Physics Lies!

I've been bashing the theists for a while, so I'm going to take a break and bash an atheist for a change.

Nancy Cartwright is the author of How the Laws of Physics Lie, and is apparently an influential philosopher. Kitcher mentions her in his Mind and Cosmos review, she is regularly included in anthologies of important papers in the philosophy of science, and her term "Dappled World" (borrowed from Manley Hopkins, according to Kitcher) seems to have become a sort of rallying point for modern philosophical views of science.

I think her work is shoddy and unconvincing.

OK, choosing that title for her book is like waving a cape in front of a bull, I suppose. But I really tried to give her a fair hearing, honest I did. Readers of this blog can, I suppose, judge how good I am at giving a fair hearing to views I disagree with. But let me try to present her argument before I tear it apart.

The Laws of Physics Lie

In Essay 3 of the book, Cartwright lays out the case that the laws of physics are not true, or, to the extent they are true, they are not interesting. In fact, they are not even approximately true. Why?

She illustrates with Newton's law of universal gravitation. For the definition of this law she quotes Feynman:

(NG1) The Law of Gravitation is that two bodies exert a force between each other which varies inversely as the square of the distance between them, and varies directly as the product of their masses.

Then she asks, and answers:

Does this law truly describe how bodies behave?

Assuredly not.

Why not? Well, she says, two bodies may have electric charges, and so the force exerted by one on the other is given neither by NG1 nor by Coulomb's law of electric force, but  by a combination of the two. Therefore

These two laws are not true: worse, they are not even approximately true.

For instance, in an atom, the law of gravitation is swamped by the Coulomb force and so the former is not even approximately true.

Notice that she is not complaining about extreme cases where Newtonian gravitation must be replaced by General Relativity. Her complaint is that the law doesn't state a fact: except, perhaps, in a universe completely empty except for two uncharged objects.

She considers an alternate version of NG1 that uses a prefatory clause to correct the deficiency:

(NG2) If there are no forces other than gravitational forces at work, then two bodies exert a force between each other which varies inversely as the square of the distance between them, and varies directly as the product of their masses.

This, she allows, may be a true law, but it is not a very interesting one, for in reality objects have both kinds of properties (mass and electric charge) and so NG2 has no (or very few) applications in reality.

From a physicist's point of view, this is all very wrong-headed. First of all, since she is talking about combining  different influences, the law she ought to be talking about is Newton's second law of motion:

(NA) The acceleration of an object is proportional to the net force and inversely proportional to its mass.

Somehow, she completely avoids mentioning this law anywhere in the chapter (though she mentions it obliquely in considering a related objection, as we will see below). Since she doesn't mention NA, I don't know whether she considers this one of laws that is not even approximately true. But you can't talk about gravitational and electric forces combining without it. With this framework in mind, there is an obvious alternative to NG1 and NG2:

(NG3) For two massive bodies, there is a contribution to the net force that varies inversely as the square of the distance between them, and varies directly as the product of their masses.

With this change, I submit, we have a law that is at least approximately true, with no further need of ceteris paribus clauses.

This is such a simple solution to Cartwright's difficulty that it's hard to believe she missed it. However, she does go on to discuss the law of vector composition of forces, and then to consider a suggestion of Lewis Creary that is similar to NG3. Let's consider what she has to say about these issues.

Vector Addition

Cartwright admits that physicists have an answer to the question of combining forces: she calls it the "vector addition story."

The vector addition story is, I admit, a nice one. But it is just a metaphor. We add forces... when we do calculations. Nature does not 'add' forces.For the component forces are not there, in any but a metaphorical sense, to be added....

For Cartwright, the individual component forces are not real. Only the net force is real.

But this is quite obviously false. Consider, for example, a spring that is subject to equal and opposite forces on its two ends:


The net force is zero, so the center of mass of the spring doesn't accelerate. But the spring is compressed - the component forces have a real, physical effect.

