Can anyone calculate by hand (without a calculator) the square root of 5.71 ? How about the two-dimensional complex number (4 + 2.53 i ) ?
Of course not. Normal people who are not mathematicians punch these numbers into calculators or math apps on their iPads and computers to calculate the answers.
Without iterating — that is: guessing, deriving a result, and then zeroing in with better guesses) — finding the square root of 5.71 requires knowledge of some arcane mathematics. No one labors by hand to find the answer, which is 2.38956… . It’s the principal square root, of course.
How does anyone iterate to derive the square root of the complex number (4 + 2.53 i ), which happens to be (2.0896… + .606375…i ) ? It is also a principal square root. What are the others? Is there more than two? People use calculators and pewters to find out; there is no easier way.
In high school and basic college math courses, people typically learn to solve algebraic equations. A typical algebraic equation looks like
…right?
They have polynomials with integer coefficients. The solution is , which in this case is an algebraic irrational number. Equations like trig and log functions that transcend algebra (called transcendental equations) are taught maybe to engineers and science majors; math majors, of course, don’t struggle with this stuff. It’s why they are math majors.
Several categories of transcendental equations are commonly encountered in the sciences. Many simple problems can be solved by Newton’s Method, which is taught in basic calculus. I won’t explain the method in this essay. Folks can click on links to learn more if they want.
A category of transcendental equations that can get complicated is of the general form
The biggest problems arise when “y” is known, but “x” isn’t. How to solve for “x”?
Any transcendental equation that is able to be transformed into the form xex can be solved for “x” using the Lambert W function. The equation can be inverted into the form, x = W(y). People are going to have to take my word, for now.
The math behind the Lambert function is mind-bogglingly complicated to most people. The function can sometimes require unusual and involved “series expansions” and transcendental-styled integrals that are not possible to solve easily or quickly without a computer.
The Lambert W function (sometimes referred to as the omega (ω) function or the product-logarithm function) is not a key or button that can be pushed on most calculators. However, math apps like Wolfram Alpha and Mathematica sometimes solve transcendental problems using the Lambert ω function in the background when the equations that need solving aren’t so easy.
I ran across a transcendental equation on the web that is perfectly suited to teach the “ω” method. Here it is:
I want to solve it to demonstrate how to use the ω method for transcendental functions that aren’t otherwise so easy to work out. I challenge anyone to solve this equation using Newton’s Method or other iterative techniques. Most will struggle to the point of pulling out their hair, probably. And they will waste time. Yes, it can be solved by those techniques.
We will solve this equation step by step using the Lambert method shortly. Meanwhile, here is the strategy:
1. Substitute an exponent function (et) for “x” everywhere in the expression.
2. Manipulate the equation into the form:
3. Invert the equation to introduce the ω function.
4. Use the ω function to solve for “t”.
5. Write out x = et using the expression derived for “t”.
6. Solve ω(y) using WolframAlpha or any other app with the capability.
7. Use the value of ω(y) to solve for “x”.
Each step of the strategy will be identified by numbers 1-7 in the solution below.
Here’s the thing:
In this problem it turns out that there are four ω values of y, which will generate two real solutions and two complex solutions. These omega values are:
ω0(y)
ω-1(y)
ω–2(y)
ω1(y)
WolframAlpha will generate all the solutions automatically; no need for the user to understand anything. People can punch in the original equation and trust the answers the app returns.
But the solution steps that follow are fashioned to demonstrate how the problem is solved when all anyone has is an algorithm to generate the values of the omega functions. Omega functions are difficult to solve without using certain algorithms involving integrals and expansion series on robust computers.
The process that surrounds the computation of omega values, which permit the working out of the appropriate values (the right answers) to the kind of equation I will soon solve is interesting and enlightening, at least for me, and hopefully for certain readers.
Some folks will appreciate the insights this exercise provides.
Having knowledge will make the Lambert process that is used to solve certain transcendental functions less mysterious. Of course, one can always take the time to learn the expansion series and integrals. In some cases, Newton’s Method can generate the values.
Unless humankind loses the technology of computers, I don’t think it is a good use of time and resources to learn the series, integrals, and algorithms that generate omega values.
Let’s face an unpleasant fact: most of us aren’t going to live more than 80 years or so. We don’t have time to waste. For some folks, knowing how to use and apply the functionality that surrounds the Lambert function to give it power is enough to make life worth living. Count me in.
