*Marcus Giaquinto*

- Published in print:
- 2007
- Published Online:
- September 2007
- ISBN:
- 9780199285945
- eISBN:
- 9780191713811
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199285945.003.0008
- Subject:
- Philosophy, Logic/Philosophy of Mathematics

This chapter shows that some general arithmetical theorems almost certainly can be discovered by visualizing. First, it describes two examples of what look like acceptable routes to general theorems ...
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This chapter shows that some general arithmetical theorems almost certainly can be discovered by visualizing. First, it describes two examples of what look like acceptable routes to general theorems in which visualization has a non-redundant role. It then canvasses and assesses two natural objections to the acceptability of these examples. Finally, focusing on a modified version of the first example, the chapter gives a positive account of how it is possible to use visual imagery together with arithmetical concepts to reach a general theorem in a reliable way.Less

This chapter shows that some general arithmetical theorems almost certainly can be discovered by visualizing. First, it describes two examples of what look like acceptable routes to general theorems in which visualization has a non-redundant role. It then canvasses and assesses two natural objections to the acceptability of these examples. Finally, focusing on a modified version of the first example, the chapter gives a positive account of how it is possible to use visual imagery together with arithmetical concepts to reach a general theorem in a reliable way.

*A. W. F. EDWARDS*

- Published in print:
- 2013
- Published Online:
- September 2013
- ISBN:
- 9780199656592
- eISBN:
- 9780191748059
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199656592.003.0008
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics, History of Mathematics

The arithmetical triangle is the most famous of all number patterns. Apparently a simple listing of the binomial coefficients, it contains the triangular and pyramidal numbers of ancient Greece, the ...
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The arithmetical triangle is the most famous of all number patterns. Apparently a simple listing of the binomial coefficients, it contains the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers that arose in the Hindu studies of arrangements and selections, and (barely concealed) the Fibonacci numbers from medieval Italy. It reveals patterns that delight the eye, raises questions that tax the number-theorists, and amongst the ‘… coefficients, there are so many relations present that when someone finds a new identity, there aren’t many people who get excited about it any more, except the discoverer!’Less

The arithmetical triangle is the most famous of all number patterns. Apparently a simple listing of the binomial coefficients, it contains the triangular and pyramidal numbers of ancient Greece, the combinatorial numbers that arose in the Hindu studies of arrangements and selections, and (barely concealed) the Fibonacci numbers from medieval Italy. It reveals patterns that delight the eye, raises questions that tax the number-theorists, and amongst the ‘… coefficients, there are so many relations present that when someone finds a new identity, there aren’t many people who get excited about it any more, except the discoverer!’