Thus communication was an important factor right from the beginning. It's importance is traditionally linked to the need for cooperation in hunting large animals. However, the fact that large animals hunt us also plays a role. To defend ourselves we clustered in groups so that we have many eyes and many ears to warn us of their approach. Since social animals need to trust one another, we established trust through mutual grooming. In his book, Grooming, Gossip, and the Evolution of Language, Robin Dunbar argues that language evolved as a sort of grooming-at-a-distance mechanism when the social group got too large for direct grooming [DUNBAR]. Gossip too serves an evolutionary function because it leads to a bad reputation and thus ostracizing of members of the group who can't be trusted. Thus, whether for attack or for defense, communication played a role right from the beginning, and, as I will argue in this book, will continue to be the major factor in the subsequent story of our species.

Despite such attempts by evolutionary psychologists like Dunbar, the origin of language in our species is still shrouded in mystery. It happened very long ago and it has left no physical record. The absence of evidence does not, however, prevent us from having theories about it. Indeed, it seems that the fewer the facts, the more the theories. Speech evolved as imitation of sounds heard in nature (ding-dong theory), as imitation of sounds made by animals (bow-wow theory), out of interjections (oof-ouch theory), to accompany strenuous group activity (yo-he-ho theory) are a few of the candidates. Speculation about the origin of language was so rife and viewed as so futile that, in 1866, SociÈtÈ Linguistique de Paris banned any further discussion in their journals [DEACON, Page 14].

An alternative approach is to identify the design features of language and determine which features distinguish human from animal communication. Charles Hockett lists three such design features of human language - displacement, productivity, and duality of patterning [HOCKETT]. While I was a graduate student at Cornell University, a campus debate developed between Charles Hockett and Karl von Frisch. Von Frisch had conducted extensive research on the "language" of bees [VON FRISCH 1950]. He had discovered that a bee could communicate the source of pollen to other bees in the hive by doing a dance in a figure-eight, in which the angle of orientation of the 8 indicated the direction and the number of wiggles in performing the figure-eight indicated distance from the hive. That is, it passed on the polar coordinates of the pollen source. Von Frisch argued that this "language of the bees" had the design feature of displacement - the bee could "talk" about things which are not here and now - and productivity - the bee could "say" things which have never been said before, when it gives precise polar coordinates never before used by its species.

Hockett argued that such communication between bees should not be described as "language" (hence the inverted commas around language in his title). It was pre-wired into the genetic code of bees - that is, it was genetic not extragenetic. This argument was vindicated by later work by von Frisch himself on dialects in bees. When North American bees were mated with European bees with a different "dialect", the sons of bees could not communicate with either parent, since their "language" was some compromise between the two dialects [VON FRISCH 1967].

In my introductory psychology textbook, I included a footnote to the third distinguishing design feature of human language - duality of patterning - which stated that I didn't understand this feature [GARDINER 1970].¹³ In the second edition, I added a footnote to this footnote, in which I stated that I had talked to Charles Hockett about this feature and I still didn't understand it. You'll be happy to know that I now finally understand it.

Language is a hierarchy of units plus rules for combining units at one level to create meaningful units at the next level. More precisely, language consists of phonemes (roughly equivalent to the letters of the alphabet), morphemes (roughly equivalent to the words in the dictionary), sentences, and discourses, plus the rules of vocabulary to combine phonemes into morphemes, of grammar to combine morphemes into sentences, and of logic to combine sentences into discourses (Figure 2-1).¹⁴

The foundation of language - the sounds which we utter and hear - are converted by our auditory system from an analog to a digital code - the language which we understand. "Duality of patterning" refers to this analog-to-digital feature of language. Our auditory system, like all our sensory systems, is an analog-to-digital convertor.¹⁵ At the upper level of language - the level of logic - it is digital in a more global sense. Since a sentence can be considered as a statement about the objective world, it can be said to be either true or false. The rules of logic enable us to determine what follows from certain assumptions about the objective world. Thus, for example, if we assume that "the shortest distance between two points is a straight line" and the other axioms of Euclidian geometry, we can deduce that "the square on the hypotenuse of a triangle is equal to the sum of the squares on the other two sides".¹⁶

In school, we learn the rules for combining phonemes into morphemes (vocabulary) and for combining morphemes into sentences (grammar) but not usually the rules for combining sentences into discourses (logic). Here's a short course in logic to help fill that gap in your education.

Let us imagine an empty universe and let us introduce into it the proposition. In the beginning, was the proposition. A proposition may take many forms - today is Tuesday, Now is the time for all good men to come to the aid of Jennifer, Kafka is a kvetch, E equals M times C squared - but they all have in common the fact that they can be said to be true or false. Our proposition - let us call it P - looks lonely in an empty universe, let us introduce another proposition - say Q - which , like all propositions, can be said to be true or false. In an empty universe containing two propositions, there are thus four possible states of affairs - they are both true, the first is true and the second is false, the first is false and the second is true, or they are both false (Figure 2-2). There exists within the English language, the means of eliminating each subset of this exhaustive set of alternatives. "Not both P & Q" eliminates the first alternative, "If P, then Q" eliminates the second alternative, and so on.