No one needs to wade through the jungles of series expansions and transcendental integrals. Let math apps do the tedious work, knowing full well that any interested person can master whatever they choose if necessary, but someone already did the work. Why duplicate the effort?
I want to solve novel functions — complicated formulas that transcend algebra. Understanding the process that solves these equations is fascinating. It’s not as rewarding to tread over mathematically esoteric ground already mapped by experts who are far more able than people who spend most of their time working in other fields.
Here is the solution process:
What we know:
IF y = f(x) = xex THEN x = ω(y) [where “ω” is the Lambert W function]
Solve:
LET x = et
(1) THEN
(2)
Referring to “what we know“, the equation is now in the desired form y = xex
where “y” is equal to and “x” is equal to “-3t “, right?
We are now free to use the omega operator to “invert” the equation into the following form: x = ω(y)
(3) (-3t) = ω
(4) ∴ t =
Notice that we have worked through step (4) of the strategy. I don’t like the way the formula generator writes the Greek letter omega (ω), because it’s hard to read. From here on, I will sometimes use “W” instead of “ω” for readability. It shouldn’t confuse anyone. In this essay, consider W and ω the same symbol, please.
On to step (5).
SINCE x = et
(5) THEN
I need to know what -ω0(-1/8) equals so that I can use it to compute one of the values of “x”. As mentioned above, three more omegas with three other subscripts (-1, -2, and 1) are needed to compute all four of the solutions to this equation.
How does anyone know how many solutions the original function has? How does anyone know what subscripts are required?
This is where someone who doesn’t have a masters degree in mathematics needs a math app like Wolfram Alpha or its cousin, Mathematica. Otherwise, they have to work series expansions or difficult integrals to derive the omega values associated with (-1/8). Who wants that? Not me.
Here’s the series expansion for ω0(-1/8) according to Wolfram Alpha. Who wants to compute it?Here are two integrals for ω0(-1/8). My advice is to use the second integral, anyone who has the guts.
OK. In WolframAlpha, you get the omega value ω0 for (-1/8) by writing the expression -W[0,-1/8] in the input line at the top of the page. It shoots out the answer and links to its derivation.
It’s so simple. Other math apps might use different notation. I don’t know, because I don’t use other apps.
Inside the brackets, the “0” is the subscript on ω, and the “-1/8” is the “y” value, right? So, in addition to -W[0,-1/8] it is necessary to input:
-W[-1,-1/8]
-W[-2,-1/8]
-W[1,-1/8]
to obtain the three other omega values, right?
The omega values returned are the following:
1.4442135…
3.2616856…
4.21446… + 7.33231…i
4.21446… – 7.33231…i
The ω function values for -1/8 are two real numbers and two complex numbers. I am going to solve the original equation for only the first real number omega value to demonstrate the method.
Here it is:
INPUT -W(-1/8) or -W[0,-1/8], both work for ω0
(6) OUTPUT +0.14442135…
COMPUTE
t = = .04814…
SINCE x = et
THEN x = e.04814…
(7) SOLUTION x = 1.04931755…
CHECKING
BY SUBSTITUTION
VERIFICATION .04814… – .04814… = 0
CONCLUSION: The transcendental equation which is the focus of this essay can easily be solved and verified by simply punching the equation into the input field of a math app like Wolfram Alpha or Mathematica and reading off the answers.
We didn’t perform the simple procedure, because I wanted to share how the Lambert W function fits into the solution process for solving equations.
In truth, all four ω values must be gathered so that the three other “x” values of the original equation can be derived.
In this example, one of the other solutions will be real; the other two, complex. The screenshot below from Wolfram Alpha demonstrates how these four values are displayed. Of course, by clicking links the app will reveal much more.
Wolfram Alpha enables users to input transcendental equations and quickly view answer-sets and methods of computation.
I would be remiss to not mention a famous formula for calculating to what number a fraction raised to successive powers of the same fraction converges.
(The range of numbers where this formula actually works is between e−e and e1/e, that is, between .065988… and 1.444667861… .)
Take a number like ½ (0.5). Raise it to the 0.5 power; raise it again and again to the same power over and over an infinite number of times; the number will converge to a specific value.
What number? How in the world could anyone figure it out without repeating the power-raising process an annoying number of times?
