Gödel argued that his incompleteness theorems imply that either “the mind cannot be mechanized” or “there are absolutely undecidable sentences.” In the precursor to this paper I examined the early arguments for the first disjunct. In the present paper I examine the most sophisticated argument for the first disjunct, namely, Penrose’s new argument. It turns out that Penrose’s argument requires a type-free notion of truth and a type-free notion of absolute provability. I show that there is a natural such system, (...) DTK. I prove a series of results which show that Gödel’s disjunction is provable in the system, Penrose’s argument is invalid in the system, there can be no proof or refutation of either disjunct in the system, the independence results are robust in that they persist when one strengthens the principles governing absolute provability, and there are reasons to believe that the situation will not improve under any plausible alteration of the underlying theory of truth. (shrink)

In this paper I address the question of whether the incompleteness theorems imply that “the mind cannot be mechanized,” where this is understood in the specific sense that “the mathematical outputs of the idealized human mind do not coincide with the mathematical outputs of any idealized finite machine.” Gödel argued that his incompleteness theorems implied a weaker, disjunctive conclusion to the effect that either “the mind cannot be mechanized” or “mathematical truth outstrips the idealized human mind.” Others, most notably, Lucas (...) and Penrose, have claimed more—they have claimed that the incompleteness theorems actually imply the first disjunct. I will show that by sharpening the fundamental concepts involved and articulating the background assumptions governing them, one can prove Gödel’s disjunction, one can show that the arguments of Lucas and Penrose fail, and one can see what likely led proponents of the first disjunct astray. (shrink)

Gödel initiated the program of finding and justifying axioms that effect a significant reduction in incompleteness and he drew a fundamental distinction between intrinsic and extrinsic justifications. Reflection principles are the most promising candidates for new axioms that are intrinsically justified. Taking as our starting point Tait’s work on general reflection principles, we prove a series of limitative results concerning this approach. These results collectively show that general reflection principles are either weak ) or inconsistent. The philosophical significance of these (...) results is discussed. (shrink)

The paper begins with an examination of Gödel's views on absolute undecidability and related topics in set theory. These views are sharpened and assessed in light of recent developments. It is argued that a convincing case can be made for axioms that settle many of the questions undecided by the standard axioms and that in a precise sense the program for large cardinals is a complete success “below” CH. It is also argued that there are reasonable scenarios for settling CH (...) and that there is not currently a convincing case to the effect that a given statement is absolutely undecidable. (shrink)

In this paper we investigate strong logics of first and second order that have certain absoluteness properties. We begin with an investigation of first order logic and the strong logics ω-logic and β-logic, isolating two facets of absoluteness, namely, generic invariance and faithfulness. It turns out that absoluteness is relative in the sense that stronger background assumptions secure greater degrees of absoluteness. Our aim is to investigate the hierarchies of strong logics of first and second order that are generically invariant (...) and faithful against the backdrop of the strongest large cardinal hypotheses. We show that there is a close correspondence between the two hierarchies and we characterize the strongest logic in each hierarchy. On the first-order side, this leads to a new presentation of Woodin's Ω-logic. On the second-order side, we compare the strongest logic with full second-order logic and argue that the comparison lends support to Quine's claim that second-order logic is really set theory in sheep's clothing. (shrink)

In 1985 the second author showed that if there is a proper class of measurable Woodin cardinals and $V^{B1} $ and $V^{B2} $ are generic extensions of V satisfying CH then $V^{B1} $ and $V^{B2} $ agree on all $\Sigma _1^2 $ -statements. In terms of the strong logic Ω-logic this can be reformulated by saying that under the above large cardinal assumption ZFC + CH is Ω-complete for $\Sigma _1^2 $ Moreover. CH is the unique $\Sigma _1^2 $ -statement (...) with this feature in the sense that any other $\Sigma _1^2 $ -statement with this feature is Ω-equivalent to CH over ZFC. It is natural to look for other strengthenings of ZFC that have an even greater degree of Ω-completeness. For example, one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for all of third-order arithmetic. Going further, for each specifiable segment $V_\lambda $ of the universe of sets (for example, one might take $V_\lambda $ to be the least level that satisfies there is a proper class of huge cardinals), one can ask for recursively enumerable axioms A such that relative to large cardinal axioms ZFC + A is Ω-complete for the theory of $V_\lambda $ . If such theories exist, extend one another, and are unique in the sense that any other such theory B with the same level of Ω-completeness as A is actually Ω-equivalent to A over ZFC, then this would show that there is a unique Ω-complete picture of the successive fragments of the universe of sets and it would make for a very strong case for axioms complementing large cardinal axioms. In this paper we show that uniqueness must fail. In particular, we show that if there is one such theory that Ω-implies CH then there is another that Ω-implies¬CH. (shrink)

