Draft. Berkeley denied the existence of abstract ideas and any faculty of abstraction. At the same time, however, he embraced innate ideas and a faculty of pure intellect. This paper attempts to reconcile the tension between these commitments by offering an interpretation of Berkeley's Platonism.
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.
This paper considers two strategies for undermining indispensability arguments for mathematical Platonism. We defend one strategy (the Trivial Strategy) against a criticism by Joseph Melia. In particular, we argue that the key example Melia uses against the Trivial Strategy fails. We then criticize Melia’s chosen strategy (the Weaseling Strategy.) The Weaseling Strategy attempts to show that it is not always inconsistent or irrational knowingly to assert p and deny an implication of p . We argue that Melia’s case for (...) this strategy fails. (shrink)
Quine rejects intensional Platonism and, with it, also rejects attributes (properties) as designations of predicates. He pragmatically accepts extensional Platonism, but conceives of classes as merely auxiliary entities needed to express some laws of set theory. At the elementary logical level, Quine develops an “ontologically innocent” logic of predicates. What in standard quantification theory is the work of variables is in the logic of predicates the work of a few functors that operate on predicates themselves: variables are eliminated. (...) This “predicate functor logic” may be conceived as a peculiar sort of Platonism - ontologically neutral, reduced to schematized linguistic forms. (shrink)
In this paper I argue against Divers and Miller's 'Lightness of Being' objection to Hale and Wright's neo-Fregean Platonism. According to the 'Lightness of Being' objection, the neo-Fregean Platonist makes existence too cheap: the same principles which allow her to argue that numbers exist also allow her to claim that fictional objects exist. I claim that this is no objection at all" the neo-Fregean Platonist should think that fictional characters exist. However, the pluralist approach to truth developed by WQright (...) in 'Truth and Objectivity' allows us to salvage our intuitions about the metaphysicial lightweightness of fictional characters: truth for discourse about fictional characters fails to exert 'Cognityive Command', whereas truth about arithmetic does. (shrink)
In his book 'Wittgenstein on the foundations of Mathematics', Crispin Wright notes that remarkably little has been done to provide an unpictorial, substantial account of what mathematical platoninism comes to. Wright proposes to investigate whether there is not some more substantial doctrine than the familiar images underpinning the platonist view. He begins with the suggestion that the essential element in the platonist claim is that mathematical truth is objective. Although he does not demarcate them as such, Wright proposes several different (...) tests for objectivity. The paper finds problems with each of these tests. (shrink)
According to Quine’s indispensability argument, we ought to believe in just those mathematical entities that we quantify over in our best scientific theories. Quine’s criterion of ontological commitment is part of the standard indispensability argument. However, we suggest that a new indispensability argument can be run using Armstrong’s criterion of ontological commitment rather than Quine’s. According to Armstrong’s criterion, ‘to be is to be a truthmaker (or part of one)’. We supplement this criterion with our own brand of metaphysics, 'Aristotelian (...) (...) realism', in order to identify the truthmakers of mathematics. We consider in particular as a case study the indispensability to physics of real analysis (the theory of the real numbers). We conclude that it is possible to run an indispensability argument without Quinean baggage. (shrink)
Psychology is dead. The self is a fiction invented by the brain. Brain plasticity isn?t all it?s cracked up to be. Our conscious learning is an observation post factum, a recollection of something already accomplished by the brain. We don?t learn to speak; speech is generated when the brain is ready to say something. False memories are more prevalent than one might think, and they aren?t all that bad. We think we?re in charge of our lives, but actually we are (...) not. On top of all this, the common belief that reading to a young child will make her brain more attuned to reading is simply untrue. (shrink)
Enrico Berti and others hold that Aquinas’s notion of God as ipsum esse subsistens conflicts with Aristotle’s view that positing an Idea of being treats being as a genus and nullifies all differences. The paper first shows how one of Aquinas’s ways of distinguishing esse from essence supposes an intimate tie between a thing’s esse and its differentia. Then it argues that for Aquinas the (one) divine essence differs from the (manifold) “essence of esse.” God is his very esse. This (...) somehow “contains” all esse, but it also transcends it, because although simple, it also “contains” all forms and differentiae. (shrink)
In this book, Balaguer demonstrates that there are no good arguments for or against mathematical platonism. He does this by establishing that both platonism and anti-platonism are defensible views. Introducing a form of platonism ("full-blooded platonism") that solves all problems traditionally associated with the view, he proceeds to defend anti-platonism (in particular, mathematical fictionalism) against various attacks, most notably the Quine-Putnam indispensability attack. He concludes by arguing that it is not simply that we do (...) not currently have any good argument for or against platonism, but that we could never have such an argument and, indeed, that there is no fact of the matter as to whether platonism is correct. (shrink)
Introduction -- What is platonism? -- Schleiermacher's pedagogical interpretation of Plato -- What's wrong with the current debate -- The romantic rediscovery of Plato's ineffable ontology -- Conclusions: Ineffability and dialogue form -- Untying Schleiermacher's gordian knot -- Metaphysical ineffability : the argument from language and human finitude -- Spiritual ineffability: the argument from self-transformation -- Existential ineffability : the argument from life choice -- Platonism reconsidered -- The context of Heidegger's interpretation of Plato -- What it all (...) means and why it matters -- Stage one: the realm of shadows -- Stage two: the fire -- Stage three: the realm of light -- The good : Heidegger's Plato is the later Heidegger -- Stage four: the return to the shadows -- The virtues of heidegger's plato -- Heidegger's crisis and opportunity -- Setting the stage -- Heidegger's crisis -- Understanding Heidegger's crisis : Nietzsche -- Heidegger as reformed madman -- Revolutionary thinker or utopian social engineer -- The Greeks and university reform -- Theoria and fundamental ontology -- A community of similarly striving researchers -- University reform and nihilism -- Back from Syracuse : four reasons to rethink Heidegger's politics -- The ontological problem -- The epistemological problem -- The moral problem -- The political problem -- What was plato doing in Syracuse -- Back from Syracuse or Eros Tyrannos -- How Heidegger should have read Plato -- Plato anticipate Heidegger's critique of technology -- Plato's problems with periclean Athens -- Alcibiades as embodiment of periclean Athens -- Alcibiades as inverted image of Socrates -- Conclusions: What Heidegger missed. (shrink)
