Mathematics and Measure

The earliest form of mathematics was no doubt counting; i.e., the invention of numbers. Like nouns, numbers are an abstraction of reality, a reduction of the infinite variability of nature to a collection of standard things. To say there are five of anything presupposes that there could possibly be more than one of any given object, thereby denying the particularity of each being in the universe. When your family sits down to dinner, you don’t have to count them to make sure everyone is there. In a society where every person is known as an individual, and where every thing is perceived in its unmediated uniqueness, number would be an absurdity. One could imagine a Paleolithic philosopher protesting, “How can you say those are three? They are one and one and one, each occupying a unique place in the world.”

It is therefore unsurprising that numerous hunter-gatherer societies have been discovered, in remote areas, who do not have words for numbers other than “one,” “two,” and “many.”[34] The modern mind typically interprets this as evidence of their childlike simplicity or their lack of cognitive development. But perhaps they simply do not have the need. They live in a concrete world. This does not mean that they cannot distinguish between five and six—reportedly even crows can perform this feat. It just means that these amounts were not subject to abstraction.

Numbering is one of the most primitive forms of measurement, which is none other than the conversion of quality into quantity, the conversion of the specific and unique into the standard and general. In numbering things, we implicitly perform an abstraction by turning a multiplicity of unique objects into just so many uniform ones. We realize today that we cannot add apples and oranges. In an earlier world that appreciated the individuality of all things and moments, we realized as well that one cannot even add apples and apples.

Numbers, even more than words, remove objects from their original cross-referential matrix and imply that they are discrete, uniform things. Eventually, as the symbolic, representational world separated us increasingly from the real, immediate world, we took the final step of assigning to numbers an ontological status more real than the things from which they were originally abstracted. Pythagoras, and Plato after him, reversed the original order of abstraction to assign primary reality to the abstractions themselves. Aristotle describes the Pythagorean view that “They supposed the elements of numbers to be the elements of all things, and the whole heaven to be a musical scale and a number.”[35] His critique highlights the danger in basing knowledge on the manipulation of abstractions:

And all the properties of numbers and scales which they could show to agree with the attributes and parts and the whole arrangement of the heavens, they collected and fitted into their scheme; and if there was a gap anywhere, they readily made additions so as to make their theory coherent, e.g., as the number ten is thought to be perfect, they say that the bodies which move through the heavens are ten, but as the visible bodies are only nine, to meet this they invent a tenth.

Already, in the sixth century BCE, scientists were selectively gathering and omitting data to prove their assumptions!

Astonishingly, Pythagoras’ elevation of abstraction apparently occurred before the Greeks had even begun using numerals.[36] His mathematics was based purely on geometry and proportion, and therefore still tangibly linked to concrete pebbles and lines in the dirt. Each new advance in mathematics widened the abstraction, made our mode of thought more symbolic, and removed it further from the reality symbolized. The concept of numeral was one such advance, the invention of decimal numbering and the zero another. With the zero, for the first time something stood for nothing—an apt statement of the generalized divorce between our symbols and the reality they purport to represent.

It is no wonder that, because number reduces the variability of concrete reality, neglected variables creep back in to wreak havoc on our attempt to control the world by extending its measurement. Since the time of Galileo, the goal and modus operandi of science has essentially been to convert the entire world of observed phenomena into numbers. Measurement converts things to numbers; then the equations of science convert these numbers into other numbers in a burgeoning tower of abstraction. The assumption seems to be that if one day we could measure everything, perfect understanding and thus perfect control would be ours. Indeed, today the hard sciences and even the social sciences offer us “data”, i.e., numbers, purporting to encapsulate just about every observable phenomena. Yet as the failed promise of “Gee whiz! The future!” demonstrates, perfect control remains elusive no matter how much of the world we quantify. We seem to have forgotten that mathematics, and therefore the science and technology built on its scaffold, by its nature as abstraction leaves something out. So far our response has been that of the technical fix: to extend measurement still further to encompass those things left out—to remedy its failures with more of the same. On a conceptual level, this program hit a brick wall in the twentieth century with the development of quantum mechanics and chaos theory. On a practical level, we have so far failed to appreciate the lesson in the repeated failure of the program to better manage reality by reducing it to numbers. Instead we call for more numbers, more data.

