons that if all four of his pals compete for Jenn Ten, only one will win her phone number, while the other three will go home with nothing. In fact, Miss Ten might reject all four of his friends. Therefore, without any communication between the four men, they all independently reason that their best chance of getting any girl’s phone number is to pursue only the fairly-attractive girls, and not “spin their wheels” in chasing “Peerless Jenn.” They each reason that “Something is better than nothing.” Thus, the individual men trade away “a high satisfaction that is highly uncertain” in exchange for “a lower satisfaction that is more certain.”
Dr. Nash also realizes that if 2, 3, or all 4 of the men go home with phone numbers, the combined satisfaction of the group will be greater than the satisfaction of the group if just one man gets the number of Jenn Ten while every other man strikes out. This is a key principle in Game Theory. By showing that rational (moderately risk-averse) thinking coupled with uncommunicated cooperation can maximize the satisfaction of the group, game theory explains one of the foundations of human cooperation.
(Note that while the cooperation does not involve conscious communication between the men, appearances and body language (including those of the women) are being communicated all the time.)
The Path of Continuing Creation says — As we have discussed in other Essays, both competition and cooperation are main sub-processes of evolution. When we take our movie example to its ultimate outcome, we see that two, three, or four mating couples usually produce more offspring for the human species than just one mating couple. Cooperation in a society can aid evolution more than pure, dog-eat-dog competition.
Complexity Mathematics (also poorly called Complexity “Theory”, or “Chaos Theory”) deals with the behavior of complex systems.
A complex system is one that has many components which are able to interact with each other. Examples of complex systems are Earth’s global weather patterns, living organisms, the human brain, stock markets, cities, governments, ecosystems, and ultimately the whole of all the interlocking processes of the Growing> Organizing> Direction of the Cosmos.
Complex systems are systems whose behavior is intrinsically difficult to model due to the dependencies, relationships, and interactions between the system’s parts, and also between the system and its environment.
Systems that are “complex” have distinct properties that arise from these relationships, such as nonlinearity, emergence, spontaneous order, adaptation, and feedback loops, among others. We describe each of those features in our Essay, Complexity Principles and Processes.
Here, we will just describe what is meant by spontaneous order. Take the weather along the eastern and southeastern U.S. as our example. As we all know, when conditions are just right, giant whirlpool storms called hurricanes (on the shore) and tornadoes (inland) appear. These storms then travel, carrying devastating high winds and/or torrential rains. While we know there is a hurricane season, and we know a number of the key variables involved in the formation of hurricanes, their interactions are so complex that no one can accurately predict when and where one of these storms will arise.
In these hurricanes and tornadoes, the emergent and spontaneous order (the pattern) is the storms’ characteristic whirlpool (vortex) shape. The mathematics describing the pattern’s generation are non-linear because a small change in one of the many inputs can produce a magnified change in an output. The result can be a kind of chain reaction, a cascade of larger changes, that produce devastating power. 39
Complex systems contain so much motion (so many elements that move) that powerful computers are required just to try to calculate all the various possibilities. That is why Complexity Mathematics (Chaos Theory) could not have emerged before the second half of the 20th century.
No One Formal Mathematical System Can Be Complete
Earlier, we saw that the parallel postulate (i.e., “two parallel lines never meet”) in Euclid’s classical geometry is not always true. It is true only for flat spaces, i.e., space which extends evenly in all directions. (In two-dimensions, we’re talking about a flat piece of paper).
We saw that when later mathematicians assumed that parallel lines could meet somewhere ahead, different kinds of geometry were developed. Elliptical geometry could handle the lines of longitude on a globe which are parallel at the equator yet meet at the north and south poles. Hyperbolic geometry was developed for “saddle-shaped” space, where lines that at parallel at one point diverge from one another as they extend out. Einstein, who was busy proving that gravity bends space itself, was able to use these new geometries to express his Theory of Relativity.
Then, in 1931, German mathematician Kurt Goedel wrote two related proofs in mathematical logic demonstrating that no mathematical system based on a set of formal axioms and deduced theorems can be both complete and consistent. These proofs established what are now called Goedel’s Incompleteness Theorems.
These proofs shook the mathematical world because mathematicians had previously thought that everything in mathematical that is true must have a mathematical proof, meaning that the system must be complete. At the same time, a mathematical system should not have contradictions, meaning that the system should not be true and false at the same time. A system that does not include contradictions is called consistent.
Gödel proved that every non-trivial formal system of mathematics or logic is either incomplete or inconsistent. Because:
- If the system is consistent, then it will be incomplete because there will be at least one statement about reality which cannot be proved to be true inside the system. Or…
- If the system is complete (i.e., fully describes reality), then it will be inconsistent because there will be at least one theorem in the system that is contradictory to the system’s axioms and/or other theorems.
Put more simply, Goedel proved that:
- If the system is complete (describes all real truths), it must have some logical inconsistency;
- If the system is consistent, it must be incomplete (i.e., it fails to describe all real truths). 40
Said still more simply, “Goedel proved that no formal system composed of a finite set of axioms and rules of inference [i.e., theorems] can ever capture the entire body of truths of mathematics.”41 Einstein said it best of all: “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
Note: Just as there’s no “complete” mathematical system that can capture all true things about reality, it is also true that no spoken-prose or poetic metaphor completely describes reality.
That sounds depressing…. But it’s not:
Mario Livio writes, “Contrary to some popular misconceptions, Goedel’s incompleteness theorems do not imply that some truths will never become known. We also cannot infer from the theorems that the human capacity for understanding is somehow limited. Rather, the theorems only demonstrate the weaknesses and shortcomings of formal systems.” In fact, “the impact on the effectiveness of mathematics as a theory-building machinery has been rather minimal.”42
For example, in the case in Euclidean geometry, you’ll remember that the parallel axiom could not be proved to be consistent, so for centuries is was just assumed to be true, and the geometry was assumed to be consistent. Then, in the mid-1800’s when the parallel postulate was viewed as inconsistent, new geometries could be developed which made the interrelated family of geometries more complete.
Human mathematicians were able to see that new and different Elliptical and Hyperbolic Geometries could be born out of the “parallel-line problem” present in standard Euclidean geometry. As Einstein once said, “In every difficulty lies opportunity.”
The Weave of Continuing Creation says — The unprovability of the parallel postulate turned out not to be a weakness of flat geometry, but a door to other regions of mathematics. When the first geometry no longer covered everything, humans invented additional geometries that cover more! This “hidden passage” phenomenon happens over and over in mathematics. Apparent “holes” in some mathematical system turn out to be opportunities to discover (or create) new realms of mathematical beauty and power.
Things Related to Mathematical Incompleteness
There are a number of interesting things that are “similar to” or “connected with” the idea that no single mathematical system can express all mathematical truths.
Some Things Can’t be Compacted or Generalized.
