Two months ago, I discovered QUORA. It’s been around since 2009.

Since 2010, Quora has enabled people to ask experts questions about topics they like; even to answer questions on subjects they claim to know something about.

Quora is a site for geeks and nerds, and so far I  like it. The people who hang out in the areas I hang out tend to be polite, kind, and smart. If they like someone, they follow them and are notified when they post. So far, ten people have signed on to follow me. It’s a start. I think most are from India.

During the first six weeks, 150 or so of my answers were viewed 35,000 times; I got nearly 175 “upvotes”, which enabled many of the answers to move to the head of the line. I wrote most answers in the wee hours between 2 AM and 7 AM when I couldn’t sleep. Insomnia inspired me.

What follows are 25 of the most popular answers I posted to the first 150 or so questions that caught my interest. They are sequenced by popularity—the most read first .

Why not read a few? How many questions can anyone answer? Not many, I’m thinking.

Who knows what you might learn?

What?  You think you know better than a pontificator with no bona-fides? I don’t think so. No way!   😉

1)   What are some of the most popular “mathematically impossible questions“?

Freeman Dyson — one of the longest-lived and most influential physicists and mathematicians of all time — argued that it is impossible to find a whole (or exact) number that is a power of two where someone can reverse its digits to create a whole number that becomes a power of five.

In other words 2^{11} = 2048, right? Reversing the digits to make 8402 does not result in an exact number that is a power of five.

In this case, 8402^{1/5} = 6.09363... plus a lot more decimals. It’s not a whole (or exact) number. Not only that, no matter how many decimal places anyone rounds-off 6.09363... , the rounded number raised to the power of five will never return 8402 exactly.

Dyson claimed that this conjecture must be true, but there is nothing in mathematics that enables anyone to write a proof. He claimed that there must be an infinite number of similar statements—-each one true, none provable.

Click the link below to learn more.


The Snowplow Problem is another “impossible” problem. My differential equations professor assigned it with the promise that anyone who solved it would receive a 4.0 grade, regardless of their performance on tests. I was the only student he ever taught who actually managed it.

The problem goes like this: It is snowing at a constant rate. A snowplow starts plowing snow at noon. By one o’clock the plow has traveled one mile. By two o’clock the plow travels an additional half mile. At what time did it start snowing?

It took me 3 days and two pages of calculations, but I got my 4.0.

Note from the Editorial Board:  Over fifty people on Quora submitted answers to Billy Lee’s Snow Plow problem. One person had the right answer, but would not produce his proof. He did point out an unusual feature of the solution that Billy Lee had not noticed before. Billy Lee characterized the feature as “very surprising.” When pressed Billy Lee refused to reveal its secret. 

2)   How much force is one Newton?

A newton is the force that an average sized apple makes on your hand when you hold it. No matter where in the universe you are; no matter on what planet you stand or how strong the gravitational field, a newton of force always feels the same.

A newton is one kilogram of mass that is accelerating at one meter per second per second. Gravity on earth accelerates everything at nearly 10 meters per second per second. A kilogram of mass feels like 2.2 pounds on earth. One tenth of 2.2 pounds is 0.22 pounds or 3.5 ounces, which is the weight of a typical apple. The weight is the force that you feel against your hand. It is one newton.

On the moon, an object with the mass of a large brick would feel as light as an apple on earth due to the moon’s lower gravity. The force of the brick in your hand would feel like one newton.

3)   x + y = 4 and x^x + y^y = 64 . What are x and y?

The simplest way to solve is to make y = (4-x) and create an equation in terms of x.

An easy version to create and solve is {x^x + (4-x)^{4-x} = 64} . You can solve it by hand using iteration or throw it into an app like Wolfram Alpha and let them solve it in a few seconds.

Either way one value for x is .606098…. The other is 3.393901… , which you can assign to y. The two numbers add to 4.000… and when substituted into both initial equations return the right results.

4)   If I had 1,000,000,000,000,000 times 1,000,000,000,000,000 hamsters floating in space in close proximity, would gravity turn them into a hamster planet?

Assuming the question is serious, it deserves a serious answer.

A typically fat hamster weighs around one ounce, which is about 0.03 kilograms of mass. The number of hamsters in your question is 10E30.

Multiplying the mass of a single hamster by this large number gives the result of 3E28 kilograms.

To compare, the mass of planet Earth is 6E24 kilograms. The mass of the proposed population of hamsters is 5,000 times the mass of the earth.

The sun contains 67 times more mass than the hamster population. If the hamsters are close enough together to hold paws, a hamster planet is almost certain. I haven’t worked out how long the process to congeal would take, but I can estimate that the hamsters would probably die of starvation before the inexorable forces of gravity completed their work.

The hamster planet would be formed mostly from three elements: hydrogen (64%), oxygen (33%), and carbon (10%). 3% would be trace elements like calcium and maybe lithium.

The most likely outcome, given enough time, is a planet-like object. The hamsters have only one-fifth of the mass to make the smallest of the smallest suns — red dwarfs, which populate 67 to 80 percent of the Milky Way Galaxy.

There are way too many hamsters to make a reasonably sized moon.

Their mass (3E28 kg) happens to fall on the border between the range of masses that are required to form celestial objects known as brown dwarfs and the less massive sub-brown dwarfs — sometimes referred to as free-floating planets.

Brown dwarfs don’t have enough mass to ignite like a star, but they do produce heat and can accept small orbiting planets. The chemistry of brown dwarfs is not well-understood and is a bit controversial.

It’s a toss-up, but my vote goes to the notion that the hamsters will eventually form a very large but ordinary planet — a free-floating planet — which I referred to earlier as a sub-brown dwarf. This hamster planet might wander through space for millions (or even billions) of years before being captured by a massive-enough star to begin to orbit.

Because the elements of hydrogen and oxygen are likely to become the constituents of frozen moisture (or water ice), there is the risk that the ice might melt into oceans and perhaps boil away if the hamster planet approaches too close to a star (or sun). In the case where the planet loses its water, a carbon planet with 50 times the mass of earth would form.

Otherwise, should the planet find itself in a far-distant future orbiting in the “goldilocks” zone around a sufficiently massive star, the water would not evaporate. Life could arise in the planet’s oceans. It’s possible.

Life-forms might very well crawl up out of the water and onto land someday where — over the eons and under  ideal conditions — they will evolve into hamsters.

5)   Why is evolution a valid scientific theory despite the fact that it can’t be conclusively proven due to the impossibility of simulating the millions-of-years processes that it entails?

Evolution is a fact that is thoroughly established by observations made in many disciplines of science. Changes in species happen fast or slow; in the lab and in the field.

The mystery is how one-celled life got established so quickly — it was solidly established within one billion years of earth’s formation. It’s taken 3.5 billion years to go from one-celled life to what we have now.

