[disinfo ed.'s note: the following is an excerpt from the forthcoming book The God Problem: How A Godless Cosmos Creates by Howard Bloom ("I have met God and he lives in Brooklyn" – Richard Metzger), courtesy of the author and Prometheus Books.]
Before we probe for clues to the God Problem, we need to equip ourselves with five tools—the five heresies. Remember the second rule of science: look at things right under your nose as if you’ve never seen them before, then proceed from there. Question your assumptions. To question your assumptions, you have to find them. And that’s the really hard part. But here are five assumptions conveniently overturned for your edification and delight. Five heresies we’ll use to crack the code of cosmic creativity.
- A does not equal a.
- One plus one does not equal two.
- The Second Law of Thermodynamics, that all things tend toward
disorder, that all things tend toward entropy, is wrong.
- The concept of randomness is mistake. These days randomness goes under the fancy name of stochasticity. But no matter how it slicks itself up with arcane terminology, there is far less randomness in this universe than today’s science believes. And far less randomness than you and I often think.
- Information theory is not really about information. Its equations cover only a tiny sliver of what the theory claims. The real core of communication is what Information Theory’s founder Claude Shannon calls “meaning.” And “meaning,” believe it or not, is not covered in Information Theory. Why is that a big mistake? Meaning is central to the cosmos. Central to quarks, protons, photons, galaxies, stars, lizards, lobsters, puppies, bees, and human beings.
And here are a few of the concepts we’ll use to peel open the robes with which nature hides the secret curves of her creativity, concepts we’ll use to probe the implications of the five heresies.
- Ur patterns, deep structures of the cosmos, patterns the cosmos repeats over and over again.
- Repetition. Better known in mathematics as iteration. When you repeat an old pattern in a new location, you sometimes make something new.
- Which leads to the concept of translation. Translation is just another word for repeating something old in a new medium. Or is it?
- Corollary generator theory. From a few basic rules you can generate a cosmos. Some call these basic rules axioms. Some call them algorithms. But don’t let the fancy names fool you. They’re just simple rules.
- Implicit versus explicit realities. Here’s a question for you. If you can generate an entire mathematical system from just a few simple rules (and you can), was that mathematical system implicit in the rules from the beginning? Was it hidden in some spooky way? Is the future hovering in your vicinity at this very minute, immanent and ghostly but just out of reach? Does every blockbuster invention that the cosmos—and that we—will someday conceive exist in a possibility space just outside the bounds of reality?
- Opposites are joined at the hip. Night and day, poisons and pleasures, innovation and destruction, are usually different facets of the very same thing. Despite the battle they wage with each other, they are Siamese twins, children of the same parents, children that have taken slightly different paths. Opposites work together in the very opposite of the way they seem—not tearing each other to bits or threatening to annihilate each other. Opposites are like the right and the left end of a football defensive line. They work together in teams.
- The bottom line? Sociality. This is a profoundly social cosmos. A profoundly conversational cosmos. In a social cosmos, a talking cosmos, a muttering, whispering, singing, wooing, and order-shouting cosmos, relationships count. Things can’t exist without each other. And the ways things relate to each other can make them radically different from their fellow things. Got that? No? Believe me, as we move forward, you will. And if the muses are with us, you’ll enjoy the ride.
+Heresy Number One: Why A Does Not Equal A
“A is A” is one of the most important assumptions underlying Western culture. Logic,11 reason, algebra,12 calculus,13 and trigonometry14 are based on the notion that “A is A.” Every equation in math has an equal sign. And every equal sign is a statement that one thing is the same as another. Every equal sign is a testament to the ubiquity of A=A.
The calculations of Newtonian science,15 of Einsteinian science, and of quantum mechanics16 are based on “A is A.” So is the software that runs medical equipment like MRI scanners. And the software that helps drug researchers sort through chemical combinations looking for cures to problems like AIDS and cancer.17 “A=A“ has taken science and mathematics a long way. A very long way indeed. Every one of the 560 active satellites in orbit around the Earth today owes the precision of its placement to “A is A.” And the manufacture of the microchips in your laptop, your iPad®, and your cellphone is also a testament to the power of “A is A.”18 But what if A does not equal A?