Goedel’s Incompleteness Theorems are paralleled by the existence of incompressibility in the branch of mathematics known as Computation Theory. Computation Theory deals with algorithms, which are the rules and steps used to solve a problem or predict a system’s behavior, particularly when using a computer.
Computer algorithms are heavily used to model the behavior of complex, chaotic, dynamic systems such as the weather, ecosystems, or the biosphere of Earth. (These are also called non-equilibrium systems, because they take in energy from the outside, i.e. from our Sun.) If one could write an algorithm complex enough to fully simulate an ecology, it would have to contain all the variables and all the interactions of the system itself. In these cases, the easiest way to see what’s going to happen is to let the system itself play out in the real world.
Professor Stuart Kauffman writes that “a well-founded theorem in the theory of computation shows that in most cases by far, there exists no shorter means to predict what an algorithm will do than to simply execute it [i.e.to run the full program], observing the succession of actions and states as they unfold. The algorithm itself is its own shortest description. It is, in the jargon of the field, incompressible.”43
Dr. Kauffman continues, “it would be of the deepest importance were it possible to establish general laws predicting the behavior of all non-equilibrium systems. Unfortunately, some believe [such laws] may never be discovered.” 44
Paradoxes are similar to the gaps in formal mathematical systems
Many mathematical systems are inherently incomplete because they are unable to refer to, or reason about, themselves.
In computation theory, the “halting problem” is analogous to Goedel’s Incompleteness Theorem because it shows that there is always some task that a computer cannot perform, namely reasoning about itself.
Goedel’s Incompleteness Theorems relate to a long tradition of mathematical paradoxes such as Russell’s paradox and Berry’s paradox, and Epimenides’ paradox. The simplest paradox is the well-known Liar’s Paradox, which consists of just one short sentence: “I am lying.” Let’s assume that the “Liar’ is named Tommy. His sentence is paradoxical because if it is true, then Tommy is lying. But if Tommy is lying, then the sentence can’t be true. And we go around and around in an endless cycle, with no resolution.
When computers run into this issue, they in fact do go around and around, in an endless loop of iterations. It takes an outside observer, perhaps a human computer operator, perhaps a supervisory computer program, to see the futility of what is going on. The human can do this because she has a larger frame of reference. She is not part of the system; she stands outside the system.
This relates to the idea in our Essay, Patterns in Continuing Creation, that nothing can have meaning apart from a frame of reference. White has no meaning if there is no black. So Non-Euclidean Elliptical geometry has no meaning for us unless we have at least a small grasp of what a closed (globe-shaped) universe is.
Other paradoxes, like the two-sided sign paradox, bounce back and forth between two mutually contradictory sentences. These are called circular reference paradoxes. Circular references are also common in the computer world, and even on simple Excel spreadsheets. Similarly, in game theory, endlessly reflective patterns can occur where two players must model each other’s mental states and behaviors, leading to infinite regress.
The Moebius Ring
Geometric figures can also be paradoxical. The Moebius Strip, which we use as the logo for our website Continuing Creation.org, is a good example. It works like this: If you cut a short strip of paper and tape its two ends together, you create a ring which you can put around your finger. If you take a second, identical, strip, but you flip over one end before taping it to the other end, you create a Moebius Strip. The Moebius Strip (or Moebius Ring) can also go around your finger.
The difference is that a regular ring has two sides – an inside and an outside, while the Moebius Ring has only one side! If you take a pencil and trace a line down the center of a Moebius-strip-ring, the track of your pencil will take you right back to your pencil’s starting point. Moreover, the pencil track on the Moebius ring is actually twice as long as the track inside the regular g! And if you cut along the pencil line all the way around, you will suddenly have two rings, chained together.
The Klein Bottle
A Klein Bottle is the three-dimensional analogue to the Moebius Ring. The Klein bottle is able to hold water, like a pitcher, yet has only one surface. This can be proven by the same pencil-track method we used for the Moebius Ring. (Note: To fill a Klein bottle with liquid, first turn it upside down. Placing a hand over the mouth where you poured the liquid in, quickly flip the bottle back over. Some of the liquid will spill out during the flip, but most will remain “inside.”)
Singularities in Mathematics and in Physics
Mathematical systems are often “incomplete” because they can’t “see” outside the bounds of their own system. The same sort of thing occurs in physics.
We are told that the Big Bang started with a Singularity – with a unity of particles and fields, of matter and forces, of time and space.
In mathematics, a “singularity” is in general a point at which a given mathematical object is not defined, or a point where it fails to be well-behaved in some particular way. For example, for real numbers, the function: f(x) = 1/x has a singularity at x = 0, where it seems to “explode” to plus-or-minus infinity. This is so mind-boggling that mathematicians just say it is “undefined.” In other words, as we learned in school, “we can’t divide by zero.” If we try, we get something that is a complete mystery, if not a complete impossibility. 45
The Path of Continuing Creation says – The Initial Singularity (Big Bang) of physics, and the mathematical singularity that leaves dividing by zero undefined, are both “discontinuities” – they are places where man’s understanding (so far) cannot go. Many people feel that these singularities are representations of the “mystery of God,” or of God himself. Are they representations of G>O>D>? Maybe, but Followers of G>O>D> choose to focus on the Creation that happens after and all around these two singularities.
Beauty in Scientific Laws, Models, and Theories
Next, we want to talk about the beauty of mathematics. But since mathematics is the language of science (particularly of physics) first describe Beauty in scietific explanations.
A scientific law, model, or theory is effective (“powerful”) and beautiful if:
- It is “elegant” — it can explain complex or multiple phenomena with simple or few equations.
- It contains few arbitrary or adjustable elements.
- It agrees with and explains all existing observations.
- It makes detailed predictions about future observations that can disprove or falsify the model if they are not borne out.
“If the modifications needed to accommodate new observations become too baroque, it signals the need for a new model.” Of course, all these criteria (e.g., “baroque-ness”) can be subjective. Even very successful models in use today do not satisfy all these criteria, which are aspirational in nature. 46
Other writers say that physics seeks to describe a vast number of experimental observations in terms of a few powerful laws expressed in compact mathematics. Some call this “power with economy,” or just “simplicity.” A fine example of powerful simplicity in physics is Einstein’s equation, E=mc2 (i.e., Energy = Mass times the Speed-of-light-squared). Simplicity is usually accompanied by three other features of good physics theory:
- Symmetry – Balance; proportion. Symmetry is also a key feature of “beauty.”
- Depth — Deep explanatory power; the ability to connect disparate fields.
- Elegance — no wasted motion or redundancy. 47
Maxwell’s Equations, which describe the rules of electromagnetism, are together a good example of all these characteristics in physics. After Michael Faraday had conducted years of experimental investigation and described his results in volumes of notes, James Clerk Maxwell was able to connect the behaviors of electricity, magnetism, and light together with a set of just four succinct equations.” 48
Beauty in Mathematics
Now, we move on to our intended topic — Beauty in Mathematics. After reading thus far in our Essay, we hope the reader already sees much of the beauty in mathematics. Demonstrating the beauty of mathematics was one of our primary goals for this Essay. (Note: At the time of this writing (3-2-18), Wikipedia has a very fine article on “Mathematical Beauty.”)
