Sunday, 12 May 2013

Science Versus Nominalism

The Indispensability Argument, developed by W.V. Quine and Hilary Putnam, and famously rebutted by Hartry Field, is fairly simple:
(1) Nominalism states that there aren't strings, formulas, numbers, sets, sequences, functions, groups, etc.
(2) Science (e.g., physics, linguistics) states that there are strings, formulas, numbers, sets, sequences, functions, groups, etc.
Therefore, nominalism is inconsistent with science.
This is not some containable inconsistency, say to do with idealization, or frictionless planes, etc. Nominalism states that there are no physical quantities. Nominalism states that Peano arithmetic doesn't exist. Nominalism states that an SU(3) gauge theory like QCD is false because there are no groups. Etc. But science states that there are quantities; that Peano arithmetic does exist, and is not finitely axiomatizable; that the gluons are associated with an SU(3) gauge symmetry; etc.

The premises (1) and (2) are justified as follows. The first (1) is how nominalists describe their view. And (2) is justified by looking at a physics textbook.

1. You've posted this gloss of the IA a couple of times now, but I have a hard time believing that you actually accept it as stated, since it is false on the face of it. Physics, of course, says nothing about the existence of abstract objects. Whatever it says about them, it *certainly* doesn't say that they are physical---i.e., amenable to investigation by the scientific method. There is just *no way* that you can actually believe it does, so I am baffled every time that you bring up this version of the IA.

2. Many thanks, A,

"Physics, of course, says nothing about the existence of abstract objects"

This is the crucial issue.
Physics (and other sciences) says a great deal about mixed mathematical objects. Consider physical laws, such as $\nabla \cdot \mathbf{B} = 0$, etc. A law usually states that certain *functions* on spacetime, taking values in certain *spaces*, have certain properties. Or it might state that certain quantity, such as a Lagrangian, say $\mathcal{L}(\phi, \partial_{\mu} \phi)$, is invariant under a certain Lie group, such as SU(3). And so on.
These quantities are typically mixed mathematical objects. This is, in part, why mathematics is applicable. To apply mathematics is to reason about mixed mathematical objects: such as sets of points, functions on spacetime, and so on.

What is a vector field, if it is not a function with values in a vector space? What is a set of points, if it is not a set of points? What is the fine structure constant, if not some real number? If there are no functions, quantities, vector spaces, Lie groups, etc., then physics is false. So, physics is inconsistent with nominalism.

To verify this, one can write down nominalism and write down a law of physics. One can deduce a contradiction. Here's a really simple example:

(Nom) There are no real numbers.
(Phys) The fine structure constant $\approx \frac{1}{137}$.
So,
(C) $\exists r \in \mathbb{R}$ such that $r \approx$ the fine structure constant.

Cheers,

Jeff

3. In your view Jeff does an analogous argument show that materialism is false?

something along the lines of;
physics says that there are mixed mathematical objects.
the mathematical parts are certainly not material in the sense that say photons are.
materialism holds that there are no things that are not material
therefore if physics is true materialism is false.

?

1. Joseph, yes, that's right.
It sounds amusing, because people like to think that physics supports materialism! But a vector field, like $\mathbf{B}$, or a vector space, like $\mathbb{R}^4$, or a gauge group, like $SU(3)$, is not a material thing!

Historically, I guess the term "materialism" was usually meant to contrast with views about minds, or spirits, or supernatural entities, etc.. Nowadays, the term "materialism" is not used much in analytic philosophy---although Marxists and religious opponents of evolutionary theory seem to enjoy using it when arguing with each other. :)

Jeff

4. I apologize in advance if what follows is wide off the mark or incoherent, as I haven't been active within the philosophical community since receiving my master's in the subject almost a decade ago. Also, what I am about to say is only restricted to my thoughts regarding mathematical discourse. Having said that, I have a hard time accepting premise #2 because it is not clear to me why we should interpret the language in which mathematics is practiced as ontologically committing the mathematician to the respective domain of discourse. Thinking back to my days as an undergraduate student in math, I never, for once, concerned myself with the existence of the objects of study, platonic or otherwise. For me, it was more a manner of speaking a certain language and whether these objects really existed in the world was not the point. Though it may not be entirely fair to extrapolate my first-person experience as an undergraduate math major to the entire mathematics community, I imagine my experience might not that be far off from the average practicing mathematician. I think the language of mathematics itself is silent on the matter of ontological commitments. I believe the debate arises only at the second-order level, where nominalism resides. Therefore, as a corollary, it might not be entirely accurate to present it as an inconsistency when you are contrasting nominalism (2nd-order) with science/mathematics (1st-order object language).

