http://www.crcsite.org/numbers.htmPerceptions of Numbers
by H. Peter Aleff
Modern questions about number
Numbers were and are astonishing entities, entirely intangible but more enduring and reliable than the fleeting reality that hides them to unaware minds. To appreciate what the ancients saw in them, it may be best to begin with the questions some modern thinkers are asking about their nature. For instance, the astronomer and cosmologist John D. Barrow muses:
“Why does the world dance to a mathematical tune? Why do things keep following the path mapped out by a sequence of numbers that issue from an equation on a piece of paper? Is there some secret connection between them; is it just a coincidence; or is there just no other way that things could be? (...) Down the centuries there have been those who saw in mathematics the closest approach we have to absolute truth (...). Its very structure forms a model for all other searches after absolute truth.”[1]
Comparably, the searcher-for-absolute-truth-mathematician yet occasionally inventive chronicler of mathematical history Eric Temple Bell discussed in his book on Pythagoras what numbers might be. He concluded that
“[Whether numbers were discovered or invented] is the oldest and the simplest of all questions regarding the nature of mathematical truths. History gives no universally accepted answer to it.” [2]
Bell was a gifted story-teller and went here for dramatic effect. The question may be simple, but old it is not.
Until the current century, people rarely, if ever, doubted the once universally accepted bedrock principle that numbers and mathematics were “out there” and not invented by humans. Here are some comments on this subject which the mathematician Bonnie Gold wrote recently in a review of two books on the philosophy of mathematics:
“In the early years of this century, Platonism (by which I mean the belief that mathematics is the science of certain mind-independent, non-physical objects with determinate properties) was dethroned as the dominant philosophy of mathematics.
Since then, there’s been a struggle to replace it with an alternative that avoids the philosophical problems of Platonism while accurately reflecting the working mathematician’s daily experiences of doing mathematical research. None of Platonism’s immediate successors -- logicism, formalism, intuitionism -- has proved satisfactory. (...) In the last 25 years, new candidates for philosophies of mathematics have become popular, including fictionalism, conventionalism, structuralism, and social constructivism.”[3]
Proponents of the last-named among these philosophies argue, for instance, that the equation 2 + 2 = 4 is only a convention our grade school teachers bullied us into accepting as a law of nature, and that mathematical objects are social entities in the same way as monetary systems or political institutions.[4]
They also propose that mathematical objects get constructed by the community of mathematicians and then only take on a sort of life on their own in the minds of its members.
This may of course all well be so, but it makes one wonder where these objects go to survive when the knowledge about say, the golden ratio and the pentagram, gets lost and then rediscovered. How long can these objects stay outside of minds, and do they get enough food, water, and air there to continue their life while waiting to snatch another mind as their new host?
Why would a different community living at a different time come up again with the very same mathematical objects that some other people had already created?
And why do we laugh about the lawgivers in Indiana who tried in 1896 to legislate a value of 3.2 for the circle constant pi since it would be easier to use than the longer and more complicated traditional number which was produced strictly by and for impractical mathematicians?[5]
Such modern thinkers may reject the Platonist belief in some abstract, non-physical and non-psychological realm of numbers and mathematics where constants and pentagrams and other such constructs exist by themselves and outside of minds.
Ancient status of NUMBER
On the other hand, people in the highly religious environment of the ancient Levant had their then unquestioned certitude: NUMBERs belonged to the divine domain. Obviously, they had to be the first things made to then serve as helpers or tools for the rest of creation. No animals with four legs or people with two of those and five toes on each foot could be shaped before those numbers existed to build them with.
We owe much of our pre-Platonic written documentation of that once common “Platonist” belief to the ancient Mesopotamians because their baked- clay tablets survived better than the parchments and papyri of their neighbors. Another source for ancient attitudes towards numbers are the traditions about the Greek Pythagoras (around 580 to 500 BCE) and his teachings because this founder of a number-venerating religion had acquired many if not all elements of his doctrine in Phoenicia, then above all in Egypt, and finally in Mesopotamia.
The apparent absence of direct evidence from Egypt or Canaan does not mean said neighbors did not share very similar basic beliefs even though they left us little or no written testimony to that effect.
All civilizations of the ancient Near East flourished on a common cultural ground from which they drew many shared ideas and core convictions. For instance, winged disks represented the sun as royal and/or divine emblem on Hittite palace portals, Egyptian stelae, and Canaanite seals, as well as in Assyrian and Babylonian art. The same image appears also in Bible verses such as Malachi 4:2:[6]
“the sun of righteousness shall rise with healing in his wings”.