Mathematics and measure are objective, in the sense that they vitiate objects of the particularity which resides in the interaction of observer and observed. They are consistent with separately existing objects that are “out there”, external to our subjectivity, denying a principle common to ancient mysticism and modern physics that “existence” is a two-place predicate, an interaction. Today the concept of objectivity is central to our world view that includes ourselves as separate, discrete individuals. It also underlies classical physics and the Scientific Method, and it informs what we mean by the very adjective “scientific”. To see how deeply it has influenced our perceptions, visualize something just “existing”. Is your picture that of something floating by itself, alone? No wonder we feel so alone ourselves. To be is to be separate. In this book, I am calling for a revolution, the deepest possible revolution—the replacement of our conception of beingness with a new equation: being equals relationship.

As with language, the abstraction of reality inherent in number has horrifying consequences. As Derrick Jensen says, “It’s easier to kill a number than an individual, whether we’re talking about so many tons of fish, so many board feet of timber, or so many boxcars of untermenschen.”[37] The logic and processes of the machine can equally accept as inputs anything that can be quantified and measured. The part left out by number’s reduction of reality does not enter into the calculations, even if that part is someone’s home, someone’s livelihood, or someone’s life. Hence the refrain, “I am not a casualty figure, I am a human being.” I don’t think the cruelty of today’s world could exist without the distancing effects of language and measure. Few people can bring themselves to harm a baby, but, distanced by the statistics and data of national policy-making, our leaders do just that, on a mass scale, with hardly a thought.

In the most extreme application, number and language combine in the ultimate expression of objectification and abstraction in the locution “one” to mean “I”. Herein, the vitiation of the particular extends even to the self, which is generalized, depersonalized, and made interchangeable, its individuality denied. The generality of this usage of “one” turns all other people into a collection of identical “ones” as well, like so many interchangeable uniform parts of a vast world-machine.

From the very beginning, the concept of number implied an objectification of the universe and a subjugation of the world to human manipulation. Tellingly, the very word number has root meanings of grasping, taking, and seizing,[38] just as the word “digit” means a finger. It is likely, therefore, that the concept of number only arose when the other forces described in this chapter—technology, language, division of labor, and most importantly agriculture—turned the world into an object of manipulation. Number and commodity are highly interdependent concepts that contributed to the replacement of sharing with exchange, commerce, and money. “This ox” became “one ox”; number became abstracted from specific objects. “Number and commodity were now cut apart forever and, as a result, most significantly, numbers could now be applied to quantify any thing around in the world. We were now able to think of the world as something which could, like grain or sheep, be inventoried, controlled, and redistributed.”[39]

The conversion of all the world to number implicit in science (and laid out explicitly by Galileo, Leibniz, and Kant) is inseparable from the program to bring the world under complete control. It is no accident that shortly after the Enlightenment scientists’ articulation of the program to convert nature into number, reformers sought to objectify measure as well, converting the old units of weight, length, and volume into new, “rational” units. The metric system replaces the human scale with a scale based on objective features of the observable universe. The old Fahrenheit scale correlates naturally to human experience, wherein zero is about as cold as it ever gets and a hundred is about as hot as it ever gets, whereas the Celsius scale is based on the freezing and boiling point of water at a given pressure. Compare also the foot with the meter, originally defined “objectively” as one forty-millionth of the earth’s circumference, and now in terms of the wavelength of a certain frequency of light.[40] Measure has been removed from its original source in the human body and everyday experience.

In the last century, the reduction of the world to numbers has only accelerated, especially in the computer age. Music is a prime example. While musical notation had set the stage long before, the mathematization of music intensified with Bach, because of whom, “The individual voice lost its independence and tone was no longer understood as sung but as a mechanical conception. Bach, treating music as a sort of math, moved it out of the stage of vocal polyphony to that of instrumental harmony, based always on a single, autonomous tone fixed by instruments, instead of somewhat variable with human voices.”[41] Today, with the digitization of music, its transformation into just so many bits of data is complete: music is, like any other “digital content”, nothing but a series of numbers. Here is a good example of the reduction that quantification entails: contrary to popular belief, the standard CD-audio format is discernibly different from analogue, particularly at the higher frequencies. Some of the original richness is gone forever; music connoisseurs sometimes speak of the “warmth” of vinyl as compared to the coldness of digital sound. The technological solution, of course, is to increase the sampling rate to the point where the lost data is below human powers of auditory discernment. The infinite will still have been made finite, the continuous made discrete, but at least, it is hoped, this semblance of reality will be “good enough”.