Why so fast to get life started; why so slow to get to human intelligence and civilization?

People have a lot of ideas, but no one is sure. Some life forms have orders of magnitude more DNA than humans. Only 2% of human DNA is used to make the proteins that shape us.

So, yes, there are lots of questions.


6)   Why do cosmologists think a multiverse might exist?

Many high-level, theoretical physicists have written about the obvious problem our universe seems to have, which is that it has too many arbitrary constants that are too tightly constrained to be explained by any theory so far. No natural cause has been found for so many constants, so it’s fertile ground for theorists.

Stephen Hawking, among others, has said that the odds of one universe having the physics that ours has is 1E500 against. He is joking in his English way, because such a large number is essentially an infinity. It’s not possible to constrain a universe like ours by chance unless there are an infinity of choices, and we happen to be in the one that supports intelligent, conscious life.

Two ways of getting to infinity are the concepts of multi-verse and the new one proposed by Paul J. Steinhardt of Princeton University in 2013, which is based on data supplied by the Planck Satellite launched in 2003. Paul is the Einstein Professor of Science at Princeton, so his opinion holds a lot of weight.

Steinhardt has proposed that the universe is ekpyrotic, or cyclic. He has asserted that the universe beats like a heart, expanding and contracting in cycles, with each cycle lasting perhaps a trillion years and repeating, on and on, forever. Each cycle produces conditions — some which are ideal for life. This heart has been beating forever and will continue to do so, forever.

Conscious Life

7)   How will we visit distant galaxies if we cannot travel faster than light? 

We will never visit distant galaxies, because they are too far away; most are moving away from us faster than our current technologies can overtake. At huge distances space itself is expanding, which adds to our problems.

The expansion of space is gradually accelerating. Any increase in performance by space vehicles over the next few thousand years is certain to be overwhelmed by the accelerating expansion of the universe.

As time goes on the amount of objects that are reachable (or even viewable) by earthlings will shrink.

On the happy side, our own solar system has at least 165 interesting places to visit that should keep folks fascinated for many thousands of years. A huge cavern has been discovered on Mars, for example, that might make a safe habitat against some forms of radiation dangers; it seems to be a place where a colony of humans might be able to live, work, and survive — perhaps even flourish.

Elon Musk is planning a mission to Mars soon.

8)   What is the mathematical proof for a+a = 2a ?

Some things that are true can’t be proved. All math systems are based on axioms, which are statements believed to be true but which, in themselves, are not provable.

This link provides a list of axioms for addition: https://sites.math.washington.edu/~hart/m524/realprop.pdf

A lot of interesting philosophical and mathematical work has been done on conjectures that are believed to be true, but can’t be proved.


9)   Can you explain renormalization in physics in simple words?

There is a problem in physics that has to do with the huge variation in scales between the very large and the very small. This problem of scales involves not only the size and mass of things, but also forces and interactions.

Philosopher Robert Pirsig believed that the number of possible explanations that scientists could invent for phenomenon were, in actual fact, unlimited.

Despite all the math and all the convolutions of math, Pirsig believed that something mysterious and intangible like quality or morality guided our explanations of the world. It drove him insane, at least in the years before he wrote his classic book, Zen and the Art of Motorcycle Maintenance.

Anyway, the newest generation of scientists aren’t embarrassed by anomalies. They have taught themselves to “shut up and calculate.” The digital somersaults they must perform to validate their work are impossible for average people to understand, much less perform. Researchers determine scales, introduce “cut-offs“, and extract the appropriate physics to make suitable matches to their experimental results.

The tricks used by physicists to zero in on pieces of a problem where sensible answers can be found have many names, but renormalization is one of the best known.

When physicists renormalize an equation, they cut away infinities and other annoying problems (like dividing by zero). They focus the range of their attention to smaller spaces where the vast differences in scales and forces don’t blow up their formulas and disrupt putative pairings of their carefully crafted mathematics to the world of actual observations.

It’s possible that the brains of humans, which use language and mathematics to ponder and explain the world, are insufficiently structured to model the complexities of the universe. We aren’t hard wired with enough power to create the algorithms for ultimate understanding.


10)   If a propeller rotates at the speed of light at half of its length, what happens to the outer parts?

Only the ends of the propeller can rotate at near light speed (in theory). At half lengths the speed of the propellers will be half the speed of their ends, because the circumference of a circle is 2πr. (There is no squared term.)

So the question is: will the propellers deform according to the rules of the Lorentz transformation along their lengths due to the difference in velocity along their lengths?

The answer is, yes.

As you move outward along the propeller, it will become thinner in the direction of rotation, and it will get more massive. A watch will tick more slowly at the end than at the middle.

I am not sure how it would look to an outside observer. Maybe such a propeller would resemble in some ways the spiral galaxies, which don’t rotate the way we think they should. Dark matter and energy are the usual postulates for their anomalous rotations. Maybe their shape and motion is related to relativity in some way. I really don’t know.

In reality, no propeller can be constructed that would survive the experiment you describe. But in theory (and ignoring the physical limitations of materials) there would be consequences.

However, no part of the propeller will move at light speed or higher. Such speeds for objects with mass are impossible.

11)   What is the fundamental concept behind logarithms?

The first thing that anyone might try to understand is that the word logarithm means exponent.

Example 1:

log 100 = 2 . What does this expression say? It says that the exponent that makes 100 is 2. What confuses people is this: exponent acting on what number?

The exponent acts on a number called the base. Unfortunately, the base is not written down in the two most common logarithm systems, which are log and ln.

The base for the log system is 10. In the example above, the exponent 2 acts on the base 10, which is not shown. In other words, 10^2 = 100 , right? The exponent that makes 100 from the base 10 (not shown) is (equals) 2.

Example 2:

ln 10 = 2.302585… .  What does this expression say? It says that the exponent that makes 10 is 2.302585… . Again, exponent acting on what number?

The base used in the ln system is 2.7182818… ,which is an irrational number that has an infinite number of decimal places. It happens to be a useful number in all branches of science and math including statistics, so mathematicians have decided to represent this difficult-to-write-down number with the letter “e”, which is known as Euler’s number.

The base for the ln system is e . In the example above, the exponent 2.302585… acts on the base e , which is not shown. In other words, e^{2.302585...} = 10 , right? The exponent on e ( which is 2.7182818… and not shown in the original equation above) that makes 10 is (equals) 2.302585… .

All other logarithmic systems express the base as a subscript to the right of the word log.

Example 3:

log_{7}49 = 2 . This expression says: The exponent on seven that makes 49 equals 2.

12)   Why do so many spiritual types have mental blocks about science and mathematics?

Everyone has mental blocks about science and math including scientists and mathematicians. Like the old song, people hear what they want to hear and disregard the rest. Einstein never accepted most of quantum physics even after it was well established and no longer controversial.

