We discuss the nature of reality in the ontological context of Penrose’s math-matter-mind triangle. The triangle suggests the circularity of the widespread view that math arises from the mind, the mind arises out of matter, and that matter can be explained in terms of math. Non-physicists should be wary of any claim that modern physics leads us to any particular resolution of this circularity, since even the sample of three theoretical physicists writing this paper hold three divergent (...) views. Some physicists believe that current physics has already found the basic framework for a complete description of reality, and only has to fill in the details. Others suspect that no single framework, from physics or other sources, will ever capture reality. Yet others guess that reality might be approached arbitrarily closely by some form of future physics, but probably based on completely different frameworks. We will designate these three approaches as the fundamentalist, secular and mystic views of the world, as seen by practicing physicists. We present and contrast each of these views, which arguably form broad categories capturing most if not all interpretations of physics. We argue that this diversity in the physics community is more useful than an ontological monoculture, since it motivates physicists to tackle unsolved problems with a wide variety of approaches. (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)

Mathematics is a critical part of much scientific research. Physics in particular weaves math extensively into its instruction beginning in high school. Despite much research on the learning of both physics and math, the problem of how to effectively include math in physics in a way that reaches most students remains unsolved. In this paper, we suggest that a fundamental issue has received insufficient exploration: the fact that in science, we don’t just use math, we (...) make meaning with it in a different way than mathematicians do. In this reflective essay, we explore math as a language and consider the language of math in physics through the lens of cognitive linguistics. We begin by offering a number of examples that show how the use of math in physics differs from the use of math as typically found in math classes. We then explore basic concepts in cognitive semantics to show how humans make meaning with language in general. The critical elements are the roles of embodied cognition and interpretation in context. Then, we show how a theoretical framework commonly used in physics education research, resources, is coherent with and extends the ideas of cognitive semantics by connecting embodiment to phenomenological primitives and contextual interpretation to the dynamics of meaning-making with conceptual resources, epistemological resources, and affect. We present these ideas with illustrative case studies of students working on physics problems with math and demonstrate the dynamical nature of student reasoning with math in physics. We conclude with some thoughts about the implications for instruction. (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)

Legal dogmatics in Continental European law (scientia iuris, Rechtswissenschaft) consists of professional legal writings whose task is to systematize and interpret valid law. Legal dogmatics pursues knowledge of the existing law, yet in many cases it leads to a change of the law. Among general theories of legal dogmatics, one may mention the theories of negligence, intent, adequate causation and ownership. The theories produce principles and they also produce defeasible rules. By means of production of general and defeasible theories, legal (...) dogmatics aims at obtaining a system of law that is both internally coherent and harmonized with its background in morality and (political) philosophy. Legal dogmatics is necessary in the context of constitutional constraints on the majority rule. Only if the courts act on the basis of Reason they can be a legitimate counterpart of the majority rule. And Reason cannot be exhausted by particular decision making. It also needs a more abstract deliberation, given by expert jurists. However, legal dogmatics has been a target of several kinds of criticism: empirical, morally-political, epistemological, logical, and ontological. The position taken in this article is to answer such criticism by mutually adjusting philosophy and the practices of the law. (shrink)

Prior research has suggested that perceptual disfluency activates analytical processing and increases the solution rate of mathematical problems with appealing but incorrect answers (i.e., the Cogn...

Lighten up about mathematics! Have fun. If you read this book, you will have to endure bad math puns and jokes and out-of-date pop culture references. You'll learn some really cool mathematics to boot. In the process, you will immerse yourself in living, thinking, and breathing logical reasoning. We like to call this proofs, which to some is a bogey word, but to us it is a boogie word. You will learn how to solve problems, real and imagined. (...) After all, math is a game where, although the rules are pretty much set, we are left to our imaginations to create. Think of this book as blueprints, but you are the architect of what structures you want to build. Make sure you lay a good foundation, for otherwise your buildings might fall down. To help you through this, we guide you to think and plan carefully. Our playground consists of basic math, with a loving emphasis on number theory. We will encounter the known and the unknown. Ancient and modern inquirers left us with elementary-sounding mathematical puzzles that are unsolved to this day. You will learn induction, logic, set theory, arithmetic, and algebra, and you may one day solve one of these puzzles. (shrink)

