This discussion focuses on psychology and the philosophy of mathematics and will contribute nothing to mathematical thought. Its aim is to introduce mathematics as a creation of psychology. Sophisticated, complex and ever evolving, but nevertheless psychology.
Mathematics translates patterns into reducible parts. These parts form theorems—incremental reasoning based on a chain of formal proofs—that conform to logic but operate beyond logic. Mathematicians argue that these patterns are universal and real and that the interconnecting system of reducible parts is what constitutes mathematics—a language of spatial positioning, geometry, numbers, volume, movement and patterns. These are complex patterns that lead to complex theorems.
Sometimes these patterns exist in reality and prove useful in terms of predicting physical events in the universe and sometimes they are the perfect embodiment of a cognitive world--true forms that exist primarily in our imagination, such as the perfect circle. Sometimes the theorems relate to patterns that are solely--as far as we know, or yet--in the realm of a group of mathematicians’ imagination. Although mathematics is not set-up, by mathematicians, to explain our reality, there is however a symbiotic relationship, in that proofs can come from within the physical experimental world.
The basis for elevating mathematics to more than just a complex system of creating theorems is the role that mathematics was given by Pythagoras (6th Century BC). Pythagoras believed that numbers were not only the way to truth, but truth itself. That mathematics not only described the work of god, but was the way that god worked. This belief, that mathematics holds an intrinsic truth remains with mathematicians today. They believe that mathematics is the language of the gods. And that is a problem if you do not believe in god or in an over-ridding principle of existence--none that we can understand anyway. Science is by definition both atheist and agnostic despite what individual scientists believe. Most mathematicians behave as deist who believe that God created the universe but that natural laws determine how the universe plays out. This is a Epicurean (341–270 BC) belief that the gods are too busy to deal with the day-to-day running of the universe but they set it in motion using mathematics.
Mathematicians therefore argue that mathematics is a higher order that is found in reality. But there are no examples of such proofs. Mathematicians argue that they are more discoverers rather than inventors. But this dichotomy also seems false. Mathematicians seem to do both, most often at the same time. The British philosopher Michael Dummett suggests that mathematical theorems are prodded into existence--he uses the term probing (Dummett, 1964). Using the analogy of the game of chess where, “It is commonly supposed … that the game of chess is an abstract entity” (Dummett, 1973). But there is certainly a sense in which the game would not have existed were it not for the mental activity of human beings. It is a delusion to believe that just because we find a pleasing pattern, a game that resonates across cultures, that the reason it is pleasing is because there is a god behind it. But mathematicians argue that chess, or theorems are not entirely products of our minds since there must already be something there to prod. But the obverse argument is equally true that mathematical “truths” are entirely dependent on us since we need to prod them to bring them into existence.
The same is true for language, art, music and other “Third World” constructs—these are incrementally evolving systems and form one of Karl Popper’s ontological tools (Carr, 1977). Third World is where the system that is developed exists beyond the creator. Language is an excellent example, although Third World also includes abstract objects such as scientific theories, stories, myths, tools, social institutions, and works of art. Language is incremental and ever evolving, and is used to help us communicate reality. Within this Third World, language as with mathematics, is also argued to be both discovered or invented.
Theory of language development has oscillated between two schools of thought. One school that argues that language is culture-bound, known as Descriptivists. And on the other side is the argument that promotes language as part of our biological makeup, known as the Generativists. As a Generativist, Chomsky (1980: p134) phrased it eloquently when he said that, “we do not really learn language; rather, grammar grows in the mind”. The analogy between formal mathematical systems and human languages is not a new or novel idea. In fact such formal language theory have already been established in its modern form by Noam Chomsky in an attempt to systematically investigate the computational basis of not just human language but has become applicable to a variety of rule-governed system across multiple domains--computer programs, music, visual patterns, animal vocalizations, RNA structure and even dance (Fitch & Friederici, 2012). This symbiotic relationship exists across all Third World constructs: mathematics and music, music and art, art and language and all other permutations. As with mathematics, we refine language with time. Future generations build upon language and mathematics and the only constraint seems to be our psychology. Mathematics similarly has this incremental nature. The last sentence of a talk given by Fine on mathematics “The only constraint is our imagination and what we find appropriate or pleasing.” (Fine, 2012: p27). What we find appropriate and pleasing is where the psychology comes in and our clue to the inception of mathematics and the description of our psychology.