People don’t like the feeling of “cognitive dissonance”. The tension between strongly held beliefs and objective facts can bring unbearable psychological pain to most people. Someone once said that genius is the ability to hold contradictory ideas inside the mind. Most people don’t do that well; they don’t like contradictions.

Here is a link to an essay called Truth that some will find interesting:


13)   Is time infinitely divisible?

Einstein said that time and space (i.e. space-time) depends on mass and energy, which are equivalent. In the absence of mass and energy, space and time are meaningless.

The most recent experiments by NASA have found no evidence that time is anything but continuous. However, the shortest time possible is the length of time it takes light to move the shortest distance possible, which is called Planck time. It is thought to be 5.39E-44 seconds.

Time can be divided into as many smaller increments as anyone wants, but nothing can happen in fewer than the number of intervals that add to 5.39E-44 seconds. Time is a variable that isn’t fundamental. It expands and shrinks in the presence of mass and energy.

Some physicists of the past suggested that the “chronon” might be the shortest interval of time. It is the time light travels past the radius of a classical (at rest) electron — an interval of 6.27E-24 seconds. Its calculation depends only on mass and charge, which can change if a particle other than an “at rest” electron is measured.

It seems to me that time is probably best thought of as being continuous. That said, it doesn’t mean that mass-energy interplay isn’t pixelated — much like a digital camera image. Pixelation is critical to a conjecture concerning the preponderance of matter over anti-matter — a conjecture described in the essay CONSCIOUS LIFE.

14)   Which is bigger:  \frac{3}{5} or \frac{1}{9} ?

Think of fractions as pies which are all the same size. The bottom number is the total number of pieces each pie is cut into. The first pie was cut into 5 pieces, which are all the same size. The second pie was cut into 9 pieces which again are all the same size.

The second pie is cut into smaller pieces than the first pie, because there are more pieces. Right?

Mice come along and eat pieces from both pies. The top number is the number of pieces they left behind; the top number is the number of pieces the mice didn’t eat.

So which pie plate has more pie on it? Is it the 5 piece pie that has 3 pieces left or the 9 piece pie that has 1 piece left?

If you think hard you will figure out that it must be the first plate that has the most pie on it. Right?  

15)   Why is a third of 30 equal to 10 and not 9.999999999, as a third of 10 is 3.33333333? 

You can make three piles of ten objects in each pile. When you count the total, it adds to exactly 30 objects. So the answer of “10” is demonstrably true, right? Three piles of ten adds to thirty.

There is no way to make three piles of any identical objects that adds to 10. Three piles of three adds to nine. Four piles of three objects adds to twelve.

We are required to make three piles of three objects and then add a piece of a fourth object to each pile that is smaller than a whole piece.

It turns out that the fourth object is 1/3 of a whole object. When these three piles of three objects plus 1/3 of an object are added up they equal exactly ten.

The problem in understanding comes from trying to grasp that 1/3 — when written as a decimal — is what mathematicians call a repeating decimal. The rules of arithmetic say that the decimal form of 1/3 is calculated by dividing “1” by “3”.

Following the rules of arithmetic when doing the division forces an answer to the problem that results in a repeating decimal — in this case, 0.333333… .

There is no way around these rules that keeps math working right, consistent, and accurate.


16)   Will we be able to have life extension through cloning? 

Cloning not only doesn’t work, it can’t work.

That said, the idea of cloning is to make a genetic replicant of an existing life-form. Extending life-span would require changes to the genome through other means involving changes to structures called telomeres, probably, which straddle the ends of chromosomes in eukaryotic cells. Here is a link:  Telomere

A short discussion of cloning is included in the essay at this link:  NO CODE

No Code is long (8,500 words), but explains in words, pics, graphics, videos, and links some of the complexities, misunderstandings, and dangers of current genetic-engineering in language suitable for undergraduates. It explains basic cell biology, protein production, and much more.

17)   Why does time slow down when we are on a massive planet or star like Jupiter? 

Gravity is equivalent to acceleration. Accelerating clocks tick slower, according to General Relativity, which has been confirmed by experiments. It has to do with the concept of space-time and the fact that all objects travel through space-time at the same rate.

To understand, it helps to read up on space-time, special relativity, and general relativity. The concepts aren’t easy. The universe is an odd place, but it can be described to a somewhat fair degree by mathematics.

Some of the underlying reasons for why things are the way they are seem to be unknowable.

18)   If the ancients had focused on science instead of religion, could we have become immortal by now? 

Immortality is not possible due to the odds of accidental death, which at the current rate makes death by age 25,000 a virtual certainty for individuals.

Worse: the odds for extinction of the human species as a whole is much higher — it’s a statistical certainty for annihilation within the next 10,000 years according to experts. It seems counterintuitive, but it’s true.


19)    How do I solve, if the temperature is given by f(x,y,z) = 3x^2 - 5y^2 + 2z^2 and you are located at (\frac{1}{3} , \frac{1}{5} , \frac{1}{2}) and want to get as cool as possible, in which direction should you set out? 

 You want to establish what the gradient is, establish its direction, then head in the opposite direction, right?

By partial differentiation the gradient is (6x – 10y + 4z), right? You don’t have to take another partial derivative and set it equal to zero to establish a maximum, because all the second derivatives of the variables are equal to one, right? You can drop the variables out and treat them as unit vectors like i, j, & k, correct?

The resulting vector points in the direction of increasing temperature, right?

Changing the signs makes a vector that points in the opposite direction toward cooler temperatures, correct? That vector is (-6, 10, -4).

The polar angle (θ) is 71.068 degrees and the azimuth angle (φ) is 120.964 degrees. The length (or magnitude) is 12.3288. Right? (We won’t use this information to solve the problem, but I wanted to write it down should I need to refer to it to respond to any comments or to help check my work graphically.)

These directions are from the origin, and you aren’t located at the origin. To determine the direction to travel to get to (-6, 10, -4), you need to subtract your current position. Again, for reference your location is .6333 from the origin at θ = 37.8636 degrees and φ = 30.9638 degrees. Right?

After subtracting your position vector from the gradient vector, the resulting vector is (-6.333, 9.8, -4.5). Agree?

This vector tells you to travel 12.506 at a polar angle (θ) of 68.9105 degrees and an azimuth angle (φ) of 122.873 degrees to intersect the gradient vector. At the intersection you must change direction to follow the gradient vector’s direction to move toward cooler temperatures at the fastest rate.

I haven’t graphed out the solution to double-check its accuracy. You might want to do this and let me know if you agree or not.

20)   What is \sqrt[3]{i} - \sqrt[3]{i} equal to?

The answer is zero, of course.

But not really. It only seems that way. Each number has three roots.

Depending on which roots are chosen the result can be one of six different sums (as well as zero if both roots are the same). We have to start somewhere so:

What is i^\frac{1}{3} ?

i = e^\frac{{i\pi}}{2} .  Right?