Rationalism, Platonism and God comprises three main papers on Descartes, Spinoza and Leibniz, with extensive responses. It provides a significant contribution to the exploration of the common ground of the great early-modern Rationalist theories, and an examination of the ways in which the mainstream Platonic tradition permeates these theories. -/- John Cottingham identifies characteristically Platonic themes in Descartes's cosmology and metaphysics, finding them associated with two distinct, even opposed attitudes to nature and the human condition, one ancient and 'contemplative', (...) the other modern and 'controlling'. He finds the same tension in Descartes's moral theory, and believes that it remains unresolved in present-day ethics. -/- Was Spinoza a Neoplatonist theist, critical Cartesian, or naturalistic materialist? Michael Ayers argues that he was all of these. Analysis of his system reveals how Spinoza employed Neoplatonist monism against Descartes's Platonist pluralism. Yet the terminology - like the physics - is Cartesian. And within this Platonic-Cartesian shell Spinoza developed a rigorously naturalistic metaphysics and even, Ayers claims, an effectually empiricist epistemology. -/- Robert Merrihew Adams focuses on the Rationalists' arguments for the Platonist, anti-Empiricist principle of 'the priority of the perfect', i.e. the principle that finite attributes are to be understood through corresponding perfections of God, rather than the reverse. He finds the given arguments unsatisfactory but stimulating, and offers a development of one of Leibniz's for consideration. -/- These papers receive informed and constructive criticism and development at the hands of, respectively, Douglas Hedley, Sarah Hutton and Maria Rosa Antognazza. (shrink)
Mathematical explanation -- What is naturalism? -- Perception, practice, and ideal agents: Kitcher's naturalism -- Just metaphor?: Lakoff's language -- Seeing with the mind's eye: the Platonist alternative -- Semi-naturalists and reluctant realists -- A life of its own?: Maddy and mathematical autonomy.
Since Benacerraf’s “Mathematical Truth” a number of epistemological challenges have been launched against mathematical platonism. I first argue that these challenges fail because they unduely assimilate mathematics to empirical science. Then I develop an improved challenge which is immune to this criticism. Very roughly, what I demand is an account of how people’s mathematical beliefs are responsive to the truth of these beliefs. Finally I argue that if we employ a semantic truth-predicate rather than just a deflationary one, there (...) surprisingly turns out to be logical space for a response to the improved challenge where no such space appeared to exist. (shrink)
Platonism in the philosophy of mathematics is the doctrine that there are mathematical objects such as numbers. John Burgess and Gideon Rosen have argued that that there is no good epistemological argument against platonism. They propose a dilemma, claiming that epistemological arguments against platonism either rely on a dubious epistemology, or resemble a dubious sceptical argument concerning perceptual knowledge. Against Burgess and Rosen, I show that an epistemological anti-platonist argument proposed by Hartry Field avoids both horns of (...) their dilemma. (shrink)
Predication is an indisputable part of our linguistic behavior. By contrast, the metaphysics of predication has been a matter of dispute ever since antiquity. According to Plato—or at least Platonism, the view that goes by Plato’s name in contemporary philosophy—the truths expressed by predications such as “Socrates is wise” are true because there is a subject of predication (e.g., Socrates), there is an abstract property or universal (e.g., wisdom), and the subject exemplifies the property.1 This view is supposed to (...) be general, applying to all predications, whether the subject of predication is a person, a planet, or a property.2 Despite the controversy surrounding the metaphysics of predication, many theistic philosophers—including the majority of contemporary analytic theists—regard Platonism as extremely attractive. At the same time, however, such philosophers are also commonly attracted to a form of traditional theism that has at its core the thesis that God is an absolutely independent.. (shrink)
Central to the philosophical understanding of music is the status of musical works. According to the Platonist, musical works are abstract objects; that is, they are not located in space or time, and we have no causal access to them. Moreover, only a particular physical occurrence of these musical works is instantiated when a performance ofthe latter takes place. But even if no performance ever took place, the Platonist insists, the musical work would still exist, since its existence is not (...) tied to spatiotemporal constraints (Kivy , and Dodd ). In this paper, I offer a critical assessment of the Platonist view. I argue that, despite some benefits, Platonism faces significant difficulties in the interpretation of music. In spite ofthe Platonist’s attempt to overcome the problem, the view ultimately doesn’t mesh well with the way we actively respond to performances and fail to respond, in any way similar, to abstract patterns. Platonism also makes knowledge of music something extremely mysterious, given that we have no access to the abstract objects that, according to the Platonist, characterize the musical works. The ability to understand how we respond to musical works is, of course, central to any interpretation of music. This ability is also crucial in explaining the role music plays in various aspects of our culture, Rom bounding with others to music therapy. Given the problems faced by Platonism, it makes more sense to adopt an altemative, non-Platonist view. I conclude the paper by sketching such a non-Platonist proposal. (shrink)
Modal Platonism utilizes “weak” logical possibility, such that it is logically possible there are abstract entities, and logically possible there are none. Modal Platonism also utilizes a non-indexical actuality operator. Modal Platonism is the EASY WAY, neither reductionist nor eliminativist, but embracing the Platonistic language of abstract entities while eliminating ontological commitment to them.
Hartry Field's formulation of an epistemological argument against platonism is only valid if knowledge is constrained by a causal clause. Contrary to recent claims (e.g. in Liggins (2006), Liggins (2010)), Field's argument therefore fails the very same criterion usually taken to discredit Benacerraf's earlier version.
