THINK ABOUT THIS:::

If computing speeds double every two years,
what happens when computer-based AIs are doing the research?
Computing speed doubles every two years.
Computing speed doubles every two years of work.
Computing speed doubles every two subjective years of work.

Two years after Artificial Intelligences reach human equivalence, their speed doubles.
One year later, their speed doubles again.

Six months - three months - 1.5 months ...
Singularity.

Plug in the numbers for current computing speeds, the current doubling time, and an estimate for the raw processing power of the human brain, and the numbers match in: 2021.

But personally, I'd like to do it sooner.

| The End of History |
It began three and a half billion years ago in a pool of muck, when a molecule made a copy of itself and so became the ultimate ancestor of all earthly life.

It began four million years ago, when brain volumes began climbing rapidly in the hominid line.

Fifty thousand years ago with the rise of homo sapiens. Ten thousand years ago with the invention of civilization. Five hundred years ago with the invention of the printing press. Fifty years ago with the invention of the computer.

In less than thirty years, it will end.

Vernor Vinge saw it first. At some point in the near future, someone will come up with a method of increasing the maximum intelligence on the planet - either coding a true Artificial Intelligence or enhancing human intelligence. An enhanced human would be better at thinking up ways of enhancing humans; would have an "increased capacity for invention". What would this increased ability be directed at? Why, creating the next generation of enhanced humans, of course.

And what would those doubly enhanced minds do? Research methods on triply enhanced humans, or build AI minds operating at computer speeds. And an AI would be able to reprogram itself, directly, to run faster - or smarter. And then our crystal ball explodes, "life as we know it" is over, and everything we know goes out the window.

"Here I had tried a straightforward extrapolation of technology, and found myself precipitated over an abyss. It's a problem we face every time we consider the creation of intelligences greater than our own. When this happens, human history will have reached a kind of singularity - a place where extrapolation breaks down and new models must be applied - and the world will pass beyond our understanding."
-- Vernor Vinge, True Names and Other Dangers, p. 47.

There are multiple paths to the Singularity. Nanotechnology - the ability to build computers atom by atom and rewire brains neuron by neuron. Artificial Intelligence, self-understanding and self-enhancing seed AI. We could bootstrap our way to the Singularity via the relatively mild enhanced humans produced by neurohacking. Direct neuron-to-silicon interfaces could improve human intelligence or computer intelligence or both. Or some completely unanticipated breakthrough could occur. A civilization with high technology is unstable; it ends when the species destroys itself or improves on itself. If the current trends continue - if we don't run up against some unexpected theoretical cap on intelligence, or turn the Earth into a radioactive wasteland, or bury the planet under a tidal wave of voracious self-reproducing nanodevices - the Singularity is inevitable. The most-quoted estimate for the Singularity is 2035 - within your lifetime! - although many, including I, think that the Singularity may occur substantially sooner.

Some terminology, due to Vinge's Hugo-winning A Fire Upon The Deep:

Power - An entity from beyond the Singularity. Transcend, Transcended, Transcendence - The act of reprogramming oneself to be smarter, reprogramming (with one's new intelligence) to be smarter still, and so on ad Singularitum. Also the metaphorical area where the Powers live, or belonging to that area. Beyond - The grey area between being human and being a Power; the domain inhabited by entities smarter than human, but not possessing the technology to reprogram themselves directly and Transcend.

| The Beyondness of the Singularity |
"I imagine bugs and girls have a dim perception that Nature played a cruel trick on them, but they lack the intelligence to really comprehend its magnitude."
-- Calvin

But why should the Powers be so much more than we are now? Why not assume that we'll get a little smarter and that's it?

Consider the sequence 1, 2, 4, 8, 16, 32. Consider the iteration of F(x) = (x + x). Every couple of years, computer performance doubles. (1) That is the proven rate of improvement as overseen by constant, unenhanced minds, progress according to mortals.

Right now the amount of computing power on the planet is equal to the power of a human brain - 10^17 ops/sec, or one hundred million billion operations per second (2) - multiplied by the number of humans, presently six billion. The amount of artificial computing power is so small as to be irrelevant, not because there are so many humans, but because of the sheer raw power of a human brain.