Conceptions of number are rooted deeply in that same shared substratum of basic ideas beneath the superficial cultural differences. It seems therefore more likely than not that the Phoenician and Egyptian beliefs about the nature of the number world had grown on similar spiritual soil as those of the ancient Mesopotamians.
Mesopotamian number gods
In those Mesopotamian beliefs, numbers were the most basic concepts of existence. Each deity had and/or was his or her number, and numbers were gods. The names of the gods could even be written as numbers. For instance, a personal name from the Third Dynasty of Ur reads “My-god-is-50.”[7]
Mathematics was above all a priestly science, embedded in and driven by a culture of number mysticism. The historian of science André Pichot describes this once common attitude towards numbers:
“Just as Mesopotamian astronomy was inseparable from astrology, Mesopotamian mathematics was inseparable from number mysticism.
(...) Numbers do not present themselves to our senses as unequivocally as geometric figures do. They are by far more abstract and are not part of our perceivable environment. Yet these abstractions exist and, moreover, they have certain regular properties.
Numbers cannot be connected with the concrete reality the way geometry is, and so they easily slide into the supernatural. They are not in nature but determine its manifestations through certain ratios and relationships; from this follows that they rank above nature. (...)
On the other hand, these “super-natural” and “nature-ruling” numbers are also a means to understand that nature. (...) [The Mesopotamians sought not mathematical rules or proofs] but technical recipes as well as certain number ratios without any direct use; this quest and this knowledge belong more to the domain of magic than to that of science. We find therefore besides the mathematical role of the numbers also purely mythical aspects which are equally interwoven with their use. (...)
This number mysticism is not to be confused with another kind of mysticism: in Greek and Hebrew writing, the numbers are represented by letters and not by symbols of their own, so that each word, particularly names, has its number, obtained, for instance, by adding the numerical values of the individual letters. (...) This [gematria] is fundamentally different from the number mysticism in which the numbers represent supernatural and nature-dominating entities; the latter can be considered a precursor of mathematical physics and even of mathematical rationalism in its broadest sense.”[8]
Many modern scholars believe that this number mysticism was the major motivation behind the Mesopotamians’ deep and ancient interest in the manipulation of numbers. The ancient sages’ search for the relationships between these supernatural but predictable entities contributed greatly to their development of mathematical skills.
Their looking for patterns in the number world anticipates the modern “alternative philosophy of mathematics”[9] which defines its subject as “the science of pattern”[10] . The search is the same even though the ancients asked different questions and looked for different patterns, and even though they interpreted some of their results in mytho-logical instead of modern-logical terms.
Here is how Pichot summarized his comments on the Mesopotamian number mysticism from which I quoted above:
"Just as the introduction of metalworking did not make chemistry any more rational but rather enriched magic by a few concepts and mythology by sundry smith gods, so the manipulation of numbers did not produce a unified corpus of rationally founded laws -- mathematics -- but remained a computing technique with a mystical background.
We should not expect rational explanations from this approach to numbers; its explanatory contribution is on the same level as that of alchemistic magic or of smith gods. It is moreover characteristic for this non-rational use of numbers and their relationships that each number is linked to a god and derives its value and position in the number system from the position of that god in the mythology. The structure of the number world corresponds therefore much more to a hierarchy of gods in heaven than to a system based on rational mathematical laws.”[11]
Pythagorean number veneration
The same ideas resonate also in the doctrine of Pythagoras where NUMBER had created and continued to rule the cosmic order. NUMBER was divine and functioned as “the principle, the source, and the root of all things”[12]. It permeated the world, intangible and unseen but as real and live to Pythagoreans as radio waves are to us.
NUMBER was even more essential to them than electromagnetic radiation is to us: electromagnetism is only one of currently four forces that hold our world together, whereas Pythagoreans held that NUMBER was the One and Only universal principle, the fabric of all there is and can be.
In Plato’s Pythagorean-influenced Timaeus, NUMBER was the World Soul from which the Demiurge had fashioned all of Creation. NUMBER wove together Form and Matter through its mathematical harmony and ruled everything, from the string lengths of musical instruments to the human emotions their sounds created and guided.