The independence results in set theory invite the search for new and justified axioms. In Chapter 1 I set the stage by examining three approaches to justifying the axioms of standard set theory and argue that the approach via reflection principles is the most successful. In Chapter 2 I analyse the limitations of ZF and use this analysis to set up a mathematically precise minimal hurdle which any set of new axioms must overcome if it is to effect a significant (...) reduction in incompleteness. In Chapter 3 I examine the standard method of justifying new axioms---reflection principles---and prove a result which shows that no reflection principle can overcome the minimal hurdle and yield a significant reduction in incompleteness. In Chapter 4 I introduce a new approach to justifying new axioms---extension principles---and show that such principles can overcome the minimal hurdle and much more, in particular, such principles imply PD and that the theory of second-order arithmetic cannot be altered by set size forcing. I show that in a sense these principles are inevitable. In Chapter 5 I close with a brief discussion of meta-mathematical justifications stemming from the work of Woodin. These touch on the continuum hypothesis and other questions which are beyond the reach of standard large cardinals. (shrink)

In this paper I examine Feferman’s reasons for maintaining that while the statements of first-order number theory are “completely clear'” and “completely definite,”' many of the statements of analysis and set theory are “inherently vague'” and “indefinite.”' I critique his four central arguments and argue that in the end the entire case rests on the brute intuition that the concept of subsets of natural numbers—along with the richer concepts of set theory—is not “clear enough to secure definiteness.” My response to (...) this final, remaining point will be that the concept of “being clear enough to secure definiteness” is about as clear a case of an inherently vague and indefinite concept as one might find, and as such it can bear little weight in making a case against the definiteness of analysis and set theory. (shrink)

The HOD Dichotomy Theorem states that if there is an extendible cardinal, δ, then either HOD is “close” to V or HOD is “far” from V. The question is whether the future will lead to the first or the second side of the dichotomy. Is HOD “close” to V, or “far” from V? There is a program aimed at establishing the first alternative—the “close” side of the HOD Dichotomy. This is the program of inner model theory. In recent years the (...) third author has provided evidence that there is an ultimate inner model—Ultimate-L—and he has isolated a natural conjecture associated with the model—the Ultimate-L Conjecture. This conjecture implies that that the first alternative holds—HOD is “close” to V. This is the future in which pattern prevails. In this paper we introduce a very different program, one aimed at establishing the second alternative—the “far” side of the HOD Dichotomy. This is the program of large cardinals beyond choice. Kunen famously showed that if AC holds then there cannot be a Reinhardt cardinal. It has remained open whether Reinhardt cardinals are consistent in ZF alone. It turns out that there is an entire hierarchy of choiceless large cardinals of which Reinhardt cardinals are only the beginning, and, surprisingly, this hierarchy appears to be highly ordered and amenable to systematic investigation, as we shall show in this paper. The point is that if these choiceless large cardinals are consistent then the Ultimate-L Conjecture must fail. This is the future where chaos prevails. (shrink)

This volume contains the proceedings of the Logic at Harvard conference in honor of W. Hugh Woodin's 60th birthday, held March 27–29, 2015, at Harvard University. It presents a collection of papers related to the work of Woodin, who has been one of the leading figures in set theory since the early 1980s. The topics cover many of the areas central to Woodin's work, including large cardinals, determinacy, descriptive set theory and the continuum problem, as well as connections between set (...) theory and Banach spaces, recursion theory, and philosophy, each reflecting a period of Woodin's career. Other topics covered are forcing axioms, inner model theory, the partition calculus, and the theory of ultrafilters. This volume should make a suitable introduction to Woodin's work and the concerns which motivate it. The papers should be of interest to graduate students and researchers in both mathematics and philosophy of mathematics, particularly in set theory, foundations and related areas. (shrink)