At the old rate of progress, when the original Singularity calculations were performed, computers were expected to reach human-equivalent levels - 10^17 floating-point operations per second, or one hundred petaflops - at around 2035. But at that rate of progress, one-teraflops machines were expected in 2000; in actual fact, one-teraflops machines were around in 1996, when this document was first written. In 1998 the top speed was 3.2 teraflops, and in 1999 IBM announced the Blue Gene project to build a petaflops machine by 2005. So the old estimates may be a little conservative.

Once we have human-equivalent computers, the amount of computing power on the planet is equal to the number of humans plus the number of computers. The amount of intelligence available takes a huge jump. Ten years later, humans become a vanishing quantity in the equation.

That doubling sequence is actually a pessimistic projection, because it assumes that computing power continues to double at the same rate. But why? Computer speeds don't double due to some inexorable physical law, but because researchers and engineers find ways to make faster chips. If some of the researchers and engineers are themselves computers...

A group of human-equivalent computers spends 2 years to double computer speeds. Then they spend another 2 subjective years, or 1 year in human terms, to double it again. Then they spend another 2 subjective years, or six months, to double it again. After four years total, the computing power goes to infinity.

That is the "Transcended" version of the doubling sequence. A Transcended version of a sequence {a0, a1, a2...} is a function where the interval between an and an+1 is inversely proportional to an. (If there's a pre-existing mathematical term for this, let me know.) So a Transcended doubling function starts with 1, in which case it takes 1 time-unit to go to 2. Then it takes 1/2 time-units to go to 4. Then it takes 1/4 time-units to go to 8. This function, if it were continuous, would be the hyperbolic function y = 2/(2 - x). When x = 2, (2 - x) = 0 and y = infinity. The behavior at that point is known mathematically as a singularity.

And the Transcended doubling sequence is also a pessimistic projection, not a Singularity at all, because it assumes that only speed is enhanced. What if the quality of thought were enhanced? Right now, two years of work - well, these days, eighteen months of work. Eighteen subjective months of work suffices to double computing speeds. Shouldn't this improve a bit with thought-sharing and eidetic memories? Shouldn't this improve if, say, the total sum of human scientific knowledge is stored in predigested, cognitive, ready-to-think format? Shouldn't this improve with short-term memories capable of holding the whole of human knowledge? A human-equivalent AI isn't merely "equivalent" - if Kasparov had had even the smallest, meanest automatic chess-playing program integrated solidly with his intuitions, he would have beat Deep Blue into a pulp. That's The AI Advantage: Simple tasks carried out at blinding speeds and without error, conscious tasks carried out with perfect memory and total self-awareness.

I haven't even started on the subject of AIs redesigning their cognitive architectures, although they'll have a far easier time of it - especially if they can make backups. Transcended doubling might run up against the laws of physics before reaching infinity... but even the laws of physics as now understood would allow one gram (more or less) to store and run the entire human race at a million subjective years per second. (3).

Let's take a deep breath and think about that for a moment. One gram. The entire human race. One million years per second. That means, using only this planetary mass for computing power, it would be possible to support more people than the entire Universe could support if biological humans colonized every single planet. It means that, in a single day, a civilization could live over 80 billion years, several times older than the age of the Universe to date.

The peculiar thing is that most people who talk about "the laws of physics" setting hard limits on Powers would never even dream of setting the same limits on a (merely) galaxy-spanning civilization of (normal) humans a (brief) billion years old. Part of that is simply a cultural convention of science fiction; interstellar civilizations can break any physical law they please, because the readers are used to it. But part of that is because scientists and science-fiction authors have been taught, so many times, that Ultimate Unbreakable Limits usually fall to human ingenuity and a few generations of time. Powered flight, faster-than-sound, space travel - all proved impossible.

We know that change crept at a snail's pace a mere millennium ago, and that even a hundred years ago it would have been impossible to place correct limits on the ultimate power of technology. We know that the past could never have placed limits on the present, and so we don't try to place limits on the future. But with transhumans, the analogy is not to Lord Kelvin, nor Aristotle, nor to a hunter-gatherer - all of whom had human intelligence - but to a Neanderthal. With Powers, to a fish. And yet, because the power of higher intelligence is not as publicly recognized as the power of a few million years - because we have no history of naysayers being embarassed by transhumans instead of mere time - some of us still sit, grunting around the fire, setting ultimate limits on the sharpness of spears; some of us still swim about, unblinking, unable to engage in abstract thought, but knowing that the entire Universe is, must be, wet.

