Problem: There is currently a great deal of mysticism, uncritical hype, and blind adulation of imaginary mathematical and physical entities in popular culture. We seek to explore what a radical constructivist perspective on mathematical entities might entail, and to draw out the implications of this perspective for how we think about the nature of mathematical entities. Method: Conceptual analysis. Results: If we want to avoid the introduction of entities that are ill-defined and inaccessible to verification, then formal systems need to (...) avoid introduction of potential and actual infinities. If decidability and consistency are desired, keep formal systems finite. Infinity is a useful heuristic concept, but has no place in proof theory. Implications: We attempt to debunk many of the mysticisms and uncritical adulations of Gödelian arguments and to ground mathematical foundations in intersubjectively verifiable operations of limited observers. We hope that these insights will be useful to anyone trying to make sense of claims about the nature of formal systems. If we return to the notion of formal systems as concrete, finite systems, then we can be clear about the nature of computations that can be physically realized. In practical terms, the answer is not to proscribe notions of the infinite, but to recognize that these concepts have a different status with respect to their verifiability. We need to demarcate clearly the realm of free creation and imagination, where platonic entities are useful heuristic devices, and the realm of verification, testing, and proof, where infinities introduce ill-defined entities that create ambiguities and undecidable, ill-posed sets of propositions. Constructivist content: The paper attempts to extend the scope of radical constructivist perspective to mathematical systems, and to discuss the relationships between radical constructivism and other allied, yet distinct perspectives in the debate over the foundations of mathematics, such as psychological constructivism and mathematical constructivism. (shrink)

A bold and all-embracing exploration of the nature and progress of knowledge from one of today's great thinkers. Throughout history, mankind has struggled to understand life's mysteries, from the mundane to the seemingly miraculous. In this important new book, David Deutsch, an award-winning pioneer in the field of quantum computation, argues that explanations have a fundamental place in the universe. They have unlimited scope and power to cause change, and the quest to improve them is the basic regulating principle not (...) only of science but of all successful human endeavor. This stream of ever improving explanations has infinite reach, according to Deutsch: we are subject only to the laws of physics, and they impose no upper boundary to what we can eventually understand, control, and achieve. In his previous book, The Fabric of Reality, Deutsch describe the four deepest strands of existing knowledge-the theories of evolution, quantum physics, knowledge, and computation-arguing jointly they reveal a unified fabric of reality. In this new book, he applies that worldview to a wide range of issues and unsolved problems, from creativity and free will to the origin and future of the human species. Filled with startling new conclusions about human choice, optimism, scientific explanation, and the evolution of culture, The Beginning of Infinity is a groundbreaking book that will become a classic of its kind. (shrink)

The prediction of protein–protein interactions based on independently obtained structural information for each interacting partner remains an important challenge in computational chemistry. Procedures where hypothetical interaction models (or decoys) are generated, then ranked using a biochemically relevant scoring function have been garnering interest as an avenue for addressing such challenges. The program PatchDock has been shown to produce reasonable decoys for modeling the association between pig alpha-amylase and the VH-domains of camelide antibody raised against it. We designed a biochemically relevant (...) method by which PatchDock decoys could be ranked in order to separate near-native structures from false positives. Several thousand steps of energy minimization were used to simulate induced fit within the otherwise rigid decoys and to rank them. We applied a partial free energy function to rank each of the binding modes, improving discrimination between near-native structures and false positives. Sorting decoys according to strain energy increased the proportion of near-native decoys near the bottom of the ranked list. Additionally, we propose a novel method which utilizes regression analysis for the selection of minimization convergence criteria and provides approximation of the partial free energy function as the number of algorithmic steps approaches infinity. (shrink)

The paper addresses Leon Hen.kin's proposition as a " lighthouse", which can elucidate a vast territory of knowledge uniformly: logic, set theory, information theory, and quantum mechanics: Two strategies to infinity are equally relevant for it is as universal and t hus complete as open and thus incomplete. Henkin's, Godel's, Robert Jeroslow's, and Hartley Rogers' proposition are reformulated so that both completeness and incompleteness to be unified and thus reduced as a joint property of infinity and of all (...) infinite sets. However, only Henkin's proposition equivalent to an internal position to infinity is consistent . This can be retraced back to set theory and its axioms, where that of choice is a key. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that some essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, these phenomena can be elucidated as both complete and incomplete, after which choice is the border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or " observer'' , or "user" to infinity implied by Henkin's proposition as the only consistent ones as to infinity. An essential area of contemporary knowledge may be synthesized from a single viewpoint. (shrink)