To begin this section, here’s a quotation from the famous mathematician Bertrand Russell, written in 1919:
“Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry.” 49
Beauty in Mathematical Method
Mathematics can be beautiful in its method, in its results, and in our experience of it.
A prime example of Beautiful Method was discussed earlier, when we described how a mathematical proof can be beautiful. A proof can be beautiful if:
- It uses a minimum of additional assumptions,
- It is succinct,
- It does its work based on new insights or using a new method, and…
- It’s method can be generalized to solve other problems in mathematics.
For a more graphic, hands-on demonstration of a beautiful and elegant proof, see Appendix A.
Beauty in Mathematical Results
This type of mathematical beauty is created when a new insight or a whole new branch of mathematics is achieved. Beauty is also created when a previously unseen connection is established between two different areas of mathematics These new inventions and connections are known as “deep.” (The opposite of “deep” is “trivial.”) We have been calling such deep connections “hidden passages.”
The most famous example of a deep connection is Euler’s Identity, which links three fundamental numbers together in one simple equation. The three numbers are: Pi (which we have discussed earlier), “e” (the base of the natural logarithm), and “i” which is the Unit of Imaginary Numbers. The equation is: (“e” to the power of [“i” times “pi”]) +1 = 0. However, we are certainly not going to attempt to explain the equation here in this Essay!
Beauty is also created when a branch of mathematics is surprisingly connected to an area of science. Concepts explored by mathematicians only for pure reasons – with absolutely no application in mind – can turn out decades later to be unexpected solutions to problems grounded in physical reality. 50
For example, the mathematician G. H. Hardy (1877-1947) was key in developing Number Theory. This branch of mathematics had no use until World War II, when it became essential for military communications and codes. And as we discussed earlier, the new non-Euclidean geometries developed by Georg Riemann et. al., in the late 1800’s turned out to be “precisely the tools that Einstein needed to explain the cosmic fabric [of space-time]” 51
Beauty in Our Experience of Mathematics
A good way to experience mathematical beauty is simply to look at illustrations (photos and graphs) of mathematical functions. For example, we refer the reader to online photographs of mineral crystals, snow-flakes, tree branches, a geometric figure called the Compound of Five Cubes, and a stunning, endlessly self-similar mathematical creation known as the Mandelbrot Set. (For more information, Google “Mandelbrot Set.”)
Just below is a second computer-generated image of the Mandelbrot Set.
Note: The original “three black spheres” shown in our earlier illustration of this set can be seen in the lower left corner of our following second image):
What Is Beauty – to Humans?
We should pause here and discuss what beauty is to human beings. A little thought and introspection results in the following list of “beauty elements.” These elements are all true aspects of Beauty, but they do often conflict with each other — “too much” of any one of them makes an object less beautiful, or even ugly. When is there “too much”? That decision is subjective, often cultural, and “a matter of artistic taste.”
The Elements of Beauty in Patterns:
Beauty happens when there is pattern. But what kinds of patterns? Patterns with:
- Uniformity, Repetition, Symmetry – A beautiful face is uniform and symmetrical.
- Proportion, Balance – Perfect symmetry (like the perfect grid on the side of a modern glass office building), can be boring. Shapes that are proportionate, as on our photo of the Parthenon’s use of the Golden Ratio at the beginning of this Essay, can be more pleasing.
- Variety, Novelty, Surprise – Japanese flower arrangements (ikebana) are never symmetrical, never perfect. An adventurous branch or blossom always ventures out from the others, seeking new territory, new possibilities.
- Naturalness, Imperfection — To be natural, a shape must be imperfect – as we saw when we discussed ideal circles versus real circles. Zen potters and flower artists nearly always introduce deliberate imperfections into their art, to connect it to Nature.
Imperfection Can Enhance Beauty
Imperfection, which is related to our inherent inability to compact, arrange, and/or explain everything, can enhance beauty if imperfection is culturally accepted.
For example, in nature, the branching (of trees, rivers, veins of copper) is almost always approximate. The pure pattern is thrown off by random fluctuations in plant growth, weather, and light blockage from higher branches. In geology, we see that crystals (including the green and gold mineral crystals pictured below), start to grow in an orderly geometry until some shift in the environment or contact with an adjacent crystal triggers a change in direction or a truncation.
Like the flower arrangers of ikebana, the great Japanese potters of the Zen tradition always leave a deliberate imperfection in their tea ceremony cups, to remind us that existence is imperfect and transitory.
Therefore, Goedel’s Incompleteness Theorem, which says that no single mathematical system can describe all of reality, presents no problem for the artist – at least not for Ikebana artists and Zen potters!
The Weave of Continuing Creation remarks that the Zen esthetic of Japan, with both pattern and variation, is a metaphor for evolution itself. Like ikebana, the biosphere has balance, repetition, novelty, imperfection (mutation) and exploration of new spaces by new species. This congruence between art and Nature confirms that humans find our natural world to be beautiful… as we should.
Even Certain Animals Can Appreciate Beauty
Humans are not the only creatures who appreciate beauty. The bird who makes patterns of seeds, buds, glass for its mate or potential mate appreciates beauty. Other birds are attracted by birdsongs and colored plumage. Bees (and hummingbirds!) appreciate beauty, because they are attracted to flowers that display symmetry, bright colors, and attractive scents.
Yale University professor Richard O. Prum, in his best-selling book, The Evolution of Beauty: How Darwin’s Forgotten Theory of Mate Choice Shapes the Animal World (2107) makes a compelling and well-documented argument that beauty is a fundamental goal of natural evolution, as are survival and reproduction. 52
What Is Mathematics?
Now we turn to the philosophical questions about mathematics that we brought up near the start of this Essay. Frankly, this section comes last because many readers (perhaps rightly) lose patience with “deep philosophy.” Appreciating the power and beauty of mathematics is more important than trying to decide what mathematics “is.”
In the last decade, there has been quite a lot of writing about “God and Mathematics.” Authors have posed questions like,
- Did mathematics arrive before space-time, the four forces, particles, and the laws of physics?
- Did God use mathematics to create the physical universe?
- Is God a Mathematician? (Livio)
- Is mathematics the language of God?
- Is God, or perhaps even reality itself, made of mathematics?
As we have said in other Essays, it can never be proven whether or not there is or is not a God, a “super-mind” who stands outside the cosmos, and who then created the cosmos. Why? Since we cannot ourselves stand outside our own universe, we cannot “see” before the Initial Singularity. It is impossible to see things that may have existed before the beginning of time itself. Trying to do that is like trying to divide by zero in the realm of mathematics – it is simply “undefined.” As we discuss in our Essay, Patterns in Continuing Creation, no logical system, including human reasoning, can step outside its own frame of reference. (Many religious people also understand this “un-provability,” which is why they say that belief in God is a matter of faith, not proof.)
