"Multiverse" cosmologies imagine our universe as just one of a vast number of others. While this idea has captivated philosophy, religion, and literature for millennia, it is now being considered as a scientific hypothesis--with different models emerging from cosmology, quantum mechanics, and string theory. Beginning with ancient Atomist and Stoic philosophies, Mary-Jane Rubenstein links contemporary models of the multiverse to their forerunners and explores the reasons for their recent appearance. One concerns the so-called fine-tuning of (...) the universe: nature's constants are so delicately calibrated that it seems they have been set just right to allow life to emerge. For some thinkers, these "fine-tunings" are evidence of the existence of God; for others, however, and for most physicists, "God" is an insufficient scientific explanation. Hence the allure of the multiverse: if all possible worlds exist somewhere, then like monkeys hammering out Shakespeare, one universe is bound to be suitable for life. Of course, this hypothesis replaces God with an equally baffling article of faith: the existence of universes beyond, before, or after our own, eternally generated yet forever inaccessible to observation or experiment. In their very efforts to sidestep metaphysics, theoretical physicists propose multiverse scenarios that collide with it and even produce counter-theological narratives. Far from invalidating multiverse hypotheses, Rubenstein argues, this interdisciplinary collision actually secures their scientific viability. We may therefore be witnessing a radical reconfiguration of physics, philosophy, and religion in the modern turn to the multiverse. (shrink)

"Multiverse" cosmologies imagine our universe as just one of a vast number of others. While this idea has captivated philosophy, religion, and literature for millennia, it is now being considered as a scientific hypothesis--with different models emerging from cosmology, quantum mechanics, and string theory. Beginning with ancient Atomist and Stoic philosophies, Mary-Jane Rubenstein links contemporary models of the multiverse to their forerunners and explores the reasons for their recent appearance. One concerns the so-called fine-tuning of (...) the universe: nature's constants are so delicately calibrated that it seems they have been set just right to allow life to emerge. For some thinkers, these "fine-tunings" are evidence of the existence of God; for others, however, and for most physicists, "God" is an insufficient scientific explanation. Hence the allure of the multiverse: if all possible worlds exist somewhere, then like monkeys hammering out Shakespeare, one universe is bound to be suitable for life. Of course, this hypothesis replaces God with an equally baffling article of faith: the existence of universes beyond, before, or after our own, eternally generated yet forever inaccessible to observation or experiment. In their very efforts to sidestep metaphysics, theoretical physicists propose multiverse scenarios that collide with it and even produce counter-theological narratives. Far from invalidating multiverse hypotheses, Rubenstein argues, this interdisciplinary collision actually secures their scientific viability. We may therefore be witnessing a radical reconfiguration of physics, philosophy, and religion in the modern turn to the multiverse. (shrink)

The maturation of the physical image has made apparent the limits of our scientific understanding of fundamental reality. These limitations serve as motivation for a new form of metaphysical inquiry that restricts itself to broadly scientific methods. Contributing towards this goal we combine the mathematical universe hypothesis as developed by Max Tegmark with the axioms of Stewart Shapiro’s structure theory. The result is a theory we call the Theory of the Structural Multiverse (TSM). The focus is (...) on informal theory development and constraint satisfaction. Some empirical consequences of the theory are worked out, in particular the feasibility of a predictive observer selection effect. The explanatory, unifying, and generative powers of the theory are found to substantially support the theory. The TSM is demonstrated to be an empirically significant scientific theory that is foundational to and continuous with the rest of the scientific image. (shrink)

Multiverse Views in set theory advocate the claim that there are many universes of sets, no-one of which is canonical, and have risen to prominence over the last few years. One motivating factor is that such positions are often argued to account very elegantly for technical practice. While there is much discussion of the technical aspects of these views, in this paper I analyse a radical form of Multiversism on largely philosophical grounds. Of particular importance will be (...) an account of reference on the Multiversist conception, and the relativism that it implies. I argue that analysis of this central issue in the Philosophy of Mathematics indicates that Radical Multiversism must be algebraic, and cannot be viewed as an attempt to provide an account of reference without a softening of the position. (shrink)

The universe appears fine-tuned for life. Bayesian confirmation theory is utilized to examine two competing explanations for this fine-tuning, namely design (theism) and the existence of many universes, in comparison with the ’null’ hypothesis that just one universe exists as a brute fact. Some authors have invoked the so-called ’inverse gambler’s fallacy’ to argue that the many-universes hypothesis does not explain the fine-tuning of ’this’ universe, but flaws in this argument are (...) exposed. Nevertheless, the hypothesis of design, being simpler, is arguably of higher prior probability and, therefore, to be preferred. The hypothesis of the single brute-fact universe is disconfirmed. (shrink)

Kurt Gödel, mathematician and logician, was one of the most influential thinkers of the twentieth century. Gödel fled Nazi Germany, fearing for his Jewish wife and fed up with Nazi interference in the affairs of the mathematics institute at the University of Göttingen. In 1933 he settled at the Institute for Advanced Study in Princeton, where he joined the group of world-famous mathematicians who made up its original faculty. His 1940 book, better known by its short title, The Consistency of (...) the Continuum Hypothesis, is a classic of modern mathematics. The continuum hypothesis, introduced by mathematician George Cantor in 1877, states that there is no set of numbers between the integers and real numbers. It was later included as the first of mathematician David Hilbert's twenty-three unsolved math problems, famously delivered as a manifesto to the field of mathematics at the International Congress of Mathematicians in Paris in 1900. In The Consistency of the Continuum Hypothesis Gödel set forth his proof for this problem. In 1999, Time magazine ranked him higher than fellow scientists Edwin Hubble, Enrico Fermi, John Maynard Keynes, James Watson, Francis Crick, and Jonas Salk. He is most renowned for his proof in 1931 of the 'incompleteness theorem,' in which he demonstrated that there are problems that cannot be solved by any set of rules or procedures. His proof wrought fruitful havoc in mathematics, logic, and beyond. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework?like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as the class of all individuals. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework?like the 1931 Gödel incompleteness paper. A pluralistic multiple- (...) class='Hi'>universe framework recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations?as in post-WWII model theory. In the early 1960s, many logicians?mistakenly, as we show?held the ?contrary alternative? that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege?Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski's philosophy underwent many other radical changes. (shrink)

