# M-Phi

A blog dedicated to mathematical philosophy.

## Saturday, 5 January 2013

### Are there causally active mixed mathematical objects?

At the heart of the indispensability arguments against nominalism lie mixed mathematical objects - henceforth MMOs. A simple example of an MMO is the set of US presidents. Or the set of eggs in a fridge. The object counts as mathematical because it is a set, and it counts as "mixed" because its elements are concrete entities. Another example would be any function $f$ from the set of US presidents to $\{0,1\}$. This function $f$ would map a president $p$ to a number $f(p) \in \{0,1\}$. Another example would be an quantity: a quantity maps concrete things to abstract values. (This is a puzzling case though. Must there be a concretum for every value? Must there be a concrete for each value of mass-in-kilograms? Possibly one should say that a quantity is really some kind of structure built of properties.) A final example would be a physical field, such as the electric and magnetic fields, $\bf{E}$ and $\bf{B}$, usually unified into the electromagnetic field (written $F_{ab}$ in tensor notation). The fields $\bf{E}$ and $\bf{B}$ are vector fields on spacetime: they are (mixed) functions which assign abstract values to points in spacetime.

The question I am interested in is whether any of these MMOs ever counts as being causally active. For it is usually (and presumably rightly) assumed that pure mathematical entities---the set of natural numbers, the sine function, $\pi$, $\aleph_{57}$, etc.---are not causally active. But it seems to me that, according to physics itself, the electromagnetic field (which is, remember, an MMO, a mixed mathematical entity) is causally active.

For example, the Lorentz force law says, for a point particle of mass $m$ and charge $q$ and position vector $\mathbf{r}(t)$,
$m \frac{d^2 \mathbf{r}}{dt^2} = q(\mathbf{E} + \frac{d \mathbf{r}}{dt} \times \mathbf{B})$
So, the motion of the particle (its acceleration) is determined by the fields, $\mathbf{E}$ and $\mathbf{B}$, which are MMOs. Consequently, it seems that there are causally active mixed mathematical objects --- namely, physical fields.

#### 24 comments:

1. This comment has been removed by the author.

2. Hi Jeff,

As far as I can see, this sort of indispensability arguments is based on conflating two distinct (putative) entities--i.e. (a) the electromagnetic field itself (if there is any such thing) and (b) the mathematical object(s) we use to represent it (if there are any such things). So, I think that the solution to the puzzle is that, while the EM field is causally active, the piece of math we use to represent it is not. Am I missing something?

(Could you please expand your comment about the quantity case? I'm not sure I see what you are trying to say)

3. Hi Gabriele,

According to physics itself, the magnetic field is a vector field, a function on spacetime. If the magnetic field is not a vector field, what is it?

Similarly, quantities are functions too.

Cheers,

Jeff

1. hi jeff,

i think we should carefully distinguish between theories and our interpretations of them. strictly speaking there are no interesting metaphysical claims that are true "according to physics itself". as far as i can see, our best interpretation of classical em is that the magnetic field is best represented mathematically as a vector field not that it is a vector field. (although some physicists sometimes talk of it as if it actually is a vector field we shouldn't take that talk liiterally). i hope you wouldn't want to claim that the compass needle points towards the north pole because vectors are making it do so! surely the vectors are there to represent the influence the earth's magnetic field has on the needle and other objects but they di not constitute the magnetic fiels!

2. sorry for all the typos--i'm writing this from my phone!

3. Hi Gabriele,

"i hope you wouldn't want to claim that the compass needle points towards the north pole because vectors are making it do so!"

Yes!!
That's precisely what I'm saying.

(The dipole's vector field aligns with the magnetic field's vectors to minimise the energy.)

Physical laws normally express relationships between physical quantities, which are usually mixed functions. I don't think the magnetic field $\mathbf{B}$ is "represented" by a vector field. I think $\mathbf{B}$ *is* a vector field. And I think this is the way physics thinks of it too (though physics could be mistaken, of course). A physical field is a mixed function that maps spacetime points to values in some space (usually a linear space).

