michael liston

A class of broadly fictionalist and arealist objections (by Balaguer, Maddy, and others) to the indispensability argument for mathematical realism denies that indispensability entails realism. This paper argues that these objections fail against Putnam’s version of the argument: scientists indispensably use mathematics to draw conclusions about, explain, and understand the limits and reliability of the computational and modeling techniques they employ, and the very content of these conclusions and explanations appears to be unintelligible without a realistic understanding of the mathematics they presuppose.

In this paper I argue three things: (1) that the interactionist view underlying Benacerraf's (1973) challenge to mathematical beliefs renders inexplicable the reliability of most of our beliefs in physics; (2) that examples from mathematical physics suggest that we should view reliability differently; and (3) that abstract mathematical considerations are indispensable to explanations of the reliability of our beliefs.

... the Old World. Moreover, claiming they meant HARBIT by 'rabbit' all along amounts to claiming rather mysteriously that they were linguistically prepared for an eventuality that was completely outside their ken. Without an under ...

Do mathematical theories describe an objective reality that is discovered by mathematicians? Or are they fictions invented by mathematicians and not really descriptive of anything? This volume tackles these and related questions. It collects 10 essays and responses contributed to two symposia (funded by the Templeton Foundation) by important mathematicians (Gowers and du Sautoy), physicists (Penrose and Polkinghorne), and philosophers (Detlefsen, Leng, Lipton, Rosen, Shapiro, and Steiner). The essays are flanked by a brief introduction by the editor and a useful bibliography and index. Three themes recur throughout the essays. First is the connection between the felt objectivity of mathematical activity and the actual objectivity of mathematical reality. As both Gowers and du Sautoy report first-hand, mathematicians indisputably feel they are operating under constraints not subject to their control and discovering prior hidden truths rather than inventing novel fictions. The interesting question is what follows. Gödel famously argued that, because ‘the axioms force themselves upon us as being true’, we must have cognitive access to an objective mathematical reality that is at least as reliable as our perceptual access to the natural world. Du Sautoy seems to agree with this inference. Gowers is skeptical, preferring to pursue an ordinary language analysis of the conditions under which we are inclined to say that a piece of mathematics is discovered or invented but resisting any inference to any philosophical conclusion about real existence. Leng argues that the felt objectivity of mathematics need amount to no more than the logical compulsion we feel to accept conclusions that logically follow from assumptions we ourselves set. Detlefsen questions Gödel’s inference and the analogy between mathematical intuition and sense perception: in sense perception false contents are sometimes forced upon us as being true due to operations of deep-seated mental dispositions rather than the properties of material objects. More generally, involuntariness of belief may not be a reliable indicator of objective content. Such questions seem legitimate given the ease with which naïve comprehension principles force themselves on us yet lead to paradox. Second is the role of inference to the best explanation (IBE) in arguments for mathematical realism. Realist contributors cite, in addition to felt objectivity, several other phenomena associated with mathematics that appear to require explanation: the coherence of mathematical beliefs arrived at by different routes and their stability over time (Polkinghorne), the unanticipated riches that turn up in mathematical investigation (Penrose, Polkinghorne, du Sautoy, and Steiner), and what Wigner called ‘the unreasonable effectiveness’of mathematics at unlocking Nature’s hidden truths (Penrose, Polkinghorne, and Steiner). Our realists then argue that mathematical realism provides the best explanation of these sundry phenomena: they are to be expected if mathematicians are exploring an independent reality but not if they are inventing fictions. Thus, by IBE mathematical realism is more plausible than antirealist contenders. Echoing a large literature, Leng and du Sautoy find these appeals

Professor Gillies is a mathematical empiricist. According to him, natural systems instantiate mathematical concepts or properties; consequently, mathematical knowledge is generated and justified in a manner not essentially different from empirical knowledge (“decimal arithmetic is a very well-confirmed theory”). Those branches of mathematics that (when conjoined with suitable auxiliary hypotheses) have implications for the physical world are ultimately empirical. Though they are not falsifiable since the results of a negative test can be blamed on failure of an auxiliary hypothesis they are confirmable (and often in practice confirmed) by the empirical predictions that they support. Much of arithmetic, geometry, analysis and set theory is, for Gillies, empirical in this way. Gillies further holds that this style of mathematical empiricism has implications for the history of mathematics: “the growth of mathematics should exhibit the same patterns of development as any other theoretical branch of natural science.” To the extent that we find supporting case histories, Gillies claims, his view will be confirmed; mutatis mutandis for refuting counterexamples.