This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti-Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions (...) that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the 'no miracles' argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific 'realists' should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism. (shrink)
It is widely believed that platonists face a formidable problem: that of providing an intelligible account of mathematical knowledge. The problem is that we seem unable, if the platonist is right, to have the causal relationships with the objects of mathematics without which knowledge of these objects seems unintelligible. The standard platonist response to this challenge is either to deny that knowledge without causation is unintelligible, or to make room for causal interactions by softening the platonism at issue. In (...) this essay I argue that the idea of causal relations with fully platonist objects is unproblematic. I would like to thank Agnes Gellen Callard, Josh Sheptow, and Palle Yourgrau for helpful discussions of the ideas presented here. (shrink)
The question posed in the title of this paper is an historical one. I am not, for example, primarily interested in the term 'Platonism' as used by modern philosophers to stand for a particular theory under discussion – a theory, which it is typically acknowledged, no one may have actually held.1 I am rather concerned to understand and articulate on an historical basis the core position of that 'school' of thought prominent in antiquity from the time of the 'founder' (...) up until at least the middle of the 6th century C.E.2 Platonism was unquestionably the dominant philosophical position in the ancient world over a period of more than 800 years. Epicureanism is perhaps the sole major exception to the rule that in the ancient world all philosophers took Platonism as the starting-point for speculation, including those who thought their first task was to refute Platonism. Basically, Platonism sent the ancient philosophical agenda. Given this fact, understanding with some precision the nature of Platonism is obviously a desirable thing for the historian of ancient philosophy. (shrink)
This article examines Gilles Deleuze’s concept of the simulacrum, which Deleuze formulated in the context of his reading of Nietzsche’s project of “overturning Platonism.” The essential Platonic distinction, Deleuze argues, is more profound than the speculative distinction between model and copy, original and image. The deeper, practical distinction moves between two kinds of images or eidolon, for which the Platonic Idea is meant to provide a concrete criterion of selection “Copies” or icons (eikones) are well-grounded claimants to the transcendent (...) Idea, authenticated by their internal resemblance to the Idea, whereas “simulacra” (phantasmata) are like false claimants, built on a dissimilarity and implying an essential perversion or deviation from the Idea. If the goal of Platonism is the triumph of icons over simulacra, the inversion of Platonism would entail an affirmation of the simulacrum as such, which must thus be given its own concept. Deleuze consequently defines the simulacrum in terms of an internal dissimilitude or “disparateness,” which in turn implies a new conception of Ideas, no longer as self-identical qualities (the auto kath’hauto), but rather as constituting a pure concept of difference. An inverted Platonism would necessarily be based on a purely immanent and differential conception of Ideas. Starting from this new conception of the Idea, Deleuze proposes to take up the Platonic project anew, rethinking the fundamental figures of Platonism (selection, repetition, ungrounding, the question-problem complex) on a purely differential basis. In this sense, Deleuze’s inverted Platonism can at the same time be seen as a rejuvenated Platonism and even a completed Platonism. (shrink)
Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and nonmental. Platonism in this sense is a contemporary view. It is obviously related to the views of Plato in important ways, but it is not entirely clear that Plato endorsed this view, as it is defined here. In order to remain neutral on this (...) question, the term ‘platonism’ is spelled with a lower-case ‘p’. (See entry on Plato.) The most important figure in the development of modern platonism is Gottlob Frege (1884, 1892, 1893-1903, 1919). The view has also been endorsed by many others, including Kurt Gödel (1964), Bertrand Russell (1912), and W.V.O. Quine (1948, 1951). (shrink)
A response is given here to Benacerraf's 1973 argument that mathematical platonism is incompatible with a naturalistic epistemology. Unlike almost all previous platonist responses to Benacerraf, the response given here is positive rather than negative; that is, rather than trying to find a problem with Benacerraf's argument, I accept his challenge and meet it head on by constructing an epistemology of abstract (i.e., aspatial and atemporal) mathematical objects. Thus, I show that spatio-temporal creatures like ourselves can attain knowledge about (...) mathematical objects by simply explaininghow they can do this. My argument is based upon the adoption of a particular version of platonism — full-blooded platonism — which asserts that any mathematical object which possiblycould exist actuallydoes exist. (shrink)
The historian of philosophy often encounters arguments that are enthymematic: they have conclusions that follow from their explicit premises only by the addition of "tacit" or "suppressed" premises. It is a standard practice of interpretation to supply these missing premises, even where the enthymeme is "real," that is, where there is no other context in which the philosopher in question asserts the missing premises. To do so is to follow a principle of charity: other things being equal, one interpretation is (...) better than another just to the extent that the one produces a better argument than the other. We show that this principle leads to paradoxical conclusions, including the following: there is no objectively correct interpretation of any real enthymeme found in the text of a major philosopher; an interpreter will not regard a real enthymeme of a major philosopher as adequately interpreted until he has found a way of reading it that makes it into a good argument; every classical philosopher is infallible and omniscient; major philosophers never disagree. These conclusions are preposterous, but there are indications that they are in fact being reached, as we show by means of a case study of recent scholarship on Plato's Third Man Argument. To avoid the overinterpretation and anachronism that result from the unrestrained use of the principle of charity, one must employ a counterbalancing principle of parsimony: to seek the simplest explanation for the text under discussion. We study the role of the principle of parsimony by means of a mathematical case study, involving the suppressed premises in Euclid's Elements. Here the principle of parsimony plays a larger role than it does in the interpretation of philosophical texts, leading to a sharper distinction between Euclid's geometry and Euclidean geometry than we find between Plato and Platonism. We conclude by comparing two models of interpretation, which we call prospective and retrospective. Although the prospective model of interpretation leads to Platonism rather than to Plato, we argue that it still has a place in Platonic scholarship. (shrink)
Mark Balaguer's Platonism and Anti-Platonism in Mathematics presents an intriguing new brand of platonism, which he calls plenitudinous platonism, or more colourfully, full-blooded platonism. In this paper, I argue that Balaguer's attempts to characterise full-blooded platonism fail. They are either too strong, with untoward consequences we all reject, or too weak, not providing a distinctive brand of platonism strong enough to do the work Balaguer requires of it.