Conventionalism about mathematics claims that mathematical truths are true by linguistic convention. This is often spelled out by appealing to facts concerning rules of inference and formal systems, but this leads to a problem: since the incompleteness theorems we’ve known that syntactic notions can be expressed using arithmetical sentences. There is serious prima facie tension here: how can mathematics be a matter of convention and syntax a matter of fact given the arithmetization of syntax? This challenge has been pressed in (...) the literature by Hilary Putnam and Peter Koellner. In this paper I sketch a conventionalist theory of mathematics, show that this conventionalist theory can meet the challenge just raised , and clarify the type of mathematical pluralism endorsed by the conventionalist by introducing the notion of a semantic counterpart. The paper’s aim is an improved understanding of conventionalism, pluralism, and the relationship between them. (shrink)

In 5, Tait identifies a set of reflection principles called equation image-reflection principles which Peter Koellner has shown to be consistent relative to the existence of κ, the first ω-Erdős cardinal 1. Tait also defines a set of reflection principles called equation image-reflection principles; however, Koellner has shown that these are inconsistent when m > 2, but identifies restricted versions of them which he proves consistent relative to κ 2. In this paper, we introduce a new large-cardinal property, the (...) α-reflective cardinals. Their definition is motivated by Tait's remarks on parameters of third or higher order in reflection principles. We prove that if κ is ℶκ + α + 1-supercompact and 0 < α < κ then κ is α-reflective. Furthermore we show that α-reflective cardinals relativize to L, and that if κ exists then the set of cardinals λ < κ, such that λ is α-reflective for all α such that 0 < α < λ, is a stationary subset of κ. We show that an ω-reflective cardinal satisfies some restricted versions of equation image-reflection, as well as all the reflection properties proved consistent in 2. (shrink)

Peter Adamson offers an accessible, humorous tour through a period of eight hundred years when some of the most influential of all schools of thought were formed. He introduces us to Cynics and Skeptics, Epicureans and Stoics, emperors and slaves, and traces the development of early Christian philosophy and of ancient science. A major theme of the book is in fact the competition between pagan and Christian philosophy in this period, and the Jewish tradition appears in the shape of (...) Philo of Alexandria. Ancient science is also considered, with chapters on ancient medicine and the interaction between philosophy and astronomy. Considerable attention is paid also to the wider historical context, for instance by looking at the ascetic movement in Christianity and how it drew on ideas from Hellenic philosophy. From the counter-cultural witticisms of Diogenes the Cynic to the subtle skepticism of Sextus Empiricus, from the irreverent atheism of the Epicureans to the ambitious metaphysical speculation of Neoplatonism, from the ethical teachings of Marcus Aurelius to the political philosophy of Augustine, the book gathers together all aspects of later ancient thought in an accessible and entertaining way. (shrink)

Film and the Emotions explores the complicated relationship between filmed entertainment, such as movies and television shows, and our capacity to feel emotions. This volume of The Midwest Studies in Philosophy covers topics such as the role of imagination in our capacity to respond emotionally to films, how emotions felt in response to films relate to emotions felt about real events, and the moral implications of responding emotionally to fictions, among others. This collection includes nineteen original articles from experts on (...) film and emotion, including Noel Carroll, Gregory Currie, Susan Feagin, Stacie Friend, Robert Hopkins, Peter Lamarque and Peter Goldie, Derek Matravers, Carl Plantinga, and Murray Smith. (shrink)

Changing Ideals in Modern Architecture revolutionized the understanding of modernism in architecture, pushing back the sense of its origin from the early twentieth century to the 1750s and thus placing architectural thought within the a broader context of Western intellectual history. This new edition of Peter Collins's ground-breaking study includes all seventy-two illustrations of the original hard cover edition, which has been out of print since 1967, and restores the large format.

