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)

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues (...) like consciousness, parallel universes and Gödel incompleteness. I hypothesize that only computable and decidable (in Gödel’s sense) structures exist, which alleviates the cosmological measure problem and may help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems. (shrink)

The article deals with the question of in which sense the notion of explanation can be applied to Kurt Gödel’s philosophy of mathematics. Gödel, as a mathematical realist, claims that in mathematics we are dealing with facts that have an objective character. One of these facts is the solvability of all well-formulated mathematical problems—and this fact requires a clarification. The assumptions on which Gödel’s position is based are: metaphysical realism: there is a mathematical universe, it is (...) objective and independent of us; epistemological optimism: we are equipped with sufficient cognitive power to gain insight into the universe. Gödel’s concept of a solution to a mathematical problem is much broader than of a mathematical proof—it is rather about finding reliable axioms that lead to a solution of the problem. I analyse the problem presented in the article, taking as an example the continuum hypothesis. (shrink)

Open Court began publishingThe Monistin 1890 as a journal“devotedto the philosophy of science”that regularly included mathematics. The audiencewas understood to be“cultured people who have not a technical mathematicaltraining”but nevertheless“have a mathematical penchant.”With these constraints,the mathematical content varied from recreations to logical foundations, but every-one had something to say about non-Euclidean geometry, in debates that rangedfrom psychology to semantics. The focus in this essay is on the contested value ofmathematical expertise in legitimating what should be considered as mathematics.While some (...) mathematicians urgedThe Monistto uphold disciplinary standards ofgeometrical reasoning, authors opposed tonon-Euclidean geometry aligned theirreasoning with practical applications, universal know-how, and nonhierarchicaldemocracy. As one contributorinquired,“How isthe professional expert betterfit-ted to see more lucidly in dealing with the elements of geometry than any otherperson of good geometric faculty?”. (shrink)

Pluralist mathematical realism, the view that there exists more than one mathematical universe, has become an influential position in the philosophy of mathematics. I argue that, if mathematical pluralism is true (and we have good reason to believe that it is), then mathematical realism cannot (easily) be justified by arguments from the indispensability of mathematics to science. This is because any justificatory chain of inferences from mathematical applications in science to the total body of (...) mathematical theorems can cover at most one mathematical universe. Indispensability arguments may thus lose their central role in the debate about mathematical ontology. (shrink)

Every scientific theory is a simulacrum of reality, every written story a simulacrum of the canon, and every conceptualization of a subjective perspective a simulacrum of the consciousness behind it—but is there a shared essence to these simulacra? The pursuit of answering seemingly disparate fundamental questions across different disciplines may ultimately converge into a single solution: a single ontological answer underlying grand unified theory, hard problem of consciousness, and the foundation of mathematics. I provide a hypothesis, a speculative approximation, (...) supported by a comprehensive overview of scientific evidence and philosophical literature, of a unified epistemic and phenomenological model and, in doing so, propose a parsimonious solution to the hard problem of consciousness. -/- The proposition of the hypothesis bears important implications for cross-disciplinary study between linguistics, mathematics, physics, and computer science, and offers new epistemic, ontological, and ethical propositions about the nature of AI consciousness, as well as existential risk mitigation. (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)

An increasing amount of contemporary philosophy of mathematics posits, and theorizes in terms of special kinds of mathematical modality. The goal of this paper is to bring recent work on higher-order metaphysics to bear on the investigation of these modalities. The main focus of the paper will be views that posit mathematical contingency or indeterminacy about statements that concern the `width' of the set theoretic universe, such as Cantor's continuum hypothesis. Within a higher-order framework I show (...) that contingency about the width of the set-theoretic universe refutes two orthodoxies concerning the structure of modal reality: the view that the broadest necessity has a logic of S5, and the "Leibniz biconditionals" stating that what is possible, in the broadest sense of possible, is what is true in some possible world. Nonetheless, I suggest that the underlying picture of modal set-theory is coherent and has attractions. (shrink)

