I just came across a reference to the work of the philosopher of mathematics Hartry Field. This is what Wikipedia has to say:

Fictionalism was introduced in 1980 when Hartry Field published Science Without Numbers, which rejected and in fact reversed Quine’s indispensability argument. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as true, Field suggested that mathematics was dispensable, and therefore should be rejected as false. He did this by giving a complete axiomatization of Newtonian mechanics that didn’t reference numbers or functions at all. He started with the “betweenness” axioms of Hilbert geometry to characterize space without coordinatizing it, and then added extra relations between points to do the work formerly done by vector fields.

Does anyone know if that is an accurate summary of Field’s argument? It seems obviously wrong to me.

Via Crooked Timber.

Thanks a lot, Walt. That is an hour of my life I won’t be getting back.

It is hard to understand how one could “characterize space”, whether or not co-ordinates were permitted. Precisely what, indeed, would such a characterization distinguish space from? Time, perhaps? Maybe the author intended to write “represent” rather than “characterize”.

I cannot rember how it really goes but this seems like a pretty bad wikipedia entry.

“…Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as true, Field suggested that mathematics was dispensable, and therefore should be rejected as false.”

The point of Quines indispensability argument, is that we should accept that mathematical entities exists beacause they are indispensable for scientific theories. We should e.g. admit that the indenumerably infinite set of reals exists (that it is not a mere fiction), because it is indispensable for physics. (The justification of the existence of math entities comes in away from the physical world.)

The argument has nothing to do with, if mathematics should be “be accepted as true” or not (whatever that means).

(I dont think that Fields suggests that mathematics is false either. Fields argument is probably that science can be done without math to some extent and that mathematics is just a very usefull tool. And that indispensability argument, is somehow because of this invalid.)

Certainly Field admits the existence of space-time points. The whole exercise is to get rid of reference to mathematical entities like numbers. So, where you thought you needed numbers to say that the distance between space-time points x and y is twice that between z and w, you find you could have rephrased it by saying there is a point u which is between x and y, such that xu is congruent to uy, and such that uy is congruent to zw.

See here for an expression of the idea that Field might have come close to some real mathematics, namely, the torsor concept.

“I dont think that Fields suggests that mathematics is false either.”

He does, but in a way which tends to baffle mathematicians and scientists. The proposition ‘2 + 2 = 4′ is strictly speaking false for him, as it could only be true if there were something that ‘2’ and ‘4’ referred to, but there aren’t. This leads to the strange conclusion that ‘There exists no natural number between 2 and 4′ is true because there just are no such things as natural numbers.

On the other hand, for Field, it’s a useful fiction to say that there are natural numbers. It’s much quicker to believe that they exist and carry on as normal. To reassure us that this won’t lead us astray we just need to show that, crudely speaking, physics + logic is every bit as powerful for our needs as physics + math + logic. So for instance we find ‘2 + 2 = 4′ useful for counting how many pieces of fruit we have when we bring together a pair of apples and a pair of oranges. But we could have said: There is one space-time region x and there is one space-time region y, and y is not x, and x is an apple and y is an apple, and if z is a space-time region which is an apple then z is either x or y. Ditto for two oranges. From which it follows given that there is no more fruit, that there a four pieces. (I’ll leave that as an exercise how to express the fact that there are exactly four pieces of fruit.)

As you may imagine, doing this for quantum field theory is even more fun. I’d be interested to hear what you mathematicians and scientists make of it all.

But if you say that ‘2 + 2 = 4′ is false, then it should be meaningful and therefore ‘2’ and ‘4’ should also refer to something. (You cannot say e.g. that ‘blue come two think from’ is false, because it does not mean anything.)

And I think that Field cannot deny that numbers really stand or coresspond to something. E.g. That 3 and 5 are somehow connected to ‘III’ and ‘IIIII’. The point must be something like that numbers stand for concrete things and that they are not free standing abstract entites.

If you have a theory that claims that ‘2 + 2 = 4′ is false, your in trouble since everybody thinks it is true.

(But I dont really know what Fields thinks, since I havent read about it.)

So for Fields the statement “the hydrogen atom has one proton” is false?

What confused me about the Wikipedia entry was somehow characterizing Hilbert’s axiomization of geometry as not mathematical. Anyway, since the real numbers are derivable from Hilbert’s axiomization, whether or not the real numbers are “real” seems a matter of taste.

I like bambam’s “you’re in trouble” argument.

This is mighty stange for me to be defending Field, whose program I see as largely pointless. You ought to try and recruit Kenny Easwaran, whose blog post I mentioned in my first comment (URL needs fixing in obvious way, webmaster).

Bambam said:

“But if you say that â€˜2 + 2 = 4â€² is false, then it should be meaningful and therefore â€˜2â€² and â€˜4â€² should also refer to something.”

I don’t think that’s right. ‘The unicorn I saw yesterday was running very fast’ is meaningful, but the term ‘the unicorn I saw yesterday’ doesn’t refer to anything. Unlike your nonsense sentence, there’s a potential for the term to refer, it’s just that it happens to fail to do so. For Field ‘2+2=4′ is meaningful. In fact it means there are some entities which ‘2’ and ‘4’ refer to, and a 3-ary relation between entities of that kind: plus(x,y,z) holds for triples such that x + y = z. But it’s false because there are no such entities.