The fine-tuning argument can be used to support the Many Universe hypothesis. The Inverse Gambler’s Fallacy objection seeks to undercut the support for the Many Universe hypothesis. The objection is that although the evidence that there is life somewhere confirms Many Universes, the specific evidence that there is life in this universe does not. I will argue that the Inverse Gambler’s Fallacy is not committed by the fine-tuning argument. The key issue (...) is the procedure by which the universe with life is selected for observation. Once we take account of the procedure, we find that the support for the Many Universe hypothesis remains. (shrink)

The answer to the fine-tuning problem of the universe has been traditionally sought in terms of either design or multiverse. In philosophy circles, this is sometimes expanded by adding the option of explanatory nihilism—the claim that there is no explanation for statements of that high level of generality: fine-tunings are brute facts. In this paper, we consider the fourth option which, at least in principle, is available to us: co-evolution of the universe and observers. Although conceptual roots (...) of this approach could be found already in ancient stoicism, it is still the least investigated explanatory option for resolving the problem of empirical fine tunings. We offer two preliminary models along which the co-evolution hypothesis could be developed further. They are still on the level of speculative metaphysics, but there are opportunities along the way to generate predictions which are in principle testable, especially in the domain of large-scale numerical simulations. (shrink)

With Fermat’s Last Theorem finally disposed of by Andrew Wiles in 1994, it’s only natural that popular attention should turn to arguably the most outstanding unsolved problem in mathematics: the Riemann Hypothesis. Unlike Fermat’s Last Theorem, however, the Riemann Hypothesis requires quite a bit of mathematical background to even understand what it says. And of course both require a great deal of background in order to understand their significance. The Riemann Hypothesis was first articulated by Bernhard (...) Riemann in an address to the Berlin Academy in 1859. The address was called “On the Number of Prime Numbers Less Than a Given Quantity” and among the many interesting results and methods contained in that paper was Riemann’s famous hypothesis: all non-trivial zeros of the zeta function, ζ(s) = ∞ n=1 n−s, have real part 1/2. Although the zeta function as stated and considered as a real-valued function is defined only for s > 1, it can be suitably extended. It can, as a matter of fact, be extended to have as its domain all the complex numbers (numbers of the form x + yi, where x and y √ −1) with the exception of 1 + 0i (at which point are real numbers and i =. (shrink)

In this article, I survey some philosophical attitudes to talk concerning `the' universe of sets. I separate out four different strands of the debate, namely: (i) Universism, (ii) Multiversism, (iii) Potentialism, and (iv) Pluralism. I discuss standard arguments and counterarguments concerning the positions and some of the natural mathematical programmes that are suggested by the various views.

This volume covers a wide range of topics in the most recent debates in the philosophy of mathematics, and is dedicated to how semantic, epistemological, ontological and logical issues interact in the attempt to give a satisfactory picture of mathematical knowledge. The essays collected here explore the semantic and epistemic problems raised by different kinds of mathematical objects, by their characterization in terms of axiomatic theories, and by the objectivity of both pure and applied mathematics. They investigate controversial (...) aspects of contemporary theories such as neo-logicist abstractionism, structuralism, or multiversism about sets, by discussing different conceptions of mathematical realism and rival relativistic views on the mathematical universe. They consider fundamental philosophical notions such as set, cardinal number, truth, ground, finiteness and infinity, examining how their informal conceptions can best be captured in formal theories. The philosophy of mathematics is an extremely lively field of inquiry, with extensive reaches in disciplines such as logic and philosophy of logic, semantics, ontology, epistemology, cognitive sciences, as well as history and philosophy of mathematics and science. By bringing together well-known scholars and younger researchers, the essays in this collection – prompted by the meetings of the Italian Network for the Philosophy of Mathematics (FilMat) – show how much valuable research is currently being pursued in this area, and how many roads ahead are still open for promising solutions to long-standing philosophical concerns. Promoted by the Italian Network for the Philosophy of Mathematics – FilMat. (shrink)

The objective of this paper is analyzing to which extent the multiverse hypothesis provides a real explanation of the peculiarities of the laws and constants in our universe. First we argue in favor of the thesis that all multiverses except Tegmark’s “mathematical multiverse” are too small to explain the fine tuning, so that they merely shift the problem up one level. But the “mathematical multiverse" is surely too large. To prove this assessment, we (...) have performed a number of experiments with cellular automata of complex behavior, which can be considered as universes in the mathematical multiverse. The analogy between what happens in some automata (in particular Conway’s “Game of Life") and the real world is very strong. But if the results of our experiments can be extrapolated to our universe, we should expect to inhabit—in the context of the multiverse—a world in which at least some of the laws and constants of nature should show a certain time dependence. Actually, the probability of our existence in a world such as ours would be mathematically equal to zero. In consequence, the results presented in this paper can be considered as an inkling that the hypothesis of the multiverse, whatever its type, does not offer an adequate explanation for the peculiarities of the physical laws in our world. (shrink)

This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page Tarski consequence-definition paper is based on a monistic fixed-universe framework—like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as ‘the class of all individuals’. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework—like the Gödel incompleteness paper. A pluralistic multiple-universe framework (...) recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations—as in post-WWII model theory. In the early 1960s, many logicians—mistakenly, as we show—held the ‘contrary alternative’ that Tarski had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege-Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski’s philosophy underwent many other radical changes. (shrink)

It is shown that, for certain classes of cosmological model which either postulate or give rise to infinitely many universes, only a measure zero subset of the set of possible universes above a given size can in fact be physically realized. It follows that claims to explain the fine tuning of our universe on the basis of such models by appeal to the existence of all possible universes fail.