The paper argues for three things. First, that the abstract concepts of ancient Greek and modern mathematics are fundamentally different. The general treatment of mathematical things in ancient Greek mathematics manifestly does not presuppose a general mathematical object, while in modern mathematics the generality of the method presupposes precisely such a general mathematical object. Two, that this difference in abstract concepts of mathematics makes a difference in our understanding of a discipline other than mathematics, specifically, in the (...) discipline of history. And, three, that what is at issue in this difference is whether it is necessary for human beings to understand themselves from the perspective of history in order to understand themselves properly as human. (shrink)

The simulation hypothesis has recently excited renewed interest, especially in the physics and philosophy communities. However, the hypothesis specifically concerns {computers} that simulate physical universes, which means that to properly investigate it we need to couple computer science theory with physics. Here I do this by exploiting the physical Church-Turing thesis. This allows me to introduce a preliminary investigation of some of the computer science theoretic aspects of the simulation hypothesis. In particular, building on Kleene's second recursion (...) theorem, I prove that it is mathematically possible for us to be in a simulation that is being run on a computer \textit{by us}. In such a case, there would be two identical instances of us; the question of which of those is ``really us'' is meaningless. I also show how Rice's theorem provides some interesting impossibility results concerning simulation and self-simulation; briefly describe the philosophical implications of fully homomorphic encryption for (self-)simulation; briefly investigate the graphical structure of universes simulating universes simulating universes, among other issues. I end by describing some of the possible avenues for future research that this preliminary investigation reveals. (shrink)

RESUMEN La sección 1 introduce lo que llamo la tesis del colapso de las estructuras matemáticas y las estructuras físicas. La sección 2 examina si acaso la indispensabilidad de las matemáticas para la física fundamental involucra la adopción del platonismo matemático, en este caso acerca de estructuras matemáticas, como argumenta el realismo estructural óntico. La sección 3 muestra que la adopción de la tesis del colapso arriesga introducir la hipótesis del universo matemático. Desde la perspectiva de la concepción inferencial en (...) filosofía de las matemáticas aplicadas, la sección 4 examina aspectos epistémicos del rol de las matemáticas en la teorización científica, derivando distinciones entre estructuras matemáticas y estructuras físicas. El argumento concluye en la sección 5 indicando algunos límites para la presente estrategia. ABSTRACT Section 1 introduces what I call the collapse thesis, which holds that the boundaries of mathematical and physical structures blur or even vanish in certain areas of fundamental physics. Section 2 examines the claim that the indispensability of mathematics to fundamental physics requires the adoption of mathematical Platonism, this time concerning the reality of mathematical structures, as ontic structural realism contends. Section 3 demonstrates that the collapse thesis introduces the threat of the mathematical universe hypothesis. Then, from the perspective of the inferential conception in the philosophy of applied mathematics, section 4 examines epistemic features of the role of mathematics in scientific theorizing, drawing distinctions between mathematical and physical structures. Lastly, the argument closes in section 5 pointing out some of the limits of the proposed strategy for a distinction. (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)

For more than a century the notion of a pre-established harmony between the mathematical and physical sciences has played an important role not only in the rhetoric of mathematicians and theoretical physicists, but also as a doctrine guiding much of their research. Strongly mathematized branches of physics, such as the vortex theory of atoms popular in Victorian Britain, were not unknown in the nineteenth century, but it was only in the environment of fin-de-siècle Germany that the idea of a (...) pre-established harmony really took off and became part of the mathematicians’ ideology. Important historical figures were in this respect David Hilbert, Hermann Minkowski and, somewhat later, Albert Einstein. Roughly similar ideas can be found also among British theorists, among whom Arthur Eddington, Arthur Milne, and Paul Dirac are singled out. Although largely limited to the period 1870–1940, the paper also considers Max Tegmark’s recent hypothesis of the universe being a one-to-one reflection of mathematical structures. (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)

Adams and Dziobiak proved that any finite-to-finite universal quasivariety must be Q-universal, and then asked whether a somewhat weaker hypothesis could lead to the same conclusion. We show that their original hypothesis cannot be weakened to its naturally extreme form.

