Kurt Gdel was the most outstanding logician of the 20th century and a giant in the field. This book is part of a five volume set that makes available all of Gdel's writings. The first three volumes, already published, consist of the papers and essays of Gdel. The final two volumes of the set deal with Gdel's correspondence with his contemporary mathematicians, this fourth volume consists of material from correspondents from A-G.

If mathematics is regarded as a science, then the philosophy of mathematics can be regarded as a branch of the philosophy of science, next to disciplines such as the philosophy of physics and the philosophy of biology. However, because of its subject matter, the philosophy of mathematics occupies a special place in the philosophy of science. Whereas the natural sciences investigate entities that are located in space and time, it is not at all obvious that this is (...) also the case with respect to the objects that are studied in mathematics. In addition to that, the methods of investigation of mathematics differ markedly from the methods of investigation in the natural sciences. Whereas the latter acquire general knowledge using inductive methods, mathematical knowledge appears to be acquired in a different way, namely, by deduction from basic principles. The status of mathematical knowledge also appears to differ from the status of knowledge in the natural sciences. The theories of the natural sciences appear to be less certain and more open to revision than mathematical theories. For these reasons mathematics poses problems of a quite distinctive kind for philosophy. Therefore philosophers have accorded special attention to ontological and epistemological questions concerning mathematics. (shrink)

Platonism about mathematics (or mathematical platonism) is the metaphysical view that there are abstract mathematical objects whose existence is independent of us and our language, thought, and practices. In this survey article, the view is clarified and distinguished from some related views, and arguments for and against the view are discussed.

Since their inception, the Perspectives in Logic and Lecture Notes in Logic series have published seminal works by leading logicians. Many of the original books in the series have been unavailable for years, but they are now in print once again. This volume, the fifteenth publication in the Lecture Notes in Logic series, collects papers presented at the symposium 'Reflections on the Foundations of Mathematics' held in celebration of Solomon Feferman's 70th birthday (The 'Feferfest') at Stanford University, California (...) in 1988. Feferman has shaped the field of foundational research for nearly half a century. These papers reflect his broad interests as well as his approach to foundational research, which emphasizes the solution of mathematical and philosophical problems. There are four sections, covering proof theoretic analysis, logic and computation, applicative and self-applicative theories, and philosophy of modern mathematical and logic thought. (shrink)

Mathematical fictionalism (or as I'll call it, fictionalism) is best thought of as a reaction to mathematical platonism. Platonism is the view that (a) there exist abstract mathematical objects (i.e., nonspatiotemporal mathematical objects), and (b) our mathematical sentences and theories provide true descriptions of such objects. So, for instance, on the platonist view, the sentence ‘3 is prime’ provides a straightforward description of a certain object—namely, the number 3—in much the same way that the sentence ‘Mars is red’ provides a (...) description of Mars. But whereas Mars is a physical object, the number 3 is (according to platonism) an abstract object. And abstract objects, platonists tell us, are wholly nonphysical, nonmental, nonspatial, nontemporal, and noncausal. Thus, on this view, the number 3 exists independently of us and our thinking, but it does not exist in space or time, it is not a physical or mental object, and it does not enter into causal relations with other objects. This view has been endorsed by Plato, Frege (1884, 1893-1903, 1919), Gödel (1964), and in some of their writings, Russell (1912) and Quine (1948, 1951), not to mention numerous more recent philosophers of mathematics, e.g., Putnam (1971), Parsons (1971), Steiner (1975), Resnik (1997), Shapiro (1997), Hale (1987), Wright (1983), Katz (1998), Zalta (1988), and Colyvan (2001). (shrink)

Contemporary philosophy’s three main naturalisms are methodological, ontological and epistemological. Methodological naturalism states that the only authoritative standards are those of science. Ontological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural. In philosophy of mathematics of the past few decades methodological naturalism has received the lion’s share of the attention, so we concentrate on this. Ontological and epistemological naturalism in the philosophy of mathematics are discussed more briefly (...) in section 6. (shrink)

Platonism about mathematics (or mathematical platonism) isthe metaphysical view that there are abstract mathematical objectswhose existence is independent of us and our language, thought, andpractices. Just as electrons and planets exist independently of us, sodo numbers and sets. And just as statements about electrons and planetsare made true or false by the objects with which they are concerned andthese objects' perfectly objective properties, so are statements aboutnumbers and sets. Mathematical truths are therefore discovered, notinvented., Existence. There are mathematical objects.

