Philosophy

"Already in the 1960s, first with Lakatos and later through a group of 'maverick' philosophers of mathematics (Kitcher, Tymoczko, and others), a strong reaction set in against the philosophy of mathematics conceived as foundation of mathematics. ...What these philosophers called for was an analysis of mathematics that was more faithful to its historical development."

"A characterization... of the main features of the maverick tradition could be..; a. antifoudationalism, i.e. there is no certain foundation... mathematics is... fallible... b. anti-logicism, i.e. mathematical logic cannot provide the tools for an adequate analysis of mathematics and its development; c. attention to mathematical practice: only detailed analysis and reconstruction of large and significant parts of mathematical practice can provide a philosophy of mathematics..."

"Philosophers and logicians have been so busy trying to provide mathematics with a "foundation" in the past half-century that only rarely have a few timid voices dared to voice the suggestion that it does not need one. I wish here to urge with some seriousness the view of the timid voices. I don't think mathematics is unclear; I don't think mathematics has a crisis in its foundations; indeed, I do not believe mathematics either has or needs "foundations." The much touted problems in the philosophy of mathematics seem to me, without exception, to be problems internal to the thought of various system builders. The systems are doubtless interesting as intelIectual exercises; debate between the systems and research within the systems doubtless will and should continue; but I would like to convince you (of course I won't, but one can always hope) that the various systems of mathematicaI philosophy, without exception, need not be taken seriously."

"By "philosophy of mathematics" I mean the specific set of concepts, categories, and theories employed, implicitly or explicitly, by philosophers and mathematicians in their discourse about mathematics. Understood in this way, philosophy of mathematics would include, among other things, some rather ethereal discussions on the nature of numbers by several hermetic philosophers, and the status of various notions, including number, space, infinity, according to the philosophers and mathematicians operating in the seventeenth century, as well as several other areas of investigation. Therefore I must introduce a qualification contained in the concept of "mathematical practice." ...I use this term as it is used today in mathematical logic and philosophy of mathematics, simply to indicate mathematics as it is done, not as it should be done according to some preconceived philosophical viewpoint. ...[F]ar from eliminating the philosophical questions, an interest in mathematical practice has actually extended their range. Addressing the issue of mathematical practice requires a detailed knowledge of the mathematical literature of the period."

"According to the dominant view, the reflection on mathematics is the task of a specialized discipline, the philosophy of mathematics, starting with Frege, characterized by its own problems and methods, and in a sense “the easiest part of philosophy”. In this view, the philosophy of mathematics “is a specialized area of philosophy... Many of the questions that arise within it... occur within the philosophy of mathematics in an especially pure, or especially simplified, form”. ...[However,] like applied mathematics, pure mathematics draws its concepts from experience, observation, scientific theories and even economics. The questions considered by the reflection on mathematics have, therefore, all the impurity and complexity of which philosophical problems are capable."

"[M]athematical apriorism... has not gone completely unquestioned. J. S. Mill attempted to argue that mathematics is an empirical science, thereby making himself the subject of Frege's biting criticism. More recently, W. V. Quine, , and Imre Lakatos have... challenged the... thesis. However, none of these have offered a systematic account of our mathematical knowledge. ...[T]he alternative...—mathematical empiricism—has never been given a detailed articulation. I shall try... I have gained much from insights of Quine and Putnam... [and] learned from Mill... My quarrel with earlier empiricists is, for the most part, that they have been incomplete rather than mistaken."

"Philosophy of mathematics appears to become a microcosm for the most general and central issues in philosophy—issues in epistemology, metaphysics, and philosophy of language—and the study of those parts of mathematics to which philosophers... most often attend (logic, set theory, aritmetic) seems designed to test the merits of large philosophical views about the existence of abstract entities of the tenability of a certain picture of human knowledge. ...[A]re [there] not other tasks ...that arise either from the current practice of mathematics or the history of the subject... the kinds of issues that occupy those who study the other branches of human knowledge... as: How does mathematical knowledge grow? What is mathematical progress? What makes some mathematical ideas (or theories) better than others? What is mathematical explanation?"

"The dialogue takes place... The class gets interested in a Problem: Is there a relation between the number of vertices V, the number of edges E and the number of faces F of a polyhedra—particularly of regular polyhedra—analogous to the trivial relation between the number of vertices and edges of polygons, namely, that there are as many edges as vertices: V = E? ...After much tiral and error they notice that for all regular polyhedra V - E + F = 2. Somebody guesses that this may apply for any polyhedron whatever. Others try to falsify [test] this conjecture... it holds good. The results corroborate the conjecture, and suggest that it could be proved. It is at this point—after the stages problem and conjecture—that we... offer a proof."

