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In this chapter, the question is addressed what the world would be like if the number zero had never been invented. This anti-historical question is dodged by asking a slightly less difficult question, namely: what would the world have been like if the so-called complex numbers (ubiquitous in many fields of science and engineering) had never been invented? [ABSTRACT FROM AUTHOR]

Zero is a very versatile entity, making its appearance in a variety of environments. It can appear as a representation of the absence of items in arithmetic to the absence of sound in the merging of words, to the zero as a number in arithmetical operations. All these aspects of zero are fascinating in their own right. However, while the idea of arithmetical zero came to the human mind very early in its intellectual journey, the human response to derivative ideas has been driven by cultures. While, on the one hand, European cultures came to the conclusion that the universe exists and therefore, by definition, a non-existent entity cannot represent reality; on the other hand, the Indian philosophers, used to the ideas of non-existent entities in other contexts, especially in the grammar of conjunction of words and sounds, were fascinated by the null and assumed that it was the null that held the key. They therefore traced the idea of the origin of the universe to this null entity, which by definition is not knowable since it is formless. This divergence meant that intellectual perspectives and studies were driven by this differing approach to zero. It allowed Indian mathematicians to explore zero much earlier than their European counterparts. Modern science, having embraced the zero in all its glory, then went on to twist its existence in different ways. In the present article we discuss some of the aspects of this journey of zero. [ABSTRACT FROM AUTHOR]

Not dividing by zero is one of the strongest prohibitions that we learn as children. Yet, as we grow in age and in competence, we realize that many mathematical constructions would be much simpler if we could divide by zero. But this implies enlarging the family of numbers and making space for an infinite number, ∞, as in the Riemann sphere, or even many infinite numbers, as the founders of calculus envisioned. [ABSTRACT FROM AUTHOR]

We investigate if philosophical ideas of nothing could have given rise to the mathematical zero, which is distinct from nothing and is an independent number with a set of basic operations defined. This investigation should be complementary to other studies, e.g., those based on epigraphic and paleographic evidence in order to probe the origin of zero. First, we argue that a practical need did not give rise to the mathematical zero. Moreover, neither a place-value number system nor a placeholder in such a system was required to conceptualize zero. Therefore, we insist that zero and its basic operations were intellectually premised, which likely required a philosophical base of nonexistence, and not that of emptiness or void as commonly assumed. The prevalence of a philosophy of nothing in ancient India and the appearance of the first clear prescription of basic operations of zero in an Indian book of the first millennium CE point to a development of the concept of zero in India. We back this up with 'living evidence' of word-numerals developed in India, and propose that the comments of the Syrian Bishop Severus Sebokht, which are often used to argue against the Indian origin of zero, do not provide evidence against such an origin. [ABSTRACT FROM AUTHOR]

This chapter is an overview of the themes regarding the concept of śūnya ('void') in Indian thought, mainly mādhyamika, with reference to apoha semantic theory, and in relation with another couple of problematic relationships, the metaphoric use of words (diffused in the milieu of poetics) and the paradox of ineffability (current in Vedantic Brahmanical circles). Did the mathematical, the grammatical, or the Buddhist philosophical meaning of śūnya come first? The chapter tries to illustrate the great deal of debate current on this question, but does not offer a unique answer. [ABSTRACT FROM AUTHOR]

This article argues that the search for a philosophy that gave rise to the number zero is misguided. No philosophy is required to invent this number. The article further shows that there are good reasons to accept that Buddhism did not play a role in this invention. It further points out that the notion of number as developed in Indian philosophy had no place for zero. [ABSTRACT FROM AUTHOR]

The world of numbers keeps us intrigued for one reason or the other. From negative numbers to imaginary numbers and from various constants to peculiar ratios, they never cease to surprise us. And our curiosity and imagination keeps on inspiring us to look for these patterns in nature. Consequently leading the way for discoveries that are important to meeting not only our material goals but also spiritual ones at the same time. However, due to an extraordinary reliance of modern applied science on numbers, we have generated more scholarly interest, especially in the last few centuries, around the application of numbers to the material realm as compared to the spiritual. Amidst our attention to the peculiar or special numbers and ratios, there are some numbers -- or rather concepts -- that still look in our face, inviting us to explore a whole new world of possibilities all over again. Though these concepts laid the foundation of modern day mathematics as we use it, yet at times we take them for granted and are not usually interested in revisiting our understanding about their nature, their past and future possibilities. In this chapter, we revisit three of these foundational concepts -- Zero, One and Infinity (ZOI) from an integrative spiritual perspective. [ABSTRACT FROM AUTHOR]