This paper considers non-standard analysis and a recently introduced computational methodology based on the notion of ①. The latter approach was developed with the intention to allow one to work with infinities and infinitesimals numerically in a unique computational framework and in all the situations requiring these notions. Non-standard analysis is a classical purely symbolic technique that works with ultrafilters, external and internal sets, standard and non-standard numbers, etc. In its turn, the ①-based methodology does not use any of these (...) notions and proposes a more physical treatment of mathematical objects separating the objects from tools used to study them. It both offers a possibility to create new numerical methods using infinities and infinitesimals in floating-point computations and allows one to study certain mathematical objects dealing with infinity more accurately than it is done traditionally. In these notes, we explain that even though both methodologies deal with infinities and infinitesimals, they are independent and represent two different philosophies of Mathematics that are not in a conflict. It is proved that texts :539–555, 2017; Gutman and Kutateladze in Sib Math J 49:835–841, 2008; Kutateladze in J Appl Ind Math 5:73–75, 2011) asserting that the ①-based methodology is a part of non-standard analysis unfortunately contain several logical fallacies. Their attempt to show that the ①-based methodology can be formalized within non-standard analysis is similar to trying to show that constructivism can be reduced to the traditional Mathematics. (shrink)

Two strategies to infinity are equally relevant for it is as universal and thus complete as open and thus incomplete. Quantum mechanics is forced to introduce infinity implicitly by Hilbert space, on which is founded its formalism. One can demonstrate that essential properties of quantum information, entanglement, and quantum computer originate directly from infinity once it is involved in quantum mechanics. Thus, thеse phenomena can be elucidated as both complete and incomplete, after which choice is the (...) border between them. A special kind of invariance to the axiom of choice shared by quantum mechanics is discussed to be involved that border between the completeness and incompleteness of infinity in a consistent way. The so-called paradox of Albert Einstein, Boris Podolsky, and Nathan Rosen is interpreted entirely in the same terms only of set theory. Quantum computer can demonstrate especially clearly the privilege of the internal position, or “observer”, or “user” to infinity implied by Henkin’s proposition as the only consistent ones as to infinity. (shrink)

Many intermediate logics, even extremely well-behaved ones such as IPC, lack the finite model property for admissible rules. We give conditions under which this failure holds. We show that frames which validate all admissible rules necessarily satisfy a certain closure condition, and we prove that this condition, in the finite case, ensures that the frame is of width 2. Finally, we indicate how this result is related to some classical results on finite, free Heyting algebras.

What should philosophical theology look like after the critique of Onto-theology, after Phenomenology, and in the age of Speculative Realism? What does Kabbalah have to say to Philosophy? Since Kant and especially since Husserl, philosophy has only permitted itself to speak about how one relates to God in terms of the intentionality of consciousness and not of how God is in himself. This meant that one could only ever speak to God as an addressed and yearned-for holy Thou, but not (...) to God as infinite creator of all. In this book-length essay, the author argues that reality itself is made up of the Holy Name of God. Drawing upon the set-theoretical ontology of Alain Badiou, the computational theory of Stephen Wolfram, the physics of Frank Tipler, the psychoanalytical theory of Jacques Lacan, and the genius of Georg Cantor, the author works to demonstrate that the universe is a computer processing the divine Name and that all existence is made of information (the bit). As a result of this ontic pan-computationalism, it is shown that the future resurrection of the dead can take place and how it may in fact occur. Along the way, the book also offers compelling critiques of several significant theories of reality, including the phenomenological theologies of Emmanuel Levinas and Jean-Luc Marion, Process Theology, and Object-Oriented Ontology. Reality in the Name of God explores how the concepts of Jewish mysticism can be articulated and deployed as philosophical theses within current metaphysical debates. It provides a new and dynamic Structural Realist ontology of information. Ultimately, the book aims to deal a death blow to the restriction of philosophy and theology in relation to elaborations of a how a believer relates to a God outside the mind and to return thought to a direct encounter with the divine nature of reality itself and its creator"--Website of the publisher, October 19, 2020. (shrink)

Numbers. Natural numbers -- Integers -- Real numbers -- Irrational and transcendental numbers -- Funny numbers. Zero -- e : the unnatural natural number -- [Phi] : the golden ratio -- i : the imaginary number -- Writing numbers. Roman numerals -- Egyptian fractions -- Continued fractions -- Logic. Mr. Spock is not logical -- Proofs, truth, and trees : oh my! -- Programming with logic -- Temporal reasoning -- Sets. Cantor's diagonalization : infinity isn't just infinity -- (...) Axiomatic set theory : keep the good, dump the bad -- Models : using sets as the LEGOs of the math world -- Transfinite numbers : counting and ordering infinite sets -- Group theory : finding symmetries with sets -- Mechanical math. Finite state machines : simplicity goes far -- The turing machine -- Pathology and the heart of computing -- Calculus : no, not that calculus- [lambda] calculus -- Numbers, booleans, and recursion -- Types, types, types : modeling [lambda] calculus -- The halting problem. (shrink)