Therefore, a third root of i is e^\frac{{i\pi}}{6} .  Right? It’s not the only root.

It’s the principal root. There are three third roots, which are equally spaced around the unit circle. Right?

It’s clear by inspection that to be equally distributed around the unit circle the other two roots must be e^\frac{{i5\pi}}{6} and -i.  Right?

Convert the three roots to rectangular coordinates and do the subtractions.

Here are the roots in rectangular form: (.86603 + .50000 i) , (-.86603 + .50000 i) , and (0.00000 -i).

Here are the six answers (in bold type) to the original question with the subtractions shown to the right:

1.7302 = (.86603 + .50000 i) – (-.86603 + .50000 i)

(.86603 +1.5 i) = (.86603 + .50000 i) – (0.00000 -i)

-1.7302 = (-.86603 + .50000 i) – (.86603 + .50000 i)

(-.86603 + 1.5 i) = (-.86603 + .50000 i) – (0.00000 -i)

(-.86603 – 1.5 i) = (0.00000 -i) – (.86603 + .50000 i)

(.86603 – 1.5 i) = (0.00000 -i) – (-.86603 + .50000 i)

These rectangular coordinates can be converted back to the Euler-form (e^{i\theta} ).  It’s easy, if you know how to work with complex variables. In Euler-form the angle in radians sits next to i.  The angle directs you to where the result lies on the unit circle. Right?

As mentioned earlier, if both roots are chosen to be the same, then in that particular case the result is zero.

21)   What is tensor analysis and how is it used in physics?

Understanding tensors is crucial to understanding Einstein’s General Theory of Relativity.

This question seems to assume that everyone knows what tensors are and how they are represented symbolically. I am guessing that some folks reading this question might want some basics to better understand the explanations of how tensors are used for analysis in physics.

If so, here are links to two videos that together will help with the basics:

22)   What is the velocity of an electron?

Electrons can move at any speed less than light depending on the strength of the electro-magnetic field that is acting on them. Inside atoms electrons seem to move around at about one-tenth of the speed of light. You might want to check me on this number. The situation is as complicated as your mind is capable of grasping.

When interacting with photons of light electrons inside atoms seem to jump into higher or lower shells or orbits instantaneously. That said, it is impossible to directly observe electrons inside atoms.

On an electrical conductor like a wire, electrons move very slowly, but they bump into one another like billiard balls or dominoes. The speed of falling dominoes can be very high compared to the speed of an individual domino, right?

So the answer is: it all depends…

23)   What exactly is space-time? Is it something we can touch? How does it bend and interact with mass? 

Space-time, according to Einstein, depends on mass and energy for its existence. In the absence of mass and energy (which are equivalent), space-time disappears.

The energy of things like bosons of light, which have no mass, is proportional to the frequencies of phenomena called electric and magnetic fields. Small packets of these oscillations are called photons.

Many kinds of oscillating fields like the electro-magnetism of light permeate (or fill) the universe. In this sense, there is no such place as nothing in the universe anywhere at any scale.

The instruments and tools of science (including mathematics) can give a misleading impression that at very small scales massive particles exist.

According to John Wheeler, mass at small scales is an illusion created by our interaction with measuring devices and sensors.

Mass is a macroscopic statistical process created by accumulations of whatever it is that exists at the very deep bottom of a reality to which we have yet to gain access. These accumulations are visible to humans; they span 46 billion light years in all directions from the vantage-point of Earth and are collected for the most part in more than one-trillion galaxies, according to the latest satellite data by NASA.

Mass interacts with everything that can be measured (including everything in the Standard Model) by changing its acceleration, which is equivalent to changing its momentum. Energy changes the momentum of everything in the Standard Model as well.

It is in this sense that mass and energy are equivalent. It is in this sense that space-time depends on mass and energy. Space-time does not act on mass and energy; it is its result.

Space-time is a concept (or model) that for Einstein at least helped him to quantify how mass and energy behave on large scales. It helped him to explain why his idea that the universe looks and behaves differently to observers in different reference frames might in fact be the way the universe on the largest of scales really works. It helped him to describe a geometric explanation for gravity that many people find compelling.


24)   Hypothetically speaking, if one could travel faster than light, would that mean you would always live in the dark?

The space in which objects in the universe swim does expand faster than light when the expansion is measured over very large distances that are measured in light-years. A light year is six trillion miles.

At distances of billions of light years the expansion of space between objects becomes dramatic enough that light begins to stretch itself out. This stretching lengthens the distance between the peaks and valleys of the electric and magnetic waves that light is made from, so its frequency drops.

The frequencies in white light can stretch so dramatically that the light begins to appear red. It’s called red-shift.

Measuring red-shift of star-light is a way to tell how far away an object like a star is. As light stretches over farther distances we lose the ability to see it.

The frequencies stretch into infra-red waves (called heat waves) and then at even farther distances stretch to very long waves called radio-waves. Special telescopes must be placed into outer-space to see these waves of light, because heat and radio waves radiating from the earth interfere with instruments placed at the earth’s surface.

Eventually the distances become so great that the amplitudes (or heights) of the waves flat-line. They flat-line because space is expanding so fast that light can’t keep up. Light loses its structure. At this distance the galaxies and stars drop out of the sight of our eyes and even our sensors and instruments. It’s a horizon beyond which the universe is not observable.

No one knows how big the universe is, because no one can see to its end. The expansion of space —  tiny over short distances — starts to get huge at distances over 10 billion light years or so. The simple, uncomplicated answer is that the lights go out at about 14.3 billion light years.

Over the next few billion years the universe we can see will get smaller, because the expansion of space is accelerating. The sphere of viewable objects is going to shrink. The expansion of space is speeding up.

The problem will be that the nearby stars that we will always be able to see (because they are close) are going to burn out over time, so the night sky is going to get darker.

Most (4 out of 5) stars in our galaxy are red dwarfs that will live pretty much forever, but we can’t see them now, so we won’t be able to see them billions of years from now either. They radiate in the infra-red, which can only be seen with special instruments from a vantage point above our atmosphere.

Stars like our sun will live another 4 or 5 billion years and then die. The not too distant future of the ageless (it seems) universe is going to fall dark to any species that might survive long enough to witness it.

25)   What does “e” mean in a calculator? 

There are two “e”s on a calculator: little “e” and big “E”.

Little “e” is a number. The number has a lot of decimals places (it has an infinite number of them), so the number is called “e” to make it quick to write down.

The number is 2.71828… . The number is used a lot in mathematics and in every field of science and statistics. One reason it is useful is because derivatives and integrals of functions formed from its powers are easy to compute.

Big “E” is not a number. It stands for the word “exponent”, but it is used to specify how many places to the right to move the decimal point of the number that comes before it.