Philosophy has been characterized (e.g., by Benson Mates) as a field whose problems are unsolvable. This has often been taken to mean that there can be no progress in philosophy as there is in mathematics or science. The nature of problems and solutions is considered, and it is argued that solutions are always parts of theories, hence that acceptance of a solution requires commitment to a theory (as suggested by William Perry's scheme of cognitive development). Progress can be (...) had in philosophy in the same way as in mathematics and science by knowing what commitments are needed for solutions. Similar views of Rescher and Castañeda are discussed. (shrink)

This is a critical analysis of the first part of Go¨del’s 1951 Gibbs lecture on certain philosophical consequences of the incompleteness theorems. Go¨del’s discussion is framed in terms of a distinction between objective mathematics and subjective mathematics, according to which the former consists of the truths of mathematics in an absolute sense, and the latter consists of all humanly demonstrable truths. The question is whether these coincide; if they do, no formal axiomatic system (or Turing machine) can comprehend the mathematizing (...) potentialities of human thought, and, if not, there are absolutely unsolvable mathematical problems of diophantine form. (shrink)

It has been suggested that certain problems may be unsolvable because of the mind's cognitive structure, but we may wonder what problems, and exactly why. The ultimate origin of the universe and the mind-body problem seem to be two such problems. As to why, Colin McGinn has argued that the mind-body problem is unsolvable because any theoretical concepts about the brain will be observation-based and unable to connect to unobservable subjective experience. McGinn's argument suggests a requirement of (...) imagability -- an observation basis -- for physical causal explanation that cannot be met for either of these problems. Acausal descriptions may be possible but not the causal analyses that provide the greatest explanatory satisfaction, a psychological phenomenon that seems tied to the strength of the underlying observation basis but is affected by other factors as well. (shrink)

Classic text considersgeneral theory of computability, computable functions, operations on computable functions, Turing machines self-applied, unsolvable decision problems, applications of general theory, mathematical logic, Kleene hierarchy, computable functionals, classification of unsolvable decision problems and more.

There is a relationship between reading and math skills, as well as comorbidity between reading and math disorders. A mutual foundation for this comorbidity could be that the quality of phonological representations is important for both early reading and arithmetic. In this study, we examine this hypothesis in a sample traced longitudinally from preschool to first grade. The results show that phonological awareness does not explain development in arithmetic, but that there is an indirect effect between phoneme awareness (...) in preschool and arithmetic in first grade via phoneme awareness in first grade. This effect is, however, weak and restricted to verbal arithmetic and not arithmetic fluency. This finding is only partly in line with other studies, and a reason could be that this study more strongly controls for confounders and previous skills than other studies. (shrink)

The mind-body problem is analyzed in a physicalist perspective. By combining the concepts of emergence and algorithmic information theory in a thought experiment employing a basic nonlinear process, it is argued that epistemically strongly emergent properties may develop in a physical system. A comparison with the significantly more complex neural network of the brain shows that also consciousness is epistemically emergent in a strong sense. Thus reductionist understanding of consciousness appears not possible; the mind-body problem does not have a reductionist (...) solution. The ontologically emergent character of consciousness is then identified from a combinatorial analysis relating to system limits set by quantum mechanics, implying that consciousness is fundamentally irreducible to low-level phenomena. In the perspective of a modified definition of free will, the character of the physical interactions of the brain's neural system is subsequently studied. As an ontologically open system, it is asserted that its future states are undeterminable in principle. We argue that this leads to freedom of the will. (shrink)

