In his Metaphysics Aristotle states:

The Pythagoreans, it seemed, somehow thought that number both is matter and describes matter, and Aristotle could not understand how this could be. That number completely describes matter is a credo of modern science, expressed by Leibniz as God is a Mathematician, and by Galileo’s statement that the book of nature is written in mathematical letters. Most people, upon some reflection, have considered even this idea--the idea that number is matter--taken literally, to be absurd. They (in agreement with Aristotle) would require no reflection at all to come to that conclusion.

Despite this, the equivalence of number to matter expressed by the Pythagorean credo indicates that he rather than Aristotle was closer to the spirit of modern physics which has mathematicized nature so thoroughly that it now only thinks of matter in terms of the symbols of mathematics. Modern physicists rarely think of matter in any way distinct from its mathematical representation, and the thought that mathematical structures representing forms of matter are not identical to the latter, has no effect on doing physics. And if so, what can the distinction between the mathematical and the sensual representation of matter mean? Which is more ‘real’, the mathematical or the sensual, and what does such reality mean?

In his essay entitled The Unreasonable Effectiveness of Mathematics in the Natural Sciences, Eugene Wigner, one of the great mathematical physicists of our time, tells of two former high school classmates talking about their jobs. One had become a statistician and was working on population trends. He showed his friend a paper with the formula for the bell shaped curve used in statistics, and indicated the mathematical symbols which designated properties of the population distribution it described–its mean, its width, and so on. His classmate was slightly incredulous and suspected that the statistician was pulling his leg. He pointed to something on the paper and asked,

End of story. The classmate could not understand why the very special ratio ¼=3.14159… expressing a geometric ratio should also pop up in formulas concerned with human population. For that indicates some relation between them, and why, indeed, should circles and populations be related? Reasonable people would agree with the classmate.

Wigner’s listeners were not reasonable people–they were mathematicians. They recognized his punch line, so he was able to just continue as follows:

The classmate’s common sense reacted to the fact that,

Wigner knew his audience so expected such unexpected connections that they would immediately smile at the naiveté of the classmate. Indeed, his goal was to remind them of how unreasonable and, as he himself put it, eerie, this situation really is:

Two things should separately surprise all of us: "The miracle of the appropriateness ... of mathematics ... ", and "[the] entirely unexpected connections [it turns up]...". In the story, the appropriateness of mathematics is perhaps not so surprising since statistics does, after all, involve numbers. But that anything might connect counting people to geometry–connect the structures of population and space–should amaze anyone.

Mathematics reveals many such amazing connections, each the product of numerous steps of perfect logic. Although each step--each elementary numerical or algebraic manipulation--is easy to perform and understand, a path made up of numerous steps quickly becomes too long to comprehend as a whole, and so the connection it creates can remain amazing forever. This is part of the answer to Wigner's story.

Any system of thought and symbol that promotes the creation of numerous steps of perfect logic becomes a form of mathematics. These systems of thought are mental machines with perfectly interlocking parts. The first were number systems, with numbers, their parts.

This section identifies and discusses, through the filter of history, the significance of number systems. In so doing it reveals some of the reasons for the truth of the Pythagorean dictum.

One of the oldest surviving archaeological records of number is a 30,000 year old Paleolithic tally stick found in Moravia. It is a wolf’s bone, 7 inches long, engraved with 55 notches. From the same period comes a bone plaque found in France, also notched with what appears to be a record of the sequential phases of the moon for two and a quarter months. A later plaque from 11,000BCE has a more complex notation and covers 3 1/2 years of lunar-solar observation. We know that the use of these did not involve ability to count for that would require remembering the names of more than a handful of integers, and that requires a system of names the first of which is known to have evolved far more recently than 11,000BCE.

Our system of names is connected to that of the symbols they name, the development of which is part of the history of writing. Much of this history has emerged from a study of over 8,000 clay artifacts called tokens--small fired clay objects from 1-3 cm across--that have been found scattered throughout the Near East. They first appeared about 11,000 years ago, at about the same time as the first cultivation of cereals. This suggests that newly expanded economy, based on the discovery of agriculture at that time, generated their use. Each token, in fact, stood for one product–one type of animal, one measure of grain, one jar of oil, and so on. Shape indicated type: ovoids represented jars of oil; spheres, measures of grain; disks, animals.

A token is an abstraction. A sheep is (1) a discrete object, (2) distinguishable from objects of other types (jars, ewes,…), and (3) a member of a set of similar objects (the set of all sheep). In this process of abstraction, these are abstracted from the set of all properties of sheep. The abstracted ones are those relevant to the intended use of the tokens; the others, properties of shape, skin, smell and so forth, are irrelevant to accounting purposes. As we shall later discuss in more detail, everyone abstracts in this way all the time; it is done consciously, unconsciously, and, significantly, it is done by the sensory system. Abstraction is biologically built into the nervous system.