I think the idea you're advocating depends on *nominalizing* the physics, a la Hartry Field, and thereby replacing the mixed physical functions with primitive spatiotemporal predicates, and then showing that a (standard) model for the resulting theory can be represented by reintroducing the mixed functions. But this simply assumes nominalism and the success of a certain kind of nominalization.

Cheers,

Jeff

4. This comment has been removed by the author.

5. This is damn interesting guys. Let's see if I manage to post something brief.

Obviously, if Jeff is right, then mixed mathematical entities (as he calls them) can be causally efficacious, since they are instantiated as concrete spatiotemporally located objects, however de-localised they may be (e.g. fields). However, I am inclined to take this as a reduction of the view Jeff is exploring and, in the spirit of Gabriele's comment, as pointing to the need to distinguish a mathematical and a physical concept. This is typically easy enough to do for any branch of experimental physics, although admittedly harder for em theory, or more generally mathematical physics, perhaps because the maths is so entrenched there. But the idea is that there is a mathematical concept of a "field", which is indeed displayed by pointed vectors in a abstract mathematical space. Then there would be a physical concept of a field which is best thought of in purely physicalistic or even operationalist terms. Something like Faraday's lines of force, or more generally force fields. Appropriately, Maxwell's work contains extended discussions of both, and how to relate them.

6. Hi Mauricio,

The thing is, the usual field quantities in physics, such as $\mathbf{B}$, $g_{ab}$, $\Psi(t)$, etc., are mixed, and so relate the two domains that one wants to keep separate (i.e., spacetime points & concreta on the one hand, and the mathematical value ranges on the other).

So, when you say the problem is to make sense of "a physical concept of a field ... thought of in physicalistic terms", I think this has to mean "thought of in nominalistic terms". ("Operational" would be going too far; usually those replying to indispensability arguments, such as Field, Melia, Leng, Sober, and others, are scientific realists.)

Hartry Field does try to achieve this in his Science Without Numbers. He agrees that a physical field quantity $\Phi$ is a function on spacetime to some abstract range; however, the usual theory T of the field $\Phi$ is nominalized, and replaced by a theory T* which uses more complicated primitive spatiotemporal predicates (e.g., "the $\Phi$-value at p1 lies between the $\Phi$ values at p2 and p3"). The success of such a view hinges on the viability of nominalized physics.

Cheers,

Jeff

4. E.g., here's Hartry Field discussing the indispensability argument:

"After all, the theories that we use in explaining various facts about the
world not only involve a commitment to electrons and neutrinos, they
involve a commitment to numbers and functions and the like. (For
instance, they say things like ‘there is a bilinear differentiable function,
the electromagnetic field, that assigns a number to each triple consisting
of a space-time point and two vectors located at that point, and it obeys
Maxwell’s equations and the Lorentz force law’.) I think that this sort
of argument for the existence of mathematical entities (the Quine-
Putnam argument, I’ll call it) is an extremely powerful one, at least
prima facie." (Field 1989, Realism, Mathematics and Modality, Introduction.)

To stress: "there is a bilinear differentiable function,
the electromagnetic field, that assigns a number to each triple consisting
of a space-time point and two vectors located at that point, and it obeys
Maxwell’s equations and the Lorentz force law"

There's no conflation here, as far as I can see.
And this is one of the premises of the indispensability argument. Consequently, Field tries to reformulate the laws without referring to such functions, just using primitive spatiotemporal predicates.

Cheers,

Jeff

1. is this an argument exi auctoritate? :-)
im sorry but i can't take field's talk literally here. the magnetic field is no more a function than hartry field is (and no hf is not a function). this is a typical case of conflation of the thing reprsented with the thing used to represent it and the fact that the mf is unobservable males the conflation harder to see.

2. i meant 'ex' and 'makes'--damn phone!

3. Hi Gabriele,

"the magnetic field is no more a function than hartry field is"

I'm quite happy to say that physical systems are complicated tensor functions. :)

"this is a typical case of conflation of the thing reprsented with the thing used to represent"

But this view will only work if you can nominalize physics. What is the thing represented? What is the thing used to represent?
Suppose we accept your view: then
(i) there is a function F;
(ii) there is a magnetic field M;
(iii) F represents M.