We informally discuss some recent results on the incompleteness of formal systems. These theorems, which are of great importance to contemporary mathematical epistemology, are proved using a variety of conceptual tools provably stronger than those of finitary axiomatisations. Those tools require no mathematical ontology, but rather constitute particularly concrete human constructions and acts of comprehending infinity and space rooted in different forms of knowledge. We shall also discuss, albeit very briefly, the mathematical intelligence both of God and of computers. (...) We hope in this manner to help the reader overcome formalist reductionism, while avoiding naive Platonist ontologies, typical symptoms of Godelitis which affected many in the last seventy years. (shrink)

The Mathematical Intelligencer recently published a note by Y. Sergeyev that challenges both mathematics and intelligence. We examine Sergeyev’s claims concerning his purported Infinity computer. We compare his grossone system with the classical Levi-Civita fields and with the hyperreal framework of A. Robinson, and analyze the related algorithmic issues inevitably arising in any genuine computer implementation. We show that Sergeyev’s grossone system is unnecessary and vague, and that whatever consistent subsystem could be salvaged is subsumed entirely within (...) a stronger and clearer system. Lou Kauffman, who published an article on a grossone, places it squarely outside the historical panorama of ideas dealing with infinity and infinitesimals. (shrink)

Our digital technologies have inspired new ways of thinking about old religious topics. Digitalists include computer scientists, transhumanists, singularitarians, and futurists. Digitalists have worked out novel and entirely naturalistic ways of thinking about bodies, minds, souls, universes, gods, and life after death. Your Digital Afterlives starts with three digitalist theories of life after death. It examines personality capture, body uploading, and promotion to higher levels of simulation. It then examines the idea that reality itself is ultimately a system of (...) self-surpassing computations. On that view, you will have infinitely many digital lives across infinitely many digital worlds. Your Digital Afterlives looks at superhuman bodies and infinite bodies. Thinking of nature in purely computational terms has the potential to radically and positively change our understanding of life after death. (shrink)

In honour of Detlef Dürr, we report on a mathematical rigorous computation of the electric vacuum polarisation current and extract the well-known expression for the second order perturbation. Intermediate steps in the presented calculation demonstrate, to the knowledge of the authors for the first time, mathematical rigorous versions of the combined dimensional and Pauli-Villars regularisation schemes. These are employed as computational tools to infer convenient integral representations during the computation. The said second order expression is determined up to a remaining (...) degree of freedom of a real number–without ill-defined terms from start to end. (shrink)

This article addresses the use of computer-assisted career guidance systems in career interventions. Major CACGS developed in the past decades were based on the trait-factor or person-environment fit approaches in their conceptualization and design. The strengths and limitations of these CACGS in addressing the career development needs of individuals are discussed. The Career Construction Theory is a promising paradigm to guide the development of new generations of CACGS. The narrative tradition, career adaptability model, and life-design interventions of CCT offer (...) rich conceptual and practical applications that could expand the scope and breadth of career exploration and identity construction through using CACGS. A digital system developed in Hong Kong called Infinity is a case-in-point of a CACGS where users could learn about career planning, engage in self and career construction through using the quantitative and qualitative assessment applications and gamified tools, take career planning actions over time, and communicate with their social supportive systems. Initial findings suggested that users of the system reported lesser decision-making difficulties, higher levels of decision clarity, and better understanding of what is career planning than non-users. Users from high academic achievement schools reported higher levels of career adaptability than their counterparts in schools of similar academic background. Users from low achievement schools reported higher intention to pursue government-supported universities than non-users from schools of similar background. Research and practice implications in schools and organizational settings are discussed. (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 subject of this paper is a characterization of the $\Sigma_1$ -definable set functions of Kripke-Platek set theory with infinity and a uniform version of axiom of choice: $KP\omega+(uniform\;AC)$ . This class of functions is shown to coincide with the collection of set functionals of type 1 primitive recursive in a given choice functional and $x\mapsto\omega$ . This goal is achieved by a Gödel Dialectica-style functional interpretation of $KP\omega+(uniform\;AC)$ and a computability proof for the involved functionals.

Computer technology has become an integral part of daily life. From online banking and shopping to email and instant messaging, cyberspace is increasingly woven into the fabric of our everyday lives. The mouse, the monitor and keyboard are all a part of the interfacing devices that over time become extensions of our bodies as we “surf” through graphical user interfaces. Icons patterned together in a mosaic on our screens link to infinite possibilities. We can visit museums, chat with family (...) and friends around the world, not to mention shop to have virtually anything we want shipped to our door. What happens to notions of infinity and transcendence when they are conflated with cyberspatial matrices? Does cyberspatial disembodiment allow us to transcend ourselves or does it blind us from the notion that transcendence in a box is not transcendence at all? (shrink)