"A proof of a mathematical theorem is a sequence of steps which leads to the desired conclusion. The rules to be followed... were made explicit when logic was formalized early in the this century... These rules can be used to disprove a putative proof by spotting logical errors; they cannot, however, be used to find the missing proof of a... conjecture. ... arguments are a common occurrence in the practice of mathematics. However... The role of heuristic arguments has not been acknowledged in the philosophy of mathematics despite the crucial role they play in mathematical discovery. ...Our purpose is to bring out some of the features of mathematical thinking which are concealed beneath the apparent mechanics of proof."

"We are still in the aftermath of the great foundationist controversies of the early twentieth century. Formalism, and , each left its trace in the form of a certain mathematical research program that ultimately made its own contribution to the corpus of mathematics... As philosophical programs, as attempts to establish a secure foundation for mathematical knowledge, all have run their course and petered out or dried up. Yet there remains, as a residue, an unstated consensus that the philosophy of mathematics is research on the foundations of mathematics. If I find [that] uninteresting or irrelevant, I conclude that I'm simply not interested in philosophy (thereby depriving myself of any chance of confronting my own uncertainties about the meaning, nature, purpose or significance of mathematical research)."

"Philosophy of mathematics has been slow to draw the analogy from the Kuhnian sea-change in philosophy of science, but during the last decade, a growing number of younger philosophers of mathematics have turned their attention to the history of mathematics and tried to make use of it in their investigations. The most exciting of these concern how mathematical discovery takes place, how new discoveries are structured and integrated into existing knowledge, and what light these processes shed on the existence and applicability of mathematical objects."

"The doctrine that mathematical knowledge is a priori—mathematical apriorism...—has been articulated in many different ways... To name only the most prominent defenders... since the seventeenth century, Descartes, Locke, Berkeley, Kant, Frege, Hilbert, Brouwer, and Carnap... Most of the disputes... conducted in our century represent internal differences... among apriorists. ...I shall offer a picture of mathematical knowledge which rejects mathematical apriorism."

"The political, ethical, social, philosophical problem of our day is not to try to liberate the individual from the state and from the state's institutions but to liberate us both from the state and from the type of individualization which is linked to the state. We have to promote new forms of subjectivity through the refusal of this kind of individuality which has been imposed on us for several centuries."

"All ideology hails or interpellates concrete individuals as concrete subjects, by the functioning of the category of the subject. ... ideology ‘acts’ or ‘functions’ in such a way that it ‘recruits’ subjects among the individuals (it recruits them all), or ‘transforms’ the individuals into subjects (it transforms them all) by that very precise operation which I have called interpellation or hailing, and which can be imagined along the lines of the most commonplace everyday police (or other) hailing: ‘Hey, you there!’"

"I am a subject that is a head, not a complement of a chicken that has no head. The subject has become the passive observer, the spectator, waiting for the curtains to fall like the guillotine. Artaud said theater can exist without the text—the production has to liberate itself from literature-and it did liberate itself—it became self-sufficient—it became a conjugation."

"Under capitalism, subjectivity can only exist antagonistically, in opposition to its own objectification. To treat the subject as already emancipated, as most mainstream theory does, is to endorse the present objectification of the subject as subjectivity, as freedom."

"All those movements which took place in the fifteenth and sixteenth centuries and which had the Reformation as their main expression and result should be analyzed as a great crisis of the Western experience of subjectivity and a revolt against the kind of religious and moral power which gave form, during the Middle Ages, to this subjectivity. The need to take a direct part in spiritual life, in the work of salvation, in the truth which lies in the Book—all that was a struggle for a new subjectivity."

"Subjectivity includes the possibility, for example, that some elements or impulses are subjectively active — they move us — without being consciously known.... It focuses on the "who I am" or, as important, the "who we are" of culture. ... Consciousness embraces the notion of a consciousness of self and an active mental and moral self-production."