One of the strangest uses of A=A is in pop philosophy. The followers of Russian-American novelist and philosophical thinker Ayn Rand, author of Atlas Shrugged and The Fountainhead, have adopted “A is A” as their slogan. These “objectivists” chant “A is A” like a mantra to ward off evil thoughts. For good reason. Says Rand in her most famous book, her emotionally compelling 1959, 1,168-page Atlas Shrugged, “A is A. Or, if you wish it stated in simpler language: You cannot have your cake and eat it, too.”19 Says Rand, “A is A” is not just an airy idea in the abstract realm of wizened graybeards speaking the incomprehensible language of academic philosophy. Ignoring the fact that “A is A,” insists Rand, is the source of “all the secret evil you dread to face.” What’s more, Rand shouts in your face that “all the disasters that have wrecked your world, came from your leaders’ attempt to evade the fact that A is A.”
Got that? Every catastrophe on the planet has come from dodging “A is A.”
But not all philosophers and mathematicians are as enthusiastic about “A=A” as Ayn Rand. Barry Mazur, the Gerhard Gade University Professor of Mathematics at Harvard University, asks, “When is one thing equal to some other thing?” The answer should be simple, right? Not really. In fact, Mazur says that, “One can’t do mathematics for more than ten minutes without grappling, in some way or other, with the slippery notion of equality.” Why slippery? Because each “A,” each “thing,” is presented to us in a different context, says Mazur. Each A is at the heart of a different network of relationships. And the very quality of A-ness is the result of an act of distortion. A violent act of reality-abuse. An act of abstraction. Says Mazur, “the general question of abstraction… is neatly packaged in the Greek verb aphairein, as interpreted by Aristotle in the later books of the Metaphysics to mean simply separation: if it is whiteness we want to think about, we must somehow separate it from white horse, white house, white hose, and all the other white things that it invariably must come along with.”20 But have you ever seen a whiteness that is not attached to some thing? To some piece of paper, to some little white house with its neat white picket fence, to a neatly pressed and folded white dress shirt, or to an albino rhinoceros? In all probability, never. So abstracting whiteness is an extremely useful trick. But it can mislead us about the nature of reality.
Mazur wrote a twenty-four-page paper on the problems with A=A. But Terrence Parsons, professor of philosophy and linguistics at UCLA, was even more bugged about A=A. He wrote an entire book on the subject, Indeterminate Identity: Metaphysics and Semantics. Parsons puts the problem of “A is A” like this:
Suppose a ship sets sail, and while at sea it is completely rebuilt, plank by plank; is the resulting ship with new parts the ship that originally set sail? What if the discarded pieces of the original ship are assembled into a ship; is that the ship that originally set sail?21
Think about this for a minute. Imagine that you are an ancient Greek ship captain. You plan a one-year voyage from the port of Piraeus near Athens to get the rarest and most expensive commodities from the Spanish colony of Empúries roughly 1,164 miles away. Because the voyage will be long, you take lumber to replace any planks of your ship that become worm-eaten or waterlogged. And you budget enough coins to pay for more lumber along the way. You have been at sea for a month when, in fact, some planks become water-sogged. So you replace them. Then you put the waterlogged planks on the deck in the sun to dry out. When they are nice and dry, you cover them with pitch to waterproof them. And when you have enough of these recycled planks, you begin to build a second ship. By the time you’ve been gone a year, you are no longer sailing just one ship. You are sailing two. The first ship is the one whose planks you’ve been replacing. And by now, you’ve replaced every single plank. Ship two, the ship you’re towing behind you, is built from the planks you’ve dried out and recycled. Now here’s the puzzle. Which A equals A?
When the two ships return to their home port, which ship is the original? Which is the ship you set sail in? Remember, the empty ship that you’re towing is really the old ship in disguise. It has every single worn-down board and plank of the original. And the ship your crew is hunkered down in has all new planks. It’s new from stem to stern. But your crew has never stopped sailing in it, sleeping in it, and eating in it. So is the ship with all new parts the original? Or is the original the ship you are towing on a rope behind you? Which ship is the real deal? Which A = A? Then Parsons poses another “A is A” mind-twister:
If a person has a brain transplant, or a memory transplant… is the resulting person the same person who antedated the operation, or has the old person ceased to exist, to be replaced by another?”22
Parsons says that philosophers have “puzzled over questions of identity” like this “throughout history.” In fact, the puzzle of the ship that is repaired en route so many times that it’s totally rebuilt is called “the ship of Theseus” dilemma. It goes all the way back to the Greek historian Plutarch, who wrote up a version of it in roughly 100 AD.23
What Parsons calls these “puzzles” of “A is A” cry out for solutions. So why have nearly two thousand years of pondering led to no answer? Because, says Parsons, “There is no answer.” “There is no answer at all.” Yes, those are Parson’s words: “no answer at all.” How could that possibly be? “Because,” says Parsons, “of the way the world is.”24 Because abstractions may be indispensable. But they don’t accurately reflect reality.
