*J. Gregor Fetterman*

- Published in print:
- 2009
- Published Online:
- March 2012
- ISBN:
- 9780195377804
- eISBN:
- 9780199848461
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780195377804.003.0016
- Subject:
- Psychology, Cognitive Psychology

The resurgence of interest in animal cognition was accompanied by the use of increasingly complex stimulus arrangements such as temporal events ...
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The resurgence of interest in animal cognition was accompanied by the use of increasingly complex stimulus arrangements such as temporal events and numerosities. The renewed emphasis on cognitive constructs and theories in animal learning resulted from a dynamic interplay of a set of variables, including the “paradigm shift” in the study of human learning and memory that preceded changes in the approaches to learning in nonhumans by 20 years or more. This chapter reviews research and theory on the ability of nonhuman animals to learn, remember, and discriminate between events that differ in duration and between those that differ in number — in each case, events with temporal extension. This property of time-based and number-based discriminations raises interesting questions for research and theories of the underlying mechanisms, such as the role of memory and the weighting of events early in the sequence versus those that occur later. The chapter begins with a brief history of attempts to understand how nonhuman animals discriminate temporal intervals and moves to a brief presentation of how animals discriminate numerosities.Less

The resurgence of interest in animal cognition was accompanied by the use of increasingly complex stimulus arrangements such as temporal events and numerosities. The renewed emphasis on cognitive constructs and theories in animal learning resulted from a dynamic interplay of a set of variables, including the “paradigm shift” in the study of human learning and memory that preceded changes in the approaches to learning in nonhumans by 20 years or more. This chapter reviews research and theory on the ability of nonhuman animals to learn, remember, and discriminate between events that differ in duration and between those that differ in number — in each case, events with temporal extension. This property of time-based and number-based discriminations raises interesting questions for research and theories of the underlying mechanisms, such as the role of memory and the weighting of events early in the sequence versus those that occur later. The chapter begins with a brief history of attempts to understand how nonhuman animals discriminate temporal intervals and moves to a brief presentation of how animals discriminate numerosities.

*Paolo Mancosu*

- Published in print:
- 2016
- Published Online:
- January 2017
- ISBN:
- 9780198746829
- eISBN:
- 9780191809095
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198746829.003.0004
- Subject:
- Philosophy, Logic/Philosophy of Mathematics, History of Philosophy

The standard Cantorian assignment of cardinal numbers to sets uses the criterion of one–one correspondence. The fruitfulness of this definition in mathematics and the lack of mathematically ...
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The standard Cantorian assignment of cardinal numbers to sets uses the criterion of one–one correspondence. The fruitfulness of this definition in mathematics and the lack of mathematically worked-out alternatives seemed to make any other approach impossible. Indeed, Gödel went as far as to offer an argument for why a generalization of arithmetic from the finite to the infinite would inevitably lead to Cantor’s theory of cardinals. Chapter 3 presents a theory of counting with infinite sets, the theory of numerosities, which assigns sizes to sets according to a different criterion from Cantor’s. The numerosities in this theory satisfy the part-whole axiom: if a set A is strictly included in a set B, then the numerosity of A is strictly smaller than the numerosity of B. The existence of a mathematical alternative to Cantor’s theory for capturing the notion of “size” for sets leads to deep mathematical, historical, and philosophical problems. In particular, Mancosu discusses in this chapter various philosophical claims that have been made in connection to Cantor’s theory of cardinal numbers (e.g. Gödel’s inevitability claim and Kitcher’s discussion of rational transitions between mathematical practices) showing that the new theory of numerosities leads to a much deeper appreciation of the issues involved.Less

The standard Cantorian assignment of cardinal numbers to sets uses the criterion of one–one correspondence. The fruitfulness of this definition in mathematics and the lack of mathematically worked-out alternatives seemed to make any other approach impossible. Indeed, Gödel went as far as to offer an argument for why a generalization of arithmetic from the finite to the infinite would inevitably lead to Cantor’s theory of cardinals. Chapter 3 presents a theory of counting with infinite sets, the theory of numerosities, which assigns sizes to sets according to a different criterion from Cantor’s. The numerosities in this theory satisfy the part-whole axiom: if a set *A* is strictly included in a set *B*, then the numerosity of *A* is strictly smaller than the numerosity of *B*. The existence of a mathematical alternative to Cantor’s theory for capturing the notion of “size” for sets leads to deep mathematical, historical, and philosophical problems. In particular, Mancosu discusses in this chapter various philosophical claims that have been made in connection to Cantor’s theory of cardinal numbers (e.g. Gödel’s inevitability claim and Kitcher’s discussion of rational transitions between mathematical practices) showing that the new theory of numerosities leads to a much deeper appreciation of the issues involved.