Tag Archives: Mathematics

To bit, or not to bit

A dualistic assessment pulses at the heart of cyber-sprawl. Our computer universe revolves around a persistent question delivered in rapid-fire succession. We call tiny, indivisible parcels of information bits. Bits answer one question, again and again: Does it exist? If it does not exist, it takes an identity based on a powerful idea often taken for granted in the rudimentary arithmetical grind.

The concept of zero arose in the wake of the agricultural revolution, most likely in South Asia. Nothingness, as a distinct mathematical entity, logged into history about the same time our ancestors first built cities. Nonexistence equates mathematically to the absence of value—zero.

Humans speciated about 300,000 year before present (ybp). We abandoned hunting and gathering approximately 10,000 ybp. The concept of zero didn’t manifest until more like 5000 ybp. The concept of zero most likely snuck into existence under the cover of flashier history: Egyptian pyramid construction, Babylonian hanging gardens, and the preponderance of pottery in human settlements across the Fertile Crescent.

We many never know zero’s precise origin in space and time. Most likely, the numerical value of nothingness arose again and again among geographically disperse cultures, throughout separate eras. Without reliable and more precise evidence, 5000 ybp serves as a sufficient estimate. Humanity has reaped the fruit of zero for roughly 2% of its time as a distinct species.

If a bit exists, we assign it the least magnitude, just enough to establish its presence. The symbol “1” represents existence.

A true or false also answers the bit’s existential query. But this is just a more complicated restatement of the answer rendered above. The use of a string of characters, a word, like “true” or “false” adds contextual meaning and alleviates the anxiety of the mathematically averse. True or false successfully answers the existence question. As simple as answering true or false may seem, imagine repeating the process hundreds, thousands, millions, or even millions of millions of times.

Many people would instinctively switch to “T” and “F” in place of writing four or five letters for each query. A single letter suffices to state the existence of something. Continue along this line of logic and some people—probably the mathematically inclined—will substitute a 1 for T, and a 0 for F. This transcends language barriers, removes ambiguity, and adds quantitative value for more complicated manipulations of our nascent data.

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photo courtesy of Mark Ordonez

The concept of zero wields power in ways that often escapes even the most mathematically gifted. Humans prefer ten digits (0-9) that repeat infinitely in a cycle up, and down, the number line. A zero initiates the process: The origin on a number line has no value. Moving up, we count to nine; further progress requires a reset of the ones place to zero, and then the addition of a one to the tens place (1 ten plus 0 ones is 10). Repeat the process until we reach nineteen (1 ten plus 9 ones is 19); now, replace the one in the tens position with a two, and reset the ones to zero—20. Without the concept of zero, we have no natural origin, or means to recycle our counting system at a convenient interval.

Computers operate with only two digits. Remember: the most basic element of a computer is a bit with two possible states—existence, or not. Humans prefer ten digits because modal humans have four fingers and a thumb on two hands. Base ten is natural for us because we typically inaugurate our mathematical experience tallying small quantities on our hands. Computers don’t have appendages. Instead, computers recognize existence or lack thereof; on or off; 1 or 0.

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photo courtesy of Chris McClanahan

Math isn’t partial to any number of digits. As long as the absence of value is expressible, only two fundamental numbers can effectively represent any quantity. Two digit numerical systems employ binary code.

Here’s a string of binaries beginning with the absence of value: 0, 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010. You don’t need master spatial perception to discern the pattern at the end of the previous sentence. Let me translate into more familiar base ten notation: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10.

The same principle governs all positional numeral systems. In binary, we recycle the process using only two digits; base ten counts nine distinct quantities before an abrupt return to the concept of zero. The first position has no value, then it does. Since we only have two possible values, we have to reset the value to zero and progress by adding value to the next position. Then we add value to the initial position. This process can repeat forever creating an infinite series of magnitudes.

Tedious repetitions, like writing and manipulating binary code, strain human attention spans and raise the probability of error. Also, we’re slow when it comes to mundane tasks. Fortunately, humans designed computers to flawlessly iterate simple instructions. According to a loose interpretation of Moore’s Law, processing speed doubles about every two years. My processor executes 2.2 billion (the same as 2.2 thousand x 1 million, or 2.2 thousand million) instructions every second.

By the way, base ten 2,200,000,000 is the same as 10000011001000010101011000000000 in binary. I think. Just transcribing a 32 digit quantity, regardless of base, carries a high probability of a simple mistake.