Multiverses are predictions based on theories. Focusing on each theory’s assumptions is key to evaluating a proposed multiverse. Although accepted theories of particle physics and cosmology contain non-intuitive features, multiverse theories entertain a host of “strange” assumptions classified as metaphysical topics such as: infinity, duplicate yous, hypothetical fields, more than three space dimensions, Hilbert space, advanced civilizations, and reality established by mathematical relationships. It is easy to confuse multiverse proposals because many divergent models exist. This (...) overview defines the characteristics of eleven popular multiverse proposals. The characteristics compared are: initial conditions, values of constants, laws of nature, number of space dimensions, number of universes, and fine tuning explanations. Future scientific experiments may validate selected assumptions; but until they do, proposals by philosophers may be as valid as theoretical scientific theories. (shrink)

Multiverses are predictions based on theories. Focusing on each theory’s assumptions is key to evaluating a proposed multiverse. Although accepted theories of particle physics and cosmology contain non-intuitive features, multiverse theories entertain a host of “strange” assumptions classified as metaphysical topics such as: infinity, duplicate yous, hypothetical fields, more than three space dimensions, Hilbert space, advanced civilizations, and reality established by mathematical relationships. It is easy to confuse multiverse proposals because many divergent models exist. This (...) overview defines the characteristics of eleven popular multiverse proposals. The characteristics compared are: initial conditions, values of constants, laws of nature, number of space dimensions, number of universes, and fine tuning explanations. Future scientific experiments may validate selected assumptions; but until they do, proposals by philosophers may be as valid as theoretical scientific theories. (shrink)

We discuss the many aspects and qualities of the number one: the different ways it can be represented, the different things it may represent. We discuss the ordinal and cardinal natures of the one, its algebraic behaviour as a neutral element and finally its role as a truth-value in logic.

Our theme is that not every interesting question in set theory is independent of ZFC. We give an example of a first order theory T with countable D(T) which cannot have a universal model at ℵ1 without CH; we prove in ZFC a covering theorem from the hypothesis of the existence of a universal model for some theory; and we prove--again in ZFC--that for a large class of cardinals there is no universal linear order (e.g. in every regular $\aleph_1 (...) < \lambda < 2^{\aleph_0}$). In fact, what we show is that if there is a universal linear order at a regular λ and its existence is not a result of a trivial cardinal arithmetical reason, then λ "resembles" ℵ1--a cardinal for which the consistency of having a universal order is known. As for singular cardinals, we show that for many singular cardinals, if they are not strong limits then they have no universal linear order. As a result of the nonexistence of a universal linear order, we show the nonexistence of universal models for all theories possessing the strict order property (for example, ordered fields and groups, Boolean algebras, p-adic rings and fields, partial orders, models of PA and so on). (shrink)

The multiverse theodicy says that because God can without cost create an infinite number of universes, the standards of acceptability that a conceivable universe must meet to be worthy of divine creation are significantly laxer than is typically supposed in discussions of the problem of evil. While the prospect of a theistic multiverse arguably helps the theist to explain suffering, I argue that it also poses a serious skeptical worry. Given the alleged laxity of the (...) standards that a universe must meet to be worthy of inclusion in a theistic multiverse, there is reason to think that God would be justified in creating a great many deceptive universes that, while good overall, are inhabited by creatures who are radically mistaken in their beliefs. And these deceptive universes would arguably be no less abundant than the nondeceptive universes. After developing this skeptical challenge, I assess some possible theistic responses. One of the more promising responses I consider argues that in order to secure the great good of true friendship between God and creatures, God has reason to exclude deceptive universes from the multiverse even when those universes have great intrinsic value. (shrink)

The universal homogeneous triangle-free graph, constructed by Henson [A family of countable homogeneous graphs, Pacific J. Math.38(1) (1971) 69–83] and denoted H3, is the triangle-free analogue of the Rado graph. While the Ramsey theory of the Rado graph has been completely established, beginning with Erdős–Hajnal–Posá [Strong embeddings of graphs into coloured graphs, in Infinite and Finite Sets. Vol.I, eds. A. Hajnal, R. Rado and V. Sós, Colloquia Mathematica Societatis János Bolyai, Vol. 10 (North-Holland, 1973), pp. 585–595] and culminating in work (...) of Sauer [Coloring subgraphs of the Rado graph, Combinatorica26(2) (2006) 231–253] and Laflamme–Sauer–Vuksanovic [Canonical partitions of universal structures, Combinatorica26(2) (2006) 183–205], the Ramsey theory of H3 had only progressed to bounds for vertex colorings [P. Komjáth and V. Rödl, Coloring of universal graphs, Graphs Combin.2(1) (1986) 55–60] and edge colorings [N. Sauer, Edge partitions of the countable triangle free homogenous graph, Discrete Math.185(1–3) (1998) 137–181]. This was due to a lack of broadscale techniques. We solve this problem in general: For each finite triangle-free graph G, there is a finite number T(G) such that for any coloring of all copies of G in H3 into finitely many colors, there is a subgraph of H3 which is again universal homogeneous triangle-free in which the coloring takes no more than T(G) colors. This is the first such result for a homogeneous structure omitting copies of some nontrivial finite structure. The proof entails developments of new broadscale techniques, including a flexible method for constructing trees which code H3 and the development of their Ramsey theory. (shrink)

The underlying claim of this essay is that we live in a multiverse, that is a universe of many universes that occupy the same space and time, not as an exotic excursion into the realms of science fiction, but as an everyday necessity that affects our social and economic interchange. Faced with such instability, the convenient way that this was managed was through an arbitrary division of labour that assigned the rational to the ‘real’ and the (...) irrational to the ‘imagined’. Recent speculation in cosmology and the science of consciousness studies has obliged us to reconsider the concept of reality as an ‘absolute given’ from which all laws can be verified. In string theory in particular, the dispute now hinges on the existence of ten or eleven dimensions in rippling membranes that discharge energy at the point of contact. In consciousness studies, similar models have appeared in order to embrace what might be understood as an apparent moment of enchantment essential to turn consciousness into awareness. The inevitable realization in scientific circles that the reality of the imagined has as an equivalent epistemological significance to the reality of the material, raises fascinating questions as it invites a sceptical reconsideration of the essential basis of knowledge and a revision of procedures. While the radical shift in scientific thought provides the moment of profound satisfaction for those artists, designers and scientists who have long argued for a transdisciplinary world-view, it also provides a moment of great challenge as we begin to consider how knowledge might be extended, codified and distributed in a multiverse, and begin to reflect on the relationships between text and world when any given world is only defined by the temporary consensus dependent on an arbitrary episteme. (shrink)

The multiverse hypothesis is the leading alternative to the competing fine-tuning hypothesis. The multiverse dispels many aspects of the fine-tuning argument by suggesting that there are different initial conditions in each universe, varying constants of physics, and the laws of nature lose their known arbitrary values; thus, making the previous single-universe argument from fine- tuning incredibly weak. The position that will be advocated will be that a form of multiverse could exist and (...) that any level of Tegmark's multiverse does not circumvent the fine-tuning argument for the existence of a fine-tuner. An argument will be presented suggesting that the multiverse could strengthen the fine-tuning argument by narrowing down the parameters of the data to a more fundamental description needed for the multiverse to function as it does since it may be the case that some physical descriptions are not the same in every universe. That is, the task of explaining how the multiverse mechanism functions cannot be accounted for by the multiverse itself. Fine-tuning by a mind serves as a sufficient and best explanation for the data. (shrink)

Many philosophers and scientists believe that we need an explanation as to why the laws of physics and the initial conditions of the universe are fine-tuned for life. The standard two options are: theism and the multiverse hypothesis. Both of these theories are extravagant and arguably have false predictions. Drawing on contemporary philosophy of mind, I outline a form of panpsychism that I believe offers a more parsimonious and less problematic explanation of cosmological fine-tuning.