"As an empiricist I continue to think of the conceptual scheme of science as a tool, ultimately, for predicting future experience in the light of past experience. Physical objects are conceptually imported into the situation as convenient intermediaries-not by definition in terms of experience, but simply as irreducible posits comparable, epistemologically, to the gods of Homer. For my part I do, qua lay physicist, believe in physical objects and not in Homer's gods; and I consider it a scientific error to believe otherwise. But in point of epistemological footing the physical objects and the gods differ only in degree and not in kind. Both sorts of entities enter our conception only as cultural posits. The myth of physical objects is epistemologically superior to most in that it has proved more efficacious than other myths as a device for working a manageable structure into the flux of experience."

"If I saw indirect explanatory benefit in positing sensibilia, possibilia, spirits, a Creator, I would joyfully accord them scientific status too, on a par with such avowedly scientific posits as quarks and black holes. What then have I banned under the name of prior philosophy?"

"What I call natural philosophy isn’t new, for it has been practiced in various ways by such distinguished philosophers as Thales, Aristotle, Epicurus, Lucretius, Bacon, Locke, Hume, Mill, Peirce, Russell (after 1920), Dewey, Quine (after 1950), and Kuhn. There are also many contemporary philosophers making progress on problems concerning the nature of knowledge, reality and ethics, without succumbing to the dogmas of analytic philosophy. Philosophy needs to be extraverted, directing its attention to real world problems and relevant scientific findings, not introverted and concerned only with its own history and techniques."

"Even postulating an unobserved Creator need be no more unscientific than postulating unobservable particles. What matters is the character of the proposals and the ways in which they are articulated and defended."

"In somewhat plainer English, what this means is this: if Carrier Naturalism (or CN) is true, then all minds, and all the contents and powers and effects of minds, are entirely caused by natural phenomena. But if naturalism is false, then some minds, or some of the contents or powers or effects of minds, are causally independent of nature. In other words, such things would then be partly or wholly caused by themselves, or exist or operate directly or fundamentally on their own."

"In short, I argue "naturalism" means, in the simplest terms, that every mental thing is entirely caused by fundamentally nonmental things, and is entirely dependent on nonmental things for its existence. Therefore, "supernaturalism" means that at least some mental things cannot be reduced to nonmental things."

"Methodological naturalism requires scientific theories to mention only natural things. One problem with this suggestion is that scientists are constantly postulating new entities, such as quantum wavefunctions, quarks, and genes. And who is to say whether or not these entities are natural? What are the defining characteristics of natural entities? The problem before us, then, is to complete the following definition:NAT: x is natural just in case…But it would be extremely difficult to complete this definition in a way that would be useful for guiding scientific practice. First, it wouldn’t be helpful to define natural entities as those mentioned by our current best scientific theories. This is because methodological naturalism would then lead to extreme conservatism about ontology: no new entities should be introduced in science. Second, it wouldn’t be helpful to define “x is natural” in terms of the words “natural,” “supernatural,” or any synonym thereof. [...] Third, it wouldn’t be helpful to define “x is natural” in terms of space, time, energy, or mass. Contemporary science already defies simple intuitions about what is natural, and we can expect future science to do so to an even greater extent."

"What’s the greatest discovery in the history of thought? Of course, it’s a silly question – but it won’t stop me from suggesting an answer. It’s Plato’s discovery of abstract objects. Most scientists, and indeed most philosophers, would scoff at this. Philosophers admire Plato as one of the greats, but think of his doctrine of the heavenly forms as belonging in a museum. Mathematicians, on the other hand, are at least slightly sympathetic. Working day-in and day-out with primes, polynomials and principal fibre bundles, they have come to think of these entities as having a life of their own. Could this be only a visceral reaction to an illusion? Perhaps, but I doubt it."

"Perhaps nobody should be particularly surprised by this, as, after all, the laws of nature physicists acknowledge also seem to be timelessly true independently of whether anyone takes them to be true. Where do they come from? Since there are somewhat mundane interpretations of what laws of nature are – including the possibility that they are accidental generalities valid in this particular universe and/or within a certain time-span – the case posed by mathematical constructs seems to be even more clear and powerful. Math, like diamonds, truly seems to be forever. If one ‘goes Platonic’ with math, one has to face several important philosophical consequences, perhaps the major one being that the notion of physicalism goes out the window. Physicalism is the position that the only things that exist are those that have physical extension [ie, take up space] – and last time I checked, the idea of circle, or Fermat’s theorem, did not have physical extension. It is true that physicalism is now a sophisticated doctrine that includes not just material objects and energy, but also, for instance, physical forces and information. But it isn’t immediately obvious to me that mathematical objects neatly fall into even an extended physicalist ontology. And that definitely gives me pause to ponder."