If you enjoyed reading this piece, check out An Electron Story or Hunter Gatherers in the Quantum Age.

Circular Political Spectrums

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Many pundits say Bernie Sanders and Donald Trump are political opposites. Bernie is a leftist and Donald is right wing. What does that mean?

Models are valuable tools that elucidate difficult concepts. Models clarify abstractions in fields riddled with uncertainties. Political scientists employ linear spectrums when modeling affiliations and governing behaviors of elected officials.

Liberals reside on the left side of the spectrum. Progressive is synonymous with liberal  because liberals favor brisk and continuous political progress. Liberals tend to promote government spending and levy more taxes to cover the cost of programs to spur progress.

Extreme liberals see business as a means to achieve humanity’s preferable end: Utopian actualization motivated by government influence in the form of funding and regulation.

The right side of the spectrum harbors conservatism. Conservatives caution for slower progress and portray government as restrictive on freedom and productivity. Conservatives believe government is fundamentally wasteful, enables sloth, and undermines civilization’s driving force—the profit motive.

Extreme conservatives abhor progress to the degree of time-reversal. They promote a return to a mythical past. Humanity cannot find its way forward because it has forgotten what brought it here; we will never achieve our destiny now that we abandoned what works best.

These fundamental philosophies clearly diverge. Using the linear spectrum model, the distance between the two dispositions widens as each political journey progresses. The effect is a palpable sense of polarization.

Where does the linear spectrum lead our disparate philosophies? Do conservatives succeed in reversing time’s arrow? Does liberalism collectively teleport us to heaven on earth?

Extreme politics, right or left, share crucial characteristics: policy ignorance and inability to compromise. Extremism relies on the force of will: Extreme personalities lead extreme politics. The end result of extremism is failure to achieve ridiculously optimistic short-term goals. Once failure is certain in the public perception, tyranny remains as the only option to loss of power.

The path of of extremists, left and right, converge in authoritarianism. The political spectrum is not a line; it is a circle.

The circumference of a circle is a linear measurement. If a circle is large enough relative to the our senses, it appears straight when traversing its surface. Straighten a meter of string. Now, shape the string into a circle. That’s how easy it is to change your thinking from linear to circular politics.

Here’s another way to look at it: Earth seems like a flat plane to our senses. We’re so small compared to our universe, and planet, that we failed to realize earth’s true geometric shape for most of humanity’s existence. A sphere is a three-dimensional circle. Before modern mathematics and the scientific method, common sense indicated walking in one direction would lead to an edge, or a walk without end. Now, we know walking in one direction on earth eventually brings us back to the starting point.

If you liked Circular Political Spectrums, try these: Poles Apart, A republic if you can keep it.

An Evolving Universe II–Energy Spreading

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This post is a continuation of An Evolving Universe I—The Greatest. Part II stands alone, but it will be difficult to appreciate and fully understand III without reading I and II first.

Entropy is a central scientific concept; it seems to have more importance in chemistry, and even more so in physics. Entropy is a measure of disorder in a closed system. Closed, in this context, means energy cannot move—maybe flow is a better word—across arbitrary barriers: the system is closed because it’s insulated, so to speak, from whatever is outside the system. The stuff and the space residing outside the barriers may as well not exist because the energy we care about can’t go there.

Energy doesn’t really move or flow. Energy exists in different arrangements; as the number of possible energy holders increases, the more complicated the system. Energy distributes itself according to strict, but simple, laws of probability. Our universe appears to contain a fixed amount of energy. That energy is allocated to a large number of particles—photons, electrons, protons, neutrons, atoms, etc. According to probability alone, energy is more likely dispersed widely across numerous particles as opposed to concentrated on a few, or just one. The level of energy concentration is the measure of order in a system; more dispersed energy makes a more disordered system.

Here’s a good example that might allow you to see the connection between order and energy distribution within a system: Imagine a series of twenty A4 papers, typed on in succession to create a short story. Stack the pages one on top of the other, chronologically. Now, take the stack of papers and throw them into the air. We know what happens: The papers fall back to the ground in a disordered way. It’s impossible to predict how or where they will fall. Reorder the papers in a stack, and then throw them into the air again. The papers fall back in a disordered, but different way than before. Each time we do this, the papers will land in a unique pattern; they will always be disordered compared to the original arrangement. There are an, essentially, infinite number of disordered arrangements of these pages and only one ordered state. Probability tells us the papers are destined to become disordered. Mathematically speaking, it’s unlikely the papers should ever be ordered in a stack that when read, top to bottom, creates a coherent story. The chronological stack is only one of a nearly infinite number of possible arrangement of the papers, and therefore, improbable.

