McLuhan discovered the work of Tobias Dantzig (1884-1956) during his travel for Project 69 (as he styled his ‘Project in Understanding New Media’):

This book [Tobias’ *Number: The Language Of Science*],* *which Einstein proclaimed “the most interesting book on the evolution of mathematics that has ever fallen into my hands”, fell into my hands at the Washington, D. C. airport. This is relevant to the present report since I was then in Washington in connection with Project 69. (*Report*)

This must have been in November 1959 or January 1960, since McLuhan’s itinerary for the project (included in his *Report*) showed him in Washington at these times.

Dantzig then joined the art historians, Hildebrand, Wölfflin, Ivins and Gombrich, as a major resource for the *Report* — and for papers and books prepared by McLuhan over the next few years extending to the 1964 *Understanding Media *(which was based on the *Report*).

*Report on Project in Understanding New Media* (1960)

- Since we consider that our way of life is rooted in literacy it concerns us deeply to know why our children will increasingly spurn it, just as our artists and physicists rejected perspective a century ago. I think the best way (…) to explain this matter further is to cite the evidence of Tobias Dantzig in his
*Number: The Language Of Science*. (…) Pages 139-147 provide all that is needed to understand why phonetic writing created Euclidean space, or the Greek miracle of perspective and naturalistic illusion. The same pages also explain why the Western world since Newton has steadily dissolved Euclidean space and pictorial illusion, and shifted to non-Euclidean geometries and non-objective art. In a word, here are all the clues to the mystery of the rise and fall of Western man, the mystery of his detribalization by literacy and his retribalization by electric communication. - More than anybody else, the mathematician is aware of the arbitrary and fictional character of this continuous, homogeneous visual space. Why? Because number, the language of science, is a fiction for re-translating the Euclidean space fiction back into auditory and tactile space. The example Dantzig uses on page 139 concerns the measurement of the length of an arc: “Our notion of the length of a curve may serve as an illustration. The physical concept rests on that of a bent wire We imagine that we have straightened the wire without stretching it; then the segment of the straight line will serve as the measure of the length of the arc. Now what do we mean by ‘without stretching?’ We mean without a change in length. But this term implies that we already know something about the length of the arc. Such a formulation is obviously a
*petitio principii*and could not serve as a mathematical definition. The alternative is to inscribe in the arc a sequence of rectilinear contours of an increasing number of sides. The sequence of these contours approaches a limit, and the length of the arc is defined as the limit of this sequence. And what is true of the notion of length is true of areas, volumes, masses, movements, pressures, forces, stresses and strains, velocities, accelerations, etc., etc. All these notions were born in a linear, ‘rational’ world where nothing takes place but what is straight, flat and uniform. Either, then, we must abandon these elementary rational notions — and this would mean a veritable revolution, so deeply are these concepts rooted in our minds; or we must adapt these rational notions to a world which is neither flat, nor straight, nor uniform.” (…) The invaluable demonstration of Dantzig is that in order to protect our vested interest in Euclidean space (i. e. literacy), Western man devised the parallel but antithetic mode of number in order to cope with all of the non-Euclidean dimensions of daily experience. He continues: “But how can the flat and the straight and the uniform be adapted to its very opposite, the skew and the curved and the non-uniform? Not by a finite number of steps, certainly! The miracle can be accomplished only by that miracle-maker, the infinite. Having determined to cling to the elementary rational notions, we have no other alternative than to regard the ‘curved’ reality of our senses as the ultra-ultimate step in an infinite sequence of flat worlds which exist only in our imagination. The miracle is that it works!” - The Graeco-Roman galaxy had a one-way road of conquest of all other cultures in the phonetic alphabet. But (…) this one-way road led from sound to sight. This was the road that hoicked Western man from the tribal space of ear and tactility to the civilized visual space of the straight, the flat and the uniform. [Two millennia later,] print, of course, gave [even]
^{1}greater stress and emphasis to the pictorial space of the straight, the flat and the uniform.**The phonetic****alphabet was the first technological medium that gave obvious salience to****the fact that all media are natural resources or staples**. Number, as Dantzig implies, is subservient to letters and meaningless without a civilized, pictorial culture to support it. - Letters translated us out of the all-at-once auditory and tactile world of pre-literate man, while
*Numbers: The Language of Science*(…) developed parallel with letters as a means of translating the visual and the literate back into the non-visual and the tactile. The artist faces the problem of responding to this type of awareness with new forms of relevance for mankind. Pre-literate society enthroned the artist as medicine man. Post-literate society (the electronic) must enthrone the artist as navigator.

