A=A is a rock on which the modern mind stands. A=A was fundamental to Aristotle. It is fundamental to logic. It is fundamental to algebra and to mathematics. It is fundamental to science. And it's even been turned into a religious slogan by the Ayn Randians. But I have sorry news to report. a=a is a false statement.

Try this bit of reasoning. If A=a, then a=A, right? Right mathematically, but not in the world of reality. There's a big difference between a big A and a small a. Size, for one thing. Shape, too. Not to mention that you read each at a different time. Which brings up the biggest difference-generator of all: time. A=A, right? Not quite. Each A is printed separately. Each "a" shows up in different territories on a piece of paper or on a computer screen. Each is composed of different stuff-slightly different inks, for example. Each is processed by your eye just a bit differently. If nothing else, the mood that wraps itself around your perception of each "a" is just a wee bit different. It may even be very different indeed. Sometimes we flick in microseconds from one feeling to another and from concentration on the girl at the next desk's hand or eye or breast to a sense of failure when we realize she's too good for us or a sense of triumph when her eyes meet ours and reveal a sense of welcome. Each "a" shows up in another emotional context.

A simply does not equal A. Why? Again, because of time. The letter "a" printed on a page at 9am is not the same as the second "a" your printer taps out at 9:01. Electrons have shifted positions in their shells, heat has moved entire empires of molecules about, the lighting of the room has shifted just a tad as the sun has changed position outside the window, the printer-desk on which the "a" rides has moved over 17 miles around the earth's axis, has sped 556 miles around the sun, and has jackrabbited 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 monitor. A is a complex social interaction. It's an interaction between the eye and that pixel or ink-shape. It's an interaction of the brain with that pixel or ink-shape. It's an interaction of the culture embedded in that brain and the shape on the screen or the page. The culture is the product of at least 35,000 years of accumulated thought--the accumulation of insights, emotions, questions, answers, and tools like language. It's the product of Lord-knows what built-in, instinctual instructions in the brain, instructions like those of linguistic deep structures.

All these things--neurons, synapses, synaptic senders and synaptic receivers--change between the reading of one a and another. Two a's on a page can be radically different because of their context--and there are myriad contexts to consider...from the electron level to the level of the cosmos. But try this context as just one example:

When, in disgrace with fortune and men's eyes,
I all alone beweep my outcast state

There are four a's in this snippet of Shakespeare. Each one is pronounced differently. That means each has to be tossed from the right brain, which recognizes words, to the left brain, which figures out how to turn the sight of a word into speech. Then it's thrown to the motor cortex, which has to send a blast of signals to millions of muscle cells in the larynx and the tongue so that those muscles can contract and relax in a way that produces a sound others will recognize as part of a word. Each "a" involves a different team of neurons and muscles. Each "a" sets up a different wave-blast in the air, the wave-blast we call sound. Each "a" has a very different meaning.

Now let's think 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 computer monitor pixels.

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 two million years or so of evolution it has taken to make language and the use of breath to make "a"-sounds rich in meaning. We will 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=a. But without the history of a one-way 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 utterly distorts reality. Is it useful? Is math a neat symbolic system with which to comprehend what's around us? Yes. Absolutely. But it is just that--a very, very simplified representation, a symbolic system that does enormous injustice to the richness of that which it attempts to represent.

Niels Bohr--the man who kidnapped quantum physics from Max Plank-- said that anything that can't be measured can't exist. Here's a quote from a book I highly recommend, John Joe McFadden. Quantum Evolution: The New Science of Life. New York: W.W. Norton & Company, 2001 (pp. 151-152): "When quizzed by a fellow physicist on what state a particle must be in, Bohr replied in exasperation, 'Be! Be! What is this be?' Bohr claimed that an 'independent reality in the ordinary physical sense can neither be ascribed to a phenomena nor to the agencies of observation."

Bohr was a reality-thief who insisted that one and only one of the many symbol sets we use to comprehend our world is real. And that symbol set--math--is so primal that no other reality exists. We have no measurements for the human feelings. We have no math for the most meaningful thoughts that pass through our brain. Measured by fmri brain-imaging, our pleasures and despairs are simply hot spots on a screen--not thoughts, not the desperate sense that:

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,
Wishing me like to one more rich in hope,
Featur'd like him, like him with friends possess'd,
Desiring this man's art and that man's scope,
With what I most enjoy contented least;
Yet in these thoughts myself almost despising.

Measured by electroencephalograms, these feelings are mere waves, not meaningful sensations. Since there were no electroencephalographs and fmri's to measure William Shakespeare's ponderings, did those ponderings exist? No. Why? Because the sonnet above was written in a symbol system that doesn't exist. It is English, it is poetry, it is language, but it is not math.

Does the letter a exist when you banish time, evolution, human culture, and the desperate dependence of humans on each other that Shakespeare brilliantly describes? Are there a's without time, mind, society, and eyes? No. In the simplest case, the case where math's the sole reality, a cannot =a because there are no a's at all.

***

In most theories of physics time is reversible because we use math to calculate time. In math, if a=2b then 2b=a. You can flip the equation either way. In other words, the equation is reversible. But that's math in a perfect world, and this world is anything but perfect. So is math missing something?

Science works with rough approximations. Physics works by simplifying things so they'll fit our math. A may equal A in algebra or logic, but no way is the slipstream between the lines of an equal sign a two-way route in the real world. There's a one-way valve hidden in the pipeline of an "=". Time and space work hand in hand to guarantee that a ain't a. And time's the valve that makes an "=" equal an arrow, Eh?

Which means that logic isn't as logical as it seems. It rules the actual stuffing out of our reality.

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