Pi Of Life The Hidden Happiness Of Mathematics Pdf

शिशुर्वेत्ति पशुर्वेत्ति वेत्ति गानरसं फणी । साहित्यरसमाधुर्यं शङ्करो वेत्ति वा न वा || śiśurvetti paśurvetti vetti gānarasaṁ phaṇī | sāhityarasamādhuryaṁ śaṅkaro vetti vā na vā || The ultimate sweetness of music is understood and enjoyed even by an infant, even by an animal and even by a snake. Indeed, we do not know whether the erudite scholar can really enjoy the sweetness of the nectar of literature, musical lyrics and the arts. The popular belief in the scientific community is that, "Mathematics is the best way of expressing the universe ...". But perhaps there is another contender, a stronger one in fact, and that might be music. We discuss how music, across cultures and time, has been the constant message of hope and happiness. Possibly, music is present in our very genetic codes giving us an innate sense of rhythm and endowing the capacity for appreciation even to newborns. We show using simple logic (isn't logic a gift from the domain of mathematics?) that one day, once we understand both mathematics and music better, we might see both of them as the same language. Until then perhaps, we should consider supplementing mathematics with music, or even replacing it if necessary, depending on the extent of comprehension in the audience. The educational (and policy) implications are to ensure that the future creators of knowledge are equally adept at both music and mathematics. This dual capability would enable researchers to encode their findings in such a way that it can be appreciated by an audience larger than the following we presently have for the mathematical transmission of knowledge.

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The Universal Language: Mathematics or Music?

Ravi Kashyap (ravi.kashyap@stern.nyu.edu)1

Estonian Business School, Tallinn, Estonia / Synesis One AG, Zug, Switzerland / City University

of Hong Kong, Hong Kong

October 25, 2021

Keywords: Mathematics; Music; Universal; Language; Multicultural; Education Policy; Don't Worry Be Happy;

Artistic Encoding of Knowledge

Mathematical Classiﬁcation Codes: 00A30 Philosophy of mathematics; 97B10 Educational research and planning;

97A80 Popularization of mathematics

Edited Version: Kashyap, R. (2021). The Universal Language: Mathematics or Music? Journal for

Multicultural Education, XX(X), XX-XX.

Contents

1 Abstract 2

2 Introduction 4

1The traceable seed for this paper was the "Culture Day: A Journal to Remember" in November, 2017 at the SolBridge International

School of Business where the author was a faculty member. There were music and dance performances from around thirty countries.

Not to mention food, which can be considered a feat, both the cooking and consumption aspects of it. It could be said that none of

the actors / actresses, both the performers and the applauders, understood more than a handful of languages (let us say, the maximum

was seven). But everyone seemed to enjoy the entire show, despite not comprehending more than a few words of the songs from other

civilizations. This enjoyment, of music and songs from unknown languages, is in stark contrast to the proceedings in most classrooms

in most universities where the lectures are held in a language which is somewhat familiar to most of the class members. This shows us

that music is indeed more universal than mathematics or perhaps any other language.

The students, faculty and other members of SolBridge have been a musically marvelous source of inspiration for this paper and many

others. Dr. Sherry Deckman, Dr. Glenn Hardaker and anonymous reviewers provided numerous suggestions to improve this manuscript.

Dr. Yong Wang, Dr. Isabel Yan, Dr. Vikas Kakkar, Dr. Fred Kwan, Dr. William Case, Dr. Srikant Marakani, Dr. Qiang Zhang, Dr.

Costel Andonie, Dr. Jeﬀ Hong, Dr. Guangwu Liu, Dr. Humphrey Tung and Dr. Xu Han at the City University of Hong Kong provided

encouragement to explore and where possible apply cross disciplinary techniques. The teachers at Prelude Music Studio, Singapore

provided a fascinating introduction to the world of music. The views and opinions expressed in this article, along with any mistakes,

are mine alone and do not necessarily reﬂect the oﬃcial policy or position of either of my aﬃliations or any other agency.

3 Challenging the Matrix (of Symbols?) 7

4 Mathematics is / as Music and Vice Versa 11

4.1 Music Plus Mathematics Equals Melody . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Musical Message, Medicine for the Mind ... Don't Worry, Be Happy 13

6 Ignoring Quirks called Quarks 14

7 Artistic Encoding of Knowledge 16

8 Conclusion 18

9 References 20

10 Figures 32

1 Abstract

śiśurvetti paśurvetti vetti g¯anarasa ˙m phan

.¯ı |

s¯ahityarasam¯adhurya ˙m śa˙nkaro vetti v¯a na v¯a ||

The ultimate sweetness of music is understood and enjoyed even by an infant, even by an animal and even by a

snake.

Indeed, we do not know whether the erudite scholar can really enjoy the sweetness of the nectar of literature,

musical lyrics and the arts.

The popular belief in the scientiﬁc community is that, "Mathematics is the best way of expressing the universe

...". But perhaps there is another contender, a stronger one in fact, and that might be music. We discuss how

music, across cultures and time, has been the constant message of hope and happiness. Possibly, music is present

in our very genetic codes giving us an innate sense of rhythm and endowing the capacity for appreciation even to

newborns. We show using simple logic (isn't logic a gift from the domain of mathematics?) that one day, once

we understand both mathematics and music better, we might see both of them as the same language. Until then

perhaps, we should consider supplementing mathematics with music, or even replacing it if necessary, depending

on the extent of comprehension in the audience. The educational (and policy) implications are to ensure that the

future creators of knowledge are equally adept at both music and mathematics. This dual capability would enable

researchers to encode their ﬁndings in such a way that it can be appreciated by an audience larger than the following

we presently have for the mathematical transmission of knowledge.

2 Introduction

Tegmark (1998; 2008) argues that with a suﬃciently broad deﬁnition of mathematics our physical world is an

abstract mathematical structure implying the "Mathematical Universe Hypothesis" (MUH). The central idea of

MUH is that our external physical reality is a mathematical structure. Mathematical existence equals physical

existence, and all structures that exist mathematically exist physically as well 2. Hofstadter (1979) considers many

aspects of mathematics, music, art, symmetry and intelligence. He provides a fascinating discussion of how cognition

emerges from hidden neurological mechanisms and how, through self-reference and formal rules, systems can acquire

meaningful context despite being made of "meaningless" elements3.

Dantzig (1930) chronicles the evolution of mathematics and shows how human endeavors, such as commerce,

war, and religion, led to advances in math. Pincock (2004; 2011) are discussions of the contribution of mathematics

to the success of science. Field (2016) argues that we can explain the utility of mathematics without assuming it

true. Waller & Flood (2016) explore whether mathematical language is a tool that transcends cultural lines and

present an argument that mathematical language exudes characteristics of worldwide understanding.

