Formal systems and interpretations for numbers

Question

In my book (Hodel's Intro To Mathematical Logic), we are given several examples of formalized mathematical theories such as group theory, Peano arithmetic, etc. But I've had this ongoing confusion: the formal system can generate theorems, but then the book goes on to talk about particular interpretations that we can provide for the symbols of the language. For example, we are given the standard interpretation N of PA for Peano Arithmetic where we're given a domain of natural numbers, a 0 interpreted as the number zero of the natural numbers, etc.

My confusion is that numbers don't exist 'out there' like tangible things do. It's easy for me to see how a sentence Pa is true in an interpretation where a=John and P='is a person', but not so much for numbers.

So: What does the book mean when we provide the standard interpretation (i.e. what is it that we are interpreting)? What do we mean when we speak of "truths of arithmetic" that lies outside of the formal theory of Peano Arithmetic? Are the numbers (and their operations) defined outside of the system or something?

p.s. this is especially confusing considering I would call myself something like a fictionalist, or formalist, or something of that nature.

Answer 1

You are correct that until we know there are numbers, we have a formal theory but not yet any domain where the theory can be interpreted.

There are two ways to go with this.

1) It's clear that we all know what the natural numbers are. Even if we don't have a formal theory about them, it's clear that they satisfy PA and are therefore a domain in which we can interpret the axioms and theorems of PA.

2) We can formalize the axioms of set theory, say Zermelo-Fraenkel (ZF). Then the concept "exists" means "can be shown to exist from the axioms of ZF." Then we can say that the empty set stands for zero, and the set containing the empty set is 1, and so forth. The objects of this construction are known as the Von Neumann ordinals.

http://en.wikipedia.org/wiki/Ordinal_number#Von_Neumann_definition_of_ordinals

Now we can verify that the numbers 0, 1, 2, ... as defined by Von Neumann, satisfy PA; and therefore we have a set in ZF that can serve as a domain of interpretation for PA.

The same thing is often done pedagogically with the real numbers. The usual procedure is to list the properties of the real numbers such as the commutativity of addition, the distributivity of multiplication over addition, and so forth. That's good enough for calculus class and lets you do everything you need to.

In a math major class called Real Analysis, they'll show that you can start with the rational numbers, and construct a model of the real numbers using Dedekind cuts or Cauchy sequences. They show that to the students once, and from then on everyone forgets about it and just uses the properties of real numbers.

In practice it turns out that most of the time nobody cares about the ontology of mathematical objects. All you need is their properties.

But ultimately you're right, if we're going to be logically comprehensive it's not enough to write down the axioms of PA; we have to show that we can formalize something that satisfies them; and that's where the Von Neumann construction comes in.

Answer 2

We have several issue "hidden" in your question ...

What does the book mean when we provide the standard interpretation (i.e. what is it that we are interpreting)?

A formal theory needs a language, made of :

• symbols (the alphabet)

• formation rules (the grammar) defining what strings of symbols are "allowed".

Thus, when we "interpret" a mathematical theory we are interpreting its language; as is the case for natural language, when we use it we assume that our statements have meaning, i.e. they "speak of" something.

What do we mean when we speak of "truths of arithmetic" that lies outside of the formal theory of Peano Arithmetic?

As you said, we can stay with the formalist or the fictionalist points of view : mathematics is only a game, like "chess".

But these points of view conflicts with the above approach : the game of chess is not a language "speaking of" something, while the "inner feeling" of mathematicians is that their theories have some sort of reference.

Are the numbers (and their operations) defined outside of the system or something?

Thus the issue is : what is the reference of a mathematical theory like arithmetic, which seemingly "speaks of" numbers ?

Where are they ? What kind of reality they have (if any) ?

The platonist and the naturalist points of view try to address - with limited success - the issue of the reality of abstract mathematical objects whose existence is independent of us and our language.

Answer 3

If you are a formalist, then you have no choice but to consider a statement like "N is a model of PA" as being merely a symbolic string expressed within your chosen foundational system, typically called a meta-system (MS) when we are discussing formal systems. Note that MS can prove this statement only if MS already has sufficient assumptions (axioms or inference rules). For example, the system ACA0 is expressive enough to talk about and perform basic reasoning about countable FOL theories and countable structures, such as being able to prove that consistency is equivalent to existence of a model, but it is unable to prove Con(PA) and hence also cannot prove the existence of a model of PA. On the other hand, ACA (which is equivalent to ACA0 plus the full induction schema) can prove both. If your MS is something like ZFC, you can define N as some specific FOL structure whose domain is ω (the subset of the inductive set given by the Infinity axiom that is contained in every inductive set) and then prove that N is in fact a model of PA. All this is done within MS, so ultimately all you get is just some string of symbols that represents a proof over MS but is devoid of meaning.

I submit that no mathematician can validly hold a (pure) formalist stand, because we have very strong empirical evidence in the meaningfulness of PA at human scales. Nevertheless, you may have quite good reason to be a formalist regarding mathematics involving higher set theory, given that there is no clear ontological justification for ZFC, as had been recognized by careful logicians including Boolos. It is extremely unfortunate that most mathematicians are unaware of the vastness of the spectrum of viable foundational systems for mathematics, that the term "formalist" almost always is meaningless. For reference, here is a simple hierarchy in drastically increasing order of (consistency) strength:

PA = ACA0 < ACA ≪ ATR ≪ Z2 ≪ BZC ≪ ZFC.

ATR = arithmetic-transfinite-recursion (roughly the predicative limit)
Z2 = full (impredicative) second-order arithmetic
BZC = bounded Zermelo set theory with choice (bounded meaning quantifier-free comprehension)

You would have to look carefully at each system to determine up to where in the spectrum you are comfortable believing its meaningfulness (i.e. realist up to that point) and from where up you cannot convince yourself to be a realist (i.e. you can be formalist from that point onwards).