more on this theme     |     more from this thinker

Single Idea 9561

[filed under theme 5. Theory of Logic / E. Structures of Logic / 6. Relations in Logic ]

Full Idea

Everything one needs to do with relations in mathematics can be done by taking a relation to be a set of ordered pairs. (Ordered triples etc. can be defined as order pairs, so that <x,y,z> is <x,<y,z>>).

Gist of Idea

The mathematics of relations is entirely covered by ordered pairs

Source

Charles Chihara (A Structural Account of Mathematics [2004], 07.2)

Book Ref

Chihara,Charles: 'A Structural Account of Mathematics' [OUP 2004], p.174

A Reaction

How do we distinguish 'I own my cat' from 'I love my cat'? Or 'I quite like my cat' from 'I adore my cat'? Nevertheless, this is an interesting starting point for a discussion of relations.

The 17 ideas from 'A Structural Account of Mathematics'

10192 We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride]
9547 Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara]
9550 We only know relational facts about the empty set, but nothing intrinsic [Chihara]
9551 What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara]
9549 The set theorist cannot tell us what 'membership' is [Chihara]
9552 Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara]
9553 Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara]
9559 If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara]
9561 The mathematics of relations is entirely covered by ordered pairs [Chihara]
9562 In simple type theory there is a hierarchy of null sets [Chihara]
9563 A pack of wolves doesn't cease when one member dies [Chihara]
9566 No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara]
9568 I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara]
9571 ZFU refers to the physical world, when it talks of 'urelements' [Chihara]
9573 The null set is a structural position which has no other position in membership relation [Chihara]
9572 Realists about sets say there exists a null set in the real world, with no members [Chihara]
9574 'Gunk' is an individual possessing no parts that are atoms [Chihara]