In this paper is constructed an analogue of the square of opposition for propositions about relations between two non-empty sets. Unlike the classical square of opposition, the proposed scheme uses all logically possible syllogistic constants, formulated in V.I. Markin’s universal language for traditional positive syllogistic theories. This scheme can be called “Logical lantern”. The basic constants of this language are representing the five basic relations between two non-empty sets: equity, strict inclusion, reversed strict inclusion, intersection and exclusion (considered are only (...) the relations between sets, but not the relations of the sets to the universe). The other relations between sets are combinations (disjunctions) of the basic ones. There are 32 constants and 32 respective propositions, and the proposed diagram expresses and/or allows to deduce the logical relations among those propositions. The constructed scheme, making it possible to visually see the logical relations among propositions about relations between two non-empty sets, allows us to suggest considering logical relations not discussed previously: many-place Aristotelian-like logical relations among propositions: exhaustive _n_-place contrariety and exhaustive _n_-place subcontrariety (_n_ is a natural number, \(n>2\) ). (shrink)

A quantified universe is a set M equipped with a Riesz space equation image of real functions on Mn, for each n, and a second order operation equation image. Metric structures 4, graded probability structures 9 and many other structures in analysis are examples of such universes. We define ultraproduct of quantified universes and study properties preserved by this construction. We then discuss logics defined on the basis of classes of quantified universes which are closed (...) under this construction. (shrink)

This thesis develops a detailed account of the emergence of for all practical purposes continuous, quasi-classical world histories from the discontinuous, stochastic micro dynamics of Minimal Bohmian Mechanics (MBM). MBM is a non-relativistic quantum theory. It results from excising the guiding equation from standard Bohmian Mechanics (BM) and reinterpreting the quantum equilibrium hypothesis as a stochastic guidance law for the random actualization of configurations of Bohmian particles. On MBM, there are no continuous trajectories linking up individual configurations. Instead, individual (...) configurations are actualized independently of each other, carving out the decoherence-induced branching structure of the universal wave function. Yet, by contrast to the Everett interpretation, branches of the universal wave function are not actualized in parallel, i.e. all at the same time. Rather, world branches, on MBM, are actualized sequentially. -/- For an introduction to MBM, the transition from BM to MBM is described, and their empirical equivalence is established. I present the conditions under which MBM can be classified as a primitive ontology (PO) theory, regarded as crucial for the acceptance of a theory as Bohmian by many proponents of BM. While rendering MBM compatible with the PO approach, it must not fall prey to the problem of communication, arising from the two-space reading of BM. -/- The issue of temporal solipsism, identified by Bell as a serious problem for his “Everett (?) theory” – the historic predecessor of MBM – is discussed. I argue that Barbour’s time capsule approach does not provide a satisfactory solution. In particular, his adopting a more than minimal psychophysical parallelism between brain processes and experience of macroscopic change is argued to be reasonable in light of our current best neuroscience, yet problematic for theories relying solely on time capsules, understood as highly structured, individual, internally static configurations. As a solution, I introduce worlds, and world histories, as key concepts in providing a link between MBM’s micro dynamics and macroscopic phenomena. Worlds, other than time capsules, are coarse-grained regions in phase- or configuration space, defined as sets of possible micro states satisfying a relation of sufficient similarity from a macroscopic perspective, with respect to a given micro state. Hence, worlds may overlap. -/- I argue that worlds thus construed are a reasonable option for replacing the disjoint macro regions of phase space, resulting from the usual way of partitioning phase space in the standard framework of Boltzmannian statistical mechanics. This move solves the issue of discontinuous change of macro variables upon the micro state crossing the boundary between different macro regions. -/- I adapt this move for MBM’s discontinuous micro dynamics in configuration space. Issues revolving around the microscopic-macroscopic distinction, particle identity, impenetrability and haecceity in light of the desideratum of particle number conservation, etc., are discussed. I provide a detailed explanation of how overlapping worlds in MBM form world histories, thereby linking up macroscopically distinct worlds. Thus, the problem of temporal solipsism is resolved in a way that is compatible with a more than minimal psychophysical parallelism. (shrink)

This treatise presents thoughts on the divide that exists in chemistry between those who seek their understanding within a universe wherein the laws of physics apply and those who prefer alternative universes wherein the laws are suspended or ‘bent’ to suit preconceived ideas. The former approach is embodied in the quantum theory of atoms in molecules (QTAIM), a theory based upon the properties of a system’s observable distribution of charge. Science is experimental observation followed by appeal to theory (...) that, upon occasion, leads to new experiments. This is the path that led to the development of the molecular structure hypothesis—that a molecule is a collection atoms with characteristic properties linked by a network of bonds that impart a structure—a concept forged in the crucible of nineteenth century experimental chemistry. One hundred and fifty years of experimental chemistry underlie the realization that the properties of some total system are the sum of its atomic contributions. The concept of a functional group, consisting of a single atom or a linked set of atoms, with characteristic additive properties forms the cornerstone of chemical thinking of both molecules and crystals and Dalton’s atomic hypothesis has emerged as the operational theory of chemistry. We recognize the presence of a functional group in a given system and predict its effect upon the static, reactive and spectroscopic properties of the system in terms of the characteristic properties assigned to that group. QTAM gives physical substance to the concept of a functional group. (shrink)

This paper focuses on a constructive treatment of the mathematical formalism of quantum theory and a possible role of constructivist philosophy in resolving the foundational problems of quantum mechanics, particularly, the controversy over the meaning of the wave function of the universe. As it is demonstrated in the paper, unless the number of the universe’s degrees of freedom is fundamentally upper bounded or hypercomputation is physically realizable, the universal wave function is a non-constructive entity in the (...) sense of constructive recursive mathematics. This means that even if such a function might exist, basic mathematical operations on it would be undefinable and subsequently the only content one would be able to deduce from this function would be pure symbolical. (shrink)

