In this paper we intend to discuss the importance of providing a physical representation of quantum superpositions which goes beyond the mere reference to mathematical structures and measurement outcomes. This proposal goes in the opposite direction to the project present in orthodox contemporary philosophy of physics which attempts to “bridge the gap” between the quantum formalism and common sense “classical reality”—precluding, right from the start, the possibility of interpreting quantum superpositions through non-classical notions. We will argue that in order to (...) restate the problem of interpretation of quantum mechanics in truly ontological terms we require a radical revision of the problems and definitions addressed within the orthodox literature. On the one hand, we will discuss the need of providing a formal redefinition of superpositions which captures explicitly their contextual character. On the other hand, we will attempt to replace the focus on the measurement problem, which concentrates on the justification of measurement outcomes from “weird” superposed states, and introduce the superposition problem which focuses instead on the conceptual representation of superpositions themselves. In this respect, after presenting three necessary conditions for objective physical representation, we will provide arguments which show why the classical representation of physics faces severe difficulties to solve the superposition problem. Finally, we will also argue that, if we are willing to abandon the presupposition according to which ‘Actuality = Reality’, then there is plenty of room to construct a conceptual representation for quantum superpositions. (shrink)

This article focuses on particular ways in which visual representations contribute to the development of mathematical knowledge. I give examples of diagrammatic representations that enable one to observe new properties and cases where representations contribute to classification. I propose that fruitful representations in mathematics are iconic representations that involve conventional or symbolic elements, that is, iconic metaphors. In the last part of the article, I explain what these are and how they apply in the (...) considered examples. (shrink)

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, namely, ZFC set theory, all mathematical objects are represented by sets, while ordinary, namely, non–set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the (...) logical strength of basic theorems named after Tietze, Heine, and Weierstrass changes significantly upon the replacement of “second-order representations” by “third-order functions.” We discuss the implications and connections to the reverse mathematics program and its foundational claims regarding predicativist mathematics and Hilbert’s program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations. (shrink)

Cet article discute le rôle des représentations dans les preuves mathématiques. Il est suggéré ici que les représentations nous permettent de diviser une preuve en plusieurs parties plus faciles à traiter. Nous illustrerons cela avec un exemple de la pratique mathématique actuelle qui consiste à trouver la valeur d'une expression en la divisant graduellement en parties plus simples. Par ailleurs, j'explique le rôle que jouent les icônes et les indices dans cette procédure. Les icônes assurent la similarité entre l'expression et (...) les résultats obtenus. En revanche, les indices agissent comme des indicateurs qui permettent de réinsérer les résultats aux emplacements appropriés. (shrink)

Cet article discute le rôle des représentations dans les preuves mathématiques. Il est suggéré ici que les représentations nous permettent de diviser une preuve en plusieurs parties plus faciles à traiter. Nous illustrerons cela avec un exemple de la pratique mathématique actuelle qui consiste à trouver la valeur d'une expression en la divisant graduellement en parties plus simples. Par ailleurs, j'explique le rôle que jouent les icônes et les indices dans cette procédure. Les icônes assurent la similarité entre l'expression et (...) les résultats obtenus. En revanche, les indices agissent comme des indicateurs qui permettent de réinsérer les résultats aux emplacements appropriés. (shrink)

Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

SummaryIn this paper we develop a general concept of a theory analogous to that of an empirical theory. It is shown that axioms can be regarded as rules for performing operations. Using this connection we give a definition of an intellectual structure, and it turns out that mathematical theories represent intellectual structures in a natural way.RésuméCet article développe un concept général de théorie analogue à celui de théorie empirique. II montre que les axiomes peuvent être considérés comme des régles pour (...) réaliser des opérations . Cette connexion permet de définir une structure intellectuelle, et il se trouve que les theories mathématiques représentent ces structures intellectuelles de façon naturelle.ZusammenfassungIn dieser Arbeit wird der allgemeine Begriff einer mathematischen Theorie entwickelt, der dem einer empirischen Theorie analog ist. Es wird gezeigt, dass sich Axiome als Regeln zur Durchführung von Operationen auffassen lassen. Unter Benutzung dieses Zusam‐rnenhangs wird eine Definition von intellektueller Struktur gegeben und es zeigt sich, dass mathe‐matische Theorien in natürlicher Weise intellektuelle Strukturen repräsentieren. (shrink)