As a guide, we have to go back to earlier (and more simple) mathematics to understand this principle of “pleasing.” Pythagoras and music is the basis for a convergence between mathematics and psychology. Pythagoras (6th century BC) observed that when the blacksmith struck his anvil, different notes were produced according to the weight of the hammer. He later discovered that the ratio of the length of two strings determines the octave "that the chief musical intervals are expressible in simple mathematical ratios between the first four integers" (Kirk & Raven, 1964: p.229). Thus, the "Octave=2:1, fifth=3:2, fourth=4:3" (p.230). These ratios harmonize, meaning that are pleasing both to the mind and to the ear. Although this mathematical system breaks down the higher we go up the scale, there was a solution by adjusting the ratio of the fifth so that it is commensurable with seven octaves. Seven octaves is 128:1, or 27. John Stillwell (2006) argues that "equal semitones" or "equal temperament" (p.21), was developed almost simultaneously in China and the Netherlnds, by Zhu Zaiyu (Chu Tsai-yü) in 1584 (during the Ming Dynasty and by the Simon Steven in 1585 and by (Ross, 2001). But the point is that a mathematical rule was developed on the basis of a harmony that we humans find pleasing.
In nature, all sounds are the same. The creator of the universe created all acoustics, all sounds are perfect. Nature cannot discriminate among them since they are all necessary and useful. As such, selecting harmonics is psychological rather than godlike. We like the separation of scales because we can psychologically compartmentalize the sound. We are creatures of order and consistency and prefer to have distinct and distinguishable sounds. In reality there is no such thing as harmonics, we look for it as humans because it is pleasing.
Such psychological preferences are automatic and require no processing and thinking on our part. This automation can be easily be disrupted by playing a tone that is ostensibly ever increasing or decreasing without end. Such a tone was developed by Roger Shepard and consists of a superposition of sine waves separated by octaves. This creates the auditory illusion of a tone that continually ascends or descends in pitch, yet remaining constant.
Not only does the Shepard Tone create dissonance because we find it difficult to understand, it also creates uneasiness as a result of this dissonance. This perceived auditory dissonance causes emotional uneasiness. We become uncomfortable when we cannot pigeon hole our perception. We need sounds that are at a prescribed distance from each other that make perception easier. Pythagoras defined the first mathematical rule for auditory perception, the definition of an octave that pleases our psychology for order and form. The fact that both European and Chinese figured this out at the same time indicates that the perception of octave generalizes across linguistic and auditory differences (for more auditory illusions see Deutsch, 2011). These psychological requirements, codified into mathematics are also found true for vision.
We like to see things in “chunks.” Mathematics was the earliest discipline to reflect this psychological need by inventing the number “one.” This basis of an “entity” forms the upside down pyramid of mathematics. Without a “one” there is no mathematics. But there are problems with the number one. There is a point at which a “one” cannot be defined mathematically, or where it fails to conform to some particular way, such as differentiability. This singularity--which is proving to be so problematic for mathematicians in explaining quantum physics for example--is only a problem for mathematicians, because an entity of “one” is the perfect creation of our mind and not nature. In fact the only way that quantum physics can explain superposition, entanglement and other quantum mechanics is by removing the “one” from the theorem. By removing the parenthesis around “one” quantum physics can be better explained, although then we have to readdress our psychology and the reliance on our perception of separate entities. From a psychological point this can be easier achieved rather than forcing quantum physics to conform to psychology.
History has been here before. Pythagoras--having traced the hand of god in how music is constructed--thought that each of the seven planets produced particular notes depending on its orbit around the earth. This was Musica Mundana and for Pythagorians, different musical modes have different effects on the person who hears them. Taking this a step further, the mathematician Boethius (480-524 AD) explained that the soul and the body are subject to the same laws of proportion that govern music and the cosmos itself. As the Italian semiotician Umberto Eco observed we are happiest when we conform to these laws because "we love similarity, but hate and resent dissimilarity" (Eco, 2002; p31).
This is not the first time that mathematicians thought they have touched the hand of god, neither will it be the last time. But what Pythagoras touched is our psychology. By focusing on pleasing patterns, similarities, and order, mathematicians are exploring the foundations of our psyche. And to do this they had to build rules and “common notions” that bind all these thoughts into a coherent language that translates into mathematics. For example if we take Euclid (4th Century BC) five "common notions” as defined in The Elements:
• Things that are equal to the same thing are also equal to one another
• If equals are added to equals, then the wholes are equal
• If equals are subtracted from equals, then the remainders are equal
• Things that coincide with one another are equal to one another
• The whole is greater than the part.
