On 2/2/2013 1:24 PM, Virgil wrote: > In article > <firstname.lastname@example.org>, > WM <email@example.com> wrote: > >> On 2 Feb., 02:56, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: >> >>> "The logicist reduction of the concept of natural number met a >>> difficulty on this point, since the definition of ?natural number¹ >>> already given in the work of Frege and Dedekind is impredicative. More >>> recently, it has been argued by Michael Dummett, the author, and Edward >>> Nelson that more informal explanations of the concept of natural number >>> are impredicative as well. That has the consequence that impredicativity >>> is more pervasive in mathematics, and appears at lower levels, than the >>> earlier debates about the issue generally presupposed." >> >> I do not agree with these authors on this point. >>> >>> So, how on earth do you know that induction is a correct >>> principle over the natural numbers? >> >> If a theorem is valid for the number k, and if from its validity for n >> + k the validity for n + k + 1 can be concluded with no doubt, then n >> can be replaced by n + 1, and the validity for n + k + 2 is proven >> too. This is the foundation of mathematics. To prove anything about >> this principle is as useless as the proof that 1 + 1 = 2. > > That inductive argument appears to be based on the very same flaws that > WM objects to in allowing actual infiniteness.
It is. That is the whole point of the quoted article.
In the last statement, WM is reasserting one of Poincare's observations.
But, the real problem is that everyone is looking at arithmetical foundations. Frege retracted his life's work and tried to direct everyone's attention back to geometry.
What foundational thinkers avoid because of "circularity" is polarity and duality in projective geometry. Embrace it. Understand it. And most of the crap falls away.