Googolplex Arithmetic

Googol is not a total joke; at least, it is a number. The name itself, suggested by a nine-year-old child and en­thu­si­as­ti­cal­ly adopted by a zilly (sic) uncle chasing after big numbers, is too absurd to be taken seriously by anyone else than people suffering from the BIC, the Big Integer Craze (and, un­der­stand­a­bly, unwilling to use one of its three quasi-scientific 'sys­tem­at­ic' names: ten duotrigintillion, ten thousand sexdecillion or ten sex­decilliard). None­the­less, 10^100 is definitely a legitimate candidate for numbership, especially among two-handed pentadactyl vertebrates that have some notion of counting. After all, the at pres­ent standard formula for number notations in a positional or 'place-value' system is a₁×b^n-1 + a₂×b^n-2 + ... + a_n-1×b¹ + a_n×b⁰, in which the most frequently used value of b (the base or 'radix') is 10, pronounced "ten". Just assign 1 to a₁, 0 to all the other as and 101 to n, and you have your googologically big number 1000000000 ... 0000000000 with a hundred zeros. You might wonder why 10^2 zeros, and not 10^1 or 10^3, or even more, but you will irritate peo­ple by asking such questions, and look insanely greedy by asking for one thousand, ten thousand or more zeros. Let's rather be glad in­stead, that the modern 'queen of science' in­tro­duced a new term which was, per­haps, not ethnically exclusivistic in that, for once, 'she' did not confine 'her' worldview to a Greco-Roman or adjacent section of the Mediterranean region on Earth. (Perhaps, because ru­mor has it that that nephew, too, was inspired by Barney Google, a comic strip character of his own time and place.)

'Googolplex', however, is an al­most entirely different matter. In this name the lack of any linguistic sense reaching beyond one's own language or family of languages, and the parts of the world where they are spoken, is varnished over with a return to the Greco-Roman vocabulary —and how! The morpheme plex derives from Latin plectere meaning (to) braid. That is why plexus is used to refer to a network or an interwoven com­bi­na­tion of parts. Un­for­tu­nate­ly, the two googol particles are nothing whatsoever to do with any network in any appropriate meaning of the word. This completely vain use of plex makes googolplex even more absurd than googol. Something of the length of a plexus was only shaped later by having ten raised, not to one googol, but to one googolplex and naming it "googolplexplex", thus opening up a gargantuan mega-avenue in­hab­it­ed by a zillion silly googolplexplex...plexplexes. (Ever met a real network by the name plexplexplex?)

But what's in a name: linguistics is about words, and mathematics about numbers, isn't it? Let's assume that never the twain shall meet (with an assumption like this one); let's focus on what googolplex is actually supposed to refer to. It is supposed to refer to ten raised to the power of one googol, that is, 10^(10^(10^2)). Ergo, it is Not a Number we are talking about (NaN in the language of computer programming). Googolplex is no more than a slogan which refers to a numerical value.

The expression, or part of an expression, 10^(10^(10^2)) just fails to fit into the scheme a₀×bⁿ + a₁×b^n-1 + ... + a_n-1×b¹ + a_n×b⁰. True, this scheme is a present-day notational convention which may have to give way to a new convention tomorrow. There ought to be one, as soon as we realize that that scheme so familiar to us does not reach any further than the second level of iteration, the one of exponentiation; in other words, the 重² or 'chong-2' level. (Read my article The Chong Operators — a universalization in arithmetic, written thirteen years ago.) To put it more boldly, tomorrow there is bound to be a different notational scheme for numbers, a 重³ or 'chong-3' notation which in­cor­po­rates the third level of iteration, the one of repeated exponentiation, in a way which still serves all num­bers equally, and not only big integers. The expression 10^(10^(10^2)) is not even an instance of such repeated exponentiation! Given the fixation on 10 (read "ten") the closest right-associative denary candidate for consideration is 10^(10^(10^10)). Clearly, it would be putting the cart before the horse by first coming up with an expression in which 2 is an incongruous quantity among the 10s and wondering later whether, perhaps, 10 makes the expression more logical or more useful here than 2 does. (And wondering later, if ever, whether right-associativeness is the correct choice, only because it 'has always been' a convention and produces the biggest num­bers in the fastest way.)

With googol's successor a pitifully arbitrary value some­where be­tween the 332nd and the 333rd terms of the chong-3-generated sequence of squares of squares was raised to the zenith of number management. Unfortunately, the 'number' is fake and its minimal significance as a numerical value only a beacon to those who des­per­ate­ly hold onto one of the worst radixes one can conceive of (apart from the odd ones). Googolplex arithmetic is shamelessly more of the same: the same ten-fingered fetishism, the same incapacity to take mathematics to a higher iterative level on a well-thought-out path, the same skewness in attention for integers and the infinite at the cost of what should be called "portions" and the infinitesimal, and the same in­ca­pac­i­ty to contribute to a sub­stan­tive universal terminology which deserves the epithet scientific.

©80aSWW, M. Vincent van Mechelen