Twentieth century über-philosopher Bertrand Russell, the man whose writings helped shoehorn you into atheism, was tortured by the paradoxes of “A is A” in his 1903 book The Principles Of Mathematics. He puzzled over whether the relationships called “=” and “is” even exist. He twisted and turned over the question, as he put it, of “whether there is such a concept at all.”25 In fact, Russell said, “It may be said, identity, cannot be a relation.”26 It can’t represent something that exists in the real world. But we have to use it. Why? It’s handy as all get out. Up to a point.
Bertrand Russell had a “friend” at Cambridge who was seventeen years his junior. A friend whose three brothers had committed suicide, leaving him and his one remaining brother to question life profoundly. In Russell’s opinion, that friend was “the most perfect example I have ever known of genius.”27 The friend’s name was Ludwig Wittgenstein. And Wittgenstein would become the airy and incomprehensible god of twentieth-century philosophy. But even Wittgenstein had his doubts about “A is A.” In his usual elliptical and indecipherable manner, Wittgenstein put “A is A” at the head of the list of “word-formations with which we feel not fully at ease.” He said this lack of ease manifests itself, e.g., in our always having found the proposition A = A to be something strange and profoundly mysterious. If we are shown a way of not coming up against this proposition, if we are offered a notation that excludes it, then we are prepared straightaway to welcome this and to abandon the law of identity’, this putative foundation of the whole of logic.28
Can we help Wittgenstein out? Can we help him escape from “A=A”? Can we show him “a way of not coming up against this… this putative foundation of the whole of logic?” Yes.
But why in the world does one “A” not equal another “A?” If we clone you and get an identical copy, why are you not your clone? Location, location, location. Location in time. Location in space. Location in a big picture. And your place in many smaller pictures nested within that big picture. Not to mention that each of the two yous is composed of different raw materials. And that each of you sets off on a different set of adventures. Being you triggers one mesh of chemical and electron flows in your brain. Looking at your clone triggers another. The two of you are not the same because of what you might call “the law of separation.” Because of what you might call “the law of differentiation.” And because of “the laws of sociality,” “the laws of the talking cosmos,” “the laws of the conversational cosmos.” Which leads us to the man who founded A=A, Aristotle.
+When Is A Frog A River—Aristotle Wrestles Heraclitus
If “A is A,” a philosopher should equal a philosopher. But that’s not the way the cosmos works. Similar things set themselves apart from each other. And that includes philosophers. What’s more, opposites are joined at the hip. Einstein says that most creative acts come from opposition. They come from pitting yourself against someone with another point of view. They come from the law of differentiation. And that was true of Aristotle and his “law of identity,” his law of non-contradiction,29 his construction of the base for “A is A.”
Aristotle came up with the idea behind “A is A”30 to fling a finger in the face of another philosopher, a philosopher who, in his words, saw “the whole of this visible nature in motion.”31 Who was Aristotle’s straw man, the thesis maker against whom Aristotle aimed his antithesis? Aristotle developed his ideas in opposition to Heraclitus, the founder of the school of perpetual transformation. Heraclitus was responsible for turning change into what Aristotle called a “dogma.” And a pernicious dogma at that. Or at least that’s the way Aristotle saw it.
Location often leads to differentiation. Athens was the home base of the Lyceum, the school that Aristotle founded in 335 B.C.E. and ran. But Heraclitus was a philosopher from the city of Ephesus, on the opposite shore of the Aegean Sea. And Heraclitus was obsessed with the shifting nature of things. “What was scattered gathers,” he said, “What was gathered blows apart.” Heraclitus tried to get that message across in slightly different terms in his best known phrase: “You can not step twice into the same river.”32 What did Heraclitus mean? The river is always changing. The water into which you put your foot the first time is no longer there the second time you dip your toes into the flow. The swirl of liquid you felt surging around your calves is now somewhere downstream. And in all probability even the patterns of the water that caressed your leg have changed as they’ve moved a few yards further toward the sea, shifting from the spiral swirl you felt around your calves to a streamlined, straight, “laminar” flow.