This paper deals with one of the basic philosophical questions in modern cosmology: can the so-called Anthropic Principle , considered as an alternative to the classical teleology of creation, be an adequate explanation of the evidence that our universe is fine-tuned for the emergence of life and consciousness. The main problem with this principle is not its presumed teleology, as it is sometimes wrongly supposed, but quite the contrary: its intention to avoid teleological explanations by including the existence of (...) many universes ( multiverse ) into extended cosmological models. After having compared logical and cosmological many-worlds concepts, this paper reaches the conclusion that the ontological reality of the multiverse is an even more problematic presupposition than some properly revised version of teleological causality. This in itself does not imply the classical theistic explanation of creation, since it also yields a pantheistic explanation of the emergence of life and consciousness in our universe. (shrink)

Among the various lecture courses that Edmund Husserl held during his time as a Privatdozent at the University of Halle (1887-1901), there was one on "Ausgewählte Fragen aus der Philosophie der Mathematik" (Selected Questions from the Philosophy of Mathematics), which he gave twice, once in the WS 1889/90 and again in WS 1890/91. As Husserl reports in his letter to Carl Stumpf of February 1890, he lectured mainly on “spatial-logical questions” and gave an extensive critique of the Riemann-Helmholtz theories. Indeed, (...) in K I 28 many lectures on this subject can be found, which are for the greater part published in Husserliana XXI. The lecture contained in K I 28/4-12 at the Husserl-Archives Leuven, however, was left out of the selection, because the lecture contained “an analysis of the concept of number” whose content is “already known” from the Philosophie der Arithmetik. Indeed, since the lecture is from the WS 1889/90, the manuscript allows a glimpse of Husserl’s ideas halfway between his Habilitationsschrift (1887) and the Philosophie der Arithmetik (1891). Among other things, in this lecture we find the first documented use of the terms "Gestalt" and "Gestaltmoment" in Husserl. (shrink)

Does the claimed fine-tuning of the constants of nature for life give reason to think that there are many other universes in which the constants have different values? Or does the inference from fine-tuning to a multiverse commit what Hacking calls the inverse gambler’s fallacy? The present paper considers two fine-tuning problems that seem promising to consider because they are in many respects analogous to the problem of the fine-tuned constants. Reasoning that parallels the inference from (...) fine-tuning to a multiverse seems prima facie adequate in these problems. However, it turns out that in both cases there are independent empirical reasons to believe the hypotheses that are analogous to the multiverse hypothesis. In the absence of such evidence, it would be coherent to raise the inverse gambler’s fallacy charge against the inference from fine-tuning to these multiverse-type hypotheses. In response to this finding, I suggest taking the possibility seriously that established standards of rationality may not allow us to decide whether the inference from fine-tuning to a multiverse is fallacious or not. The paper concludes by sceptically assessing the prospects for obtaining independent empirical evidence for concrete multiverse theories. (shrink)

I develop the decision-theoretic approach to quantum probability, originally proposed by David Deutsch, into a mathematically rigorous proof of the Born rule in (Everett-interpreted) quantum mechanics. I sketch the argument informally, then prove it formally, and lastly consider a number of proposed ``counter-examples'' to show exactly which premises of the argument they violate. (This is a preliminary version of a chapter to appear --- under the title ``How to prove the Born Rule'' --- in Saunders, Barrett, Kent and Wallace, (...) "Many worlds? Everett, quantum theory and reality", forthcoming from Oxford University Press.). (shrink)

Many professionals, mainly scientists have told me that a divine being is not necessary to explain why there is something rather than nothing. Theology and even metaphysics are redundant disciplines, they say. To me to reject a discipline a priori is irrational, whereas the methodology of science itself is basically rational. Why do Newton's equations say time is symmetrical, Einstein's relativity and quantum mechanics say time is an illusion and now Smolin says time is reborn - the present moment (...) is a fundamental aspect of reality. Why has matter been the carrier of reality over centuries, then matter dissolves away becoming an ephemeral quantum activity and now information is the carrier of reality? The laws of physics were presumed to be eternal, now Smolin says they are evolving like in biology - natural selection. Why do cosmologists like Brian Greene and Leonard Susskind postulate that there are nearly infinite number of replicas of each and every one of us - doppelgangers - in an infinite number of universes when the gut feeling of each and every one of us that we are unique in this universe? For Newton, gravity was an instantaneous attraction between two masses, for Einstein it was an inherent potential even with a single mass. For Einstein, gravity acts at the speed of light. Plato says that what is observed is phenomena. Reality resides in Platonic Forms. Descartes says that knowledge is acquired through perception and deduction. No reality without the mind. Kant synthesized rationalism and empiricism. Knowledge can be derived without empirical observation. Free will touches almost everything that human beings value. But many scientists postulate that free will is an illusion. Christian theology posits that God created the universe out of nothing. Cosmologists also now say the same thing, but the definition of 'nothing' is disputed. Christian theology once postulated that there is no salvation outside the Church. Now it is not so. Why these confusions? Why these contradictions? On the present methodology of acquisition of knowledge, answers may never be possible. The Theory of Everything (TOE) is more a distant dream than reaching for the stars. However, the discipline of Zygology which is introduced into the academia, a rational explanation why this is not possible has been postulated. Knowledge is holistic - a seamless web runs through all disciplines. Necessity of a new epistemology of reality and existence is inevitable....Zygology may have the answers!". (shrink)