To convey the rate of progress driven by smarter researchers, I needed to invent a function more complex than the doubling function used above. We'll call this new function T(n). You can think of T(n) as representing the largest number conceivable to someone with an n-neuron brain. More formally, T(n) is defined as the longest block of 1s produced by any halting n-state Turing Machine acting on an initially blank tape. If you're familiar with computers but not Turing Machines, consider T(n) to be the largest number that can be produced by a computer program with n instructions. Or, if you're an information theorist, think of T(n) as the inverse function of complexity; it produces the largest number with complexity n or less.

The sequence produced by iterating T(n), S{n} = T(S{n - 1}), is constant for very low values of n. S{0} is defined to be 0; a program of length zero produces no output. This corresponds to a Universe empty of intelligence. T(1) = 1. This corresponds to an intelligence not capable of enhancing itself; this corresponds to where we are now. T(2) = 3. Here begins the leap into the Abyss. Once this function increases at all, it immediately tapdances off the brink of the knowable. T(3) = 6? T(6) = 64?

T(64) = vastly more than 1080, the number of atoms in the Universe. T(1080) is something that only a Transcendent entity will ever be able to calculate, and that only if Transcendent entities can create new Universes, maybe even new laws of physics, to supply the necessary computing power. Even T(64) will probably never be known to any strictly human being.

Now take the Transcended version of S{n}, starting at 2. Half a time-unit later, we have 3. A third of a time-unit after that, 6. A sixth later - one whole unit after this function started - we have 64. A sixty-fourth later, 10^80. An unimaginably tiny fraction of a second later... Singularity.

Is S{n} really a good model of the Singularity? Of course not. "Good model of the Singularity" is an oxymoron; that's the whole point; the Singularity will outrun any model a human could have formulated a hundred years ago, and the Singularity will outrun any model we formulate. Also, if we wanted a function that really modeled the way things are, T(10^17), or T(human), should presently equal 10^12, or the power of a computer, and S{n} should equal S{n-1} + T(S{n -1}).

The main objection, though, would be that S{n} is an ungrounded metaphor. The Transcended doubling sequence models faster researchers. It's easy to say that S{n} models smarter researchers, but what does smarter actually mean in this context?

| The Definition of Smartness |
Smartness is the measure of what you see as obvious, what you can see as obvious in retrospect, what you can invent, and what you can comprehend. To be more precise about it, smartness is the measure of your semantic primitives (what is simple in retrospect), the way in which you manipulate the semantic primitives (what is obvious), the structures your semantic primitives can form (what you can comprehend), and the way you can manipulate those structures (what you can invent). If you speak complexity theory, the difference between obvious and obvious in retrospect, or inventable and comprehensible, is like the difference between NP and P.

All humans who have not suffered neural injuries have the same semantic primitives. What is obvious in retrospect to one is obvious in retrospect to all. (Four notes: First, by "neural injuries" I do not mean anything derogatory - it's just that a person missing the visual cortex will not have visual semantic primitives. If certain neural pathways are severed, people not only lose their ability to see colors; they lose their ability to remember or imagine colors. Second, theorems in math may be obvious in retrospect only to mathematicians - but anyone else who acquired the skill would have the ability to see it. Third, to some extent what we speak of as obvious involves not just the symbolic primitives but very short links between them. I am counting the primitive link types as being included under "semantic primitives". When we look at a thought-sequence and see it as being obvious in retrospect, it is not necessarily a single semantic primitive, but is composed of a very short chain of semantic primitives and link types. Fourth, I apologize for my tendency to dissect my own metaphors; I really can't help it.)

Similarly, the human cognitive architecture is universal. We all have the same sorts of symbolic structures. The nature of these structures is not known, no more than we know what symbols are made of, but our ability to communicate with each other indicates that, whatever we are communicating, it is the same on both sides. If any two humans share a set of symbols, any structure composed of those symbols that is understood by one will be understood by the other.

