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)
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) 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.
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)
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) 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.
Kurt Gödel (1906 - 1978) was the most outstanding logician of the twentieth century, famous for his hallmark works on the completeness of logic, the incompleteness of number theory, and the consistency of the axiom of choice and the continuum hypothesis. He is also noted for his work on constructivity, the decision problem, and the foundations of computability theory, as well as for the strong individuality of his writings on the philosophy of mathematics. He is less well known for his (...) discovery of unusual cosmological models for Einstein's equations, in theory permitting time travel into the past. The Collected Works is a landmark resource that draws together a lifetime of creative thought and accomplishment. The first two volumes were devoted to Gödel's publications in full (both in original and translation), and the third volume featured a wide selection of unpublished articles and lecture texts found in Gödel's Nachlass. These long-awaited final two volumes contain Gödel's correspondence of logical, philosophical, and scientific interest. Volume IV covers A to G, with H to Z in volume V; in addition, Volume V contains a full inventory of Gödel's Nachlass. L All volumes include introductory notes that provide extensive explanatory and historical commentary on each body of work, English translations of material originally written in German (some transcribed from the Gabelsberger shorthand), and a complete bibliography of all works cited. Kurt Gödel: Collected Works is designed to be useful and accessible to as wide an audience as possible without sacrificing scientific or historical accuracy. The only comprehensive edition of Gödel's work available, it will be an essential part of the working library of professionals and students in logic, mathematics, philosophy, history of science, and computer science and all others who wish to be acquainted with one of the great minds of the twentieth century. (shrink)
P. Kyle Stanford (2000) attempts to offer a truth-linked explanation of the success of science which, he thinks, can be welcome to antirealists. He proposes an explanation of the success of a theory T1 in terms of its predictive similarity to the true theory T of the relevant domain. After raising some qualms about the supposed antirealist credentials of Stanford's account, I examine his explanatory story in some detail and show that it fails to offer a satisfactory explanation (...) of the success of science. (shrink)
The prevailing pedagogical approach in business ethics generally underestimates or even ignores the powerful influences of situational factors on ethical analysis and decision-making. This is due largely to the predominance of philosophy-oriented teaching materials. Social psychology offers relevant concepts and experiments that can broaden pedagogy to help students understand more fully the influence of situational contexts and role expectations in ethical analysis. Zimbardo's Stanford Prison Experiment is used to illustrate the relevance of social psychology experiments for business ethics instruction.
In this short letter to Ed Zalta we raise a number of issues with regards to his version of Neo-Logicism. The letter is, in parts, based on a longer manuscript entitled “What Neo-Logicism could not be” which is in preparation. A response by Ed Zalta to our letter can be found on his website: http://mally.stanford.edu/publications.html (entry C3).
Notice: This PDF version was distributed by request to members of the Friends of the SEP Society and by courtesy to SEP content contributors. It is solely for their fair use. Unauthorized distribution is prohibited. To learn how to join the Friends of the SEP Society and obtain authorized PDF versions of SEP entries, please visit https://leibniz.stanford.edu/friends/.
A comparison of the engineering schools at UC Berkeley and Stanford during the 1940s and 1950s shows that having an excellent academic program is necessary but not sufficient to make a university entrepreneurial (an engine of economic development). Key factors that made Stanford more entrepreneurial than Cal during this period were superior leadership and a focused strategy. The broader institutional context mattered as well. Stanford did not have the same access to state funding as (...) public universities (such as Cal in the period under consideration) and some private universities (such as the Massachusetts Institute of Technology and the Johns Hopkins University in their early histories). Therefore, in order to gather resources, Stanford was forced to become entrepreneurial first, developing business skills (engaging with high-tech industry) at the same time Cal was developing political skills (protecting and increasing its state appropriation). Stanford’s early development of entrepreneurial business skills played a crucial role in the development of Silicon Valley. (shrink)
The origin of my article lies in the appearance of Copeland and Proudfoot's feature article in Scientific American, April 1999. This preposterous paper, as described on another page, suggested that Turing was the prophet of 'hypercomputation'. In their references, the authors listed Copeland's entry on 'The Church-Turing thesis' in the Stanford Encyclopedia. In the summer of 1999, I circulated an open letter criticising the Scientific American article. I included criticism of this Encyclopedia entry. This was forwarded (by Prof. Sol (...) Feferman) to Prof. Ed Zalta, editor of the Encyclopedia, and after some discussion he invited me to submit an entry on 'Alan Turing.'. (shrink)
The Stanford Encyclopedia of Philosophy is an open access, dynamic reference work designed to organize professional philosophers so that they can write, edit, and maintain a reference work in philosophy that is responsive to new research. From its inception, the SEP was designed so that each entry is maintained and kept up to date by an expert or group of experts in the field. All entries and substantive updates are refereed by the members of a distinguished Editorial Board before (...) they are made public. (shrink)
It is widely argued that, in the United States, the Department of Defense dictated the intellectual contours of academic science and engineering during the Cold War. However, in important ways, American science was also deeply influenced by industry. Between 1955 and 1985, Stanford University embraced three waves of industrial innovation in solid state technology (transistors, integrated circuits, and VLSI systems). As this essay shows, it was these transfers that enabled Stanford engineers to make significant contributions to the expanding (...) fields of microelectronics and computing. (shrink)
Feelings and experiences vary widely. For example, I run my fingers over sandpaper, smell a skunk, feel a sharp pain in my finger, seem to see bright purple, become extremely angry. In each of these cases, I am the subject of a mental state with a very distinctive subjective character. There is something it is like for me to undergo each state, some phenomenology that it has. Philosophers often use the term ‘qualia’ (singular ‘quale’) to refer to the introspectively accessible, (...) phenomenal aspects of our mental lives. In this standard, broad sense of the term, it is difficult to deny that there are qualia. Disagreement typically centers on which mental states have qualia, whether qualia are intrinsic qualities of their bearers, and how qualia relate to the physical world both inside and outside the head. The status of qualia is hotly debated in philosophy largely because it is central to a proper understanding of the nature of consciousness. Qualia are at the very heart of the mindbody problem. (shrink)
Notice: This PDF version was distributed by request to members of the Friends of the SEP Society and by courtesy to SEP content contributors. It is solely for their fair use. Unauthorized distribution is prohibited. To learn how to join the Friends of the..