Notions such as śūnyata, catuskoti, and Indra's net, which figure prominently in Buddhist philosophy, are difficult to readily accommodate within our ordinary thinking about everyday objects. Famous Buddhist scholar Nāgārjuna considered two levels of reality: one called conventional reality, and the other ultimate reality. Within this framework, śūnyata refers to the claim that at the ultimate level objects are devoid of essence or 'intrinsic properties', but are interdependent by virtue of their relations to other objects. Catuskoti refers to the claim that four truth values, along with contradiction, are admissible in reasoning. Indra's net refers to the claim that every part of a whole is reflective of the whole. Here we present category theoretic constructions that are reminiscent of these Buddhist concepts. The universal mapping property definition of mathematical objects, wherein objects of a universe of discourse are defined not in terms of their content, but in terms of their relations to all objects of the universe is reminiscent of śūnyata. The objective logic of perception, with perception modeled as [a category of] two sequential processes (sensation followed by interpretation), and with its truth value object of four truth values, is reminiscent of the Buddhist logic of catuskoti. The category of categories, wherein every category has a subcategory of sets with zero structure within which every category can be modeled, is reminiscent of Indra's net. Our thorough elaboration of the parallels between Buddhist philosophy and category theory can facilitate better understanding of Buddhist philosophy, and bring out the broader philosophical import of category theory beyond mathematics. [ABSTRACT FROM AUTHOR]

Although the presence of zero has been traced to many ancient civilizations including India, the question of its origin is still a moot point among historians and mathematicians. As such, in the present chapter a humble attempt will be made to gather data from the Vedic and post-Vedic Sanskrit literature to understand the origin of zero; the philosophy behind its origin; various denominations of zero used in various phases of Indian civilization; and the philosophy and epistemology behind the symbolic representation of zero (0) in India. [ABSTRACT FROM AUTHOR]

This is an exploration of the mathematics, history, and philosophy of zero. It looks at zero and its development from three perspectives: purely as a sign (syntactic); its meaning (semantics); and its social contexts, roles, and uses (pragmatic). Adopting the meaning as use theory, it argues that the meaning of zero was not fully established until the time of Brahmagupta, at around 628 CE, for Brahmagupta established the role of zero within the network of number relations in the integers. Most notably, zero is the additive identity. This synthesis is analogous for number theory with Euclid's geometry. Thus, it is posited that there are stages in the development of the concept of zero. First, its predecessor serves as a placeholder in an empty column. Second, zero is understood as a sign for nothing (nought), resting on the concept of the empty void. Third, its role as additive identity is understood and accepted. The conjecture that the idea of void is a prerequisite for the development of the number zero is confirmed positively in Indian and Mayan cultures. But the case of Ancient Egypt, which has the concept of void but not zero shows that having the concept of void appears, not of itself, sufficient. [ABSTRACT FROM AUTHOR]

The aim of this chapter is to highlight some conceptual connections between the concept of zero and the philosophical concept of śūnyatā. Both ideas first arose in the Indian subcontinent, and I will try to suggest that this geographical congruence is more than a coincidence. As such, this will likely be a more speculative contribution to this volume than others, but I will attempt to substantiate my proposal by stressing the conceptual advantages deriving from such a syncretic enterprise. I will begin with an overview of the historical development of the number and concept of zero, followed by some considerations of its treatment in contemporary mathematics, as well as an examination of its less-evident semiotic properties. I will then proceed to re-examine the concept of śūnyatā, identifying its central meontological significance in the philosophy of Nāgārjuna. In this context, I will examine the idea of the 'trace' in the thought of Jacques Derrida, viewing it as another actor in the assemblage of concepts that I attempt in this chapter. Finally, I undertake a synthetic reading of these three 'voids' -- of zero, of śūnyatā, and of the trace -- aimed at a speculative rearrangement of these heterogeneous concepts, in a philosophical exercise that I see as both a possible and a necessary evolution of 'comparative philosophy'. [ABSTRACT FROM AUTHOR]

Since recent datings of the Bakhshali manuscript confirm our understanding that the zero is indeed an Indian creation, what is it in Indian culture that could have influenced an invention of this kind? From data relating to the Bakhshali manuscript, it is possible to recognize the influence of Buddhism and, from this, to try to see how this movement could have played a role in the theory leading to the inception of the placeholder zero. Moreover, with Brahmagupta this first form and use of zero was developed to the point of becoming a number, and thus adopting an operative role. While the writer of the Brāhmasphuṭasiddhānta was not a Buddhist, it is still possible to show in that document how Buddhist thinking, particularly its speculative developments in the meaning of śūnyatā as described by the great writers of Mahāyāna Buddhism, may provide a favorable setting for theories about the numerical concept of zero, even to the point where it is perceived as a condition of its 'thinkability' or conceptualization. [ABSTRACT FROM AUTHOR]