Cheers,
Wing

1. Hello Wing,

That's a perfectly coherent reply, thanks!

There'd be three things I suggest in reply.
The first is that we're concerned with what's the case if the statements/axioms/theorems of maths are true, rather whether any person believes them to be true. (Cf., a physicist may spend years studying Dirac's equation and not have any interest if it's true or not.) So, e.g., if "There is a set $x$ containing $\varnothing$ and closed under $x \mapsto x \cup \{x\}$" is true, then there is an infinite set. Consequently, if there are no infinite sets, then this statement is not true. Second, for purely mathematical purposes, I agree that it usually needn't matter if this axiom/theorem is true or not. All that matters is whether it is an axiom or a theorem of some fairly well-defined framework. I.e., can one prove $\phi$, given the assumptions already accepted? This view is called "deductivism" or "if-thenism". (I.e., mathematical knowledge consists in true claims of the form "$\phi$ is a theorem of framework $F$").
However, when we examine applied mathematics and the laws of physics, the initial assumptions usually do need to be assumed to be true (or approximately true). So, e.g., f we accept scientific theories as approximately true (realism), then we accept the law "$\nabla \cdot \mathbf{B} = 0$" as saying something true -- i.e., the magnetic field $\mathbb{B}$ has zero divergence. But since $\mathbb{B}$ is a function, this cannot be accepted if one also denies that there are functions, vector fields, etc. If there are no vector fields, groups, etc., then a scientific theory which asserts that there are vector fields, groups, etc., must be untrue.

Third thing is the issue of syntax: this makes things even more problematic if one is dead set against admitting abstract entities. Suppose you imagine formalizing your knowledge of numbers, sets, functions, etc., in some standard system -- e.g., Zermelo-Fraenkel set theory $ZFC$. In doing this, you need to talk about a language $\mathcal{L}_{\in}$, and in fact, this *language* contains infinitely many variables, infinitely many terms and infinitely many formulas. So, the meta-theory which deals with the syntactic entities is true only if there are $\aleph_0$-many variables, terms, formulas, etc.

Cheers,

Jeff

5. Hi there. I"m no expert on philosophy, but I'm terribly bothered by the grouping of strings and formulas along with abstract entities such as numbers, sets, functions and groups.

A string or a formula is just a word in a language just like the words on the page. If I say "I had a tuna sandwich for lunch," that string of symbols has the same ontological status as "1 + 1 = 2".

Whether there "really are" things such as numbers and sets in some Platonic realm of math; or whether these are just fictional abstractions of our mind, is a reasonable question.

But surely nobody can doubt the existence of strings of symbols in a human language. I don't understand this at all.

6. Hi A, on the strings issue, one can distinguish between symbol types (e.g., the letter "a") and symbol tokens (e.g., all the inscriptions of the letter "a" that may appear on a sheet of paper). Tokens are usually thought of as physical entities, while types are abstract entities. A nominalist then says that there are tokens, but there are no symbol types, as they're abstract.

Strings from an alphabet are usually defined to be finite sequences (of symbol types). For example, the string

"aaaaaaaaaaaabbbaaaab"

is a function $f: \{1,\dots 20\} \to \{a', b'\}$ such that:

$f(n) = a'$ for $1 \leq n \leq 12$
$f(n) = b'$ for $13 \leq n \leq 15$
$f(n) = a'$ for $16 \leq n \leq 19$
$f(20) = b'$.

In principle, one can have a string of any length one likes, say $10^{10^{10}}$. But a nominalist doubts that there are strings in this sense, because they are finite sequences. A classic paper on nominalism, "Steps Toward a Constructive Nominalism" by Quine and Goodman (1947) actually does doubt that there are strings in this sense. It admits only string tokens (i.e., physical inscriptions). But this leads to trouble, because there are strictly finitely many such strings. Consequently, one lacks certain closure properties (e.g., under concatenation). A nominalist then has to renounce a lot of syntax. For example, even basic theorems of meta-theory will be false on this interpretation (e.g., the completeness theorem, which states that every valid formula $\phi$ has a derivation).