Heraclitus proved his proposition that all things change in a rather abrupt way. He died ninety-one years before Aristotle was born. His flesh scattered just as he’d implied it would. However, location in time is another source of differentiation. And there was a ninety-one-year gap between Heraclitus and Aristotle. But like a whirlpool in a stream, Heraclitus’ ideas survived. In fact, they thrived. Heraclitus’ concepts were so pervasive that another Athenian philosopher, Cratylus, took Heraclitus’ notion of perpetual, second-by-second change a step farther.33 According to University of Pennsylvania philosopher Charles H. Kahn, “Cratylus denied that you could even step in the river once, since you are changing too.”34
The result, says Aristotle, was that the “most extreme”35 followers of Heraclitus said it was impossible to fix a name to anything. Is this little green creature hopping across your kitchen table after your trip to a summer pond a frog? According to the Heraclitan change-zealots, you can’t say yes or no. Why? In Aristotle’s words, the Heraclitans “considered that verification was not a thing that is possible.”36 OK, but once again, why? Because the frog is changing. A year ago it was an egg. Two weeks ago it was a tadpole. And by the end of the summer it could well be digested into the muscles and bones of your frog-eating dog. This, says Aristotle led to an “extreme opinion” among some of the change-enthusiasts—the opinion that “one ought to speak of nothing.” Cratylus was the change-extremist who Aristotle used as a prime example. And Aristotle says that Cratylus “was of [the] opinion that one ought to speak of nothing, but moved merely his finger.“37 In other words, Cratylus reduced all philosophy to helpless hand waving. Or, as Aristotle put it, the change-obsessives’ argument meant that you couldn’t even consider things “as existing.”
This was intolerable for the hard-minded Aristotle. He wanted things to stand still and stay the same long enough to allow him to use reason on them. He was sufficiently generous of mind to admit “that there is some foundation in reason”38 for the dogma of change. But Aristotle wanted to trounce it nonetheless. The result? Aristotle put forth a principal that would remain fundamental to philosophy, mathematics, and logic for the next 2,300 years. Formally it’s called “the law of non-contradiction.”39 Here’s how Aristotle put it in his Metaphysics: “The same attribute cannot at the same time belong and not belong to the same subject and in the same respect.”40
Like much of the language of philosophy, Aristotle’s “law of non- contradiction” was in dire need of simplification. It needed an interpreter with a gift for straight talk. And that’s what it got. Two thousand years later. In the form of a diplomat for the royal family of Hanover, Germany,41 a man who helped Hanover’s George I become king of England in 1714. A man who met in Hanover with Russia’s six-foot-six-inch tsar Peter the Great. A man who also dropped in on the philosopher Spinoza, who became a collaborator with one of the fathers of the wave theory of light, Christian Huygens, and a man who spent time with the father of the microscope, Anton van Leeuwenhoek. In 1673, a hundred years before the American Revolution, this multi-talent was sent on a geopolitical mission to England. While he was there, he showed off a calculating machine he had invented to Britain’s Royal Society, and was promptly made a member. Meanwhile he came up with another breakthrough, calculus. Then he had his reputation smeared by Sir Isaac Newton, who wanted total credit for the invention of calculus for himself.
The man we’re talking about is Gottfried Wilhelm Leibniz. And Leibniz became the great simplifier of Aristotle’s concept of non-contradiction. Aristotle told us that “the same attribute cannot at the same lime belong and not belong to the same subject and in the same respect.”42 That’s a nearly incomprehensible statement. But Leibniz put it a bit more clearly. He came up with “A is A.”43 Either “A” is “A” or it is not “A.”44 There is no Mr. In Between. Much easier to understand. Right?
However you phrase it, Aristotle put the law of non-contradiction, the law of identity—“A is A”—at the very center of his philosophy and at the very heart of something else that Aristotle tried to codify45—logic. Aristotle promoted identity as the most basic and incontrovertible law in this cosmos. Here are a few of the things that the philosopher to beat all philosophers, Aristotle, said about the law of non-contradiction, his precursor of “A is A.” It is “the most certain principle of all things.”46 It is a principle “regarding which it is impossible to be mistaken.” It is “the best known” of all principles. It is not just a guess. It is absolutely “non-hypothetical.” It is “a principle which every one must have who understands anything that is.” Why? Because of all the principles on the planet, this one is the topper, “the most certain of all.” Look, says Aristotle, let’s be frank, “It is impossible for any one to believe the same thing to be and not to be.”47
So what about Heraclitus, whose principles, says Aristotle, seem to imply that opposites can coexist—that a can be “a” and “not a” all at once? Heraclitus’ principles imply that a frog can be a former tadpole, a terrific jumper, and a future doggie dinner all at the same time. They imply that if you looked closely for a week or two, you’d see the frog change before your very eyes. Heraclitus may have said things of this sort, Aristotle says, but “what a man says, he does not necessarily believe.”48 Heraclitus, in Aristotle’s opinion, could not possibly have really felt deep down that “A” is sometimes not “A.” Why? Because it is “impossible for the same man at the same lime to believe the same thing to be and not to be.” Case closed.