5E6 is the number 5,000,000, for example. The way to say the number is, “five times ten raised to the sixth power”. It’s basically a form of shorthand that means 5 multiplied by 10^6 .

Sometimes the number after E can be negative. 5E-6 would then specify how many places to the left to move the decimal point. In this case the number is 0.000005, which is 5 multiplied by 10^-6 .

Bonus Question:  What difficulties lie in finding particles smaller than quarks, and in theory, what are possible solutions? 

The Standard Model is complete as far as it goes. Unfortunately, it covers only 5% of the matter and energy believed to exist in the universe.

And humans can only see 10% of the 5% that’s out there. We are blind to 99.5% of the universe. We can’t see energy, and we can’t see most stars, because they radiate in the infra-red, which is invisible to us.

The Standard Model doesn’t explain why anti-matter is missing. It doesn’t explain dark matter and energy, which make up the majority of the material and energy in the universe. It doesn’t explain the accelerating expansion of the universe.

Probing matter smaller than quarks may require CERN-like facilities the size of our solar system, or if we’re unlucky, larger still.

We are approaching the edge of what we can explore experimentally at the limits of the very small. Some theorists hope that mathematics will somehow lead to knowledge that can be confirmed by theory alone, without experimental confirmation.

I’m not so sure.

The link below will direct readers to an essay about the problem of the very small.


Bonus Question 2 – What if science and wisdom reached a point of absolute knowledge of everything in the universe, how would this affect humanity?   

Humanity has reached a tipping point where more knowledge increases dramatically the odds against species survival. Absolute knowledge will result in absolute assurance of self-destruction.

Astronomers have not yet detected advanced civilizations. The chances are excellent that they never will.

Humans are fast approaching an asymptotic limit to knowledge, which when reached will bring catastrophe — as it apparently has to all life that has gone before in other parts of the universe.

Everywhere we look in the universe the tell-tale signatures of advanced civilizations are missing.


We hope readers enjoyed these questions. Follow Billy Lee on Quora where you will find answers to thousands of unusual and interesting questions on every subject imaginable.  The Editorial Board  

If it don’t roast, I don’t post.

Billy Lee

What is e exp (-i π) ?

What is e^{-i\pi} ?

I posted a long answer on Quora.com where it sort of didn’t do well. Answers given by others were much shorter but they seemed, at least to me, to lack geometric insights. After two days my answer was ranked as the most read, but for some reason no one upvoted it. It did receive a few positive replies, though.

I can’t help but believe that there must be nerds in cyberspace who might enjoy my answer. Why not post it on my blog? Maybe someday one of my grandkids will get interested in math and read it. Who knows?

I added some graphics here that wouldn’t post on Quora. The site lacks graphics functionality, apparently. Either that or I’m too dense to figure out how to insert something.

NOTE: 24 Oct 2017: Today Billy Lee finally figured out how to post media on Quora.com. After admonishment and chastisement by the staff, he added a pic and a GIF to his “answer.” Unfortunately, Billy Lee never did get the GIF to run right, so he took it down. Otherwise, Billy Lee is doing good. He really is. THE EDITORIAL BOARD

Anyway, this pic and a working GIF below make a big difference in understanding, I hope. And anyone who doesn’t understand something can always click on a link for more information. (No one ever clicks on links, but I spend a lot of time adding them—maybe so I can click on them myself during times when remembering my name or where I live seems to lie just outside my intellectual skill-set.)

Here is the drawing I added and the answer:

This diagram is excellent but contains a mystery point not on the unit circle—{i^i}. The point is shown at .2078… on the real number line. It has 3 other values on the horizontal line that are not shown: -.2078… and +/- 4.8104… . Its solution can be reviewed in this video link, which Billy Lee has carefully reviewed and confirmed to be sound. An imaginary number raised to the power of an imaginary number yields a result that is four real numbers. How can that be? It’s something to ponder; something to think about. The Editorial Board 

What is e^{-i\pi} ?

The expression evaluates to minus one; the answer is (-1). Why?

Numbers like these are called complex numbers. They are two-dimensional numbers that can be drawn on graph-paper instead of on a one-dimensional number line, like the counting numbers. They are used to analyse wave functions—i.e. phenomenon that are repetitive—like alternating current in the field of electrical engineering, for example.

“e” is a number that cannot be written as a fraction (or a ratio of whole numbers). It is an irrational number (like π, for instance). It can be approximated by adding up an arbitrary number of terms in a certain infinite series to reach whatever level of precision one wants. To work with “e” in practical problems, it must be rounded off to some convenient number of decimal places.

Punch “e” into a calculator and it returns the value 2.7182…. The beauty of working with “e” is that derivatives and integrals of functions based on exponential powers of “e” are easy to calculate.

What is “e” raised to the power of (-iπ) ?

A wonderful feature of the mathematics of complex numbers is that all the values of expressions that involve the number “e” raised to the power of “i” times anything conveniently lie on the edge (or perimeter) of a circle of radius one. This happy fact makes understanding the expressions easy.

I should mention that any point in the complex plane can be reached by adding a number in front of e^{i\theta} to stretch or shrink the unit circle of values. We aren’t going to go there. In this essay “e” is always preceded by the number one, which by the familiar convention is never shown.

The number next to the letter “i” is simply the angle in radians where the answer lies on the circle. Draw a line from the center of the circle at the angle specified in the exponent of “e” and it will intersect the circle at the value of the expression. What could be easier?

For the particular question we are struggling to answer, the number in the exponent next to “i” is (-π), correct?

“π radians” is 180 degrees, right? The minus sign is simply a direction indicator that says to move clockwise around the unit circle—instead of counter-clockwise if the sign was positive.

After drawing a unit circle on graph paper, place your pencil at (1 + 0i)—located at zero radians (or zero degrees)—-and trace 180 degrees clockwise around the circle. Remember that the circle’s radius is one and its center is located at zero, which in two dimensional, complex space is (0 + 0i). You will end up at the value (-1 + 0i) on the opposite side of the circle, which is the answer, by the way.

[Trace the diagram several paragraphs above with your finger if you don’t have graph paper and a pencil. No worries.]

Notice that +π radians takes you to the same place. The value you land on is (-1 + 0i), which is -1. The answer is minus one.

Imagine that the number next to “i” is (π/2) radians. That’s 90 degrees, agreed? The sign is positive, so trace the circle 90 degrees counter-clockwise. You end at (0 + i), which is straight up. “i” in this case is a distance of one unit upward from the horizontal number line, so the number is written (0 + i)—-zero distance in the horizontal direction and “plus one” distance in the “i” (or vertical) direction.

So, the “i” in the exponent of “e” says to “look here” to find the angle where the value of the answer lies on the unit circle; on the other hand, the “i” in the rectangular coordinates of a two-dimensional number like (0 + i) says “look here” to find the vertical distance above or below the horizontal number line.