All these arguments lead to the commonly held belief in the scientiﬁc community that, "Mathematics is the best

way of expressing the universe. If we choose to ignore the murky waters of elementary logic, mathematics becomes

the language of the universe simply because it has to be"4. Before we explore mathematics, music and much more

in greater depth, we provide the following deﬁnition to set the stage for the rest of this article.

Deﬁnition 1. A universal language is any medium of communication to express and share any concept about any

aspect of the universe. Despite the fact that there can be many contenders for this designation, the instrument has

to be endowed with the capacity to be recognized and appreciated by many while being usable with the utmost

2MUH holds that the physical universe is not merely described by mathematics, but is mathematics (speciﬁcally, a mathematical

structure, see below). Mathematical Universe Hypothesis, Wikipedia Link

In mathematics, a structure is a set endowed with some additional features on the set (e.g. an operation, relation, metric, or topology).

Often, the additional features are attached or related to the set, so as to provide it with some additional meaning or signiﬁcance.

Mathematical Structure, Wikipedia Link; "The Deﬁnitive Glossary of Higher Mathematical Jargon — Mathemaical Structure"

3Douglas Hofstadter (in his 1979 book, Gödel, Escher, Bach: An Eternal Golden Braid, also known as GEB) also discusses what

it means to communicate, how knowledge can be represented and stored, the methods and limitations of symbolic representation, and

even the fundamental notion of "meaning" itself. In response to confusion over the book's theme, Hofstadter emphasized that GEB is

not about the relationships of mathematics, art, and music. Gödel, Escher, Bach, Wikipedia Link

4Mathematics Universal Language Link One

relative ease (compared to other possibilities).

Criterion 1. It can be seen that it makes sense for this means of expression to be related to how we receive stimuli

through our sense organs. Though, this method of articulation needs to be able to transcend any particular sense

faculty: known, yet to be known, known to be unknown, known to be unknowable, unknown unknown senses or other

combinations of how we perceive the world around us. It is also apparent that there is a great degree of subjectivity

when it comes to being recognized, appreciated or used. Hence, we emphasize that the deﬁnition is meant to provide

a benchmark to compare diﬀerent prospects. In this article we assess music and mathematics, but the deﬁnition

makes it clear that many new (or old but modiﬁed?) potential mechanisms might emerge over time.

The following descriptions of mathematics chronicle some reasons for why mathematics can be considered the

universal language. These provide some additional guidance as to what constitutes a universal language and what

must be the criteria that would need to be satisﬁed for such a claim. We provide corresponding properties of music

as well below.

1. "There is no simpler, more fundamental way of expressing the universe than through the basic ideas of equality

and inequality, which in turn lead to the concept of quantiﬁcation, which lead to the concept of value and

numbers, and once we have numbers, the rest of mathematics seems to bloom from all around us"5.

Pitch is a perceptual property of sounds that allows their ordering on a frequency-related scale, or more

commonly, pitch is the quality that makes it possible to judge sounds as "higher" and "lower" in the sense

associated with musical melodies (Klapuri 2006; Plack, Oxenham & Fay 2006).

2. "Mathematics has been the language of science for thousands of years, and it is remarkably successful"6.

Music has been the language of society for thousands of years. To think of the success of any society without

considering the corresponding music would be incomplete since music is remarkably intertwined with the

5Mathematics is the most fundamental type of logic possible (in physics anyway), and therefore it is easy to reason that mathematics

is the best way of expressing the universe. Mathematics Universal Language Link One

6If we ever establish contact with intelligent aliens living on a planet around a distant star, we would expect some problems

communicating with them. However, the surest common culture would be mathematics. Mathematics Universal Language Link Two

history of humanity. For discussions of music and society see: Supicic (1987); Leppert & McClary (1989);

Small (1996).

3. "Pi is still approximately 3.14159 regardless of what country you are in"7.

Lesser (2015); Chu, Urtasun & Fidler (2016) are discussions of how Pi can be used to create pleasing music

(albeit arguably). Benson (2006); Garland & Kahn (1995); van Gend (2014) explore the relationship between

mathematics and music using many mathematical concepts.

4. "The consensus is that advanced technology is near impossible without mathematics"8.

No advanced civilization has been without music and perhaps music can be a gauge of the extent of progress

in civilizations. Sachs (2008) has a discussion of the rise of music in the ancient world. Lehman (1989) surveys

the unsatisfactory state of arts education indicating that we are yet to make many advances as a civilization.

Throughout the rest of this article we discuss many aspects of mathematics and music. Section (3) considers how

music, across cultures and time, has been the constant message of hope and happiness. This suggests that music

could be a challenger for mathematics and a potential candidate for the title "The Universal Language". Section

(4) considers similarities between mathematics and music and how the study of one might aid in enhancing one's

abilities of the other. Section (5) considers the primary objectives of any engagement with music, including the many

therapeutic beneﬁts. Section (6) looks at complexity in mathematics, computing science and also in the universe

around us despite the simplicity that underlies their existence. This could oﬀer hope to those of us looking for a

universal language. Section (7) makes a case for combining musical training along with mathematical education, so

that knowledge can be encoded in musical form along with mathematical proofs.

We show using simple logic (isn't logic a gift from the domain of mathematics?) that one day, once we under-

stand both mathematics and music better, we might see both of them as the same language. Until then perhaps,

7Mathematics is the only language shared by all human beings regardless of culture, religion, or gender. With this universal language,

all of us, no matter what our unit of exchange, are likely to arrive at math results the same way. Mathematics Universal Language Link

Three

8The use of prime numbers in communication is talked about in Prime Numbers and the Search for Extraterrestrial Intelligence:

Pomerance (2004). If they contacted us - or we simultaneously met - chances are good that they would have access to mathematics.

Mathematics Universal Language Link Four

we should consider supplementing mathematics with music, or even replacing it if necessary, depending on the

extent of comprehension in the audience. The educational policy implications, for all ﬁelds, are to ensure that the

future creators of knowledge are equally adept at both music and mathematics. This dual capability would enable

researchers to encode their ﬁndings in such a way that it can be appreciated by an audience larger than the following

we presently have for the mathematical transmission of knowledge.