In this paper we ask the question: to what extent do basic set theoretic properties of Loeb measure depend on the nonstandard universe and on properties of the model of set theory in which it lies? We show that, assuming Martin's axiom and κ-saturation, the smallest cover by Loeb measure zero sets must have cardinality less than κ. In contrast to this we show that the additivity of Loeb measure cannot be greater than ω 1 . Define $\operatorname{cof}(H)$ as (...) the smallest cardinality of a family of Loeb measure zero sets which cover every other Loeb measure zero set. We show that $\operatorname{card}(\lfloor\log_2(H)\rfloor) \leq \operatorname{cof}(H) \leq \operatorname{card}(2^H)$ , where card is the external cardinality. We answer a question of Paris and Mills concerning cuts in nonstandard models of number theory. We also present a pair of nonstandard universes $M \preccurlyeq N$ and hyperfinite integer H ∈ M such that H is not enlarged by N, 2 H contains new elements, but every new subset of H has Loeb measure zero. We show that it is consistent that there exists a Sierpiński set in the reals but no Loeb-Sierpiński set in any nonstandard universe. We also show that it is consistent with the failure of the continuum hypothesis that Loeb-Sierpiński sets can exist in some nonstandard universes and even in an ultrapower of a standard universe. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:RUSSELLS ARGUMENT AGAINST FREGE'S SENSE-REFERENCE DISTINCTION PAWEL TURNAu Philosophy I Jagiellonian University Cracow, Poland I n "On Denoting"l Russell argued that Frege's theory of sense and reference was an "inextricable tangle", but, ironically, many readers found the argument even more knotry. In an effort to make sense of it, commentators were often driven to attribute to Russell quite obvious and simple fallacies. A different approach was taken by (...) Peter Geach, who suggested that Russell's argument could be given a consistent reading if it were construed as arguing, not against Frege's theory, but against Russell's own earlier theory, which he put forward in The Principles of Mathematics. It is not an implausible hypothesis, considering how prone Russell was to misrepresent' other philosophers' views and read his own powerful ideas into their writings. In order to justifY Geach's hypothesis properly, one would have to show the crucial difference between Russell's and Frege's theories that made only one of them susceptible to the attack in "On Denoting ". Russell explicitly considered the two theories to be "very nearly the same"; this self-interpretation would have to be questioned. Geach's hypothesis would be further strengthened by providing an actual exegesis of the "On Denoting" argument as directed against The Principles ofMathematics theory. In this paper 1 will try to accomplish both of these tasks. Russell is often accused of confusing use and mention. The accusation is not fair, however. It is true that he does not insist on this distinction; he does not employ it extensively. But he is certainly Russell's Argument against Frege's Distinction 53 aware of it; he simply does not think that it is an important logical distinction. Today this view may seem almost incomprehensible, but that only shows the strength of Frege's influence on our ideas. There are two striking symptoms of Russell's lack of concern for the use-mention distinction: firstly, he does not observe the convention which says that putting an expression in quotation marks produces a name of that expression; secondly, he does not feel any need to employ the pair of terms "sentence" and "proposition", but contents himself with the latter only. The claim about Russell's use of inverted commas needs to be proved; I will come to that later. Russell's assertions are often ambiguous between use and mention of expressions. For example, he asserts that "propositions do not contain words" (PaM, p. 47). However, a few pages later, he speaks of propositions that contain the phrase "any number" (p. 53). If this is not to be self-contradictory, then either the term "proposition" must sometimes signifY a linguistic expression, or the term "phrase" must sometimes be understood without implying that a phrase consists of words. This ambiguity is openly admitted in a later article, where he writes: [we shall be concerned with] the distinction between verbs and substantives, or, more correctly, between the objects denoted by verbs and the objects denoted by substantives. Since this more correct expression is long and cumbrous, I shall generally use the shorter expression to mean the same thing. Thus, when I speak of verbs, I mean the object denoted by verbs, and similarly for substantives.2. Russell's attitude towards the distinction between use and mention is not unjustified. It has its reason in his view oflanguage and the subject -matter of logic. In Russell's view, the subject-matter of logic has nothing to do with language. Words, both spoken and written, are entirely outside its realm. Logic, just as other sciences (except linguistics ), is concerned with the entities indicated by words, and not with words themselves. Even the relation of indicating, which holds between words and things in the world, is of no interest to it: "mean1 In Lf(, pp. 39-56. russ.lI: the Journal of the Bertrand Russell Archives McMaster Univecsity Library Press n.s. 11 (summer 1991): 51- 66 ISSN 0036-o16~ 2 B. Russell, "On the Relations of Universals and Particulars", Proceedings of the Aristotelian Society, n.s. 12 (r9II-12): 1-24; in Lf(, pp. 103-24 (at 107-8). 54 PAWEL TURNAU Russell's Argument... (shrink)

We prove a number of results motivated by global questions of uniformity in computabi- lity theory, and universality of countable Borel equivalence relations. Our main technical tool is a game for constructing functions on free products of countable groups. We begin by investigating the notion of uniform universality, first proposed by Montalbán, Reimann and Slaman. This notion is a strengthened form of a countable Borel equivalence relation being universal, which we conjecture is equivalent to the usual notion. With this (...) additional uniformity hypothesis, we can answer many questions concerning how countable groups, probability measures, the subset relation, and increasing unions interact with universality. For many natural classes of countable Borel equivalence relations, we can also classify exactly which are uniformly universal. We also show the existence of refinements of Martin’s ultrafilter on Turing invariant Borel sets to the invariant Borel sets of equivalence relations that are much finer than Turing equivalence. For example, we construct such an ultrafilter for the orbit equivalence relation of the shift action of the free group on countably many generators. These ultrafilters imply a number of structural properties for these equivalence relations. (shrink)

Competent speakers of natural languages can borrow reference from one another. You can arrange for your utterances of ‘Kirksville’ to refer to the same thing as my utterances of ‘Kirksville’. We can then talk about the same thing when we discuss Kirksville. In cases like this, you borrow “ aboutness ” from me by borrowing reference. Now suppose I wish to initiate a line of reasoning applicable to any prime number. I might signal my intention by saying, “Let p (...) be any prime.” In this context, I will be using the term ‘p’ to reason about the primes. Although ‘p’ helps me secure the aboutness of my discourse, it may seem wrong to say that ‘p’ refers to anything. Be that as it may, this paper explores what mathematical discourse would be like if mathematicians were able to borrow freely from one another not just the reference of terms that clearly refer, but, more generally, the sort of aboutness present in a line of reasoning leading up to a universal generalization. The paper also gives reasons for believing that aboutness of this sort really is freely transferable. A key implication will be that the concept “set of natural numbers” suffers from no mathematically significant indeterminacy that can be coherently discussed. (shrink)

