Meaning and Almostness
By Jim Chaffee
Essence and existence. For certain believers in God and such, essence precedes existence. For some unbelievers, most famously Jean-Paul Sartre or Simone de Beauvoir, the statement is reversed. Neither of these notions is anything more than a repetition of the point they are attempting to drive home, however, and as far as existential dilemmas go, the God-no God question is at best meaningless and really a yawner. Besides which all the arguments end in begging.
Consider instead a chilling ontological-epistemological cocktail with the potential for profound existential hangover.
Access a calculator with a square root function and apply that operation to 2. That ought to get the decimal expansion of some number bigger than 1 but less than 2. Ask for as many digits as the calculator will provide. You should have some subsequence of 1.41421 35623 73095 04880 16887 24209 69807 85696 71875 37694 80731 76679 73799, assuming decimal output.
Now ask yourself, Is this the decimal (or whatever base your calculator is providing) expansion of the thing named √2? Without even seeing your numbers, it can be said with universal and everlasting truth and certainty that the answer is no. Neither the numbers listed above nor those displayed on your computing machine are the expansion of the thing named √2.
In fact, no matter what calculator you use, the answer will always be no. Some dweeb-ass engineer will say, Close enough. To which the appropriate response is, Close enough to WHAT? Exactly what the fuck is this thing named √2? That is the question. And here's the existential dilemma: The essence of this bitch is to be the solution to x² -2 = 0, hence its alternate name "square root of two" (more an operational title than a name, perhaps, like calling the CEO the chief ceremonial fuck-off). Yet whatever its essence, it has no existence in the sensate world.
Consider: It can be given a sort of physical essence as the "length" of the diagonal of the "unit square" (in Euclidean space, for those who are picky), whatever that is. I have never seen one of these unit squares so far as I can tell. If anyone found one, they would not know it since they could not measure the sides well enough to be certain they were exactly the same length. In fact, to fob this essence thing off onto such a phantasm is to end up begging again. After all, it isn't as if we are defining something by physical example as with the meter, once a metal bar somewhere in Paris and more recently the distance light travels in some interval of time as measured by a cesium clock, which last quasi-physicality amounts to begging again. At any rate, such a trick of hardware would be akin to bypassing the essence for a handy thingamabob, as when Plato's Academy defined man to be a featherless biped. A famous wag of the day tossed a plucked chicken over the hallowed Academy's wall.
Yet mathematicians, being bizarre creatures, imagine a phantasmal continuum of something they call the real numbers that stretches on forever as a line in two directions; a continuum that without this thingamabob named √2 would have a hole in it. If you don't believe me, read the beginning of Walter Rudin's elementary textbook Principles of Mathematical Analysis.
So the difficulty here is that we are faced with a thing, ideative and ostensibly a number, which cannot be displayed. Its existence is not concrete. No human will ever see this number in all its decimal (or whatever integer base one chooses) glory.
How do we know it can't be seen? That's simple. It can be demonstrated in a manner that is universally and timelessly true that any even (positive) integer with a fraction for a square root is an integral multiple of 4. This rules out 2 as such an integer. It can also be demonstrated just as conclusively that any number with a terminating or recurring pattern for its decimal expansion is a fraction. This means that only fractions present themselves to the world with a terminating message, that is to say with finite information; an irrational number requires an infinite expansion without possibility of termination, equivalent to a message without end, an infinite-information hobgoblin. Thus is consigned the thing named √2 to eternal imperceptibility. And worse, actually, as will be seen.
To show this is no bluff, consider a simple argument. If an even integer, let's call it N, had a square root that was a fraction or rational number (as fractions are also called) it could be written as a N = (A/B)², where the integers we are denoting by A and B are both positive and contain no common factors. (If they contained a common factor, we could divide it out.) This means, in particular, that not both of them are even since we could divide out two until at least one of them was no longer even. Since N is even, it can written as 2X where X is an integer, and so A² = 2XB². This means that A² is an even integer. Moreover, if A² (= AA) is an even integer, then A is even and therefore we can write A = 2Y where Y represents an integer. Then we can write 4Y² = 2XB² and divide both sides by 2 which shows that 2Y² = XB². Now we know that B is not even, since A is even and not both A and B can be even as they have no common factors. So X must be even, leading us to conclude that X is 2Z for an integer Z. Since N = 2X, we are forced to conclude that N = 4Z for an integer Z.
Logical necessity has forced the conclusion that if N represents an even integer with rational square root, then N must be a multiple of 4, and of course this precludes 2 from having a rational root. Interesting that such logical necessity, without grounding in any empirical knowledge, informs regarding the physical world. (For those with a more abiding interest in philosophy, you might find it fun to formalize this argument in first-order predicate logic. Be sure to specify the domains of your quantifiers.)
To keep from beating this dead horse any further, it is best to avoid the argument that only rational numbers have integer expansions (like decimal expansions) that either terminate in a suitably chosen number base or repeat after some point with a definite pattern. This too is a logical necessity following from the nature of these prehensive beasts as surely as the logical necessity that even integers with rational square roots are integer multiples of 4, or as surely as the logical necessity that 2 has no rational square root.
So what, you ask, if the square root of two is not a fraction? Well, this is the "so what." It's a number that can never be written down, so does not exist in a physical (temporal, sensate, empirical) state. And yet it has an essence that makes it far more useful than, say, a unicorn or vampire-slayer or motion picture star. The essence of this never-seen noetic bitch makes it crucial for daily life, though you may not be aware of it.
Take care here not to confuse numbers as seen before your eyes with some abstract things. The number 2 is a real thing. It is that symbol itself. Just like 2.4142 is a real thing, that sequence of symbols you see before your very eyes now. No need for any stinking abstract number system.