Other than the educational connotations, the message from this article is to create a blueprint for a society with

a vibrant musical and artistic environment. Such an atmosphere, including the pedagogical aspects, would have to

highly inclusive and multicultural given the wide assortment of musical and artistic developments across diﬀerent

communities (Cain 2010; Nethsinghe 2012; Balakrishnan 2017). This would be an additional way towards boosting

policy eﬀorts aimed at creating a more tolerant, united and peaceful society (Brevetti & Ford 2017; Arsal 2019).

3 Challenging the Matrix (of Symbols?)

While the beauty and utility of mathematics are perhaps not to be debated, the training required to discern

this loveliness is non-trivial. Rota (1997); McAllister (2005) try to uncover the sense of the term beauty as it is

currently used by mathematicians. Hersh & Ekeland (1997); Courant & Robbins (1941); Harel (2008); Dummett

(1994) consider what mathematics really is? 9. Berch & Mazzocco (2007) consider why math is hard for some

children. De Veaux & Velleman (2008) reason about the reasons regarding why statistics became harder to teach,

due to changes in the methods of teaching statistics, in the recent past. They have suggestions on superior ways

to teach math and many remarkable yet related insights on literature, chess, music and philosophy. Maloney &

Beilock (2012) discuss how many of the techniques employed to reduce or eliminate the link between math anxiety

and poor math performance involve addressing the anxiety rather than math training itself. Wang et al. (2014)

investigated the genetic and environmental factors contributing to the observed diﬀerences in the anxiety people

feel when confronted with mathematical tasks (Figure 1).

It might very well be the case that the dialect of mathematics we have adopted, with abstruse notation and

omitted details, might not be that conducive to grasp meanings with. Chinn (2013) is a comprehensive discussion

of problems when dealing with math problems and how in maths another key communication factor is the use of

9Mathematics (from Greek máth¯ema, "knowledge, study, learning") includes the study of such topics as quantity, structure, space,

and change. Mathematicians seek and use patterns to formulate new conjectures; they resolve the truth or falsity of conjectures by

mathematical proof. When mathematical structures are good models of real phenomena, then mathematical reasoning can provide

insight or predictions about nature. Mathematics, Wikipedia Link

symbols, which introduces another layer the need to relate symbols to vocabulary and to concepts. Hoﬀert (2009);

Whiteford (2009) are discussions about teaching mathematics, supposedly the universal language, to students

with deﬁciencies in the medium of instruction such as English. Tevebaugh (1998); Cavanagh (2005) oﬀer counter

viewpoints that mathematics is not necessarily the universal language that we have always thought it to be.

Devlin (2000) argues that we might have an innate ability for mathematical thinking even though the develop-

ments in higher math have been in the last 400 years or so, a time span which is as short as the ﬂap of a butterﬂy's

wings on an evolutionary scale. Human beings have evolved from the ﬁrst single celled organisms that appeared

around 3.5 billion years ago on Earth (Shubin 2008). Even with a very conservative estimate, we have appeared on

the horizon, in human form, very recently, around 2 million years ago. The bulk of the small amount of time that we

have been around has been spent trying to survive. Perhaps, as survival pressures eased, songs and stories began to

blossom, well before more advanced mathematical concepts started taking shape. Hence it is safe to conjecture that

music has been around a lot longer than formal mathematics (Burney 1789; Nettl 2013; Wallin, Merker & Brown

2001; Seidenberg 1978; Eves 1983; Ifrah 2000; Boyer & Merzbach 2011). This suggests that recent developments in

mathematics are far from becoming second nature to most of us.

Music, on the other hand, is something to which even newborn babies respond. There is ample evidence that

this response, which is positive, starts well before being born during the process of birth and extends through

childhood and beyond. Peretz & Hyde (2003) suggest that there is increasing evidence that humans are born

with musical predispositions. They also point out that recent ﬁndings indicate that the brain is equipped with

music-speciﬁc neural networks. Possibly, music is present in our very genetic codes giving us an innate sense of

rhythm and endowing the capacity for appreciation even to newborns. For the growing literature between music

and babies, including during pregnancy, see: Chang, Chen & Huang (2008); Schwartz & Ritchie (1999); Marjanen

(2009); Arabin (2002); Whitwell (1999); Browning (2000); Lind (1980); Standley (1991); Fox (2000); Fridman

(2000); Blood & Zatorre (2001).

This raises the possibility that there is another contender for the title "The Universal Language", a stronger one

in fact, and that might be music. We list a few sources from an exhaustive literature that can equip the reader with

a better understanding of music. Alperson (2010); Dorrell (2005) provide an in-depth exploration of the seemingly

simple question: "What is music?"10 . Bowman (1998) explores diverse accounts of the nature and value of music.

Jackendoﬀ & Lerdahl (2006) study the capacity and abilities for music including a discussion of how listeners might

10 Music (from Greek mousike; "art of the Muses, the inspirational goddesses of literature, science, and the arts") is an art form and

cultural activity whose medium is sound organized in time. General deﬁnitions of music include common elements such as pitch (which

governs melody and harmony), rhythm (and its associated concepts tempo, meter, and articulation), dynamics (loudness and softness),

and the sonic qualities of timbre and texture (which are sometimes termed the "color" of a musical sound). Music, Wikipedia Link

acquire these capabilities.

Having considered the eﬀect of music on newborns, it is essential that we look at the mathematical counterpart.

We could only ﬁnd some research papers on babies and mathematics (Dehaene 2011; Skemp 1987). That said, there

is a wealth of literature on the mathematical thinking in young children starting with the inﬂuential work of Piaget

(1953) and later works in developmental psychology (Piaget 1964; Fisher et al. 2012). More research is needed

to explore this particular connection between mathematics in the womb and beyond. The one concern we have is

that many worried parents might object to math experiments on babies, especially since, as discussed earlier, many

adults show signs of panic when it comes to mathematics11.

The sleep inducing eﬀect of mathematics, observed in many older humans within classrooms across the world,

should provide us with some clues. We can only conjecture that babies might also respond to mathematics by falling

asleep immediately. Or in certain other cases, not even extreme ones given the many anxieties many adults have

when it comes to math, by crying and displaying signs of being frightened, possibly getting concerned about which

world and life they have ended up in. This therapeutic eﬀect (sleep) and its opposite (anxiety) are not observed

in all of us showing that there might be some of us made for math, or, just more ready for it. Kisner, Colby &

Borstad (2017) provide a deﬁnition of therapeutic exercises12. For anxiety in math and elsewhere, see: Ashcraft &

Ridley (2005); Wood (1988); Tobias (1993); Spielberger (2013); Kierkegaard (2013).