Well, not quite entirely closed. Look, says Aristotle, “if a man” was foolish enough to claim that “A” is not “A,” “he would have contrary opinions at the same time.”49 And, says Aristotle, no sensible man would walk around denying his own claims and making himself seem idiotic. Right?
The result? Aristotle says that “A is A,” the law of non-contradiction, is the most fundamental of all the propositions in philosophy and in daily life. It is “the starting point even for all the other axioms.” According to Aristotle’s way of thinking, “A is A” is a notion that we take for granted every time we open our mouth to say, “What are we going to feed this frog? And whose bedroom is it going to sleep in?” (Note that in those two sentences we just took it for granted that the bewildered beast is a frog. And that it will be the same frog no matter who it sleeps with.) “A is A” is an assumption that we take for granted every time we grab the frog and put it back in its shoebox. Our decision to reach out our hand and gently grip a blob of green shows that we believe the frog is actually there. And that the frog we manage to get our hands around is the same frog we’ll see the next morning when we open the shoebox again. What’s more, we assert that “A is A” every time we Google “frog food” and take it for granted that we can feed this frog the same sorts of things that our quick Google shows other amphibian lovers have fed theirs.
“A=A” is fundamental to logic. It is fundamental to mathematics. It is fundamental to science. And it is fundamental to the care and feeding of frogs. But I have sorry news to report. “A=A” is false. It is sometimes a good approximation. But in the end, it’s not one hundred percent true. Why? Because Aristotle was right. But so was Heraclitus. Opposites can be true simultaneously. In fact, they usually are.
It all goes back to location, location, location. It all goes back to differentiation.
Try this bit of reasoning.
A does not equal A because of location. For example, location in time. The letter “a” printed by your computer on a page at 9 a.m. is not the same as the second “a” your printer zips out at 9:01. Electrons have shifted positions in their shells, heat has moved entire empires of molecules around. The lighting of your room has shifted as the Sun has changed position outside your window. The printer-desk on which the “a” rides has moved over seventeen miles around the Earth’s axis, has sped 556 miles around the Sun, and has jack-rabbited thousands or millions of miles around the core of our galaxy. No way are the two “a”s printed at slightly different times the same.
“A” is not simply a shape represented by ink on the mulched and pressed tree pulp we know as paper or on the pixels of a computer screen. And it is not just a logical abstraction. “A” is a complex social interaction. It’s an interaction between your eye and a patch of pixels or an ink-shape. It’s an interaction of your brain with that pixel or ink-shape. And it’s an interaction of the culture embedded in your brain with the squiggles on the screen or on the page. Your culture is the product of 2.5 million years of accumulated thought—the accumulation of insights, emotions, questions, answers, and tools like language. Tools like the alphabet. Tools like a, b, c, and d. Your culture is the product of built-in, instinctual instructions in your brain, instructions like those that linguist Noam Chomsky50 and his pupil Steven Pinker refer to as your linguistic deep structures and your language instinct.51
All these things—neurons, synapses, synaptic senders, synaptic receivers, and the facets of culture in the cloud of your mind, a cloud that shifts from second to second —change between the reading of one “A” and another. Your mind is like Heraclitus’ river. Your mind, in fact, is like nineteenth- century father of psychology William James’ “stream of consciousness,”52 a bubbling, babbling brook. Your mind constantly produces different currents of associations, different swirls of thought, and different moods.
Then there is the change that location makes in the network of relationships that comes to mind around each “A.” The location of each “A” is different in a gestalt, different in a large scale structure. Try this big picture structure to get a feel for how different the mesh of relationships can be mere fractions of an inch apart:
When, in disgrace with fortune and men’s eyes, I all alone beweep my outcast state
And trouble deaf heaven with my bootless cries And look upon myself and curse my fate,
There are twelve a’s in this well-known snippet of Shakespeare. Each one is pronounced differently. That means each “a” has to be tossed from the primary visual cortex in the back of your head to the temporal and frontal lobes up front, where some sense of what in the world it is begins to become clear.53 Then the “a” is thrown to your motor cortex, which figures out how to send a blast of signals to billions of muscle cells in your larynx54 and your tongue so that those muscles can contract and relax in a way that produces a sound that others will recognize as part of a word. Or so that your motor cortex can “say” each “a” silently in your head. That’s a staggering web of relationships. And it is different for each “a” that you pronounce.