Different humans may have different degrees of the ability to manipulate and structure symbols; different humans may see and invent different things. The great breakthroughs of physics and engineering did not occur because a group of people plodded and plodded and plodded for generations until they found an explanation so complex, a string of ideas so long, that only time could invent it. Relativity and quantum physics and buckyballs and object-oriented programming all happened because someone put together a short, simple, elegant semantic structure in a way that nobody had ever thought of before. Being a little bit smarter is where revolutions come from. Not time. Not hard work; although hard work was usually necessary, others had worked far harder without result. Raw smartness.

Now think about the Singularity. Think about a chimpanzee trying to understand integral calculus. Think about the people with damaged visual neurology who cannot remember what it was like to see, who cannot imagine the color red or visualize two-dimensional structures. Think about a visual cortex with trillions of times as many neuron-equivalents. Think about twenty thousand distinct colors in the rainbow, none a shade of any other. Think about rotating fifty-dimensional objects. Think about attaching semantic primitives to the pixels, so that one could see a rainbow of ideas in the same way that we see a rainbow of colors.

Our semantic primitives even determine what we can know. Why does anything exist at all? Nobody knows. And yet the answer is obvious. The First Cause must be obvious. It has to be obvious to Nothing, present in the absence of anything else, a substance formed from -blank-, a conclusion derived without data or initial assumptions. What is it that evokes conscious experience, the stuff that souls are made of? We are made of conscious experiences. There is nothing we experience more directly. How does it work? We don't have a clue. Two and a half millennia of trying to solve it and nothing to show for it but "I think therefore I am." The solutions operate outside the representations that can be formed with the human brain.

Our descendants, successors, future selves will figure out the semantic primitives necessary and alter themselves to perceive them. The Powers will dissect the Universe and the Reality until they understand why anything exists at all, analyze neurons until they understand qualia. And that will only be the beginning. It won't end there. Why should there be only three hard problems? After all, if not for humans, the Universe would apparently contain only one or two hard problems, for how could a non-conscious thinker formulate the hard problem of consciousness? Might there be states of existence beyond mere consciousness - transsentience? Might solving the nature of reality create the ability to create new Universes, manipulate the laws of physics, even alter the kind of things that can be real - ontotechnology? That's what the Singularity is all about.

So before you talk about life as a Power or the Utopia to come - a favorite pastime of transhumanists and Extropians is to discuss the problems of uploading, life after being uploaded, and so on - just remember that you probably have a much better chance of solving all three hard problems than you do of making a valid statement about the future. This goes for me too. I'll stand by everything I said about humans, including our inability to understand certain things, but everything I said about the Powers is almost certainly wrong. "They'll figure out the semantic primitives necessary and alter themselves to perceive them." Wrong. "Figure out." "Semantic primitives." "Alter." "Perceive." I would bet on all of these terms becoming obsolete after the Singularity. There are better ways and I'm sure They - or It, or [sound of exploding brain] will "find them".

| Perceptual Transcend |
I would like to introduce a unit of post-Singularity progress, the Perceptual Transcend or PT.

[Brief pause while audience collapses in helpless laughter.]

I'm not trying to get it right, just make a point.

A Perceptual Transcend occurs when all things that were comprehensible become obvious in retrospect, and all things that were inventable become obvious. A Perceptual Transcend occurs when the semantic structures of one generation become the semantic primitives of the next. To put it another way, one PT from now, the whole of human knowledge becomes perceiveable in a single flash of experience, in the same way that we now perceive an entire picture at once.

Computers are a PT above humans when it comes to arithmetic - sort of. While we need to manipulate an entire precarious pyramid of digits, rows and columns in order to multiply 62305 by 10358, a computer can spit out the answer - 645355190 - in a single obvious step. These computers aren't actually a PT above us at all, for two reasons. First of all, they just handle numbers up to two billion instead of 9; after that they need to manipulate pyramids too. Far more importantly, they don't notice anything about the numbers they manipulate, as humans do. If you multiply 23704 by 14223, using the wedding-cake method of multiplication, you won't multiply 23704 by 2 twice in a row; you'll just steal the results from last time. If one of the interim results is 12345 or 99999 or 314159, you'll notice that, too. The way computers manipulate numbers is actually less powerful than the way we manipulate numbers.