One of the many virtues of Martin Seel’s Aesthetics of Appearing is that it lays its cards on the table at the very outset. The final three chapters consist in a series of complex digressions from the main discussion: one on the aesthetic significance of ‘resonating’(p. 139), one organized around the metaphysics of pictures, and one charged with defending the implausible claim that the artistic representation of violence is uniquely capable of revealing ‘what is violent about violence’ (p. 191). But (...) the thesis of the book and its main arguments are stated in the preface, preceding even the acknowledgements. Seel writes, ‘[t]his book makes the proposal of having aesthetics begin not with concepts of being‐so or semblance but with a concept of appearing’ (p. xi). This might initially seem opaque, as though reducing aesthetics to a subtlety involving the meaning of the Greek word phainomai, but Seel immediately clarifies the stance that he wishes to advance. Seel’s position is that aesthetics is distinguished by attention to the indeterminable particularity of sensory experience; aesthetics so considered comprises the philosophy of art as well as non‐art experience; aesthetic experience is a legitimate mode of world‐encounter by virtue of its immediacy or ‘presence’ (p. xi)(Gegenwart – i.e., a contrary of ‘past’, rather than of ‘absence’); and because the presence of our experience reveals the presence of our lives, aesthetic experience constitutes an important form of self‐knowledge. The subsequent chapters are devoted to explicating this position in extraordinary detail. Seel’s position depends on a somewhat implicit account of subjectivity. In this account, what we fundamentally perceive, conditioned by conceptual activity but transcending any possible determinate content, is a ‘play’ of sensuous qualities (p. 47). Since it is ‘unfettered’ (p. 51) by theoretical interest, this form of perception is far qualitatively richer than our more structured experiences: here one is ‘able to perceive.... (shrink)
This book introduces a new approach to the issue of radical scientific revolutions, or "paradigm-shifts," given prominence in the work of Thomas Kuhn. The book articulates a dynamical and historicized version of the conception of scientific a priori principles first developed by the philosopher Immanuel Kant. This approach defends the Enlightenment ideal of scientific objectivity and universality while simultaneously doing justice to the revolutionary changes within the sciences that have since undermined Kant's original defense of this ideal. Through a modified (...) Kantian approach to epistemology and philosophy of science, this book opposes both Quinean naturalistic holism and the post-Kuhnian conceptual relativism that has dominated recent literature in science studies. Focussing on the development of "scientific philosophy" from Kant to Rudolf Carnap, along with the parallel developments taking place in the sciences during the same period, the author articulates a new dynamical conception of relativized a priori principles. This idea applied within the physical sciences aims to show that rational intersubjective consensus is intricately preserved across radical scientific revolutions or "paradigm-shifts and how this is achieved. (shrink)
Logical AI involves representing knowledge of an agent’s world, its goals and the current situation by sentences in logic. The agent decides what to do by inferring that a certain action or course of action is appropriate to achieve the goals. We characterize brieﬂy a large number of concepts that have arisen in research in logical AI. Reaching human-level AI requires programs that deal with the common sense informatic situation. This in turn requires extensions from the way logic has been (...) used in formalizing branches of mathematics and physical science. It also seems to require extensions to the logics themselves, both in the formalism for expressing knowledge and the reasoning used to reach conclusions. A large number of concepts need to be studied to achieve logical AI of human level. This article presents candidates. The references, though numerous, to articles concerning these concepts are still insuf- ﬁcient, and I’ll be grateful for more, especially for papers available on the web. This article is available in several forms via http://www-formal.stanford.edu/jmc/conceptsai.html. (shrink)
The story goes that Epimenides, a Cretan, used to claim that all Cretans are always liars. Whether he knew it or not, this claim is odd. It is easy to see it is odd by asking if it is true or false. If it is true, then all Cretans, including Epimenides, are always liars, in which case what he said must be false. Thus, if what he says is true, it is false. Conversely, suppose what Epimenides said is false. Then (...) some Cretan at some time speaks truly. This might not tell us anything about Epimenides. But if, to make the story simple, he were the only Cretan ever to speak, and this was the only thing he ever said, then indeed, he would have to speak truly. And we would then have shown that if what he said was false, it must be true. (shrink)