The earliest tokens are found at Uruk, the first and foremost Sumerian city on the ancient delta of the Euphrates, situated between Ur of the Chaldees and Babylon, in southern Mesopotamia. These were plain tokens, found in 16 unmarked shapes. Complex, or marked, tokens began appearing after about 4 millennia. The changeover came with the first efflorescence of manufacture; the greatly expanded range of products the tokens could represent extended from textiles and garments to bread and trussed ducks.

Marking was an easy way to create the hundreds of types of complex tokens that facilitated a dramatic increase in economic complexity. The increase in token complexity continued for about another 500 years, until about 5600 years ago.

Marks were precursors of written words. A word would have represented an object, by itself, whereas a marking merely distinguished between different types of tokens. Thus, people still thought a single solid object–a token–should represent another single solid object. They did not utilize (perhaps they did not realize) the fact that this solidity was irrelevant to a token's true purpose.

Like the first step towards writing, the next was also generated by economics. Clay envelopes containing tokens served as bills of lading accompanying shipments of goods. Originally, they had to be broken open to see their contents, but about 5600 years ago Sumerian accountants began pressing tokens onto the outer wet clay surface of the envelope before filling, sealing, and baking them. The external impressions made it unnecessary to break open the envelope to know what was inside; once the irrelevance of what was inside became obvious, writing began its birth. It took yet another five centuries for first the envelopes and then tokens themselves to disappear.

By 5100 years ago, pictographic script traced with a stylus on a clay tablet came into its own. This final, full abstraction of the written symbol led to the next characteristic step of reduction--systematization--organization according to uniformly applied rules; and this implies in turn a set of objects subject to uniform rules. Physical tokens and inscriptions on them being very different things, relatively few rules can be uniformly applied to both; thus the advantage in their separation through abstraction.

Similarly, since symbols for quantity are subject to a powerful set of rules that are inapplicable to symbols for things and qualities, it is advantageous to abstract the former from the latter. An single incised symbol initially stood for three combined concepts such as one+jar+oil. This was soon analyzed into its component parts, and ease of incision allowed symbols for them to be generated. When the symbol for ‘one’ became distinct it became the first purely numerical symbol.

The separation of symbols of different kinds is an example of reduction through combinatorics. Suppose a merchant sells 10 spices A, B, C,…,J, in 3 grades, low, medium and high. Without analyzing an item into its type and grade 30 unrelated symbols would be needed to represent all possible items of merchandise. With analysis, we create three symbols, one for each grade, and ten symbols, one for each spice. Using them, 30 different symbols are reduced to combinations of 13 symbols (e.g.A1,A2,A3, B1,…,J3).

This modest reduction of magnitude 30/13 grows quickly if one supposes up to 1000 of 100 different commodities were traded. The comparison would then be between 100+1000=1100 words and 100´ 1000=100,000 word combinations. Instead of scribes being required to invent and remember 100,000 separate symbols, they need do so for 1100, which is about a hundred times less effort. The data reduction is 100,000/1100, or about 100.

This form of reduction is found everywhere. The alphabet provides a particularly dramatic instance of its power. All combinations of the 26 letters of the alphabet arranged into, say, not more than 5 letter words would produce 12 million different words. Writing itself, combinations of words, even more so: a few hundred thousand words in different combinations are still producing an endless supply of literature. Atoms and genes provide important natural examples of how combinations of few things produce many.

The central problem appears to have concerned zero. The notation treats zero and the other ciphers, 1 through 9, equally, whereas for the longest time the former was not viewed as a number at all. This view created a 'box' out of which it was very hard to see--so hard, in fact, that even after the adoption of this notation, zero was given only grudging acceptance as a number. Thus, in an early book on arithmetic its use in subtraction is described as follows:

The author here does not say that when nothing is left over in, say, the third decimal place, write a little circle to signify the number 0x100. That little circle, 0, is not yet a number–it is merely a placeholder, a reminder. And even as late as the 15^th century zero was described as a symbol that merely causes trouble and lack of clarity.

The problem with zero that troubled thinkers was that they thought of number as describing a multiplicity of units of things whereas zero described, literally, no-thing. In fact, they were even troubled by the idea that one was a number, since that too does not describe a multiplicity. Some quotes to this effect, from early Renaissance writers are as follows:

The shift to the modern view of zero, and to number in general, can be traced back to what is called the 'Algorithmic Revolution'.