In this paper, I will discuss the need for a theory of essences within Object-Oriented Ontology (OOO) and then formulate one. I will do so by drawing on Graham Harman’s work on OOO and Martin Heidegger’s thought on the essence of being, presented in his Introduction to Metaphysics. Harman touches on essences, describing them as the tension between a withdrawn object and its withdrawn qualities, but fails to distinguish between essential and inessential qualities within this framework. To fill in the (...) gaps, I will turn to Heidegger’s explication of phusis in order to show that an essential aspect of being is how one enters into causal relations and continually reveals oneself to other beings. In bringing OOO and Heidegger together, I will find that each object has a unique way of exerting itself in the world and that the domestic relations that make up this unique profile are essential to it, while other domestic relations, those that do not influence its particular way of exerting itself, are inessential. Thus, the essence will be found to be the set of domestic relations that make up the determinate form, or unique causal profile, of the object. (shrink)

The incredible achievements of modern scientific theories lead most of us to embrace scientific realism: the view that our best theories offer us at least roughly accurate descriptions of otherwise inaccessible parts of the world like genes, atoms, and the big bang. In Exceeding Our Grasp, Stanford argues that careful attention to the history of scientific investigation invites a challenge to this view that is not well represented in contemporary debates about the nature of the scientific enterprise. The historical (...) record of scientific inquiry, Stanford suggests, is characterized by what he calls the problem of unconceived alternatives. Past scientists have routinely failed even to conceive of alternatives to their own theories and lines of theoretical investigation, alternatives that were both well-confirmed by the evidence available at the time and sufficiently serious as to be ultimately accepted by later scientific communities. Stanford supports this claim with a detailed investigation of the mid-to-late 19th century theories of inheritance and generation proposed in turn by Charles Darwin, Francis Galton, and August Weismann. He goes on to argue that this historical pattern strongly suggests that there are equally well-confirmed and scientifically serious alternatives to our own best theories that remain currently unconceived. Moreover, this challenge is more serious than those rooted in either the so-called pessimistic induction or the underdetermination of theories by evidence, in part because existing realist responses to these latter challenges offer no relief from the problem of unconceived alternatives itself. Stanford concludes by investigating what positive account of the spectacularly successful edifice of modern theoretical science remains open to us if we accept that our best scientific theories are powerful conceptual tools for accomplishing our practical goals, but abandon the view that the descriptions of the world around us that they offer are therefore even probably or approximately true. (shrink)

Scientific realists have suggested that changes in our scientific communities over the course of their history have rendered those communities progressively less vulnerable to the problem of unconcieved alternatives over time. I argue in response not only that the most fundamental historical transformations of the scientific enterprise have generated steadily mounting obstacles to revolutionary, transformative, or unorthodox scientific theorizing, but also that we have substantial independent evidence that the institutional apparatus of contemporary scientific inquiry fosters an exceedingly and increasingly theoretically (...) conservative form of that inquiry. I conclude that contemporary scientific communities are actually more vulnerable to the problem of unconceived alternatives than their historical predecessors, and I briefly suggest how we might seek to pursue scientific inquiry in a less theoretically conservative way. (shrink)

Scientific realists have suggested that changes in our scientific communities over the course of their history have rendered those communities progressively less vulnerable to the problem of unconcieved alternatives over time. I argue in response not only that the most fundamental historical transformations of the scientific enterprise have generated steadily mounting obstacles to revolutionary, transformative, or unorthodox scientific theorizing, but also that we have substantial independent evidence that the institutional apparatus of contemporary scientific inquiry fosters an exceedingly and increasingly theoretically (...) conservative form of that inquiry. I conclude that contemporary scientific communities are actually more vulnerable to the problem of unconceived alternatives than their historical predecessors, and I briefly suggest how we might seek to pursue scientific inquiry in a less theoretically conservative way. (shrink)

The historical record of scientific inquiry, Stanford suggests, is characterized by what he calls the problem of unconceived alternatives. Past scientists have routinely failed even to conceive of alternatives to their own theories and lines of theoretical investigation, alternatives that were both well-confirmed by the evidence available at the time and sufficiently serious as to be ultimately accepted by later scientific communities. Stanford supports this claim with a detailed investigation of the mid-to-late 19th century theories of inheritance and (...) generation proposed in turn by Charles Darwin, Francis Galton, and August Weismann. He goes on to argue that this historical pattern strongly suggests that there are equally well-confirmed and scientifically serious alternatives to our own best theories that remain currently unconceived. Moreover, this challenge is more serious than those rooted in either the so-called pessimistic induction or the underdetermination of theories by evidence, in part because existing realist responses to these latter challenges offer no relief from the problem of unconceived alternatives itself.Stanford concludes by investigating what positive account of the spectacularly successful edifice of modern theoretical science remains open to us if we accept that our best scientific theories are powerful conceptual tools for accomplishing our practical goals, but abandon the view that the descriptions of the world around us that they offer are therefore even probably or approximately true."Stanford has genuinely advanced the philosophical discussion about scientific realism with his careful articulation of the problem of unconceived alternatives."-- The Review of Metaphysics "Stanford's book deserves to be widely read. Its central argument is clearly stated, its conclusion is radical, it engages in a productive fashion with detailed case studies, and it lays down several substantial challenges to scientific realism. Lastly, it is consistently thought-provoking."-Science. (shrink)