Try telling this guy that the component forces aren't real:


Cartwright is essentially saying "'two apples plus one apple equals three apples' can't be true, because if the two apples and the one apple are real, and the three apples are real, then I would have six apples, not three." But this is just silly: two apples plus one apple equals three apples because that's simply what addition means, when applied to apples. Similarly, in the vector addition of forces, two real, occurrent forces can be added to make a real net force, because that's simply what it means to combine two forces.

For this, Cartwright deserves an award for Worst Misuse of Mathematics By a Professional Philosopher Not Named Craig.

Causal Action

Cartwright then turns to Creary, who claims that there are two types of physical laws: laws of causal influence and laws of causal action. Though she doesn't say so, in Newton's mechanics, the law of causal action is good old F = ma (NA).

On Creary's account, Coulomb's law and the law of gravity come out true because they correctly describe what influences are produced.... The vector addition law then combines the separate influences to predict what motions will occur. 

 This seems to me to be a plausible account of how a lot of causal explanation is structured. But as a defence of the truth of fundamental laws, it has two important drawbacks. First, in many cases there are no general laws of interaction... In fact, classical mechanics may well be the only discipline where a general law of action is always available.

Apparently Cartwright doesn't know about quantum mechanics, where Schroedinger's equation gives a general law, or quantum field theory, where the Lagrangian path integral does the same.

Anyway, it makes no sense to claim that there are no true fundamental laws of physics, except for those few cases where the laws are both fundamental and true. Naturally, if there are general laws of action, and if physics is a more or less unified subject, then we would expect the general laws to be few. The fact that there are only a few such fundamental laws actually shows the strength of the reductionist thesis rather than the opposite.

On the other hand, if her point is just that most of the time, working physicists are not dealing with the fundamental laws, but with some approximations to them, or phenomenological laws that are not fundamental, then I would agree with her, but find the point trite and uninteresting. These physicists wouldn't ever claim that their approximations are true in all cases.

Actually, Cartwright does know about quantum mechanics, because in the very next section she discusses a quantum example: the spectrum of a carbon atom. What she says here is so pathetic that I can't bear to grind through it point by point. Essentially, she again ignores the general law of action (Schroedinger's equation) in order to make her point that there is no general law of action.

At first I thought Cartwright had simply not bothered to understand the physics before she began drawing philosophical conclusions. But later in the book she shows a detailed understanding of some much more difficult areas of physics, so that doesn't seem to be the problem.

The only other explanation I can think of for these egregious errors is that Cartwright is working with a pre-conceived agenda, such that all the examples are contorted so as to support her thesis.

This is no way to ground a philosophical world view.

A Natural Reduction

One of the great advantages of the naturalist world view is how it all hangs together. The workings of the mind can be explained in terms of the workings of the brain, which can be explained in terms of the workings of the brain cells, which can be explained in terms of the electrical and chemical properties of molecules, which can be explained in terms of the physical properties of the particles of which those molecules are composed. And the same goes for anything else in the universe: stars, moons, clovers....

Now, I will admit that some of the links in that proposed chain of explanation are not as strong as others: the mind-brain link, for example. But the overall scheme seems sound, and the naturalistic world view can count innumerable successes as evidence of its truth: the technologies of transportation, agriculture, communication, medicine, and psychiatry, to mention just a few. Put this against the abject failure of alternative ways of thinking: what did Christianity (just to pick on one alternative) accomplish in the 1500 years before scientific thinking arose?

At any rate, even when precise explanations are lacking, there doesn't seem to be any strong argument why the gaps cannot be filled out in a naturalistic way.

It seems, though, that many atheist philosophers are no longer satisfied with this reductionistic picture.  The recent book Mind and Cosmos, by atheist philosopher Thomas Nagel, argues that there are aspects of subjective experience that can't be explained by reductionistic means. (Short version here.) Another atheist philosopher, Philip Kitcher, whom I have the greatest respect for, disagrees with Nagel but seems to agree that the reductionist program has not accomplished the task it set out to do. "Unity fails at both ends," writes Kitcher.

For once, I agree with Professor Feser: if naturalism fails to give us a unified picture of everything, then it is time to abandon naturalism and seek a different explanation, rather than to cling to a failed program.

