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.

Lots of tourists flock to this part of Baker Street in London, and there is even now a house on the street very close to 221b dedicated to Holmes, and which purports to be “his” house. This fact seems to me to suggest that the “you’re in trouble” argument is easily refuted, or else we are forced to accept as true all manner of statements simply on the basis of popular misconceptions. It is even possible for mathematicians to hold misconceptions (despite their common belief otherwise), and to do so for long periods of time, as the history of the calculus demonstrates.