I develop an account of predictive similarity that allows even Antirealists who accept a correspondence conception of truth to answer the Realist demand (recently given sophisticated reformulations by Musgrave and Leplin) to explain the success of particular scientific theories by appeal to some intrinsic feature of those theories (notwithstanding the failure of past efforts by van Fraassen, Fine, and Laudan). I conclude by arguing that we have no reason to find truth a better (i.e., more plausible) explanation of a theory's (...) success than predictive similarity, even of its success in making novel predictions. (shrink)

In the concluding chapter of Exceeding our Grasp Kyle Stanford outlines a positive response to the central issue raised brilliantly by his book, the problem of unconceived alternatives. This response, called "epistemic instrumentalism", relies on a distinction between instrumental and literal belief. We examine this distinction and with it the viability of Stanford's instrumentalism, which may well be another case of exceeding our grasp.

One of the most intriguing features of mathematics is its applicability to empirical science. Every branch of science draws upon large and often diverse portions of mathematics, from the use of Hilbert spaces in quantum mechanics to the use of differential geometry in general relativity. It's not just the physical sciences that avail themselves of the services of mathematics either. Biology, for instance, makes extensive use of difference equations and statistics. The roles mathematics plays in these (...) theories is also varied. Not only does mathematics help with empirical predictions, it allows elegant and economical statement of many theories. Indeed, so important is the language of mathematics to science, that it is hard to imagine how theories such as quantum mechanics and general relativity could even be stated without employing a substantial amount of mathematics. (shrink)

Deductivism says that a mathematical sentence s should be understood as expressing the claim that s deductively follows from appropriate axioms. For instance, deductivists might construe “2+2=4” as “the sentence ‘2+2=4’ deductively follows from the axioms of arithmetic”. Deductivism promises a number of benefits. It captures the fairly common idea that mathematics is about “what can be deduced from the axioms”; it avoids an ontology of abstract mathematical objects; and it maintains that our access to mathematical truths requires nothing (...) beyond our ability to make logical deductions. Sections 1 and 2 define and motivate deductivism in more detail. Section 3 covers four authors (Russell, Hilbert, Pasch, Curry) who have endorsed deductivism at some point. Section 4 aims to clarify what semantic claim deductivists make. Sections 5–9 review objections to deductivism and possible replies. (shrink)

I argue for accepting a pluralist approach to species, while rejecting the realism about species espoused by P. Kitcher and a number of other philosophers of biology. I develop an alternative view of species concepts as divisions of organisms into groups for study which are relative to the systematic explanatory interests of biologists at a particular time. I also show how this conception resolves a number of difficult puzzles which plague the application of particular species concepts.

Evidence is presented to support the hypothesis that there is a set of basically similar phenomena or characteristics of physics, biology, and sociology. Six of these are identified. Five of them are usually associated with quantum mechanics. They are the existence of eigenstates, transform domains, bosons and fermions, particles and antiparticles, and complementarity. The sixth, namely alternation of generation, is usually associated with biology. The hypothesis leads to some new points of view and interpretations in biology, sociology, and physics.

The cell theory of Schleiden and Schwann is generalized to the effect that throughout the natural world, in physics, biology, and sociopsychology, there is a widespread phenomenon of the existence of organized cells, whose organization is usually protected by barriers. These barriers exist not only in space, but in time and even in other domains. These barriers typically not only protect the organization within the cell from external disturbance, but they actively participate in reducing the internal disorganization. It appears that (...) the quantum phase cell or its equivalent, the discrete quantum state, may be a physical analog of the organized biological cell. The sampling and ambiguity theorems of communication theory throw new light on phase cells. The quantum phase cell has both configuration and momentum elements, which are analogous respectively to the somatic elements in the cytoplasm of a cell and the genetic elements in the DNA. A biological cell or organism, a physical quantum state, and a human being as a sociopsychological entity each has a natural individuality. A natural individuality is a somewhat mysterious thing, since in addition to specificity, it also involves stochasticity and complementarity. There are some striking and unexplained spin-analog characteristics in biology which have a relation to boson- and fermion-analog characteristics in biology that corresponds with the known relation in physics. The hypothesis suggested in our earlier paper, that quantum mechanical entropy is diminished in interaction, is clarified and sharpened in this paper. (shrink)