Followers of Continuing Creation say that if a super-mind did create everything, then he, she, or it likely created only the earliest things — time and space, the fundamental forces, the first elementary particles, and the laws of physics. We think mathematics should also be included in that group, for a total of five. After those five things were “born,” they interacted with each other over vast spans of time, and without further intervention from the “super-mind,” to evolve everything else – including us humans.
The idea that God actively created only the earliest framework and physical laws of the universe, and then “stepped to the side” after that, is known as Deism. Deism became a widespread belief among educated Western people during the Enlightenment of the 18 th century. Deists have included Voltaire, Benjamin Franklin, John Locke, Thomas Edison, Mark Twain, and John Muir. Deism is very much a forerunner of the Path of Continuing Creation. See our Essay, Forerunners of the Book of G>O>D>.
The Way of Continuing Creation elects to make the Process of G>O>D>, the Growing> Organizing> Direction of the Cosmos, the focus of our Spiritual Path. We do not focus on a God who may or may not have initiated the Process of G>O>D>; we focus on the Process of G>O>D> itself. Since we can’t see “before” the Initial Singularity, we choose not to waste time speculating about it. There is more than enough Creation since the Big Bang to provide us with spiritual sustenance.
So, the first thing we need to do is re-phrase those questions above and add a few additional ones. This gives us the following set of questions about mathematics, along with our “short answers,” in bold.
- When did mathematics arrive? Likely with or very soon after the Initial Singularity.
- Does G>O>D> use mathematics to create the physical universe? Yes, every day.
- Is G>O>D> a Mathematician? No – because G>O>D> is not a human being.
- Is mathematics the language of G>O>D>? It is one of the Languages.
- Is G>O>D> (or reality itself), made of mathematics? Matter & energy are also required.
- Is mathematics more real than the physical universe? A perfect circle is ideal, not real.
- Is mathematics fundamental in the physical universe? Yes, along with the laws of physics.
- Is there only One mathematics, or many different mathematics? One, but with branches.
- Is mathematics something we humans discover, or something we invent? Both!
Opposing Arguments About the Nature of Mathematics
All these questions, can be sorted into two generally opposing groups, or two opposing arguments, both of which have been and still are championed by different camps of mathematicians. These two opposing argument-groups are as follows:
- Objectivism: There is only one mathematics (although it has many branches). Its truths are objective and fundamental. We discover them by looking at reality. Since humans discover mathematics, branch by branch, then mathematics comes from reality, and mathematics is objectively true. Mathematics has an existence outside the human mind.
- Subjectivism: There are many different mathematics. They are all subjective models. We create them in our minds and fit them onto reality as best we can. Their “truths” are partial and provisional. Since humans invent mathematics, and each mathematical system is a subjective construct devised by the brain, then other mathematical systems that are “just as good” may be devised by humans later. Mathematics exists only in the human mind.
Answers to these questions are slippery. Even among modern mathematicians there is no firm consensus about them. As we will see, The Book of Continuing Creation comes down sometimes on the side of Group A, sometimes on the side of Group B, and sometimes in between the two. We have done the best we can, for now. Here are more complete answers to those bullet-pointed questions above that warrant a more detailed treatment:
Is Mathematics the Language of G>O>D>?
No. Mathematics is only one of the languages of G>O>D>, along with French, Arabic, music, the genetic code, and diagrams of all kinds.
Spoken languages have developed words to describe and evoke sight, sound, and smell making those languages suitable for literature and poetry. Spoken languages likely arose before mathematics, although the first mathematics – counting – was probably not far behind.
The most basic idea of mathematics is that the quantity, the count, is divorced from the things being counted. So, 2+2=4. So, 2 rocks +2 rocks = 4 rocks. And 2 dogs + 2 dogs = 4 dogs. But 2 rocks + 2 dogs = 4 things. If you can’t generalize from rocks and dogs to things, you can’t count.
The Earth is home to non-human animals that can also count, at least in an approximate way. These include primates, parrots, and even chickens. 53
We are able to count things only because they are separated and/or different from each other. We saw in earlier Essays that differentiation was the very first process of Continuing Creation, happening in the micro-seconds after the Big Bang. We’ve also seen that one of the basic requirements for life is having a boundary wall – a membrane that separates one single-celled organism from another. Our ability to count is a simple, evolved extension of our ability to perceive and appreciate separateness. Today, cognitive scientists hold that “Mathematical insights [continue to] develop by borrowing mind tools used for building language.” [Mario Livio, Ibid., p. 232[/note]
Counting is likely also related to time perception, because counting requires a brain that perceives one thing at a time. If we focused totally on just the now (like Buddhist monks aim to do, and jellyfish can’t help but doing), we would not count (or classify, or analyze) anything. With no sense of time, no memory of a “first” thing, we could never perceive a “second” thing.
Mathematics is a Language with Unique and Powerful Properties
The human brain invented language make thought-models of reality, mentally simulating what reality does or is expected to do. Language was also invented to communicate these simulations between people.
Language evolved when the brain acquired the ability to function off-line, effectively running simulations (models) of actions without reflexively generating immediate actions. 54
Spoken languages have verbs that stand for actions, and nouns that stand for things. We use verbs to name the actions of the four fundamental forces, and we use nouns to name the matter that the verbs act upon. The verbs and nouns in spoken languages are an evolved outgrowth of the energy and matter in our universe.
Similarly, scientists use mathematical symbols (or English and Greek letters) to stand for different kinds of energy and matter, to denote shapes and quantities, and to describe functions. Mathematical languages (e.g., algebra, set theory, and calculus) fulfill roughly the same purposes as spoken languages.
Both spoken languages and mathematical languages have concepts like equals, abstraction, negation, metaphors, hypotheticals, and modifiers (e.g., adverbs and adjectives). 55
However, mathematics has developed/evolved into a particularly precise, compact, and logical language (and modeling tool), used by humans and computers as agents of Continuing Creation to describe the behaviors of matter and energy; and to predict and manage events that happen in our world. 56
Our brains evolved to deal with the physical world, so it is no wonder our spoken and mathematical languages also evolved to deal with the physical world. 57. Mathematics, just like spoken language, evolves. It has progressed from simple counting to geometry, algebra, calculus, and all the various “maths” on the Map of Mathematics.
On the other hand, as we’ve noted, the language of mathematics lacks context and connotation, and is not well-suited to literature and poetry.
Are the Processes of G>O>D>, or Is Reality Itself, “Made” of Mathematics?
No. They are not composed out of mathematics only. Both the processes of G>O>D and reality itself are made out of the four fundamental forces, energy, and matter… using mathematics and the laws of physics as governing “blueprints.”
Is Mathematics More Real than the Physical Universe?
