What is a mathematical theory? We may define a theory as a set of axioms. Of course there needs to be more to a theory than that. What do we do with this set of axioms? Firstly, we need the rules of inference. These rules of inference tell us what we can and can't infer or derive from the set of axioms.
What can we infer or derive from the set of axioms? Theorems. That is, a theorem is any formula which follows from the axioms by repeated application of the rules. The theorems grow out of the axioms (as it were). This must mean that whatever is in the derived or inferred theorems must have already have been in the axioms. Nothing in the theorems which was not first in the axioms – even if hidden or disguised. This makes one ask: How, exactly, do the theorems ‘follow’ from the axioms? What do we mean by ‘follow’, ‘derive’ and ‘infer’?
However, despite the fact that we have a set of axioms for a mathematical theory, this theory "is empty until the axioms have been provided with an interpretation" (394). Intuitively it seems strange that we can ‘interpret’ axioms at all. After all, don’t we create the axioms? Or was Plato right – do we actually discover them? In any case, how do we interpret axioms? We "need to assign values to the primitive terms, and to show how the values of formulae may be derived from the values of their parts" (394). This makes mathematical meta-theory seems like semantics. In semantics too we ‘need to assign values’ to the predicates and phrases found in a truth-valued sentence. Not only that: we also have the other basic Fregean insight that we "show how the values of formulae may be derived from the values of their parts" (394). What are the ‘values of the primitive terms’? What are the primitive terms?
In any case, the interpretation of the axioms, or their primitive terms,
"is not something done by the theory itself, but something done by us, through another theory – the meta-theory" (394).
Clearly the mathematical theory can't interpret itself. It can't even be done by us through the theory itself. We must construct another theory, a meta-theory, in order to offer interpretations of the lower-level theory. That simply means that the theory exists, or can exist, even before an interpretation of its axioms is given. That is why lower-level mathematicians need not concern themselves with meta-theory at all – indeed, they rarely do.
One of the main things mathematical meta-theories do to theories is provides proofs of particular formulas in the theory (or, in Gödel’s case, a proof that a theorem can't in fact be proved). The important point is that the theory itself can't provide a proof of a formula within itself.
What would count as a proof of a formula? If
"we can show that the axioms generate all truths of arithmetic, and that p is such a truth, then we have proved that p is provable’"(395).
It's hard to imagine that the axioms of a particular theory can ‘generate all truths of arithmetic’; so perhaps this means many or all such theories in mathematics. In addition, if these axioms generate all the truths of arithmetic, then this must mean that the axioms, and perhaps the theory to which they belong, must exist before arithmetic. Or is it the case, instead, that arithmetic exists though it doesn't have its ‘truths’ proved until the meta-theory comes along? However, on a platonistic reading, the truths still exist before the proofs. The proofs are for us (as it were). It isn't that p, in the theory, isn't true until it's proved by the meta-theory. It's just the case that the meta-theory proves that it's true – for us. It would still be true without proof and it was true before proof.
It's often said that a mathematical theory or system can't be complete and fully consistent or fully consistent and complete. This is a question of proof as to consistency and completeness. That is, ‘there can be no proof of the completeness of arithmetic which permits a proof of its consistency and vice versa’ (395). So we either have a proof of arithmetic’s consistency without a proof of its completeness, or a proof of its completeness without a proof of its consistency.
Why is it, precisely, that we can't have both? More to the point, Gödel provided us with a proof that we can't have proofs of both completeness and consistency; though we can have a proof of either one or the other. Again we must stress that this is primarily a question of the set of axioms which generate arithmetic. That is, "we cannot know, of some system of axioms which is sufficient to generate arithmetic, that it is both complete and consistent" (395). Why can’t we show both completeness and consistency? The end result of this is that ‘there may be formulae of arithmetic which are true, but not provable’ (395). So, in this case at least, truth need not come along with proof. Isn’t that what the intuitionists are against? That is, if we have no proof of an equation or formula, then that formula, quite simply, can't be true. The proof brings its truth into existence; just as it's also said that proofs bring the actual numbers of arithmetic into existence (or at least their ‘constructions’).
What has all this to do with the failure of the logicist programme? We can now say that "no logical system, however refined, will suffice to generate the full range of mathematical truths" (395). Isn’t it the case that the logicists tried to reduce mathematics to logic? Here it is said that ‘no logical system will suffice to generate mathematics’? Does this amount to the same thing? That is, is the generation of mathematics from logic the same as the reduction of mathematics to logic? That is, if mathematics is genuinely generated from logical principles, even if that is not fully known, then clearly it will also be the case that mathematics can be reduced to the logic. If logic generates maths, then maths can be reduced to logic.
There also seems to be a connection being made here between the logicist programme, or between logic itself, and mathematical provability. It ‘follows too that we cannot treat mathematics as Hilbert had wished, merely as strings of provable formulae: the theory of “formalism” is false’ (395). That is, if mathematics is reducible to logic, then all mathematical formulae must be capable of being proved. This means that proof is essential in logic. However, as Gödel and others discovered, not all mathematical statements can be proved. Thus, because proof isn’t everything in mathematics, then maths must be something above and beyond logic, in which, we can say, proof is everything. This lack of (complete) provability in mathematics, in fact, makes it very different from logic.
Does this make mathematics non-a priori and therefore a posteriori? Or is it non-a priori without also being a posteriori – if that is possible? Whatever the truth is, mathematics, or its propositions, outstrip provability and, one must conclude, the principles of logic too!
This conclusion has a very platonistic ring to it. That is, ‘if there can be unprovable truths of mathematics, then mathematics cannot be reduced to the proofs whereby we construct it’ (395). This conclusion not only goes against Hilbert’s attempted reduction, as well as the logicist programme, but also against the intuitionists who believe that truth and proof are intimately connected. Indeed that one can't have truth without proof.
Not only that: the intuitionists also believed that the truths of mathematics were generated by the proofs whereby we construct them. As for the platonistic conclusion to all this, we can say that if truth is genuinely divorced from proof, and that mathematical truths can be true without thereby being proved to be true, then they must be true regardless of our means of proving or even knowing them. Thus they must be mind-independent in some substantial sense.
We can conclude that there ‘is a realm of mathematical truth, whether or not we can gain access to it through our own intellectual procedures’ (395). This is not just a question of the rejection of the truth = proof principle; but of the necessity of mathematical truth being necessarily connected to any of ‘our own intellectual procedures’ (395). Thus it isn't necessary that we have any kind of access to the ‘realm of mathematical truth’! We certainly don’t need causal or epistemic access to it. We may have access to this mathematical realm; though it isn't necessary that we do so. It would still exist, mind-independently, even if we didn't know that it in fact exists. This platonic realm doesn't even care about being known by us. It exists happily without being known by us.
Not only did Gödel prove that certain truths within mathematics can't be proved to be true, he also proved that this platonic realm of unprovable and provable mathematical truths exists. He proved that p can't be proved. However, p is still true; though not provably true.
The question remains as to whether or not Gödel needed this platonic realm in order to believe, or prove, what he said about the nature of mathematics. Can Godelianism be non-platonic or must it be platonic? Can we accept the notion of mathematics’ essential incomplete provability and yet not be a Platonist as well? Or do all these conclusions about mathematics come along with a commitment to Platonism?