That music is appreciated not just by humans but by many (all?) living creations is to be studied further13 . Patel,

11 Another concern is that if such experiments start becoming popular they might get classiﬁed under torture methods by adults

with math anxiety, due to the possibility of leading to severe childhood trauma, attracting the attention of human rights activists or

not getting ethics board approvals.

12 Therapeutic eﬀect refers to the response(s) after a treatment of any kind, the results of which are judged to be useful or favorable.

This is true whether the result was expected, unexpected, or even an unintended consequence. Therapeutic Eﬀect, Wikipedia Link

13 The eﬀects of music on animals has been observed since antiquity as shown in the below ancient Indian verse. This subhashita

(sayings with wisdom) is found in many places in part or full though we have not been able to trace its exact source or origin. It is found

in Subhashita Ratna Bhandagara (Sharma 1952). Some ascribe this verse to Kumarila Bhatta (Sharma 1980; Bhatt 1989; Bhatta 1985;

Sheridan 1995; Kum¯arila Bha{

.t}{

.t}a, Wikipedia Link) and some relate part of this shloka to Bharata Muni (Dace 1963; Mehta 1995;

Lidova 2014; Bharata Muni, Wikipedia Link). The ﬁrst part of the verse is the opening line for the musical movie Sankarabharanam

(Murthy 2014; Sankarabharanam (1980 Film), Wikipedia Link).

śiśurvetti paśurvetti vetti g¯anarasa ˙m phan

.¯ı |

s¯ahityarasam¯adhurya ˙m śa ˙nkaro vetti v¯a na v¯a ||

The ultimate sweetness of music is understood and enjoyed even by an infant, even by an animal and even by a snake.

Indeed, we do not really know whether the erudite scholar can enjoy the sweetness of the nectar of literature, musical lyrics and the

Iversen, Bregman & Schulz (2009) report experimental evidence by manipulating the tempo of a musical excerpt

across a wide range and showing that a sulphur-crested cockatoo spontaneously adjusts the tempo of its rhythmic

movements to stay synchronized with the beat. Rickard, Toukhsati & Field (2005) propose that understanding how

music aﬀects animals provides a valuable conjunct to human research and may be vital in uncovering how music

might be used to enhance cognitive performance. Krause (2012); Wallin, Merker & Brown (2001) discuss the origins

of music including links to wild places and music in the natural world14. The study of music origins provides a fresh

perspective to the study of human evolution since the evolution of languages and cultural behavior are strongly

connected to the evolution of music.

It is important here to clarify the extensive use of notation in the formal methods of teaching, learning and

practicing both mathematics and music (Cajori 1993; Mazur 2014; Williams 1903; Kelly 2014). We wish to empha-

size that notation and complexities can materialize in any ﬁeld of study, over time, due to the necessity of ensuring

precise means of replication and propagation. Music and mathematics have much in common, in this regard and in

other aspects as well, as discussed in later sections. The main diﬀerence is the natural ease and relative simplicity

with which the majority of life, as we know it, can associate to contemporary versions of music when compared to

current mathematics.

This widespread admiration of music makes it a contender for the "matrix of symbols" into which we seem to

have slipped or maybe even relapsed. It is certainly possible that we have entered such an arena for the ﬁrst time or

perhaps, we have entered it before without being aware of our previous experiences within such a construct. Matrix

of symbols could mean at-least two things. The ﬁrst meaning is mathematical in terms of our present awareness of

mathematics. Bernstein (2005) provides details about a mathematical construct called Matrix. This ﬁrst reference

hints at our present status quo with regards to the extensive use of mathematical symbols. The second meaning

could refer to a simulated reality created with or without mathematics. Bostrom (2003); Moravec (1993; 1999);

Hut, Alford & Tegmark (2006) are discussions about whether we are living in a computer simulation15 . Wachowski

arts.

14 It is prudent not to leave out the eﬀect of music on those, so called, non-living things. Just because we don't see a connection,

does not mean, that there is no connection (Kashyap 2020). This brings up the topic of Questions & Answers, Q&A. In this case, the

question is: What is a Living creature? Biology, we suppose provides an answer, based on some Deﬁnitions and Assumptions, D&A

(Woese, Kandler & Wheelis 1990). But if we change those D&A, we might get diﬀerent Q&A, even telling us that Q&A and D&A

might be in our very DNA (Alberts 2017).

Deoxyribonucleic acid (DNA) is a molecule composed of two chains that coil around each other to form a double helix carrying the

genetic instructions used in the growth, development, functioning, and reproduction of all known living organisms and many viruses.

DNA, Wikipedia Link

15 Simulated reality is the hypothesis that reality could be simulated—for example by quantum computer simulation—to a degree

& Wachowski (1999); Wright (2000) describe a story about a contestant, Neo, who ﬁghts "the matrix", which is a

simulated reality created using numbers, computational concepts and related mathematical ideas16.

4 Mathematics is / as Music and Vice Versa

There are numerous studies that probe the association between music and mathematics. Graziano, Peterson &

Shaw (1999) put forth the hypothesis that early music training would enhance spatial-temporal reasoning which

could lead to better learning of certain math concepts. Music improves the development of our brains and this

enhances our abilities in other subjects such as reading and mathematics. Harkleroad (2006) explores the math-

related aspects of music and points out that mathematics has been used for centuries to describe, analyze and

create music. From simple sums to complex functions, mathematical concepts form a part of the world of music.

Mathematics can be used: 1) to analyze musical rhythms; 2) to study the sound waves that produce musical notes;

3) to explain why instruments are tuned; and 4) to compose music (Garland & Kahn 1995). The relationship

between mathematics and music can be observed through proportions, patterns, Fibonacci numbers or the Golden

Ratio, geometric transformations, trigonometric functions, fractals, and other mathematical concepts (Benson 2006;

Kravanja 2008) .