Gottlob Frege is often called a "platonist". In connection with his philosophy we can talk about platonism concerning three kinds of entities: numbers, or logical objects more generally; concepts, or functions more generally; thoughts, or senses more generally. I will only be concerned about the first of these three kinds here, in particular about the natural numbers. I will also focus mostly on Frege's corresponding remarks in The Foundations of Arithmetic (1884), supplemented by a few asides on Basic Laws (...) of Arithmetic (1893/1903) and "Thoughts" (1918). My goal is to clarify in which sense the Frege of Foundations and Basic Laws is a platonist concerning the natural numbers.1.. (shrink)
Mark Balaguer argues for full blooded platonism (FBP), and argues that FBP alone can solve Benacerraf's familiar epistemic challenge. I note that if FBP really can solve Benacerraf's epistemic challenge, then FBP is not alone in its capacity so to solve; RFBP—really full blooded platonism—can do the trick just as well, where RFBP differs from FBP by allowing entities from inconsistent mathematics. I also argue briefly that there is positive reason for endorsing RFBP.
Draft version of essay. ABSTRACT: Benjamin Whichcote developed a distinctive account of human nature centered on our moral psychology. He believed that this view of human nature, which forms the foundation of “Cambridge Platonism,” showed that the demands of reason and faith are not merely compatible but dynamically supportive of one another. I develop an interpretation of this oft-neglected and widely misunderstood account of human nature and defend its viability against a key objection.
It is argued here that mathematical objects cannot be simultaneously abstract and perceptible. Thus, naturalized versions of mathematical platonism, such as the one advocated by Penelope Maddy, are unintelligble. Thus, platonists cannot respond to Benacerrafian epistemological arguments against their view vias Maddy-style naturalization. Finally, it is also argued that naturalized platonists cannot respond to this situation by abandoning abstractness (that is, platonism); they must abandon perceptibility (that is, naturalism).
In this paper, we develop an alternative strategy, Platonized Naturalism, for reconciling naturalism and Platonism and to account for our knowledge of mathematical objects and properties. A systematic (Principled) Platonism based on a comprehension principle that asserts the existence of a plenitude of abstract objects is not just consistent with, but required (on transcendental grounds) for naturalism. Such a comprehension principle is synthetic, and it is known a priori. Its synthetic a priori character is grounded in the fact (...) that it is an essential part of the logic in which any scientific theory will be formulated and so underlies (our understanding of) the meaningfulness of any such theory (this is why it is required for naturalism). Moreover, the comprehension principle satisfies naturalist standards of reference, knowledge, and ontological parsimony! As part of our argument, we identify mathematical objects as abstract individuals in the domain governed by the comprehension principle, and we show that our knowledge of mathematical truths is linked to our knowledge of that principle. (shrink)
David Bloor and Crispin Wright have argued, independently, that the proper lesson to draw from Wittgenstein's so-called rule-following considerations is the rejection of meaning Platonism. According to Platonism, the meaningfulness of a general term is constituted by its connection with an abstract entity, the (possibly) infinite extension of which is determined independently of our classificatory practices. Having rejected Platonism, both Bloor and Wright are driven to meaning finitism, the view that the question of whether a meaningful term (...) correctly applies to a given entity is not determined in advance of anyone's judgement about the matter. I argue that the two views do not form a dichotomy - there is room for a middle position which can account for the correct applications existing in advance of anyone's judgements without being committed to meaning Platonism. Furthermore, I will show how such a middle position arises quite naturally from the view that our competence with semantically basic terms is response-dependent. (shrink)
The expression 'platonism in mathematics' or 'mathematical platonism' is familiar in the philosophy of mathematics at least since the use Paul Bernays made of it in his paper of 1934, 'Sur le Platonisme dans les Mathématiques'. But he was not the first to point out the similarities between the conception of the defenders of mathematical realism and the ideas of Plato. Poincaré had already stressed the 'platonistic' orientation of the mathematicians he called'Cantorian', as opposed to those who (like (...) himself) were 'pragmatist' ones. I examine in this paper some very perplexing aspects of the use which is made at that time of a number of concepts, particularly 'idealism' (which generally designates what we would call 'mathematical realism') and 'empiricism' (which can designate almost any form of antirealism, even if, like for example intuitionism, it is not empiricist at all). There are, of course, historical reasons that may explain why it was for a time so easy and natural to use the words and the concepts in a way that may seem now very strange and to treat as if they were equivalent the two oppositions: realism/antirealism and idealism/empiricism. (shrink)
Book Information Knowledge, Cause, and Abstract Objects: Causal Objections to Platonism. Knowledge, Cause, and Abstract Objects: Causal Objections to Platonism Colin Cheyne , Dordrecht: Kluwer Academic Publishers , 2001 , xvi + 236 , £55 ( cloth ) By Colin Cheyne. Dordrecht: Kluwer Academic Publishers. Pp. xvi + 236. £55.
This paper responds to Colin Cheyne's new anti-platonist argument according to which knowledge of existential claims—claims of the form such-tmd-so exist—requires a caused connection with the given such-and-so. If his arguments succeed then nobody can know, or even justifiably believe, that acausal entities exist, in which case (standard) platonism is untenable. I argue that Cheyne's anti-platonist argument fails.