This essay is a contribution to the historical phenomenology of science, taking as its point of departure Husserl’s later philosophy of science and Jacob Klein’s seminal work on the emergence of the symbolic conception of number in European mathematics during the late sixteenth and seventeenth centuries. Sinceneither Husserl nor Klein applied their ideas to actual theories of modern mathematical physics, this essay attempts to do so through a case study of the conceptof “spacetime.” In §1, I sketch Klein’s account (...) of the emergence of the symbolic conception of number, beginning with Vieta in the late sixteenth century. In §2,through a series of historical illustrations, I show how the principal impediment to assimilating the new symbolic algebra to mathematical physics, namely, thedimensionless character of symbolic number, is overcome via the translation of the traditional language of ratio and proportion into the symbolic language of equations. In §§3–4, I critically examine the concept of “Minkowski spacetime,” specifically, the purported analogy between the Pythagorean distance formula and the Minkowski “spacetime interval.” Finally, in §5, I address the question of whether the concept of Minkowski spacetime is, as generally assumed, indispensable to Einstein’s general theory of relativity. (shrink)

I argue that mathematical representations can have heuristic power since their construction can be ampliative. To this end, I examine how a representation introduces elements and properties into the represented object that it does not contain at the beginning of its construction, and how it guides the manipulations of the represented object in ways that restructure its components by gradually adding new pieces of information to produce a hypothesis in order to solve a problem.In addition, I defend an ‘inferential’ (...) approach to the heuristic power of representations by arguing that these representations draw on ampliative inferences such as analogies and inductions. In effect, in order to construct a representation, we have to ‘assimilate’ diverse things, and this requires identifying similarities between them. These similarities form the basis for ampliative inferences that gradually build hypotheses to solve a problem.To support my thesis, I analyse two examples. The first one is intra-field, that is, the construction of an algebraic representation of 3-manifolds; the second is inter-fields, that is, the construction of a topological representation of DNA supercoiling. (shrink)

The Enhanced Indispensability Argument (Baker [ 2009 ]) exemplifies the new wave of the indispensability argument for mathematical Platonism. The new wave capitalizes on mathematics' role in scientific explanations. I will criticize some analyses of mathematics' explanatory function. In turn, I will emphasize the representational role of mathematics, and argue that the debate would significantly benefit from acknowledging this alternative viewpoint to mathematics' contribution to scientific explanations and knowledge.

Often philosophers, logicians, and mathematicians employ a notion of intended structure when talking about a branch of mathematics. In addition, we know that there are foundational mathematical theories that can find representatives for the objects of informal mathematics. In this paper, we examine how faithfully foundational theories can represent intended structures, and show that this question is closely linked to the decidability of the theory of the intended structure. We argue that this sheds light on the trade-off between (...) expressive power and meta-theoretic properties when comparing first-order and second-order logic. (shrink)

This paper distinguishes between 3 meanings of reversal, all of which are mathematically equivalent in classical mechanics: velocity reversal, retrodiction, and time reversal. It then concludes that in order to have well defined velocities a primitive arrow of time must be included in every time slice. The paper briefly mentions that this arrow cannot come from the Second Law of thermodynamics, but this point is developed in more details elsewhere.

It is often supposed that we can use mathematics to capture the time evolution of any physical system. By this, I mean that we can capture the basic truths about the time evolution of a physical system with a set of mathematical assertions, which can then be used as premises in arbitrary mathematical arguments to deduce more complex properties of the system. ;I would like to argue that this picture of the role of mathematics in physics is incorrect. (...) Specifically, I shall assert: ;The Deduction Failure Thesis: Bodies of knowledge in physics are generally not closed under otherwise valid mathematical argument forms. ;The Representation Failure Thesis: We cannot assume that the state of any system, together with its fundamental laws, can be captured by some set of mathematical assertions or equations. In fact, it is more likely that the world is not representable by a set of mathematical assertions or equations than that it is. ;The dissertation largely consists of arguments for these two theses. (shrink)

Critical notice of C. Pincock's Mathematics and Scientific Representation (2012).