There is an unambiguous relationship with classic Euclidian mathematics and Gestalt psychology. Gestalt psychology has rules that mirror these Euclidian common notions (Lagopoulos & Boklund-Lagopoulou, 1992). But there have been further developments. The prolific Swiss psychologist Jean Piaget (1896–1980) while investigating children’s conception of space discovered highly abstract mathematical structures in the child’s primordial conception of space. He argues that the further development of geometric space should not be understood as reflecting the capacity of the child’s developing physiological functions, but as a product of the child’s interaction with the world. The child constantly builds up specific structures of perception and reorganizes spatial conception. Accordingly, Euclid’s elements and the topological properties of shapes have their origin neither in the world nor in the history of sciences, but in cognitive schemes that we build up in our daily interaction with objects.
The same understanding—that there are mathematical structure embedded in our cognitive processes—precludes the need for either mathematical or language. These theorems exist independent because that is how the brain is structured. A good example of this pre-mathematics and pre-linguistic ability is provided by a tribe that does not have a concept of numbers in its language. Dan Everett’s description of the Pirahã language of the southern Amazon basin exposes the tangled relationship between mathematical constructs and our cognitive capacity (Everett 2012). The Pirahã language has no clause subordination (e.g. after, because, if) at all, indeed it has no grammatical embedding of any kind, and it has no quantifier words (e.g. many, few, none); and it has no number words at all (e.g. one, two, many). But they can still count and perform quit complex mathematical comparisons despite the lack of linguistic structure. The main deficit is that they cannot memorize these functions. So they can perform mathematical functions only for the immediate situation. In Popperian terms, they do not have a Third World construct of mathematics to enable them to retain an abstract representation of numbers which mathematicians, through their use of mathematical language can. And mathematicians have created this language, this mathematics where “one” forms the foundation.
Mathematics however has evolved and built upon this concept of “one.” It would be naïve to assume that mathematics has stood still as a discipline. Although the early conception of “one” is very restrictive number, in which ‘number’ means ‘natural number’ mathematics evolved to adopt a less restrictive conception of “one” in which it means ‘integer’; then meaning rationals; then reals, and then complex numbers. With such creations, there is a more nuanced appreciation of the finite interpretations of “one”. In psychology we might distinguish a human being (aka one), and then talk about aggregate or composite features such as family, community, or head, eyes, nose (reals), and then complex numbers such becoming a millionaire, getting divorced, losing a limb, becoming blind (complex numbers.) Mathematics has not extended the domain of numbers, but liberalized what we mean by ‘number’ and as a collinearity what we mean by “one.” Our presumption that there is a single number “one” and that, in extending the number system we simply add and perform “functions” to the numbers that were already there is not what mathematics has become. There are as many number “ones” as there are types of numbers. But by redefining the meaning we are creating a new definition of “one”. One that is less suspect to investigation and study, and bears less of a relationship with anything tangible (Fine, 2012).
We think in very complex ways that is still not understood, continues to be misrepresented and remains misunderstood. The human brain has more synaptic transmissions than we have stars in the universe. The capacity for human thought is immense. Clues are emerging that we think in very abstract ways that mirror the development of theorems in mathematics. Holographic theory of thinking is just one crude method of representing this universe of thought. It is plausible that mathematics could be a portal to understanding our psyche, our art and our behavior. We could learn our limitations, and our attributes and allow for the exploration of a process that we do not yet know and cannot know. We grow up developing our thinking as theorems--despite that in some cases our language does not accommodate such thinking--we still use innate mathematics to develop our sense of numbers and patterns. Mathematics is our way of thinking. We simply grow out of it, as do mathematicians who simply grow out of being brilliant mathematicians and converge into cultural thought (language, roles and cultural morals.) Mathematicians have a short life of brilliance since their natural thought processes are eventually taken over by pragmatic concerns. Such is the final objective of our brain, survival in the real experiential world. Survival in a sentient world—a world dominated by feeling and experiencing. But mathematics can form the basis of formalizing theories of our thinking processes, mind sensations and feelings. We need to see beyond the silos of disciplines and view our humanity as more than pitting humans against the hand of god, and simply see the hand of god as our own genius waiting to be acknowledged.
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I am indebted to David Edwards, emeritus professor of mathematics from Georgia University for discussing with me the subtleties of some of these thoughts. Having such a knowledgeable and challenging adversary promoted the thinking of this argument and produced a much clearer thesis. However, all misrepresentations, deficiencies and shortfalls are purely my responsibilities.
© USA Copyrighted 2015 Mario D. Garrett