# Return to Answer

 2 added 459 characters in body edited Apr 23 '17 at 20:13 Conifold 29.7k22 gold badges6161 silver badges153153 bronze badges If you are unwilling to accept on faith some sort of meta-theory (which has to be stronger than the Peano arithmetic itself) the answer is no, the "intended interpretation", a.k.a. the "standard interpretation", of arithmetic is ephemeral. But this should not be surprising, your own phrasing "namely, the one that consist of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard", i.e. only these and nothing else, can not be formalized in the first order logic. At a minimum you need the second-order arithmetic, and if you formalize such "intuitions" there (i.e. assume "full semantics"), then that's the "meta-theory" required to prove existence and uniqueness (up to isomorphism) of a well-founded model of Peano arithmetic, see discussion under Why do we know that Gödel sentences are true in the standard model of set theory, but do not know if the continuum hypothesis is? But even with all the meta-theory one wants, there is no one "standard model" of set theory (ZFC), so one should wonder how much stock to put into the alleged "intuitions" about arithmetic as well. I do not see however, how external world is of much help in "rigorously defining" addition of units. The usual view of those who see mathematical objects as non-fictions (whether platonists or intuitionists) is that arithmetic, and mathematics in general, is eternal/a priori, and while it can be applied empirically it depends on no such application. The "meta-theory" in such views is seen as formalizing either platonist mindsight (Frege, Gödel) or constructive intuition of the subject (Poincare, Brouwer), not anything empiricalexternal. Even Hilbert, who was a fictionalist about most of mathematics, still admitted such intuitions about symbols in meta-theory, see Was there a Kantian influence on Hilbert's formalist programme? Indeed, the whole idea of "one true arithmetic", and set-theoretic models of formal theories in general, is platonist in origin. At the root it is based on the naive, but psychologically powerful, stereotype that nouns talked about in mathematics actually refer to something out there, the way desks and chairs do. Already intuitionists weaken this stereotype to a point where their meta-theory is insufficient to prove that the "standard interpretation", or indeed any complete interpretation, of arithmetic exists. By denying the law of excluded middle they explicitly assert the opposite. If you are unwilling to accept on faith some sort of meta-theory (which has to be stronger than the Peano arithmetic itself) the answer is no, the "intended interpretation", a.k.a. the "standard interpretation", of arithmetic is ephemeral. But this should not be surprising, your own phrasing "namely, the one that consist of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard", i.e. only these and nothing else, can not be formalized in the first order logic. At a minimum you need the second-order arithmetic, and if you formalize such "intuitions" there (i.e. assume "full semantics"), then that's the "meta-theory" required to prove existence and uniqueness (up to isomorphism) of a well-founded model of Peano arithmetic, see discussion under Why do we know that Gödel sentences are true in the standard model of set theory, but do not know if the continuum hypothesis is? But even with all the meta-theory one wants, there is no one "standard model" of set theory (ZFC), so one should wonder how much stock to put into the alleged "intuitions" about arithmetic as well. I do not see however, how external world is of much help in "rigorously defining" addition of units. The usual view of those who see mathematical objects as non-fictions (whether platonists or intuitionists) is that arithmetic, and mathematics in general, is eternal/a priori, and while it can be applied empirically it depends on no such application. The "meta-theory" in such views is seen as formalizing either platonist mindsight (Frege, Gödel) or constructive intuition of the subject (Poincare, Brouwer), not anything empirical. Indeed, the whole idea of "one true arithmetic", and set-theoretic models of formal theories in general, is platonist in origin. At the root it is based on the naive, but psychologically powerful, stereotype that nouns talked about in mathematics actually refer to something out there, the way desks and chairs do. If you are unwilling to accept on faith some sort of meta-theory (which has to be stronger than the Peano arithmetic itself) the answer is no, the "intended interpretation", a.k.a. the "standard interpretation", of arithmetic is ephemeral. But this should not be surprising, your own phrasing "namely, the one that consist of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard", i.e. only these and nothing else, can not be formalized in the first order logic. At a minimum you need the second-order arithmetic, and if you formalize such "intuitions" there (i.e. assume "full semantics"), then that's the "meta-theory" required to prove existence and uniqueness (up to isomorphism) of a well-founded model of Peano arithmetic, see discussion under Why do we know that Gödel sentences are true in the standard model of set theory, but do not know if the continuum hypothesis is? But even with all the meta-theory one wants, there is no one "standard model" of set theory (ZFC), so one should wonder how much stock to put into the alleged "intuitions" about arithmetic as well. I do not see however, how external world is of much help in "rigorously defining" addition of units. The usual view of those who see mathematical objects as non-fictions (whether platonists or intuitionists) is that arithmetic, and mathematics in general, is eternal/a priori, and while it can be applied empirically it depends on no such application. The "meta-theory" in such views is seen as formalizing either platonist mindsight (Frege, Gödel) or constructive intuition of the subject (Poincare, Brouwer), not anything external. Even Hilbert, who was a fictionalist about most of mathematics, still admitted such intuitions about symbols in meta-theory, see Was there a Kantian influence on Hilbert's formalist programme? Indeed, the whole idea of "one true arithmetic", and set-theoretic models of formal theories in general, is platonist in origin. At the root it is based on the naive, but psychologically powerful, stereotype that nouns talked about in mathematics actually refer to something out there, the way desks and chairs do. Already intuitionists weaken this stereotype to a point where their meta-theory is insufficient to prove that the "standard interpretation", or indeed any complete interpretation, of arithmetic exists. By denying the law of excluded middle they explicitly assert the opposite. 1 answered Apr 23 '17 at 19:58 Conifold 29.7k22 gold badges6161 silver badges153153 bronze badges If you are unwilling to accept on faith some sort of meta-theory (which has to be stronger than the Peano arithmetic itself) the answer is no, the "intended interpretation", a.k.a. the "standard interpretation", of arithmetic is ephemeral. But this should not be surprising, your own phrasing "namely, the one that consist of all numbers that we can get by repeatedly adding 1 together; any other numbers would be considered non-standard", i.e. only these and nothing else, can not be formalized in the first order logic. At a minimum you need the second-order arithmetic, and if you formalize such "intuitions" there (i.e. assume "full semantics"), then that's the "meta-theory" required to prove existence and uniqueness (up to isomorphism) of a well-founded model of Peano arithmetic, see discussion under Why do we know that Gödel sentences are true in the standard model of set theory, but do not know if the continuum hypothesis is? But even with all the meta-theory one wants, there is no one "standard model" of set theory (ZFC), so one should wonder how much stock to put into the alleged "intuitions" about arithmetic as well. I do not see however, how external world is of much help in "rigorously defining" addition of units. The usual view of those who see mathematical objects as non-fictions (whether platonists or intuitionists) is that arithmetic, and mathematics in general, is eternal/a priori, and while it can be applied empirically it depends on no such application. The "meta-theory" in such views is seen as formalizing either platonist mindsight (Frege, Gödel) or constructive intuition of the subject (Poincare, Brouwer), not anything empirical. Indeed, the whole idea of "one true arithmetic", and set-theoretic models of formal theories in general, is platonist in origin. At the root it is based on the naive, but psychologically powerful, stereotype that nouns talked about in mathematics actually refer to something out there, the way desks and chairs do.