For a much more interesting discussion of mathematics and reality why don’t you visit my blog, e.g., here and here. (Let’s hope these URLs show up right.)

On re-reading it, I see that the article is a bit confusing. I actually wrote much of that paragraph, but I did it fairly quickly I think. I see that there’s a distinction between saying that mathematical entities exist, and saying that mathematical statements are true, but it really wasn’t clear to me which was a clearer notion. I stuck with truth, rather than existence, but perhaps that’s just because the indispensability argument seems to me to work better for truth than existence.

Field cannot deny that numbers really stand or coresspond to something. E.g. That 3 and 5 are somehow connected to â€˜IIIâ€™ and â€˜IIIIIâ€™.He does deny this. I would have thought that no one thinks numbers stand for those particular strokes up there, because they certainly don’t stand for any particular collection, but rather (we like to think) for something that any collection of a certain type has in common. I’m pretty sure Field doesn’t think numerals stand for concrete things (though John Burgess and Gideon Rosen have suggested in

A Subject with no Objectthat this might be a better way to read him) – rather, he thinks numerals don’t stand for anything real at all, but for fictional objects, that we pretend to use to count things.The paragraph quoted doesn’t suggest that Hilbert’s axioms aren’t mathematical – it just says they “characterize space without coordinatizing it”, and don’t “reference numbers or functions”, and avoids vector fields.

Real numbers are only derivable from Hilbert’s axioms in the sense that one can find an isomorphic structure by making a few arbitrary choices – I think this is what Burgess and Rosen want to do with Field, but it seems slightly inelegant to me. And it doesn’t get at what we normally think the real numbers are, but merely representatives of some sort.

The “you’re in trouble” argument is the biggest reason people have criticized Field, I think. His response is just that “2+2=4″ is true in exactly the same way that “Sherlock Holmes lived at 22b Baker Street” is true. Literally speaking, according to Field, both are false, but it is very easy and natural to use them to mean something that is “true according to the fiction”, whatever that means. The former is no worse than the latter, and since the story of mathematics is a more widely shared one (and more useful one) than the Sherlock Holmes stories, it may in fact be even better off while still being “merely fictional”.

Anyway, maybe I should edit that article some more, to include more of this discussion? Thanks for pointing out where it seems to be misleading.

I think I agree. But my point was that you cannot say that ‘a + b = c’

is false (or true) if you dont give the terms ‘a’, ‘b’ and ‘c’ any meaning.

The sentence is just incomplete and all depends on what you mean by ‘a’, ‘b’

and ‘c’. (If a=b=2 and c=4, then it happens to be true, but if a=b=4 and c=2

then it is false and if a=b=c=’the moon’ then it is nonsense).

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.

Saying that ‘2 + 2 = 4′, is an instance of predicate p(x,y,z), which is

somehow meant to stand for ‘x + y = z’, seems rader dubious.

Because whatever intepretation or meaning you give p(x,y,z) it cannot have

much todo with the common sense meaning of ‘x + y = z’,

which it is meant to model, since it is always false no matter what you

place in for ‘x’, ‘y’ and ‘z’ (because there are no x, y, and z that it

holds for).

I Field says that ‘2 + 2 = 4′ is meaningful and that it is false. Then ‘2 + 2 =4′ means for him something entirely different than to us. No senseable reinterpretaion or restatement of ‘2 + 2 = 4′ can make it false.

â€œSherlock Holmes lived at 22b Baker Streetâ€

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.

My last comment refered to dcorfields posts, not to Kennys which I had not read.

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.

You’re right that it’s somewhat odd for Field to say that these claims about non-existent objects (as he sees them) are false. But just seeing them as false doesn’t prevent them from being a conservative extension of a non-mathematical physical theory. Conservativity just means that adding the new statements doesn’t let you formally deduce anything expressible in the old language that you couldn’t already deduce – and this is possible both for true theories and false theories. This is very easy to see in cases where the new statements added are in an entirely new language with no shared non-logical symbols, as long as the new theory is consistent. But it’s also possible in other cases as well.

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.)

Well ok. Your right about the conservativity.

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.

Field’s argument, that mathematics is a conservative extension of physical reality, to me seems an argument that mathematics is true, not a useful fiction. Sherlock Holmes is not a conservative extension of anything.

Bambam said:

“You (dcorfield) seemed to argue that Field should regard â€˜2 + 2 = 4â€² as false, since there is nothing that â€˜2â€²and â€˜4â€² 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.

Rewritting statements is a nice way of eliminating the need for mysterious non-real entities such as unicorns and ‘the present king of france’ (but it might not be a good idea). But I don’t see how all statements containing e.g. unicorns should be regarded as false, even if there are no unicorns.

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’?

Yep. ‘All unicorns are white’ and ‘All unicorns are pink’ are both true. And so, in Field’s view, ‘All numbers are even’ is also true. Not ‘true within the fiction’, but strictly true.

Firstly, then some mathematical statements are true, so it is not the case that ‘mathematics is a body of falsehoods’, since e.g. ‘All primes > 2, are not even’ is true.

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).

I guess he would agree with you that mathematical entities cannot be proved not to exist. But he will still feel he has done us a great service by showing us how we can consistently carry on as normal, while believing that they don’t exist. One of the advantages of the latter belief is that we then don’t need to concoct an account of how we know about non-spatiotemporal entities, when there’s no causal means to know about them.

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