In his account of Plato’s ideas in the first book of the “Transcendental Dialectic”, “On the concepts of pure reason”, Kant, in describing how for Plato ideas were “archetypes of things themselves”, adds that these ideas “flowed from the highest reason, through which human reason partakes in them”.1 Later, in the section of the Transcendental Dialectic treating the “ideals of pure reason”, he again attributes to Plato the notion of a “divine mind” within which the “ideas” exist. An “ideal”, Kant (...) says, “was to Plato, an idea in the divine understanding”.2 But as the editors of the Cambridge University Press translation of the Critique of Pure Reason point out, the idea of a divine mind as container of the ideas was not Plato’s and did not originate until the “syncretistic Platonism from the period of the Middle Academy”. From there it “was later adopted by Platonists as diverse as Philo of Alexandria, Plotinus and St Augustine, and became fundamental to later Christian interpretations of Platonism”. (shrink)
According to standard mathematical platonism, mathematical entities (numbers, sets, etc.) are abstract entities. As such, they lack causal powers and spatio-temporal location. Platonists owe us an account of how we acquire knowledge of this inaccessible mathematical realm. Some recent versions of mathematical platonism postulate a plenitude of mathematical entities, and Mark Balaguer has argued that, given the existence of such a plenitude, the attainment of mathematical knowledge is rendered non-problematic. I assess his epistemology for such a profligate (...) class='Hi'>platonism and find it unsatisfactory because it lacks an adequate semantics, in particular, an adequate account of reference. (shrink)
del's appeal to mathematical intuition to ground our grasp of the axioms of set theory, is notorious. I extract from his writings an account of this form of intuition which distinguishes it from the metaphorical platonism of which Gödel is sometimes accused and brings out the similarities between Gödel's views and Dummett's.
In Science without numbers Hartry Field attempted to formulate a nominalist version of Newtonian physics?one free of ontic commitment to numbers, functions or sets?sufficiently strong to have the standard platonist version as a conservative extension. However, when uses for abstract entities kept popping up like hydra heads, Field enriched his logic to avoid them. This paper reviews some of Field's attempts to deflate his ontology by inflating his logic.
This essay gives an extensive treatment of Heidegger's confrontation (Auseinander-setzung) with Nietzsche' thought. It argues that Heidegger's confrontation entails situating what Heidegger calls Nietzsche's "transformed" understanding of the sensuous outside the metaphysics of both Plato and Platonism. The essay establishes, by the end of the second section, that Heidegger's confrontation with Nietzsche's thought culminates with the insight that for Nietzsche sensuousness is metaphysical. The third section of the essay takes as its point of departure Heidegger's intimation at the conclusion (...) of The Will to Power as Art, where he advances the inference that Nietzsche's new grounding of the metaphysical in sensuousness brings along with it "readiness for the gods." The essay offers explicit support for Heidegger's intimation through an analysis of three essential steps, outlined by Nietzsche in The Birth of Tragedy, in which sensuousness proves to be indicative of a way of access to the gods, the dual gods, Apollo and Dionysus, at the origin of Greek tragedy. (shrink)
Plato's Theaetetus is an acknowledged masterpiece, and among the most influential texts in the history of epistemology. Since antiquity it has been debated whether this dialogue was written by Plato to support his familiar metaphysical doctrines, or represents a self-distancing from these. David Sedley's book offers a via media, founded on a radical separation of the author, Plato, from his main speaker, Socrates. The dialogue, it is argued, is addressed to readers familiar with Plato's mature doctrines, and sets out to (...) show how these doctrines, far from being an abandonment of his Socratic heritage, are its natural outcome. The Socrates portrayed here is the same Socrates as already portrayed in Plato's early dialogues. While not a Platonist, he is exhibited - to put it in terms of an image made famous by this dialogue - as having been Platonism's midwife. In a comprehensive rereading of the text, Sedley tracks the ways in which Socrates is shown unwittingly preparing the ground for Plato's mature doctrines, and reinterprets the dialogue's individual arguments from this perspective. The book is addressed to all readers interested in Plato, and does not require knowledge of Greek. (shrink)
Jody Azzouni has offered the following argument against the existence of mathematical entities: if, as it seems, mathematical entities play no role in mathematical practice, we therefore have no reason to believe in them. I consider this argument as it applies to mathematical platonism, and argue that it does not present a legitimate novel challenge to platonism. I also assess Azzouni's use of the ‘epistemic role puzzle’ (ERP) to undermine the platonist's alleged parallel between skepticism about mathematical entities (...) and external-world skepticism. I conclude that ERP fails to undermine this parallel. (shrink)
Olszewski claims that the Church-Turing thesis can be used in an argument against platonism in philosophy of mathematics. The key step of his argument employs an example of a supposedly effectively computable but not Turing-computable function. I argue that the process he describes is not an effective computation, and that the argument relies on the illegitimate conflation of effective computability with there being a way to find out . ‘Ah, but,’ you say, ‘what’s the use of its being right (...) twice a day, if I can’t tell when the time comes?’ Why, suppose the clock points to eight o’clock, don’t you see that the clock is right at eight o’clock? Consequently, when eight o’clock comes round your clock is right. Lewis Carroll. (shrink)
"Platonism" is not only an example of this movement, the first "in" the whole history of philosophy. It commands it, it commands this whole history. [But the "whole" of this history is conflictual, heterogenous; it gives place to only relatively stabilizable hegemonies. Thus, it is never totalized, never totalizes itself.] A philosophy as such (an effect of hegemony) would henceforth always be "Platonic." Hence the necessity to continue to try to think what takes place in Plato, with Plato, what (...) is shown there, what is hidden, so as to win there or lose there.1. (shrink)
This essay attempts to reflect on Bergson’s contribution to the reversal of Platonism. Heidegger, of course, had set the standard for reversing Platonism. Thus the question posed in this essay, following Heidegger, is: does Bergson manage not only to reverse Platonism but also to twist free of it. The answer presented here is that Bergson does twist free, which explains Deleuze’s persistent appropriations of Bergsonian thought. Memory in Bergson turns out to be not a memory of an (...) idea, or even of the good, which is one, but a memory of multiplicity. Therefore Bergson’s memory is really, from a Platonistic standpoint, forgetfulness or, even, a counter-memory. (shrink)