It has been widely discussed whether the existence of other universes than the known universe is a purely metaphysical consideration. This paper argues to the contrary that the existence of other universes is a meaningful hypothesis, which can be tested based on observable data. The paper applies the perspective of observer bias as an approach for assigning probabilities to the mutually excluding hypotheses of universe vs. multiverse, i.e., whether the known universe is the (...) only universe, or just one universe in an ensemble of universes, a so-called multiverse. The basic idea of the paper relies on the following consideration. In a multiverse, the vast majority of observers would live in universes that are more life friendly. Thus, conditional on there being a multiverse, we should expect to find ourselves in a universe with values of the fundamental parameters that provide particularly fertile grounds for life. In contrast, if there is only a single universe, it may well be the case that there is still a few observers even if the parameters are far from ideal for life, and in that case, those observers will find values of the parameters that are not ideal for intelligent life. It may well be the case that, among all parameter configurations that allow life at least somewhere, those that are not ideal for life far outnumber those that are ideal for life. Based on this elementary consideration, the paper proceeds to propose a quantitative framework to determine probabilities for either hypothesis. In particular, it is described how a future ab initio determination of some of the factors in the Drake equation may be used to infer which one of the two hypotheses is the more likely. A quantitative approach to this end is proposed. Expressing these factors of the Drake equation in the context of the two hypotheses, a general likelihood approach is first described. Then, to support intuition, example computations are provided, illustrating how an actual hypothesis test would work in practice. (shrink)

Complexity and life as we know it depend crucially on the laws and constants of nature as well as the boundary conditions, which seem at least partly “fine-tuned.” That deserves an explanation: Why are they the way they are? This essay discusses and systematizes the main options for answering these foundational questions. Fine-tuning might just be an illusion, or a result of irreducible chance, or nonexistent because nature could not have been otherwise (which might be shown within a fundamental theory (...) if some constants or laws could be reduced to boundary conditions or boundary conditions to laws), or it might be a product of selection: either observational selection (weak anthropic principle) within a vast multiverse of many different realizations of physical parameters, or a kind of cosmological natural selection making the measured parameter values quite likely within a multiverse of many different values, or even a teleological or intentional selection or a coevolutionary development, depending on a more or less goal-directed participatory contribution of life and intelligence. In contrast to observational selection, which is not predictive, an observer-independent selection mechanism must generate unequal reproduction rates of universes, a peaked probability distribution, or another kind of differential frequency, resulting in a stronger explanatory power. The hypothesis of Cosmological Artificial Selection (CAS) even suggests that our universe may be a vast computer simulation or could have been created and transcended by one. If so, this would be a far-reaching answer – within a naturalistic framework! – of fundamental questions such as: Why did the big bang and fine-tunings occur, what is the role of intelligence in the universe, and how can it escape cosmic doomsday? This essay critically discusses some of the premises and implications of CAS and related problems, both with the proposal itself and its possible physical realization: Does CAS deserve to be considered as a convincing explanation of cosmic fine-tuning? Is life incidental, or does CAS revalue it? And are life and intelligence ultimately doomed, or might CAS rescue them? (shrink)

I briefly defend the following claims in response to my critics: (1) We cannot make a principled division between features of contingent reality that do and features that don’t "cry our for explanation." (2) The physical data indicating fine-tuning provide confirmation of the hypothesis of a personal necessary cause of the universe over against an impersonal necessary cause, notwithstanding the fact that the probability of either hypothesis, if true, would be 1. (3) Theism that commits to God’s (...) necessary existence makes more sense than theism that denies it. (4) God is likely to have created an infinity of universes, and this conclusion helps with (though does not solve) the many problems of evil. (shrink)

The multiverse view in set theory, introduced and argued for in this article, is the view that there are many distinct concepts of set, each instantiated in a corresponding set-theoretic universe. The universe view, in contrast, asserts that there is an absolute background set concept, with a corresponding absolute set-theoretic universe in which every set-theoretic question has a definite answer. The multiverse position, I argue, explains our experience with the enormous range of set-theoretic possibilities, (...) a phenomenon that challenges the universe view. In particular, I argue that the continuum hypothesis is settled on the multiverse view by our extensive knowledge about how it behaves in the multiverse, and as a result it can no longer be settled in the manner formerly hoped for. (shrink)

If the physical constants, initial conditions, or laws of nature in our universe had been even slightly different, then the evolution of life would have been impossible. This observation has led many philosophers and scientists to ask the natural next question: why is our universe so "fine-tuned" for life? The debates around this question are wide-ranging, multi-disciplinary, complicated, technical, and heated. This study is a comprehensive investigation of these debates and the many metaphysical and epistemological questions (...) raised by cosmological fine-tuning. Waller's study reaches two significant and controversial conclusions. First, he concludes that the criticisms directed at the "multiverse hypothesis" by theists and at the "theistic hypothesis" by naturalists are largely unsuccessful. Neither of these options can plausibly be excluded. Choosing between them seems to turn on primitive metaphysical intuitions. Second, in order to break the philosophical deadlock, Waller moves the debate from the level of universes to the level of possible worlds. Arguing that possible worlds are also "fine-tuned" in an important and interesting sense, Waller concludes that the only plausible explanation for the fine-tuning of the actual world is to posit the existence of some kind of "God-like-thing.". (shrink)

Peter Byrne tells the story of Hugh Everett III (1930-1982), whose "many worlds" theory of multiple universes has had a profound impact on physics and philosophy. Using Everett's unpublished papers (recently discovered in his son's basement) and dozens of interviews with his friends, colleagues, and surviving family members, Byrne paints, for the general reader, a detailed portrait of the genius who invented an astonishing way of describing our complex universe from the inside. Everett's mathematical model (called (...) the "universal wave function") treats all possible events as "equally real", and concludes that countless copies of every person and thing exist in all possible configurations spread over an infinity of universes: many worlds. Afflicted by depression and addictions, Everett strove to bring rational order to the professional realms in which he played historically significant roles. In addition to his famous interpretation of quantum mechanics, Everett wrote a classic paper in game theory; created computer algorithms that revolutionized military operations research; and performed pioneering work in artificial intelligence for top secret government projects. He wrote the original software for targeting cities in a nuclear hot war; and he was one of the first scientists to recognize the danger of nuclear winter. As a Cold Warrior, he designed logical systems that modeled "rational" human and machine behaviors, and yet he was largely oblivious to the emotional damage his irrational personal behavior inflicted upon his family, lovers, and business partners. He died young, but left behind a fascinating record of his life, including correspondence with such philosophically inclined physicists as Niels Bohr, Norbert Wiener, and John Wheeler. These remarkable letters illuminate the long and often bitter struggle to explain the paradox of measurement at the heart of quantum physics. In recent years, Everett's solution to this mysterious problem - the existence of a universe of universes - has gained considerable traction in scientific circles, not as science fiction, but as an explanation of physical reality. (shrink)