(And then it is required that the reference to the function F can be nominalized away.) But what is this function F a function on? Spacetime? What is its range? A vector space?
What is this entity M? What are its properties? If M somehow "like" a vector field on spacetime?
What does "represents" mean?
Is the usual magnetic field $\mathbf{B}$ the same as F or M?

Cheers,

Jeff

4. Damn good questions. Of course, if one is by instinct or default a nominalist, one would start from the other end of Jeff's starting point - and what is surprising there is the effectiveness of maths in representing physics. From Jeff's non-nominalist perspective, there can hardly be a surprise there - the maths is the physics. Maybe the nominalist view makes more sense for areas of physics, such as quantum mechanics, or experimental physics, where there is de facto underdetermination of maths by physics. In em theory, or even worse, spacetime theory, it is much harder to see how such nominalist distinctions play a role

5. "But this view will only work if you can nominalize physics. [(a)] What is the thing represented? [(b)] What is the thing used to represent?"

I have no idea of what people mean when they talk of "nominalizing" physics because I find that both the rules and the object of the game are unclear. As far as I can see, physical theories are not committed to the existence of anything other than concrete objects. The fact that they employ (putative) mathematical objects to describe (the behaviour of) those concrete objects is beside the point.

Anyway, here are my quick answers to your questions:

(a) the target of the representation (the thing that is represented) is a concrete entity (e.g. the EM field).
(b) the vehicle of the representation (the thing that is used to represent the target) is (typically) a fictional object (e.g. a vector field).

Of course, I would need to give an account of fictional objects that does not appeal to abstract entities in order for this to work but I think this can be done (along the lines of Kendall Walton's pretense account of fiction).

If I have just nominalized physics, clearly it wasn't that hard. If I haven't, I just can't possibly understand what would count as "nominalizing" physics.

"I think the idea you're advocating depends on *nominalizing* the physics, a la Hartry Field, and thereby replacing the mixed physical functions with primitive spatiotemporal predicates, and then showing that a (standard) model for the resulting theory can be represented by reintroducing the mixed functions. But this simply assumes nominalism and the success of a certain kind of nominalization."

With regards to the alleged question-begging, as far as I can see this is (an admittedly oversimplified sketch of) the dynamic of the dialecitc. Pre-theoretically we all accept that the existence of concrete objects such as chairs, trees, and magnetic needles. But we don't all accept the existence of "abstract entities" such as number, vectors, etc. However, it turns out that talking and thinking as if there are abstract entities is very useful. Is that a reason to believe that those entities exist? I don't see why, if we can make-do with the fictionalist story I just sketched. Is this begging the question in favour of nominalism? I don't think so, for it seems thatthe burden of proof is on those who think that the fictionalist story I sketched is not enough and that we should add abstract entities to our ontology beside the concrete ones we used to believe in (if we still believe in those once we start philosophizing that is).

1. Hi Gabriele,

Do you think that, e.g.,"$\nabla \cdot \mathbf{B} = 0$" does not imply that there is a function mapping spacetime points to vectors, whose divergence is 0. But this is surely just a logical implication. Or do you deny the existence of $\mathbf{B}$? But this then implies that Maxwell's equations are false, which scientific realists wish to resist.

"(a) the target of the representation (the thing that is represented) is a concrete entity (e.g. the EM field).
(b) the vehicle of the representation (the thing that is used to represent the target) is (typically) a fictional object (e.g. a vector field)."

But now I'm confused about what you think the EM field is. You deny the existence of $\mathbf{B}$, the function from spacetime to a vector space. But you say that the target is the EM field, so let's call it M. Is M not $\mathbf{B}$? Isn't M a vector-like field on spacetime? If not, then I can't make sense of what your entity M is ... it seems strangely noumenal (but I think it preserves some operational properties).

Also, your vehicle of representation also is a certain abstract object, and yet you deny that there are abstract objects. Hence, there is no vehicle of representation.

The only way I can see anything like this working is to take Maxwell's theory T and nominalize it, as T*, also giving certain representation theorems relating models of T* to models of T. Additionally, one gives certain conservativeness theorems guaranteeing that one can introduce mixed functions and sets without disturbing what one says about concreta.