Machine generated contents note: Introduction: Plethon and the notion of Paganism; Part I. Lost Rings of the Platonist Golden Chain: 1. Underground Platonism in Byzantium; 2. The rise of the Byzantine Illuminati; 3. The Plethon affair; Part II. The Elements of Pagan Platonism: 4. Epistemic optimism; 5. Pagan ontology; 6. Symbolic theology: the mythologising of Platonic ontology; Part III. Mistra versus Athos: 7. Intellectual and spiritual utopias; Part IV. The Path of Ulysses and the Path of Abraham: 8. (...) Conclusion; Epilogue: 'Spinozism before Spinoza', or the pagan roots of modernity. (shrink)
In this paper, I challenge those interpretations of Frege that reinforce the view that his talk of grasping thoughts about abstract objects is consistent with Russell's notion of acquaintance with universals and with GÃ¶del's contention that we possess a faculty of mathematical perception capable of perceiving the objects of set theory. Here I argue the case that Frege is not an epistemological Platonist in the sense in which GÃ¶del is one. The contention advanced is that GÃ¶del bases his Platonism (...) on a literal comparison between mathematical intuition and physical perception. He concludes that since we accept sense perception as a source of empirical knowledge, then we similarly should posit a faculty of mathematical intuition to serve as the source of mathematical knowledge. Unlike GÃ¶del, Frege does not posit a faculty of mathematical intuition. Frege talks instead about grasping thoughts about abstract objects. However, despite his hostility to metaphor, he uses the notion of âgraspingâ as a strategic metaphor to model his notion of thinking, i.e., to underscore that it is only by logically manipulating the cognitive content of mathematical propositions that we can obtain mathematical knowledge. Thus, he construes âgraspingâ more as theoretical activity than as a kind of inner mental âseeingâ. (shrink)
The expression 'platonism in mathematics' or 'mathematical platonism' is familiar in the philosophy of mathematics at least since the use Paul Bernays made of it in his paper of 1934, 'Sur le Platonisme dans les Math?matiques'. But he was not the first to point out the similarities between the conception of the defenders of mathematical realism and the ideas of Plato. Poincar? had already stressed the 'platonistic' orientation of the mathematicians he called 'Cantorian', as opposed to those who (...) (like himself) were 'pragmatist' ones. I examine in this paper some very perplexing aspects of the use which is made at that time of a number of concepts, particularly 'idealism' (which generally designates what we would call 'mathematical realism') and 'empiricism' (which can designate almost any form of antirealism, even if, like for example intuitionism, it is not empiricist at all). There are, of course, historical reasons that may explain why it was for a time so easy and natural to use the words and the concepts in a way that may seem now very strange and to treat as if they were equivalent the two oppositions: realism/antirealism and idealism/empiricism. (shrink)
In his Realism, Mathematics, and Modality, Hartry Field attempted to revitalize the epistemological case against mathematical platontism by challenging mathematical platonists to explain how we could be epistemically reliable with regard to the abstract objects of mathematics. Field suggested that the seeming impossibility of providing such an explanation tends to undermine belief in the existence of abstract mathematical objects regardless of whatever reason we have for believing in their existence. After more than two decades, Field’s explanatory challenge remains among the (...) best available motivations for mathematical nominalism. This paper argues that Field’s explanatory challenge misidentifies the central epistemological problem facing mathematical platonism. Contrary to Field’s suggestion, inexplicability of epistemic reliability does not act as an epistemic defeater. The failure to explain our epistemic reliability with respect to the existence and properties of abstract mathematical objects is simply one aspect of a broader failure to establish that we are epistemically reliable with respect to abstract mathematical objects in the first place. Ultimately, it is this broader failure that is the source of mathematical platonism’s real epistemological problems. (shrink)
Emmanuel Levinas's concept of "the face of the Other" involves an ethical mandate that is presumably transcultural or, in his terms, "precultural." His essay "Meaning and Sense" provides his most explicit defense of the idea that the face has a meaning that is not culturally relative, though it is always encountered within some particular culture. Levinas identifies his position there as a "return to Platonism." Through a careful reading of that essay, exploring Levinas's use of religious terminology and the (...) (sometimes implicit) relationships of the essay to the work of other phenomenologists and of Saussure, the author seeks to clarify (1) what Levinas retains and what he rejects in returning to Platonism "in a new way," (2) the sense in which this return constitutes an "overcoming" of relativism, and (3) the nature of the phenomenological warrant that he offers for his position. (shrink)
This paper concerns the role of intuitions in mathematics, where intuitions are meant in the Kantian sense, i.e. the “seeing” of mathematical ideas by means of pictures, diagrams, thought experiments, etc.. The main problem discussed here is whether Platonistic argumentation, according to which some pictures can be considered as proofs (or parts of proofs) of some mathematical facts, is convincing and consistent. As a starting point, I discuss James Robert Brown’s recent book Philosophy of Mathematics, in particular, his primarily examples (...) and analogies. I then consider some alternatives and counterarguments, namely John Norton’s opposite view, that intuitions are just pictorially represented logical arguments and are superfluous; and the Kantian transcendental theory of construction in imagination, as it is developed in the works of Marcus Giaquinto and Michael Friedman. Although I support the claim that some intuitions are essential in mathematical justification, I argue that a Platonistic approach to intuitions is partial and one should go further than a Platonist in explaining how some intuitions can deliver new mathematical knowledge. (shrink)