Would the Powers settle for less? A PT above us, multiplication is carried out automatically but with full attention to interim results, numbers that happen to be prime, and the like. If I were designing one of the first Powers [and I am - '99], I would create an entire subsystem for manipulating numbers, one that would pick up on primality, complexity, and all the numeric properties known to humanity. A Power would understand why 62305 times 10358 equals 645355190, with the same understanding that would be achieved by a top human mathematician who spent hours studying all the numbers involved. And at the same time, the Power will multiply the two numbers automatically.

For such a Power, to whom numbers were true semantic primitives, Fermat's Last Theorem and the Goldbach Conjecture and the Riemann Hypothesis might be obvious. Somewhere in the back of its mind, the Power would test each statement with a million trials, subconsciously manipulating all the numbers involved to find why they were not the sum of two cubes or why they were the sum of two primes or why their real part was equal to one-half. From there, the Power could intuit the most basic, simple solution simply by generalizing. Perhaps human mathematicians, if they could perform the arithmetic for a thousand trials of the Riemann Hypothesis, examining every intermediate step, looking for common properties and interesting shortcuts, could intuit a formal solution. But they can't, and they certainly can't do it subconsciously, which is why the Riemann Hypothesis remains unobvious and unproven - it is a conceptual structure instead of a conceptual primitive.

Perhaps an even more thought-provoking example is provided by our visual cortex. On the surface, the visual cortex seems to be an image processor. In a modern computer graphics engine, an image is represented by a two-dimensional array of pixels (4). To rotate this image - to cite one operation - each pixel's rectangular coordinates {x, y} are converted to polar coordinates {theta, r}. All thetas, representing the angle, have a constant added. The polar coordinates are then converted back to rectangular. There are ways to optimize this process, and ways to account for intersecting and empty pixels on the new array, but the essence is clear: To perform an operation on an entire picture, perform the operation on each pixel in that picture.

At this point, one could say that a Perceptual Transcend depends on what level you're looking at the operation. If you view yourself as carrying out the operation pixel by pixel, it is an unimaginably tedious cognitive structure, but if you view the whole thing in a single lump, it is a cognitive primitive - a point made in Hofstadter's Ant Fugue when discussing ants and colonies. Not very exciting unless it's Hofstadter explaining it, but there's more to the visual cortex than that.

For one thing, we consciously experience redness. (If you're not sure what conscious experience a.k.a. "qualia" means, the short version is that you are not the one who speaks your thoughts, you are the one who hears your thoughts.) Qualia are the stuff making up the indescribable difference between red and green.

The term "semantic primitive" describes more than just the level at which symbols are discrete, compact objects. It describes the level of conscious perception. Unlike the computer manipulating numbers formed of bits, and like the imagined Power manipulating theorems formed of numbers, we don't lose any resolution in passing from the pixel level to the picture level. We don't suddenly perceive the idea "there is a bear in front of me"; we see a picture of a bear, containing millions of pixels, every one of which is consciously experienced simultaneously. A Perceptual Transcend isn't "just" the imposition of a new cognitive level; it turns the cognitive structures into consciously experienced primitives.

"To put it another way, one PT from now, the whole of human knowledge becomes perceiveable in a single flash of experience, in the same way that we now perceive an entire picture at once."

Of course, the PT won't be used as a post-Singularity unit of progress. Even if it were initially, it won't be too long before "PT" itself is Transcended and the Powers jump out of the system yet again. I exerted all my ability to write an even briefly plausible description of progress beyond the Singularity, and yet the Singularity is as far beyond me as it is beyond any other human, and my PTs will be as worthless a description as the doubling sequence discarded so long ago. Even if we accept the PT as the basic unit of measure, it simply introduces a secondary Singularity. Maybe the Perceptual Transcends will occur every two consciously experienced years at first, but then will occur every conscious year, and then every conscious six months - get the picture?

It's like the "Birthday Cantatatata..." in Hofstadter's book Godel, Escher, Bach. You can start with the sequence {1, 2, 3, 4 ...} and jump out of it to w (omega), the symbol for infinity. But then one has {w, w + 1, w + 2 ... }, and we jump out again to 2w. Then 3w, and 4w, and w2 and w3 and ww and w^(ww) and higher towers of w until we jump out to the ordinal e0, which includes all exponential towers of ws.