In the third century b.c.e., Eratosthenes of Cyrene made a famous measurement of the circumference of the Earth. This was not the first such measurement, but it is the earliest for which significant details are preserved. Cleomedes gives a short account of Eratosthenes’ method, his numerical assumptions, and the final result of 250,000 stades. However, many ancient sources attribute to Eratosthenes a result of 252,000 stades. Historians have attempted to explain the second result by supposing that Eratosthenes later made better (...) measurements and revised his estimate or that the original result was simply rounded to 252,000 to have a number conveniently divisible by 60 or by 360. These explanations are speculative and untestable. However, Eratosthenes’ estimates of the distances of the Sun and Moon from the Earth are preserved in the doxographical literature. This essay shows that Eratosthenes’ result of 252,000 stades for the Earth’s circumference follows from a solar distance that is attributed to him. Thus it appears that Eratosthenes computed not only a lower limit for the size of the Earth, based on the assumption that the Sun is at infinity, but also an upper limit, based on the assumption that the Sun is at a finite distance. The essay discusses the consequences for our understanding of his program. (shrink)

A practical viewpoint links reality, representation, and language to calculation by the concept of Turing (1936) machine being the mathematical model of our computers. After the Gödel incompleteness theorems (1931) or the insolvability of the so-called halting problem (Turing 1936; Church 1936) as to a classical machine of Turing, one of the simplest hypotheses is completeness to be suggested for two ones. That is consistent with the provability of completeness by means of two independent Peano arithmetics discussed in Section I. (...) Many modifications of Turing machines cum quantum ones are researched in Section II for the Halting problem and completeness, and the model of two independent Turing machines seems to generalize them. Then, that pair can be postulated as the formal definition of reality therefore being complete unlike any of them standalone, remaining incomplete without its complementary counterpart. Representation is formal defined as a one-to-one mapping between the two Turing machines, and the set of all those mappings can be considered as “language” therefore including metaphors as mappings different than representation. Section III investigates that formal relation of “reality”, “representation”, and “language” modeled by (at least two) Turing machines. The independence of (two) Turing machines is interpreted by means of game theory and especially of the Nash equilibrium in Section IV. Choice and information as the quantity of choices are involved. That approach seems to be equivalent to that based on set theory and the concept of actual infinity in mathematics and allowing of practical implementations. (shrink)

We investigate in detail the electromagnetic fields of a uniformly accelerated charge, in order to ascertain whether such a charge does ‘emit’ radiation, especially in view of the Poynting flow computed at large distances and taken as an evidence of radiation emitted by the charge. In this context, certain important aspects of the fields need to be taken into account. First and foremost is the fact that in the case of a uniformly accelerated charge, one cannot ignore the velocity fields. (...) This then leads to other equally vital points. The net field energy turns out to be exactly the same as that of a non-accelerated charge having a uniform velocity equal to the instantaneous velocity of the uniformly accelerated charge. Further, the Poynting vector, seen with respect to the ’present’ location of the uniformly accelerated charge, during the deceleration phase, possesses everywhere a radial component pointing inward toward the charge, becoming nil when the charge becomes momentarily stationary, and during the acceleration phase, points away from the charge position. Last, but not least, when the leading spherical front of the relativistically beamed Poynting flux, advances forward at a large time t to a far-off distance \, the charge too is not lagging far behind. In fact, these relativistically beamed fields, increasingly resemble fields of a charge moving in an inertial frame with a uniform velocity \, with a convective flow of fields in that frame along with the movement of the charge. There is no other Poynting flow in the far-zones that could be termed as radiation emitted by the charge which, in turn, is fully consistent with the absence of radiation reaction and is also fully conversant with the strong principle of equivalence. (shrink)

Ordinal polytheism is motivated by the cosmological and design arguments. It is also motivated by Leibnizian–Lewisian modal realism. Just as there are many universes, so there are many gods. Gods are necessary concrete grounds of universes. The god-universe relation is one-to-one. Ordinal polytheism argues for a hierarchy of ranks of ever more perfect gods, one rank for every ordinal number. Since there are no maximally perfect gods, ordinal polytheism avoids many of the familiar problems of monotheism. It links theology with (...) counterpart theory, mathematics and computer science. And it entails that the system of universes has an attractive axiological structure. (shrink)

An introduction to the philosophy of mathematics grounded in mathematics and motivated by mathematical inquiry and practice. In this book, Joel David Hamkins offers an introduction to the philosophy of mathematics that is grounded in mathematics and motivated by mathematical inquiry and practice. He treats philosophical issues as they arise organically in mathematics, discussing such topics as platonism, realism, logicism, structuralism, formalism, infinity, and intuitionism in mathematical contexts. He organizes the book by mathematical themes--numbers, rigor, geometry, proof, computability, incompleteness, (...) and set theory--that give rise again and again to philosophical considerations. (shrink)

