Mancosu - Philosophy of Mathematical Practice (Oxford, 2008).pdf

56 marcus giaquintosay that the structure of the mental number line is that of a well-ordered setwith a single initial element and no terminal element. I will call this structureN. My proposal is twofold. First, in becoming aware in the indirect way justdescribed of the representational content of a visual category specification forthe mental number line, we have a grasp of a type of structured set, namely, aset of number marks on a line endless to the right taken in their left-to-rightorder of precedence. Secondly, we can have knowledge of the structure N asthe structure of a ‘number line’ of this type.Mathematical logicians are understandably preoccupied by the fact thatthe set of first-order Dedekind–Peano axioms for number theory has nonisomorphicmodels. So those axioms taken together fail to determine aunique structure as the structure of the natural numbers¹³ (under successor,addition, multiplication, and 0). This raises the question of how we determinethat structure, how the mind succeeds in picking out models of just oneisomorphism type (which we call ‘the standard model’ when identifying inthought isomorphic models.) One response is that our concept of the systemof natural numbers is essentially second-order; if we replace the first-orderinduction axiom by the second-order induction axiom of Dedekind’s originalpresentation, the result is an axiom set whose models are all isomorphic, asDedekind showed. The worry about this response is that in order to understandthe second-order induction axiom we would already need a cognitive grasp ofthe totality of sets of natural numbers; thus, if this response were adequate, ourgrasp of the set of natural numbers would depend on a prior grasp of its powerset—hardly a plausible position.I am inclined to think that there are two mutually reinforcing sourcesof comprehension of the natural number structure. One comes from ourunderstanding of the natural numbers as the denotations of the number-wordexpressions in our natural language, and (later) as the denotations of our writtennumerals. We pick up algorithms for generating the number-words/numerals,and we think of a number as what such an expression stands for. The numbersystem thus has the structure of the number-word system and the numeralsystem. So we can grasp the structure of the set of natural numbers under theirnatural ‘less-than’ ordering as the structure of the set of number-words (ornumerals) under their order of precedence. The second comes from the visualcategory specification described earlier. This in turn depends on representationsof space, time, and motion that cannot incorporate infinite bounded lengths,¹³ Models of these axioms are sets with functions for successor, plus, times and constant zero; a‘less-than’ relation can be defined in terms of these. The models of the first-order axioms vary withrespect to the induced ‘less-than’ relation: many, but not all, have infinite receding subsets.