"One cannot escape the feeling that these mathematical formulas have an independent existence and intelligence of their own, that they are wiser than we are, wiser even than their discoverers, that we get more out of them than was originally put into them."

"I don’t call myself a “Platonist,” since I don’t even know what it even means to say that mathematical truths “exist in a Platonic realm.” Indeed, that concrete way of putting things even strikes me as a step backwards—as if astronomers could one day send probes to Plato’s realm to find out what’s in it, or prove that it never existed. The concepts of “existence” and “reality” that we use when discussing physical objects are just flat-out irrelevant here. I instead call myself an “anti-anti-Platonist”: someone who’s not literally committed to a “Platonic realm” of mathematical truth, but who given the choice, would much rather that people believe in such a realm than that they spout the self-evident garbage that goes with denying math’s intellectual autonomy."

"How is it that mathematical ideas can be communicated in this way? I imagine that whenever the mind perceives a mathematical idea, it makes contact with Plato's world of mathematical concepts. ... When one 'sees' a mathematical truth, one's consciousness breaks through into this world of ideas, and makes direct contact with it ('accessible via the intellect'). I have described this 'seeing' in relation to Gödel's theorem, but it is the essence of mathematical understanding. When mathematicians communicate, this is made possible by each one having a direct route to truth, the consciousness of each being in a position to perceive mathematical truths directly, through this process of 'seeing'. (Indeed, often this act of perception is accompanied by words like 'Oh, I see'!) Since each can make contact with Plato's world directly, they can more readily communicate with each other than one might have expected. The mental images that each one has, when making this Platonic contact, might be rather different in each case, but communication is possible because each is directly in contact with the same externally existing Platonic world!"

"It's a feature of new paradigms that ... pitchforks come out with new paradigms."

"Molecular biology had a smal number of facts and a very, very strong paradigm, so you didn't have to know any facts. Unlike botany, which had nothing but facts and no paradigm. And since I have a bad memory for facts and a very good understanding of paradigms, it was ideal for me. I found it very simple."

"Almost always the men who achieve these fundamental inventions of a new paradigm have either been very young or very new to the field whose paradigm they change."

"The shared idea in the mind of society, the great big unstated assumptions, constitute that society's paradigm, or deepest beliefs about how the world works. [...] people who have managed to intervene in systems at the level of paradigm have hit a leverage point that totally transforms systems."

"T. S. Kuhn's notion of a paradigm has replaced the positivist account of theories in many discussions, particularly in the social sciences. Most generally, a paradigm is a conceptual scheme representing a group's shared commitments and providing them with a way of looking at phenomena (Kuhn, 1970b). This notion is flexible enough to have much practical and historical applicability, but it is too vague to help with philosophical problems about explanation, justification, and meaning. Despite a professed desire to avoid total subjectivity, Kuhn has not succeeded in describing how paradigms can be rationally evaluated, or how different paradigms can relate to the same world, or even what it is for a paradigm to be used in solving a problem."

"The most effective way to approach the problem of consciousness would be to use the descriptions of psychologists and cognitive scientists and attempt to map different aspects of their models onto what is known about the neuroanatomy and neurophysiology of the brain. Naturally we have attempted to do this but have not found it as useful as one might hope, although such models do point to the importance of attention and short-term memory and suggest that consciousness should have easy access to the higher, planning levels of the motor system. A major handicap is the pernicious influence of the paradigm of the von Neumann digital computer."

"Though one can question the extent to which Kuhn's cyclic theory of scientific revolution fits what we know of the history of science, in itself this theory would not be very disturbing, nor would it have made Kuhn's book famous. For many people, it is Kuhn's reinvention of the word "paradigm" that has been either most useful or most objectionable. … But the quarrel over the word "paradigm" seems to me unimportant. Kuhn was right that there is more to a scientific consensus than just a set of explicit theories. We need a word for the complex of attitudes and traditions that go along with our theories in a period of normal science, and "paradigm" will do as well as any other."

"Paradigms, especially old ones, die harder than Bruce Willis."