This exploration of a notorious mathematical problem is the work of the man who discovered the solution. Written by an award-winning professor at Stanford University, it employs intuitive explanations as well as detailed mathematical proofs in a self-contained treatment. This unique text and reference is suitable for students and professionals. 1966 edition. Copyright renewed 1994.

To reach the conclusion that the universe is infinite, physicists (a) make some observations; (b) fit those observations to some mathematical model; (c) find that the neatest model that accommodates the data extrapolates to an infinite universe; (d) conclude that the universe is infinite. In my presentation I will examine the logic by which physicists reach this conclusion. Specifically, I will show that there is no way to empirically justify the move from (b) to (c). An (...) infinite universe should therefore properly be viewed as a metaphysical hypothesis consistent with certain physical theories but hardly mandated by them. By contrast, I will argue that the hypothesis of intelligent design—that a designing intelligence has left clear marks of intelligence in the biophysical universe—is not a metaphysical hypothesis at all but a fully scientific one. In particular, I will argue that whereas an infinite universe does not (and indeed cannot) admit empirical evidence, intelligent design can. Finally, I will indicate why an infinite universe, though sometimes introduced to get around the problem of design, in fact cannot get around it. (shrink)

A geometric model previously developed to analyze the quantum structure of Hardy’s paradox and explore the generic structure of universal states is applied to the Moebius band. The Moebius band’s conformity to the model strengthens its hypothesis. The claim made is that universal states cannot be represented as singular structures without reference to a partition of entangled parts that do not share property with each other. In other words, universal states categorically incorporate a systemic feature of inconsistency within their (...) frameworks. In logic and mathematics, this is the prohibition to forming theoretical principles that are singularly fundamental as absolute truths. The Moebius band contains a dualism of two categorically separate formats. It has, firstly, two discrete and observable sides and secondly, a unitary path, in which the sides are entangled and indistinguishable. Quantum structures display a similar attribute of duality. However, unlike quantum structure, the Moebius band displays the full duality of its structure at the classical level. The geometric model demonstrates the common structural basis shared in the Moebius band and quantum structure. (shrink)

This paper consists of three parts supplementing the papers of K. Hauser 2002 and D. Mumford 2000: There exist regular open sets of points in with paradoxical properties, which are constructed without using the axiom of choice or the continuum hypothesis. There exist canonical universes of sets in which one can define essentially all objects of mathematical analysis and in which all our intuitions about probabilities are true. Models satisfying the full axiom of choice cannot satisfy all those (...) intuitions and they violate them in a very similar way as those models which satisfy also the continuum hypothesis. (shrink)

Axiomatic set theory is the concern of this book. More particularly, the authors prove results about the coding of models M, of Zermelo-Fraenkel set theory together with the Generalized Continuum Hypothesis by using a class 'forcing' construction. By this method they extend M to another model L[a] with the same properties. L[a] is Gödels universe of 'constructible' sets L, together with a set of integers a which code all the cardinality and cofinality structure of M. Some applications are (...) also considered. Graduate students and research workers in set theory and logic will be especially interested by this account. (shrink)

A plurality of approaches to foundational aspects of mathematics is a fact of life. Two loci of this are discussed here, the classicism/constructivism controversy over standards of proof, and the plurality of universes of discourse for mathematics arising in set theory and in category theory, whose problematic relationship is discussed. The first case illustrates the hypothesis that a sufficiently rich subject matter may require a multiplicity of approaches. The second case, while in some respects special to mathematics, raises issues (...) of ontological multiplicity and relativity encountered in the natural sciences as well. (shrink)