The PTs may introduce a second Singularity, and a third Singularity, and a fourth, until Singularities are coming faster and faster and the first w-Singularity is imminent -

Or the Powers may simply jump beyond that system. The Birthday Cantatatata... was written by a human - admittedly Douglas Hofstadter, but still a human - and the concepts involved in it may be Transcended by the very first transhuman.

The Powers are beyond our ability to comprehend.

Get the picture?

| Great Big Numbers |
It's hard to appreciate the Singularity properly without first appreciating really large numbers. I'm not talking about little tiny numbers, barely distinguishable from zero, like the number of atoms in the Universe or the number of years it would take a monkey to duplicate the works of Shakespeare. I invite you to consider what was, circa 1977, the largest number ever to be used in a serious mathematical proof. The proof, by Ronald L. Graham, is an upper bound to a certain question of Ramsey theory. In order to explain the proof, one must introduce a new notation, due to Donald E. Knuth in the article Coping With Finiteness. The notation is usually a small arrow, pointing upwards, here abbreviated as ^. Written as a function:

int arrow (int num, int power, int arrownum) { int answer = num; if (arrownum == 0) return num * power; for (int i = 1; i < power; i++) answer = arrow(num, answer, arrownum - 1); return answer; } // end arrow 2^4 = 24 = 16. 3^^4 = 3^(3^(3^3)).

7^^^^3 = 7^^^(7^^^7).

3^3 = 3 * 3 * 3 = 27. This number is small enough to visualize.

3^^3 = 3^(3^3) = 3^27 = 7,625,597,484,987. Larger than 27, but so small I can actually type it. Nobody can visualize seven trillion of anything, but we can easily understand it as being on roughly the same order as, say, the gross national product.

3^^^3 = 3^^(3^^3) = 3^(3^(3^(3^...^(3^3)...))). The "..." is 7,625,597,484,987 threes long. In other words, 3^^^3 or arrow(3, 3, 3) is an exponential tower of threes 7,625,597,484,987 levels high. The number is now beyond the human ability to understand, but the procedure for producing it can be visualized. You take x=1. You let x equal 3^x. Repeat seven trillion times. While the very first stages of the number are far too large to be contained in the entire Universe, the exponential tower, written as "3^3^3^3...^3", is still so small that it could be stored on a modern supercomputer.

3^^^^3 = 3^^^(3^^^3) = 3^^(3^^(3^^...^^(3^^3)...)). Both the number and the procedure for producing it are now beyond human visualization, although the procedure can be understood. Take a number x=1. Let x equal an exponential tower of threes of height x. Repeat 3^^^3 times, where 3^^^3 equals an exponential tower seven trillion threes high.

And yet, in the words of Martin Gardner: "3^^^^3 is unimaginably larger than 3^^^3, but it is still small as finite numbers go, since most finite numbers are very much larger."

And now, Graham's number. Let x equal 3^^^^3, or the unimaginable number just described above. Let x equal 3^^^^^^^(x arrows)^^^^^^^3. Repeat 63 times, or 64 including the starting 3^^^^3.

Graham's number is far beyond my ability to grasp. I can describe it, but I cannot properly appreciate it. (Perhaps Graham can appreciate it, having written a mathematical proof that uses it.) This number is far larger than most people's conception of infinity. I know that it was larger than mine. My sense of awe when I first encountered this number was beyond words. It was the sense of looking upon something so much larger than the world inside my head that my conception of the Universe was shattered and rebuilt to fit. It felt as I imagine a mountain looks to people who don't appreciate nuclear weapons: Something forever beyond us to subdue. All theologians should face a number like that, so they can properly appreciate God. My happiness was completed when I learned that the actual answer to the Ramsey problem that gave birth to that number - rather than the upper bound - was probably six.

Why was all of this necessary, mathematical aesthetics aside? Because until you understand the hollowness of the words "infinity", "large" and "transhuman", you cannot appreciate the Singularity. You must know that even appreciating the Singularity is as far beyond us as visualizing that number is to a chimpanzee. Farther beyond us than that. No human analogies will ever be able to describe the Singularity, because we are only human.