Long division as learned in grade school illustrates what is meant by an algorithm. It is a set of rules for performing a calculation mechanically, without real thought, by rote: a procedure that can be done by a machine, or by a relatively few lines burned on a computer chip. Algorithm is a word derived from the name of the Arab mathematician Al-Kwarizmi who wrote the first book describing how to calculation mechanically using the Indian notational system. These methods became known collectively as algorithmic arithmetic--history's first example of methods classed as symbolic manipulation. Competing with it were older sets of methods, most prominently the set developed for computing on an abacus--a system of beads strung on wires--token manipulation. Practitioners were known as algorithmi and abaci respectively.

As each abacus bead is a token of unity, the changeover from abaci to algorithmi, is analogous to that between users of clay tokens and of inscriptions on clay tablets. Both led to unforeseeably large changes in human culture. The immediate reasons for the changeovers were also analogous: a growth of international commerce increased the demand for people with arithmetic skills, and algorithmic methods allowed ordinary people to do calculations that had previously been beyond them.

The simplicity of algorithmic methods is based on the uniform perfection of the notational system. In comparison, for example, the imperfect Roman system makes long division difficult. Algorithmic long division can be done semi-consciously, whereas in Roman notation it requires constant careful alertness.

Alfred N. Whitehead commented on the significance of this as follows

By the mid 13^th century algorithmic arithmetic was established in Italy well in advance of anywhere else in Europe, the immediate reason for this being the commercial pre-eminence of the northern Italian cities. Dissemination was slow: although by the 15^th century one could learn algorithmic addition and subtraction in Germany, instruction in the arts of multiplication and division could still only be found in Italy. By the beginning of the 16^th century, just before the scientific revolution, the Germans were still 200 years behind!

The algorithmic revolution made zero into a number by expanding the meaning of number, a process driven by expanding its use.

Before the advent of algorithmic arithmetic, a calculation required continual thought: remembering numbers, rules, and relations, adding, multiplying, and so on. It was a mental stream of concepts and operations in which occasionally a number might be written, but only as an aid to memory. The written symbol sat beside the stream.

In the switch to algorithmic arithmetic, calculation became symbolic manipulation, and a stream of symbols replaced a stream of thought. In the symbolic stream, 0 and 1, were very similar to the other (multiplicity denoting) ciphers. The algorithmi began to see the ciphers 0,…,9 as merely the atoms out of which the symbols for all integers were formed, and all integers began to be viewed in terms of their roles in symbolic manipulation.

The practice of the algorithmi effectively established a dual meaning for number, and all are now computational symbols, while only some also are quantifiers (the natural numbers quantify discreta, the real numbers, continua). We now switch between the two meanings without much thought, and naturally this has caused confusion over meaning. In 250 CE, the Greek mathematician Diaphantos, had already stated general rules for manipulating negative numbers: Minus multiplied by minus gives plus, minus times plus gives minus…, yet in 1545, the eminent mathematician Cardan still called negative numbers ‘absurd’. Similarly, today, physicists often represent time in a computation as an imaginary number, and some will say "time is imaginary". Without specifying meaning, this also appears absurd.

Algorithmic arithmetic depends on the ability of infinite sets of numbers to transform one another into one another: 1 transforms itself into 2 through the operation of addition, 7 transforms 9 into 63 through multiplication, and so on. A set of numbers plus a set of operations transforming them into members of the same set (allowed operations) defines a number system.

For example, the natural number system (1,2,...) allows only subtraction of lesser from greater. If you break this rule, the result is not part of the system; within its context, the result is meaningless. As a result, if you create an algorithm that uses the naturals, but requires subtractions, it is hard to ensure that an undefined symbol (1-2) will not appear during the calculation. The integer number system (…,-2, -1, 0, 1, 2,…) does not have this problem; it permits any subtraction.

Similarly, while multiplication is allowed for all pairs of natural numbers, division is not. As a result, if you create an algorithm that uses the naturals, but requires divisions, it is hard to ensure that an undefined symbol (such as 1/2) will not appear during the calculation. The rational number system does not have this problem.

Similarly, you are allowed to take the square root of 5 if you are working in the real number system but not in the rationals because Ö 5 is not rational (it was called an '(ab)surd' number). And again, you are allowed the square root of -5 in the complex number system, but not in the real because Ö -5 is not real (it is called 'imaginary').

The number systems (the natural numbers, integers, rationals, reals, complex, quaternions) were created as people expanded the computational use of number. Like algorithms, they were created to eliminate unnecessary brain-work. Each system defines its allowed operations, and guarantees that they work together smoothly, like gears in a machine; they never lead to mathematical inconsistencies. No matter how long, a calculation is guaranteed to be correct as long one respects only its few simple prohibitions (e.g. in the complex number system you are allowed to take any complex root of any complex number except zero).