Iris Murdoch’s Metaphysics as a Guide to Morals ranges wide over the field of Western philosophical thought. Throughout the work, Murdoch proposes and enacts a form of philosophical inquiry that she believes supports a moral philosophy based on the idea of the good. One of her attempts, partly inspired by Paul Tillich and J. N. Findlay, centers on her critique and appropriation of the structure of the so-called “ontological argument” in Anselm’s Proslogion. This study assesses Murdoch’s accomplishment and the tenability (...) of the kind of Platonism she proposes against Anselm’s argument about the good in both the Monologion and the Proslogion. My claim is that Anselm’s conception of the good simply does not permit the kind of interpretation that Murdoch puts on the “ontological argument.”. (shrink)
In this paper I consider what it would take to combine a certain kind of mathematical Platonism with serious presentism. I argue that a Platonist moved to accept the existence of mathematical objects on the basis of an indispensability argument faces a significant challenge if she wishes to accept presentism. This is because, on the one hand, the indispensability argument can be reformulated as a new argument for the existence of past entities and, on the other hand, if one (...) accepts the indispensability argument for mathematical objects then it is hard to resist the analogous argument for the existence of the past. (shrink)
Jim Brown (1991, viii) says that platonism, in mathematics involves the following: 1. mathematical objects exist independently of us; 2. mathematical objects are abstract; 3. we learn about mathematical objects by the faculty of intuition. The same is being claimed by Jerrold Katz (1981, 1998) in his platonistic approach to linguistics. We can take the object of linguistic analysis to be concrete physical sounds as held by nominalists, or we can assume that the object of linguistic study are psychological (...) or mental states which presents the conceptualism or psychologism of Chomsky and that language is an abstract object as held by platonists or realists and urged by Jerrold Katz hinlself.I want to explicate Katz’s proposal which is based on Kant’s conception of pure intuition and give arguments why I find it implausible. I also present doubts that linguists use intuitive evidence only. I conclude with some arguments against the a prioricity of intuitive judgements in general which is also relevant for Jim Brown’s platonistic beliefs. (shrink)
In the first half of the first century BC the Academy of Athens broke up in disarray. From the wreckage of the semi-sceptical school there arose the new dogmatic philosophy of Antiochus, synthesised from Stoicism and Platonism, and the hardline Pyrrhonist scepticism of Aenesidemus. With his extensive knowledge of the ways in which Plato was read and invoked as an authority in late antiquity Dr Tarrant builds a most impressive reconstruction of Philo of Larissa's brand of Platonism and (...) of its arrival in Middle Platonism, particularly that of Plutarch, long after the Academy's institutional demise. Particularly valuable is his exploitation for this purpose of a text barely discussed since its publication 80 years ago - a commentary on Plato's Theaetetus whose unidentified author Dr Tarrant has cogently argued to be a follower of Philo, perhaps Eudorus of Alexandra. Among many other achievements, Dr Tarrant throws much light on the relation of Aenesideman scepticism to the Academy. (shrink)
Contemporary scholars have generally labelled Adolf Reinach, a founding member of early phenomenology’s Göttingen Circle, a Platonist. Because Reinach conceives of states of affairs as neither real nor ideal, as involved with timeless essences and necessary logical laws, many have hastily concluded that states of affairs are Platonic entities. In this essay, I analyse Barry Smith’s argument that Reinach is a Platonist. Smith’s widely accepted argument often becomes utilised to show that Reinach and other phenomenologists, including Husserl, are Platonic realists (...) (or, simply, Platonists). A closer look at Reinach’s text indicates, however, that he is notcommitted to Platonic realism. (shrink)
The philosophy of mathematics of the later Wittgenstein is normally not taken very seriously. According to a popular objection, it cannot account for mathematical necessity. Other critics have dismissed Wittgenstein's approach on the grounds that his anti-platonism is unable to explain mathematical objectivity. This latter objection would be endorsed by somebody who agreed with Paul Benacerraf that any anti-platonistic view fails to describe mathematical truth. This paper focuses on the problem proposed by Benacerraf of reconciling the semantics with the (...) epistemology for mathematics. It is claimed that there is a way of solving Benacerrafs problem along the lines suggested by Wittgenstein's later remarks on mathematics. This will require demonstrating that a satisfactory conception of mathematical objectivity can be extracted from his mature philosophy. (shrink)
The following paper argues that Blaise Pascal, in spite of his famous opposition between the God of the Philosophers and the God of “Abraham, Isaac and Jacob“ has significant affinities with the tradition of Renaissance Platonism and is in fact a Platonist in his overall outlook. This is shown in three ways. Firstly, it is argued that Pascal's skeptical fideism has roots in the notion of faith developed in post-Plotinian neo-Platonism. Secondly, it is argued that Pascal makes considerable (...) use of the Platonic notion of an indefinite dyadic principle. Thirdly, it is argued that Pascal's religious psychology gives a centrality to the body that brings it close to the theurgical standpoint of figures like Iamblichus. Pascal is then contrasted to figures like Cusanus and Pico in that a dyadic principle of opposition is more prominent in his work than a triadic logic of mediation. (shrink)
The Handbook of Platonism or Didaskalikos, attributed to Alcinous (long identified with the Middle Platonist Albinus, but on inadequate grounds), is a central text of later Platonism. In Byzantine times, in the Italian Renaissance, and even up to 1800, it was regarded as an ideal introduction to Plato's thought. In fact it is far from being this, but it is an excellent source for our understanding of Platonism in the second century AD. Neglected after a more accurate (...) view of Plato's thought established itself in the nineteenth century, the Handbook is only now coming to be properly appreciated for what it is.It presents a survey of Platonist doctrine, divided into the topics of Logic, Physics, and Ethics, and pervaded with Aristotelian and Stoic doctrines, all of which are claimed for Plato. -/- John Dillon presents an English translation of this work, accompanied by an introduction and a philosophical commentary in which he disentangles the various strands of influence, elucidates the complex scholastic tradition that lies behind, and thus reveals the sources and subsequent influence of the ideas expounded. (shrink)
A study of the influence of Platonism on two central areas of Early Christian doctrine, the relation of God the Son to the Father, and the mutual relations of the persons of the Trinity. In the former case, logos-theory and the figure of the demiurge are important; the latter, particularly Porphyry’s theory of the relation between Being, Life and Mind.