Cheers,

Jeff

2. This comment has been removed by the author.

3. Hi Gabriele,

I forgot to answer your question about quantities, "(Could you please expand your comment about the quantity case? I'm not sure I see what you are trying to say)"

Take the case of the quantity Mass.

The thing is, the range of this quantity is, in some sense (up to isomorphism), all positive reals, $\mathbb{R}_{\ge 0}$. And given some mass scale $m$ (say, mass-in-kg), then, for certain kinds of physical system c, we have $m(c) = r$.

But there is not always a physical system $c$ for each $r \in \mathbb{R}_{\ge 0}$. So, there are some mass properties---having mass of $r$ kg---for which there is no actual instantiation.

This suggests that the mass properties are basic.

Another reason is connected to acceptable scale ("gauge") transformations. The property of having mass $1$ kg is the same property as the property of having mass $1000$ g. So, we want to identify these properties, even though they're indexed by different real numbers.

This allows us to understand quantity values as what Carnap and Quine called "impure numbers", such as $100 \text{ g}$, $25^{\circ}C$, etc. And we can write equations between them, like

$150 \text{ g} = 0.15 \text{ kg}$

Cheers,

Jeff

6. "Do you think that, e.g.,"∇⋅B=0" does not imply that there is a function mapping spacetime points to vectors, whose divergence is 0. But this is surely just a logical implication."

"∇⋅B=0" is a statement about a fictional object along the lines "Sherlock Holmes lives on Baker Street" and neither implies the existence of the object it is supposedly about. However, both are true within the relevant fiction and entail that there are such-and-such objects in the relevant fiction.

"Or do you deny the existence of B? But this then implies that Maxwell's equations are false, which scientific realists wish to resist.

Maxwell's equations are a piece of math and, as such, in and of themselves are neither true nor false of the world because they don't say anything about the world--they just describe a fictional object. However, those very equations can be used to describe certain dependency relations between physical quantities in the real world. Those descriptions, unlike the equations, are capable of being true or false, but they are purely about concrete physical stuff. To take a simpler example, the equation "F = G(Mm/r^2)" is just a piece of math (and, as an equation, it is neither true nor false). It is only when the equation is taken to express something about the relation between the magnitude of the gravitational force between two concrete objects, the masses of those two objects and the distance between them that it becomes part of a physical description of the world and can be true or false. But the math is simply providing us with a langauge that allows us to express stuff that we cannot express in ordinary language.

7. This comment has been removed by the author.

8. "But now I'm confused about what you think the EM field is. You deny the existence of B, the function from spacetime to a vector space. But you say that the target is the EM field, so let's call it M. Is M not B? Isn't M a vector-like field on spacetime? If not, then I can't make sense of what your entity M is ... it seems strangely noumenal (but I think it preserves some operational properties)."

No, M is not B. Take this magnet on my desk (I always keep one there to make realistic philosophical examples ;-)). The magnet is surrounded by a magnetic field, which is as concrete as the magnet itself. The fact that the magnetic field is unobservable makes it harder for us to distinguish the field itself from its mathematical representation, but the field is no less of a concrete entity than the magnet and, if you n fact if you sprinkle some iron dust around the magnet, you'll see the effects of the magnetic field. You claim that my magnetic field (M) is noumenal. I have to say I think this is just philosophical name-calling, but why is it so? What makes your magnetic field (B)less noumenal? If I understand your position correctly, you seem to think that what makes the iron dust form those patterns is the vectors that make up the magnetic field; I think it's the magnetic force the magnet exerts on the iron dust. (Sorry this is very rough but I don't have the time to get into the details at the moment.) Why would your position be less noumenal? You are postulating the existence of causally efficacious abstract entities!!! What could be more noumenal than that?!?

Furthermore, you call whatever causes the pattern of the iron dust an abstract entity (B), I say no it's just a concrete entity (M), which can be represented mathematically by B. But what arguments do you have for the cause of the pattern to be an abstract object? Both nominalists and platonists seem to agree that causal efficacy is the hallmark of the concrete and that abstract entities are causally inefficacious, so what would make your causally efficacious abstract entities abstract in the first place? And what would make them vectors?

"Also, your vehicle of representation also is a certain abstract object, and yet you deny that there are abstract objects. Hence, there is no vehicle of representation."