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least (and plausibly for others), scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is (...) on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts. (shrink)

Representations of monadic MV -algebra, the characterization of locally finite monadic MV -algebras, with axiomatization of them, definability of non-trivial monadic operators on finitely generated free MV -algebras are given. Moreover, it is shown that finitely generated m-relatively complete subalgebra of finitely generated free MV -algebra is projective.

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least, scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track (...) with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts. (shrink)

The first English translation of Hans Reichenbach's lucid doctoral thesis sheds new light on how Kant’s Critique of Pure Reason was understood in some quarters at the time. The source of several themes in his still influential The Direction of Time, the thesis shows Reichenbach's early focus on the interdependence of physics, probability, and epistemology.

The philosophy of mathematical practice sometimes investigates the social constitution of mathematics but does not always make explicit the philosophical-normative framework that guides the discussion. This chapter investigates some recent proposals in the philosophy of mathematical practice that compare social facts and mathematical objects, discussing similarities and differences. An attempt will be made to identify, through a comparison with three different perspectives in social ontology, the kind of objectivity attributed to mathematical knowledge, the type of representational or non-representational semantics (...) adopted, and the justificatory or coordinative role entrusted to axiomatics. After a brief introduction to key issues in social ontology, Sect. 3 of the chapter offers a survey of contributions by Feferman, Ferreirós, and Cole, highlighting a difference between approaches based on rules, practices, and intentional states, respectively. Section 4.1 also discusses results by Carter, Collin, Giardino, and Pantsar, focusing on differences between the objectivity of knowledge and objectivity of objects and showing that many authors combine a realist, structuralist, and pragmatist perspective, as well as the idea that objectivity comes in degrees. Section 4.2 focuses on the kind of semantics that is adopted in socially oriented approaches. A representational semantics is preferred by authors grounding their views on mental states, whereas a non-representational semantics best fits with views based on practices. Section 5 considers how axiomatics can be understood in a social perspective. Axiomatics generally plays a justificatory role also in theories that aim to explain the social constitution of mathematical knowledge. Yet, if one shifts from approaches based on mental states to approaches anchored on rules or habits, axiomatics can be viewed as an institution based on obligations, functions, coordination problems, and agent’s actions and roles, thus playing an organizational and coordination role. (shrink)

Epistemology is the study of knowledge and justified belief. Belief is thus central to epistemology. It comes in a qualitative form, as when Sophia believes that Vienna is the capital of Austria, and a quantitative form, as when Sophia's degree of belief that Vienna is the capital of Austria is at least twice her degree of belief that tomorrow it will be sunny in Vienna. Formal epistemology, as opposed to mainstream epistemology (Hendricks 2006), is epistemology done in a formal way, (...) that is, by employing tools from logic and mathematics. The goal of this entry is to give the reader an overview of the formal tools available to epistemologists for the representation of belief. A particular focus will be the relation between formal representations of qualitative belief and formal representations of quantitative degrees of belief. (shrink)

We study, in an abstract and general framework, formal representations of dependence and groundedness which occur in semantic theories of truth. Our goals are: (a) to relate the different ways in which groundedness is defined according to the way dependence is represented; and (b) to represent different notions of dependence as instances of a suitable generalisation of the mathematical notion of functional dependence.