Loy (2011) provides a tour of the mathematics of music suggesting that mathematics can be as eﬀortless as

humming a tune, if we know the tune. Southgate & Roscigno (2009) point out that music and math were one and the

same to the Pythagoreans (ancient Greeks). They assess the eﬀects of music involvement on math, ﬁnding improved

achievement. Vaughn (2000) is a report on three meta-analysis investigating the positive relationship between music

and mathematics. Beer (1998) oﬀers the insight that music often has some mathematical characteristics and, more

importantly, that artistic aspects can be found in mathematics as well. Santos-Luiz (2007) highlights that it is

possible to establish a positive correlation between participation/performance in music and cognitive development

indistinguishable from "true" reality. It could contain conscious minds that may or may not know that they live inside a simulation. This

is quite diﬀerent from the current, technologically achievable concept of virtual reality, which is easily distinguished from the experience

of actuality. Simulated reality, by contrast, would be hard or impossible to separate from "true" reality. Simulated Reality, Wikipedia

Link

16 The Matrix is a 1999 science ﬁction action ﬁlm written and directed by The Wachowskis. It depicts a dystopian future in which

humanity is unknowingly trapped inside a simulated reality called the Matrix, created by thought-capable machines (artiﬁcial beings)

to control humans while using their bodies as an energy source. This simulation is a construction that uses mathematical entities and

computers, or machines, that are sentient. Needless to say, if we tweak our D&A, Deﬁnitions and Assumptions, we might fall under the

category of machines;The Matrix, Wikipedia Link

in mathematics. An, Ma & Capraro (2011); An, Capraro & Tillman (2013); An & Tillman (2015); An, Kulm & Ma

(2008); An, Tillman & Lesser (2018) are studies about teaching and learning mathematics through music wherein

music activities are integrated into regular mathematics lessons. These studies emphasize the connections between

mathematics and music by contextualizing the mathematics learning process within musical experiences.

Rothstein (1995) suggest that mathematics and music share common origins in cult and mystery and have been

linked throughout history: one is a science, the other an art; one useful, the other seemingly decorative. Considering

Art and Science as disparate, might be overlooking the liaison that "Art is Science that we don't know about; and

Science is Art restricted to a set of symbols governed by a growing number of rules" Kashyap (2017). Maybe if we

continue to deal with the symbolic version of mathematics, and remain contented with our acceptance of playing

with puzzles that have missing pieces, evolution (Darwin 1859; Dawkins 1976)17, or mother nature, will ensure

that one day we might be equally adept at glancing at symbols and comprehending what is conveyed as we are at

listening to tunes and taking in the tidings .

Here we need to consider the fact that ideas expressed in mathematics stay the same once understood, irrespective

of who is conveying the idea, who is trying to understand it, where and when this communication is happening.

This requires making precise our deﬁnitions and assumptions (D&A) depending on the circumstances, and being

open to changing our D&A as situations change, as we attempt to ﬁnd answers to the question: what is the true

universal language? If the universal language is one that gives the same meaning irrespective of the circumstance

under which it is used then, to the best of our knowledge, mathematics is best placed for that. Though the risk we

run into, by using this particular approach, is that no meaning might be conveyed since it might not be understood

at all. But perhaps, with lots of time and all the rules worked out it can be solved like a puzzle. On the other hand,

it could be argued that music has a simple message no matter how it is understood and complex messages, though

varied, should be welcome along with this simple message. The case we are really making for is to supplement

mathematics with music, which seems to be understood and certainly more widely appreciated than mathematics.

4.1 Music Plus Mathematics Equals Melody

We are currently working on other projects that are taking a closer empirical look at the relationship between

music and mathematics. The study we are conducting seeks to answer the question whether, "Music Plus Math-

ematics Equals Melody". Our approach to this problem involves converting diﬀerent types of sound and hence

music, songs, poems, speeches etc., in diﬀerent languages and varieties as applicable and from as many sources

17 Evolution is change in the heritable characteristics of biological populations over successive generations. These characteristics are

the expressions of genes that are passed on from parent to oﬀspring during reproduction. Evolution, Wikipedia Link

as possible, into a numerical form. We then analyze the numerical representation, of the various sound forms,

by looking for mathematical patterns within them. The hypothesis we are testing is that sounds that are more

melodious, or pleasing to more of us, have certain mathematical properties embedded within them or they have a

disproportionate share of mathematical patterns ingrained within them.

5 Musical Message, Medicine for the Mind ... Don't Worry, Be Happy

The main message of music is simple. MacDonald, Kreutz & Mitchell (2013) consider the many beneﬁcial eﬀects

of music on health and well-being. McClellan (2000); Horden (2017) are detailed accounts of the healing abilities of

music through time and across many cultures. Bicknell (2009) tries to comprehend the extraordinary power music

has to move us. Hesmondhalgh (2013) investigates how music can enrich our lives (Figure 2). The multicultural

aspects of music, in terms of both therapeutic and educational elements, cannot be emphasized enough (Elliott

1989; Moreno 1988). Despite the many cultural nuances, music is a medicine for the mind to forget worries that

might seem overbearing and focus on rejoicing the miracle that has created us and given us ample things to be

happy about18.

Of course, we should be open to the possibility of complex messages, or thoughts, which could result due to

the stimulation music produces in our minds as discussed in Section (3). But this can be an optional path only for

someone who wishes to go down that route. Simply enjoying the music should suﬃce for the uninterested. This

directive is best exempliﬁed in English by the magical composition, "Don't Worry, Be Happy", by the Master Bobby

McFerrin 19 . A purely musical rendition of this song, without any language as we currently seem to perceive, is

bound to produce the same amount of delight. This might also be an indication that understanding is unnecessary

at times, even overrated perhaps, as long as the desired end goal, joy in this case, is accomplished.

A deeper exploration of this topic will be pursued in other venues (Kashyap 2016). But for now, it suﬃces for us

to realize that a deterministic world can be made to look stochastic easily. Few simple components can succeed in

creating complexity quite eﬀortlessly. So instead of puzzling over the many elements of uncertainty we encounter,

one of which is languages we don't comprehend, our eﬀorts might be well spent marveling the creations that cross

our lives.

18 We can call this force of creation, Mother Nature, God, Evolution or whatever. Mariner (1977); Hayward (1998); Scott & Branch,

G. (2003); Moore, Decker & Cotner (2009) have a discussion of the evolution versus creation controversy.

19 "Don't Worry, Be Happy" is a popular worldwide hit song by musician Bobby McFerrin. It was released in September 1988 (Don't

Worry ... Be Happy, Wikipedia Link). The complete song "Don't Worry, Be Happy" can be accessed at this link: Don't Worry ... Be

Happy, YouTube Link

November 14, two days after the ﬁrst draft of the paper was completed, is celebrated as children's day in certain

parts of the world. In the spirit of this occasion, gloriﬁed on other days across the globe, it becomes our responsibility

to ensure that we are providing the right training for all children all over the cosmos. It is worth reminding ourselves

that, all of us are just kids at heart, though sometimes, we let our concerns overtake our curiosity, which cloud the

sense of wonder we have about the world around us. Here, we implicitly use the deﬁnition that childhood ends,

when concerns start dominating our curiosity. This requires introducing children to the techniques of producing

tantalizing tunes along with other tools such as mathematics. Also, it is worth reminding ourselves that, it is never

too early (or late?) to start leading an extraordinary life since all living (and perhaps, non-living too?) entities are

simply fantastic20

6 Ignoring Quirks called Quarks

To create any universe similar to the one we physically dwell in, all we need are three building blocks: protons,

electrons and neutrons. This view overlooks other fundamental particles and is accurate only with a certain level of

understanding (Harari 1979; Hooft 1996; Davies 1982; Lederman & Teresi 1993; Turner 2009; Emsley 2011; Pohl et

al. 2010). Given this similarity in our fundamental constitution and the marvelous diﬀerences we see both within

our behavior and around us, we need to marvel at the amazing diversity that nature, or whoever or whatever, has

created using just diﬀerent combinations of the same basic building blocks21.