Who invented zero? Interestingly, there is no one-line answer. And the acceptability of any attempted answer generally depends on the perception of the seeker, as the connotation of the word 'zero' can be perceived from several different but interrelated perspectives. Many ancient civilizations, including India, had their own version of zero or a zero-like concept or symbol as a representative of 'nothingness'; some as a philosophical conundrum, elsewhere some others even in a practical sense, such as a filler or a gap on their counting board; but except in India, none of these early and somewhat hesitant initial concepts did ultimately mature to its true mathematical potential. Today it is generally accepted worldwide that this peerless concept of a decimal place value system of enumeration in tandem with the true zero of our present-day mathematics, evolved in ancient India. Initially philosophically nurtured and analyzed during the early periods of oral tradition in India by several different schools of thought, eventually at some point of time the concept reflected in the Sanskrit word śūnya was transformed into a numeral for mathematical expression of 'nothing'. Perhaps at a later period, it bloomed into its full potential as a number, kha, an integer on which mathematical operations can be performed. Indeed, this unique feature makes kha, the Indian zero, the true progenitor of our modern mathematical zero. However, the exact time frame of this gradual evolution is still hotly debated, a recent controversy in this direction being the outcome of attempted radiocarbon dating of the famous Bakhshali manuscript by the Bodleian Library of Oxford. Going in the other direction, some scholars suggest that a trace of this concept, if not in total operational perspective, might have had a Greek origin that traveled to India during the Greek invasion of the northern part of the country in the pre-Mauryan period. A relatively recent third view professes the Chinese origin of the concept of zero as a placeholder, which might have traveled with the traders from China to the far eastern parts of Asia, to places like Cambodia, then under Buddhist influence that spread from mainland India, where it got the shape of a 'bold dot', the earliest known written form of zero. Scholars belonging to this school of thought want to credit mainland India only for 'garlanding' this concept of zero toward its modern shape, sometime around the eighth or ninth century CE. Embracing all these paradigms of claims and counterclaims about the origin of zero made till date, this article is a nutshell version of an odyssey, an unparalleled journey from a concept to a number in its own right, perhaps the most significant creation of the human mind ever. [ABSTRACT FROM AUTHOR]

Many cultures contributed to the invention of zero, but how to apportion credit? This article explores a new way to do so: imagine royalties were due for every usage of zero, and then figure out how these should be divided among present-day countries. We report on what such a divided allocation might look like, and give reactions, both positive and negative, to it. We also explore how such ideas might be relevant in terms of potential exploitation of the Indian cyber market by foreign corporations. [ABSTRACT FROM AUTHOR]

Authors of medieval Arabic books on arithmetic often solved problems with 'nothing', and in calculation with Indian (i.e., 'Arabic') numerals, they frequently operated on zero while at the same time claiming that the zero is a sign indicating the absence of a number. We investigate how these authors conceived of nothing and zero from the many remarks they made in their books, and in the context of their understanding of number as an amount or measure of some material or noetic unit. [ABSTRACT FROM AUTHOR]

If mathematics is the language the universe was written in and mathematics is discovered rather than invented, then Brahmagupta's 628 CE Sanskrit text first defines a zero as old as the universe. Brahmagupta's definition of zero as a sum of equal and opposite negative and positive quantities is the first scientific definition of zero found to be consistent with laws of motion and particle physics. However, having originated in the East, the full power of India's symmetric and scientific zero failed to migrate West via the medieval Arabic world to Renaissance Europe. Thus, only a trivial mathematical concept of zero emerged, representing either nothing on its own or an arithmetical placeholder used alongside other numbers. Zero as an arbitrary midpoint for measurement purposes within a single quantity is similarly trivial, whether it be centuries BCE or CE, or a measure above or below a surveyor's foundation line as used for construction in ancient Egypt. Thus, the oldest original extant definition of zero compatible with laws of physics describing our universe originated with Brahmagupta in India 628 CE. Yet long lost in transmission and translation, the sad aftermath in classrooms today is a zero in which negatives to its left are arbitrarily less than positives to its right. [ABSTRACT FROM AUTHOR]

As various articles in this volume demonstrate, zero has many forms and finds expression in a whole host of contexts, each as mystifying as the other. Each culture seems to have a very specific reaction to the number, and the reaction varies significantly from culture to culture. From the great hatred in Europe to the near obsession of their cousins in the Indian subcontinent, the entire spectrum of reactions can be found. Egyptians, Sumerians, and the Chinese managed to do some interesting algebra while circumventing zero completely. In this chapter we will investigate why cultures even with deeply common roots reacted so differently to zero. We take the example of the Indo-European cultures and their diametrically opposite reactions to zero. We suggest that this may be related to their experiences of nature post-separation, which were further amplified by their way of life. We then study the wider context of zero in its different manifestations and explore the relation between its different forms. We show that music seems to make a connecting link between these different aspects. [ABSTRACT FROM AUTHOR]