Technology, the Media, and Culture (1960)

- In his book
*Number: The Language of Science*, Tobias Dantzig tells us that “The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics.” Today, that crisis is occurring on a massive cultural scale. Our rational Euclidean world of continuous and homogeneous space, extrapolated by the phonetic alphabet from the resonating tribal world, has now to face the electronic challenge of its own irrelevance and superfluousness. l think Dantzig can help us some more to get our bearings here. Just before the passage already quoted he is explaining the crucial use made in mathematics of the Renaissance concept of the “infinite process.” If this concept does not derive from the new perception of perspective or vanishing point, it is at least parallel to it. “The prototype of all infinite processes,“ says Dantzig, “is repetition.” And this is a facet of the concept of convergence, recession, vanishing point, perspective, infinity which is inseparable from Gutenberg technology. For uniformity and repeatability are as basic to print as visuality to the phonetic alphabet. - Dantzig continues: “The importance of infinite processes for the practical exigencies of technical life can hardly be overemphasized. Practically all applications of arithmetic to geometry, mechanics, physics and even statistics involve these processes directly or indirectly. . . . Banish the infinite process, and mathematics pure and applied is reduced to the state in which it was known to the pre-Pythagoreans.” That is to say, without the minute segmentation, whether of alphabet or of the infinitesimal calculus, there can be no translation, no bridge from the tactile, resonating, tribal world, to the rational, flat, visual world. Dantzig simply points out that number aided by infinite process can measure our world by translating visual, Euclidean space created by the phonetic alphabet back into the tactile modalities of touch and sound. One of the many prices we paid for abstracting ourselves from the tribal, multi-sensuous world was that we came to rely more and more on number to get us back into relation to that tribal world. It is not surprising therefore that number, the servant of letters, finally outgrew its master, civilization. For pushed all the way, number or tactile measurement gave us the new electric media which restore the resonating, tactile world as an immediate datum and all-embracing matrix of culture.
- “Our notion of the length of an arc of a curve,” says Dantzig, “may serve as an illustration. The physical concept rests on that of a bent wire. We imagine that we have straightened the wire without stretching it; then the segment of the straight line will serve as the measure of the length of the arc. Now what do we mean by “without stretching?” We mean without a change in length. But this term implies that we already know something about the length of the arc. Such a formulation is obviously a
*petitio principii*and could not serve as a mathematical definition. The alternative is to inscribe in the arc a sequence of rectilinear contours of an increasing number of sides. The sequence of these contours approaches a limit, and the length of the arc is defined as the limit of this sequence.” Calculus, that is to say, is a means of translation of one kind of space into another — especially of visual into tactile and auditory fields of measurement. “And what is true of the notion of length is true of areas, volumes, masses, moments, pressures, forces, stresses, and strains, velocities, accelerations, etc., etc. All these notions were born in a “linear,” “rational” world where nothing takes place but what is straight, flat, and uniform. Either, then, we must abandon these elementary rational notions — and this would mean a veritable revolution, so deeply are these concepts rooted in our minds; or we must adapt those rational notions to a world which is neither flat, nor straight, nor uniform. But how can the flat and the straight and the uniform be adapted to its very opposite, the skew and the curved and non-uniform? Not by a finite number of steps, certainly! The miracle can be accomplished only by that miracle-maker, the infinite. Having determined to cling to the elementary rational notions, we have no other alternative than to regard the “curved” reality of our senses as the ultra-ultimate step in an infinite sequence of flat worlds which exist only in our imagination.” The same navigational techniques of adaptation, compensation, and correction for distortion, which the mathematician provides for the sciences, the artist provides for sensibilities distorted by social technologies and media change.

*The Gutenberg Galaxy* (1962)