When evaluating “e” raised to the power of “i” times anything, the angle next to “i”—call it “θ”—can be transformed into rectangular coordinates by using this expression: [cos(θ) + i sin(θ)].

For example: say that the exponent of “e” is i(π/3). (π/3) radians is 60 degrees, right? The cosine of 60 degrees is 0.5 and the sine of 60 degrees is .866….

So the value of “e” raised to the power of i(π/3) is by substitution (0.5 + .866… i ). It is a two-dimensional number. And it lies on the unit circle.

The bigger the exponent on “e” the more times someone will have to trace around the circle to land at the answer. But they never leave the circle. The answer is always found on the circle between 0 and 2π radians (or 0 and 360 degrees) no matter how large the exponent.

It’s why these expressions involving “e” and “i” are ideal for working with repetitive, sinusoidal (wave-like) phenomenon.

In this essay Billy Lee uses θ in place of the Greek letter φ shown in this GIF. “r” equals “one” in a unit circle. The Editorial Board

Some readers might wonder about what radians are. A radian is the radius of a circle, which can be lifted and bent to fit perfectly on the edge of the circle. It takes a little more than three radius pieces (3.14159… to be more precise) to wrap from zero degrees to half-way around any circle of any size. This number—3.14159…—is the number called “π”. 2π radians are a little bit more than six-and-a-quarter radians (radius pieces) and will completely span the perimeter (or circumference) of a circle.

A radian is about 57.3 degrees of arc. Multiply 3.1416 by 57.3 to see how close to 180 degrees it is. I get 180.01… . The result is really close to 180 degrees considering that both numbers are irrational and rounded off to only a few decimal places.

One of the rules of working with complex numbers is this: multiplying any number by “i” rotates that number by 90 degrees. The number “i” is always located at 90 degrees on the unit circle by definition, right? By the rule, multiplying “i” by “i” rotates it another 90 degrees counter-clockwise, which moves it to 180 degrees on the circle.

180 degrees on the unit circle is the point (-1 + 0i), which is minus one, right?

So yes, absolutely, “i” multiplied by “i” is equal to -1. It follows that the square root of minus one must be “i”. Thought of in this way, the square root of a minus one isn’t mysterious.

It is helpful to think of complex numbers as two dimensional numbers with real and imaginary components. There is nothing imaginary, though, about the vertical component of a two-dimensional number.

The people who came up with these numbers thought they were imagining things. The concept of two-dimensional numbers was too radical at the time for anyone to believe that numbers could exist on a plane as naturally as they do on a simple number line of one dimension.

Visit my website for an illustrated and concise explanation of these topics titled, “What is Math?” It is easily found by making a simple entry into the search box on theBillyLeePontificator.com .

Thanks, I was confused about the -i part, so it’s not – sqrt(-1)? My calculator gives me an answer of -1.

Good answer, I approve 🙂


Billy Lee


Truth 3

Is it possible for humans to tell the truth always; to never lie?  Psychologists say it is not possible; most reasonably informed people agree. It’s a trait that can distinguish humans from some forms of artificial intelligence, which engineers at Google and other companies are working furiously to bring on-line.

People’s ideas—their belief systems—are inconsistent, incomplete, and almost always driven by logically unreliable, emotionally-laden content, which is grounded in their particular life experiences and even trauma. Does anyone disagree? 

Cognitive dissonance is the term psychologists use to describe the painful condition of the mind that results when people are unable to achieve consistency and completeness in their thinking. Every person suffers from it, to one degree or another. An unhealthy avoidance of cognitive dissonance can drive people into rigid patterns of thought. Political and religious extremists are examples of people who probably have a low tolerance for it.  

Kurt Godel
Kurt Friedrich Gödel (1906-1978) mathematician, logician, philosopher. Kurt trusted no one but his wife to feed him; not even himself. He never ate another meal after his wife died. He starved to death.

Decades ago, mathematicians like Kurt Gödel proved that any math-based logic-system that is consistent can never be complete; it always contains truthful assertions—including but not limited to foundational truths, called axioms—which are impossible to prove.

Whenever an idea or a conjecture that seems to be self-evident can’t be proved, it seems reasonable, at least to me, to assume that some people might feel compelled to disbelieve it; they might even believe they are trapped in what could turn out to be a lie, because no one should be expected to embrace a set of unprovable truths, right?  Axioms that can’t be proved are nothing more than assertions, right? Certainly any theorems built-up from unprovable assertions (axioms) must carry some inherent risk of falsifiability, shouldn’t they?  

Someone unable to convince themselves that an assertion or axiom they believe is true actually is true might necessarily feel uncomfortable; even incomplete. Folks often teach themselves to not examine too closely those things they believe to be true, which they can’t prove. It helps them avoid cognitive dissonance.

I’m not referring to science by the way. No easy way exists for non-technical folks to confirm claims by scientists that the earth is round, for example. The earth looks flat to most people, but scientists who have the right tools and techniques can reach beyond the grasp of non-scientists to prove to themselves that the earth is round.

Reasonable people agree that the truth of science is discoverable to any group of humans who have the resources and training to explore it. Most agree that those who are scientifically well-qualified are fully capable of passing the torch of scientific truth to the rest of humankind.

But this essay isn’t really about science; it’s about truth itself—a concept far more mysterious and elusive than any particular assertion a scientist might make that the Earth is not the center of the Universe, or that the Moon is not made of cheese.

All logically consistent ways of reasoning that we know about have been invented—some say, discovered—by human beings who live on planet Earth. Humans can and often have argued that the unprovable assertions which form the basis of any consistent way of thinking are an Achilles heel, which can be attacked to bring down whatever logical structure has been erected.

But it isn’t only the few foundational axioms of any mathematically logical system that are by definition true, but unprovable. Very complicated conjectures about the nature of numbers, for example, are always being discovered, which everyone believes they know to be true, but will never be proved, because they can’t be.

Freeman Dyson, British mathematician and physicist (born Dec 15, 1923)

Freeman Dyson—one of the longest-lived and most influential physicists and mathematicians of all time—argues that it is impossible to find a whole (or exact) number that is a power of two where someone can reverse its digits to create a whole number that becomes a power of five.

In other words 2^{11} = 2048, right?  Get out the calculator, those who don’t believe it. Reversing the digits to make 8402 does not result in an exact number that is a power of five. In this particular case, 8402^{1/5} = 6.09363... plus a lot more decimals. It’s not a whole (or exact) number. Not only that, no matter how many decimal places anyone rounds-off 6.09363... to, the number raised to the power of five won’t return 8402 exactly. A calculator will confirm it.

Dyson asserts that no number which is a power of two can ever be manipulated in this way to yield an exact number that is a power of five, no matter how large or unlikely the number might be. Freeman Dyson and all other super-intelligent beings—perhaps aliens living in faraway galaxies—will never be able to prove this conjecture, even though they all know for certain inside their own logical brains that this particular statement must be true.