Various criticisms have been brought against a Platonistic construal of the musical work: that is, against the view that the musical work is a universal or kind or type, of which the performances are instances or tokens. Some of these criticisms are: (1) that musical works possess perceptual properties and universals do not; (2) that musical works are created and universals cannot be; (3) that universals cannot be destroyed and musical works can; (4) that parts of tokens of the same (...) type can be interchanged and still yield tokens of that type, whereas we cannot interchange parts of performances of the same work and still get performances of the work. Of these claims, (1) and (2) seem to be true, but are not incompatible with a Platonistic construal of the musical work, whereas (3) and (4) just seem to be false and, therefore, of no concern to the musical Platonist. (shrink)
The Cambridge Platonists were a group of religious thinkers who attended and taught at Cambridge from the 1640s until the 1660s. The four most important of them were Benjamin Whichcote, John Smith, Ralph Cudworth, and Henry More. The most prominent sentimentalist moral philosophers of the Scottish Enlightenment – Hutcheson, Hume, and Adam Smith – knew of the works of the Cambridge Platonists. But the Scottish sentimentalists typically referred to the Cambridge Platonists only briefly and in passing. The surface of Hutcheson, (...) Hume, and Smith's texts can give the impression that the Cambridge Platonists were fairly distant intellectual relatives of the Scottish sentimentalists – great great-uncles, perhaps, and uncles of a decidedly foreign ilk. But this surface appearance is deceiving. There were deeply significant philosophical connections between the Cambridge Platonists and the Scottish sentimentalists, even if the Scottish sentimentalists themselves did not always make it perfectly explicit. (shrink)
The paper raises doubts concerning tenability of the platonistic conception of linguistic meaning. It gives examples of some problems that philosophers who employ entities from the realm of platonic objects as a kind of unexplained explainer tend to neglect.
Many philosophers posit abstract entities – where something is abstract if it is acausal and lacks spatio-temporal location. Theories, types, characteristics, meanings, values and responsibilities are all good candidates for abstractness. Such things raise an epistemological puzzle: if they are abstract, then how can we have any epistemic access to how they are? If they are invisible, intangible and never make anything happen, then how can we ever discover anything about them? In this article, I critically examine epistemological objections to (...) belief in abstract objects offered by Paul Benacerraf, Colin Cheyne and Hartry Field. (shrink)
The literature on mathematics suggests that intuition plays a role in it as a ground of belief. This article explores the nature of intuition as it occurs in mathematical thinking. Section 1 suggests that intuitions should be understood by analogy with perceptions. Section 2 explains what fleshing out such an analogy requires. Section 3 discusses Kantian ways of fleshing it out. Section 4 discusses Platonist ways of fleshing it out. Section 5 sketches a proposal for resolving the main problem facing (...) Platonists—the problem of explaining how our experiences make contact with mathematical reality. (shrink)
If anything is taken for granted in contemporary metaphysics, it is that platonism with respect to a discourse of metaphysical interest, such as fictional or mathematical discourse, affords a better account of the semantic appearances than nominalism, other things being equal. This belief is often motivated by the intuitively stronger one that the platonist can take the semantic appearances “at face-value” while the nominalist must resort to apparently ad hoc and technically problematic machinery in order to explain those appearances (...) away. -/- In this paper, I argue that, on any natural construal of “face-value”, the platonist, like the nominalist, does not in general seem to be able to take the semantic appearances at face-value. And insofar as the nominalist is forced to adopt apparently ad hoc and technically problematic machinery in order to explain those appearances away, the platonist is generally forced to adopt machinery which is at least prima facie ad hoc and technically problematic as well. One moral of the story is that the thesis that platonism affords a better account of the semantic appearances than nominalism, other things being equal, is not trivial. Another is that we should rethink our methodology in metaphysics. (shrink)
In this paper, I explore the origins of the ‘problem of universals’. I argue that the problem has come to be badly formulated and that consideration of it has been impeded by falsely supposing that Platonic Forms were ever intended as an alternative to Aristotelian universals. In fact, the role that Forms are supposed by Plato to fulfill is independent of the function of a universal. I briefly consider the gradual mutation of the problem in the Academy, in Alexander of (...) Aphrodisias, and among some of the major Neoplatonic commentators on Aristotle, including Porphyry and Boethius. (shrink)
In a previous paper, Thomas V. Morris and I sketched a view on which abstract objects, in particular, properties, relations, and propositions (PRPs), are created by God no less than contingent, concrete objects. In this paper r suggest a way of extending this account to cover mathematical objects as well. Drawing on some recent work in logic and metaphysics, I also develop a more detailed account of the structure of PRPs in answer to the paradoxes that arise on a naive (...) understanding of the structure ofthe abstract universe. (shrink)
Philosophers of mathematics agree that the only interpretation of arithmetic that takes that discourse at 'face value' is one on which the expressions 'N', '0', '1', '+', and 'x' are treated as proper names. I argue that the interpretation on which these expressions are treated as akin to free variables has an equal claim to be the default interpretation of arithmetic. I show that no purely syntactic test can distinguish proper names from free variables, and I observe that any semantic (...) test that can must beg the question. I draw the same conclusion concerning areas of mathematics beyond arithmetic. This paper is a greatly extended version of my response to Stewart Shapiro's paper in the conference 'Structuralism in physics and mathematics' held in Bristol on 2–3 December, 2006. (shrink)
Mathematics is about numbers, sets, functions, etc. and, according to one prominent view, these are abstract entities lacking causal powers and spatio-temporal location. If this is so, then it is a puzzle how we come to have knowledge of such remote entities. One suggestion is intuition. But `intuition' covers a range of notions. This paper identifies and examines those varieties of intuition which are most likely to play a role in the acquisition of our mathematical knowledge, and argues that none (...) of them, singly or in combination, can plausibly account for knowledge of abstract entities. (shrink)
This paper sees me clarify, elaborate, and defend the conclusions reached in my ‘Musical Works as Eternal Types’ in the wake of objections raised by Robert Howell, R. A. Sharpe, and Saam Trivedi. In particular, I claim that the thesis that musical works are discovered rather than created by their composers is obligatory once we commit ourselves to thinking of works of music as types, and once we properly understand the ontological nature of types and properties. The central argument of (...) the paper is ‘the argument from the eternal existence of properties’, its moral being that types are eternal entities because they inherit their existence conditions from their eternally existent property-associates. The two key premises in this argument—that properties exist eternally, and that a type exists just in case its property-associate exists—are motivated and then defended at length. (shrink)