Orderly, concentrated energy eventually becomes disordered, dispersed. Actually, the idea of energy being ordered guarantees misunderstandings. Let’s go back the stack of papers. Not only is it unlikely that the papers will ever exist in chronological order, the papers should rarely be in the same proximity. There are just too many scenarios where the papers are scattered. If the papers are concentrated initially; eventually, they spread out. Our intuition should tell us this is true but we blame it on influences like a person intentionally throwing them into the air. The stack of papers was always doomed to scatter, not because of malicious design or lack of maintenance; the papers will scatter because there’s so many more ways for the paper to scatter than to be stacked chronologically.

Energy prefers to be spread out, dispersed, in much the same way our stack of papers. It’s not that energy strives for disorder; energy tends to spread out on more particles in a system, not because it’s herded along to that end by some shepherd. There’s just so many more ways for energy to spread itself over many particles than to be concentrated on a few, or one.

When energy is concentrated, it’s more probable that it will spread out. Energy’s tendency to move from concentrated to dispersed gives rise to something we are all painfully familiar with: the one-way flow of time. We tend to see time as the master. Time makes our stack of papers scatter. Time disperses energy concentrations. It might be more correct to say the perception of time is created by the overwhelming probability that energy should be spread out, as opposed to concentrated. Time might be the way a conscious being processes the myriad of possibilities present continuously in the now—whatever “now” means. In other words, time is simply a vessel to explore an infinite universe.

Click here to go back to Part I. Here’s Part III.

An Evolving Universe I—The Greatest

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Isaac Newton was the greatest, most influential scientist.

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This is a fact but not a really scientific fact. There aren’t really any facts—even in science—because the scientific method (question, hypothesis, experiment, analysis, conclusion, evaluation) dictates all ideas must carry some degree of uncertainty. The scientific method never rests. It does get tired after many iterations. If exhaustive repetitions fail to uncover evidence against—scientists attempt to falsify, not support their predictions—a hypothesis becomes a theory: a scientific fact is born. Keep in mind all facts—theory is probably a better moniker; a fact and a theory are essentially the same—are subject to ongoing review.

Any evidence against a theory compels at least a modification, or even abandonment of that theory. The idea that facts don’t exist confuses the general public; it often confounds people with advanced degrees. Most realize the universe is continuously changing, evolving. Facts are part of the universe. Assuming ideas are manifestations of the physical universe, facts should be subject to evolution too.

Why was Newton the greatest scientist? His influential accomplishments were many. In order of estimated decreasing importance, here is what Newton revealed: the nature of light (he even hypothesized light came in particles called corpuscles, a precursor of photons, but he conducted no experiments regarding this belief), the universal nature of gravitation and the laws of motion. He invented (should we say discovered?) calculus too.

Calculus would be a more significant achievment but another bright chap, Gottfried Leibniz, created the same branch of mathematics about the same time as Newton. Had Newton died in the plague—he fled London when the pandemic ravaged the British Isles—calculus would have been Leibniz’s baby, so to speak.

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It’s unlikely another scientist would have discovered (should we say invented?) the other three ideas within a few decades. Newton’s color theory of light might have taken a century or more before another scientist discovered it.

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If Newton were alive today, he wouldn’t claim to be history’s first scientist; Newton would most likely defer to Galileo. Galileo seems to be the first person we know of to test his ideas. Newton didn’t really do anything distinctly different from Galileo. Newton just took Galileo’s practices to another level.

I’ve never heard an argument that any scientist surpasses Newton’s greatness. Albert Einstein is often considered as Newton’s competition. Einstein was the first scientist to compel a modification to Newton’s gravitation law; it was a cosmetic adjustment really, and it only modified it in extreme conditions. But Einstein’s General Theory of Relativity did do something Newton couldn’t do: Einstein explained the true nature of gravity—a distortion of space-time caused by the presence of matter.

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It’s appropriate that we distinguish between laws and theories. It’s likely most people believe laws are superior to theories. Unfortunately, the word theory is often mistakenly applied when the word hypothesis should be employed. A hypothesis is an educated guess; a theory is system of ideas backed up by a vast and complicated reservoir of experiments. In short, and once again, a theory is what we commonly call a scientific fact.