According to Slote's ``agent-based'' virtue ethics, the rightness orwrongness of an act is determined by the motive it expresses. Thistheory has a problem with cases where an agent can do her duty onlyby expressing some vicious motive and thereby acting wrongly. In sucha situation, an agent can only act wrongly; hence, the theory seemsincompatible with the maxim that `ought' implies `can'. I argue thatSlote's attempt to circumvent this problem by appealing to compatibilism is inadequate. In a wide range of psychologically (...) realistic cases, an agent's effective choice will be between failingto do her duty and doing it from inferior motives. Then anythingshe can do will be wrong, according to the agent-based theory,contrary to the maxim Slote wishes to preserve. (shrink)

The uniform halting problem (UH) can be stated as follows:Give a decision procedure which for any given Turing machine (TM) will decide whether or not it has an immortal instantaneous description (ID).An ID is called immortal if it has no terminal successor. As it is generally the case in the literature (see e.g. Minsky [4, p. 118]) we assume that in an ID the tape must be blank except for some finite number of squares. If we remove this restriction the (...) UH becomes the immortality problem (IP). (shrink)

"A valuable collection both for original source material as well as historical formulations of current problems."-- The Review of Metaphysics "Much more than a mere collection of papers . . . a valuable addition to the literature."-- Mathematics of Computation An anthology of fundamental papers on undecidability and unsolvability by major figures in the field, this classic reference opens with Godel's landmark 1931 paper demonstrating that systems of logic cannot admit proofs of all true assertions of arithmetic. Subsequent papers (...) by Godel, Church, Turing, and Post single out the class of recursive functions as computable by finite algorithms. Additional papers by Church, Turing, and Post cover unsolvable problems from the theory of abstract computing machines, mathematical logic, and algebra, and material by Kleene and Post includes initiation of the classification theory of unsolvable problems. Suitable for graduate and undergraduate courses. 1965 ed. (shrink)

This book offers a unique analysis of how ideas about science and technology in the public and scientific imaginations (in particular about maths, logic, the gene, the brain, god, and robots) perpetuate the false reality that values and politics are separate from scientific knowledge and its applications. These ideas are reinforced by cultural myths about free will and individualism. Restivo makes a compelling case for a synchronistic approach in the study of these notoriously 'hard' cases, arguing that their significance reaches (...) far beyond the realms of science and technology, and that their sociological and political ramifications are of paramount importance in our global society. This innovative work deals with perennial problems in the social sciences, philosophy, and the history of science and religion, and will be of special interest to professionals in these fields, as well as scholars of science and technology studies. (shrink)

The tripartite account of propositional, fallibilist knowledge that p as justified true belief can become adequate only if it can solve the Gettier Problem. However, the latter can be solved only if the problem of a successful coordination of the resources (at least truth and justification) necessary and sufficient to deliver propositional, fallibilist knowledge that p can be solved. In this paper, the coordination problem is proved to be insolvable by showing that it is equivalent to the ''''coordinated attack'''' problem, (...) which is demonstrably insolvable in epistemic logic. It follows that the tripartite account is not merely inadequate as it stands, as proved by Gettier-type counterexamples, but demonstrably irreparable in principle, so that efforts to improve it can never succeed. (shrink)

Philosophers and historians of the French Revolution have seen Rousseau's influence as the decisive link between the doctrines of the Enlightenment and the practice of its revolutionary disciples. Professor Leigh here addresses the bibliographical foundations of that question, without which all attempts to settle it in the past have lacked authority. Introducing the most advanced techniques to identify variant and pirate editions of Rousseau's writings, he establishes that there were at least 28 separate imprints and an additional 12 reprints of (...) the Contrat Social in collective editions between 1762 and 1783. Professor Leigh shows also that Rousseau's life and thought excited a fascination and interest in the last years of the Ancien Regime which was nursed by the publishers of his Oeuvres, who sought to satisfy an apparently ceaseless demand by extending their editions, while at the same time attempting to ward off both their creditors and their imitators. (shrink)