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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,
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.
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,
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
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