On 2/28/2014 4:05 PM, email@example.com wrote: > On Friday, 28 February 2014 16:15:18 UTC+1, fom wrote: >> On 2/28/2014 2:07 AM, firstname.lastname@example.org wrote: >> >>> On Thursday, 27 February 2014 23:15:28 UTC+1, fom wrote: >> >>> >> >>> >> >>>> >> >>>> What exactly do you mean when you distinguish "+1" in >> >>>> >> >>>> the manner you have given above? >> >>> >> >>> Either you know it or you don't. It is the first step of mathematical understanding. It cannot be formalized without assuming other primitives which however are not as primitive, in the development of the human race as well as in the development of the single individual. ( Ontogenesis repeats die phylogenesis, haeckel) >> >> >> >> Well, I disagree with that. > > That makes it not become false. >> >> >> >> But, it is ridiculously difficult. >> > It is very simple. Everybody learns to count and then to add, first 1, later larger numbers. From that all mathematics is evolving. All formalization of these foundations is not necessary (I see that every student knows the natural numbers before I invent them) but at most a hobby of logicians. They may do so, if they like, but they should ntz pretend that insufficient formalization, like Peanos's, is superior to the real foundation of mathematics.
Yes. But that is a different matter.
Although I never found the quote, Husserl said that Bolzano held logic to be good only for instruction. And, that is what it is used for in mathematics. Proofs are used to teach and proofs are used to communicate that one statement follows another.
Questions concerning foundations are justified, but, they should not lead to ontologies.
I gave an answer on math.stackexchange.com to a gentleman asking about whether or not the idea that a real vector space is a set had been axiomatized. The other responses gave some minor explanations reflecting how simply one could define them. I outlined the full construction from natural numbers.
In his reply, he stated that he fully understood the construction. Yet, it did not answer his question. At that point I understood.
I explained that when one does a geometry problem, one does "analysis" by working the problem both backward and forward. To work backward, one assumes the solution and seeks antecedents.
Once analysis is complete, one sits down and writes the entire proof synthetically.
The process is inherently circular. Moreover, our foundational theories are arrived at in the same manner.
But, in synthesis, one is merely constructing what one already knows.
He accepted that answer.
I am assuming that this is the nature of your point here.