Peter Adamson presents the first full history of philosophy in the Islamic world for a broad readership. He traces its development from early Islam to the 20th century, ranging from Spain to South Asia, featuring Jewish and Christian thinkers as well as Muslim. Major figures like Avicenna, Averroes, and Maimonides are covered in great detail, but the book also looks at less familiar thinkers, including women philosophers. Attention is also given to the philosophical relevance of Islamic theology and mysticism--the (...) Sufi tradition within Islam, and Kabbalah among Jews--and to science, with chapters on disciplines like optics and astronomy. The first part of the book looks at the blossoming of Islamic theology and responses to the Greek philosophical tradition in the world of Arabic learning, the second discusses philosophy in Muslim Spain, and a third section looks in unusual detail at later developments, touching on philosophy in the Ottoman, Mughal, and Safavid empires. (shrink)

Peter Adamson presents a lively introduction to six hundred years of European philosophy, from the beginning of the ninth century to the end of the fourteenth century. The medieval period is one of the richest in the history of philosophy, yet one of the least widely known. Adamson introduces us to some of the greatest thinkers of the Western intellectual tradition, including Peter Abelard, Anselm of Canterbury, Thomas Aquinas, John Duns Scotus, William of Ockham, and Roger Bacon. And (...) the medieval period was notable for the emergence of great women thinkers, including Hildegard of Bingen, Marguerite Porete, and Julian of Norwich. Original ideas and arguments were developed in every branch of philosophy during this period - not just philosophy of religion and theology, but metaphysics, philosophy of logic and language, moral and political theory, psychology, and the foundations of mathematics and natural science. (shrink)

School leaders are optimally positioned to identify teachers who would make strong and inspiring administrators. Today's Teachers, Tomorrow's Leaders: A Guide to Identifying and Developing Future Administrators by Peter Marshall guides leaders on how to spot teachers with the potential to lead and how to model the skills and behaviors that will make them successful. Marshall describes ten Ways of Being of the impactful school leader and presents his Situational Leadership Model to help leaders identify teachers who exhibit strong (...) leadership qualities and to inspire them to take the leap. Thoroughly researched with real-life applications, Today's Teachers, Tomorrow's Leaders is a practical guide full of sample meeting planners, planning pages, and other useful resources to streamline the selection process. (shrink)

This book explores the political, economic, social, and environmental health relations and politics of the global tech and electronics industry. Peter Little argues that, in the digital age, we need greater synthesis of political ecology, ethnography, and technocapital critique.

The mystical and spiritual authors of the thirteenth to eighteenth centuries provided rich inspiration for Kierkegaard's religious thought. Kierkegaard owned numerous works by these authors, who are associated with the spiritual traditions of Rheno‐Flemish mysticism, Devotio Moderna, post‐Tridentine and Baroque Catholicism, and Reformed Pietism. The accurate spiritual diagnostics and the apt methods of spiritual formation found in (Pseudo‐)Tauler, Theologia Deutsch, Abraham a Sancta Clara, and François Fénelon deeply impressed Kierkegaard. He adopted and further developed motifs from the mystical and spiritual (...) authors in a variety of his writings, and his journals contain abundant reflections on themes from Thomas à Kempis, Ludovicus Blosius, and Gerhard Tersteegen, such as the imitation of Christ, spiritual guidance, or divine pedagogy. The best testimony to the importance of spiritual literature for Kierkegaard is the fact that he sought counsel in it in a personal crisis when he was deciding on the future of his religious authorship. (shrink)

In this paper, I provide an explication and defense of Merleau-Ponty’s theory of concept formation. I argue that at the core of this theory is a distinction between concepts proper and the kinds of generalities characteristic of perceptual experience, which I call “pre-conceptual generalities.” According to Merleau-Ponty, concepts are developed through a two-stage process: first, the establishment of such pre-conceptual generalities, and second, the clarification of these generalities into concepts. I provide phenomenological evidence for the existence of pre-conceptual generalities and (...) explain how they can serve to ground concept formation. To motivate Merleau-Ponty’s account, I consider objections raised by Husserl, Cassirer, and Sellars to the classical philosophical account of empirical concept formation, abstraction. I show that Merleau-Ponty’s distinction between concepts and pre-conceptual generalities provides a philosophically viable account of the formation of empirical concepts while avoiding the problems faced by accounts that rely on abstraction. (shrink)

In three exemplary essays, author Peter Bornedal promotes Deconstruction as a cogent analytical method whose distinctive critical object is foundational knowledge. In this, he wants to restore Deconstruction as a rational discourse, while continuing to emphasize it as a critique of metaphysics.