People don’t want to believe it, but Russia can win a conventional war against Ukraine without nuclear weapons. It doesn’t mean they will.
Outcomes of conventional wars are simply not predictable… too many variables, too many unknowns. Another problem is that all sides of a conflict tend to lie—both to themselves and others. During war, it’s hard to know what’s true.
One thing thinking people seem to know for sure: war always turns into a mess, which ruins the lives of people who experience it up close and personal. People who fight on the ground in war say that combat is loud, remote, confused, and incomprehensible. They are unlikely to meet anyone on the other side until they drive over their corpses.
And that is on a good day.
War wears folks down, weakens physical and mental health, and imprints on warrior-minds vivid sights, smells, and sounds they don’t forget.
A cousin who fought in Vietnam shakes like a leaf when in public spaces. He is unable to stand in lines, even to buy a sandwich. Somehow, crowds, and people standing in lines, trigger bad memories he can’t set aside.
Of course, he’s stone deaf as well—his ears ring like bell towers. The constant roar doesn’t stop until he sleeps. A Marine helicopter gunner, he blew out his ears with the sound of his own fires.
As for nuclear war, the most important rumor I’ve heard is that the USA modernized its nuclear fleet during the Obama years. Though classified, reliable people have published reports, I’m told, that money spent on upgrades soared to as high as two-trillion dollars. If true, Americans can be assured that the USA nuclear fleet is modernized and probably works damn good. It’s a staggering state of affairs, which adversaries are wise to take to account.
Maintaining nukes is not cheap or easy, right? It requires frequent replenishing of triggering elements which, unfortunately for war-planners, possess half-lives which last months, not years. It is why nuclear power plants are essential.
As for power plants, do readers know that neutron-emitting elements tend to poison fuel rods? After rods expend a mere 5% or so of their fuel, neutron-generating elements called poisons gum-up fuel rods to make them useless for power generation. Spent fuel rods typically are stored on-site in cooling ponds until their poisons can be extracted to boost quickly decaying trigger elements essential for bomb detonation.
Need for neutron-emitting trigger materials is why every country that builds nuclear bombs builds nuclear power plants. The secret keys to success hide in spent fuel rods.
It’s obvious, right?
Why would any right-minded country risk building future Chernobyls and Fukishimas unless it had really good reasons? One good reason is trigger materials. Isn’t it true that almost anyone can make cheap electricity from almost anything? — water, coal, sun, oil, natural gas, wind, or whatever, right?
Why nuclear?
Yes, plutonium lasts for millennia, but triggers don’t. To keep nuclear weapons viable for years and decades, not weeks or months, bombs must be continuously resupplied with reliable sources of neutron-emitting elements, which can be harvested most easily from fuel rod poisons.
It is easy to understand.
Most Americans assume USA concentrates its intelligence resources to keep up to date on the status of others’ nuclear inventories. What do they have, where is it hidden, how well is it maintained? Some understand that USA super-computers connect with hi-tech surveillance systems to track every piece of nuclear hardware and every kilogram of plutonium and bomb-grade uranium anywhere on planet Earth.
It’s true.
The technology is not new.
Here is a video from 2017 that lays it out. No excuses to anyone for choosing ignorance. Try to understand.
Some don’t believe blanket surveillance is possible. Listen folks. The system is already built. Artificial intelligence assesses threats and then tells leaders who it will be who prevails after a nuclear exchange on any given day.
Hint: most days, it doesn’t seem to be Russia or China.
Prolly not the USA, either.
Joe Biden insists that winning nuclear war is simply impossible. I have no way to know; neither does any civilian or any other person who lacks access to the super AI technologies of the United States.
Is there anyone out there who believes delusional dictators can be trusted to safely possess weapons of mass destruction? These strongmen threaten to unleash hell, presumably, using anything from viruses to chemical poisons to nuclear fires to cyber implosions that collapse civilizations. Who’s going to tell them, no…
Shouldn’t humanity be working to outlaw dictatorships? Individuals who have accumulated enough wealth and unchecked power to make war are a clear and present danger to humanity and all non-human life on Earth as well.
A line needs to be established; anyone who crosses becomes a felon — subject to arrest by the international community. The line divides those who have seized unlawful wealth and excessive power. The line must be defined and defended by international courts established by free nations. How hard can it be?
Limits are necessary or humanity is destined to be driven to extinction by a handful of folks driven mad by reckless power.
