Please help. Great confusion about the "two levels of discourse'' mathematical logic.

Question

I regard mathematics as being build up in the following way: We have some collections of symbol and rules (which are and have to be described in a natural language) to manipulate these symboles. If we now fix certain string of symbols (which can turn out to be, for example, the ZFC axioms), we are able to derive via our rules all of what we consider to be mathematics.

So for me, any piece of mathematics, may it be a proposition or a definition, is just a deductible string (I probably am a formalist, although I don't have a clear graps on the different philosphical orientations a mathematician can adopt), so I don't have any issues with things like "absolute truth" since I only believe in my system of deduction and my intuition which helps me accept a proof as correct without writing out/reading the complete deduction of it, but rather only the "cornerstone" deductions steps, which then enable me to "write out the complete proof if I wanted" (which of course I would never do - I deliberately exagerated a little bit, in hope to clarify my view, since I feel that "platonistic" mathematicians, when talking about things like these, often misunderstand me).

Here is now a list of questions, which are all interconnected, and I think are most due to the fact that I confuse in which settings we talk about which objects (hence the title) :

Is my view of mathematics correct ?

Why is it that in many books on mathematical logic (for example im Ebbinghaus, Flum and Thomas' book) the first chapters, which describe how to we can manipulate these symbols (i.e. the syntax), often the word "set" is used and mathematical operations (like assignements) are performed, where we at this point don't have sets and functions and so on at our disposition (meaning although of course I intuitively know what they are, we just don't have them at disposal right know) ?

If I understand all appearances of the word "set" and assignements in those chapter to mean not a set in the sense of ZFC and just a collection of symbols and all uses of functions as applications of the simple rules of manipulations of those symbols, which permit me to replace some symbols with others, I can make sense of the syntax part. But when semantics come into play and we suddenly deal with structures like $\left(\mathbb{N},R^{\mathbb{N}}\right)$, I am totally thrown. We don't even have ZFC yet, so how can we talk about $\mathbb{N}$ ?

Last but not least I read that ZFC can be used as a basis for first-order logic ? But how can this be if we needed first-order logic in the first place to be able to talk about the strings which make up ZFC ?

I'm hoping very much for detailed answers, since these questions have bugged me for a long time and I am tired reading introductions in different logic books without getting these answers.

Answer 1

There are various philosophical approaches to this problem, and it's probably not possible to make any interesting claim about it that every bona-fide logician will agree with.

However, I think a reasonably mainstream attitude is that the formal game with symbols and rules that texts in mathematical logic describes is not Mathematics itself. Rather, the formal game is a mathematical model of the kind of reasoning actual working mathematician accept as valid proofs, in the same way that differential equations can be a model of projectile trajectories or graph theory can model electricity distribution grids.

First-order logic in general and ZFC in particular comprise a remarkably successful model of mathematical reasoning, in that most arguments employed by actual mathematicians can be modeled exactly in ZFC in a reasonably direct manner (except for some category-theoretic arguments which need some arguably clumsy workarounds to be shoehorned into ZFC), and that most mathematicians would agree that any argument that can be modeled in ZFC is a valid "mainstream mathematics" argument. But still, the model is not the thing itself -- that mathematics can be modeled in ZFC doesn't mean that mathematics is ZFC.

What conventional introductions to mathematical logic do is assume that you already have a workable intuitive understanding of ordinary mathematical reasoning, what a valid proof is, how the integers work, and so forth. Then they show you how, using these preexisting tools, you can build a model of mathematical reasoning and use that model to understand it better.

Most modern texts will also assume that your existing mathematical education has introduced you to simple "well-typed" uses of sets in mathematical arguments, and therefore feel free to use that in the construction of a model. If you want to minimize the amount of intuitive mathematical concepts you need in order to build the model, you can do it without any concept of "set" -- Gödel showed that arithmetic on the integers is in some technical sense enough, but in practice it can be argued that you need some intuitive sense of what finite symbol strings are, how they concatenate and so forth.

However it is not possible to entirely eliminate all prerequisites about mathematical reasoning, because then you wouldn't be able to start saying anything on page 1.

Answer 2

The opening statement sketches a kind of naive formalism. For an exploration of some of the troubles in articulating a coherent kind of formalism, and the difficulties which attend such a position, you can't do much better than start here: http://plato.stanford.edu/entries/formalism-mathematics/

I'm not sure what it can mean to say that you don't have numbers and functions etc. at your disposition when you come to reading advanced texts on mathematical logic. After all, your high school maths classes told you a great deal about such things! 19th century mathematics tells us a vast deal more, before ZFC was ever dreamt of. (And few working mathematicians today care tuppence about ZFC: I'm willing to bet that 90% of a maths department's lecturers and professors couldn't even tell you what the axioms are. But they presumably have numbers and functions and indeed some sets of those at their disposal! To be sure, you can cook up proxies for real numbers, functions, vector spaces, etc. etc. in ZFC if that's to your taste or suits your purpose: but most mathematicians get on perfectly well without.)

And I don't know what it means to say that ZFC is a basis for first-order logic. Basis in what sense? A standard first order natural deduction system in no way depends on ZFC, and can be shown to be sound and complete with respect to the standard semantics without appeal to it.

Answer 3

Is my view of mathematics correct?

I don't believe books on mathematical logic completely explain mathematics. In fact most of mathematics does not necessarily build on set theory. The early 20th century mathematics was a lot about the axiomatic foundations, e.g. see Bertrand Russell's work and Hilbert's program, etc. A lot of 20th analytical philosophy deals with the logical foundations of mathematics, so there is a lot to say about these kinds of questions, which usually are not discussed in books on mathematical logic.

where we at this point don't have sets and functions and so on at our disposition ?

Yes, the definition of sets are somewhat circular. In a way natural numbers are not more advanced concepts than sets, although set theory is build in this way. Perhaps a way to think about this is a finite state machine which accepts natural numbers is not incorporating the axioms ZFC. I would agree, that these kinds of questions are very hard. Mathematical textbooks are usually not philosophical, they have a different modus operandi.

We don't even have ZFC yet, so how can we talk about ℕ ?

A good text book on set theory should explain what the progression from ZFC to N is. ZFC is indeed needed as the basis for ℕ, see the Axiom of Infinity, http://en.wikipedia.org/wiki/Axiom_of_infinity . Remember, that sets don't have an order, that is ordinal numbers have much richer structure than just sets. For example, let's assume you want to count notes in a piece of music. A method for counting is based on the ability to distinguish between different notes. There is actually the view, that set theory introduces a certain view of the world, which does not cover every domain. That some mathematicians invented category theory. A lot can be said about the hierarchy of ontology induced by the notions of set theory. By the way, music is actually a very good example where set theory offers not the appropriate semantics, so this is a good way to test the concepts.

Last but not least I read that ZFC can be used as a basis for first-order logic ?

No, it's the other way round. We need formula such as: there exists a thing such that X, for all X: ..., logical operators (and, or). Perhaps everything should be clearer if you read the following: http://en.wikipedia.org/wiki/Hilbert_system. By the way, if you have the time I would suggest reading the historical texts as well.