Advocates have sought to prove that underdetermination obtains because all theories have empirical equivalents. But algorithms for generating empirical equivalents simply exchange underdetermination for familiar philosophical chestnuts, while the few convincing examples of empirical equivalents will not support the desired sweeping conclusions. Nonetheless, underdetermination does not depend on empirical equivalents: our warrant for current theories is equally undermined by presently unconceived alternatives as well-confirmed merely by the existing evidence, so long as this transient predicament recurs for each theory and body (...) of evidence we consider. The historical record supports the claim that this recurrent, transient underdetermination predicament is our own. (shrink)

Advocates have sought to prove that underdetermination obtains because all theories have empirical equivalents. But algorithms for generating empirical equivalents simply exchange underdetermination for familiar philosophical chestnuts, while the few convincing examples of empirical equivalents will not support the desired sweeping conclusions. Nonetheless, underdetermination does not depend on empirical equivalents: our warrant for current theories is equally undermined by presently unconceived alternatives as well-confirmed merely by the existing evidence, so long as this transient predicament recurs for each theory and body (...) of evidence we consider. The historical record supports the claim that this recurrent, transient underdetermination predicament is our own. (shrink)

Most commonly, the scientific realism debate is seen as dividing those who do and do not think that the striking empirical and practical successes of at least our best scientific theories indicate with high probability that those theories are ‘approximately true’. But I want to suggest that this characterization of the debate has far outlived its usefulness. Not only does it obscure the central differences between two profoundly different types of contemporary scientific realist, but even more importantly it serves to (...) disguise the most substantial points of actual disagreement between these two kinds of realists and those who instead think the historical record of scientific inquiry itself reveals that such realism is untenable in either form. (shrink)

A range of empirical findings is first used to more precisely characterize our distinctive tendency to objectify or externalize moral demands and obligations, and it is then argued that this salient feature of our moral cognition represents a profound puzzle for evolutionary approaches to human moral psychology that existing proposals do not help resolve. It is then proposed that such externalization facilitated a broader shift to a vastly more cooperative form of social life by establishing and maintaining a connection between (...) the extent to which an agent is herself motivated by a given moral norm and the extent to which she uses conformity to that same norm as a criterion in evaluating candidate partners in social interaction generally. This connection ensures the correlated interaction necessary to protect those prepared to adopt increasingly cooperative, altruistic, and other prosocial norms of interaction from exploitation, especially as such norms were applied in novel ways and/or to novel circumstances and as the rapid establishment of new norms allowed us to reap still greater rewards from hypercooperation. A wide range of empirical findings is then used to support this hypothesis, showing why the status we ascribe to moral demands and considerations exhibits the otherwise puzzling combination of objective and subjective elements that it does, as well as showing how the need to effectively advertise our externalization of particular moral commitments generates features of our social interaction so familiar that they rarely strike us as standing in need of any explanation in the first place. (shrink)

This is a further development of the author's paper “A Unified Theory of Biology and Physics.” It is found that male and female in biology, as well as particle and antiparticle in physics, are analogues of symmetrical sideband pairs in communication theory. This gives a new point of view from which to investigate the significance and characteristics of these different paired entities.These findings are intimately related to the fact that there are two transform domains of representation of entities in all (...) the cases involved. They are the somatic and the genetic domains in biology, the configuration domain and the domain of conserved observables in physics, and the time and frequency domains in communication. (shrink)

Critics who use historical evidence to challenge scientific realism have deployed a perfectly natural argumentative strategy that has created a profoundly misguided conception of what would be required to vindicate that challenge. I argue that the question fundamentally in dispute in such debates is neither whether particular terms in contemporary scientific theories will be treated as referential nor whether particular existential commitments will be held true by future scientific communities, but whether the future of science will exhibit the same broad (...) pattern of repeated, profound, and unpredictable changes in theoretical orthodoxy that such historicist critics argue characterizes its past. (shrink)

Realists have responded to challenges from the historical record of successful but ultimately rejected theories with what I call the selective confirmation strategy: arguing that only idle parts of past theories have been rejected, while truly success‐generating features have been confirmed by further inquiry. I argue first, that this strategy is unconvincing without some prospectively applicable criterion of idleness for theoretical posits, and second, that existing efforts to provide one either convict all theoretical posits of idleness (Kitcher) or stand refuted (...) by detailed consideration of the very examples (optical/electromagnetic ether, caloric fluid) to which they appeal (Psillos). I also argue that available avenues for improving on these proposals are unpromising. (shrink)

Reply in Book Symposium on James Ladyman, Don Ross: 'Everything must go: metaphysics naturalized', Oxford: Oxford University Press, 2007.