Can we face some unpleasant facts? Russian and Chinese elites hold at least 1.2 billion people under some kind of house arrest inside their territories, right? It’s a style of living that seems alien to Americans. Doesn’t everyone long to live free and unafraid? Doesn’t everyone want to shout whatever they think is true without fear of prison or worse?
If Putin and his ilk continue down the path they are on—threatening others with nuclear terror on their Fox-equivalent state media—maybe those who oppose their catastrophic ambitions might, with a little luck and a lot of bravery, prevail to someday see dreams of freedom and peaceful living come true at last.
It’s gotta be possible, right?
Who disagrees?
Does anyone want to live in a post-apocalyptic world—especially one where one side “wins bigly?”
Some folks say nuclear Armageddon is impossible. One side wins, and life goes on as it always does.
My problem with such views is fear that survivors of nuclear war might look like jellyfish and live at the bottom of oceans.
It’s why monsters who possess WMDs should be given no quarter in a world where folks plan to live free someday without fear.
For decades now, Nick Bostrom has defended his view that the reality of existence can be described by one of only three possible states:
1 – Life is rare in the universe; what life does exist always perishes before it reaches “technological maturity.”
2 – Life is rare in the universe; some life reaches techno-maturity, but all advanced life decides to avoid the temptations and the consequences presented by its mastery over artificial super-intelligence and other high-technologies.
3 – Life is abundant in the universe, but it is simulated. A few technologically mature civilizations yielded to temptation; their thirst for knowledge and entertainment pulled them into a spawning-orgy to artificially inseminate faked-life within an ever-increasing globe of stars and planets, galaxies and clusters — perhaps throughout all space.
Bioethicist and philosopher of artificial intelligence, Nick Bostrom.
This injection of simulated-life (and the infrastructure to sustain it) serve the research and entertainment needs of the original civilizations who created the simulations and then broadcast them into the cosmos like farmers throwing flower-seeds into empty gardens.
By now, faked life is pervasive. By now — after at least three generations of star formations — the number of simulations is hundreds-of-millions; perhaps billions; perhaps hundreds-of-billions.
Humans forced to wager on the odds that they themselves are artificial — that they are in fact simulated — must place their bets knowing that the odds could be as high as a billion to one, maybe more.
Humans are machines; they aren’t real; they aren’t what emerged from the chemistry of the universe but were instead invented in the imaginations of super-computers programmed by an ancient civilization whose address and time may forever remain unknowable.
If human civilization is faked, so might be its history, what it has been allowed to know, and what technologies it feels compelled to develop. A simulated, artificial civilization won’t necessarily know what is the time or era or eon in the real universe that exists beyond its view. It might never be able to understand how large or old is the real universe where its creators live and play.
Simulations might be created by life-forms curious about how certain scenarios they can imagine play out. Simulation might be a mature-tech version of television where advanced life-forms unveil the vagaries of their visions for the entertainment of vast audiences.
Some simulations could be simple games designed for small children to entertain themselves while mommy does laundry and dad mows the lawn.
Simulations might be simple algorithms developed by quantum super-computers to test the limits of their power.
Even the rules-of-play embedded in some releases might be undiscoverable — hidden by super-intelligent gamers who perhaps don’t really care about us; they are sure one day to lose interest and unplug the simulation.
What are the odds?
People who think like Nick believe the odds make countless simulations a near-certainty. If it were not so, then it is equally certain that human civilization will implode like all those civilizations that came before; humans will become the victim of their own technological march into the high-risk skill sets that lead inexorably to oblivion.
It’s what the Fermi Paradox is all about, right? Astronomers assume that life is common — pervasive perhaps — but they search the universe in vain for our companions.
Where is everybody?
Is it possible that the evidence will forever be that in this cosmos humans are alone and on their own?
If life — pervasive intelligent life — makes itself known, can anyone be sure that it will be authentic and not a simulation created by a life-form they will never meet? Can anyone trust that this newly encountered life is conscious?
Or is consciousness simulated so that no one real can discern who is genuine, what is authentic, who is faking true love, or what might behave in blind obedience to rules that render impossible any prejudices against faked life, which does not care?
What are the odds?
During his #1350 podcast of 11 September 2019 Joe Rogan asked Nick Bostrom why it cannot be true that we humans are the first to broach the limits of the technologies that are spread before us. Why cannot humans know for sure that they are real, not simulated?
Why is it not realistic to assume that human civilization is on the cusp of becoming the first creator of simulations and artificial life instead of being itself one more simulation added to a long line of simulations that have spanned the cosmos during the past billions of years?
