From the excerpt we have of Science Without Numbers, I don't see why mathematics has to be conservative. On page 13, Field says that "it would be extremely surprising if it were to be discovered that standard mathematics implied that there are at least 10^6 non-mathematical objects in the universe, or that the Paris Commune was defeated." But I don’t see why this is true. By 'standard mathematics' does he mean to rule out any application of mathematics to non-mathematical objects? I realize pure mathematics is independent of applications to non-mathematical objects, but even pure mathematics can be useful to non-mathematical enterprises. I take it that major mathematical discoveries are important partly because we expect them to be applied to non-mathematical projects someday.

So, why does Field say that "Good mathematics is conservative" (13)? I think that good mathematics is essential to genuinely new discoveries about the world. I know I'm just missing something important, though. (Obviously, good mathematics is consistent, and if consistency in pure mathematics requires conservativeness, then I must be wrong. But I don't quite understand the relationship between consistency and conservativeness, either.)

-- HelenHabermann - 26 Nov 2006

"I realize pure mathematics is independent of applications to non-mathematical objects." That's the point: we would regard it as a defect in an applicable mathematical theory if it, on its own, had consequences about the Paris Commune or the number of nonmathematical objects in the universe. The worst offenders are inconsistent theories: an inconsistent theory, for example, has as a consequence that the Paris Commune was defeated and there are at least 10^6 non-mathematical objects in the universe, since an inconsistent theory has every sentence as a consequence. Thus, if a theory is inconsistent, it is not conservative. The converse is not true: As Field explains, set theory plus the axiom that there is a set of all nonmathematical objects is a standard theory of applicable mathematics—the axiom that there is a set of all nonmathematical objects guarantees that the nonmathematical objects can be counted, and the theory is conservative If, however, one adds the axiom that every set is finite, the resulting theory is consistent (it is the standard theory of finite sets), but not conservative—it has as a consequence that the set of all nonmathematical objects is finite.

-- ShaughanLavine - 26 Nov 2006

Let's take a mathematical theory that often gets applied, say, number theory, %$ P$%. Let's take a theory %$ T$% to which we wish to apply it. The vocabulary of %$ T$% is, in the normal case, disjoint from that of number theory. Consider the obvious theory in which to formulate applications: %$ T\cup P$%. _ It isn't completely clear what that should mean, since %$ T$% and %$ P$% have disjoint domains. We have to patch things up a bit: add predicates %$ \mathbb{P}$% and %$ \mathbb{T}$% for the domains of the two theories, and restrict the quantifiers appropriately. Call that theory %$ P+T$%. That theory has no applications of numbers to the domain of %$ T$%. %$ P$% is not, contrary to what you have been told all your life, an applicable theory. What is the minimal usual application of numbers? We use them to count things. What does that mean? It means we put sets of numbers into correspondence with sets of things. We need to augment %$ P+T$% with a theory of sets and of correspondences. Luckily, we have a theory that covers both (at least we are usually said to), set theory, %$ S$%. So, we need to work in %$ P+T+S$%. Of course, that is just as useless as %$ P+T$% was. This is bad. What we need is a new theory, applicable set theory, which I'll call %$ A$%. That theory allows that things other than sets can be members of sets. It also allows definitions of sets that involve the relations, constants, and functions of the theories of the things other than sets. Example, the following is a set: %$ \{x:{\mathbb{P}(x)}\land (\exists y)y+y=x\}.$% In addition, we want to be able to count the things that are not sets. What we count is sets, and so that means that we need to assume that all the things that are not sets form a set. In order to form the sets we want, we also need that we can separate out subsets of a set using any definable properties using the full vocabulary of %$ P+T+S$%, not just that of %$ S$%. That does it.

The theory %$ A$% is consistent if and only if %$ S$% is, since we can just take the model in which there are no nonsets. But that isn't good enough: in order to always be able to count things, we want to be sure that, if a theory %$ T$% is consistent, then so is the theory formed by adding %$ A$% to %$ T$%. That is conservativeness.

-- ShaughanLavine - 28 Nov 2006