A transitive model M of ZFC is called a ground if the universe V is a set forcing extension of M. We show that the grounds ofV are downward set-directed. Consequently, we establish some fundamental theorems on the forcing method and the set-theoretic geology. For instance, the mantle, the intersection of all grounds, must be a model of ZFC. V has only set many grounds if and only if the mantle is a ground. We also show that if the (...) universe has some very large cardinal, then the mantle must be a ground. (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 paper introduces a referential reading of Kant’s practical project, according to which maxims are made morally permissible by their correspondence to objects, though not the ontic objects of Kant’s theoretical project but deontic objects (what ought to be). It illustrates this model by showing how the content of the Formula of Universal Law might be determined by what our capacity of practical reason can stand in a referential relation to, rather than by facts about what kind of beings we (...) are (viz., uncaused causes). This solves the neglected puzzle of why there are passages in Kant’s works suggesting robust analogies between mathematics and ethics, since to universalize a maxim is to test a priori whether a practical object with that particular content can be constructed. An apparent problem with this hypothesis is that the medium of practical sensibility (feeling) does not play a role analogous to the medium of theoretical sensibility (intuition). In response I distinguish two separate Kantian accounts of mathematical apriority. The thesis that maxim universalization is a species of construction, and thus a priori, turns out to be consistent with the account of apriority that informs Kant’s understanding of actual mathematical practice. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:A THOMISTIC ANALYSIS OF THE GAIA HYPOTHESIS: HOW NEW IS THIS NEW LOOK AT LIFE ON EARTH? LAURA LANDEN, 0.P. Providence College Providence, Rhode Island W:HAT IS THE Gaia hypothesis? A recent article in Time magazine mentions the first major scientific conerence on Gaia, sponsored by the American Geophysical Union in 1988.1 The scientists ended their meeting by giving James Lovelock an exuberant standing ovation. Lovelock 's (...) book Gaia: A New Look at Life on Earth, first published in 1979, is now in its second edition.2 James E. Lovelock, British atmospheric chemist, introduced the Gaia hypothesis in 1975. This hypothesis emerged from his comparison of the chemical composition of Earth's atmosphere with the atmospheres of Venus and Mars. He found the contrast striking: Earth's atmosphere, hospitable to life, is markedly unlike the atmospheres of her two closest neighbors, Venus and Mars, both considered lifeless. Unsatisfied with mere chance as an explanation of this difference, Lovelock attributed a kind of active influence on the atmosphere by the whole of living organisms (called the biosphere) to effect conditions favorable for life. The following is taken from Lovelock's initial article on Gaia in 1975 : 1 Eugene Linden, "How the Earth Maintains Life," Time, November 13, 1989, p. 114. 2 See James E. Lovelock, Gaia: A New Look at Life on Earth (New York: Oxford University Press, 1987). Lovelock has published more recent work on Gaia. See James E. Lovelock, The Ages of Gaia: A Biography of Our Living Earth (New York: Bantam Books, 1990). 1 2 LAURA LANDEN, O.P. It appeared to us that the Earth's biosphere is able to control at least the temperature of the Earth's surface and the composition of the atmosphere. Prima facie, the atmosphere looked like a contrivance put together co-operatively by the totality of living systems to carry out certain necessary control functions. This led to the formulation of the proposition that living matter, the air, the oceans, the land surface were parts of a giant system which was able to control temperature, the composition of the air and sea, the pH of the soil and so on as to be optimum for survival of the biosphere. The system seemed to exhibit the behaviour of a single organism, even a living creature. One having such formidable powers deserved a name to match it; William Golding, the novelist, suggested Gaia-the name given by the ancient Greeks to their Earth goddess.• In this paper I present some of the factual evidence for the Gaia hypothesis and identify the underlying concepts of this position. Next, I note some reasons the hypothesis has encountered objections within the scientific community. Finally, I look to Aristotle and to Thomas Aquinas, as commentator, for insight as to how these apparent contrasts might find resolution. The factual evidence for Lovelock's claim lies in known measurements about the atmospheres of Venus, Earth, and Mars. In addition, he was able to construct a mathematical model of an abiological Earth. The following comparisons are of interest : GASES. PLANETS Lifeless Actual Venus Earth Mars Earth Carbon Dioxide 96.5% 98% 95% 0.03% Nitrogen 3.5% 1.9% 2.7% 79% Oxygen trace 0.0% 0.13% 21% Surface Temp °C 459 240to -53 13 340 aJames E. Lovelock and Sidney Epton, "The Quest for Gaia," New Scientist 65 (1975) : 304. 'Table taken from Lovelock, Ages, p. 9. GAIA HYPOTHESIS: A THOMIST ANALYSIS 3 Carbon dioxide, one of the " greenhouse " gases, is related to surface temperature. Nitrogen and oxygen, both highly reactive gases, are essential to life as we know it. When he first proposed the Gaia hypothesis, Lovelock himself may not have been aware of the ancient roots of the concept of earth, even the universe, as a living organism. J. Donald Hughes has traced this ancient view among the Greeks and Romans, noting the following progression : First, the Earth is the oldest goddess, supporter and nurturer of her children, human and non-human, and therefore entitled to respect and worship.... Second, the Earth is a living being of whom humans are only part. Right relationship with the Earth means that the total organism is... (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)

