That doesn’t even make sense. It makes no sense to say “there are no natural numbers”. You have to have an appropriate domain of discourse. For example, saying “there are no natural numbers in the set of all natural numbers” at least makes sense, even though it is false. Hartry Field’s claim doesn’t even make sense. It cannot be rationally attacked or defended because it is incoherent. To be agnostic about it is also an incoherent position. Hartry Field has neglected one of the basic duties of the philosopher: to make sense.

Secondly, ontological gibberish aside, Hartry Field’s axiomatization does require some numerical notions to understand it. It doesn’t explicitly mention 2 or the name of any number, but nor does Peano Arithmetic, it mentions only ss0, etc. The point is, even to understand the notion of “a point” requires one to have the notion of 1. He also uses relations. The concept of a relation between two objects is incomprehensible unless one realises that they must be two and only two. Thus one also needs a notion of 2. Thus right off the bat, you need a notion of 1 and a notion of 2, to interpret the axiomatization.

I heard Hartry Field tried to construct the Periodic Table without appealing to the notion of number, but he got stuck on atomic numbers.

It seems to me that he should also want to rewrite to avoid reliance on other abstract entities.

I feel (eventhough I know very little about and will soon have to go to the library) that Fields argument is (or can at lest be understood as) an argument that shows that it is possible that science can be done without the explicit use of mathematics. Not as a proof that mathematical entities do not exisits. Which means that one can remain agnostic about the strange world of mathematical entities (with indenumerable sets and uncomputable numbers floating around).

The existence of unicorns is an entirely empirical matter, so it might have been that you actually saw a unicorn yesterday. If ‘2 + 2 = 4′ is true or not, when taken in the ‘there exists an x…’ form, depends on wheter there are really numbers.

If Field appaels to Ockham’s razor to get rid of mathematical entites, then he does not really give any absolute argument why there might not be mathematical entities. We have just no reason to assume them, and should therefore not assume there existence (but there still might be mathematical entities, and therefore we should remain agnostic).

What about a sentence like ‘All unicorns are white’, if there are no unicorns then sentence comes out true.

Or should it be converted to ‘there is unicorns and all unicorns are white’? And if so should a sentence like ‘all even prime numbers > 2, are > 2′, be converted to ‘there are even prime numbers > 2, and all even prime numbers > 2, are > 2′?

“You (dcorfield) seemed to argue that Field should regard ‘2 + 2 = 4′ as false, since there is nothing that ‘2′and ‘4′ refer to. But this only makes the sentence at best incomplete, not false.”

No ’should’ about it. I was just trying to convey how he does argue. That there’s any plausibility to this argument comes from an analogy with our everyday world. In the tradition to which Field belongs a statement such as “I saw a unicorn yesterday” would be parsed as “there exists x such that x is a unicorn and such that I saw x yesterday”. “x is a unicorn” is “x is an animal, rather like a white horse, but with a horn sticking out of its head” (perhaps one needs to add in some extra properties). Then, so the reasoning goes, “I saw a unicorn yesterday” is meaningful and complete, but false, because
“there exists x such that x is a unicorn and such that I saw x yesterday” is false. I saw plenty of things yesterday, but none were unicorns. You need then to think of ‘2 + 2 = 4′ as starting out something like ‘There exists x, there exists y, x is 2, y is 4,…’.

As to whether the rendering of statements into logic to reveal their existential commitments is a good idea, now that’s a different matter.

What I meant was something like that if mathematics is in some sense really false, that is false in this model (that is reality or the physical world), then mathematics cannot be a conservative extension of a physical theory, since we get a contradiction and then anything follows (if that makes sense).

Fields has to translate claims about mathematical statements into statements about space-time regions. But I cannot see how he can claim that the mathematical statements are really false, because then the translations into his space-time scheme would also be false.

The theory of the real numbers in the language of fields is complete, so any consistent extension will be conservative. Therefore, consider one extension including PA+Con(PA), and another extension including PA+~Con(PA) - presumably, the former is true and the latter false, but both are conservative since both are consistent. (Assuming Peano Arithmetic actually is consistent.)

Just a few short comments, even if they get me to deep in a discussion about something I dont know that much about (that is Fields position).

To Kenny:

It was clumsy of me to say that numbers stand for strokes, e.g. that 3 stands for ‘III’. But this does not really affect my point, which was that Field has bind statements to about numbers to the real world. This seems to be a part of his project.

He tries to eliminate mathematical statements from science, by rewritting them using logic and scientific concepts that are not mathematical. Mathematics does not add anything new to this core theory, it just makes things easier. Regular physics is just a conservative extension of the non mathematical core of physics.

I’am having trouble accepting that Field could consider that mathematics

“…should be considered as a body of falsehoods not talking
about anything real” [from the new wikipedia article]

according to Field. The part’ not talking about anything real’ is ok, but the ‘body of falsehoods’ part is not. The sentence

“Sherlock Holmes lived at 22b Baker Street”

is in an intuitive sense neither true or false, because there is no Sherlock Holmes. So it does not follow that mathematics is false, even if Field claims that mathematics is a fiction.

The claim that mathematics is false, would moreover make the idea, that adding mathematics to a physical (non-mathematical) theory yields a conservative extension of this theory impossible, since we would be adding a body of falsehoods to the theory.