The number above was forged of the human mind. It is nothing but a finite positive integer, though a large one. It is composite and odd, rather than prime or even; it is perfectly divisible by three. Encoded in the decimal digits of that number, by almost any encoding scheme one cares to name, are all the works ever written by the human hand, and all the works that could have been written, at a hundred thousand words per minute, over the age of the Universe raised to its own power a thousand times. And yet, if we add up all the base-ten digits the result will be divisible by nine. The number is still a finite positive integer. It may contain Universes unimaginably larger than this one, but it is still only a number. It is a number so small that the algorithm to produce it can be held in a single human mind.

The Singularity is beyond that. We cannot pigeonhole it by stating that it will be a finite positive integer. We cannot say anything at all about it, except that it will be beyond our understanding.

If you thought that Knuth's arrow notation produced some fairly large numbers, what about T(n)? How many states does a Turing machine need to implement the calculation above? What is the complexity of Graham's number, C(Graham)? Probably on the order of 100. And moreover, T(C(Graham)) is likely to be much, much larger than Graham's number. Why go through x = 3^(x ^s)^3 only 64 times? Why not 3^^^^3 times? That'd probably be easier, since we already need to generate 3^^^^3, but not 64. And with the extra space, we might even be able to introduce an even more computationally complex algorithm. In fact, Knuth's arrow notation may not be the most powerful algorithm that fits into C(Knuth) states.

T(n) is the metaphor for the growth rate of a self-enhancing entity because it conveys the concept of having additional intelligence with which to enhance oneself. I don't know when T(n) passes beyond the threshold of what human mathematicians can, in theory, calculate. Probably more than n=10 and less than n=100. The point is that after a few iterations, we wind up with T(4294967296). Now, I don't know what T(4294967296) will be equal to, but the winning Turing machine will probably generate a Power whose purpose is to think of a really large number. That's what the term "large" means.

| Smarter Than We Are |
It's all very well to talk about cognitive primitives and obviousness, but again - what does smarter mean? The meaning of smart can't be grounded in the Singularity - I haven't been there yet. So what's my practical definition?

"The toughest challenge for a writer is a character brighter than the author. It's not impossible. Puzzles the writer needs months to solve, or to design, the character may solve in moments. But God help the writer if his abnormally bright character is wrong!"
-- Larry Niven

"Of course, I never wrote the 'important' story, the sequel about the first amplified human. Once I tried something similar. John Campbell's letter of rejection began: 'Sorry - you can't write this story. Neither can anyone else.'"
-- Vernor Vinge

Smartness is that quality which makes it impossible to write a story about a character smarter than you are. You can write about super-fast thinkers, eidetic memories, lightning calculators; characters who learned a dozen languages in a week, who can read a textbook in an hour, or who can invent all kinds of wonderful stuff - as long as you don't have to produce the actual invention. But you can't write a character with a higher level of emotional maturity, a character who can spot the obvious solution you missed, a character who knows (and can tell the reader) the Meaning Of Life, a character with superhuman self-awareness. Not unless you can do these things yourself.

Let's take a concrete example, the story Flowers for Algernon (later the movie Charly), by Daniel Keyes. (I'm afraid I'll have to tell you how the story comes out, but it's a Character story, not an Idea story, so that shouldn't spoil it.) Flowers for Algernon is about a neurosurgical procedure for intelligence enhancement. This procedure was first tested on a mouse, Algernon, and later on a retarded human, Charlie Gordon. The enhanced Charlie had the standard science-fictional set of superhuman characteristics; he thought fast, learned a lifetime of knowledge in a few weeks, and discussed arcane mathematics (not shown). Then the mouse, Algernon, gets sick and dies. Charlie analyzes the enhancement procedure (not shown) and concludes that the process is basically flawed. Later, Charlie dies.

That's a science-fictional enhanced human. A real enhanced human, of course, would not have been taken by surprise. A real enhanced human would realize that any simple intelligence enhancement will be a net evolutionary disadvantage - if enhancing intelligence were a matter of a simple surgical procedure, it would have long ago occurred as a natural mutation. This goes double for a procedure that works on rats! (As far as I know, this never occurred to Keyes. I selected Flowers, out of all the famous stories of intelligence enhancement, because, for reasons of dramatic unity, this story shows what happens to be the correct outcome.)

Note that I didn't dazzle you with an abstruse technobabble explanation for Charlie's death; my explanation is two sentences long and can be understood by someone who isn't an expert in the field. That's the difference between a fictional genius and an actual enhanced human such as myself. I wouldn't have been taken by surprise, and for that matter, I wouldn't have been that dramatically upset if I had. Normal author, no enhancement. Real enhancement, no story.