Each “a” involves a different team of axons, dendrites, electrons, and muscles. If you speak the lines of Shakespeare out loud, each “a” sets up a different wave-blast in the air, the wave-blast we call sound. And, most important, each “a” has a very different meaning. Take a look at just this super-short phrase, a phrase with two “a”s in very different contexts doing very different jobs:
Small as this phrase is, large-scale structure, big-picture structure, gives each “a” a radically different role. And large scale structure makes each “a” a part of a very different team. The three-letter “all” team makes a very different sound and meaning from the five letter team of “alone.”
Let’s shift “A=A” to physics for a second. A proton = a proton, right? Two protons are identical, n’est-ce pas? Not quite. Like the letter “a” in a Shakespearean sonnet, every proton has a unique place in big picture structures. And that place in the big picture changes the proton’s role in the cosmos. Protons are participants in social processes. And those social processes help generate the radical differences between the swatches of space and the clumps of matter in this universe. In the minutes after the Big Bang, all protons were almost equal. But not quite. Some clumped together in dense zones, zones in which they bounced around, colliding head on and ricocheting at manic speed. Others were just a tad more spread out. And just a tad more leisurely in their crash, smash, slam, and bang. The great UNequalizer was what Nobel Prize winning astrophysicist George Smoot calls “quantum mechanical fluctuations—tiny wrinkles in space-time.”55 Smoot should know. He’s the man who headed the team of one hundred scientists on the COBE project, the cosmic background radiation project that discovered the modern traces of these primordial quantum-mechanical wrinkles, wrinkles that stretched and pinched the space time manifold into a spotty pattern like the patches of color on a spotted cow’s back. Just how different were these patches of newborn space-time from each other? Sufficiently different, in the words of the Department of Energy Office of Science News and Information , to form the “the primordial seed from which, over billions of years, the galaxies and large structures of the present-day universe grew.”56
Let’s go back to our café table at the beginning of the universe. From the Big Bang to roughly 300,000 years ABB (after the Big Bang), protons were part of a hot soup, a plasma. But that plasma surged with pressure waves like a stormy sea. And each proton played a different role. If you were a proton, you might be bunching shoulder to shoulder with other protons to make a peak in the pressure wave. I might be off somewhere doing the opposite, putting distance between myself and my neighbors to make one of the pressure wave’s dips and gullies, one of the pressure wave’s troughs. In addition, you might be participating in the formation of a dense patch of space-time and matter from which a galaxy would eventually grow. And I might be part of the more widely separated slam dance of protons that would someday produce the lacey macramé of empty space between gangs of galaxies. I might be dancing out the early shapes of the spacing pattern that makes the universe on a very large scale look like a lace, like the tracery of a dishwashing detergent foam.
A billion years down the line, you might be surrounded by the spherical surge of a moving electron. You might be the nucleus of a hydrogen atom. And you might be captured by an evolving star, you might be forced to emit light as your electron is excited then is left alone to calm down again, or as your electron is stripped away. Your electron might be turned into a photon that goes on a multi-light-year trip as a kind of sentence, a kind of sonnet— in a very distinct set of frequencies, the unique visual cry of distressed hydrogen.
Meanwhile, if I was a proton, I might be part of a molecule of water, freezing with a mass of my fellow water molecules into a spicule of ice way out in the cold darkness of space.
Both of us would be protons, right? “A is A.” A=A. But we’d each be different. Like the “a”s in a Shakespearian sonnet, we’d play different roles even if we were side by side. Just as you and I can be side by side in a poker game, but each play a different hand, and each play a very different part in the social drama of the night. Big picture structure counts. Your unique place in the social mesh changes your role. So does mine. And big picture structure and positioning in the social mesh are location. Or, to put it in real estate terms, big picture structure and positioning are location, location, location. And in the end, location, location, location gives every fleck and fiber of this cosmos a different role in a massive weave, a massively shifting, changing, and self-upgrading tapestry.
There’s a bit more to this business of my frog is not equal to your frog. Yes, your frog and mine both eat the same kind of food. And they look very much the same. But each one is unique. Each has a different life history and a different future. Each is composed of different molecules of raw material, and each has a different place in your home and mine.