A law is a mathematical system which allows us to make predictions. Laws are powerful scientific tools. Laws have a profound weakness: they don’t explain what’s actually happening, physically. We just know that, as long as we realize the necessary constraints, laws yield reliable predictions.

Niels Bohr is a dark horse candidate to compete with Newton for greatest scientist. What did he do? Bohr was a father of quantum theory. Why not the father of quantum theory? There are many fathers of quantum theory: Max Planck and Einstein to name two more—there are others we should consider too—but neither really accepted the fundamental weirdness that goes with quantum theory.

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Bohr was the first scientist to embrace the weirdness, the probabilistic nature of the universe, at the root of quantum theory. Once Bohr convinced the scientific community—not all scientists we on board with Bohr, Einstein was stubborn and never accepted the dicey nature of quantum theory—a vast array of successive quantum theorists continued to build the most explicative theoretical system in the history of science: quantum mechanics.

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Bohr is not the father of quantum theory, but he’s the first on the list of potential fathers. Since quantum theory is the most successful scientific system of ideas, it makes sense that the first on the list of fathers is one of the greatest scientists.

It will be nearly impossible to knock Newton of his lofty perch. He had the advantage of getting in at the start of the game. Science didn’t really exist in an organized way when he was born.

The whole discipline rests on a foundation he constructed. Thanks to Newton, the base of science is strong. The only way to supersede Newton may be to discover a new characteristic of the foundation, or something we had not thought about how the foundation rests on whatever is supporting it. In my opinion, there is one possibility for another scientist to take the title of greatest scientist from Isaac Newton.

Click here to go to Part II. Here’s Part III.

An Electron Story

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We humans are victims of common sense. If something is common to me then it must be common to all of us. Unfortunately, vast–and also tiny–sectors of the universe are uncommon to our senses.

The first step to develop a working knowledge of quantum physics is to abandon your belief that anything valuble should fall within the bounds of common sense. Do not worry that I will burden you with an explanation of Schrodinger’s Cat. You probably would not understand me. Honestly, I do not think I understand it. No one really understands Schrodinger’s Cat.

Quantum means small, subatomic small. The word subatomic is a self contradiction. Atom means smallest. According to its most basic definition an atom cannot be broken into smaller parts. Essentially, subatomic means smaller than the smallest.

The word atom represents an influential philosophical breakthrough. Atom arose from the Ancient Greek word, atomos. In the 21st century, we know atoms are not smallest. But just the idea that all matter is made of tiny, indivisible things was a tremendous realization.

Fundamental is probably the best word to represent these tiny, indivisible things.

We continue to search for fundamental entities. These fundamental entities are often called particles in modern physics. The study of particles is the essence of quantum physics. Unfortunately, when we observe the smallest things, classical physics breaks down.

(Classical physics is precisely predictable and mostly algebra-based. Isaac Newton revealed the foundations of classical physics about three centuries ago. Calculus applies marvellously to classical physics too, but algebra is enough to communicate the principal ideas of the subject. By the way, Isaac Newton invented calculus. Classical physics is pretty much all that is taught at the high school level. It is a bit boring after you learn modern physics, so let us get back to the more exciting stuff.)

The most likely candidates for particles of matter appear to be quarks and electrons. There are other particles of matter, but quarks and electrons will serve as excellent representatives for all matter particles during this post.

Our common sense often informs us that nature is fluid and infinitely divisible; continuous may be the best word for this apparent fluidity of the universe. Water running from a hose seems to be a continuous flow of stuff that we can keep dividing forever.

A water stream is effectively modeled using a basic mathematical concept: the number line. According to basic geometric definitions a line can be divided into an infinite number of segments. We could also say a line has an infinite number of points or positions.

A number line is an abstraction. The abstraction only applies under ideal conditions. Experimental evidence indicates that a water stream can be divided into a finite number of discrete particles called water molecules.

Those molecules are made of atoms. Atoms are made of protons, neutrons electrons.

Electrons are the smallest of the three. It would take nearly 2000 electrons to equal the mass as a proton or neutron. Electrons have a mass of 0.00000000000000000000000000000091 kg. Although there are about a billion billion billion electrons in a human, there is nothing common to our senses about an individual electron.

An electron is so small that looking at one would change its nature. We see objects when photons reflect off those objects: the photons deliver information to our brain via our eyes.

Photons are light-energy particles.