Do I hear the audience demanding an explanation? Well, a full explanation is elsewhere; the brief version is that I am a Specialist, with a neurological perturbation (best guess on location: pathway from the right mammillary body or amygdala) that indirectly resulted in cognitive resources being over-allocated to a few favored abilities, including causal analysis and combinatorial design. This is a net evolutionary disadvantage, in accordance with Algernon's Law, since an evenly balanced set of abilities is the design optimum.

From my perspective, of course, I'm perfectly normal. Other humans have this odd blind spot or slowness when it comes to seeing certain "obvious" answers, although they have no trouble understanding them. And I also have blind spots; there are some classes of problem where I can understand solutions, but I can't formulate solutions myself. (The specifics of which forms of cognition were affected, causal analysis and symbol formation and so on, are not really germane; but be assured that they're there.) The point is...

It's hard to convey what the term smarter means to someone who's never seen their own or someone else's blind spot. The analogy usually used is the difference between modern civilization and the bad old days of ignorance, although a more personal analogy might be between your current self and you at age fourteen. The essential argument is the same: If we couldn't have gotten it right then, what makes us think we can get it right now? But both of these analogies deal not with the magic of intelligence, but the magic of knowledge, or of lack of stupidity.

We're all familiar with individual variations in human intelligence, distributed along the great Gaussian curve; this is all the referent most of us have for "smarter". But precisely because these variations fall within the design range of the human brain, they're nothing out of the ordinary. One of the very deep truths about the human mind is that evolution designed us to be stupid - be blinded by ideology, refuse to admit we're wrong, think "the enemy" is inhuman, be affected by peer pressure. Variations in intelligence that fall within the normal design range don't directly affect this stupidity. That's where we get the folk wisdom that intelligence doesn't imply wisdom, and within the human range this is mostly correct (5). The variations we see don't hit hard enough to make people appreciate what "smarter" means.

I am the most fanatic Singularitarian on the planet because I am a neurohack; because I have some slight, infinitesimal, actual experience with how intelligence enhancement works. Because I have a visceral appreciation of how utterly, finally, absolutely impossible it is to think like someone even a little tiny bit smarter than you are. I know that we are all missing the obvious, every day. There are no hard problems, only problems that are hard to a certain level of intelligence. Move the smallest bit upwards, and some problems will suddenly move from "impossible" to "obvious". Move a substantial degree upwards, and all of them will become obvious. Move a huge distance upwards...

And I know that my picture of the Singularity will still fall short of the truth. I may not be modest, but I have my humility - if I can spot anthropomorphisms and gaping logical flaws in every alleged transhuman in every piece of science fiction, it follows that a slightly higher-order Specialist (much less a real transhuman!) could read this page and laugh at my lack of imagination. Call it experience, call it humility, call it self-awareness, call it the Principle of Mediocrity. I know, in a dim way, just how dumb I am.

I've tried to show the Beyondness of the Singularity by brute force, but it doesn't take infinite speeds and PTs and ws to place something utterly beyond us. All it takes is a little tiny bit of edge, a bit smarter, and the Beyond stares us in the face once more. I've never been through the Singularity. I've never been to the Transcend. I, like any Specialist, just staked out an area of the Low Beyond. This page is devoted to communicating a sense of awe that comes from personal experience.

From my cortex, to yours; every concept here was born of a mere human - and any impression it has made on you was likewise born of a mere human. Someone who has devoted a bit more thought, or someone a bit more fanatic; it makes no difference. Whatever impression you got from this page has not been an accurate picture of the far future; it has, unavoidably, been an impression of me. And I am not the far future. Had this page been written by a Power, you would have gained an accurate impression. But it isn't, and wasn't, and so you didn't. Take whatever emotion this page evoked, and associate it not with the Singularity; associate it with me, the mild, quiet-spoken fellow infinitesimally different from the rest of humanity. Don't bother trying to extrapolate beyond that. You can't. Nobody can - not you, not me.

2035. Probably earlier.

Want more stuff like this? if so log onto Long Now . com and find the selection dubbed 'The Low Beyond'.