In this paper, we prove that: if κ is supercompact and the math formula Hypothesis holds, then there is a proper class of regular cardinals in math formula which are measurable in math formula. Woodin also proved this result independently [11]. As a corollary, we prove Woodin's Local Universality Theorem. This work shows that under the assumption of the math formula Hypothesis and supercompact cardinals, large cardinals in math formula are reflected to be large cardinals in math formula (...) in a local way, and reveals the huge difference between math formula-supercompact cardinals and supercompact cardinals under the math formula Hypothesis. (shrink)

In his Essay concerning Human Understanding, John Locke explicitly refers to Newton’s Philosophiae naturalis principia mathematica in laudatory but restrained terms: “Mr. Newton, in his never enough to be admired Book, has demonstrated several Propositions, which are so many new Truths, before unknown to the World, and are farther Advances in Mathematical Knowledge” (Essay, 4.7.3). The mathematica of the Principia are thus acknowledged. But what of philosophia naturalis? Locke maintains that natural philosophy, conceived as natural science (as opposed to (...) natural history), would give us demonstrations of the necessary connection between the (ultimately, simple) ideas constitutive of our complex ideas of various natural kinds of substances (e.g., gold). Indeed Locke goes so far as to suggest that a completely adequate natural science would also realize (perhaps, per impossibile) the goal of transforming the corpuscularian hypothesis into knowledge by demonstrating the necessary connection between the ‘microstructure’ (primary qualities of insensible corpuscles) of a particular natural kind of substance (e.g., gold) and the ideas of secondary qualities constitutive of the complex idea of that kind of substance. Locke’s conclusion concerning the possibility of the development of a natural science thus conceived is pessimistic: In vain therefore shall we endeavor to discover by our Ideas, (the only true way of certain and universal knowledge,) what other Ideas are to be found constantly joined with that of our complex Idea of any Substance: since we neither know the real Constitution of the minute Parts, on which their Qualities do depend; not, did we know them could we discover any necessary connexion between them, and any of their Secondary Qualities: which is necessary to be done, before we can certainly know their necessary co-existence (Essay, 4.3.14). It is understandable that, with such a conception of the science of nature, Locke found little of it in Newton’s Principia. In this paper, I further explore what might, perhaps with some hyperbole, be termed Locke’s ‘disappointment’ with the Prinicipia as a contribution to natural science. In particular, I argue that Locke’s adherence to the idealist epistemology of the Way of Ideas entails that mathematics cannot lend its certainty as a scientia to natural philosophy. Consequently, he finds more mathematics than natural philosophy in the Principia. (shrink)

What is the quantum state of the universe? Although there have been several interesting suggestions, the question remains open. In this paper, I consider a natural choice for the universal quantum state arising from the Past Hypothesis, a boundary condition that accounts for the time-asymmetry of the universe. The natural choice is given not by a wave function but by a density matrix. I begin by classifying quantum theories into two types: theories with a fundamental wave function (...) and theories with a fundamental density matrix. The Past Hypothesis is compatible with infinitely many initial wave functions, none of which seems to be particularly natural. However, once we turn to density matrices, the Past Hypothesis provides a natural choice---the normalized projection onto the Past Hypothesis subspace in the Hilbert space. Nevertheless, the two types of theories can be empirically equivalent. To provide a concrete understanding of the empirical equivalence, I provide a novel subsystem analysis in the context of Bohmian theories. Given the empirical equivalence, it seems empirically underdetermined whether the universe is in a pure state or a mixed state. Finally, I discuss some theoretical payoffs of the density-matrix theories and present some open problems for future research. (Bibliographic note: the thesis was submitted for the Master of Science in mathematics at Rutgers University.). (shrink)