No. The perfect circle is not real, it is ideal. Here’s what we mean:
True, in ancient Greece, the followers of Pythagoras and Plato did argue that the concept of a perfect circle (the set of points on a flat plan equidistant from a single central point on the plane) is more real than, say, the circle of a sawn tree trunk. But this distorts the everyday meaning of “real.” Most people would say that the imperfections in a tree trunk are what proves that the tree trunk is real.
Besides, where would the Greeks’ perfect circle reside? In some off-world heaven or “realm”?” In the human mind or in the “collective mind of our culture”? Maybe so. But if the perfect circle resides in our minds, and if our minds reside in our brain-matter, which resides the physical world, then the perfect- circle-in-our-minds is part of the physical world, and not out there is some spooky, heavenly realm.
In any case, human-built machines are increasingly able to make near-perfect circles, and all these increasingly-perfect circles reside in reality, here on Earth.
Is Mathematics Fundamental, (in some way Controlling), in the Physical Universe?
Yes… maybe. Both mathematics and the laws of physics play a central role in Continuing Creation. If we use construction of a building as our metaphor, we could say that matter is the building material, the laws of physics are the construction workers and their machines, the food inside the workers and the gasoline inside the machines provide the energy, and mathematics is the blueprint.
So, did the perfect circle, the mathematical concept of the circle, come before the Big Bang? Did it come – like “God” — from somewhere outside our universe? Is it a “blueprint” drawn by some outside “intelligence”? Maybe –there is no way for us to know, since we cannot stand outside the universe.
But instead, the circle may be the result of the laws of laws of physics, because it is the shortest line which allows a living thing to enclose a given area of substance. So, the cross-section tree trunks are circular. Similarly, the laws of gravity operate equally in all directions, causing new planets to coalesce into spherical shapes. And so on.
A wonderful 2012 book called Design in Nature, by Adrian Bejan and J. Peder Zane (both holding doctorates in engineering) makes the case that the shapes found in nature – the branching of plants, the meanderings of rivers, the V-shape of formations of flying geese, etc. – are the direct result of the laws of physics as expressed in the principles and equations of engineering.
In any case, we know that mathematics is central in Continuing Creation just by looking at mineral crystals found in the Earth. Their forms are clearly geometric, clearly formed according to a mathematical blueprint. These crystals are certainly not just “subjective models created in the minds of humans,” because Nature forms them without any human agency. This is also true of the regular branchings of certain plants, of snowflakes, and of cylinder-shaped tree trunks.
All these geometric patterns arise from the shapes of the underlying molecules of the matter that composes them. Their formation occurs when environmental conditions – energy flow, pressure, temperature – are in the right balance. Slight variations in the environment cause the infinite variety of snowflakes, and the imperfect individuality of tree trunks and crystals.
The simple circle also plays an orchestrating role in Creation. We discussed this earlier, in our section called The Circle and The Number Pi. Here, let’s just add one more example, an example taken from sociology: When hikers defend their camp against wolves, they do so by forming a circle, because the circle gives them the shortest perimeter for defending a given area of land.
The Weave of Continuing Creation concludes that mathematics is as fundamental as the four fundamental forces and the laws of physics, all of which likely co-evolved at or near the time of the Initial Singularity.
We agree with mathematician Shing-Tung Yau, who writes, “most scientific facts are based on things we cannot see with the naked eye or hear with our ears or feel with our hands. Many of them are described and guided by mathematical theory. Is the sphere part of nature or is it a mathematical artifact [invented by the human mind?]… Perhaps the abstract mathematical concept is [itself] actually a part of nature.” 58
Are There Many Different Mathematics, or Only One Mathematics?
There is one mathematics, but it has many interconnected and interrelated branches.
To prove that mathematics is One, we exhibit three pieces of evidence:
Evidence Exhibit #1: We don’t have several geometries, we have several variations of geometry. All of them deal with lines and they all deal with angles. Geometry-as-a-whole incorporates and subsumes both Euclidean Geometry and Non-Euclidean Geometry, just as the Theory of Relativity incorporates and subsumes Newtonian physics.
The Map of Mathematics shows about 40 different branches or fields. The branches that are most related are grouped together in about 14 different regions on the Map. However, there are also important relationships between branches that are “distant” from each other.
As we have mentioned, theorems in one language (such as geometry) can often be proven in another language (e.g. in algebra).
Evidence Exhibit #2: Trigonometry draws on (is connected to) algebra and geometry; and calculus draws on trigonometry, algebra, and geometry. Still, trigonometry and calculus each go in new directions of their own, and answer questions that geometry and algebra do not address. As we have said earlier in this Essay, no one mathematical system can completely explain all of reality.
Of course, different branches of mathematics are better suited to different aspects of our real world. Counting works for pebbles in a jar, but not for drops of water in a jar. For the latter, we need weighing and/or volume. So: Different mathematics for different aspects of reality, but all correct for their particular set of phenomena. 59
Evidence Exhibit #3: As far as anyone knows, there are no correct contradictions anywhere in mathematics. Apparent contradictions turn out to be correctable errors or they are resolved by new mathematical syntheses. While conventional Euclidean geometry proved to be universally incomplete, it is quite complete for flat surfaces. Non-Euclidean geometry is complete for curved surfaces. Together, geometry-as-a-whole completely addresses all surfaces. (See https://math.stackexchange.com/questions/753997/are-there-contradictions-in-math.)
Mathematics Something We Discover in the World, or Something We Invent in the Mind?
The question is the same as asking, “Does mathematics have an existence that is entirely independent of the human mind, or, is mathematics nothing but a human invention?” Or, “Is mathematics objective or subjective?”
The Weave of G>O>D concludes that we both discover and create mathematics. Sometimes we do the first, sometimes the second, and sometime the two are interwoven. After all, both the outer cosmos and our inner brains are parts of G>O>D>, and they are interrelated and co-evolved.
Historically, sometimes new mathematics precedes any usefulness in the real world. That was the case in Non-Euclidean geometry. But in other cases, we see something in the real world which prompts us to invent a new mathematics that only later turns out to describe a part of reality Here’s an example of each case:
- A real-world discovery that preceded mathematics: Until the 1980’s, it was thought that mineral crystals can only take one of 230 shapes, because that number is all you can assemble by connecting squares, rectangles, triangles parallelograms and hexagons. You can’t connect pentagons, heptagons, octagons and all other regular polygons because they always leave open spaces between them. However, in 1984 a new man-made alloy of aluminum and manganese was found to have a unique crystalline shape – a “quasi-periodic crystal,” or “quasicrystal.” By 1984, other quasi-crystals were made in the laboratory. To see a quasi-crystal made out of pentagons, google “quasicrystal.” Then, further study revealed that these new crystals are mathematically related to the beautiful aperiodic (i.e. not-regularly repeating) patterns called “Penrose tiles.”