According to a grand narrative that long ago ceased to be told, there was a seventeenth century Scientific Revolution, during which a few heroes conquered nature thanks to mathematics. This grand narrative began with the exhibition of quantitative laws that these heroes, Galileo and Newton for example, had disclosed: the law of falling bodies, according to which the speed of a falling body is proportional to the square of the time that has elapsed since the beginning of its fall; (...) the law of gravitation, according to which two bodies are attracted to one another in proportion to the sum of their masses and in inverse proportion to the square of the distance separating them -- according to his own preferences, each narrator added one or two quantitative laws of this kind. The essential feature was not so much the examples that were chosen, but, rather, the more or less explicit theses that accompanied them. First, mathematization would be taken as the criterion for distinguishing between a qualitative Aristotelian philosophy and the new quantitative physics. Secondly, mathematization was founded on the metaphysical conviction that the world was created pondere, numero et mensura, or that the ultimate components of natural things are triangles, circles, and other geometrical objects. This metaphysical conviction had two immediate consequences: that all the phenomena of nature can be in principle submitted to mathematics and that mathematical language is transparent; it is the language of nature itself and has simply to be picked up at the surface of phenomena. Finally, it goes without saying that, from a social point of view, the evolution of the sciences was apprehended through what has been aptly called the "relay runner model," according to which science progresses as a result of individual discoveries. Grand narratives such as this are perhaps simply fictions doomed to ruin as soon as they are clearly expressed. In any case, the very assumption on which this grand narrative relies can be brought into question: even in the canonical domain of mechanics, the relevant epistemological units crucial to understanding the dynamics of the Scientific Revolution are perhaps not a few laws of motion, but a complex set of problems embodied in mundane objects. Moreover, each of the theses just mentioned was actually challenged during the long period of historiographical reappraisal, out of which we have probably not yet stepped. Against the sharp distinction between a qualitative Aristotelian philosophy and the new quantitative physics, numerous studies insist that Rome wasn't built in a day, so to speak. Since Antiquity, there have always been mixed sciences; the emergence of pre-classical mechanics depends on both medieval treatises and the practical challenges met by Renaissance engineers. It is indeed true that, for Aristotle, mathematics merely captures the superficial properties of things, but the Aristotelianisms were many during the Renaissance and the Early Modern period, with some of them being compatible with the introduction of mathematics in natural philosophy. In addition, the gap between the alleged program of mathematizing nature and its effective realization was underlined as most natural phenomena actually escaped mathematization; at best they were enrolled in what Thomas Kuhn began to rehabilitate under the appellation of the "Baconian sciences," i.e., empirical investigations aiming at establishing isolated facts, without relating them to any overarching theory. Hence, mathematization of nature cannot pretend to capture a historical fact: at most, it expresses an indeterminate task for generations to come. On top of these first two considerations, and against the thesis of the neutrality of the mathematical language, it was urged that mathematics is not "only a language" and that, exactly as other symbolic means or cognitive tools, it has its own constraints. For example, it has been thoroughly explained that the Euclidean theory of proportions both guides and frustrates the Galilean analysis of motion; its shortages were particularly clear with respect to the expression of continuity, which is crucial in the case of motion. Consequently, when calculus was invented and applied to the analysis of motion, it was not a transposition that left things as they stood. Even more clearly than in the case of a translation from one natural language to another, the shift from one symbolic language to another entails that certain possibilities are opened while others are closed. The cognitive constraints imposed by established mathematical theories, as seen in the theory of proportions or calculus, were not the only ones to be studied in relation to mathematization. Certain schemes dependent on the grammar of natural languages, e.g., the scheme of contrariety, or certain symbolic means of representation, e.g. geometrical diagrams and numerical tables, were also subject to such scrutiny. Lastly, it was insisted that, even if we concede the existence of scientific geniuses, mathematics is largely produced by intellectual communities and embedded within social practices. More attention was consequently paid to the forms of communication in given mathematical networks, or to the teaching of the discipline in, for example, Jesuit colleges and universities. The set of mathematical practices specific to specialized craftsmen, highly-qualified experts and engineers began to be studied in its own right. All these reflections may have helped us change our perspectives on the question of mathematization. It seems, however, that they were instead set aside, both because of a general distrust towards sweeping narratives that are always subject to the suspicion that they overlook the unyielding complexity of real history, and because of a shift in our interests. The more obscure and idiosyncratic they are, an alchemist, a patron of the sciences or a lunatic collector is nowadays honored in journals of the history of sciences. As for the general issues involved in the question of mathematization, they are rejected as obsolete, or reserved for specialized journals in the history of mathematics. Consequently, before presenting the essays of this fascicle, I would like to say a few words in favor of a renewed study of the forms of mathematization in the history of the early sciences. (shrink)

