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Computers in mathematical inquiry
 in The Philosophy of Mathematical Practice
, 2008
"... Computers are playing an increasingly central role in mathematical practice. What are we to make of the new methods of inquiry? In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character, ..."
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Computers are playing an increasingly central role in mathematical practice. What are we to make of the new methods of inquiry? In Section 2, I survey some of the ways that computers are used in mathematics. These raise questions that seem to have a generally epistemological character,
Probabilistic Proofs and Transferability
 Philosophia Mathematica
"... One of the central questions in the philosophy of mathematics concerns the nature of mathematical knowledge. The version of this question familiar from [Benacerraf, 1973] asks how knowledge of any mathematical proposition could be consistent with any picture of the semantics of mathematical language ..."
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One of the central questions in the philosophy of mathematics concerns the nature of mathematical knowledge. The version of this question familiar from [Benacerraf, 1973] asks how knowledge of any mathematical proposition could be consistent with any picture of the semantics of mathematical language (and in particular with the apparently abstract and acausal nature of mathematical objects). However, there is a further question even granting existing knowledge of mathematical propositions, one may wonder what exactly it takes for a mathematician to come to know yet more propositions. To begin to address this question, I note that there is some extremely close connection in mathematics between knowledge and proof. Mathematicians often say that a claim is not known until a proof has been given, and an account somewhat like this is presupposed in some naturalistic discussions of mathematical knowledge (see [Horsten, 2001, pp. 1869], where he concedes that other means may provide knowledge of mathematical propositions, but suggests that proof must underlie a notion of “mathematical knowledge”).
Epistemic Value Theory and Judgment Aggregation
"... A foolish consistency is the hobgoblin of little minds. – Ralph Waldo Emerson We frequently make judgments about the world. Juries make judgments about whether defendants are guilty. Umpires make judgments about whether pitches are strikes. ..."
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A foolish consistency is the hobgoblin of little minds. – Ralph Waldo Emerson We frequently make judgments about the world. Juries make judgments about whether defendants are guilty. Umpires make judgments about whether pitches are strikes.
11 Computers in Mathematical Inquiry
"... Computers are playing an increasingly central role in mathematical practice. What are we to make of the new methods of inquiry? In Section 11.2, I survey some of the ways in which computers are used in mathematics. These raise questions that seem to have a generally epistemological ..."
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Computers are playing an increasingly central role in mathematical practice. What are we to make of the new methods of inquiry? In Section 11.2, I survey some of the ways in which computers are used in mathematics. These raise questions that seem to have a generally epistemological
Press. Mathematical Induction and Induction in Mathematics
"... However much we many disparage deduction, it cannot be denied that the laws established by induction are not enough. Frege (1884/1974, p. 23) At the yearly proseminar for firstyear graduate students at Northwestern, we presented some evidence that reasoning draws on separate cognitive systems for a ..."
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However much we many disparage deduction, it cannot be denied that the laws established by induction are not enough. Frege (1884/1974, p. 23) At the yearly proseminar for firstyear graduate students at Northwestern, we presented some evidence that reasoning draws on separate cognitive systems for assessing deductive versus inductive arguments (Rips, 2001a, 2001b). In the experiment we described, separate groups of participants evaluated the same set of arguments for deductive validity or inductive strength. For example, one of the validity groups decided whether the conclusions of these arguments necessarily followed from the premises, while one of the strength groups decided how plausible the premises made the conclusions. The results of the study showed that the percentages of “yes ” responses (“yes ” the argument is deductively valid or “yes ” the argument is inductively strong) were differently ordered for the validity and the strength judgments. In some cases, for example, the validity groups judged Argument A to be valid more often than Argument B, but the strength groups judged B inductively strong more often than A. Reversals of this sort suggest that people do not just see arguments as ranging along a single
Randomized Arguments are Transferable
, 2009
"... Easwaran has given a definition of transferability and argued that, under this definition, randomized arguments are not transferable. I show that certain aspects of his definition are not suitable for addressing the underlying question of whether or not there is an epistemic distinction between rand ..."
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Easwaran has given a definition of transferability and argued that, under this definition, randomized arguments are not transferable. I show that certain aspects of his definition are not suitable for addressing the underlying question of whether or not there is an epistemic distinction between randomized and deductive arguments. Furthermore, I demonstrate that for any suitable definition, randomized arguments are in fact transferable. Kenny Easwaran [Easwaran2008] has recently given a definition of transferability of mathematical proofs and has attempted to use this notion to epistemically differentiate between traditional deductive proofs and certain inductive arguments based on randomized trials (which I will call randomized arguments; more on this below). However, I will show that there are problems with Easwaran’s definition and further demonstrate that, for any suitable definition of the term, transferability does not epistemically distinguish between deductive proofs and randomized arguments. 1
ARISTOTELIAN REALISM
"... Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as rat ..."
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Aristotelian, or nonPlatonist, realism holds that mathematics is a science of the real world, just as much as biology or sociology are. Where biology studies living things and sociology studies human social relations, mathematics studies the quantitative or structural aspects of things, such as ratios, or patterns, or complexity,
Towards a theory of mathematical argument
"... Abstract. In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, an ..."
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Abstract. In this paper, I assume, perhaps controversially, that translation into a language of formal logic is not the method by which mathematicians assess mathematical reasoning. Instead, I argue that the actual practice of analyzing, evaluating and critiquing mathematical reasoning resembles, and perhaps equates with, the practice of informal logic or argumentation theory. It doesn’t matter whether the reasoning is a fullfledged mathematical proof or merely some nondeductive mathematical justification: in either case, the methodology of assessment overlaps to a large extent with argument assessment in nonmathematical contexts. I demonstrate this claim by considering the assessment of proofs, probabilistic evidence, computeraided proofs, and the acceptance of axioms. I also consider Jody Azzouni’s ‘derivation indicator ’ view of proofs because it places derivations – which may be thought to invoke formal logic – at the center of mathematical justificatory practice. However, when the notion of ‘derivation ’ at work in Azzouni’s view is clarified, it is seen to accord with, rather than to count against, the informal logical view I support. Finally, I pose several open questions for the development of a theory of mathematical argument.
Transferability and Proofs
, 2007
"... [Grice, 1957] argues for the following account of “nonnatural meaning ” (i.e., ordinary linguistic meaning): “A meantNN something by x ” is roughly equivalent to “A uttered x with the intention of inducing a belief by means of the recognition of this intention. ” [Grice, 1957, p. 384] In particular ..."
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[Grice, 1957] argues for the following account of “nonnatural meaning ” (i.e., ordinary linguistic meaning): “A meantNN something by x ” is roughly equivalent to “A uttered x with the intention of inducing a belief by means of the recognition of this intention. ” [Grice, 1957, p. 384] In particular, he adds, “A’s intending that the recognition should play this part implies... that he does not regard it as a foregone conclusion that the belief will be induced in the audience whether or not the intention behind the utterance is recognized.” The motivation for this particular condition is a series of examples in which A utters x with the intention of inducing a belief, and with the intention that this intention be recognized, but where (because the recognition of the intention plays no essential role in the formation of the belief) we don’t want to say that A “means ” anything by x. These examples include Herod presenting Salome with the head of John the Baptist (to get her to believe that John the Baptist