""Graphene" is the name given to a single-layer hexagonal lattice of carbon atoms, and extended two-dimensional lattice of benzene rings, devoid of hydrogen atoms. This one-atom-thick material has recently been found to be robust, if not completely planar, in samples tens of micrometers up to 30 inches in extent, on a supporting substrate. Graphene is a contender in the new information technology (and other) applications, beyond being a scientific breakthrough and curiosity. As we will see, electrons in graphene display properties similar to photons and neutrinos, never before observed in a condensed-matter environment. ... The discovery of graphene extends, beyond some theoretical predictions, what useful forms matter can take. It is truly a new paradigm."

"... we rarely think from first principles, even first principles we ourselves sincerely endorse, but rather from sentiments instilled in us by our culture. Nowhere is this more obvious than in the case of humanity’s moral attitudes toward non-human animals. Even most utilitarians, who by their own ideals ought to consider the suffering of all beings important, are still in fact exceptionally anthropocentric in their attitudes. The insights of Darwin have not yet trickled fully into our moral consciousness, not even among those whose moral views demand it. Such is the heavy momentum of culture, which is reflected in every facet of modern politics and political thought. The anthropocentrism of most political philosophy is, to put it mildly, a massive failure."

"The founder of modern political philosophy is Machiavelli. He tried to effect, and he did effect, a break with the whole tradition of political philosophy. He compared his achievement to that of men like Columbus. He claimed to have discovered a new moral continent. His claim is well founded; his political teaching is "wholly new." The only question is whether the new continent is fit for human habitation."

"A major danger in using highly abstractive methods in political philosophy is that one will succeed merely in generalizing one’s own local prejudices and repackaging them as demands of reason. The study of history can help to counteract this natural human bias."

"In attempting to teach the prince how to achieve, maintain, and expand power, Machiavelli made his fundamental and celebrated distinction between "the effective truth of things" and the "imaginary republics and monarchies that have never been seen nor have been known to exist." The implication was that moral and political philosophers had hitherto talked exclusively about the latter and had failed to provide guidance to the real world in which the prince must operate. This demand for a scientific, positive approach was extended only later from the prince to the individual, from the nature of the state to human nature. Machiavelli probably sensed that a realistic theory of the state required a knowledge of human nature, but his remarks on that subject, while invariably acute, are scattered and unsystematic."

"The ideas of economists and political philosophers, both when they are right and when they are wrong, are more powerful than is commonly understood. Indeed the world is ruled by little else. Practical men, who believe themselves to be quite exempt from any intellectual influence, are usually the slaves of some defunct economist. Madmen in authority, who hear voices in the air, are distilling their frenzy from some academic scribbler of a few years back. I am sure that the power of vested interests is vastly exaggerated compared with the gradual encroachment of ideas. Not, indeed, immediately, but after a certain interval; for in the field of economic and political philosophy there are not many who are influenced by new theories after they are twenty-five or thirty years of age, so that the ideas which civil servants and politicians and even agitators apply to current events are not likely to be the newest. But, soon or late, it is ideas, not vested interests, which are dangerous for good or evil."

"Ask me if you choose if a Cynic shall engage in the administration of the State. O fool, seek you a nobler administration that that in which he is engaged? Ask you if a man shall come forward in the Athenian assembly and talk about revenue and supplies, when his business is to converse with all men, Athenians, Corinthians, and Romans alike, not about supplies, not about revenue, nor yet peace and war, but about Happiness and Misery, Prosperity and Adversity, Slavery and Freedom? Ask you whether a man shall engage in the administration of the State who has engaged in such an Administration as this? Ask me too if he shall govern; and again I will answer, Fool, what greater government shall he hold than he holds already?"

"Is the ordinary person incompetent? No judgment is more decisive for one's political philosophy. It was perhaps the single most important difference in judgment between Plato and Aristotle."

"There is the same difference between political theory and constitutional laws as there is between poetics and poetry. The illustrious Montesquieu is to Lycurgus, in the intellectual hierarchy, what Batteux is to Homer or Racine. Moreover, these two talents positively exclude each other, as can be seen by the example of Locke, who fumbled badly when he presumed to give laws to the Americans."

"Today political science is often said to be ‘descriptive’ or ‘empirical,’ concerned with facts; political philosophy is called ‘normative’ because it expresses values. But these terms merely repeat in more abstract form the difference between political science, which seeks agreement, and political philosophy, which seeks the best."

"At its heart, political philosophy is about power: who has it, what they do with it, and toward what ends."

"The annual produce of the land and labour of any nation can be increased in its value by no other means, but by increasing either the number of its productive labourers, or the productive powers of those labourers who had before been employed."