Humbug! Russell so went way out on the limb of "pure formalism", well before the essence of mathematics was discerned. "Pure" mathematics entails descriptions of KINDS of things that are being attended, together with whatever logically substantial concepts and conclusions are derived from those descriptions. That is how mankind created mathematics, and how it has progressively refined mathematics, to become a specialized *art of learning.* Outrageous formalism has done far more to inhibit mankind's mathematical progress than to facilitate it.
Cordially, Clyde - -------------------------------------------------- From: "Jonathan Crabtree" <email@example.com> Sent: Friday, July 05, 2013 5:54 PM To: <firstname.lastname@example.org> Subject: Re: Is logic part of mathematics - or is mathematics part of logic?
> Pure Mathematics is the class of all propositions of the form 'p implies > q' where p and q are propositions containing one or more variables, the > same in the two propositions, and neither p nor q contains any constants > except logical constants. > > And logical constants are all notions definable in terms of the following: > Implication, the relation of a term to a class of which it is a member, > the notion of such that, the notion of relation, and such further notions > as may be involved in the general notion of propositions of the above > form. > > In addition to these, mathematics uses a notion which is not a constituent > of the propositions which it considers, namely the notion of truth. > > Source: Principles of Mathematics Bertrand Russell 1903 > http://archive.org/stream/principlesofmath01russ#page/n35/mode/2up