*amateur*scientist), a

*natural*division of mathematics is the following list (pp. 42-3 of Five Questions):

......Symmetry, Invariance and Language

......Counting, with Numbers and Language

......Order

......Proof

......Computation and Complexity

......Paradoxes and Meta-theorems

......Probability

......Games and Information Dynamics

......Prediction and Dynamical Systems

And the following, from Feferman's Answers (pp. 131-2) to those Five Questions (on PoM), is, I think, very true.

Discovery in mathematics is one of the highest exercises of creative intelligence. But confirmation of mathematical discoveries requires rigorous calculation and demonstration, and in this respect mathematics is logical at its core. Moreover, mathematics is progressive, it builds on what came before. Thus, since there can be no infinite regress, from the point of view of logic mathematics must rest ultimately on some sort of axiomatic foundations. While mathematicians may accept this in principle, there is a sharp dichotomy between the logicians’ conception of mathematics and that of the practicing mathematician. The latter pays little or no attention to logical or foundational axioms, even if he or she subscribes to some overall foundational viewpoint such as that of axiomatic set theory. And in fact, the logical picture of mathematics bears little relation to the logical structure of mathematics as it works out in practice. The use of certain basic structures like the natural numbers and the real numbers (and of structures built directly from them like the integers, rationals and complex numbers) is ubiquitous, and there is constant appeal to such principles as proof by induction and definition by recursion on the naturals and of the lub principle for the real numbers. But these are not viewed from an axiomatic point of view, e.g. from that of the Peano Axioms for the naturals. The essential difference is that the language of PA is limited to a fixed vocabulary, whereas induction and recursion can be applied in any subject in which natural numbers play some sort of role. For example, the operation x^n is defined in any (multiplicative) semi-group for every element x and natural number n, and its properties are proved by induction on n. So even where the practicing mathematician invokes the basic axioms of the natural numbers, that is done without restriction to a fixed vocabulary. According to the current set-theoretical point of view, all such concepts that the mathematician might want to use in addition to those expressed in PA are defined in the language of ZFC, so we need only look no further in order to give full logical scope to what underlies daily mathematics. It seems however, that if we accept the language of set theory we ought to accept notions not defined in that language, such as the notion of truth in the set-theoretical universe. Moreover there are informal outlying notions that have mathematical coherence, but are not (as given) defined within set theory. [... Feferman proposes] an informal framework to account for mathematical practice and its actual and future possible applications in a more direct way than through the use of the various formal systems currently dominating logical work. This is work in progress, as an extension of my earlier work on unfolding of open-ended schematic systems. An essential new feature is the introduction of a quite general underlying “proto-mathematical” framework for operations and properties; that allows for the interaction of basic schematic systems like those for the natural numbers, real numbers, and subsets of any domain.Feferman is an antiplatonist, but his points appeal to me nonetheless (other promising logics were mentioned in that book, e.g. by Hintikka, Visser, Weir and Zalta).

......By contrast, although Hellman favours "an objective, broadly "realist" view of mathematics" he fails to notice how widespread

*open*quantification is in real mathematics. So he tries to distinguish (p. 162) between "absolutely every object" and "anything we would ever come to recognise as an object" (i.e. between 'all' and 'any'). Now, I liked the way he nicely listed (pp. 158-160) our collective failure to justify

*the axiom of infinity*, but not his distinguishing between quantifying over a given

*set*of ordinals, and over

*all*ordinals (p. 163):

The cases are, after all, entirely different. In the former, we areBut in ordinary maths 5 + 7 = 12 basically means any five things plus another seven things are twelve things, and those things can be anything whatsoever, e.g. possibilities, feelings, even proper classes. And "for all" just means "for any" in ordinary maths, of course. Even when we move to ordinals (starting from 1), the finite ordinals are given by a rule (keep adding 1), and the ordinals need another (include limit ordinals), but other sets of ordinals need yet another (stop somewhere), so Hellman's distinction is not really that natural.givena set and then asked to consider all subsets ofit, or all functions defined on it with values in some othergivenset (e.g. {0, 1}). That is, these are limited notions, "already restricted" as it were. Moreover, they are commonplace in mathematical practice. [...] But "all sets" or "all ordinals" in some putatively absolute sense are supposed to be entirely unlimited and unrestricted, and they raise suspicions in much the same way "absolutely all objects" does—or should, at any rate! Not surprisingly, they are quite foreign to ordinary mathematical practice.

......A more obvious distinction is between

*finite sequences*and those simply infinite

*endless sequences*that might well be expected to be indefinitely extensible, in a way slightly analogous to, but basically quite unlike (since arising from endlessness), vagueness (due to predicates not being well defined) or fuzziness (due to objects being not logical). And as he had already shown, we have no reason to assume that they are

*not*indefinitely extensible (and there is also this reason to assume they are).

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