- Tobias D. Dantzig points out in his
*Number: The Language of Science*(pp. 141-2): “The attempt to apply rational arithmetic to a problem in geometry resulted in the first crisis in the history of mathematics. The two relatively simple problems, the determination of the diagonal of a square and that of the circumference of a circle, revealed the existence of new mathematical beings for which no place could be found within the rational domain (…) A further analysis showed that the procedures of algebra were generally just as inadequate. So it became apparent that an extension of the number field was unavoidable (…) And since the old concept failed on the terrain of geometry, we must seek in geometry a model for the new. The continuous indefinite straight line seems ideally adapted for such a model.” (*GG*81) - Dantzig explains why the language of number had to be increased to meet the needs created by the new technology of letters. (…) Tobias Dantzig, in his
*Number: The Language of Science*, has provided a cultural history of mathematics which led Einstein to declare: “This is beyond doubt the most interesting book on the evolution of mathematics that has ever fallen into my hands.” The explanation of the rise of Euclidean sensibility from the phonetic alphabet was given in the early part of this book. Phonetic letters, the language and mythic form of Western culture, have the power of translating or reducing all of our senses into visual and “pictorial” or “enclosed” space. More than anybody else, the mathematician is aware of the arbitrary and fictional character of this continuous, homogeneous visual space. Why? Because number, the language of science, is a fiction for retranslating the Euclidean space fiction back into auditory and tactile space. The example Dantzig uses on page 139 concerns the measurement of the length of an arc: “Our notion of the length of an arc of a curve may serve as an illustration. The physical concept rests on that of a bent wire. We imagine that we have straightened the wire without stretching it; then the segment of the straight line will serve as the measure of the length of the arc. Now what do we mean by ‘without stretching’? We mean without a change in length. But this term implies that we already know something about the length of the arc. Such a formulation is obviously a petitio principii and could not serve as a mathematical definition. The alternative is to inscribe in the arc a sequence of rectilinear contours of an increasing number of sides. The sequence of these contours approaches a limit, and the length of the arc is defined as the limit of this sequence. And what is true of the notion of length is true of areas, volumes, masses, movements, pressures, forces, stresses and strains, velocities, accelerations, etc., etc. All these notions were born in a ‘linear’, ‘rational’ world where nothing takes place but what is straight, flat, and uniform. Either, then, we must abandon these elementary rational notions — and this would mean a veritable revolution, so deeply are these concepts rooted in our minds; or we must adapt those rational notions to a world which is neither flat, nor straight, nor uniform.” (*GG*176-177) - The invaluable demonstration of Dantzig is that in order to protect our vested interest in Euclidean space (i.e., literacy) Western man devised the parallel but antithetic mode of number in order to cope with all of the non-Euclidean dimensions of daily experience. He continues (p. 140): “But how can the flat and the straight and the uniform be adapted to its very opposite, the skew and the curved and the non-uniform? Not by a finite number of steps, certainly! The miracle can be accomplished only by that miracle-maker the infinite. Having determined to cling to the elementary rational notions, we have no other alternative than to regard the ‘curved’ reality of our senses as the ultra-ultimate step in an infinite sequence of flat worlds which exist only in our imagination.” The miracle is that it works! (
*GG*178) - today number is as obsolete as the phonetic alphabet as a means of endowing and applying experience and knowledge. We are now as post-number as we are post-literate in the electronic age. There is a mode of calculation that is pre-digital Dantzig points out (p. 14): “There exists among the most primitive tribes of Australia and Africa a system of numeration which has neither 5, 10, nor 20 for base. It is a binary system, i.e., of base two. These savages have not yet reached finger counting. They have independent numbers for one and two, and composite numbers up to six. Beyond six everything is denoted by ‘heap’.” Dantzig indicates that even digital counting is a kind of abstraction or separation of the tactile from the other senses, whereas the yes-no which precedes it is a more “whole” response. Such, at any rate, are the new binary computers that dispense with number, and make possible the structuralist physics of Heisenberg. (
*GG*178-179) - Dantzig explains in his
*Number: The Language of Science*a great step in numeration and calculation taken by the Phoenicians under commercial pressure: “The ordinal numeration in which numbers are represented by the letters of an alphabet in their spoken succession.” But using letters, Greek and Roman alike never got near a method suited to arithmetical operations: “This is why, from the beginning of history until the advent of our modern positional numeration, so little progress was made in the art of reckoning.” That is, until number was given a visual, spatial character and abstracted from its audile-tactile matrix it could not be separated from the magical domain. “A man skilled in the art was regarded as endowed with almost supernatural powers. . . . even the enlightened Greeks never completely freed themselves from this mysticism of number and form.” It is easy to see with Dantzig how the first crisis in mathematics arose with the Greek attempt to apply arithmetic to geometry, to translate one kind of space into another before printing had given the means of homogeneity: “This confusion of tongues persists to this day. Around infinity have grown up all the paradoxes of mathematics: from the arguments of Zeno to the antinomies of Kant and Cantor.” It is difficult for us in the twentieth century to realize why our predecessors should have had such trouble in recognizing the various languages and assumptions of visual as opposed to audile-tactile spaces. It was precisely the habit of being with one kind of space that made all other spaces seem so opaque and intractable. From the eleventh to the fifteenth centuries the Abacists fought the Algorists. That is, the literate fought the numbers people. In some places the Arabic numerals were banned. In Italy some merchants of the thirteenth century used them as a secret code. Under manuscript culture the outward appearance of the numerals underwent many changes and, says Dantzig: “In fact, the numerals did not assume a stable form until the introduction of printing. It can be added parenthetically that so great was the stabilizing influence of printing that the numerals of today have essentially the same appearance as those of the fifteenth century.” (*GG*180) - The great sixteenth century divorce between art and science came with accelerated calculators. Print assured the victory of numbers or visual position early in the sixteenth century. By the later sixteenth century the art of statistics was already growing. Dantzig writes: “The late sixteenth century was the time when in Spain figures were printed giving the population of provinces and the population of towns. It was the time when the Italians also began to take a serious interest in population statistics — in the making of censuses. It was the period when in France a controversy was carried on between Bodin and a certain Monsieur de Malestroit concerning the relations of the quantity of money in circulation to the level of prices.” There was soon great concern with ways and means of speeding up arithmetical calculations: “It is hard for us to realize how laborious and slow were the means at the disposal of medieval Europeans for dealing with calculations ‘which seem to us of the simplest character’. The introduction of Arabic numbers into Europe provided more easily manipulated counters than the Roman numbers, and the use of Arabic numbers seems to have spread rapidly towards the end of the sixteenth century, at least on the Continent. Between about 1590 and 1617 John Napier invented his curious ‘bones’ for calculating. He followed this invention with his more celebrated discovery of logarithms. This was widely adopted all over Europe almost at once, and in consequence arithmetical calculations were immensely accelerated.” (GG 181)
- “Strike flat the thick rotundity of the world,” cries Lear as a curse to snap “the most precious square of sense.” And the striking flat, the isolation of the visual is the great achievement of Gutenberg and the Mercator projection. And Dantzig notes: “Thus the alleged properties of the straight line are of the geometer’s own making. He deliberately disregards thickness and breadth, deliberately assumes that the thing common to two such lines, their point of intersection, is deprived of all dimension … but the assumptions themselves are arbitrary, a convenient fiction at best.” It is easy for Dantzig to see how fictional classical geometry was. It got huge nourishment from printing after being engendered by the alphabet. (GG 182)