But I am not as pessimistic as these philosophers. Perhaps that's merely ignorance on my part. But some of the anti-reductionist arguments I've come across seem just silly, and, as I already said, there don't seem to be any good arguments for the impossibility of naturalistic explanation. I remain a hard-core reductionist, and I'm going to try to defend that view.

Next: Do the laws of physics lie?

Determined by What?

Kripke is central to Ross's argument, and it is certainly true that both Ross and Kripke take his point to be a metaphysical (not just an epistemological) one, so it is fair of Professor Feser to require a more detailed argument that the one I gave in my first post on Ross. I still think that what I wrote there was basically correct, and that Feser has not adequately countered my objection. But let me try to say it again, more clearly and (I hope) convincingly.

I was helped immensely in my understanding of the structure of Kripke's argument by a critical response to Kripke by Scott Soames. In an intricate bit of philosophical analysis, Soames shows that Kripke is equivocating between two different meanings of "determine." I think Ross is making a similar, but more basic, mistake, as I will explain.

What does it mean for a set of facts (F) to determine another set of facts (G)? This is the fundamental issue of determinacy. In order to be clear about Ross's argument, we need to know what he thinks F is, what he thinks G is, and what he means by "determine." For Soames, the problem lies in the last of these. For Ross, it lies in the other two.

The first thing we have to get clear is that Ross is not talking about the indeterminateness of meaning, Feser's claims notwithstanding. If he were, he would have to discuss the meaning of "meaning", as Kripke (?) and Soames do. Also, because Kripke's argument leads to skepticism about whether humans ever mean anything, as well as machines, Ross would owe us an account of how he can avoid the skeptical conclusion about humans while affirming skepticism about machines. But he does none of this. Indeed, as one commenter noted at the beginning of the discussion, Ross never mentions "meaning" in the article. Furthermore, neither Ross nor the naturalist thinks an adding machine means anything when it performs an operation, so if meaning were the issue, the entire discussion about the adding machine would be beside the point.

So the short reply to Ross' use of Kripke is that Ross has divorced the quaddition argument from the Kripkean context. The result is that quaddition becomes simply another version of the problem of limited data. And we have already seen that the problem of limited data helps Ross not at all. If you are convinced of this point, you can skip the rest of this too-long post. What follows simply expands and explains this point.

Ross never discusses meaning - his discussion is entirely about whether the machine is executing a function, and whether the machine's future outputs are determinate. Let's look again at the way he begins his argument:

Whatever the discriminable features of a physical process may be, there will always be a pair of incompatible predicates, each as empirically adequate as the other, to name a function the exhibited data or process "satisfies." That condition holds for any finite actual "outputs," no matter how many. That is a feature of physical process itself, of change. There is nothing about a physical process, or any repetitions of it, to block it from being a case of incompossible* forms ("functions"), if it could be a case of any pure form at all. That is because the differentiating point, the point where the behavioral outputs diverge to manifest different functions, can lie beyond the actual, even if the actual should be infinite; e.g., it could lie in what the thing would have done, had things been otherwise in certain ways. For instance, if the function is x(*)y = (x + y, if y < 10⁴⁰ years, = x + y + 1, otherwise), the differentiating output would lie beyond the conjectured life of the universe.

 And later on:

Secondly, opposed functions that are infinite (that is, are a "conversion" of an infinity of inputs into an infinity of outputs) can have finite sequences, as large as you like, of coincident outputs; they can even have subsequences that are infinitely long and not different (e.g., functions that operate "the same" on even numbers but differently on odd numbers). So for a machine process to be fully determinate, every output for a function would have to occur. For an infinite function, that is impossible. The machine cannot physically do everything it actually does and also do everything it might have done.

And from Thought and World

If the machine is not really adding in the single case, no matter how many acutal outputs seem "right," there might eventually be nonsums.

[Emphasis added]

I interpret Ross to be saying that what is not determined - his G - is what function the system is computing. Further, on the basis of the preceding quotes, I take it that he cashes out this G in terms of 
   a) what the system might output at a future time, and
   b)  what the system would have output for inputs it might have had, but did not have.

Let's consider a system with two inputs, x and y. Then the question is, "Is the output z determined for every possible x and y?"