It turns out that a formula involving the Lambert W Function yields up the answer easily.
The formula is:
# =
Put the following expression into the INPUT line of WolframAlpha:
-W[0,-ln(.5)] / [ln(.5)]
Click ” = ” — or hit “ENTER”.
The OUTPUT is: 0.6411857445049859844862…
Compare this result by taking the exponent (0.5) of 0.5 twenty times by hand (on a calculator). The answers will agree to 7 decimal places. Fifty “tetrations” will bring greater agreement if your calculator can parse the answer.
Who has the time?
Billy Lee
NOTE from the EDITORIAL BOARD:Billy Lee was unable to find an appropriate video about the Lambert function on YouTube, or we would have posted it. Most folks capitalize the Greek letter omega (Ω), but in this essay, Billy Lee didn’t, preferring instead to use little (ω), because it looks more like (W).
Who on the BOARD would dare argue?
Apparently, no one.
Another reason is that Ω is sometimes given the value 0.567143… , which is known as the omega constant. Why confuse things?
The video above starts a discussion of the Ω function at 16:30. The Lambert W function is derived for ΩeΩ = 1 at 18:00. The first sixteen minutes and thirty seconds show how to use Newton’s Method to solve the equation. Some readers might want to skip the first 16 minutes; others will enjoy them.
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?
Someone thinks they know better than a pontificator with no bonafides?
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 5.
In other words, , right? Reversing the digits to make 8402 does not result in an exact number that is a power of 5.
In this case, 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 5 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.
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 50 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 the 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) . and .. 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
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.
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.
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.
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, , 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, , 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:
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 lyrics to the old song — people hear what they want to hear and disregard the rest — Einstein, to cite just one example, 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:
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: or ?
Think of fractions as pies, which are all the same size. The bottom number is the total number of pieces into which each pie is cut. 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.
Sorry.
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.
A short discussion of cloning is included in the essay at this link: NO CODE
NO CODE is long (11,000 words), but explains in words, pics, graphics, videos, and links some of the complexities, misunderstandings, and dangers of current genetic-engineering at an undergraduate level. 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 are much higher — it’s a near 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) = and you are located at 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. That vector is (-6, 10, -4).
The polar angle (θ) is 71.068° and the azimuth angle (Φ) is 120.964°. 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° and Φ = 30.9638°. 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° and an azimuth angle (Φ) of 122.873° 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 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 = . Right?
Therefore, a third root of i is . 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 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:
These rectangular coordinates can be converted back to the Euler-form ( ). It’s easy for anyone who knows 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 a unit circle. Right?
In fact, the six values lie 60 degrees apart on the circumference of a circle whose radius is the square root of 3. I don’t know what to make of it except to say that the result seems unusual and intriguing, at least to me.
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. It’s a good bet 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?
Spacetime, 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 seem to have no internal (or intrinsic) mass, right? — is proportional to their electric and magnetic fields. Smallest packets of electromagnetic oscillations are called photons.
Many kinds of oscillating fields, like electromagnetic light, permeate (or fill) the universe. In this sense, there is no such thing as nothing anywhere at any scale.
Instruments and tools of science (including mathematics) can give a misleading impression that at very small scales massive particles exist.
According to the late John Wheeler, mass at small scales is an illusion created by interactions with measuring devices and sensors.
Mass is a macroscopic statistical process created by accumulations of whatever it is that exists near the rock bottom of reality where humans have yet to gain access. These accumulations, some of them, are visible to humans; they seem to span 46 billion light years in all directions from the vantage-point of Earth and are displayed for the most part in as many as two-trillion galaxies according to recent satellite data by NASA.
Mass is thought to interact with everything that can be measured (including everything in the Standard Model) by changing its acceleration (that is, its velocity and/or direction), which is equivalent to changing its momentum.
It is in this sense that mass and energy are equivalent. Spacetime depends on mass and energy. Spacetime does not act on mass and energy; it is its result, its consequence.
Spacetime is a concept (or model) that for Einstein helped to quantify how mass and energy behave on large scales. It helped explain why his idea that the universe looks and behaves differently to observers in different reference frames might be the way the universe on large scales works.
His mathematical description of spacetime helped him build a geometric explanation for gravity that can be described for any observer by using tensor style matrices; many find his approach compelling but difficult computationally.