In a simulation, which of its fake creatures is able to determine how old is the authentic universe it will never see? What avatar is able to determine how long or short-lived will be the simulation where it is trapped?
Why can it not be more certain that the civilization that survives and prevails will be ourselves, the species human, who will be first to spawn false populations and fake technologies to coat with lies the cosmos whose lifespan is likely to last trillions of years?
Nick Bostrom went through the numbers with Joe. He described how the probabilities of his ideas are constructed; he explained that if humans are not fake; not simulated; not artificially created, it is more likely — much more likely — that homo-sapiens won’t make it into the future.
We will suffer the extinction of every advanced civilization that went before, no matter where in space and time they were once located — if ever there were any.
Neither Nick nor Joe seemed interested to discuss more than perfunctorily the validity of the second listed possibility — the existence of technologically mature civilizations who refuse to extend their capabilities to logical conclusions.
The idea that civilizations might forego the use of artificial super-intelligence to secure their grip on the universe seemed a boring and unrealistic option. Nick included the possibility on the list of three only because it is possible to imagine that dozens of civilizations might decide, perhaps independently, to lay down their powers for some higher, universal moral-order.
Such a scenario defies common sense, does it not?
So the choices seem to have collapsed from three to two: self-annihilation or a successful breach of the barrier that enables breachers to create new, simulated worlds — to raise their status, finally and forever, to the heights of what the ancient-world called “gods”, the creators of worlds.
Is the ancient-world even real? — or is it another fabrication by simulators?
What do we know and when did we know it?
Why does science and history make no sense?
What are the odds?
Every theoretical physicist seems to be saying that quantum mechanics and general relativity cannot be fundamental. A reality underlies these systems of physics that seems to lie beyond our reach.
Physicists today admit that at least for now they are stuck on stupid. They wait for la seconde venue d’Einstein — Albert Einstein, part two.
How smart and creative can a simulation be when it can’t answer basic questions like:
What time is it?
Where am I ?
Is anyone in charge?
Why do the simplest things make no sense?
Are simulated life-forms — necessarily separated by their natures from reality and truth — always insane?
Are simulations evil when they challenge the authentic life that created them? Is authentic-life virtuous when it destroys the faked-lives of its troublesome simulations?
Which deserve to feel the emotions of existence more intensely — real-life? or artificial super-intelligence? or the billions-of-simulations, which Bostrom’s probabilities argue flow from them both?
Billy Lee
Warning from the EDITORIAL BOARD: Billy Lee sometimes “pontificates” to try out ideas, which to our minds are absurd.
For one thing, Billy Lee seems to imply in FAKED LIFE that the absence of evidence for intelligent life in the Universe is exactly what a civilization locked inside a simulation would experience.
Simulated life hidden behind the walls of a game constructed by “Super-Intelligence” will inevitably come to believe it is alone and dependent on a Supreme Being who loves only the “simulants”, because no one else is “out there” for God to love.
Billy Lee postulates in FAKED LIFE that the universe and its history make no sense, because it isn’t real. Maybe Billy Lee is dumb; maybe he doesn’t get things, because he can’t. Did the possibility that he is stupid ever cross his mind?
Perhaps “pity” is what Billy Lee deserves.
Billy Lee has written in the past that desperate folks might want to trust God to explain things they don’t understand; otherwise, they will miss chances to make the world a fairer and more loving place for the billions of people who live in misery — the weak and impoverished, right?
According to Billy Lee, the Bible says that God created people; God is love; He promised to never abandon the poor, the sad, the humble, the strivers for what is right, the merciful, the pure-hearted, the peacemakers, the persecuted.
God’s power is that He cannot change.
So, according to the Bible, it’s all true. We are simulants who will never be unplugged. The rules of the game are simple: love God who gave us our lives; love each other as much as we can.
Who will do it?
We are as real as our Creator made us to be. God decides who is real and who is fake, who lives and who dies. In His eyes, we aren’t fake — even though some of us say He is.
Added May 15, 2026:Sample problems with solutions provided at end of essay.The Editors
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 called 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 use it, sometimes to solve transcendental problems in the background when equations that need solving aren’t so easy.
Omega functions are almost always many valued, because once an equation is inverted there are usually many paths that lead back to the original equation. The useful solution requires picking the principal solution, which is the number identified by subscript 0.