An electron is so small that when a photon bounces off an electron, the electron’s nature is changed: the reflected photon delivers outdated information about the electron to our eyes.

Classically speaking, an electron is not observable. But when not observing an electron it is not really there. An Electron is everywhere in space-time when not observed. It is more likely to be in some places and times compared to others but we can never know for certain. Once observed, the electron becomes what the observation compels!

If you enjoyed this post you might also like: Binary Fraud or De-frag Brain.

hunter gatherer

Hunter Gatherers in the Quantum Age

Fifteen thousand years ago it’s probable that all humans banded together in hunter-gatherer clans of 50 to 100. That’s the way we survived for thousands of generations. Subsistence in permanent settlements is relatively novel for our species. Although we have spread world wide on the waves of an agricultural revolution, we thrive on the heart of a fundamentally nomadic species.

Most human brains can’t maintain more than 100 concurrent relationships. Apparently, this is the number when alpha male rivalry drove apart prehistoric nomadic clans. (Try a little test: write down all the people you interact with, face to face, in an average month. I bet you’ll struggle on your pass 50.)

Social media may be a development on par with the printing press because it allows us to engage in hundreds (thousands?) of concurrent relationships, and get past evolutionary cognitive barriers. Ultimately, this new connectedness could generate a hyper-level of creativity. Add this connectivity to the advent of quantum computers—they should be  available in about 30 years—and we might become a completely interconnected species.

What is a quantum computer? Ordinary computers communicate via binary mathematics: All instructions are coded as ones and zeroes, value and no value.

For an obvious reason, humans prefer to use a numerical base of ten symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. We start repeating these symbols in different positions and arrangements to represent any quantity on the number line.

All the familiar mathematical operations are possible using only 0 and 1. For example, a base ten 7 is equivalent to 111 in binary. Seventeen is 10001 in base two. I won’t explain how to translate from base two to base ten or visa versa. Just take my word for it.

Computers only have two fingers or I guess you could say the computer alphabet only has two letters. Computers make up for this weakness by processing the 0’s and 1’s rapidly. For example, my computer can do 2,660,000,000 actions every second.

Each 0 or 1 represents a bit that is or is not. Quantum computers have qubits. A qubit is allowed to occupy both value and no value simultaneously. Don’t feel bad if you don’t completely understand; no one really understands quantum physics. Here’s a good example to help you understand the power of quantum computers: If you wanted to find your way out of a complicated maze and, try all options until you discover a correct path. That’s what ordinary computers do, but they do it faster than humans.

I’m sure you can imagine a maze so complicated that my 2.66 GHz processor will get bogged down and take a long time to find a solution. The perfect solution to escape the maze is to try all paths simultaneously. That’s what a quantum computer would do; I guess you could say exponential technological growth becomes essentially vertical.

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If you enjoyed this post, try Binary Fraud or De-frag Brain.

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Binary Fraud

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Humans tend to substitute duality when it’s clear unity provides the best description.

Consider the concepts of light and dark. Ordinary language indicates these two things are separate entities and work in opposition. Try to define darkness without using the concept of light. Good luck. Perhaps a clever wordsmith will succeed, but I doubt it.

Darkness is the absence of light. Darkness does not exist independently of light. Darkness cannot overcome the light. When light appears, darkness vanishes.

Hot and cold is another false duality. Scientifically, hot and cold don’t represent distinct  physical conditions. Hot objects possess relatively high temperatures; cold things have correspondingly low temperatures.

This duality inspires misunderstanding of one of the central concepts in science, temperature. The faster molecules move in a substance, the higher it’s temperature. The amount of stuff in each molecule is important for temperature too, but temperature essentially represents the measure of molecular motion in a substance.

It’s not a question of “Are the molecules moving fast or slow?” because this is another false duality. The true question: how much motion does it have?

Duality’s power is apparent in binary numbering systems: computers have transformed humanity. Computers operate according to binary mathematics. All operations in binary math use a language based on 0 and 1. Humans prefer a base ten system: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.

Actually, binary math isn’t really binary. The concept of zero is based on the absence of value. A one represents value. Value vs. the absence of value.

Humanity is on the cusp of constructing quantum computers. The fundamental strength of quantum computing is each bit is allowed to occupy value and no value simultaneously. Quantum computers will one day change humanity in ways that cannot be predicted, or described with current language. Quantum computers are based on a unifying principle, probability.

If you enjoyed this post, you might also like Hunter Gatherers in the Quantum Age, De-frag Brain.