All logically consistent methods of reasoning which can be modeled by simple (or not so simple) mathematics have these Achilles heels. Gödel proved this truth beyond all doubt; he proved it using a method he invented, which allowed him to circumvent the dilemmas posed by the unprovable truths of the system of thinking he contrived to demonstrate his discoveries.

I’m not going to get into the details of Gödel’s Incompleteness Theorems; books have been written about them; most people don’t have the temperament to wade through the structures he built to make his point. He basically assigned simple numbers to logical statements—some being very complex statements encoded by very long strings of numbers—so that he could perform garantuan operations of logic using rules of simple arithmetic on ordinary whole numbers. Take my word, his method requires traveling over unfamiliar mathematical roads; it takes getting used to.  

But it should amaze non-mathematicians that truths abound in mathematics, which not only haven’t been proved; they never will be proved, because no proof is possible. A logical path to the truth of these statements does not exist; indeed, it cannot exist. But it is useful and necessary to believe or at least accept these statements to make progress in mathematics.

Paul Joseph Cohen (1937-2007) Stanford mathematician

The late mathematician Paul Cohen—at one time a friend to Gödel—said that Gödel once told him he wondered if it might be true that any and all conjectures in mathematics could be solved if only the right set of axioms could be collected to construct the proofs. Cohen is best known, perhaps, for showing that indeed—in the case of the Continuum Hypothesis, at least—he could collect two reasonable, self-evident, and distinct sets of axioms that led to logically consistent and useful proofs. One small problem, though—the proofs were completely contradictory. One proved the conjecture was true; the other proved it was false.

His result is often explained this way: the consistency of any system of mathematical reasoning cannot be proved by its foundational axioms alone. If it can, the system must necessarily be incomplete; its conjectures—many of them—undecidable.

Cohen showed that a consistent and sound axiomatization of all statements about natural numbers is unachievable. Many such statements, in his view could be true, but not provable. Cohen introduced the concept that all systems of logic built on numbers have embedded within them some combination of ambiguity, un-decidability, inconsistency, and incompleteness.

People who want their thinking to be consistent must believe things that cannot be proved. But believing logical statements that can’t be proved always renders their thinking incomplete, even when it might be flawlessly consistent. What they believe to be true depends fundamentally on what they believe to be self-evident: it depends on statements no one can prove: on axioms, and a little bit more.

For those who decide to believe and accept only statements that can be proved, their thinking will necessarily unravel to become inconsistent or incomplete; most likely both. Their assertions become undecidable. It can’t be any other way, according to Gödel, whose proof has withstood the test of eighty years of intense scrutiny by the smartest people who have ever lived.

Paul Cohen jumped onto the dilemma-pile by showing that the incompleteness made necessary by a particular choice of axioms can turn a logically consistent proof to rubble when a mathematician tampers-with or swaps-out the foundational axioms. A sufficiently clever mathematician can prove that black is white; and vice-versa.

Some might be tempted to say that Gödel‘s Incompleteness Theorems apply only to formal, math-based logic-structures—not the minds of human beings, because minds and the way they operate when analyzed are always found to be inconsistent and incomplete. But such talk makes the point. Think about it.  


So again, we ask: what is truth?  How do folks determine that a particular statement is true, if it happens to be one of those assertions that lies beyond the reach of logic, which no one—no matter how smart—will ever be able to prove? 

What good do collections of so-called self-evident axioms serve, if different collections can lead to contradictions in theorems?

Most important: how do we avoid believing lies?

Billy Lee

Post Script: This post is one of those living-essays Billy Lee writes that is likely to be modified and added to as he thinks about the subject and learns more. Check back for updates from time to time.

Click on this link to view a short movie clip, where Jesus, played by Robert Powell, is asked by Pontius Pilate, What is truth?  The Editorial Board

[added April 3, 2016]  I found a 2013 essay by Derek Abbott, the Australian Electrical Engineer and Physicist, who argued that our mathematics is invented, not discovered; anthropological, not universal. It enables us to simplify truth to enable our limited minds to find ways to do and understand simple things. Click this link for a good read. 

I don’t know if Derek’s view is correct, but offer it as fodder for readers who are interested in why Truth and mathematics seem connected somehow in the minds of so many thinkers like Plato, for example; and why these thinkers could be dead wrong, at least in Derek Abbott’s opinion. He offers Clifford’s Geometric Algebra as a practical example of a useful system of mathematics used by robotics engineers, which is built very differently than the mathematics most people use.  

[added February 20, 2017] I will offer this opinion: if mathematics is anthropological; if math is just another way our minds work and not the golden key to a deeper reality beyond ourselves, then it can tell us nothing new about the mysteries of existence; we are not going to be able to calculate our way along a path to truth; we will have to do the hard physical experiments to figure out just what the heck is going on.

Based on what the smartest humans know to be true today, we can’t build the kind of instruments required to answer the mysteries of the very large and the very small. Getting answers will take detectors the size of galaxies; it will demand the energy supply of thousands of suns.

If mathematics lacks a symbiotic connection to realty; if God is not a mathematician, if He doesn’t play dice after all, as Einstein insisted, well, we won’t get to a deeper understanding of how the universe works—or why it exists—through clever use of mathematics. It just isn’t going to happen—not now; not anytime soon; not ever.

The 18th century German playwright and philosopher, Friedrich Schiller, wrote …truth lies in the abyss.  

Pray that he’s wrong.

Billy Lee

Microwave Coffee

coffee 2
I drink the high octane stuff.

My wife and I are coffee drinkers. Beverly Mae used to drink a dozen cups of fully caffeinated coffee everyday.  By her third cup she could boogie with the best of them.  But that was a long time ago, before we got old. Now she drinks about two cups, and it’s decaffeinated. Me, on the other hand, well, I still imbibe the high-octane stuff.  I love it.

69050_Coffee Maker.37173901_std
My wife and I have one coffee maker between us. It’s not enough.


If your marriage is anything like ours, you probably own one coffee-maker, but you and your spouse drink different brands, flavors or styles of coffee.  For us, it means we have to store the contents of at least one of our coffee-pots off-site, away from the coffee-maker, in containers and carafes; perhaps cups or bowls or glasses or whatever is handy.  The coffee gets cold.

We have two microwaves. It is one of the very few things which seem to have made a difference in our marriage.
Two microwaves: they can sometimes save a marriage.

Only one of us at a time can store coffee in the coffee-pot.  But since we have two microwave-ovens, we don’t really need to keep our coffee hot.  We can turn off the coffee-maker after we brew each pot to save energy.  And so, everyday, we reheat our coffees in the microwaves, more than once, and thanks to having two of them, we don’t wait in line.