All information in the vastness of the universe can be represented using just two symbols 0 and 1. This includes

knowledge and wisdom without getting into the speciﬁcs of the corresponding deﬁnitions and assumptions regarding

how they might diﬀer. Kashyap (2017) has a discussion of this distinction. Boole (1847); Shannon (1948) are starting

points for the use of the binary code in computers and other practical applications22 . Shannon (1956) showed that

20 All that is needed for a phenomenal experience in this test called life is below (Figure 3):

Recipe for a Remarkable Life (???)

Trust No One, but Love Everyone;

Travel Everywhere, and Respect Everything;

Remember "e", the Exponential Function, Naturally Stands for Everyone, Everywhere and Everything (including Evolution, Ethics,

Education and Examinations) ...

As You Sense and Rejoice, Do Your Best;

Relax !!! Someone Else Will Take Care of the Rest ...

21 Life would be quite mundane without this variation everywhere: variety is not just the spice of life, variety is life.

22 A binary code represents text, computer processor instructions, or any other data using a two-symbol system. The two-symbol

system used is often "0" and "1" from the binary number system (Lodder 2008). Binary Code, Wikipedia Link

two symbols were suﬃcient to duplicate a Turing Machine, the model for a general computer, so long as enough

states were used or vice versa (Turing 1937; Sipser 2006; Woods & Neary 2009)23.

The universes that we are creating, the so called virtual worlds and hence a simpliﬁcation compared to our world,

will someday perhaps rival our own universe in complexity (Castronova 2008; Shaw & Warf 2009; Belk 2013; Wolf

2014; Billieux et al. 2013). These virtual worlds are developed using computers and related technologies built with

just two basic components: zero and one. This suggests that a universal language does indeed exist for computers

and the corresponding virtual world made of just two syllables: 0 and 1. Hence, despite the many numerous

languages used to create computer programs, there is a simple fundamental structure that they all share24 . The

existence of a universal language for computers should give hope to those of us looking for a universal language for

the, so called, real world we live in.

Mathematics is built on one simple operation, addition, making it a fractal with addition as its starting point

(Mandelbrot 1982; Briggs 1992; Falconer 2004; Kashyap 2017). This being the case, we have ended up in a situation

where we seem to use hundreds of symbols in our mathematical endeavors. When someone begins the journey of

mathematical learning, perhaps, it is this relentless use of notation without clariﬁcation, results without steps and

stories without a narrative that cause confusion and frustration. Confusion and Frustration, both of which though

scary and ugly to begin with, can be powerful motivators as long as we don't let them bother us (Kashyap 2020)25.

In mathematics and digital electronics, a binary number (Lytton 2002) is a number expressed in the base-2 numeral system or binary

numeral system, which uses only two symbols: typically "0" (zero) and "1" (one). Binary Number, Wikipedia Link

The bit (a portmanteau of binary digit) is a basic unit of information used in computing and digital communications (Mackenzie

1980). A binary digit can only have one of two values, and may be physically represented with a two-state device. These state values

are most commonly represented as either a 0 or 1. Bit, Wikipedia Link

23 A Turing machine is a mathematical model of computation that deﬁnes an abstract machine, which manipulates symbols on a

strip of tape according to a table of rules. Despite the model's simplicity, given any computer algorithm, a Turing machine capable of

simulating that algorithm's logic can be constructed (Sipser 2006). Turing Machine, Wikipedia Link

24 Bergin & Gibson (1996); Wexelblat (2014) discuss the history of programming languages. McCartney (2002) is about a computer

language for music. A computer language is a system of communication with a computer. Computer Language, Wikipedia Link

A programming language is a formal language, which comprises a set of instructions used to produce various kinds of output.

Programming languages are used in computer programming to create programs that implement speciﬁc algorithms (Knuth 2014; Van-

Roy & Haridi 2004). Programming Language, Wikipedia Link

25 Confusion is the beginning of Understanding. Necessity, is the mother of all creation (or innovation or invention), but the often

forgotten father is Frustration. What we learn from the story of, Beauty and the Beast (De Beaumont 1804), is that we need to love

the beasts to ﬁnd beauty. Hence, if we start to love these monsters (Confusion and Frustration), we can unlock their awesomeness and

ﬁnd truly stunning solutions.

Kashyap (2018) considers the possibility that languages that have many symbols, such as Chinese, could be ideal

cultural training for being adept at mathematics. Until our eﬀorts to make strange symbols not so strange start

bearing fruit, it might be beneﬁcial to take extra care to ensure that there is a good degree of acclimatization with

the unfamiliar shapes that we need to encounter. As we strive to make sense of symbols, combining melody with

math might prove to be highly rewarding.

7 Artistic Encoding of Knowledge

If science is the name given to our eﬀorts to understand the world around us, mathematics then becomes the

language of science. We have been embarking on attempts to converse with extraterrestrials, and share our scientiﬁc

ﬁndings, using mathematics based on our belief that it is the universal language (Sagan 1980; Musso 2004; Ornes

& Tarter 2012; Pomerance 2004). Should we not ask ourselves ﬁrst, how many of the seven billion human beings

actually understand any of the messages that have been created using mathematics. This of course, does not count

the many more numerous lifeforms on this planet that seem to be more receptive to music than mathematics.

Surely, we must be open to the possibility that alien life forms can be similar to creatures unlike us at all.

There might be intelligence that cannot be represented by the mathematics known to us, though there could be

fundamental mathematical principles behind that intelligence that we are not yet aware of. That we are not

receiving any replies from aliens might be due to the fact that we are not being heard, understood, or, we are

showing signs of not being advanced enough to see what is more omnipresent, music or mathematics, or whether

both are equally ubiquitous. It is reassuring that many of our best musical compositions were also sent into space

along with the mathematical messages that we deemed worthy enough to represent us (Fraknoi 2007). But the

next step is discover, or fabricate, some way to present our scientiﬁc ﬁndings in musical form, which we discuss

next. As one source of inspiration, Fraknoi (2002; 2003; 2007) and Ronan (1976) explores the relationship between

astronomy and music including the use of poetry and science ﬁction in the astronomy classroom and present a list

of popular music inspired by astronomical ideas.