Now we come to central question: determined by what? What is Ross's set F - the facts that (fail to) determine the output, z?

Since we are considering a purely physical system, a prime candidate for F would be the set

   F1: All the physical facts about the system.

In this case, the question being asked becomes "Is the output z determined by the physical facts about the system, for all possible inputs x and y?" But this is nothing more nor less than the question of physical determinism. In a deterministic world, the set F1 certainly does determine the possible outputs, z, even for cases that the machine hasn't actually computed. Setting aside issues of quantum indeterminacy (which Ross never mentions), it seems that all outputs are determined by F1.

But F1 is not what Ross has in mind. He never attempts any discussion of physical determinism. Instead, he seems to have in mind something like

   F2: The physical facts that are known about the system at some time T.

I take this from his talk of the "discriminable features of a physical process" in the first quote above and from his talk of "empirical adequacy," though I have to say that Ross is extremely vague about this.

If this  is Ross's argument - if he is saying "What the system might have done, or will do, is not determined by the physical facts that we know about the system" - then we should simply reply, "So what?" As we saw already with the problem of limited data, there is no way to argue from an epistemological lack to a metaphysical conclusion.

There is another possibility: suppose that, instead of the physical facts that are known about the system, what Ross really means is

   F3: All the physical facts that can be known about the system.

But now we have to be careful. What does is mean to say something "can be known"? Does this mean all the physical facts that can in principle be known about the system? Then F3 is the same as F1 - all the physical facts about the system can in principle be known (barring quantum uncertainty). In that case, there is no reason to think the outputs are undetermined. However, in actual fact we can never know all the physical facts about the system, no matter what set of observations we make. Thus, any given set of observations, no matter how detailed, is consistent with incompatible functions. Does this  justify Ross's conclusion? No, because in that case, there is again only an epistemological lack.

Let me explain the last remark using the example of a computer. The computer seems at first to be a simple counterexample to Ross's claim that the outputs z are not determined by the physical features of the system. For I can look at the program the computer is running (it is encoded physically somewhere in the computer's memory) and see  what the function is: I can deduce, for example, what the output would have been for some inputs x and y that the computer has not actually calculated. But (as Feser points out) a wire might burn out or a transistor go bad inside the computer, so that the actual output is not what the computer program would lead us to believe. This is true, but it doesn't really answer the objection. For suppose I insist on a more detailed physical description of the computer: one so detailed that the failure of the wire/transistor is predictable by this description and so is accounted for. Then we see that the indeterminacy was only apparent: the output is in fact determined by this more detailed set of facts. (In this case the computer would be executing something like quaddition, rather than addition.) If we dig deep enough, we will always find some set of physical facts that do determine the output.

We can now see why Ross is wrong to say that "for a machine process to be fully determinate, every output for a function would have to occur." A sufficiently detailed set of physical facts about the system determines not only what outputs will occur at a future time, but also what outputs would have occurred for other inputs that were not actually submitted to the machine.

I have gone into this at length because I think it shows both why Ross's argument is so appealing and why it is wrong. For any given set of physical facts about the system, there are infinitely many inequivalent functions compatible with the behavior of the system. But the set of possible observations is not fixed: if we find a particular set of physical facts leaves the outcome undetermined, we can always ask for a more detailed description of the system. For a sufficiently detailed set of physical facts, we find the outcome is determined by those facts. What makes the argument seem reasonable is the slide from a given set of facts, to any possible set of facts, to all physical facts.

To summarize:

• Ross's argument is not about  whether meanings are determined by the physical facts about the system, but whether a functional form is determined.
• Whether a functional form is determined is cashed out in terms of whether future or counterfactual outputs are determined.
• Ross is unclear about what F it is that fails to determine the functional form.
• If we take F to be the set of all physical facts about the system, then all outputs are determined, and Ross's argument fails.
• If we take F to be the set of known physical facts, then outputs are indeed undetermined, but this is only an epistemological issue. For a sufficiently detailed set of physical facts, the output is determined.
• Thus, all three of Ross's main arguments: quaddition, grue, and the problem of limited data, only point to a lack of knowledge about the system.
• These epistemological concerns are not enough to draw the conclusion that the system is "physically and logically" indeterminate.