Genic selectionism holds that all selection can be understood as operating on particular genes. Critics (and conventional biological wisdom) insist that this misrepresents the actual causal structure of selective phenomena at higher levels of biological organization, but cannot convincingly defend this intuition. I argue that the real failing of genic selectionism is pragmatic – it prevents us from adopting the most efficient corpus of causal laws for predicting and intervening in the course of affairs – and I offer a Pragmatic (...) account of causation itself which ultimately bears out the claim that genic selectionism misrepresents the causal structure of selective contexts. (shrink)

Some scientific realists suggest that scientific communities have improved in their ability to discover alternative theoretical possibilities and that the problem of unconceived alternatives therefore poses a less significant threat to contemporary scientific communities than it did to their historical predecessors. I first argue that the most profound and fundamental historical transformations of the scientific enterprise have actually increased rather than decreased our vulnerability to the problem. I then argue that whether we are troubled by even the prospect of increasing (...) theoretical conservatism in science should depend on the position we occupy in the ongoing debate concerning scientific realism itself. (shrink)

Fish, S. Georgics of the mind: Bacon's philosophy and the experience of his Essays.--Brett, R. L. Thomas Hobbes.--Watt, I. Realism and the novel.--Tuveson, E. Locke and Sterne.--Kampf, L. Gibbon and Hume.--Frye, N. Blake's case against Locke.--Abrams, M. H. Mechanical and organic psychologies of literary invention.--Ryle, G. Jane Austen and the moralists.--Schneewind, J. B. Moral problems and moral philosophy in the Victorian period.--Donagan, A. Victorian philosophical prose: J. S. Mill and F. H. Bradley.--Pitcher, G. Wittgenstein, nonsense, and Lewis Carroll.--Bolgan, A. C. (...) The philosophy of F. H. Bradley and the mind and art of T. S. Eliot: an introduction.--Davie, D. Yeats, Berkeley, and Romanticism.--Ross, M. L. The mythology of friendship: D. H. Lawrence, Bertrand Russell, and "The Blind man".--Rosenbaum, S. P. The philosophical realism of Virginia Woolf.--Bibliography (p. 357-360). (shrink)

When Stanford M. Lyman authored The Seven Deadly Sins: Society and Evil in 1978 it was hailed by Alasdair MacIntyre as 'a book of absorbing interest and importance_[that] places us all in his debt.' By Nelson Hart as 'a masterful and thought-provoking book_[that] is the only scholarly treatment of sin that is so well-informed by the best of ancient through modern perspectives.' By James A. Aho as a work whose 'abstract hardly does justice to the scholarly and detailed analysis (...) of sin.' And by Harry Cohen as a 'book_[that] stands as a beautiful illustration of what holistic, idiosyncratic, interdisciplinary, and creative thinking and writing can bring to bear on the age-old problem of society and evil.' The American Sociological Association's section on the Sociology of the Emotions selected this book as one of the works that laid the foundations for the study of pride, lust, envy, and anger—basic sentiments embedded in the social process. For this revised and expanded edition Lyman has written a new chapter, 'Sentiments, Sin, and Social Conflict: Toward a Sociology of the Emotions.' The new edition will be a valuable work for courses in social psychology, ethics, deviance, and the sociology of morals and of religion. (shrink)

I respond to the many thoughtful suggestions and concerns of my commentators on a wide variety of questions. These include whether moral norms form a unified category, whether they have a distinctive phenomenology, and/or whether moral normativity is a cultural construct; whether moral externalization is necessary for correlated interaction or human prosociality; precisely how such externalization generates correlated interactions among prosocial agents; and whether there are any convincing alternative explanations for it.

We present a theoretical and an empirical challenge to Cushman's claim that rationalization is adaptive because it allows humans to extract more accurate beliefs from our non-rational motivations for behavior. Rationalization sometimes generates more adaptive decisions by making our beliefs about the world less accurate. We suggest that the most important adaptive advantage of rationalization is instead that it increases our predictability as potential partners in cooperative social interactions.