Our eﬀorts to understand the universe, and pass on the communique to everyone, can also be known as knowledge

creation or research. Knowledge creation and dissemination involve putting our ﬁndings, at-least as many of them

as possible, in a mathematical format. Would it be such a stringent requirement to ask that every journal article,

Beauty and the Beast (French: La Belle et la Bête) is a fairy tale written by French novelist Gabrielle-Suzanne Barbot de Villeneuve

and published in 1740 in The Young American and Marine Tales (French: La Jeune Américaine et les contes marins). Her lengthy

version was abridged, rewritten, and published ﬁrst by Jeanne-Marie Leprince de Beaumont in 1756. Beauty and the Beast, Wikipedia

Link

which has ample amounts of mathematical proofs, or other thoughts, also be accompanied by a musical encoding?

This might be for the beneﬁt of the knowledge presented in that composition since music might ensure greater

survival of that concept and the corresponding passage of the ideas to later generations more easily. We need

to be realistic in realizing that mathematical (also logical, theoretical, empirical, conceptual, etc.) masterpieces

merged with music might not start getting composed right away. As more people get better acquainted with music,

better techniques to musically represent knowledge will develop. This chain of meditation holds a lesson for all

curriculum designers that we need to consider music classes to be as important as math courses and start infusing

this juxtaposition at all levels of formal and informal education.

Any guide for policy making is likely to be far from complete in terms of providing actionable items on getting

appropriate resolutions implemented and listing out possible issues that might crop up once any change is enacted.

Kashyap (2016) suggests that a trial and error based approach, without too many rules and mandatory prescriptions,

would be prudent. Eisner (1998) ﬁnds that the eﬀects of the arts appear to be greatest when the arts are intentionally

used to raise academic achievement. Though he cautions that setting such priorities could undermine the value of

art's unique contributions to education. Eisner (2000); Grey (2009); Russell-Bowie (2011) provide many interesting

perspectives on policy attempts with regards to art education both historically and geographically. Sabol (2013);

Laszlo (2019); Shaw (2019) examine policy issues based on a selected number of contemporary developments in

the general ﬁeld of education including an attempt to understand how policy frameworks could aﬀect arts policy

development. Heilig, Cole & Aguilar (2010); Allina (2018) provide a discussion of the many intended and unintended

consequences including debates that might ﬂare up from the proponents of a particular discipline, such as science

versus arts. Gigliotti (2001); Gouzouasis (2006); Choi & Piro (2009); Patton & Buﬃngton (2016); Hebert (2016)

consider arts education policy challenges in today's (and tomorrow's?) digital and technological world.

The earlier discussion, about encoding knowledge using music, brings up a related question of how to encode

our knowledge as musical objects. To address this conundrum we point out that the more important question

is to answer what needs to be done? Once we are clear about what is necessary, how to accomplish something

will become clearer. There are also many ways in which a particular goal can be reached but the fundamental

aspect is to be certain about where we need to be headed. The mechanism through which the knowledge and

wisdom of ancient Indians and Greeks has been immortalized as poems and passed on for generations should oﬀer

us further guidance (Knox & Fagles 1990; Narayan 2000). Tiwari et al. (2008); Tirtha & Agrawala (1992); Kumar

& Charishma (2012); Chandler (1987); Hagelin (1989); Brown (2012) are detailed accounts of musical and poetic

renderings of mathematical and scientiﬁc knowledge from early civilizations.

On the practical side, encoding knowledge using musical language would require non-trivial expertise and skills.

Here, we would like to draw a clear distinction between someone who creates musical pieces and others who receive

them. Hence, it becomes apparent that even though the creation of mathematics and music are both demanding,

musical reception is much less exacting when compared to mathematics. In addition, there are many cultural

dimensions and nuances to music. Trying to represent knowledge across the many diﬀerent forms of music can be

an arduous task. To begin with, any representation of knowledge using music is likely to be well received. Hence,

trying to have multiple formulations, while helpful, might not always be required. Another subtle message hidden

with this chain of thoughts is that all knowledge does not have to have a musical encoding. It would be great to

have such a complete representation, but it is not always necessary. What would suﬃce in many instances would

be to ensure that a story captures the main elements of knowledge. Human beings have evolved thus far to relate

to the stories of others. What would ensure that we grasp the main lessons from any endeavor is to ensure that we

have an interesting enough narrative to transmit it to others.

Our paper makes the case that, along with every mathematical encoding of knowledge, we need to have musical

renderings or other artistic means of communicating the message to ensure that the knowledge is transmitted to a

greater audience. All of what we have discussed is perhaps, not new. This is because, as someone said, everything

has already been said but maybe not by everyone and to everyone. This scenario is plausible, and perhaps necessary,

since many times we forget what we have said and need to be reminded. Our main hope is that this paper will

convince everyone, acting as a reminder to those who have forgotten and becoming a stimulus for action to others,

that we need to develop musical talents along with mathematical aptitude. Once we better understand music, and

the world including mathematics, we might ﬁnd ways to express the same thing in diﬀerent forms of music, creating

inﬁnitely many new ways to make the same thing novel and enjoyable.

8 Conclusion

We have discussed the human obsession for creating symbols without limits. Limits are again a notion from

mathematics (Rudin 1964). While it is okay to cross the limits in mathematics, it might not be that wise to do so

in other areas of life as we know it. We have also pondered over the logic of sending messages to aliens that are

not understood by most creatures on this planet. We mean no disrespect to any such attempts. One reason why

such unwanted outcomes creep up is because we live in a world that requires around 2000 IQ points to consistently

make correct decisions. But the smartest of us has only a fraction of that intelligence (Kashyap 2021)26 . Hence,

we need to rise above the urge to ridicule the seemingly obvious blunders of others since without those marvelous

mistakes the path ahead will not become clearer for us.

26 Taleb (2012); Ismail (2014) mention that knowledge can give you a little bit of an edge, but tinkering (trial and error) is the

equivalent of 1,000 IQ points. It is tinkering that allowed the industrial revolution. Nassim Taleb and Daniel Kahneman discuss Trial

and Error / IQ Points, among other things, at the New York Public Library on Feb 5, 2013.