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)
Inverting the equation to introduce the omega function creates solution branches, some real, some complex, which are layered sequentially and labeled by integers from minus ∞ to plus ∞. In the example, real solutions lie on grey and yellow spiral surfaces numbered 0 and -1. Complex solutions lie on red and green spiral surfaces numbered -2 and 1. Omega subscripts in the list above this graphic identify the spiral layer where the solution for its omega value is located.
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.
Who knows?
ADDED April 25, 2026
Recently Facebook, in its feed, published math puzzles like this one:
xx = 9
What does “x” equal?
X is exponent in transcendental equations like these, which are easily solved using Lambert.
xx = 9
x ln x = ln 9
LET x = et
SUBSTITUTING et ln et = ln 9 {~2.1972, right?}
THEREFORE et t = 2.1972
OR t et = 2.1972
LAMBERT SAYS t = W [0, 2.1972]
ENTER W [0, 2.1972] into Wolfram Alpha to get ~.8965
t = .8965, right?
BUT x = et
THEREFORE x = e.8965 = ~2.7183.8965 = 2.451
ANSWER x = 2.451
CHECK xx = 2.4512.451 = 9
QED
Here’s another perhaps perplexing problem. What does k equal?
49k = k14
k ln 49 = 14 ln k
k ln 72 = 14 ln k
2k ln 7 = 14 ln k
= ln k
= ln k
LET k = ex
ex = ln ex
ex = x
ex = x
1 = x / ex
-1 = -x e-x
= -x e-x
LET -x = t
= t et
equation is now in correct form.
t = W
PUT W[0, ] INTO WOLFRAM ALPHA
t = -.42536…
BUT x = -t
x = .42536…
BUT k = ex
THEREFORE k = e .42536
ANSWER k = 1.53014
CHECK 49 1.53014 = 1.53014 14
385.688… = 385.688…
QED
Should be mentioned that there are many solutions to this equation as anyone would assume after reading essay above. Wolfram will provide them. However, an integer answer exists that can be solved by inspection.
k ln 49 = 14 ln k
Make the base 49 to make ln 49 disappear.
k ln49 49 = 14 ln49 k
k = 14 ln49 k
Since by definition logs are exponents acting on a base (in this case 49), a sharp-eyed reader might notice that base 49 is the square of the number 7. The square root of a number is its 1/2 power, right?
491/2 = 7
IF ln k = 1/2
THEN 14 * 1/2 =7 = k
ANSWER k = 7
CHECK 497 = 714 = 6.78223…E11
QED
Here’s a trick to solve problems like the following: 3x + x = 7
3x + x – 7 = 0
Create variable y = 7 – x
Then x = 7 – y
Substitute into original equation
37-y + (7 – y) = 7
37-y = 7-7+y
37-y = y
By exponent rules
37-y =
So = y
37 = y3y
2187 = y3y
By log rules
eln3 = 3
Substituting for 3
2187 = ye(ln3)y
I have a lot to say about renormalization; if I wait until I’ve read everything to know about it, my essay will never be written; I’ll die first; there isn’t enuf time.
Click this link and the one above to read what some experts argue is the why and how of renormalization. Do it after reading my essay, though.
Here is a snippet rewritten from a Quora post: Renormalization is a set of techniques used to resolve infinities that pop up when calculating. It is based on the idea that system behavior depends on scale. Cut away problematic pieces that scale badly to extract finite, testable portions where infinities don’t appear.
Let’s move on.
Our guess is that this graphic will be incomprehensible to the typical reader of Billy Lee’s blog. So, don’t worry about it. Billy Lee isn’t going to explain it, anyway. More important things need to be told that everyone can understand, and they will. The Editorial Board
There’s a problem inside the science of science; there always has been. Facts don’t match the mathematics of theories people invent to explain them. Math seems to remove important ambiguities that underlie all reality.
People noticed the problem as soon as they started doing science. The diameter of a circle and its circumference was never certain; not when Pythagoras studied it 2,500 years ago or now. Number π is the problem. It’s irrational, not a fraction; it’s a number with no end and no pattern — 3.14159…forever into infinity.
More confounding, π is a number which transcends all attempts by algebra to compute it. It is a transcendental number that lies on the crossroads of mathematics and physical reality, a curious number at the heart of creation because using it means that diameters, surface areas, and volumes of spheres cannot be calculated precisely.
For a circle, either the circumference or the diameter can be rational (written as a fraction) but not both. Perfect circles and spheres cannot exist in nature. Why? ”π” is irrational. It can’t be written like a fraction — a ratio — where one integer divides another.