And Santa's got a beard that's long and white and yet I deny that he exists. Hence, there is no person who puts presents under the tree. :-)

No, seriously, I don't think I can answer this question satisfactorily within the limits of a blog comment, but if you are interested in my take on these issues, I can send you a copy of my book on models and representation as soon as it's out sometime this year (sorry for the little ad!) and then we can talk about it over a pint of real ale next time I'm in your neck of the woods. Deal? ;-)

9. Nice one, Gabriele. I too agree that the magnetic field is not a mathematical entity, but a physical one, which is in turn appropriately represented by a mathematical entity, namely the vector field B. This disposes of Jeff's worry regarding the causal efficacy of mathematical entities. On my view they are abstract, do not live in spacetime and have no causal powers. Jeff observes that the field B is a function that ascribes vectorial quantities to each spacetime point, which suggests to him that B lives in spacetime after all, and may be causally efficacious. But this seems to me to assume that the space and time parameters that appear in the equations that define B (implicit in the use of time and space derivatives, dt, dx, etc) are the real concrete physical points of spacetime, while I think they are just one more mathematical representation of it.

Where I part company with you is in your insistence to read Maxwell's equations as fictional. In fact you seem to want to defend that the source of any representation (the 'vehicle' as you call it) is fictional. Admittedly this is a popular view nowadays, but I think it is both mistaken and not particularly enlightening (it seems to me to raise more questions than it answers). My view, by contrast, is that in most mathematical sciences the maths represents the physics directly, without any detour via any further entity, whether fictional or not. There are no genuine questions regarding "the fiction that Maxwell's equations are true within". This is just philosophical gobbledegook that physicists have no time for. The equations are true in the sense that they appropriately represent the em field - and there are no further issues of truth or semantics involved here.

My fictionalism rather emphasizes how in scientific representation, typically targets (not sources!) are fictional or highly idealized constructs. Maxwell provides a beautiful example since throughout his life he thought he was representing the ether through his equations. (I have a forthcoming paper on this topic, if you'd like to look at it ....) Cheers, Mauricio.

1. Hi Mauricio,

"But this seems to me to assume that the space and time parameters that appear in the equations that define B (implicit in the use of time and space derivatives, dt, dx, etc) are the real concrete physical points of spacetime, while I think they are just one more mathematical representation of it."

Ok, but then there are two objects here!

(i) the magnetic field $\mathbf{B}: M \to \mathbb{R}^3$, a mixed (axial) vector field, defined on spacetime $M$;
(ii) the co-ordinate representation $\mathbf{B} \circ \phi^{-1}$ (where $\phi : M \to \mathbb{R}^4$ is a co-ordinate chart) which is purely mathematical function from $\mathbb{R}^4$ to $\mathbb{R}^3$.

So, I think the magnetic field just is the spacetime field $\mathbf{B}$, which is a mixed vector field, whereas $\mathbf{B} \circ \phi^{-1}$ is indeed a purely mathematical entity. Then I suggest that the mixed vector field $\mathbf{B}$ is causally efficacious. I don't mean that the co-ordinate representation $\mathbf{B} \circ \phi^{-1}$ is causally efficacious, as that would conflate two separate entities, one mixed and the other pure.

But, if I understand it right, Gabriele's view involves actually denying the existence of the magnetic field $\mathbf{B}$ on spacetime, as well as its co-ordinate representation $\mathbf{B} \circ \phi^{-1}$. What there is, "physically", is some other entity, M, but whose properties seem to me ineffable: for example, M isn't a vector field, with zero divergence, whose vector product with velocity of a point particle gives the local force vector. All we know about this entity M is

(R) $\mathbf{B}$ represents M.

But then, given that "represents" is defined in terms of the existence of isomorphic mappings on Gabriele's theory, I now suspect that this view will face a severe Newman problem, so that Maxwell's theory, thus construed, collapses to its empirical consequences.

So, provisionally, it seems that we have a form of scientific anti-realism with a Newman problem.

Cheers,

Jeff

2. Hi Gabriele and Mauricio,

Thanks chaps for a very good discussion of these topics.
Gabriele - I'll look forward to the book and definitely have a chat if you're nearby!

Cheers,

Jeff

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