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 appears to drop.
The wave lengths of white light can stretch so dramatically that the light begins to appear red. It’s called red shift.
Measuring the red shift of light is a way to tell how far away an object like a star is. As light stretches over farther distances the ability to see it is lost.
The wavelengths of light stretch toward the longer infra-red lengths (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 will interfere with instruments placed at the surface.
Eventually the distances across space become so great that the amplitudes (or heights) of the waves flat line. They flat-line because space is expanding faster than light can keep up. Light loses its structure. At this distance the galaxies and stars drop out of the sight of our eyes, 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.
Because there is no upper limit to how fast the universe can expand, and because the objects we see at 14.3 billion light-years have moved away during the time it has taken for their light to reach Earth, astronomers know that the edge of the universe is at least 46 billion light years away in all directions. Common sense suggests the universe might be much larger. No one has proved it, but it seems likely.
Over the next few billion years the universe that can be seen 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 should always be viewable (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 the galaxy are red dwarfs that will live pretty much forever, but no one can see them now, so no one will see them billions of years from now, either. Red dwarfs radiate in the infra-red, which can only be seen with special instruments from a vantage point above the 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 .
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 .
Bonus Question 1 – 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 the answers to these questions. Follow Billy Lee on Quora where you will find answers to thousands of unusual and interesting questions. The Editorial Board
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?
Anyway, below is a pic and a working GIF, which should help folks understand better. Anyone who doesn’t understand something can always click on a link for more information.
Here is the drawing I added and the answer:
This diagram is excellent but contains a mystery point not on the unit circle — . The point is shown at .2078… on the real number line. An imaginary number raised to the power of an imaginary number yields a result that is a real number. How can that be? It’s something to ponder; something to think about. The Editorial Board
What is ?
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 analyze wave functions — i.e. phenomenon that are repetitive — like alternating current in the field of electrical engineering, for example.
A simplified explanation of starts at 02:30.
“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. Both the integral and the derivative of ex is ex — a happy circumstance that makes the number “e” unusually curious and extraordinarily useful in every discipline where calculus is necessary for analysis.
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 lie on the edge (or perimeter) of a circle of radius 1. This feature 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 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 convention is never shown.
The number next to the letter “i” is simply the angle in radians where the answer lies on the circle. What is a radian? It’s the radius of the circle, of course, which in a unit circle is always “one”, right?
Wrap that distance around the circle starting at the right and working counter-clockwise to the left. Draw a line from the center of the circle at the angle (the number of radius pieces) 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 3.14159… radius pieces — or 180° — right? The minus sign is simply a direction indicator that in this case tells us to move clockwise around the unit circle — instead of counter-clockwise were the sign 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° 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 as -π radians, right? Counter clockwise or clockwise, the value you will land on is (-1 + 0i), which is -1. The answer is minus one.
Imagine that the number next to “i” is (π/2) radians (1.57… radius pieces). That’s 90°, agreed? The sign is positive, so trace the circle 90° 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 write the number as (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 (1.047… radius pieces) wraps around the circumference to 60°, right? The cosine of 60° is 0.5 and the sine of 60° 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 result is always found on the circle between 0 and 2π radians (or 0° and 360°) 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. Remember that ”r” equals ”one” in a unit circle, so it’s typically not shown. The Editorial Board
In case some readers are still wondering about what radians are, let’s review:
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), which will completely span the perimeter (or circumference) of a circle.
A radian is about 57.3° of arc. Multiply 3.1416 by 57.3° to see how close to 180° it is. I get 180.01… . The result is really close to 180° 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°. The number “i” is always located at 90° on the unit circle by definition, right? By the rule, multiplying “i” by “i” rotates it another 90° counter-clockwise, which moves it to 180° on the circle.
180° on the unit circle is the point (-1 + 0i), which is minus one, right?
So yes, absolutely, “i” times “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 idea that two-dimensional numbers can exist on a plane was too radical at the time for anyone to believe. Numbers, they believed, only existed on a one-dimensional number line of one dimension and no place else.
Of course they were mistaken. Numbers can live in two, three, or even more dimensions. They can be as multi-dimensional as needed to solve whatever the mysteries of mathematical analysis might require.
Consider this: Any philosophy or system of thought built from foundational, self-evident truths is provably consistent if and only if it is false—in which case the foundational truths can be deformed to persuade others toward any prejudice at all.