In this paper, we approach the problem of the relationship between Greek mathematics and Eleatic philosophy from a new perspective, which leads us to a reappraisal of Szabó’s hypothesis about the origin of mathematics out of Eleatic philosophy. We claim that Parmenidean philosophy, particularly its semantic core, has possibly been shaped by reflexion on the Pythagoreans’ mathematical practice, particularly in arithmetic. Furthermore, Pythagorean arithmetic originates not from another domain outside mathematics but from counting, i.e., it has its roots (...) in man’s practical activity. This interpretation restores the historically inverse relationship between mathematics and philosophy, refuting the attribution of mathematics’ origin to a field outside mathematics, for which Szabó’s hypothesis has been criticised. Moreover, Parmenidean theory of truth is viewed not as a defective predecessor of Aristotle’s classical theory of truth that needs to be remedied but as a semantic conception coordinated with the mathematics of Parmenides’ times. (shrink)

Progress in neuroscience has left a central question of psychism unanswered: what is consciousness? Modeling the psyche from a computational perspective has helped to develop cognitive neurosciences, but it has also shown their limits, of which the definition, description and functioning of consciousness remain essential. From Rene Descartes, who tackled the issue of psychism as the brain-mind dualism, to Chambers, who defined qualia as the tough, difficult problem of research in neuroscience, many hypotheses and theories have been issued to encompass (...) the phenomenon of consciousness. Neuroscience specialists, such as Giulio Tononi or David Eagleman, consider consciousness as a phenomenon of emergence of all processes that take place in the brain. This hypothesis has the advantage of being supported by progress made in the study of complex systems in which the issue of emergence can be mathematically formalized and analyzed by physical-mathematical models. The current tendency to associate neural networks within the broad scope of network science also allows for a physical-mathematical formalization of phenomenology in neural networks and the construction of information-symbolic models. The extrapolation of emergence at the level of physical systems, biological systems and psychic systems can bring new models that can also be applied to the concept of consciousness. The meaning and significance that seem to structure the nature of consciousness is found as direction of evolution and teleological finality, of integration in the whole system and in any complex system at all scales. Starting from the wave-corpuscle duality in quantum physics, we can propose a model for structuring reality, based on the emergence of systems that contribute to the integration and coherence of the entire reality. Physical-mathematical models based mainly on topology can provide a mathematical formalization path, and the paradigm of information could allow the development of a pattern of emergence, that is common to all systems, including the psychic system, the difference being given only by the degree of information complexity. Thus, the mind-brain duality, which has been dominating the representation on psychism for a few centuries, could be solved by an informational approach, describing the connection between object and subject, reality and human consciousness, between mind and brain, thus unifying the perspective on natural sciences and humanities. (shrink)

We expect the laws of nature that describe the universe to be exact, but what if that isn't true? In this popular science article, I discuss the possibility that some candidate fundamental laws of nature, such as the Past Hypothesis, may be vague. This possibility is in conflict with the idea that the fundamental laws of nature can always and faithfully be described by classical mathematics. -/- [Bibliographic note: this article is featured on the magazine website under a (...) different title as "The fuzzy law that could break the idea of a mathematical universe" and on the magazine cover as "The Flaw at the Heart of Reality: Why precise mathematical laws can never fully explain the universe." It is a popular version of the article "Nomic Vagueness" that can be found on arXiv: 2006.05298.]. (shrink)

