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An Interdisciplinary Perspective
Zero has been axial in human development, but the origin and discovery of zero has never been satisfactorily addressed by a comprehensive, systematic and above all interdisciplinary research program. In this volume, over 40 international scholars explore zero under four broad themes: history; religion, philosophy & linguistics; arts; and mathematics & the sciences. Some propose that the invention/discovery of zero may have been facilitated by the prior evolution of a sophisticated concept of Nothingness or Emptiness (as it is understood in non-European traditions); and conversely, inhibited by the absence of, or aversion to, such a concept of Nothingness in the West. But not all scholars agree. Join the debate.
An Interdisciplinary Perspective
Zero has been axial in human development, but the origin and discovery of zero has never been satisfactorily addressed by a comprehensive, systematic and above all interdisciplinary research program. In this volume, over 40 international scholars explore zero under four broad themes: history; religion, philosophy & linguistics; arts; and mathematics & the sciences. Some propose that the invention/discovery of zero may have been facilitated by the prior evolution of a sophisticated concept of Nothingness or Emptiness (as it is understood in non-European traditions); and conversely, inhibited by the absence of, or aversion to, such a concept of Nothingness in the West. But not all scholars agree. Join the debate.
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Investigating zero in ancient Egypt raises different questions and covers about 3,000 years of cultural history. Starting with the Egyptian language (which was easily able to express the idea of non-existence), I first provide some general information about Egyptian number writing, followed by multiplicative number writing and then a critical assessment of the traditional opinion that the Egyptian word nfr was ‘zero’. After that are considered, in turn, Egyptian expressions for ‘nothingness’ and missing objects (from the third millennium BCE), the absence of entries in bookkeeping (from the second millennium BCE), the absence of numbers and placeholder signs (from the second half of the first millennium BCE). Of particular importance was iwty/iwṱ which was incorporated into Greek (in the first century CE) and finally into Arabic (in the eleventh century) astronomical texts in sexagesimal notations.
The Egyptians did not understand zero as a numerical value. They did not need it for expressing any number and they did not calculate using zero. The question is finally raised as to which criteria one should accept as a firm and definite proof for the existence of the concept of zero as a numerical/mathematical value in a culture, not only the Egyptian.
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Two of the characteristics of our modern concept of zero find equivalents in Ancient Mesopotamia. First, the Babylonian sexagesimal positional notation from its invention in the late third millennium BCE required a means of indicating a ‘placeholder’ in a number, a missing power of the base. Secondly, the needs of mathematical astronomy, developed in the first millennium BCE, had to deal with a numerical concept of ‘nothing’ as a full-fledged number, capable of entering into arithmetical operations.
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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.
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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.
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Attempts to explain the importance of inventing zero and the application thereof in the context of the history of culture, philosophy, and the exact sciences bring to the forefront a range of problems in relation to the overlapping of challenging issues (and terminology) in science and culture. The symbol of zero is qualified both as a scientific discovery (or invention) and an empirically found solution for the satisfaction of certain practical needs of humans. Moreover, its invention is also based on references to certain religious and philosophical teachings. All this makes the explanation of this phenomenon extremely difficult, taking into account that explanations frequently connect fundamentally different fields that actually address completely different areas of research. Furthermore, when looking more closely into the materials related to scientific-technical and religious-philosophical explanations, we see that the emergence of zero (or the idea of ‘emptiness’, ‘nothingness’) on the religious-philosophical side does not at all indicate the existence of this phenomenon in exact scientific use, and vice versa. There might be a discussion about ‘emptiness’ or ‘nothingness’ within a specific historical period in a certain society, and at the same time zero might not show up at all in the mathematical theory and practice known to that society, as was the case, for example, in medieval Europe. Likewise, the introducers of zero in mathematical practice may theoretically disagree with the idea of ‘nothingness’ per se at the religious-philosophical level, as was seen in the Europe of the early modern period.
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The article looks at the concept of zero, from the philosophical and mathematical sides, in the Jewish tradition from the Bible to the modern age. With regard to philosophy, three aspects will be considered: (1) ex nihilo creation; (2) the definition of the deity as the Naught and the creation out of the Naught; (3) the existence of the vacuum in the material world after the Creation. Two mathematical aspects will be examined: (1) the Hebrew system for designating numbers (numerals) from the Bible until the adoption of the Arabic numerals; (2) the notion of zero as a numerical value. The sources for both discussions are mainly the Bible, rabbinic texts of the Talmudic age, Sefer ha-Yezirah, the medieval philosophic literature (including the work of Abraham Ibn Ezra, the very first Hebrew author to deal with zero), and medieval Kabbalah.
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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.
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Since Brahmagupta (seventh century), the reciprocal 1/0 of the number 0 was used in mathematical operations. The ideas were not limited to the notions of limits in calculus but appear to be unusual algebraic constructs which have no parallel in history of mathematics. We present an analysis using ideas from modern Algebra. We also discuss numerous similar structures in modern mathematics.
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