- Mathematics that preceded a real-world discovery: An abstract branch of mathematics called “Knot Theory” deals with shape relationships depending on position alone. It is a kind of topology. Topology is the branch of mathematics that shows what happens when a figure is stretched or deformed, without tearing off pieces or poking holes. Knot theory has turned out to have “extensive modern applications in topics ranging from the molecular structure of DNA to string theory in physics. 61
Mathematics Can Be Culturally Dependent When It Is First Discovered and/or Invented
In Western civilization, Pythagoras started with the “sacred” number 5, drew a pentagram, and found the number phi. Later, (but still in Europe), Fibonacci found his “Sequence,” and still later Kepler linked the Fibonacci Sequence to the number Phi. However, the eastern civilizations of China and India never focused on pentagrams, never found Phi, and never made the connection to the Fibonacci Sequence. 62
On the other hand, the concept that zero could be used as a number originated in ancient India and then spread into the pre-Islamic Middle East. 63 By 1779 BC, the Egyptians had a symbol for zero in their accounting texts. 64 Fibonacci (1170-1250), who grew up on North Africa, is credited with introducing the number zero to western civilization.
As we have discussed, Pythagoras, Plato, and their disciples thought mathematics was something we discover. They reasoned that no one has ever seen a perfect circle in the physical world, but we can define a perfect circle in our heads, using mathematics. When we define that a circle is “the set of all points on a plane that are equidistant from one central point,” we have discovered the perfect circle, according to the objectivists.
The Modeling of Everything
Modern mathematical objectivism starts with the idea that everything in the universe can be modeled in mathematics. Here are four supporting points:
- The code of DNA can describe, and direct the assembly of, every one of Earth’s plants and animals.
- [Fourier’s Theorem also said that] any complex pattern, whether in time or space, can be described as a series of overlapping sine waves of multiple frequencies and various amplitudes…In effect, any complex physical event can be reduced to the mathematical simplicity of sine waves. As Bruce Hood has written, “It doesn’t matter whether it is Van Gogh’s ‘Starry Night,’…Chanel’s No. 5, … or a Waldorf salad.” 65
- Stephen Wolfram created a new math out of cellular automata instead of the differential equations mostly used today. 66
- Starting in the mid-1970s was a new discovery that there was a close correspondence between quantum field theory and advanced geometry. A Rosetta Stone of sorts called The Wu-Wang Dictionary was developed to translate the tenets of quantum field theory over to the tenets of advanced geometry. Before, the math of quantum physics was a “train wreck.” Now, “Quantum field theory, despite its name, turns out to really be a piece of pure mathematics developed by ingenious amateurs.” All known physical phenomena can now be recognized as fashioned from the pure marble of geometry…. The source code of the universe is likely to be a purely geometric operating system written in a single programming language.” 67
This viewpoint holds that no one body of mathematics, no one rigorous model, can fully describe reality. We have no problem with that, as we have said above. But Model-dependent Realism goes a lot farther and says that reality itself depends on the model we use to detect it.
Model-dependent realism claims that it is meaningless to talk about “true reality” because we can never be absolutely certain of anything. The only meaningful thing is the usefulness of the model. The term “model-dependent realism” was coined by Stephen Hawking and Leonard Mlodinow in their 2010 book, The Grand Design.” 68
The Book of G>O>D holds that this view ignores the fact that all mathematical models are interrelated, because all types of mathematics are interrelated branches of one mathematics. New connections between and syntheses of mathematical branches happen all the time. And the same is true of models in physics. Mathematics may be different from the laws of physics, but it is clearly tightly woven with them. The relationship between physics and mathematics is dynamic – it co-evolves, like the creative interplay of yang and yin.
Model-dependent Realism in Physics
In physics, whether a subatomic particle exists depends on our trying to observe it. Common sense says that if we don’t look for a particle, we won’t see a particle. But Model-dependent Realism says that if we don’t look for a particle, there is no particle. The idea relates to whether light is a “wave” or a stream of discrete photons. When our detectors look for photons, they find them, and at all other times, we see only light waves.
The Book of Continuing Creation suggests that the phenomenon may be both wave and a virtual particle (a.k.a. an “energy-bundle” or a “transient fluctuation”). Sometimes we see its wave aspect, and sometimes we see its energy-bundle aspect. Also, we cannot predict when an energy bundle is going to pop up out of the energy field or wave; we can only know its probability of doing so.
Model-dependent Realism in the Social Sciences:
The social sciences have their own versions of model-dependent reality. Many sociologists and anthropologists hold that all cultures are equal. Therefore, there are no “bad” cultural traits (or no general cultural traits at all). Or, social scientists will say that a cultural trait (e.g. torturing all prisoners captured in tribal warfare among American Plains Indians) is not bad, because it is not perceived as being bad inside those tribal cultures.
The Program of G>O>D> rejects this absurdity. It seems as ridiculous as a totalitarian dictator making no distinction between real news and fake news: if the “Precious Leader” says something is true, then it is true. Captors who torture their prisoners for “fun” are indulging their human ability to be evil.
The Mathematical Universe Hypothesis
Then, in 2104, MIT’s neo-platonist cosmologist Max Tegmark went still farther than Model-dependent realism. In his book, Our Mathematical Universe, Dr. Tegmark argues that:
- Mathematics can describe (simulate) everything in the real world,
- Every possible computable mathematical structure exists,
- However, our human brains only perceive the particular structures that we inhabit, and
- mathematics itself is the ultimate reality, because every mathematical system must have created a universe parallel to our own, because all the mathematical systems all have the same creative power that ours has clearly had.
Tegmark calls his idea the Mathematical Universe Hypothesis. It combines the platonic concept that ideal forms (like the circle) are more real than Earthly incarnations of those forms (a sawn tree trunk), with the new idea that all reality can be modeled and described with mathematics. He goes on to say that “in those [universes] complex enough to contain self-aware structures” [like we human “structures” are here are on Earth], those self-aware structures “will subjectively perceive themselves as existing in a physically ‘real’ world”. 69
The Book of G>O>D> says – Tegmark’s theory is “a bridge too far.” We say that Reality consists of forces and matter, patterned by algorithms. The algorithms (the patterns) are not themselves the reality; they cannot exist without the forces and the matter. Without forces and matter, there is no reality as humans understand reality. A simulation of reality can never be reality. Besides, there’s no evidence that other universes exist – we have only abstract theory’s suggestion that they might. If other universes do exist, they surely don’t exist for us. Therefore, they have no role in our practical or spiritual lives.
Our view from the Path of G>O>D> is that the best word to describe the progress of mathematical knowledge is not exclusively “discovered,” or “invented.” The best description is that mathematics evolves and grows. This position is consistent with humankind’s purpose within G>O>D>, which is that we should explore, learn, and create the new. The creative interplay of yang and yin, of differentiation and combination, of energy and matter, of science and mathematics provides us with plenty of inspiration and challenge in our own universe-reality, without speculating about the possible existence of “parallel” universe-realities.
APPENDIX A – A Proof of the Pythagorean Theorem that Everyone Can Understand:
Over the centuries there have been hundreds of different proofs of the Pythagorean Theorem!