None of this addresses Professor Feser's point about the meaning of a physical process, which I will have to address (I hope!) another time.

* I take Ross to mean "inequivalent" rather than "incompossible" here and throughout - see Richard's remarks in the previous post.

Guest Post: Purity of Form and Function

While I'm gearing up for my assault on Mount Kripke, here's a guest post from Richard Wein.

Hi everyone. Robert's invited me to make a guest post on the subject of James Ross's paper "Immaterial Aspects of Thought". The resulting post is rather long, partly because there's a lot of linguistic confusion to be cleared up. I hope I can dispel a little of that confusion.

At the core of Ross's argument is his insistence that our logical thinking must involve "pure forms". He then argues that physical processes can't have such forms, and so logical thinking must involve more than just physical processes. I see no good reason to accept that we need any such forms.

Ross's concept of "pure forms" is hardly explained, and remains mysterious to me. He says, for example, that squaring involves thinking in the form "N X N = N^2". He doesn't seem to mean that we must think such words to ourselves. He seems to have in mind some unseen form, possibly Platonistic. In Section III, he talks briefly about "Platonistic definitions", and perhaps this example is one such.

It may help here if I briefly give my own physicalist view, so that I can consider Ross's in contrast to it. I say that the only verbal forms that exist are the sorts that we observe, such as those in writing, speech and conscious thought. These observed verbal forms are produced by non-conscious, non-verbal physical cognitive processes. Of course there's a lot more to be said about how this happens, and particularly about consciousness, but these are not issues that Ross raises. He is not, for example, making an argument from consciousness. Nor is he making an inference to the best explanation, where we must consider the relative merits of his explanation and a physicalist one. He is making a purely eliminative argument, and so the onus is on him to eliminate physicalist alternatives, not on me to elaborate on them.

The claim that our thinking must take the form of definitions in this sense seems to lead to a problem of infinite regress. If squaring is defined in terms of a more basic operation, multiplying, then how is multiplying defined? And so on. But I won't dwell on this point, because Ross's "pure forms" are so mysterious that I doubt I could make any specific positive criticism stick. My point is that we just don't need anything of the sort Ross is insisting on. He gives us no reason to think that the sorts of physical processes I've mentioned are incapable of producing everything that we actually observe. I don't think he even tries to show that. His only response to views like mine seems to be that, on such views, the actual processes we currently call "squaring" wouldn't be "real" squaring, but only "simulated" squaring.

This response is a confused use of language, mistaking an empty verbal distinction for a substantive one. First, regardless of whether we call such operations "real" or "simulated", if they're sufficient to deliver everything we actually observe--and Ross doesn't seem to argue the contrary--then there's no reason to think we need anything more. That in itself should suggest some confusion on Ross's part.

The distinction Ross was originally making was between processes that involve "pure forms" and those that don't. If the distinction he's now making between "real" and "simulated" processes is just a translation of the original distinction into different words, then the translation achieves nothing. He's just re-asserting his unsupported claim that we need such pure forms, but doing it in confusing new words. If, on the other hand, the new distinction were genuinely different from the original one, Ross would actually have to demonstrate that denying pure forms entails denying real squaring. He would have to make a substantive argument, and he wouldn't be able to do that without clarifying the meaning of his new distinction. In fact he makes no such argument (or clarification). He simply puts the words "we only simulate" into the mouth of the denier, as if it's indisputable that denying pure forms entails denying real squaring. So it's pretty clear that this is just a confusing terminological switch, masquerading as a substantive argument. The appearance of having achieved something arises through conflation of a weaker sense of the words (in which they are just a translation of the original claim) with a stronger sense (which has the appearance of a more irresistible claim). To accept Ross's conclusion on this basis is to commit a fallacy of equivocation.

Unlike Ross's denier, I don't say that we don't "really" square. Neither do I say that we do "really" square. The word "really" is misleading here. If denying that we "really" square is to be taken as just another way of saying that we don't think in "pure forms", then I prefer to say--more directly--that we don't think in pure forms.