Generalizations about frogs, generalizations that “my frog=your frog,” are extremely useful. Without them the vet would not be able to operate on your frog or mine. Without them, books on the care and feeding of frogs would be useless. But “A=A” is a generalization. It is not precise. It is half a truth. The whole truth? “A is A.” But each “A” is different. Aristotle was right. And so was Heraclitus. Opposites are joined at the hip.
Now let’s go back to thinking like physicists and mathematicians for a second. To simplify things, we will strip away the context of the cosmos, its galaxies, its photon-floods, and its gamma rays. We will strip away the context of language. We will strip away the context of the human brain. We will strip away the passage of time and its impact on the movement of atoms, molecules, the aging of paper, the aging of ink, and the flow of fresh electron messages through the pixels of your laptop, your iPad®, your Kindle®, and your brain.
We will strip away the 3.85 billion years of evolution it took to make a human being. And we will strip away culture and the 2.5 million years or so of evolution it has taken to make language and the use of breath that make the sound of “A” rich in meaning. We’ll ignore your change in moods and associations as you move from reading one word to reading another. And we’ll also strip away the phonetic alphabet and its history and evolution.
If, indeed, there is no cosmos, no evolution, no humans, no culture, and if time stands still or is reversible. then “A” may equal “A”. But without the history of the cosmos, without evolution, without humans, without the brain, and especially without language there is no “A” at all. None!
So “A=A” is a simplification, one so radical that it sometimes utterly distorts reality. It skins reality alive. Is “A=A” useful? Does logic come in handy? Is math a magnificent symbolic system with which to comprehend what’s around us? And is math based on “A=A”? Yes. Absolutely. But math and logic are just that—very, very simplified representations. Symbolic systems with massive powers. But symbolic systems that sometimes do enormous injustice to the richness of that which they attempt to represent. Symbol systems that sometimes do enormous injustice to science’s greatest mystery, cosmic creativity.
One frog is never identical to another frog. And the very same frog is a slightly different frog ten minutes from now. Aristotle and Heraclitus were both right. A equals A. But A does not equal A.
11 Alfred Tarski, Introduction to logic and to the methodology of deductive sciences (New York: Oxford University Press), p. 54. Amita Chatterjee, “Identity Statements,” in Pranab Kumar Sen, ed. Logic, Identity And Consistency: Studies in Philosophical and Non-Standard Logic II (Mumbai: Allied Publishers, 1998).
A.N. Kolmogorov, A. P. Yushkevich, eds., Mathematics of the 19th century: mathematical logic, algebra, number theory, probability theory (Boston: Birkhäuser, 2001) p. 20.
+Heresy Number One: Why A Does Not Equal A
12 Frank Ayres, Schaum’s outline of theory and problems of modern algebra (New York: Macmillan, 1965), p. 71. Seymour Lipschutz, Marc Lars Lipson, 2000 solved problems in discrete mathematics (New York: McGraw-Hill, 1992), p. 19. George Lakoff, Rafael Núñez, Where mathematics comes from: how the embodied mind brings mathematics into Being (New York: Basic Books, 2000), p. 113.
13 Morris Kline , Calculus: an intuitive and physical approach (New York: Wiley, 1967). 14 Maxime Bôcher and Harry Davis Gaylord, Trigonometry (New York: Henry Holt, 1914). Earl William Swokowski, Jeffery Alan Cole, Algebra and trigonometry with analytic geometry (Belmont, CA, Brooks/Cole, 2006). 15 Linus Pauling, Edgar Bright Wilson, Introduction to quantum mechanics: with applications to chemistry (New York: McGraw-Hill, 1935). 16 Walter Greiner, Relativistic quantum mechanics: wave equations (New York: Springer, 2000). 17 Michael William Lutz, Terrence P. Kenakin, Quantitative molecular pharmacology and informatics in drug discovery (West Sussex, UK: Wiley, 1999) p. xiv. 18 Peter Van Zant, Microchip fabrication: a practical guide to semiconductor processing (McGraw-Hill, 1990). 19 Ayn Rand, Atlas shrugged, (New York: Signet, 1992), p. 1016. 20 Barry Mazur, “When is one thing equal to some other thing?” in Bonnie Gold, Roger A. Simons, eds. Proof and other dilemmas: mathematics and philosophy (Washington,DC:MathematicalAssociationofAmerica,2008). 21 Terence Parsons, Indeterminate identity: metaphysics and semantics (Oxford: Oxford University Press, 2000), p. 1. 22 Ibid. 23 Plutarch, Lives of illustrious men, John Dryden et al trans., A. H. Clough, ed. (New York: American Book Exchange, 1881), p. 50. 24 Parsons, Indeterminate identity, pp. 2-4. 25 Bertrand Russell, The principles of mathematics, Volume 1 (Cambridge, UK: Cambridge University Press, 1903), p. 63. 26 Ibid. 27 Brian McGuinness, Wittgenstein: a life : young Ludwig 1889-1921 (Berkeley, CA: University of California Press, 1988), p. 118. 28 Ludwig Wittgenstein and Friedrich Waismann, The Voices of Wittgenstein: The Vienna Circle, Gordon P. Baker ed. (London: Routledge, 2003), p. 73.