Mathematicians’ use of external representations, such as symbols and diagrams, constitutes an important focal point in current philosophical attempts to understand mathematical practice. In this paper, we add to this understanding by presenting and analyzing how research mathematicians use and interact with external representations. The empirical basis of the article consists of a qualitative interview study we conducted with active research mathematicians. In our analysis of the empirical material, we primarily used the empirically based frameworks provided by distributed (...) cognition and cognitive semantics as well as the broader theory of cognitive integration as an analytical lens. We conclude that research mathematicians engage in generative feedback loops with material representations, that they use representations to facilitate the use of experiences of handling the physical world as a resource in mathematical work, and that their use of representations is socially sanctioned and enabled. These results verify the validity of the cognitive frameworks used as the basis for our analysis, but also show the need for augmentation and revision. Especially, we conclude that the social and cultural context cannot be excluded from cognitive analysis of mathematicians’ use of external representations. Rather, representations are socially sanctioned and enabled in an enculturation process. (shrink)

The [lambda]-calculus can be represented topologically by assigning certain spaces to the types and certain continuous maps to the terms. Using a recent result from category theory, the usual calculus of [lambda]-conversion is shown to be deductively complete with respect to such topological semantics. It is also shown to be functionally complete, in the sense that there is always a ‘minimal’ topological model in which every continuous function is [lambda]-definable. These results subsume earlier ones using cartesian closed categories, as well (...) as those employing so-called Henkin and Kripke [lambda]-models. (shrink)

This paper is about pairing relation algebras as well as fork algebras and related subjects. In the 1991-92 fork algebra papers it was conjectured that fork algebras admit a strong representation theorem . Then, this conjecture was disproved in the following sense: a strong representation theorem for all abstract fork algebras was proved to be impossible in most set theories including the usual one as well as most non-well-founded set theories. Here we show that the above quoted conjecture can still (...) be made true by choosing an appropriate set theory as our foundation of mathematics. Namely, we show that there are non-well-founded set theories in which every abstract fork algebra is representable in the strong sense, i.e. it is isomorphic to a set relation algebra having a fork operation which is obtained with the help of the real pairing function. Further, these non-well-founded set theories are consistent if usual set theory is consistent. Finally, we will discuss related developments in propositional multi-modal logics of quantification and substitution, in algebraic logic e.g. cylindric algebras, the so called finitization problem, and applications to a logic introduced and studied in Tarski-Givant [42]. In particular, representable, weakly higher order cylindric algebras are finitely axiomatizable in a set theory which admits the axiom of foundation for finite sets. (shrink)

In this paper we define sheaf spaces of BL-algebras (or BL-sheaf spaces), we study completely regular and compact BL-sheaf spaces and compact representations of BL-algebras and, finally, we prove that the category of non-trivial BL-algebras is equivalent with the category of compact local BL-sheaf spaces.

In his book The Value of Science Poincaré criticizes a certain view on the growth of mathematical knowledge: “The advance of science is not comparable to the changes of a city, where old edifices are pitilessly torn down to give place to new ones, but to the continuous evolution of zoological types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past” (Poincaré (...) 1958, p. 14). The view criticized by Poincaré corresponds to Frege’s idea that the development of mathematics can be described as an activity of system building, where each system is supposed to provide a complete representation for a certain mathematical field and must be pitilessly torn down whenever it fails to achieve such an aim. All facts concerning any mathematical field must be fully organized in a given system because “in mathematics we must always strive after a system that is complete in itself” (Frege 1979, p. 279). Frege is aware that systems introduce rigidity and are in conflict with the actual development of mathematics because “in history we have development; a system is static”, but he sticks to the view that “science only comes to fruition in a system” because “only through a system can we achieve complete clarity and order” (Frege 1979, p. 242). He even goes so far as saying that “no science can be so enveloped in obscurity as mathematics, if it fails to construct a system” (Frege 1979, p. 242). By ‘system’ Frege means ‘axiomatic system’. In his view, in mathematics we cannot rest content with the fact that “we are convinced of something, but we must strive to obtain a clear insight into the network of inferences that support our conviction”, that is, to find “what the primitive truths are”, because “only in this way can a system be constructed” (Frege 1979, p. 205). The primitive truths are the principles of the axiomatic system. Frege’s stress on the role of systems also determines his views on the growth of mathematical knowledge.. (shrink)

In this paper we give new criterions for left distributive posets to have neatest representations. We also illustrate a construction that would embed left distributive posets into representable semilattices.