There ends my main response to Ross's argument. But I'd likely briefly to address some other aspects of his paper which are liable to cause confusion.

Ross uses the term "pure forms" interchangeably with "pure functions", and I'm afraid this translation may have led to a conflation of these concepts of his own with the concept of a mathematical function in the ordinary sense of that term. Mathematical functions are purely abstract, and don't exist in anything like the sense that physical objects do. Pairs of mathematical functions like addition and quaddition are correctly called "non-equivalent". To call them "incompossible" would be a kind of category error, mistakenly implying that it makes any sense to ask whether they can co-exist. Ross's talk of incompossible pure forms/functions is further support for the conclusion that he sees these as having a more real sort of existence than do mathematical functions (in the ordinary sense).

I have no idea what it could mean for the process of squaring to take the form of a mathematical function. The messy, fallible real-world processes that we call "squaring" are quite a different thing from the abstract function that mathematicians call "f(x)=x^2". Talk of a process taking the form of a mathematical function seems to me like a category error, an attempt to transfer properties inappropriately between pure abstractions and real processes. Of course, during the process of performing an arithmetic operation, some definition of a function (some form of words) might be produced, e.g. in conscious thought. But that's a production of the process, and not the process itself or the form of the process. Moreover, simple arithmetic doesn't always involve giving ourselves any definitions, rules or instructions for how to proceed. The answers can come to mind (or speech) as the result of non-verbal non-conscious processes, without any verbal reasoning. That's why there's no infinite regress of definitions, rules or instructions.

You may have noticed that I haven't mentioned determinacy. Ross's argument is primarily made in terms of pure forms. But at times he translates into the language of determinacy, and his summary argument is expressed in such language. This translation into the language of determinacy serves no useful purpose, but creates further opportunities for confusion, because Ross's "indeterminacy" is easily conflated with other senses of the word, and Ross himself encourages such conflation by appealing to work on other sorts of indeterminacy (and even "underdetermination") which have little to do with his argument.

Since I don't think we need any "realization" of "pure forms", and I would question whether the the concept is even coherent, there's no point in my addressing Section II of the paper in detail. But I think it would be useful briefly to give a clearer account of the addition/quaddition scenario and "indeterminacy". Ross employs a variant of Kripke's quaddition function, where the differentiating point (instead of 57) is set to a number of years greater than the lifetime of the universe. That example seems peculiar to me, as I can see no reason why a system can't calculate a number of years greater than that lifetime. So I'll take a slightly different example of my own. Let the differentiating point be a number greater than any that can be represented by a given calculator. Then there's a sense in which the calculator equally well "realizes" addition and quaddition. That sense is that the calculator gives the answers for quaddition as well as it gives them for addition. As long as we don't confuse ourselves with talk of "pure forms", there's nothing remarkable about this. Ross wants to say that the calculator can't realize two different functions, so it must realize neither. But in the sense of "realize" that I've just used, there's no problem with saying that it realizes both, and the fact that it realizes both has no substantive significance.

Given that we're not talking about epistemological or quantum indeterminacy, any indeterminacy lies just in the fact that often our categories are not sufficiently well-defined for us to be able to assign a given state of affairs to a single category. For example, for some people there is no fact of the matter as to whether they are best described as children or adults. That's just a limitation of language. Indeterminacy is significant for our understanding of how language works, and for making sure we use language in ways that don't cause confusion, but it doesn't have any substantive, non-linguistic significance. Some people are reading far too much into indeterminacy.

Grue Some More

The second point Ross brings up in support of the indeterminacy of the physical is Goodman's Grue Argument. This one is easily dealt with.

Goodman defines something as "grue" if it is first observed before Jan1, 2025 (say) and is green, or is first observed after Jan 1, 2025 and is blue. He uses this to make a point about induction: any evidence we cite as evidence for the proposition "all emeralds are green" is also evidence for the proposition "all emeralds are grue." Thus, the grue problem casts doubt on the rationality of inductive conclusions.