If strings are defined as finite sequences, then their concatenations exist, though the definition is a bit messy. More generally, it turns out that there's an intimate relationship between:

(i) the theory of strings with its binary concatenation operation.
(ii) the theory of natural numbers with $+$ and $\times$.

This relationship is part of what lies behind Godel's incompleteness results.

Cheers,

Jeff

7. Some points/questions:

1) Looking just at the case of the fine structure constant, it seems to me that you've cheated by choosing a dimensionless quantity. If an astronomer says something like 'the speed of this planet is 100km/s', that doesn't require us to invent a mathematical object to which '100km/s' refers. (It also doesn't require us to invent an object to which 'the speed of this planet' refers, but since I note that you, quite reasonably, don't explicitly treat 'the fine structure constant is r' as an identity, you probably weren't tempted to do that anyway. But if that's the case, why treat mathematical terms differently from terms like 'the speed'? And, since if 'the constant is r' is read literally it has the form of an identity, but you are willing to admit that it doesn't really have that form, why insist that it still does have a logical form that requires r to be a referring term?) But the fine structure constant is just a ratio of quantities that do require units; either you treat it like '100km/s', or you analyse claims about it as claims about a ratio, in which case it doesn't pose any problem for nominalists that they don't get from ordinary mathematics.

2) On a tangent, is there a non-contentious way of proving the existence of pure sets and functions from concrete ones? Is there room for a sort of semi-nominalist, who accepts that e.g. there is a set containing all and only those planets closer to the Sun than Saturn, but doesn't accept the existence of an empty set or other pure sets?

3) What are the foundations of mixed mathematics? For instance, we have axioms of pure set theory that let us avoid known sources of inconsistency in the rules for set formation. Are there such rules determining when concrete sets do and do not exist? I'm struggling to think of plausible rules that would give us infinite sets of concreta but not a sort of naive comprehension for them (and even if the latter were consistent when restricted to concreta, since it isn't consistent for pure sets, we'd need to explain why we should treat concrete and abstract things differently when forming sets of them).

4) Aren't you at risk of making it too easy for nominalists to just accept that physics is false, but only in a way that doesn't matter (that doesn't affect its success as a science at all)? Anyone can make a game of interpreting other people's claims too literally, in a way that makes them strictly false but still entirely successful as assertions. If I have a friend who I believe to be honest, and who turns up late for an engagement with the excuse 'I got a puncture', and you pipe up that they can't be speaking the truth because human beings don't even have tyres, that shouldn't make me doubt my friend's honesty at all. Likewise, if you interpret physics as committed to abstracta, you allow the nominalist to accept that it is strictly false without losing any of their trust in or deference to it, i.e. without really paying a price. (I'm guessing that one of the reasons you picked physics for this argument is that you think we should have more trust in/deference to physics than most other intellectual fields.) After all, no-one becomes a nominalist who isn't already disposed to think that there are lots of important facts that could be strictly false in a way that doesn't really matter (likewise, mereological nihilists). If you want to refute nominalism, you have to show that it can't multiply falsehoods without paying an unexpected price. On its own, showing that physics is committed to abstracta just provides another opportunity for nominalists to practise their game of rendering important facts false without harm to themselves (it's like kids and fire - it's fun for precisely as long as they believe they can burn stuff without getting hurt).

1. Hello A,

"you ... don't explicitly treat 'the fine structure constant is r' as an identity, you probably weren't tempted to do that anyway."

Hey, I certainly do! I treat

(i) the fine structure constant $= r$

as an identity. The left-hand-side is a singular term defined using certain physical quantities. The right-hand side is a variable ranging over $\mathbb{R}$.

To do not so seems hard to make sense of. Is "the fine structure constant = x" simply a primitive predicate? But then, why does it satisfy certain axioms of identity? For

(ii) the fine structure constant = $1$
(iii) the fine structure constant = $2$

jointly imply

(iv) 1 = 2.

Jeff

2. HI A,

"It also doesn't require us to invent an object to which 'the speed of this planet' refers ..."

