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.

__References__
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Diana Deutsch Accessed 8/20/2015::
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Dummett M (1964) Bringing about the
past. Philosophical Review 73: 338–59.

Eco U (2002). Art and beauty in the
middle ages. Yale University Press.

Everett C (2012). A Closer Look at A
Supposedly Anumeric Language 1. International Journal of American Linguistics,
78(4), 575-590.

Fine K (2012). Mathematics: Discovery
of Invention? Think, 11, pp 11-27

Fitch WT & Friederici AD (2012).
Artificial grammar learning meets formal language theory: an overview. Philosophical
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Accessed 8/20/2015: http://doi.org/10.1098/rstb.2012.0103

Hockenbury DH & Hockenbury SE
(2006). Psychology. New York: Worth Publishers.

Kirk GS & Raven JE (1964). The
Presocratic Philosophers, Cambridge University Press.

Lagopoulos, A. P., &
Boklund-Lagopoulou, K. (1992). Meaning and geography: The social conception of
the region in northern Greece (No. 104). Walter de Gruyter.

Ross
KL (2011) Mathematics & Music, after Pythagoras. Accessed 8/20/2015: http://www.friesian.com/music.htm

Stillwell J (2006). Yearning for the
impossible: The surprising truths of mathematics A. K. Peters, Ltd.

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