The diameter of a circle is multiplied by π to calculate its circumference, and vice-versa. No one can know exactly everything about a circle, because π is uncertain, undecidable, and in truth unknowable.
Back in the day, people learned to use the fraction 22 /7or, for more accuracy, 355/113. These fractions gave the wrong value for π but they were easy to work with and close enough to do engineering problems.
Fast forward to Isaac Newton, English astronomer and mathematician, who studied the motion of planets. Newton published Philosophiæ Naturalis Principia Mathematica in 1687. I have a modern copy in my library. It’s filled with formulas and derivations. Not one works to explain the real world — not one.
Newton’s equation for gravity describes the interaction between two objects — the strength of attraction between Sun & Earth, for example, and the resulting motion of Earth.
Moon, Mars, Venus, and other bodies warp space-time waters in the pool where Earth & Sun swim. No formula predicts the future of multi-bodied systems. Numerical methods, 2-body approximations among others are used.
This simple 3-body problem cannot be solved using a single equation. System is chaotic.
In 1887 Henri Poincare and Heinrich Bruns proved that such formulas cannot be written. The three-body problem (or any N-body problem, for that matter) cannot be solved by a single equation. Fudge-factors must be introduced by hand, Richard Feynman once complained. Powerful computers combined with numerical methods seem to work well enough for some problems.
Perturbation theory was first proposed in the 17th century. By 1960, physicists used it to solve problems in quantum physics. It helped. Space exploration depends on perturbation techniques. It’s not perfect. A fudge factor called rectification is sometimes used to update changes as systems evolve. When NASA lands probes on Mars, no one can know exactly where they are relative to reference points on the Earth.
Even when using the signals from constellations of six or more Global Positioning Systems (GPS) in high earth-orbit, it’s not possible to know exactly where anything is. Beet farmers out west combine the GPS systems of at least two countries to hone the courses of their tractors and plows. On a good day farmers can locate a row of beets to within an eighth of an inch. That’s plenty good, but the several GPS systems they depend on are fragile and cost billions per year. In beet farming, an eighth inch isn’t perfect, but it’s close enough.
Although a general method of perturbations can be described in mathematics, problems emerge when quantum and astral scales overlap. Relativity & quantum mechanics do not work well together, at least mathematically. Infinities appear that seem unresolvable.
Quantum physics is a frontier of knowledge that presents unique roadblocks to precision. Physicists have invented more excuses for why they can’t get anything exactly right than probably any other group of scientists. Quantum physics is maybe 100-years old, but problems today seem more difficult than ever.
The sub-atomic world seems to be smeared and messy. Vast numbers of particles — virtual and actual — makes the use of mathematics problematic. This pic is an artist’s conception. Concepts such as ”looks like” have no meaning at sub-atomic scales, because small things can’t be resolved by any frequency of light that enables them to be visualized realistically by humans.
Insurmountable?
Well, interaction of sub-atomic particles with themselves combined with, I don’t know, interactions with swarms of virtual particles might disrupt the expected correlations between theories and experimental results. The mismatches can be spectacular. They dwarf N-body problems of astronomy.
Worse — the problem of scales. For one thing, electrical forces are a billion times a billion times a billion times a billion times stronger than gravitational forces at sub-atomic scales. Forces appear to manifest themselves according to the distances across which they interact. It’s odd.
Measuring the charge on electrons produces different results depending on their energy. High energy electrons interact strongly; low energy electrons, not so much. So again, how can experimental results lead to theories that are both accurate and predictive? Divergent amplitudes that lead to infinities aren’t helpful.
An infinity of scales piles up to produce troublesome infinities in the math which tend to erode predictive usefulness of formulas and diagrams. Once again, researchers are compelled to fabricate fudge-factors. Renormalization is the buzzword for several popular methods.
Probably the best-known renormalization technique was described by Shinichiro Tomonaga in his 1965 Nobel Prize speech. According to the view of retired Harvard physicist Rodney Brooks, Tomonaga implied that …replacing the calculated values of mass and charge, infinite though they may be, with the experimental values… is the adjustment necessary to make things right, at least sometimes.
Isn’t such an approach akin to cheating? — at least to working theorists worth their salt? Well, maybe… but as far as I know results are all that matter. Truncation and faulty data mean that math can never match well with physical reality, anyway.
Folks who developed the theory of quantum electrodynamics (QED) used perturbation methods to bootstrap their ideas to useful explanations. Their work produced annoying infinities until they introduced creative renormalization techniques to chase them away.