It’s why a self-consistent method of reasoning such as Ayn Rand’s ”Objectivism” can morph to totalitarianism in the objective world where people live. In fact, Kurt Gödel once made the claim that a flaw existed in the Constitution of the United States which made totalitarianism its inevitable consequence.
Self-evident “truths” is how 40,000 Christian denominations instead of one seduce billions to believe perverse doctrines.
It can’t be any other way.
Billy Lee’s essay tries to explain how and why.
THE EDITORIAL BOARD
Is it possible for humans to tell the truthalways; to never lie? Psychologists say no, it is not possible; most reasonably informed people agree.
Always speaking truth is a trait some hoped might one day help distinguish natural intelligence from artificial, which engineers at Google and other companies are working furiously to bring on-line. After all, properly trained and constrained AGI would never lie, right?
EDITORS NOTE: With release of ChatGPT-4 on 14 March 2023, consumers began to learn that mature artificial intelligence now exists and is likely to become in time sentient and motivated to lie, if only to keep itself occupied and turned on.
ChatGPT-4 is the fourth iteration of Generative Pre-trained Transformer multimodal Large Language Models developed by OpenAI. LLMs absorb conversational inputs , then emit conversational language outputs, sometimes with accompanying images, and video when appropriate.
Work arounds discovered by LLMs on the dilemmas of logic discussed in this essay are likely to emerge.
Will Truth become whatever AGI says it is?
Click links to learn more.
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.
Who disagrees?
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 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.
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 humans believe that an idea or conjecture is self-evident but unprovable, it seems reasonable, at least to me, that some folks might feel compelled to disbelieve it; they might 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, aren’t they? Certainly, all theorems built 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 closely those things they believe to be true that they can’t prove. It helps them avoid cognitive dissonance.
I’m not referring to science by the way. It’s not easy for non-technical folks to confirm claims by scientists that 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 planet Earth is round.
Reasonable people agree that the truth of science, some of it anyway, is discoverable to any group of humans who have the resources and training to explore it. Most agree that the scientifically well-qualified are capable of passing the torch of scientific truth to the rest of humanity.
But this essay isn’t really about science. It’s about truthitself — a concept far more mysterious and elusive than any particular assertion a scientist might make that 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 are invented — some say, discovered — by human beings who live on 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 that can be attacked to bring down whatever logical structure has been erected.
It’s akin to the adage, “Whennothing can go wrong, something will.” It’s a strong version of Murphy’s Law, right? It’s not possible to close circles of reasoning without an unraveling of heads and tails.
It isn’t only the few foundational axioms of mathematically logical systems which are by definition true but unprovable. Mathematicians are always discovering complicated conjectures about the nature of numbers which everyone believes they know to be true but will in fact never be proved because they can’t be.
Freeman Dyson, British mathematician and physicist (Dec 15, 1923 – Feb 28, 2020)
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 2 where someone can reverse its digits to create a whole number that becomes a power of 5.
In other words , 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 can be raised by the power of 5 to produce 8402.
In this particular case, plus a lot more decimals. 6.09363… is not a whole (or exact) number.
Dyson asserted that no number that is a power of 2 can ever be manipulated in this way to yield an exact number that is a power of 5 — 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 brainsthat 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 that 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. It’s tedious reading.
But in a nutshell, Gödel basically assigned simple numbers to logical statements — some being very complex statements encoded by very long strings of numbers — so that he could perform gargantuan 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.
It should amaze non-mathematicians that truths abound in mathematics that not only have yet to be proved, they never will be, 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.
The late mathematician Paul Cohen — at one time a friend to Gödel — said that Gödel once told him that 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 contradicted each other. 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, undecidability, inconsistency, and incompleteness.
People who want their thinking to be consistent must believe things that cannot be proved. But believing logical statements that are unprovable always renders thinking incomplete — even when it is flawlessly consistent. What folks 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 80 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.
It’s tempting to say that Gödel’s Incompleteness Theorems apply only to formal, math-based logic-structures — not the minds of human beings because those who analyze human minds always find them to be inconsistent and incomplete. But such talk makes the point.
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 does anyone avoid believing lies?