So we see that Goodman's point was about induction, not indeterminacy. But this is really unimportant for Ross's argument, because Ross doesn't actually use the grue argument in any essential way. Rather, he either cites grue as an example of the problem of limited data, or as an analogy to Kripke's quaddition argument. For instance, Ross writes:

A decisive reason why a physical process cannot be determinate among incompossible abstract functions is "amplified grueness": a physical process, however short or long, however few or many outputs, is compatible with counterfactually opposed predicates; even the entire cosmos is. Since such predicates can name functions from "input to output" for every change, any physical process is indeterminate among opposed functions. This is like the projection of a curve from a finite sample of points: any choice has an incompatible competitor.

 But the  problem of limited data, as we have seen, is irrelevant for the indeterminacy question. So the grue point devolves onto the Kripke/quaddition point, which I will consider next.

A Pointless Point about Data Points

I've been, and continue to be, busy with my real work, so I have to apologize if these posts dribble out slowly, a bit at a time, rather than in one long, well-thought-out post the way Prof. Feser does. But maybe it will actually be an advantage to try to clear up one point at a time.

Ross gives three main arguments to support his claim B: "No physical system is determinate." These are:

1. Kripke's addition/quaddition argument.
2. Goodman's grue argument.
3. The problem of limited data.

Kripke is the central point of Ross's argument, and it is the most difficult to tackle. I'm going to start at the other end, with (3).

The problem of limited data (PLD) is pretty easy to state. Suppose I have some system from which I can take data, and I am trying to determine what function the system is following in order to produce the data. If I take a limited number (say a finite number) of input-output pairs, there will be an infinite number of functions which fit the given data points. For example, suppose I have only three data points. Then there is exactly one quadratic function that will exactly fit those three points. But I could also fit the data with a cubic function, or a quartic function, or a polynomial of any higher degree, or an exponential function times an appropriate polynomial, etc.

Now, how does this help Ross establish his (B)? Actually, it doesn't help at all. The mere fact that I have a limited amount of data doesn't tell me anything about the process that is producing that data. Since the PLD applies equally to determinate and indeterminate systems, it can't. This seems completely obvious to me, but since it seems to be a point of contention, I will spell it out.

Suppose I get some set of N input-output pairs from a determinate process. (For Ross, this means having a human compute them.) Let someone else give me N input-output pairs from an indeterminate process that is simulating (in Ross's sense) the first process. (For Ross, this could be a computer.) Now, since Ross allows that the indeterminate process can simulate the determinate process very closely, the two data sets will be identical. (This is easy to see if we say the process in question is addition: the computer and the human will give the same outputs for the same inputs. Unless, of course, the human makes a mistake.)

Since the N input-output pairs are identical whether I get them from a determinate or an indeterminate process, there is obviously no way I can tell from the data which sort of process produced that data.

Now let's introduce the PLD. Clearly, it applies to both processes. So if (as Ross claims) the PLD provides support for the claim that the purely physical process is indeterminate, then it also provides support for the claim that the human-generated process is indeterminate. So the PLD strengthens Ross's case for B to the exact extent that it weakens his case for A (that humans are capable of determinate processes). In other words, it doesn't help him at all.

This is what I meant when I wrote that Ross's arguments don't go beyond epistemology. The PLD says I can have only limited knowledge about the process that produced the data. But it says nothing at all about the metaphysical properties of that process.

Goodbye, Hilda

I'd like to thank Prof. Feser for his continued patience in responding to my critique of Ross's argument. I've been very busy, but I finally had some time to look at his most recent response. He misconstrued the A,  B, C, of my previous post (understandably, since I hadn't spelled them out clearly), and I began a long post carefully laying out the logic of my argument and why Feser's response didn't answer it. Then I realized that it did answer it, in spite of the misunderstanding about A, B, and C. The "purely physical" assumption is indeed the critical assumption in Ross's argument that I wasn't taking into account, and it does eliminate the Hilda objection in a non-question-begging way. I apologize to Prof. Feser for the unwarranted and unnecessary snark in my last post. I am hereby giving Hilda the boot.

I hope to return to my original epistemological objection (as time permits), but I wanted to get this apology out in a timely manner.

Crow is a dish best eaten warm.