Why absolute certainty is impossible in science In today's unpredictable and chaotic world, we look to science to provide certainty and answers—and often blame it when things go wrong. The Blind Spot reveals why our faith in scientific certainty is a dangerous illusion, and how only by embracing science's inherent ambiguities and paradoxes can we truly appreciate its beauty and harness its potential. Crackling with insights into our most perplexing contemporary dilemmas, from climate change to the global financial meltdown, this (...) book challenges our most sacredly held beliefs about science, technology, and progress. At the same time, it shows how the secret to better science can be found where we least expect it—in the uncertain, the ambiguous, and the inevitably unpredictable. William Byers explains why the subjective element in scientific inquiry is in fact what makes it so dynamic, and deftly balances the need for certainty and rigor in science with the equally important need for creativity, freedom, and downright wonder. Drawing on an array of fascinating examples—from Wall Street's overreliance on algorithms to provide certainty in uncertain markets, to undecidable problems in mathematics and computer science, to Georg Cantor's paradoxical but true assertion about infinity—Byers demonstrates how we can and must learn from the existence of blind spots in our scientific and mathematical understanding. The Blind Spot offers an entirely new way of thinking about science, one that highlights its strengths and limitations, its unrealized promise, and, above all, its unavoidable ambiguity. It also points to a more sophisticated approach to the most intractable problems of our time. (shrink)

Pāṇini's fourth century BC Sanskrit grammar uses rewrite rules utilizing an explicit formal language defined through a semi-formal metalanguage. The grammar is generative, meaning that it is capable of expressing a potential infinity of well-formed Sanskrit sentences starting from a finite symbolic inventory. The grammar's operational rules involve extensive use of auxiliary markers, in the form of Sanskrit phonemes, to control grammatical derivations. Pāṇini's rules often utilize a generic context-sensitive format to identify terms used in replacement, modification or deletion (...) operations. The context-sensitive rule format is itself defined using Pāṇini's more general method of auxiliary markers, the latter used to define many dozens of linguistic categories and rules controlling derivations of Sanskrit sentences through the manipulation of ‘non-terminal’ and ‘terminal’ symbols. This technique for controlling formal derivations was rediscovered by Emil Post in the 1920s and later shown by him to be ca... (shrink)

Initially coined by Weizenbaum in 1976, ‘alien' thought refers to the radical difference with which ‘thinking machines’ approach the process of thinking. The contemporary paradox of over-determination and indeterminacy—caused largely by algorithmic decision-making in the civic realm—makes these differences both more entangled and more difficult to navigate. In this essay, I trace over-determination to Leibniz and Turing’s axiomatic procedures and to instrumental rationality, and I trace indeterminacy to the mid-twentieth century co-development of computers and neurosciences to advance the following proposition: (...) understanding alien thought requires understanding incomputability, temporal swarming, and inscriptive-significational errance. Understanding these phenomena in turn requires understanding thinking by doing, distributed thinking, and ontological indeterminacy. All are present in machinic operations as well as in the twentieth century experimental artistic practices of artists such as Duchamp, Cage, and Xu. These practices rely on indeterminate procedures and function as diagrammatic machines. A diagrammatic machine is neither abstract nor particular; neither an idea that is determining in the supreme instance, nor an infrastructure that is determining in the last instance, but rather instantiates a real yet to come (Deleuze and Guattari in A thousand plateaus: capitalism and schizophrenia: Trans. Massumi B, University of Minnesota Press, Minneapolis, 1987). In this essay, indeterminate artistic practices are used as an entry into alien thought and its correlates—infinity and complexity—by way of aesthetic analogy. (shrink)

In his most comprehensive book on the subject , Roger Penrose provides arguments to demonstrate that there are aspects of human understanding which could not, in principle, be attained by any purely computational system. His central argument relies crucially on oft-cited theorems proven by Gödel and Turing. However, that key argument has been the subject of numerous trenchant critiques, which is unfortunate if one believes Penrose's conclusions to be plausible. In the present article, alternative arguments are offered in support of (...) Penrose-like conclusions . It is argued here that a purely computational agent, which lacked conscious awareness, would be incapable of possessing crucial concepts and of understanding certain kinds of geometrically-based proofs. Specifically, it is argued that the acquisition of human-like concepts of countable and non-denumerable infinities, and human-like comprehension of a particular geometrically motivated proof does require conscious apprehension of the subject matter involved. This does not preclude the possibility that a computational agent might come to possess the requisite consciousness, but it is argued that if this consciousness does arise within the agent, it does so, at best, as an emergent, contingent side-effect of the underlying processes involved. (shrink)