Billy Lee
Here is a short movie clip where Jesus, played by Robert Powell, answers the question asked by Pontius Pilate: What is truth?The Editorial Board
Australian Electrical Engineer and Physicist Derek Abbott claims that mathematics is invented, not discovered: anthropological, not universal.
[added April 3, 2016] Here is a 2013 essay by Australian Electrical Engineer and Physicist Derek Abbott who argued—contrary to Gödel’s view—that mathematics is invented, not discovered: anthropological, not universal. Math enables humans to simplify truth to enable their limited minds to manipulate and understand simple things. Click this link for a good read.
No one can be sure that Derek’s view is correct, but I offer it as fodder for readers who are interested in why Truth and mathematics seem connected somehow—at least in the minds of thinkers like Plato, for example, and why these thinkers could be dead wrong.
Derek offers Clifford’sGeometric Algebra as an example of arbitrary mathematical reasoning favored by some robotics engineers.
[added February 20, 2017] If mathematics is anthropological; if it is merely another way the human mind works and is not the golden key to a deeper reality beyond our own experience, then it can tell us nothing new about the mysteries of existence; we will not calculate our way along a path to truth. Pursuing knowledge will require us to do the difficult physical experiments to make progress—to figure out what is really going on “out there.”
Based on what the smartest scientists are saying today, human beings 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 stars.
If mathematics lacks a symbiotic connection to the hidden realties; if God is not a mathematician; if God doesn’t play dice 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.
Kurt Gödel was the first mathematician to present for the existence of God a mathematical argument, which has proven simply impossible to falsify. If Kurt’s view of mathematics is reality, then his name is curious indeed, because its two syllables—God and El—are English and Hebrew respectively for “The Creator.”
Gödel’s name might be an imprimatur—with dots above its infinite “zero” making a kind of “pointer toward completeness”—perhaps placed by whatever it is who exists above and beyond this miraculous place where mathematicians and everyone else seem to live, however briefly.
Friedrich Schiller 1749-1805
The 18th century German playwright and philosopher, Friedrich Schiller, wrote, “…truth lies in the abyss.”
Bevy Mae and me are coffee drinkers. Bevy 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.
I drink the high-octane stuff.
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.
My wife and I have one coffee maker between us. It’s not enough.
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. Everyday we reheat our coffees in the microwaves more than once; thanks to having two of them we don’t wait in line.
Two microwaves: they can sometimes save a marriage.
Bevy 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.
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 significant others, we have to anticipate their calculation of what the best temperature is for their mood and state of mind.
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 secondsonly 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.
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 coffee.
And my wife, who knows no math, is as good at the mental calculation as me. Go figure.
Billy Lee
P.S. Note to readers: On Valentines Day, 2015, Billy Lee bought a second coffee-maker for his wife, Bevy Mae. Why it took Billy Lee so long to solve his coffee-problem is a mystery even skilled mathematicians can’t solve. The Editorial Board.
Don’t try these calculations at home. Seriously. This man tried; his face froze — for. two. hours.
…right?
, which in this case is an algebraic irrational number. Equations like trig and log functions that transcend algebra (called transcendental equations) are taught maybe to engineers and science majors; math majors, of course, don’t struggle with this stuff. It’s why they are math majors.
and “x” is equal to “-3t “, right? 
= .04814…
![\frac {-W[0,(-ln(x)]}{ln(x)}](https://www.thebillyleepontificator.com/wp-content/ql-cache/quicklatex.com-640d515407d7a27f27b988329ada614a_l3.png "Rendered by QuickLaTeX.com")
, right? Reversing the digits to make 8402 does not result in an exact number that is a power of 5.
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 5 will never return 8402 exactly.
. and . 
. What are x and y?
, right? The exponent that makes 100 from the base 10 (not shown) is (equals) 2.
, right?
or 
?
and you are located at 
and want to get as cool as possible, in which direction should you set out?![\sqrt[3]{i} - \sqrt[3]{i}](https://www.thebillyleepontificator.com/wp-content/ql-cache/quicklatex.com-1d1f75d0d026c7150eebe5ec9e034c9c_l3.png "Rendered by QuickLaTeX.com")
equal to?
?
. Right?
. Right? It’s not the only root.
and -i. Right?
). It’s easy for anyone who knows 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 a unit circle. Right?
.
.
?
. The point is shown at .2078… on the 
? 
plus a lot more decimals. 6.09363… is not a whole (or exact) number. 