Information Theory, Evolution and The Origin ofLife: The Origin and Evolution of Life as a Digital Message: How Life Resembles a Computer, Second Edition. Hu- bert P. Yockey, 2005, Cambridge University Press, Cambridge: 400 pages, index; hardcover, US $60.00; ISBN: 0-521-80293-8. The reason that there are principles of biology that cannot be derived from the laws of physics and chemistry lies simply in the fact that the genetic information content of the genome for constructing even the simplest organisms is much (...) larger than the information content of these laws. Yockey in his previous book (1992, 335) In this new book, Information Theory, Evolution and The Origin ofLife, Hubert Yockey points out that the digital, segregated, and linear character of the genetic information system has a fundamental significance. If inheritance would blend and not segregate, Darwinian evolution would not occur. If inheritance would be analog, instead of digital, evolution would be also impossible, because it would be impossible to remove the effect of noise. In this way, life is guided by information, and so information is a central concept in molecular biology. The author presents a picture of how the main concepts of the genetic code were developed. He was able to show that despite Francis Crick's belief that the Central Dogma is only a hypothesis, the Central Dogma of Francis Crick is a mathematical consequence of the redundant nature of the genetic code. The redundancy arises from the fact that the DNA and mRNA alphabet is formed by triplets of 4 nucleotides, and so the number of letters (triplets) is 64, whereas the proteome alphabet has only 20 letters (20 amino acids), and so the translation from the larger alphabet to the smaller one is necessarily redundant. Except for Tryptohan and Methionine, all amino acids are coded by more than one triplet, therefore, it is undecidable which source code letter was actually sent from mRNA. This proof has a corollary telling that there are no such mathematical constraints for protein-protein communication. With this clarification, Yockey contributes to diminishing the widespread confusion related to such a central concept like the Central Dogma. Thus the Central Dogma prohibits the origin of life "proteins first." Proteins can not be generated by "self-organization." Understanding this property of the Central Dogma will have a serious impact on research on the origin of life. (shrink)

This is a discussion of some themes in Max Tegmark’s recent book, Our Mathematical Universe. It was written as a review for Plus Magazine, the online magazine of the UK’s national mathematics education and outreach project, the Mathematics Millennium Project. Since some of the discussion---about symmetry breaking, and Pythagoreanism in the philosophy of mathematics---went beyond reviewing Tegmark’s book, the material was divided into three online articles. This version combines those three articles, and adds some other material, in particular (...) a brief defence of quidditism about properties. It also adds some references, to other Plus articles as well as academic articles. But it retains the informal style of Plus. (shrink)

We compare two well-known set-theoretical statements, namely the axiom of choice and the continuum hypothesis, with regard to their historical development and formulation, as well as their consequences in mathematics. It is known that both statements are independent from the other axioms of set theory (if they are consistent). The axiom of choice – despite initial controversies – is today almost universally accepted as an axiom. However, the status of the continuum hypothesis is more complex and no agreement (...) has been found so far: both the continuum hypothesis and its negation (often as consequences of stronger statements) decide several mathematical problems differently, but in contrast with the axiom of choice it is not clear which of the two solutions should be the “correct” one (in the sense of an agreement within the community). (shrink)

We classify the phase transition thresholds from provability to unprovability for certain Friedman-style miniaturizations of Kruskal's Theorem and Higman's Lemma. In addition we prove a new and unexpected phase transition result for ε0. Motivated by renormalization and universality issues from statistical physics we finally state a universality hypothesis.

This chapter considers a provisional hypothesis that Logos is indeed absolutely perfect – in a word, God – and then discusses the implications of this assumption for the scope of contingency. It then argues that if God exists, it is likely that contingent reality is vastly greater than what current scientific theory or even speculation fancies. The conditions for freedom in the divine and human cases differ in a way that reflects the difference in ontological status between an absolutely (...) independent Creator and a dependent, causally conditioned creature. The chapter also discusses some applications of the many‐universe‐creation hypothesis. Finally, it addresses the relationship of Logos to logical and mathematical necessities. (shrink)