+When Is A Frog A River—Aristotle Wrestles Heraclitus
29 Graham Priest, J. C. Beall, Bradley P. Armour-Garb, The law of non- contradiction: new philosophical essays (Oxford: Oxford University Press, 2004). John Watson, An outline of philosophy: with notes, historical and critical. (Glasgow: James Maclehose, 1898), pp. 376-377.
30 Abraham Edel, Aristotle and his philosophy (New Brunswick, NJ: Transaction, 1996), pp. 108-110. 31 Aristotle, The metaphysics of Aristotle, John Henry M’Mahon, trans. (London: George Bell, 1896), p. 101.
32 This quote is reported by Aristotle (Aristotle, Metaphysics, M’Mahon trans., p. 101). For a discussion of Heraclitus’ original words, see: Heraclitus, The art and thought of Heraclitus: an edition of the fragments with translation and commentary, Charles H. Kahn, editor (Cambridge: Cambridge University Press, 1979), p. 168.
33 Aristotle, Metaphysics, M’Mahon, p. 101. 34 Heraclitus, The art and thought of Heraclitus, p. 168. 35 Aristotle, Metaphysics, M’Mahon, pp. 100-101. 36 Ibid. 37 Ibid. 38 Ibid, p. 101. 39 Aristotle, Prior analytics, Robin Smith, trans. (Indianapolis, IN: Hackett, 1989), p. 212. 40 Aristotle, Metaphysics, W. D. Ross, trans. (Lawrence, KA: Digireads.com, 2006), p. 35. Edel, Aristotle and his philosophy p. 108-110.
41 Leibniz worked for the House of Brunswick.
42 Aristotle, Metaphysics, W. D. Ross trans. p. 35.
43 Gottfried Wilhelm Leibniz, Logical papers By Gottfried Wilhelm Leibniz,
George Henry Radcliffe Parkinson, ed. (Oxford: Oxford University Press, 1966), p. 82. Gottfried Wilhelm Leibniz, Metaphysics and its foundations Volume I, R. S. Woolhouse ed., (London: Routledge, 1994), p. 297. 44 Or, as Leibniz put it, “Everything is what it is.” Gottfried Wilhelm Leibniz, Discourse on metaphysics and related writings, R. Niall Martin, Stuart Brown, eds. (Manchester, UK: Manchester University Press, 1988), p. 132. 45 Augusto Vera, Introduction to speculative logic and philosophy (St. Louis, MO: Gray, Baker, 1875), p. 16. 46 Aristotle, Metaphysics, Ross, p. 35. 47 Aristotle, Metaphysics, Ross, p. 35. 48 Ibid. 49 Ibid. 50 Noam Chomsky, Language And Mind (New York: Cambridge University Press, 2006), p. 144.
51 Steven Pinker, The Language Instinct: How the Mind Creates Language, (New York: William Morrow, 1994). 52 William James, Text-book of psychology (London: Macmillan, 1892), p. 151.
53 Stanislas Dehaene, Reading in the brain: the science and evolution of a human invention (New York: Penguin, 2009). 54 Andrew Blitzer, Mitchell F. Brin, Lorraine O. Ramig, Neurologic disorders of the larynx (New York: Thieme, 2009), p. 18. 55 George Smoot, Keay Davidson, Wrinkles in time: witness to the birth of the universe (New York: HarperCollins, 2007), p. 187. 56 U.S. Department of Energy, Office of Science, “Detecting the Afterglow of the Big Bang Anisotropy in the cosmic microwave background radiation,” http://www.er.doe.gov/accomplishments_awards/Decades_Discovery/41.ht ml (accessed August 13, 2010).