The idea that incompatibilism is intuitive is one of the key motivators for incompatibilism. Not surprisingly, then philosophers who defend incompatibilism often claim that incompatibilism is the natural, commonsense view about free will and moral responsibility (e.g., Pereboom 2001, Kane Journal of Philosophy 96:217–240 1999, Strawson 1986). And a number of recent studies find that people give apparently incompatibilist responses in vignette studies. When participants are presented with a description of a causal deterministic universe, they tend to deny (...) that people are morally responsible in that universe. Although this suggests that people are intuitive incompatibilists, Eddy Nahmias and Dylan Murray, in a recent series of important papers, have developed an important challenge to this interpretation. They argue that people confuse determinism with bypassing, the idea that one’s mental states lack causal efficacy. Murray and Nahmias present new experiments that seem to confirm the bypassing hypothesis. In this paper, we use structural equation modeling to re-examine the issue. We find support instead for an incompatibilist explanation of the bypassing results, i.e., incompatibilist judgments seem to cause bypassing judgments. We hypothesize that this phenomenon occurs because people think of decisions as essentially indeterministic; thus, when confronted with a description of determinism they tend to think that decisions do not even occur. We provide evidence for this in three subsequent studies which show that many participants deny that people make decisions in a deterministic universe; by contrast, most participants tend to allow that people add numbers in a deterministic universe. Together, these studies suggest that bypassing results don’t reflect a confusion, but rather the depth of the incompatibilist intuition. (shrink)

A multiverse is comprised of many universes, which quickly leads to the question: How many universes? There are either finitely many or infinitely many universes. The purpose of this paper is to discuss two conceptions of infinite number and their relationship to multiverses. The first conception is the standard Cantorian view. But recent work has suggested a second conception of infinite number, on which infinite numbers behave very much like finite (...) numbers. I will argue that that this second conception of infinite number is the correct one, and analyze what this means for multiverses. (shrink)

Among the various lecture courses that Edmund Husserl held during his time as a Privatdozent at the University of Halle (1887-1901), there was one on Ausgewählte Fragen aus der Philosophie der Mathematik (Selected Questions from the Philosophy of Mathematics), which he gave twice, once in the WS 1889/90 and again in WS 1890/91. As Husserl reports in his letter to Carl Stumpf of February 1890, he lectured mainly on “spatial-logical questions” and gave an extensive critique of the Riemann-Helmholtz theories. Indeed, (...) in K I 28 many lectures on this subject can be found, which are for the greater part published in Husserliana XXI. The lecture contained in K I 28/4-12 at the Husserl-Archives Leuven, however, was left out of the selection, because the lecture contained “an analysis of the concept of number” whose content is “already known” from the Philosophie der Arithmetik. Indeed, since the lecture is from the WS 1889/90, the manuscript allows a glimpse of Husserl’s ideas halfway between his Habilitationsschrift (1887) and the Philosophie der Arithmetik (1891). From an exam- ple Husserl gives in the lecture: “Der wievielte Januar ist heute?” (The how-many-th of January is it today?), it can be estimated to be from January 1890, placing it in close relation to the letter to Stumpf and the views and doubts expressed therein. The bundle of papers K I 28 is wrapped in a blue cover, bearing only “Vor- lesungen” (Lectures) as a title. The lecture “On the concept of Number” in K I 28/4-12 is contained again in a separate cover, without any title. As in the case of most of his lectures, Husserl folded the paper in half, using the right half as a margin for annotations. Before the lecture there is a single page (K I 28/3) with some annotations on the treatment of imaginary numbers as assumptions in calculation. This text is published here as an Appendix to the lecture, because, while it is not part of the lecture itself, it is closely related to the subjects discussed therein. (shrink)

From the end of the 19th century until his death, one of history's most brilliant mathematicians languished in an asylum. The Mystery of the Aleph tells the story of Georg Cantor (1845-1918), a Russian-born German who created set theory, the concept of infinite numbers, and the "continuum hypothesis," which challenged the very foundations of mathematics. His ideas brought expected denunciation from established corners - he was called a "corruptor of youth" not only for his work in mathematics, but for (...) his larger attempts to meld spirituality and science. (shrink)

The paper undertakes three interdisciplinary tasks. The first one consists in constructing a formal model of the basic arithmetic competence, that is, the competence sufficient for solving simple arithmetic story-tasks which do not require any mathematical mastery knowledge about laws, definitions and theorems. The second task is to present a generalized arithmetic theory, called the arithmetic of indexed numbers (INA). All models of the development of counting abilities presuppose the common assumption that our simple, folk arithmetic encoded linguistically in (...) the mind is based on the linear number representation. This classical conception is rejected and a competitive hypothesis is formulated according to which the basic mature representational system of cognitive arithmetic is a structure composed of many numerical axes which possess a common constituent, namely, the numeral zero. Arithmetic of indexed numbers is just a formal tool for modelling the basic mature arithmetic competence. The third task is to develop a standpoint called temporal pluralism, which is motivated by neo-Kantian philosophy of arithmetic. (shrink)

It has long been the habit of philosophers, and is still a common failing of ordinary playgoers, to see tragedy through the coloured spectacles of an acquired philosophical or religious outlook, and to commend or condemn rather from the standpoint of partiality for a certain view about life in general than from that of one assessing the intrinsic merits of a work of art. Because we all, whether laymen or specialists, theorize about the nature and destiny of that mysterious (...) class='Hi'>universe which, for a brief span, our minds illumine, and because the working hypothesis so engendered is inevitably tinged by our personal idiosyncrasies of temperament and taste, there have been—and, building on such a foundation, there must be—almost as many different theoretical explanations of the tragic as theoreticians speculating on this high theme. Besides introducing that personal equation which has always been anathema to science, the distorted vision for which philosophical preconceptions are responsible brings with it minor illusions that are scarcely less injurious; the reading into a play or a novel of notions that had never entered the poet's brain, or the interpolation of history with incidents that never really occurred, the discarding of much that is genuinely tragic because it happens not to fit into the mould shaped by our own ideas, and the acceptance of much that is not merely untragic, but actually inartistic—pacifist, or communist, or feminist, or religious propaganda—because it happens to display the native hue of our own convictions. (shrink)