The theorem was named after Pythagoras because he was the first to prove it. He probably used a dissection type of proof similar to the following in proving this theorem.
Although it is often argued that knowledge of the theorem predates him,70 the theorem is named after the ancient Greek mathematician Pythagoras (~570–495 BC), as it is he who, by tradition, is credited with its first recorded proof. 71 Mesopotamian, Indian and Chinese mathematicians all discovered the theorem independently and, in some cases, provided proofs for special cases.
The theorem has been proven in hundreds of different ways – possibly the most for any mathematical theorem. They are very diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years.
The theorem can be generalized in various ways, including higher-dimensional spaces, to spaces that are not Euclidean, to objects that are not right triangles, and indeed, to objects that are not triangles at all, but n-dimensional solids. The two little videos showing the proof by triangle rotation and triangle movement are really cool. And we have presented it using words, below.
First — An Example
First, let’s see if it really works using an example — a right triangle with sides that are 3, 4, and 5 units long. Using Legos, we can see that the Pythorean Theorem turns out to be true for this particular example. The red square of 25 units (“dots”) equals the sum of the yellow square (9 dots) and the blue square (16 dots).
Second — A “Cut-and-Paste” Proof of the Pythagorean Theorem
Now we’ll present what is reputed to be Pythagoras’ own proof of the Pythagorean Theorem, proving that the theorem is true for right triangles of every size:
First, we recall that a “right”-triangle is one where one of its angles is a 90-degree angle.
Then, let a, b, c denote the legs and the hypotenuse (the longest side) of a given right-triangle.
Consider the two squares in the accompanying figure, each having a+b as its side. The first square is dissected into six pieces-namely, the two squares on the legs and four right triangles congruent to the given triangle. The second square is dissected into five pieces-namely, the square on the hypotenuse and four right triangles congruent to the given triangle. By subtracting equals from equals, it now follows that “the square on the hypotenuse is equal to the sum of the squares on the two smaller sides.”
Consider the following two figures. Start with the figure on the right.
- Consider the right triangle with little sides a and b, and big side (hypotenuse) c.
- Draw a square on the hypotenuse c.
- Now draw in triangles on the 3other side of the square that are the same as our original triangle. This gives the drawing on the right side, above.
- Staying inside the big square, move the triangles around so they make the left-hand drawing, like the big square on the left, above.
- Having done that, the left-hand big square now shows the squares on the little two sides of the original triangle: aXa and bXb.
- The areas of the two big squares are equal.
- If we subtract the same amount of area from both Left and Right Big Squares, the remaining areas on the Left and Right must still be equal.
- So, take away the four triangles from the Left Big Square and the four triangles from the Right Big Square. (Note we have removed the same amount of area from the Left as from the Right.)
- We are left with the two little squares (aXa and bXb) on the Left, EQUALLING the one remaining square cXc on the right.
Therefore: The area sum of the squares on the two smaller sides of every right triangle equals the area of the square on the longest side (the hypotenuse) of each of those right triiangles.
Better: the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.
Q. E. D.
(Note: “Q.E.D.” stands for “Which Was To Be Proved,” in Latin. QED is typically placed at the End of every successful Proof, much the same way that the number “30” is written at the end of finalized newspaper stories ready to go to press.)
We can do this same proof using algebra rather than geometry:
- The area of the first square is given by (a+b)^2 or 4(1/2ab)+ a^2 + b^2.
- The area of the second square is given by (a+b)^2 or 4(1/2ab) + c^2.
- Since the squares have equal areas we can set them equal to another and subtract equals.
- The case (a+b)^2=(a+b)^2 is not interesting. Let’s do the other case.
- 4(1/2ab) + a^2 + b^2 = 4(1/2ab)+ c^2
- Subtracting equals from both sides we have…
Q. E. D.… Giving us a second version – an algebraic version — of Pythagoras’ proof. Of course, this proof depends on the reader knowing and trusting in the algebraic steps that were used; but those steps can also be demonstrated with geometric figures. (We won’t take the time and space to do that here).
Our doing a Pythagorean Theorem proof in both geometry and algebra shows how the two math languages can be translated one to the other. The Pythagorean Theorem can also be proven using the systems of Calculus or Trigonometry. (See: https://en.wikipedia.org/wiki/Pythagorean_theorem#Other_forms_of_the_theorem)
On the other hand, many theorems in mathematics have been proved in only one way. Even more important is the fact that a good number of mathematical “conjectures” remain unproven. For example, as we discussed in the body of our Essay, there is no proof, so far, for Goldbach’s Conjecture.
- Mario Livio, The Golden Ratio: The Story of Phi, the World’s Most Astonishing Number, 2002, Broadway Books, Random House, p. 14.
- References: http://en.wikipedia.org/wiki/Golden_ratio. Also, PBS Nova program “Secrets of the Parthenon” at http://video.pbs.org/video/980040228/
- Mario Livio, Is God a Mathematician?, 2009, Simon & Schuster, p. 236.
- “The Great Math Mystery,” NOVA, Public Broadcasting System, aired April, 2015.
- Mario Livio, Is God a Mathematician?,2009, Simon & Schuster, p. 237.
- Padovan, Richard (1999). Proportion: Science, Philosophy, Architecture. London: Taylor & Francis. pp. 305–306. ISBN 0-419-22780-6.
- Alan Turing, “The Chemical Basis of Morphogenesis,” Philosophical Transactions of the Royal Society, 1952. B. 237 (641): 37–72.
- Grzegorz Rozenberg & Arto Salomaa, The Mathematical Theory of L Systems, 1980, Academic Press, ISBN 0-12-597140-0.
- Benoît B. Mandelbrot, The Fractal Geometry of Nature, 1983, Macmillan. ISBN 978-0-7167-1186-5
- Shing-Tung Yau, “The Shape of Inner Space,” Quoted in , This Explains Everything, John Brockman, Ed., 2013, Edge Foundation, Inc., Harper Perennial, p. 34
- Quoted in J. Neihardt & P. Deloria, Black Elk Speaks, Being the Life Story of a Holy Man of the Oglala Sioux,
State University of New York Press, 1932; 2008.
- Steven Strogatz, “Why Pi Matters,” The New Yorker, 3-13-15.
- Steven Strogatz, “Why Pi Matters,” The New Yorker, 3-13-15.
- Strogatz, Ibid.
- Steven Strogatz, Ibid.
- Hans-Henrik Stølum, “River Meandering as a Self-Organization Process,” Science, March 22, 1996). (5256): 1710–1713. Bibcode:1996Sci…271.1710S. doi:10.1126/science.271.5256.1710. See also Natalie Wolchover, “What Makes Pi So Special,” on Twitter @nattyover or Life’s Little Mysteries @llmysteries.
- Steven Strogatz, Ibid.
- Steven Strogatz, Ibid.
- “Cosmic Code Breakers,” WHJJ, (PBS), Dec, 2012.