Allowed operations are also uniform: e.g. the rule for adding fractions (i.e. rational numbers) is the same for all. Lacking this property, the important algorithms of arithmetic (for adding long numbers, long division, for extracting a root), all built out of simple cycles, indefinitely repeated, could not be guaranteed. Calculations that had been long and often tricky were reduced to being merely long: a few simple steps mechanically repeated. This achievement of uniformity was made possible by the prior abstraction: the characteristics of number were reduced to only those relevant to their computational use. Striped away were the irrelevant properties associated with their origin as symbols of multiplicity.

Systems that lack the foregoing properties are complex. They are characteristically composed of many dissimilar parts that operate on each other in many dissimilar ways; in contrast, the infinitely many integers are all similar--each, just "one more of the same"--and they operate on each other in just a few, simple ways. A paradigmatic example of a complex system is a large scale computer application; typically, it can never be guaranteed to be bug free, whereas computational applications of number systems can be. One can say that a complex system is never completely understood, whereas an arithmetic algorithm is, always. As suggested, reasons for such differences reduce to the concept of reduction.

A properly running machine has all it parts moving exactly according to plan, without the necessity of an outside hand reaching in to fix or adjust--without a deus ex machina. A God or prime mover may be needed to supply power (and to design and build the machine), but otherwise its motion self regulating. Each gear and lever regulates the motion of the other. Such a machine can be said to run autonomously.

The atoms of numerical calculation are the ciphers 0,1,…,9. The molecules that determine material properties are the numbers formed from the ciphers, be they the integer, real, or complex. The gears and levers are the basic operations defining transformations amongst the numbers. The machine design is the algorithm. The source of power (but not intelligence) is the human computer (or the electric power running the electronic computer).

Once numbers to be input to an algorithm are set, and the power is turned on, then, without thought or intervention, a stream of symbols appears. Just as gears and levers change each other, so do the numbers produced during the calculation. The numbers transform autonomously according to the sequence of operations allowed by system and prescribed by algorithm.

As we shall see, this kind of symbolic autonomy is essentially unique to number, and it is the key to its role in the physical sciences.

We can imagine many sets of discrete elements with these properties: the set of all instants in time starting from the first, the set of all humans as ordered by birth starting with Adam, the set of all theorems of arithmetic alphabetically ordered, and so on. No notational system is needed to imagine their structure.

But we could number them if we wished (first, second,…). They are all numerable. This suffices to show that their structures are the same.

We can superimpose additional structure on these sets. In fact, it is natural to think of humans and theorems in terms of multiple ancestors--parents in one case, predicates in the other. In principle, a notation reflecting ancestry could be constructed out of the natural numbers. In fact, all types of number can be constructed from the naturals, even non-denumerable numbers such as the reals. Thus, one can say that the naturals form the simplest and most basic number set. In this sense, they are unique.

The foregoing, when supplemented by another important idea, can provide an understanding of why Pythagoras was correct in stating that all is number. The supplementary idea--remarkably, found already throughout early Greek natural philosophy--states that physics cannot tell us what is but only how things change. Here, change refers to both properties and place, i.e. it includes motion. 'What is', in this context, refers to composition but not conformation. It asks for components, but not how they conform themselves. Conformation falls under the heading of change: how components arrange themselves in positions and orbits.

Thus, if one asks, what is water? The answer is Hydrogen and Oxygen. Then, what is Hydrogen? A proton and an electron?…then what is an electron? An elementary particle?…then what is that? That just is; 'elementary' means we cannot say of what it is made.

If you claim that the electron is not elementary and that you know of what it is made (e.g.strings, energy,…?) you just get to the same question down a level. Eventually, there is an end to it; a final name. It could be some other elementary particle (or basic matter, field, or energy)--any kind of basic stuff. Whatever it is called, it just labels a set of associated properties, and these just specify interactions. (For example, an electron's properties of mass and charge help specify its interactions with gravitational and electromagnetic fields.) But as the sole purpose of interactions is to determine change, asking what something is, leads only to information about how it changes; it cannot lead to qualitatively different information.

In sum, the endpoint of a physical inquiry into what something is, can be no more than a catalog of its constituent elementary parts--a catalog of names whose meaning consists solely in information about how things change.

The word 'interaction' combines internal and action; when things interact, the action is internal, and internally determined. There is no external determinate (external to the set of interacting things) of change. Interacting things therefore change autonomously: according to their own (natural) law. This (standard) use of the word interaction by Physics reflects its sole concern with natural change, nature being definable as just that part of existence that changes autonomously, whereas supernatural means beyond nature and its laws. Autonomy is a powerful idea. It instructs us that natural change is regulated without reference to anything external to nature, and this includes, in particular, without reference to gods, or to God.

If, then, the basic stuff of nature changes autonomously, then the scientific symbols representing that stuff must do so also. The central role of autonomous change in physics, and the essential uniqueness of symbolic autonomy to number, together, form part of the reason that All is Number. Further parts will emerge as we proceed.