We have suggested that music is as much, if not more, of a universal language than mathematics. We have made

a case for introducing music education in the curriculum starting right at a very early stage of formal schooling. This

holds a strong message for formulating policies to accommodate such an educational environment. To complement

such an eﬀort more exposure to wider forms of artistic endeavors other than music can be encouraged. Frith (1990)

attempts to tackle the fundamental question: what is good music? Schmidt (1998) probes the question, what is

good music teaching? Williams (2013) consider the qualities of a good music department, which can cater to the

various needs of a wide range of pupils27.

Despite the complexities involved in the formal teaching of music, musical systems can provide a solid foundation

for understanding other complex systems. To feel and experience the beauty of a musical composition is the primary

motivation for recreating and understanding the complex musical notation. The instructional emphasis in music

makes clear the reasons for why the complexities have arisen and why they are needed. Mathematics education,

and other formal training programs, can be modeled after the pedagogical aspects of music. We have hinted at the

possibility that many seemingly complex entities (including our universe, virtual computer worlds, mathematical

operations etc.), are made up of combinations of much simpler building blocks. Familiarity with any complex

element of life, without getting ﬂustered, is bound to produce remarkable results in other such endeavors as well.

Perhaps, as we improve our cognition of the Cosmos, (Sagan 1980), we might conclude that mathematics and

music are Avatars of one another (Lochtefeld 2002; Parrinder 1971; Sheth 2002; Figure 4)28. We might also have

27 People adept at mathematics are good at solving problems phrased in certain ways. But their eﬀorts are not noticed if there are no

challenging problems for them to solve. If there are no problems, there need be no eﬀorts at all to solve problems. In contrast, musical

problem solving can simply be about getting the attention and evoking pleasant sensations in the audience. Hence, problem creators

are just as important, if not more important, than problem solvers. This tells us that as we crave challenges, to make our existence

interesting and exciting, the ones that are creating those predicaments deserve our gratitude. These problem creators are ourselves, in

most cases, due to the unintended consequences we discussed earlier. Continuing this line of thoughts, perhaps enlightenment is the

solution to the marvelous riddle created for us to be engaged and entertained, which is simply our Universe. Kashyap (2016) has a

deeper discussion of these topics.

28 An avatar (Sanskrit: avat¯ara), a concept in Hinduism that means "descent", refers to the material appearance or incarnation of a

deity on earth.Avatar, Wikipedia Link

The incarnation of a Hindu deity (such as Vishnu); an incarnation in human form; an embodiment (as of a concept or philosophy)

often in a person. Avatar, Webster Dictionary

Avatar, marketed as James Cameron's Avatar, is a 2009 American epic science ﬁction ﬁlm directed, written, produced, and co-edited

by James Cameron. The ﬁlm is set in the mid-22nd century, when humans are colonizing Pandora, a lush habitable moon of a gas giant

in the Alpha Centauri star system, in order to mine the mineral unobtanium, a room-temperature superconductor. The expansion of

the mining colony threatens the continued existence of a local tribe of Na'vi – a humanoid species indigenous to Pandora. The ﬁlm's

title refers to a genetically engineered Na'vi body operated from the brain of a remotely located human that is used to interact with

the natives of Pandora. Avatar Movie, Wikipedia Link

realized by then that thoughts alone can suﬃce to commune everything with everyone everywhere including other

species on our planet and also with extraterrestrials. It might have dawned upon us by then that thinking (musical

and mathematical, simultaneously) might be the truly universal mechanism for communicating in the Multiverse

we dwell in (Carr 2007)29.

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10 Figures

Figure 1: Row upon Row of Symbols: Comprehend That!!!

Figure 2: Musical Message, Medicine for the Mind: Don't Worry, Be Happy ...

Figure 3: Recipe for a Remarkable Life

Figure 4: The Universal Language: Mathematics or Music?

... Mathematics has long been regarded as a universal language since it relies on numbers and consistent operations, although this terminology has been met with a degree of a practical challenge since comprehension is dependent upon literacy [21 -23]. Connections have been recognized between music and mathematics in terms of logic and organization [4,24,25]. Basic mathematical ideas can be recognized within music and, conversely, mathematical consideration of patterns can expose organizing principles of music [26]. Mathematics has been studied in the context of artificial neural networks [27]. ...

Objective Audio files of spontaneous signal streams generated by ex vivo neuronal networks cultured on multi-electrode arrays generated an oscillating sine wave with an inherent musical quality. This was not anticipated considering that synaptic signals are "all - or – none", and therefore digital, events. Methods These findings may provide insight into why music can be perceived as pleasurable and invoke a calm mood despite that music is ultimately perceived and stored as a series of digital signals; it is speculated that music may reinforce and/or enhance this spontaneous digital stream. Results and Conclusion These findings also support the relationship between music and mathematics.

Ravi Kashyap

This article discusses preliminary results from two experiments and puts forth the notion that the development of sensory systems might be more geared toward discerning differences rather than for spotting similarities. The article presents the possibility that the necessity to spot differences might have evolved to ensure the survival of the organism, which suggests numerous other experiments to assess the response of participants to various stimuli. The article considers our present state of affairs, wherein the need is to thrive and not merely survive, which requires us to spot similarities around us. Some suggestions are provided on how this attribute can be developed, which includes mathematical education. This article concludes with an alternate measure for intelligence, termed the Involvement Quotient (also, IQ), which gauges the level of involvement of the sense organs to whatever is happening around the individual.

Ravi Kashyap

We formulate one methodology to put a value or price on knowledge using well accepted techniques from finance. We provide justifications for these finance principles based on the limitations of the physical world we live in. We start with the intuition for our method to value knowledge and then formalize this idea with a series of axioms and models. To the best of our knowledge this is the first recorded attempt to put a numerical value on knowledge. The implications of this valuation exercise, which places a high premium on any piece of knowledge, are to ensure that participants in any knowledge system are better trained to notice the knowledge available from any source. Just because someone does not see a connection does not mean that there is no connection. We need to try harder and be more open to acknowledging the smallest piece of new knowledge that might have been brought to light by anyone from anywhere about anything.

Richard R. Skemp

This classic text presents problems of learning and teaching mathematics from both a psychological and mathematical perspective. The Psychology of Learning Mathematics, already translated into six languages (including Chinese and Japanese), has been revised for this American Edition to include the author's most recent findings on the formation of mathematical concepts, different kinds of imagery, interpersonal and emotional factors, and a new model of intelligence. The author contends that progress in the areas of learning and teaching mathematics can only be made when such factors as the abstract and hierarchical nature of mathematics, the relation to mathematical symbolism and the distinction between intelligent learning and rote memorization are taken into account and instituted in the classroom. © 1987 by Lawrence Erlbaum Associates, Inc. All rights reserved.