On Saturday, September 30, 2017 at 8:12:47 PM UTC+2, John Gabriel wrote: > On Saturday, 30 September 2017 12:58:40 UTC-5, Me wrote: > > On Saturday, September 30, 2017 at 5:50:19 PM UTC+2, John Gabriel wrote: > > > > > > Peano assumes the prior existence of natural numbers. > > > > > Of course. But his axioms describe the PROPERTIES of those numbers, idiot. > > > Can you please reach some sort of agreement among your fellow [...] > academics? Because many of them don't think the same way. For example > [...] Dan Christensen is still under the belief that Peano's Crapaxioms > define the natural numbers.
There's no need for a general agreement concerning all (mathematical) questions. Certain disagreements concerning foundational questions are natural and healthy.
I personally prefer the view that the Penao Axioms /characterise/ the natural numbers (by stating their fundamental properties).
> Now as for properties: Thank goodness the Ancient Greeks didn't need the > [gineous] Peano to state the properties of numbers because <etc. etc.>
Their approach was admirable. But it does not satisfy modern requirements concerning formal precision.
> > > The successor function requires that the natural numbers are in place. > > > > > Sure. And its domain is IN, right. > > > > Nothing new, really. :-) > > > Oh boy, you have made an enemy out of [...] Dan Christensen.
Not sure about that. He may just POSTUALTE the existence of IN, such that ...
> > You can find a discussion of theses matters in Russell's "Introduction > > to Mathematical Philosophy" (Chapter I): > > > > https://people.umass.edu/klement/imp/imp-ebk.pdf > > > No thanks. If you don't mind, I have a very low opinion of Russell...
Well, Russell... It's indeed rather ironic that *he* and Whitehead got all the fame, and not Frege. Actually, in the preface of PM they write: "In all questions of logical analysis, our chief debt is to Frege."
Much later people started to recognice that:
"It is to be regretted that this first comprehensive and thorough-going presentation of a mathematical logic and the derivation of mathematics from it [is] so greatly lacking in formal precision in the foundations [...] that it represents in this respect a considerable step backwards as compared with Frege." (Kurt Goedel, Russell's mathematical logic, 1944)