*Understanding Media* (1964)

- So far as Tobias Dantzig is concerned in his
*Number: The Language of Science,*the progress from the tactile fingering of toes and fingers to “the homogeneous number concept, which made mathematics possible” is the result of visual abstraction from the operation of tactile manipulation. (*UM*113) - Dantzig, having made clear that the idea of homogeneity had to come before primitive numbers could be advanced to the level of mathematics, points to another literate and visual factor in the older mathematics. “Correspondence and succession, the two principles which permeate all mathematics—nay, all realms of exact thought—are woven into the very fabric of our number system,” he observes. So, indeed, are they woven into the very fabric of Western logic and philosophy. We have already seen how the phonetic technology fostered visual continuity and individual point of view, and how these contributed to the rise of uniform Euclidean space. Dantzig says that it is the idea of correspondence which gives us cardinal numbers. Both of these spatial ideas — lineality and point of view — come with writing, especially with phonetic writing; but neither is necessary in our new mathematics and physics. Nor is writing necessary to an electric technology. Of course, writing and conventional arithmetic may long continue to be of the utmost use to man, for all that. Even Einstein could not face the new quantum physics with comfort. Too visual a Newtonian for the new task, he said that
*quanta*could not be handled mathematically. That is as much as to say that poetry cannot be properly translated into merely visual form on the printed page. (*UM*113-114) - Dantzig develops his points about number by saying that a literate population soon departs from the abacus and from finger enumeration, though arithmetic manuals in the Renaissance continued to give elaborate rules for calculating on the hands. It could be true that numbers preceded literacy in some cultures, but so did visual stress precede writing. For writing is only the principal manifestation of the extension of our visual sense, as the photograph and the movie today may well remind us. And long before literate technology, the binary factors of hands and feet sufficed to launch man on the path of counting. (
*UM*114) - Dantzig reminds us also that in the age of manuscript there was a chaotic variety of signs for numerals, and that they did not assume a stable form until printing. (
*UM*114)

- McLuhan: “much”. ↩