- Hideki Uematsu, Cosmic Code Breakers: The Secrets of the Prime Numbers, WHJJ (PBS) Dec, 2012. http://www.archive.pariscience.fr/en/showing/260/cosmic-code-breakers-the-secrets-of-prime-numbers/.
- “Cosmic Code Breakers,” WHJJ, (PBS), Dec, 2012.
- Mario Livio, Is God a Mathematician?, 2009, Simon & Schuster Paperbacks, p. 37.
- Robert A. Bix and H.J. D’Souza, “Analytic Geometry,” Encyclopedia Britannica. Retrieved 3-19-18.
- Saunders MacLane & Garrett Birkhoff, Algebra (First ed.), 1967, Macmillan. pp. 1–13.
- Hartley Rogers Jr., Theory of Recursive Functions and Effective Computability, 1987, The MIT Press. ISBN 0-262-68052-1.
- Chad Perrin, “ITLOG Import: Elegance” Chad Perrin: SOB, 8-16-2006. Also see: Joel Spolsky, “Elegance,” Website: Joel on Software, 12-15-2006.
- Mario Livio, Is God a Mathematician?”, p. 38.
- D.M.Y. Somerville, The Elements of Non-Euclidean Geometry, 2005, Dover Publications.
- Mario Livio, Is God a Mathematician? pp. 123-4.
- Jerry Adler, describing research by Lobo and Bettencourt, “X and the City,” Smithsonian, May 2013 pp 74.
- Mario Livio, Is God a Mathematician, p. 135.
- Amanda Onion, “Why Have Americans Stopped Growing Taller?”, ABC News, 7/3/17. https://abcnews.go.com/Technology/story?id=98438.
- Jerry Adler, “X and the City,” Smithsonian, May 2013 p. 74.
- Rudy Rucker, “Inverse Power Laws,” in This Explains Everything, John Brockman, Ed., 2013, Edge Foundation, Harper Collins, pp. 367-369.
- Rudy Rucker, Inverse Power Laws, in This Explains Everything, John Brockman, Ed., 2013, Edge Foundation, Harper Collins, pp. 367-9.
- John Cassidy, “American Inequality in Six Charts,” The New Yorker, 11-18-2013.
- Rudy Rucker, “Inverse Power Laws,” in This Explains Everything, John Brockman, Ed., 2013, Edge Foundation, Harper Collins, p. 369.
- See: http://www.abarim-publications.com/ChaosTheoryIntroduction.html#.WmkqxKinFaQ.
- Ernest Nagel, J.R. Newman, H& Douglas Hofstadter, Douglas, (1958) ,2002, Gödel’s Proof, revised ed. IBSN 0-8147-5816-9ISBN 0-8147-5816-9. Via Wikipedia article on incompleteness theorems.
- Mario Livio, Is God a Mathematician, pp. 196-7.
- Mario Livio, Is God a Mathematician, p. 201
- Stuart Kaufman, At Home in the Universe: The Search for the Laws of Self-Organization and Complexity, 1995, Oxford University Press, pp. 153-4.
- Stuart Kaufman, At Home in the Universe: The Search for the Laws of Self-Organization and Complexity, 1995, Oxford University Press, pp. 21-22.
- “Singularity (mathematics),” in Wikipedia, accessed 3-26-18.
- Stephen Hawking & Leonard Mlodinow, The Grand Design (1st ed.), 2010, Bantam Books (a division of Random House, Inc.), pp. 52-53.
- Frank Wilczekm “Simplicity,” in This Explains Everything, John Brockman, Ed., 2013, Edge Foundation, Harper Collins, 35-37.
- Mario Livio, Is God a Mathematician? p. 4. See also: Nancy Forbes & Basil Mahon, Faraday, Maxwell, and the Electromagnetic Field, 2014, Prometheus Books.
- Bertrand Russell, 1919). “The Study of Mathematics”. Mysticism and Logic: And Other Essays, 1919, Longman. p. 60.
- Mario Livio, Is God a Mathematician?, p.5.
- Mario Livio, Is God a Mathematician p.5.
- The Evolution of Beauty: How Darwin’s Forgotten Theory of Mate Choice Shapes the Animal World, 2017, Doubleday.
- Michael Tennesen, “More Animals Seem to Have Some Ability to Count,” Scientific American, September, 2009.
- Keith Devlin, “Language and Natural Selection,” in This Explains Everything: Deep, Beautiful, and Elegant Theories of How the World Works, John Brockman, Ed.. 2013, Harper Perennial, pp. 99-101.
- Mario Livio, Is God a Mathematician, p. 240.
- Keith Devlin, “Language and Natural Selection,” in This Explains Everything: Deep, Beautiful, and Elegant Theories of How the World Works, John Brockman, Ed.. 2013, Harper Perennial, pp. 99-101.
- Mario Livio, Ibid., p. 244
- Shing-Tung Yau, “Mathematical Object or Natural Object?” in This Explains Everything, John Brockman, Ed., 2013, The Edge Foundation, Harper Collins, p. 34.
- Livio, Ibid., pp. 247-249.
- “Quasicrystal,” article in Wikipedia, accessed 3-30-18. https://en.wikipedia.org/wiki/Quasicrystal
- Livio, Is God a Mathematician, pp. 208; 216-17
- Livio, Is God a Mathematician?, pp. 234-40.
- Smithsonian Institution, Oriental Elements of Culture in the Occident, p. 518, at Google Books, Annual Report of the Board of Regents of the Smithsonian Institution; Harvard University Archives.
- George G. Joseph, The Crest of the Peacock: Non-European Roots of Mathematics (Third Edition), 2011, Princeton. p. 86. ISBN 978-0-691-13526-7.
- Bruce Hood, “Complexity out of Simplicity, in This Explains Everything, John Brockman, Ed., 2013, The Edge Foundation, Harper Collins, p. 204.
- Mario Livio, Is God a Mathematician?, 2009, Simon & Schuster, p. 240.
- Eric Weinstein, “Einstein’s Revenge: The New Geometric Quantum,” in This Explains Everything, John Brockman, Ed., 2013, The Edge Foundation, Harper Collins, pp. 144-147.
- Stephen Hawking & Leonard Mlodinow, The Grand Design, 2010, Bantam Books (a division of Random House), pp. 39–59. See also: Andrew Zimmerman Jones, “What Is Model-Dependent Realism?” at https://www.thoughtco.com/what-is-model-dependent-realism-2699404.
- Max Tegmark, “The Mathematical Universe,” Foundations of Physics, February, 2008, pp. 101-150.
- Alfred Posamentier, The Pythagorean Theorem: The Story of Its Power and Beauty, 2010, Prometheus Books, p. 23
- George Johnston Allman, Greek Geometry from Thales to Euclid (Reprinted by Kessinger Publishing LLC 2005 ed.). Hodges, Figgis, & Co. p. 26. ISBN 1-4326-0662-X.