ACCORDING to DE Over ('Effective and non-effective referiVence', ANALYSIS 43.2, March 1983, pp. 85-91), Donnellan's distinction between referential and attributive uses of definite descriptions is best explained in terms of the notion of an effective method. The core of ...

russell’s views about the proper logical and epistemological treatment of names conspired to lead him to set aside considerations that support the claim that names are not definite descriptions. Though he appreciated those considerations, he famously argued that ordinary names are truncated definite descriptions. Nevertheless, his appreciation of the distinctive semantic behavior of ordinary names combined with his view that acquaintance comes in degrees led him to attempt to secure a semantically privileged status for ordinary names: only special kinds of descriptions can go proxy for ordinary names “used as names”. The paper attempts to tell this story, filling in gaps where russell doesn’t provide sufficient elaboration, and to draw some general conclusions about acquaintance-based approaches to names and singular thoughts.

it is commonly presupposed that all instances of the deflationary reference schema ‘f’ applies to x if and only if x is ‘f’ are correct. This paper argues, mainly on the basis of concrete example, that we have little reason to be confident about this presupposition: our tendency to believe the instances is based on local successes that may not be globally extendible. There is a problem of semantic projection, i argue, and standard accounts that would resolve or dissolve the problem are problematic.

This book is an important contribution to the philosophy of mathematics. It aims to clarify and answer questions about realism in connection with mathematics, in particular whether there exist mathematical objects (ontological realism) and whether all meaningful mathematical ...

Duhem used historical arguments to draw philosophical conclusions about the aim and structure of physical theory.  He argued against explanatory theories and in favor of theories that provide natural classifications of the phenomena.  This paper presents those arguments and, with the benefit of hindsight, uses them as a test case for the prevalent contemporary use of historical arguments to draw philosophical conclusion about science.  It argues that Duhem provides us with an illuminating example of philosophy of science developing as a contingent, though natural, response to problems arising in a particular scientific context and under a particular understanding of the history of science in that context.  It concludes that the history of science provides little support for interesting theses about the present or future state of science.

... 1 is prime and infers from it the true belief that there's a prime number less than 3. Bertie appears to ... Cheyne presents critiques of mathematical intuition (Gödel), apriorism (Hale and Wright's Neo-Fregean program, Katz's realistic rationalism), indispensability arguments (Putnam ...

... Second, the link between physics and reality via intuition was severed with the arrival of quantum mechanics in ... that, among a family of consistent histories plotting its position as a function of time, the histories ... Though they follow quantum laws, they appear to follow classical laws ...

... 1 is prime and infers from it the true belief that there's a prime number less than 3. Bertie appears to ... Cheyne presents critiques of mathematical intuition (Gödel), apriorism (Hale and Wright's Neo-Fregean program, Katz's realistic rationalism), indispensability arguments (Putnam ...

Duhem used historical arguments to draw philosophical conclusions about the aim and structure of physical theory.  He argued against explanatory theories and in favor of theories that provide natural classifications of the phenomena.  This paper presents those arguments and, with the benefit of hindsight, uses them as a test case for the prevalent contemporary use of historical arguments to draw philosophical conclusion about science.  It argues that Duhem provides us with an illuminating example of philosophy of science developing as a contingent, though natural, response to problems arising in a particular scientific context and under a particular understanding of the history of science in that context.  It concludes that the history of science provides little support for interesting theses about the present or future state of science.

Duhem used historical arguments to draw philosophical conclusions about the aim and structure of physical theory.  He argued against explanatory theories and in favor of theories that provide natural classifications of the phenomena.  This paper presents those arguments and, with the benefit of hindsight, uses them as a test case for the prevalent contemporary use of historical arguments to draw philosophical conclusion about science.  It argues that Duhem provides us with an illuminating example of philosophy of science developing as a contingent, though natural, response to problems arising in a particular scientific context and under a particular understanding of the history of science in that context.  It concludes that the history of science provides little support for interesting theses about the present or future state of science.

ACCORDING to DE Over ('Effective and non-effective referiVence', ANALYSIS 43.2, March 1983, pp. 85-91), Donnellan's distinction between referential and attributive uses of definite descriptions is best explained in terms of the notion of an effective method. The core of ...