Many books explain what is known about the universe. This book investigates what cannot be known. Rather than exploring the amazing facts that science, mathematics, and reason have revealed to us, this work studies what science, mathematics, and reason tell us cannot be revealed. In The Outer Limits of Reason, Noson Yanofsky considers what cannot be predicted, described, or known, and what will never be understood. He discusses the limitations of computers, physics, logic, and our own thought processes. Yanofsky describes (...) simple tasks that would take computers trillions of centuries to complete and other problems that computers can never solve; perfectly formed English sentences that make no sense; different levels of infinity; the bizarre world of the quantum; the relevance of relativity theory; the causes of chaos theory; math problems that cannot be solved by normal means; and statements that are true but cannot be proven. He explains the limitations of our intuitions about the world -- our ideas about space, time, and motion, and the complex relationship between the knower and the known. Moving from the concrete to the abstract, from problems of everyday language to straightforward philosophical questions to the formalities of physics and mathematics, Yanofsky demonstrates a myriad of unsolvable problems and paradoxes. Exploring the various limitations of our knowledge, he shows that many of these limitations have a similar pattern and that by investigating these patterns, we can better understand the structure and limitations of reason itself. Yanofsky even attempts to look beyond the borders of reason to see what, if anything, is out there. (shrink)

For over a decade, the hypercomputation movement has produced computational models that in theory solve the algorithmically unsolvable, but they are not physically realizable according to currently accepted physical theories. While opponents to the hypercomputation movement provide arguments against the physical realizability of specific models in order to demonstrate this, these arguments lack the generality to be a satisfactory justification against the construction of any information-processing machine that computes beyond the universal Turing machine. To this end, I present a more (...) mathematically concrete challenge to hypercomputability, and will show that one is immediately led into physical impossibilities, thereby demonstrating the infeasibility of hypercomputers more generally. This gives impetus to propose and justify a more plausible starting point for an extension to the classical paradigm that is physically possible, at least in principle. Instead of attempting to rely on infinities such as idealized limits of infinite time or numerical precision, or some other physically unattainable source, one should focus on extending the classical paradigm to better encapsulate modern computational problems that are not well-expressed/modeled by the closed-system paradigm of the Turing machine. I present the first steps toward this goal by considering contemporary computational problems dealing with intractability and issues surrounding cyber-physical systems, and argue that a reasonable extension to the classical paradigm should focus on these issues in order to be practically viable. (shrink)

In this compendium of essays, some of the world's leading thinkers discuss their conceptions of space and time, as viewed through the lens of their own discipline. With an epilogue on the limits of human understanding, this volume hosts contributions from six or more diverse fields. It presumes only rudimentary background knowledge on the part of the reader. Time and again, through the prism of intellect, humans have tried to diffract reality into various distinct, yet seamless, atomic, yet holistic, independent, (...) yet interrelated disciplines and have attempted to study it contextually. Philosophers debate the paradoxes, or engage in meditations, dialogues and reflections on the content and nature of space and time. Physicists, too, have been trying to mold space and time to fit their notions concerning micro- and macro-worlds. Mathematicians focus on the abstract aspects of space, time and measurement. While cognitive scientists ponder over the perceptual and experiential facets of our consciousness of space and time, computer scientists theoretically and practically try to optimize the space-time complexities in storing and retrieving data/information. The list is never-ending. Linguists, logicians, artists, evolutionary biologists, geographers et cetera, all are trying to weave a web of understanding around the same duo. However, our endeavour into a world of such endless imagination is restrained by intellectual dilemmas such as: Can humans comprehend everything? Are there any limits? Can finite thought fathom infinity? We have sought far and wide among the best minds to furnish articles that provide an overview of the above topics. We hope that, through this journey, a symphony of patterns and tapestry of intuitions will emerge, providing the reader with insights into the questions: What is Space? What is Time? Chapter [15] of this book is available open access under a CC BY 4.0 license. (shrink)

With mind-brain identity theories no longer dominant in philosophy of mind in the late 1950s, scientific materialists turned to functionalism, the view that the identity of any mental state depends on its function in the cognitive system of which it is a part. The philosopher Hilary Putnam was one of the primary architects of functionalism and was the first to propose computational functionalism, which views the human mind as a computer or an information processor. But, in the early 1970s, (...) Putnam began to have doubts about functionalism, and in his masterwork _Representation and Reality_, he advanced four powerful arguments against his own doctrine of computational functionalism. In Gödel, Putnam, and Functionalism, Jeff Buechner systematically examines Putnam's arguments against functionalism and contends that they are unsuccessful. Putnam's first argument uses Gödel's incompleteness theorem to refute the view that there is a computational description of human reasoning and rationality; his second, the "triviality argument," demonstrates that any computational description can be attributed to any physical system; his third, the multirealization argument, shows that there are infinitely many computational realizations of an arbitrary intentional state; his fourth argument buttresses this assertion by showing that there cannot be local computational reductions because there is no computable partitioning of the infinity of computational realizations of an arbitrary intentional state into a single package or small set of packages. Buechner analyzes these arguments and the important inferential connections among them -- for example, the use of both the Gödel and triviality arguments in the argument against local computational reductions -- and argues that none of Putnam's four arguments succeeds in refuting functionalism. _ Gödel, Putnam, and Functionalism_ will inspire renewed discussion of Putnam's influential book and will confirm Representation and Reality as a major work by a major philosopher. Jeff Buechner is Director of the Bioethics Institute and Lecturer in Philosophy at Rutgers University-Newark. (shrink)