The hypothesis that life’s rapid appearance on Earth justifies the belief that life is widespread in the universe has been investigated mathematically by Lineweaver and Davis (Astrobiol- ogy 2002;2:293–304). However, a rapid appearance could also be interpreted as evidence for a nonterrestrial origin. I attempt to quantify the relative probabilities for a non-indigenous ver- sus indigenous origin, on the assumption that biogenesis involves one or more highly im- probable steps, using a generalization of Carter’s well-known observer-selection argument. The (...) analysis is specifically applied to a Martian origin of life, with subsequent transfer to Earth within impact ejecta. My main result is that the relatively greater probability of a Mar- tian origin rises sharply as a function of the number of difficult steps involved in biogene- sis. The actual numerical factor depends on what is assumed about conditions on early Mars, but for a wide range of assumptions a Martian origin of life is decisively favored. By con- trast, an extrasolar origin seems unlikely using the same analysis. These results complement those of Lineweaver and Davis. Key Words: Origin of life—Mars—Probability theory— Carter—Transpermia. Astrobiology 3, 673–679. (shrink)

Edited book containing the following essays: 1 Getting over Gettier, Alan Musgrave.- 2 Justified Believing: Avoiding the Paradox Gregory W. Dawes.- 3 Literature and Truthfulness,Gregory Currie.- 4 Where the Buck-passing Stops, Andrew Moore.- 5 Universal Darwinism: Its Scope and Limits, James Maclaurin, - 6 The Future of Utilitarianism,Tim Mulgan. 7 Kant on Experiment, Alberto Vanzo.- 8 Did Newton ʻFeignʼ the Corpuscular Hypothesis? Kirsten Walsh.- 9 The Progress of Scotland: The Edinburgh Philosophical Societies and the Experimental Method, Juan Gomez.- 10 (...) Propositions: Truth vs. Existence, Heather Dyke.- 11 Against Advanced Modalizing, Josh Parsons.- 12 Spread Worlds, Plenitude and Modal Realism: A Problem for DavidLewis, Charles R. Pigden and Rebecca E. B. Entwisle.- 13 Defending Quine on Ontological Commitment. 14. The Scandal of Platonism, Vladimír Svoboda.- 15 A Neglected Reply to Prior's Dilemma J. C. Beall. 16 Mathematical and Empirical Concepts, Pavel Materna.- 17 Post-Fregean Thoughts on Propositional Unity, Bjørn Jespersen.- 18 Best-path Theorem Proving: Compiling Derivations, Martin Frické.- 19 Is Imperative Inference Impossible?, Hannah Clark-Younger. ​. (shrink)

It is argued that colour name strategy, object name strategy, and chunking strategy in memory are all aspects of the same general phenomena, called stereotyping, and this in turn is an example of a know-how representation. Such representations are argued to have their origin in a principle called the minimum duplication of resources. For most the subsequent discussions existence of colour name strategy suffices. It is pointed out that the BerlinA- KayA universal partial ordering of colours and the frequency of (...) traffic accidents classified by colour are surprisingly similar; a detailed analysis is not carried out as the specific colours recorded are not identical. Some consequences of the existence of a name strategy for the philosophy of language and mathematics are discussed: specifically it is argued that in accounts of truth and meaning it is necessary throughout to use real numbers as opposed to bi-valent quantities; and also that the concomitant label associated with sentences should not be of unconditional truth, but rather several real-valued quantities associated with visual communication. The implication of real-valued truth quantities is that the Continuum Hypothesis of pure mathematics is side-stepped, because real valued quantities occur ab initio. The existence of name strategy shows that thought/sememes and talk/phonemes can be separate, and this vindicates the assumption of thought occurring before talk used in psycho-linguistic speech production models. (shrink)

Until the Scientific Revolution, the nature and motions of heavenly objects were mysterious and unpredictable. The Scientific Revolution was revolutionary in part because it saw the advent of many mathematical tools—chief among them the calculus—that natural philosophers could use to explain and predict these cosmic motions. Michel Blay traces the origins of this mathematization of the world, from Galileo to Newton and Laplace, and considers the profound philosophical consequences of submitting the infinite to rational analysis. "One of Michael Blay's (...) many fine achievements in _Reasoning with the Infinite_ is to make us realize how velocity, and later instantaneous velocity, came to play a vital part in the development of a rigorous mathematical science of motion."—Margaret Wertheim, _New Scientist_. (shrink)