Even abstract concepts such as light and dark or good and bad were NUMBERS, and of course, so were the planetary motions that produced by their numerical ratios the famous though inaudible “Music of the Spheres”. NUMBER was the unchanging reality behind the illusion of the ever-changing phenomena we perceive.
Indo-European number rituals
The mystic status of numbers which led to this intense concern with their properties and relationships seems to have existed also, even before Sumerian times, in the beliefs of the neighboring Indo-Europeans.
Modern scholars interpret similarities between word roots as signs of deep and original connections, just as ancient sages had long done with similarities between the sounds of words, or between the ways to write them. Based on this principle, some of the moderns have shown that the religious view of numbers among speakers of Indo-European languages goes back to the prehistoric period when the words for their relationships formed.
Here is what David R. Fideler says about these early word roots in his introduction to Guthrie's “The Pythagorean Sourcebook and Library”:
“Cameron, in his important study of Pythagorean thought, observes that harmonia in Pythagorean thought inevitably possesses a religious dimension. He goes on to note that both harmonia -- there is no “h” in the Greek spelling -- and arithmos appear to be descended from the single root “ar”.
This seems to ‘indicate that somewhere in the unrecorded past, the Number religion, which dealt in concepts of harmony or attunement, made itself felt in Greek lands. And it is probable that the religious element belonged to the arithmos - harmonia combination in prehistoric times, for we find that ritus in Latin comes from the same Indo-European root’.”[13]
Such traces of early reverence for invisible but knowable Numbers suggest that if some ancient mathematicians were aware of the major constants, they might have ranked these mysterious “super-numbers” even higher than the natural numbers. They would have assigned them important religious and symbolic roles, and they would have explored their properties and permutations as a means to understand the relationships among the gods they represented or were.
Copyright © 2003 bv H. Peter Aleff
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Footnotes :
[1] John D. Barrow: “Pi in the Sky - Counting, Thinking, and Being”, Oxford University Press, Oxford, 1992, page 4.
[2] Erik Temple Bell: “The Magic of Numbers”, 1946 and 1974, edition consulted Dover, New York, 1991, pages 20 and 21.
[3]) Bonnie Gold: Review of “Social Constructivism as a Philosophy of Mathematics” by Paul Ernest and of “What is Mathematics, Really” by Reuben Hersh, The American Mathematical Monthly, April 1999, pages 373 to 380.
[4] For instance, Paul Ernest in: “Social Constructivism as a Philosophy of Mathematics”, State University of New York Press, 1998, per Bonnie Gold in the above review, page 375.
[5] Jan Gullberg: “Mathematics, From the Birth of Numbers”, W.W. Norton & Co., New York, 1997, pages 95 and 96.
[6] See, for instance, the biblical king Hezekiah’s stamp on jar handles mentioned in Frank Moore Cross: “King Hezekiah’s Seal Bears Phoenician Imagery”, Biblical Archaeology Review, March/April 1999, pages 42 to 45 and 60, see page 45 left.
[7] Simo Parpola: “The Assyrian Tree of Life: Tracing the Origins of Jewish Monotheism and Greek Philosophy”, Journal of Near Eastern Studies, Volume 52, July 1993, Number 3, pages 161-208. see page 182 left and note 87.
[8] André Pichot: “La naissance de la science”, Gallimard, Paris, 1991, edition consulted “The Birth of Science -- from the Babylonians to the early Greeks”, Wissenschaftliche Buchgesellschaft, Darmstadt, 1995, pages 92 to 94. (In German, my translation)
[9] so dubbed by Bonnie Gold, in the above review, American Mathematical Monthly, April 1999, page 379.
[10] as promoted in another book title by “Making the Invisible Visible” author Keith Devlin: “Mathematics: The Science of Pattern: The Search for Order in Life, Mind, and the Universe”, Scientific American Library, 1997.
[11] André Pichot: “The Birth of Science ...”, page 92. (in German, my translation)
[12]) David R. Fideler, in his “Introduction” to Kenneth Sylvan Guthrie, compiler and translator: “The Pythagorean Sourcebook and Library”, Phanes Press, Grand Rapids, Michigan, 1988, page 21 bottom, quoting Theon of Smyrna.
[13] David R. Fideler: “Introduction” to Kenneth Sylvan Guthrie: “The Pythagorean Sourcebook and Library”, Phanes Press, Grand Rapids, Michigan, 1987, citing Alister Cameron: “The Pythagorean Background of Recollection”, see note 30 on page 51 middle.