The four logic problems set for you in Figure 2-2 focus on "if P, then Q". Check your answers as follows. The premises (the propositions after "Suppose you know that") eliminate some subset of the alternatives in a universe containing two propositions, and the conclusion (the proposition after "Then would this be true?") is evaluated in terms of the remaining alternatives. If it is contained in all the remaining alternatives, the answer is "yes"; if it is contained in some of the alternatives, the answer is "maybe"; if it is contained in none of the alternatives, the answer is "no".

A certain subset of propositions (e.g. I am a Scotsman, this is a pipe, etc.) states that a particular element is a member of a particular set. Because such propositions occur so often, the cumbersome "If x is a member of set A, then x is a member of set B" is condensed to "All As are Bs" The statements "Some As are Bs" and "No A's are Bs" can be generated in the same way. Class reasoning, involving those relations of All, Some and No, permits us to place the objects in our world into categories and consider the relationships between them within those categories. In the same way, the propositions stating the position of an element with respect to a dimension generates the relations "is greater than", "is equal to", and "is less than". Ordinal reasoning, involving those relations, permits us to place the objects in our environment along dimensions and consider the relationships between them along those dimensions.

Having placed objects along dimensions, we can conclude that, if A is greater than B and B is greater than C, then A is greater than C. In order to be more precise - to say how much greater A is than C, we have to invent the natural numbers. Thus, if A is 8 units, B is 6 units, and C is 3 units, then A is 5 units greater than C.

Within this system, you can always add and get an answer. However, if you reverse this operation and subtract you can't always get an answer. You have to invent 0 to provide an answer when you subtract a number from itself and negative numbers when you subtract a number from a smaller number. Within this system, you can always multiply and get an answer. However, if you reverse this operation and divide, you don't always get an answer. You have to invent fractions or decimals to provide an answer when the division is not even. Within this system, you can always square and get an answer. However, when you reverse this operation and take the square root, you don't always get an answer. You have to invent irrational numbers to provide an answer when the number is not a perfect square.

Thus the entire superstructure of mathematics is erected in order to permit closure under various mathematical operations (Figure 2-3). No matter how sophisticated it gets however, it is still based on class and ordinal reasoning, which are special cases of propositional reasoning, which is an aspect of everyday language.

This "nutshell" history of mathematics is, of course, grossly oversimplified. Two recent books [KAPLAN, SIEFE] describe the long, complex, and arduous process of discovering only one element in this system - zero. The important point is that those elements are discoveries rather than inventions. They are all implied in the conception-day gift. It just takes us a long time to unwrap it. My Ph. D. thesis was a study of the development of understanding of the meanings of such "logical operators". John Roberts, an anthropologist on my thesis committee, replicated my study with Indian children. He assured me that every North American Indian language he knew also contained a full set of such logical operators. All languages are equally complex. They all contain the means of logic and thus of mathematics. The conception-day gift is a gift to all members of our species.

Thus, logic is the link between everyday English (or Ethiopian or Estonian or whatever) and mathematics. In school, we learn English in English class and mathematics in mathematics class. Because of the missing link of logic, we seldom realize that they are different aspects of the same thing. Thus, a wedge is driven between the artists, who excel in English class, and the scientists, who excel in maths class. Everyday language and mathematics are part of the conception-day gift of the wisdom of our species. This gift culminates in the magnificent invention of speech. Using language, we can share our experience of the world. Mathematics simply allows us to talk more precisely. It also allows us to talk to people who are not part of our linguistic community. Cosmonauts and astronauts both got into the same orbit around the same planet, despite the fact they spoke different languages, because they shared the common language of mathematics.

13   My editor was initially horrified. Textbook writers are supposed to know everything about their subject. However, he kindly permitted me to include the footnote, on the grounds that this textbook writer was an exception.

14   I say "roughly" because there is not perfect phoneme-grapheme correspondence and there are some morphemes - the smallest linguistic unit which has meaning of its own - which are not words. For example "-ed" at the end of a verb means past tense.

15   During an argument in a bar between two of my students about the relative merits of analog and digital media, I suddenly realized that the argument was familiar. It was an argument I had had with myself. After fifteen years of concentrated study and teaching, I spent a decade in California to get "out of my mind and back to my senses". I had become too digital and needed to restore my analog-digital balance.

16   I flunked mathematics all the way through High School. My first teaching assignment was to teach maths to all the High School grades in a small school. In desperation, I sat down and read the preface to Euclid's Geometry . Euclid said, lets pretend the following statements are true and listed his axioms - The shortest distance between two points is a straight line, etc. Now, he said, lets see what follows and proceeded to build up his system of theories on this foundation. I learned Euclidian Geometry in an evening. In four years of High School, I couldn't learn it because no one told me that it was a game of let's pretend and let's see what happens. The Pythagorean Society considered those theorems sacred and killed anyone who divulged them to outsiders. This secret society is still alive and well with cells in High School common rooms!