In this paper I present a novel supertask in a Newtonian universe that destroys and creates infinite masses and energies, showing thereby that we can have infinite indeterminism. Previous supertasks have managed only to destroy or create finite masses and energies, thereby giving cases of only finite indeterminism. In the Nothing from Infinity paradox we will see an infinitude of finite masses and an infinitude of energy disappear entirely, and do so despite the conservation of energy in all collisions. (...) I then show how this leads to the Infinity from Nothing paradox, in which we have the spontaneous eruption of infinite mass and energy out of nothing. I conclude by showing how our supertask models at least something of an old conundrum, the question of what happens when the immovable object meets the irresistible force. (shrink)

Quantum computer is considered as a generalization of Turing machine. The bits are substituted by qubits. In turn, a "qubit" is the generalization of "bit" referring to infinite sets or series. It extends the consept of calculation from finite processes and algorithms to infinite ones, impossible as to any Turing machines (such as our computers). However, the concept of quantum computer mets all paradoxes of infinity such as Gödel's incompletness theorems (1931), etc. A philosophical reflection on how (...) quantum computer might implement the idea of "infinite calculation" is the main subject. (shrink)

The problem of infinities in quantum field theory (QFT) is a longstanding problem in particle physics. To solve this problem, different renormalization techniques have been suggested but the problem persists. Here we suggest another approach to the elimination of infinities in QFT, which is based on non-Diophantine arithmetics – a novel mathematical area that already found useful applications in physics, psychology, and other areas. To achieve this goal, new non-Diophantine arithmetics are constructed and their properties are studied. In addition, non-Diophantine (...) integration is developed in these arithmetics. These constructions allow using constructed non-Diophantine arithmetics for computing integrals associated with Feynman diagrams. Although in the conventional QFT such integrals diverge, their non-Diophantine counterparts are convergent and rigorously defined. As the result, QFT becomes consistent with quantum experiments. (shrink)

In this paper, I address the controversy between Henry More and René Descartes on the indefinite extension of the world. I provide a new reading of Descartes’ famous final answer of 15 April 1649. I read the entire debate in the terms of a disagreement concerning the epistemological status of the necessity of our judgement about the extension of the universe. Accordingly, the disagreement on the infinity of the world constitutes a case of a more general disagreement on the (...) nature of the necessity of the theorems of Cartesian Physics. In particular, as concerns Descartes’ last reply, I argue that his assertion that a finite world is contradictory should be interpreted as a reply to More’s claim that the thesis of the infinity of the world, in so far as it cannot be grounded on the identity between matter and extension, does not express a logical necessity. Descartes’ assertion of the logical impossibility of a finite world, far from being, as it has always been read, a concession he made under the pressure of More’s objections, expresses the more radical element of the entire debate about the extension of the universe. (shrink)

The problem of multiple-computations discovered by Hilary Putnam presents a deep difficulty for functionalism (of all sorts, computational and causal). We describe in out- line why Putnam’s result, and likewise the more restricted result we call the Multiple- Computations Theorem, are in fact theorems of statistical mechanics. We show why the mere interaction of a computing system with its environment cannot single out a computation as the preferred one amongst the many computations implemented by the system. We explain why nonreductive (...) approaches to solving the multiple- computations problem, and in particular why computational externalism, are dualistic in the sense that they imply that nonphysical facts in the environment of a computing system single out the computation. We discuss certain attempts to dissolve Putnam’s unrestricted result by appealing to systems with certain kinds of input and output states as a special case of computational externalism, and show why this approach is not workable without collapsing to behaviorism. We conclude with some remarks about the nonphysical nature of mainstream approaches to both statistical mechanics and the quantum theory of measurement with respect to the singling out of partitions and observables. (shrink)

The goal of the present article is to contribute to the epistemology and methodology of computer simulations. The central thesis is that the process of simulation modeling takes the form of an explorative cooperation between experimenting and modeling. This characteristic mode of modeling turns simulations into autonomous mediators in a specific way; namely, it makes it possible for the phenomena and the data to exert a direct influence on the model. The argumentation will be illustrated by a case study (...) of the general circulation models of meteorology, the major simulation models in climate research. (shrink)

