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The Logic of Number
In The Logic of Number , Neil Tennant defines and develops his Natural Logicist account of the foundations of the natural, rational, and real numbers. Based on the logical system free Core Logic, the central method is to formulate rules of natural deduction governing variable-binding number-abstraction operators and other logico-mathematical expressions such as zero and successor. These enable 'single-barreled' abstraction, in contrast with the 'double-barreled' abstraction effected by principles such as Frege's Basic Law V, or Hume's Principle.
Natural Logicism imposes upon its account of the numbers four conditions of First, one must show how it is that the various kinds of number are applicable in our wider thought and talk about the world. This is achieved by deriving all instances of three respective Schema N for the naturals, Schema Q for the rationals, and Schema R for the reals. These provide truth-conditions for statements deploying terms referring to numbers of the kind in question. Second, one must show how it is that the naturals sit among the rationals as themselves again, and the rationals likewise among the reals. Third, one should reveal enough of the metaphysical nature of the numbers to be able to derive the mathematician's basic laws governing them. Fourth, one should be able to demonstrate that there are uncountably many reals.
Natural Logicism is realistic about the limits of logicism when it comes to treating the real numbers, for which, Tennant argues, one needs recourse to geometric intuition for deeper starting-points, beyond which logic alone will then deliver the sought results, with absolute formal rigor. The resulting program enables one to delimit, in a principled way, those parts of number theory that are produced by the Kantian understanding alone, and those parts that depend on recourse to (very simple) a priori geometric intuitions.
Natural Logicism imposes upon its account of the numbers four conditions of First, one must show how it is that the various kinds of number are applicable in our wider thought and talk about the world. This is achieved by deriving all instances of three respective Schema N for the naturals, Schema Q for the rationals, and Schema R for the reals. These provide truth-conditions for statements deploying terms referring to numbers of the kind in question. Second, one must show how it is that the naturals sit among the rationals as themselves again, and the rationals likewise among the reals. Third, one should reveal enough of the metaphysical nature of the numbers to be able to derive the mathematician's basic laws governing them. Fourth, one should be able to demonstrate that there are uncountably many reals.
Natural Logicism is realistic about the limits of logicism when it comes to treating the real numbers, for which, Tennant argues, one needs recourse to geometric intuition for deeper starting-points, beyond which logic alone will then deliver the sought results, with absolute formal rigor. The resulting program enables one to delimit, in a principled way, those parts of number theory that are produced by the Kantian understanding alone, and those parts that depend on recourse to (very simple) a priori geometric intuitions.
376 pages, Hardcover
Published May 25, 2022
7 people want to read
About the author
Neil Tennant
15 books12 followersThere is more than one author by this name in the Goodreads data base.
For the Philosophy author, see: Neil Tennant
Neil Francis Tennant is an English musician, singer, songwriter, music journalist and co-founder of the synthpop duo Pet Shop Boys, which he formed with Chris Lowe in 1981. He also was a journalist for Smash Hits, and was assistant editor for the magazine for a period in the mid-1980s.
In 1975, having completed a degree in history at North London Polytechnic (now London Metropolitan University), Tennant worked for two years as London editor for Marvel UK, the UK branch of Marvel Comics. He also wrote occasional features for the comics, including interviews with pop stars Marc Bolan and Alex Harvey. In 1977, he moved to Macdonald Educational Publishing where he edited The Dairy Book Of Home Management and various illustrated books about cookery, playing the guitar, and other home interests. Then he moved to ITV Books where he edited TV tie-in books. [wikipedia]
For the Philosophy author, see: Neil Tennant
Neil Francis Tennant is an English musician, singer, songwriter, music journalist and co-founder of the synthpop duo Pet Shop Boys, which he formed with Chris Lowe in 1981. He also was a journalist for Smash Hits, and was assistant editor for the magazine for a period in the mid-1980s.
In 1975, having completed a degree in history at North London Polytechnic (now London Metropolitan University), Tennant worked for two years as London editor for Marvel UK, the UK branch of Marvel Comics. He also wrote occasional features for the comics, including interviews with pop stars Marc Bolan and Alex Harvey. In 1977, he moved to Macdonald Educational Publishing where he edited The Dairy Book Of Home Management and various illustrated books about cookery, playing the guitar, and other home interests. Then he moved to ITV Books where he edited TV tie-in books. [wikipedia]
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