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stoutfellow) wrote2009-02-15 02:43 pm
stoutfellow) wrote2009-02-15 02:43 pm
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Ramble, Part 67: Refoundation
With the emergence of the new geometries and the new algebras, and with the general increase in abstraction that mathematics underwent in the nineteenth century, there came a need to find new foundations for the subject. If Euclid's certainties were to be dispensed with, where were mathematicians to find secure footing? Three main schools of thought emerged; the first, Logicism, saw in the work of Cantor a new beginning.
One of the first major attempts to refound mathematics on purely logical grounds was that of Gottlob Frege. His work rested on Cantor's ideas, although, for some reason, Cantor took a dislike to it and savaged it in his reviews.
On the one hand, Dedekind and Cauchy had shown that, if one accepts the natural numbers (here taken as beginning with 0), all the rest of the apparatus of mathematics followed: the integers, the rationals, the real numbers, Euclidean geometry, non-Euclidean geometry, calculus - the whole ball of wax. On the other, Cantor's discussion of cardinality opened the door to a purely logical definition of the natural numbers.
Frege began by defining numbers to be "the extensions of concepts". In this he lay somewhere between Leibniz' algebra of concepts and Cantor's new set theory; the extension of a concept was, essentially, the cardinality of the collection of items satisfying that concept. A concept had extension 0 if it was self-contradictory - if there were no items satisfying it. A concept had extension 1 if it did not have extension 0 and had the following property: if x satisfies the concept and y satisfies the concept, then x and y are identical.
This is a start, but obviously one couldn't explicitly define every one of the natural numbers. Instead, Frege took advantage of another idea of Dedekind: if we know what "0" means and, whenever we know what "n" means, we can define the successor of n - "n+1" - then we have implicitly defined all of the natural numbers. What, then, is the successor of n? Says Frege, it is the extension of the concept "member of the series of natural numbers ending with n". (Of course, this concept required careful definition, which Frege gave.) That series consists of the natural numbers 0, 1, ..., n - and there are (by the usual definition) n+1 of them!
And there we go; Frege had, implicitly, defined all of the natural numbers, on sound logical foundations, and had thus given a firm basis to all of mathematics. Not for nothing did David Hilbert say, "No one shall drive us from the paradise which Cantor has created for us".
There was a snake in the garden.
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One of the first major attempts to refound mathematics on purely logical grounds was that of Gottlob Frege. His work rested on Cantor's ideas, although, for some reason, Cantor took a dislike to it and savaged it in his reviews.
On the one hand, Dedekind and Cauchy had shown that, if one accepts the natural numbers (here taken as beginning with 0), all the rest of the apparatus of mathematics followed: the integers, the rationals, the real numbers, Euclidean geometry, non-Euclidean geometry, calculus - the whole ball of wax. On the other, Cantor's discussion of cardinality opened the door to a purely logical definition of the natural numbers.
Frege began by defining numbers to be "the extensions of concepts". In this he lay somewhere between Leibniz' algebra of concepts and Cantor's new set theory; the extension of a concept was, essentially, the cardinality of the collection of items satisfying that concept. A concept had extension 0 if it was self-contradictory - if there were no items satisfying it. A concept had extension 1 if it did not have extension 0 and had the following property: if x satisfies the concept and y satisfies the concept, then x and y are identical.
This is a start, but obviously one couldn't explicitly define every one of the natural numbers. Instead, Frege took advantage of another idea of Dedekind: if we know what "0" means and, whenever we know what "n" means, we can define the successor of n - "n+1" - then we have implicitly defined all of the natural numbers. What, then, is the successor of n? Says Frege, it is the extension of the concept "member of the series of natural numbers ending with n". (Of course, this concept required careful definition, which Frege gave.) That series consists of the natural numbers 0, 1, ..., n - and there are (by the usual definition) n+1 of them!
And there we go; Frege had, implicitly, defined all of the natural numbers, on sound logical foundations, and had thus given a firm basis to all of mathematics. Not for nothing did David Hilbert say, "No one shall drive us from the paradise which Cantor has created for us".
There was a snake in the garden.
Previous Next
Ramble Contents