On the contrary. The speed of an object is an abstract entity. For what is the mass of a speed? For example, in Measurement Theory, and in discussions of the structure of physical quantities. Physical quantities possess an abstract value range, often isomorphic to the ordered positive reals. For further information, read this article by Terry Tao,

http://terrytao.wordpress.com/2012/12/29/a-mathematical-formalisation-of-dimensional-analysis/

Jeff

3. Hi A,

"Is there room for a sort of semi-nominalist, who accepts that e.g. there is a set containing all and only those planets closer to the Sun than Saturn, but doesn't accept the existence of an empty set or other pure sets?"

This is pretty much the position explicitly denied by nominalists (e.g., Field). Comprehension for predicates applicable to concrete things is forbidden, even if restricted to non-empty predicates.

Still, there's hope for the semi-nominalist on this line of thought. The Polish logician Lesniewski tried to develop a kind of replacement theory, along these lines: mereology. There is a technical sense in which a mereological extension of a theory behaves just like assuming comprehension for the objects of a theory. So, if we think of the planets as "atoms", then the fusion of them (assuming there is at least one) behaves rather like the set of them.
Some of the technical aspects of this are explained in this paper,

Jeff

4. A,

"What are the foundations of mixed mathematics?"

Basically, I'd say,
(i) Set theory with ur-elements (e.g., $ZFU$).
(ii) Second-order logic, with Hume's Principle, and perhaps other abstraction principles.

A physicist who works in GR for example, will happily consider point sets, regions, geodesics, etc., definable in various ways in his spacetime models $(M, g, \dots)$. The physicist doesn't, in principle, care what the elements of $M$ are, but they're representing physical spacetime points somehow; and so regions, geodesics, etc., are mixed; neighbourhoods and open sets are too. The hard-working nominalist might get round this by treating spacetime regions mereologically. This, more or less, is Field's approach in his Science Without Numbers (1980). How to deal with quantities, on Field's approach, is tricky, and uses "Representation Theorems" from the theory of fundamental measurement, along with similar representation results for Tarski's geometry.

Jeff

5. Hi A,

On (4), "Likewise, if you interpret physics as committed to abstracta, you allow the nominalist to accept that it is strictly false without losing any of their trust in or deference to it, i.e. without really paying a price."

But I'm not sure what "trust" and "deference" mean here. For example, what does

trusting "$\nabla \cdot \mathbf{B} = 0$"

mean? Does it mean believing that this law makes empirically correct predictions? That then reduces to an instrumentalist view of scientific theories.

Also, for example, real number variables in physics range over $\mathbb{R}$. So, it's unclear what interpretation one might mean. Do they range over a finite set? A countable set? How would one get continuous symmetries?

"After all, no-one becomes a nominalist who isn't already disposed to think that there are lots of important facts that could be strictly false in a way that doesn't really matter"

I am sceptical about the existence of nominalists. :)

Cheers,

Jeff

8. Another point: there are sometimes two (or more) mathematical formulations of the same theory (e.g. Lagrangian vs. Hamiltonian mechanics; the Heisenberg picture vs. the Schroedinger picture), both of which are accepted by physicists as correct. Since the two formulations may have different (mixed-) mathematical ontologies (admittedly, the examples I gave don't differ a lot, but they could have done), which one are we committed to? And doesn't the fact that physicists are willing to accept both descriptions, yet accept each as if it were perfectly literal, suggest that physicists' literal use of mathematics shouldn't be taken to require their acceptance of its ontology?

9. Hello A,

"Since the two formulations may have different (mixed-) mathematical ontologies ..."

I don't think they have different ontologies. Mixed quantities on spacetime, like $H$ and $L$, are physical quantities, usually interdefinable. We accept both of these quantities. Perhaps one is more "basic" than the other in some more refined sense of "naturalness". But, even so, the more-basicnesss of the function $L$, say, doesn't entail the *non-existence* of $H$.

Perhaps analogously, there is a "unit of selection" debate in biology, which, in a sense, is concerned which kind of entity plays a certain natural/causal role in selection. But no matter how this is settled, this will not imply the non-existence of species, or individuals, or genes.

"And doesn't the fact that physicists are willing to accept both descriptions, yet accept each as if it were perfectly literal, suggest that physicists' literal use of mathematics shouldn't be taken to require their acceptance of its ontology?"

I think physics simply *presupposes* mathematics (including comprehension for mixed sets/functions, etc.) as true and uses it. How much, is an interesting question.

Jeff

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