This book is part of a major project undertaken by the Centre for Studies in Civilizations , being one of a total of ninety-six planned volumes. The author is a statistician and computer scientist by training, who has concentrated on historical matters for the last ten years or so. The book has very ambitious aims, proposing an alternative philosophy of mathematics and a deviant history of the calculus. Throughout, there is an emphasis on the need to combine history and (...) philosophy of mathematics, especially in order to evaluate properly the history of mathematics in India, in particular the history of the calculus.The pivotal goals of the book are to oppose the Eurocentric account of the history of science in general and mathematics in particular; to avoid the usual philosophical idea of the centrality of proof for mathematical knowledge, in favour of the traditional Indian notion of pramāṇa [validation] encompassing empirical elements and emphasizing calculation; to analyze the thousand-year-long development of infinite series in India, starting in the fifth century; and to show ‘how and why the calculus was imported into Europe’ from about 1500. The result is a picture in which inputs from the Indian subcontinent and epistemology are the driving forces of the history of mathematics, as people in the European subcontinent struggle to adopt new calculation techniques from the East in spite of Western philosophico-religious biases . Thus, a ‘first math war’ involved the algorismus de numero indorum, adopted for practical reasons, which forced Europeans to modify their epistemology of number and quantities. A ‘second math war’ revolved around infinite series and the calculus, since the background of Western epistemology created ‘spurious difficulties’ about infinities and infinitesimals, partially resolved with theories of real numbers. And a ‘third math war’ is …. (shrink)

This book is part of a major project undertaken by the Centre for Studies in Civilizations , being one of a total of ninety-six planned volumes. The author is a statistician and computer scientist by training, who has concentrated on historical matters for the last ten years or so. The book has very ambitious aims, proposing an alternative philosophy of mathematics and a deviant history of the calculus. Throughout, there is an emphasis on the need to combine history and (...) philosophy of mathematics, especially in order to evaluate properly the history of mathematics in India, in particular the history of the calculus.The pivotal goals of the book are to oppose the Eurocentric account of the history of science in general and mathematics in particular; to avoid the usual philosophical idea of the centrality of proof for mathematical knowledge, in favour of the traditional Indian notion of pramāṇa [validation] encompassing empirical elements and emphasizing calculation; to analyze the thousand-year-long development of infinite series in India, starting in the fifth century; and to show ‘how and why the calculus was imported into Europe’ from about 1500. The result is a picture in which inputs from the Indian subcontinent and epistemology are the driving forces of the history of mathematics, as people in the European subcontinent struggle to adopt new calculation techniques from the East in spite of Western philosophico-religious biases . Thus, a ‘first math war’ involved the algorismus de numero indorum, adopted for practical reasons, which forced Europeans to modify their epistemology of number and quantities. A ‘second math war’ revolved around infinite series and the calculus, since the background of Western epistemology created ‘spurious difficulties’ about infinities and infinitesimals, partially resolved with theories of real numbers. And a ‘third math war’ is …. (shrink)

Computer simulation and thought experiments seem to produce knowledge about the world without intervening in the world. This has called for a comparison between the two methods. However, Chandrasekharan et al. argue that the nature of contemporary science is too complex for using TEs. They suggest CS as the tool for contemporary sciences and conclude that it will replace TEs. In this paper, by discussing a few TEs from the history of science, I show that the replacement thesis about (...) TE is a failure. The paper is divided into three sections. The first section discusses the arguments of Chandrasekharan et al. and demonstrates the three distinct aspects of the replacement thesis. The second section examines the argument against TE and shows that they are inadequate to prove the withering of TE from science. The third section discusses Albert Einstein’s Magnet and Conductor TE and demonstrates that replacing such TE with CS yield no advantage. (shrink)

The book is devoted to the contemporary stage of quantum mechanics – quantum information, and especially to its philosophical interpretation and comprehension: the first one of a series monographs about the philosophy of quantum information. The second will consider Be l l ’ s inequalities, their modified variants and similar to them relations. The beginning of quantum information was in the thirties of the last century. Its speed development has started over the last two decades. The main phenomenon is entanglement. (...) The subareas are quantum computer, quantum communication (and teleportation), and quantum cryptography. The book offers the following main conceptions, theses and hypotheses: – dualistic Phythagoreanism as a new kind among the interpretations of quantum mechanics and information: arithmetical, logical, and metamathematical one; – Gödel ’ s first incompleteness theorem is an undecidable proposition, and consequently the second one,too. – a partial rehabilitation of Hilbert ’ s program for the self-foundation of mathematics; – the dual-foundation of mathematics; – Skolemian relativity between: Cantor ’s kinds of infinity, finiteness and infinity, discreteness and continuity, completeness and incompleteness, etc.; – information is a physical quantity representing the non-reducibility of a system to its parts, particularly nonaddtivity; – there exist pure relations «by itself», which cannot be reduced to predications; – energy conservation can and should be generalized; – Einstein’ s «general covariance» or «principle of relativity» can and should be generalized to cover discrete morphisms where the notion of velocity does not make sense. (shrink)