Digitization is applied to images as well, and potentially to anything capable of “analysis.” The motions of the human body, for example, can be converted into a set of numerical coordinates in 3-dimensional space; the sounds of human speech to just so many sine waves. The supposedly inherent reducibility (and notice the naturalness of the word “reducibility”) of the world into numbers, and the assumption that either nothing significant is lost, or that we’ll never notice what is lost provided enough numbers are used, motivates the ultimate technological separation of humans from reality called “virtual reality” (VR). VR is the penultimate step in the substitution of manufactured reality for natural reality. With VR, the separate human realm will be complete except for one, final step. If, as science implies, the entire universe is reducible to numbers, then we too are so reducible; hence the science-fiction scenarios of one day achieving immortality by uploading our consciousness onto a computer, where we could enjoy the best, most pleasurable of artificial experiences forever.[42] The Technological Program of complete control over the world we experience would be fulfilled. Or so goes the fantasy.

Science fiction writers such as Neal Stephenson and Vernor Vinge have described futures in which people live almost entirely in digital representations of reality, or in wholly constructed realities. Such scenarios are already taking shape in the individual RFID labeling of all products sold, setting the stage for the Internet to become nothing less than a virtual copy of the entire planet:

The present IP, offering 32-bit data labels, can now offer every living human a unique online address, limiting direct access to something like 10 billion Web pages or specific computers. In contrast, IPv6 will use 128 bits. This will allow the virtual tagging of every cubic centimeter of the earth’s surface, from sea level to mountaintop, spreading a multidimensional data overlay across the planet. Every tagged or manmade object may participate, from your wristwatch to a nearby lamppost, vending machine or trash can — even most of the discarded contents of the trash can.[43]

As presaged by naming and number, the potential exists for us to attempt the conversion of every object and person in the world into a dataset.

Whether in the digitization of music or the quantization of reality implicit in VR, the insanity of our ferocious strivings to manufacture a reality almost as good as the real thing should be plain. We are exerting tremendous effort to make an inferior version of a freely available original, similar to our fevered attempt to recreate the original affluence, in which “work” was not yet a concept, through ever more efficient labor-saving devices. That manufactured reality is inferior is demonstrated by the increasing intensity of our simulations, which seek to compensate for the lost richness and intensity of unmediated experience. No matter how large a set of numbers we produce to describe reality, something of its infinitude is lost. Instead we must settle for “close enough”, the lesser lives we live in the age of separation. The systems of representation—number, language, image, and so forth—we interpose between ourselves and reality are always a reduction of what they purport to represent. The Scientific Program of a complete mathematization of the universe is yet another Tower of Babel, aiming to attain the Infinite by finite means.

 

[34] Levy-Bruhl, Lucien, How natives think, New York: Humanities Press, 1926. p. 181

 

[35] Aristotle, Metaphysics. Cited by J.B. Wilbur and H.J. Allen, The Worlds of the Early Greek Philosophers, p. 86.

 

[36] Wilbur, J.B. and H.J. Allen, The Worlds of the Early Greek Philosophers, p. 87.

 

[37] Jensen, Derrick, A Language Older than Words, Context Books, 2000. p. 41

 

[38] Zerzan, p. 48

 

[39] Burke and Ornstein, p. 45.

 

[40] The original definition of the meter was abandoned when it was realized that the circumference of the earth is not absolute and unchanging. Now we are beginning to suspect the same may be true of the physical “constants” of the universe. This might be another example of the doomed quest for certainty described in Chapter Three.

 

[41] Zerzan, p. 57

 

[42] David Deutsch demonstrates convincingly the impossibility of perfect reality-simulation on a classical (non-quantum) computer; however on a quantum computer the very uncertainties of uncontrolled reality creep back in in subtle ways.

 

[43] Brin, David, “Three Cheers for the Surveillance Society!” Salon , 8/4/2004. www.salon.com/tech/feature/2004/08/04/mortal_gods/index1.html