Mathematical concepts play at least three roles in the application of mathematics: an inferential role, a representational role, and an expressive role. In this paper, I argue that, despite what has often been alleged, platonists do not fully accommodate these features of the application of mathematics. At best, platonism provides partial ways of handling the issues. I then sketch an alternative, anti-realist account of the application of mathematics, and argue that this account manages to accommodate these features (...) of the application process. In this way, a better account of mathematical applications is, in principle, available. (shrink)

We investigate the use of coalgebra to represent quantum systems, thus providing a basis for the use of coalgebraic methods in quantum information and computation. Coalgebras allow the dynamics of repeated measurement to be captured, and provide mathematical tools such as final coalgebras, bisimulation and coalgebraic logic. However, the standard coalgebraic framework does not accommodate contravariance, and is too rigid to allow physical symmetries to be represented. We introduce a fibrational structure on coalgebras in which contravariance is represented by indexing. (...) We use this structure to give a universal semantics for quantum systems based on a final coalgebra construction. We characterize equality in this semantics as projective equivalence. We also define an analogous indexed structure for Chu spaces, and use this to obtain a novel categorical description of the category of Chu spaces. We use the indexed structures of Chu spaces and coalgebras over a common base to define a truncation functor from coalgebras to Chu spaces. This truncation functor is used to lift the full and faithful representation of the groupoid of physical symmetries on Hilbert spaces into Chu spaces, obtained in our previous work, to the coalgebraic semantics. (shrink)

The article investigates the role of symbolic means of knowledge representation in concept development using the historical example of medieval diagrams of change employed in early modern work on the motion of fall. The parallel cases of Galileo Galilei, Thomas Harriot, and René Descartes and Isaac Beeckman are discussed. It is argued that the similarities concerning the achievements as well as the shortcomings of their respective work on the motion of fall can to a large extent be attributed to their (...) shared use of means of knowledge representation handed down from antiquity and the Middle Ages. While the interpretation of medieval diagrams was unproblematic in the scholastic context from which they arose, in the early modern context, which was characterized by the confluence of natural philosophy and practical mathematics, it became ambiguous. It was the early modern mathematicians’ work within this contradictory framework that brought about a new conceptualization of motion which, in particular, eventually led to an infinitesimal concept of velocity. In this process, the diagrams themselves remained largely unchanged and thus functioned as a catalyst for concept development. (shrink)

A defining characteristic of modern science is its ability to make immensely successful predictions of natural phenomena without invoking a putative god or a supernatural being. Here, we argue that this intellectual discipline would not acquire such an ability without the mathematical zero. We insist that zero and its basic operations were likely conceived in India based on a philosophy of nothing, and classify nothing into four categories—balance, absence, emptiness and nonexistence. We argue that zero is a tangible representation of (...) nonexistence and constitutes all nonzero numbers, which together represent existence. It appears that zero’s journey out of India somewhat separated its mathematical and philosophical aspects, with the former being more valued by some cultures and the latter by others. The European culture, in which modern science grew, largely ignores a philosophy of nothing due to a deep-rooted Greek philosophical base, although this science relies on the notion of nonexistence through zero. Consequently, zero is a mere number of convenience without its foundational philosophy in science, and techniques to circumvent zero are developed. We insist that, while such techniques contribute to the progress of science and mathematics within the current framework, a tendency to avoid zero and its philosophy leads to approximations and may hinder a deeper understanding. Finally, we argue that nonexistence may notionally constitute existence, and hence may be the fundamental. This implies that, if a supernatural being exists, it is not the fundamental. The independence of modern science from a supernatural being is consistent with this. (shrink)