So as to whether or not there is some Platonic world where such strings exist as abstract beings is an irrelevant diversion. When computing with a calculating engine, a string of digits is the desired outcome, not some abstract number. That is the end goal and that is the thing that exists. Yet we have an essence of something, namely that something which when multiplied by itself gives the number 2.
Aha, you say. What if my calculating machine takes the finite sequence and multiplies it by itself and gets 2? Fine, but there is a machine with more digits (bits) that will not get 2. Or you (a type of computing engine?) can multiply that number by itself, carefully, and you will certainly never get 2. That is the point of the theorem in a concrete setting.
To object that neither does 2 exist in a physical (temporal, sensate, empirical) state misses the point. The thing 2 of interest here is the representation itself, not some idea that it somehow symbolizes. And that representation resides everywhere around you. Check the fixed point representation in your computer, for example. The difficulty is that unlike most things, 2 is both a name and a thing (unlike 10 in binary, which is the name for the number ten but is also the number two for binary arithmetic; or 1.01101010000010011010101010100011 or 1.6A09AAA3AD18, which ought to be the binary, respectively hexadecimal, representation for the decimal 1.41421).
In my garden grows blue kale, and the words blue kale are not something I can cut and cook and may represent an idea (of that stuff growing in my garden). More or less. But 2 is not only the name for a number, it is the number in an operational sense and can be added to or multiplied by another fraction, whether it is represented as a decimal or a ratio of integers. And this can be done on a computer (though likely not as a decimal). But the computer cannot multiply √2 by 2 or anything else. And in fact, neither can you, though you can write 2√2. But with the same problem as before. Now you have a name for a thing that does not exist temporally since you cannot write it down.
Of course, it is theoretically possible to get around this problem if you could apply some algorithm and compute the first digit in a decimal (or other) expansion for √2 in 1/2 second, the second digit in 1/4 second, the third digit in 1/8 second, the fourth in 1/16 second, and so on. In such a way, you could write the entire decimal expansion for √2 in 1 second. So it is possible to bring time into the discussion.
Return your attention to the dweeb of an engineer who says that some decimal, or other integer expansion, is close enough. She means she knows she'll never get to the end of it, literally, since no matter how many integers in the expansion she writes, there is always an infinity of them she will never see (akin to the realization that it is not that life is so short, but instead that you are dead for so long) or imagine (since they don't repeat forever with a pattern). But she doesn't care. She can get enough of an expansion so that when she squares the expansion it is almost the integer 2.
She might be right, in some mundane and practical, perhaps ugly, sense. There is a recursive formula, for example, that starts with any positive value and gets closer to the thing (number?) named √2 with each iteration. There are a lot of them actually, but one that moves along quickly is to take the nth recursion to be Xn = 1/2( Xn-1 + 2/Xn-1). In words, starting from some arbitrary positive initial guess and computing repeatedly to a value using the formula, the next computed value is given by dividing the value in hand into two and adding that to the value in hand, and then dividing the whole mess by two. It can be shown, again by an application of logic, that this gets you as close to that thingamabob named √2 as you have the patience to get, without ever actually getting there. After the initial guess, the fractions you compute will decrease and home in on that doohickey named √2 while eventually bypassing all nearby numbers greater than it.
The logic one needs to demonstrate this necessity is not particularly interesting, but is again ironclad. It is perhaps more enlightening for the empiricist to compute a few values, though that is not particularly fun and provides no logical necessity. However, if you start from 1 for the initial guess, then the first value you compute is 3/2, the second is 17/12, the third is 577/408, the fourth is 665857/470832. For fun, you might try to get the fractional form of the hundredth computation and compare it to the one listed in the third paragraph. No decimal cheating. Get each and every value as a fraction, if you can.
Admittedly, this could be terminally boring and perhaps is not even possible given the limitations of your favorite computing engine, which could be your own brain. It would be less punishing and more enlightening to compute decimal representations by dividing the denominators into the numerators of the fractions listed above. Use a calculating machine. Compare these numbers progressively with that given in the third paragraph.
Okay, so it has been established that the thing named √2 has an essence we can all relate to, namely that when multiplied by itself it provides the familiar integer 2. And with only a modicum of punishment, it has also been established that this thingamabob does not reside in any temporal world in all its decimal (or binary or octal or hexadecimal or duodecimal or any other such) glory. We mortals cannot experience it. There is no empirical verification that it exists in a concrete mode. But we can see some of it, and in principle as much of it as we want to see, but always only a finite chunk. Its infinite extension is forever beyond us. It almost exists, this thing we call √2, somewhere between being and nothingness, neither substantial nor insubstantial. In a state of almostness.
Where does one go from here? What can be said about the essence of the "number" called π? Or worse, what can we say about randomness, also precisely defined in mathematics? Or the mathematical fact that there are more irrational "numbers" than rational numbers (though there are the same number of integers as there are fractions as there are even positive integers as there are even integers as there are odd integers...)? It is perhaps best for now to let these sleeping dogs lie, though the π question takes one into more difficult questions regarding essence.
As for Sartre and de Beauvoir and their turned phrase "existence precedes essence," we can toss in a monkey wrench. We can throw in information and turn the phrase back to what the God-intoxicated crave, namely essence precedes existence. And we can do so while kicking out God. Is essence, for example, encoded in genetic information? Does that information precede existence? Do we have a chicken and egg thing here?
Perchance this is what Norbert Wiener meant when he wrote in his classic Cybernetics: Or Control and Communications in the Animal and the Machine (MIT Press), "Information is information, not matter or energy. No materialism that does not admit this can survive at the present day."
© Jim Chaffee 2008