At first physicists felt uncomfortable discarding the infinities that showed up in their equations; they hated introducing fudge-factors. Maybe they felt they were smearing theories with experimental results that weren’t necessarily accurate. Some may have thought that a poor match between math, theory, and experimental results meant something bad; they didn’t understand the hidden truth they struggled to lay bare.
Philosopher Robert Pirsig believed the number of possible explanations scientists could invent for phenomena were in fact unlimited. Despite all the math and convolutions of math, Pirsig believed something mysterious & intangible like quality or morality guided human understanding of the Cosmos. He saw an infinity of notions floating inside his mind. It drove him insane, at least in the years before he wrote his iconic Zen and the Art of Motorcycle Maintenance.
The newest generation of scientists aren’t embarrassed by anomalies. N. David Mermin said, “shut up and calculate.” Digital somersaults are executed to validate work difficult for most to understand, much less perform. Researchers determine scales, focus on relevant portions, cut-away irrelevant parts, and extract appropriate physics to make suitable matches of math to experiment. They put horse before the cart more times than not, some critics have said.
Apologists say, no. Renormalization is simply a reshuffling of parameters in a theory to prevent its failure. Renormalization doesn’t sweep infinities under the rug; it is a set of techniques scientists use to make useful predictions in the face of divergences, infinities, and blowup of scales which might otherwise wreck progress in quantum physics, condensed matter physics, and even statistics. From YouTube video above.
It’s not always wise to question smart folks, but renormalization seems a bit desperate, at least to my way of thinking. Is there a better way?
The complexity of the language scientists use to understand and explain the world of the very small is a convincing clue that they could be missing pieces of puzzles, which might not be solvable by humans regardless how much IQ any petri-dish of gametes might deliver to brains of future scientists.
It’s possible that humans, who use language and mathematics to ponder and explain, are not properly hardwired to model complexities of the universe. Folks lack brainpower sufficient to create algorithms for ultimate understanding.
Perhaps Elon Musk’s Neuralink add-ons will help someday.
Nick Bostrom, author of SUPERINTELLIGENCE – Paths, Dangers, Strategies
The smartest thinkers — people like Nick Bostrom and Pedro Domingos (who wrote The Master Algorithm) — predict artificial super-intelligence will be developed & hardwired with hundreds or thousands of levels — each loaded with trillions of parallel links — to digest all meta-data, books, videos, and internet information (a complete library of human knowledge). Armies of trained computers will discover paths to knowledge unreachable by puny humanoid intelligence.
Super-intelligent computer systems might achieve understanding in days or weeks that all humans working together over millennia might never acquire. The risk of course is that such intelligence, when unleashed, might enslave us.
Another downside might involve communication between human and machine. Think of a father — a math professor — teaching calculus to the family cat. It’s hopeless, right?
Imagine an expert in AI & quantum computation joining forces with billionaire Musk who possesses the rocket launching power of a country. Right now, neither is getting along, Elon said. They don’t speak. It could be a good thing, right?
What are the consequences?
Entrepreneurs don’t like to be regulated. Temptations of military power added to science-knowledge provided by super-intelligence might corrupt these men and push them & humanity toward unmitigated… what’s the word I’m looking for?
I heard Elon say he doesn’t like regulation, but he wants to be regulated. He believes super-intelligence will be civilization ending. He’s planning to put a colony on Mars to escape its power and ensure human survival.
Elon Musk
Is Elon saying he doesn’t trust himself, that he doesn’t trust people he knows like Larry? Are these guys demanding governments save Earth from themselves?
I haven’t heard Larry ask for anything like that. He keeps a low profile. God bless him as he gathers to himself in cyberspace everything anyone says & does.
Think about it.
Think about what it means.
We have maybe 10 years, tops, maybe less. Maybe it’s ten days. Maybe the worst has already happened, but no one told us. Somebody, think of something — quick.
Who thought laissez-faire capitalism might someday spawn an airtight autocracy that enslaves the world?
Humans might want to renormalize their aspirations — their civilizations — before infinities of misery wreck Earth and unfree futures emerge that no one wants.
…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")
= ln k
= ln k
= ln ex
= -x e-x
![\frac{W[0, 2402.655]}{ (ln3)}](https://www.thebillyleepontificator.com/wp-content/ql-cache/quicklatex.com-4683ae80481f2a5a55f689e159f461a8_l3.png "Rendered by QuickLaTeX.com")