Beverly Mae and I have an eclectic collection of coffee mugs gathered together over decades of marriage. I’ve often wondered: how is it that no matter how big the coffee cup or how tiny; how robust the mug or how dainty; how full or empty we fill—or even which microwave we choose—my wife and I almost never set the timer to reheat our coffee more than once? We always seem to set the microwave to exactly the right number of minutes and seconds to heat our coffee to exactly the right temperature.

coffee 4
                She got the calculation right.

Think about it. Can there be any doubt that the mathematics required to accurately set the timer must be beyond the capabilities of 99% of the people who set these timers to reheat their coffee everyday?

The size of the cup, its thickness and material; the amount of coffee in the cup—these are important variables that are required to be taken into account when setting the timer. Not only these variables, but there is the subjective calculation: how hot do I want this coffee to be today? Real hot? Tepid? Mildly warm? There are many tricky variables to track and put into an equation. And, if we are reheating coffee for our spouses, we have to anticipate their calculation of what the best temperature is for their mood and state of mind.

differential equations EngMath_DifferentialEq_Terminology_02
It takes a matrix of differential and difference equations plus a super computer to get microwave coffee right.

It really takes a sophisticated matrix populated with complex differential and difference equations to work out what the results might be under all the possible scenarios. And it might require a government super-computer to crunch the numbers. Of course, I’ve never done the actual work of creating the matrix or the equations. Even if I had, I would have faced the daunting task of isolating all the relevant variables, the tedium of tracking all the units to make sure I ended up with seconds only in the answer, and the exhaustive testing of results to see if they match up with my expectations and experience. Successful creation and application of a workable calculus might involve a lot of tweaking.

drinking coffee
He didn’t bother with the calculation.

Come to think of it, why would I do that? Guessing seems to work better and it’s a lot faster. But I wonder. Am I really guessing? Or is my brain, somehow, doing the math in some far away place inside my brain, behind the scenes and beyond my conscious scrutiny? It’s kind of mysterious, being right all the time, about something as complicated as getting the number of minutes and seconds correct when setting the timer to reheat my coffee.

And my wife, who knows no math, is just as good at the mental calculation as me. Go figure.

Billy Lee

P.S.  Readers take note: on Valentines Day, 2015, Billy Lee bought a second coffee-maker for his wife, Beverly Mae. Why it took so long for Billy Lee to solve his coffee-problem is a mystery even skilled mathematicians won’t solve.  The Editorial Board.

microwave coffee math graphic 6
Don’t try these calculations at home. Seriously. This man tried; his face froze—for two hours.

What is Math?

math with color

People seem to think mathematics is something special—a kind of magic language that, when tinkered with properly, makes it possible for mortals to stumble upon answers to all kinds of mysteries about the Universe hitherto known only to God.

I see it differently. Mathematics isn’t a language per se. Although mathematics can be (and is) explained by language, math itself is a collection of rules and symbols that makes it possible to avoid the encumbrances, flourishes, and ambiguities of language. It accomplishes this feat by defining things and their relationships in strictly limited—but important—ways.

euler formula hatEuler Identity – Khan Academy

Math involves symbols and rules that aren’t explained inside the equations. It is the lack of words that gives math its mysterious and magical reputation. But once everything is defined and understood, applying the contrived but logical rules of mathematics enables folks to manipulate equations to uncover previously hidden and non-intuitive relationships among the things they have defined.

What am I saying exactly? I am saying that it is possible to use words alone to describe the process of solving and manipulating an equation, which can lead to insights into the relationship of the things in the equations. But these words will make the process of computation cumbersome, impractical, and confusing.

Spoken language contains noise and nuances that interfere with the manipulation of carefully defined relationships between narrowly defined variables. Yes, the no-nonsense logic and bare bones precision of mathematics, as well as the reduction of things to a few carefully chosen attributes, enables mathematicians to apply rules to discover consequences that might otherwise remain undiscovered.

But the tightness of mathematical construction makes it a tool which is almost useless for describing and analyzing many subtle yet vivid experiences of a conscious mind—like beauty, the feel of an orgasm, or the experience of grief. For these realities of conscious experience, mathematics has a reputation for being irrelevant.

euler ring     Euler Identity – Wikipedia

Spoken language gives conscious humans the messy modeling mechanism they need to connect with each other to share and understand the more nuanced experiences of life. The messiness and ambiguity of spoken language makes the unique intimacies of human communication possible. Mathematics, despite its elegance, doesn’t do intimacy well.

The Euler Identity, illustrated above, is sometimes presented as an example of the mysterious power of mathematics. But if you think about it, what does the equation say?  It says that minus one plus one equals zero.

Complex PlaneThe explanation is easy.   -1 can be rewritten as e raised to the power of i times π because of simple rules, which place on a circle of radius 1 all the values of e raised to the ith power times anything.  

The number that sits next to i is the angle in radians where the result lies. In this case, an angle of π radians (180°) takes us from the value 1 (at 0°, or 0 radians) to half-way around the circle to the value -1. 

Despite the reputation of equations for precision, it turns out that physicists and other scientists struggle to make mathematics match the results of real world measurements. The non-technical public is unaware, for the most part, that astronomical observations involving the movement of stars, planets, and other celestial bodies—or the results of observations made of the subatomic world (no matter how carefully contrived)—as often as not fail to provide results sufficiently in agreement with mathematics to be of any practical use.

Fudge factors are a big component of doing real science. People have won Nobel prizes for inventing fudge factors to fix things. It’s true.

Renormalization, perturbation theory (for phenomenon both small and large), Green’s functions, propagators, Feynman diagrams, and many other adjustments and tweaks make up the contortions and modifications that scientists overlay onto their beautiful equations to make them work.

They say they have good reasons for all the tinkering; it’s complicated down there among the quarks or up there, among the quasars; there are nuances and messiness and ambiguity in the underlying reality of nature that no one can see or fully understand—not now; not anytime soon; perhaps not ever.

At subatomic scales, a tangled mess of virtual particles which come into and out of existence more or less spontaneously often gets the blame for the mismatch between mathematical elegance and the cold reality of experimental results. On the scale of the universe, dark matter and energy (which can’t be detected) are sometimes blamed for anomalies. Click on the links in this paragraph to learn more.

It’s possible that no system involving mathematics can be contrived by humans to bring the satisfaction of knowing everything for certain; nothing we are able to invent will bring a tranquil end to the pain of cognitive dissonance that seems to drive our species to wonder and explore.

The more deeply people travel into the complexities of mathematics and science the more elusive truth becomes; perhaps God is not a mathematician; maybe Einstein was right when he said, God does not play at dice. A die is cast into the lap, yes, but its decision is from the LORD, according to an old proverb of Solomon.

Can it really be true that understanding the world is beyond the limitations of all life on the earth—beyond the abilities of the most brilliant minds that have lived or ever will live?

If so, it’s time to kneel.

Billy Lee