Michael Potter
- Published in print:
- 2002
- Published Online:
- May 2007
- ISBN:
- 9780199252619
- eISBN:
- 9780191712647
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780199252619.003.0008
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
In the previous chapter, it was shown that the Tractatus accounted for only a limited part of arithmetic at best. When Russell read Wittgenstein's manuscript, he was convinced that this was a gap ...
More
In the previous chapter, it was shown that the Tractatus accounted for only a limited part of arithmetic at best. When Russell read Wittgenstein's manuscript, he was convinced that this was a gap that needed to be filled. Curiously, it was not the omission of an account of real numbers that he found egregious, but that of a general account of cardinals. Russell was not willing simply to abandon the development of mathematics from the theory of types. He accepted, though, some of Wittgenstein's criticisms of the account he had given in Principia. He took the opportunity presented by the publication of a second edition of Principia to prepare a new Introduction indicating how mathematics could be based on a new theory of types consonant with the parts of Wittgenstein's account that Russell agreed with.Less
In the previous chapter, it was shown that the Tractatus accounted for only a limited part of arithmetic at best. When Russell read Wittgenstein's manuscript, he was convinced that this was a gap that needed to be filled. Curiously, it was not the omission of an account of real numbers that he found egregious, but that of a general account of cardinals. Russell was not willing simply to abandon the development of mathematics from the theory of types. He accepted, though, some of Wittgenstein's criticisms of the account he had given in Principia. He took the opportunity presented by the publication of a second edition of Principia to prepare a new Introduction indicating how mathematics could be based on a new theory of types consonant with the parts of Wittgenstein's account that Russell agreed with.
Victor J. Katz, Menso Folkerts, Barnabas Hughes, Roi Wagner, and J. Lennart Berggren (eds)
- Published in print:
- 2016
- Published Online:
- January 2018
- ISBN:
- 9780691156859
- eISBN:
- 9781400883202
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691156859.001.0001
- Subject:
- Mathematics, History of Mathematics
Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the ...
More
Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the time. This book presents original Latin, Hebrew, and Arabic sources of medieval mathematics, and shows their cross-cultural influences. Most of the Hebrew and Arabic sources appear here in translation for the first time. Readers will discover key mathematical revelations, foundational texts, and sophisticated writings by Latin, Hebrew, and Arabic-speaking mathematicians, including Abner of Burgos's elegant arguments proving results on the conchoid—a curve previously unknown in medieval Europe; Levi ben Gershon's use of mathematical induction in combinatorial proofs; Al-Muʾtaman Ibn Hūd's extensive survey of mathematics, which included proofs of Heron's Theorem and Ceva's Theorem; and Muhyī al-Dīn al-Maghribī's interesting proof of Euclid's parallel postulate. The book includes a general introduction, section introductions, footnotes, and references.Less
Medieval Europe was a meeting place for the Christian, Jewish, and Islamic civilizations, and the fertile intellectual exchange of these cultures can be seen in the mathematical developments of the time. This book presents original Latin, Hebrew, and Arabic sources of medieval mathematics, and shows their cross-cultural influences. Most of the Hebrew and Arabic sources appear here in translation for the first time. Readers will discover key mathematical revelations, foundational texts, and sophisticated writings by Latin, Hebrew, and Arabic-speaking mathematicians, including Abner of Burgos's elegant arguments proving results on the conchoid—a curve previously unknown in medieval Europe; Levi ben Gershon's use of mathematical induction in combinatorial proofs; Al-Muʾtaman Ibn Hūd's extensive survey of mathematics, which included proofs of Heron's Theorem and Ceva's Theorem; and Muhyī al-Dīn al-Maghribī's interesting proof of Euclid's parallel postulate. The book includes a general introduction, section introductions, footnotes, and references.
J. Levesque Hector
- Published in print:
- 2012
- Published Online:
- August 2013
- ISBN:
- 9780262016995
- eISBN:
- 9780262301411
- Item type:
- chapter
- Publisher:
- The MIT Press
- DOI:
- 10.7551/mitpress/9780262016995.003.0004
- Subject:
- Computer Science, Artificial Intelligence
This chapter discusses how to write the sorts of Prolog programs that will be used in the rest of the book. Section 1 examines what it means for a program to be fully correct. Section 4.2 introduces ...
More
This chapter discusses how to write the sorts of Prolog programs that will be used in the rest of the book. Section 1 examines what it means for a program to be fully correct. Section 4.2 introduces a new program in a blocks world and studies how it was written. This leads to a discussion of recursion in Section 4.3, and to its companion, mathematical induction, in Section 4.4. Section 4.5 considers the issue of programs that run forever and how to avoid writing them. Section 4.6 looks at a more complex predicate as it appears in the blocks-world program. Finally, Section 4.7 introduces the issue of program efficiency.Less
This chapter discusses how to write the sorts of Prolog programs that will be used in the rest of the book. Section 1 examines what it means for a program to be fully correct. Section 4.2 introduces a new program in a blocks world and studies how it was written. This leads to a discussion of recursion in Section 4.3, and to its companion, mathematical induction, in Section 4.4. Section 4.5 considers the issue of programs that run forever and how to avoid writing them. Section 4.6 looks at a more complex predicate as it appears in the blocks-world program. Finally, Section 4.7 introduces the issue of program efficiency.
Neil Tennant
- Published in print:
- 2022
- Published Online:
- March 2022
- ISBN:
- 9780192846679
- eISBN:
- 9780191939167
- Item type:
- chapter
- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780192846679.003.0011
- Subject:
- Philosophy, Logic/Philosophy of Mathematics
We prove in the metalanguage, by induction on natural numbers n, that derivations can be given in free Core Logic of all instances of Schema N (#xΦx = n ⊣⊢ ∇nxΦx). We also prove all of the ...
More
We prove in the metalanguage, by induction on natural numbers n, that derivations can be given in free Core Logic of all instances of Schema N (#xΦx = n ⊣⊢ ∇nxΦx). We also prove all of the Dedekind‒Peano postulates for successor arithmetic, including the Principle of Mathematical Induction. To this end one needs only the rules of free Core Logic itself, the rules for 0, s and # set out in Chapter 10, and certain logical rules about one-one mappings that are clearly stated. Among the lemmas proved along the way to the Dedekind‒Peano postulates is the result known as ‘Frege’s trick’: any natural number is the number of naturals preceding it. All derivations are given with complete formal rigor, but with additional commentary in ‘logician’s English’ to convey the gist of the formal work.Less
We prove in the metalanguage, by induction on natural numbers n, that derivations can be given in free Core Logic of all instances of Schema N (#xΦx = n ⊣⊢ ∇nxΦx). We also prove all of the Dedekind‒Peano postulates for successor arithmetic, including the Principle of Mathematical Induction. To this end one needs only the rules of free Core Logic itself, the rules for 0, s and # set out in Chapter 10, and certain logical rules about one-one mappings that are clearly stated. Among the lemmas proved along the way to the Dedekind‒Peano postulates is the result known as ‘Frege’s trick’: any natural number is the number of naturals preceding it. All derivations are given with complete formal rigor, but with additional commentary in ‘logician’s English’ to convey the gist of the formal work.
