Post-Goedelian mathematics as an empirical inquiry

Once one absorbs the lessons of the Goedel incompleteness theorems, a formalist view of mathematics as just about logical relationships such as provability becomes unsupportable (for me the strongest indication of this is the independence of logical validity). Platonism thereby becomes more plausible (but even Platonism is not unproblematic, because mathematical Platonism tends towards plenitude, and given plenitude it is difficult to identify which natural numbers we mean).

But there is another way to see post-Goedelian mathematics, as an empirical and even experimental inquiry into the question of what can be proved by beings like us. While the abstract notion of provability is subject to Goedelian concerns, the notion of provability by beings like us does not seem to be, because it is not mathematically formalizable.

We can mathematically formalize a necessary condition for something to be proved by us which we can call “stepwise validity”: each non-axiomatic step follows from the preceding steps by such-and-such formal rules. To say that something can be proved by beings like us, then, would be to say that beings like us can produce (in speech or writing or some other relevantly similar medium) a stepwise valid sequence of steps that starts with the axioms and ends with the conclusion. This is a question about our causal powers of linguistic production, and hence can be seen as empirical.

Perhaps the surest way to settle the question of provability by beings like us is for us to actually produce the stepwise valid sequence of steps, and check its stepwise validity. But in practice mathematicians usually don’t: they skip obvious steps in the sequence. In doing so, they are producing a meta-argument that makes it plausible that beings like us could produce the stepwise valid sequence if they really wanted to.

This might seem to lead to a non-realist view of mathematics. Whether it does so depends, however, on our epistemology. If in fact provability by beings like us tracks metaphysical necessity—i.e., if B is provable by beings like us from A₁, ..., A_n, then it is not possible to have A₁, ..., A_n without B—then by means of provability by beings like us we discover metaphysical necessities.

Nitpicking about the causal exclusion argument

Exclusion arguments against dualism, and sometimes against nonreductive physicalism, go something like this.

1. Every physical effect has a sufficient microphysical cause.

2. Some microphysical effects have non-overdetermined mental causes.

3. If an event E has two distinct causes A and B, with A sufficient, it is overdetermined.

4. So, some mental causes are identical to microphysical causes.

But (3) is just false as it stands. It neglects such cases of non-overdetermining distinct causes A and B as:

5. A is a sufficient cause of E and B is a proper part of A, or vice versa. (Example: E=window breaking; A=rock hitting window; B=front three quarters of rock hitting window.)

6. A is a sufficient cause of B and B is a sufficient cause of E, or vice versa, with these instances of sufficient causation being transitive. (Example: E=window breaking; A=Jones throwing rock at window; B=rock impacting window.)

7. B is an insufficient cause of A and A is a sufficient cause of B, with these instances of causation being transitive. (Example: E=window breaking; B=Jones throwing rock in general direction of window; A=rock impacting window.)

8. A and B are distinct fine-grained events which correspond to one coarse-grained event.

To take care of (6) and (7), we could replace “cause” with “immediate cause” in the argument. This would require the rejection of causation by a dense sequence of causes (e.g., the state of a Newtonian system at 3 pm is caused by its state at 2:30 pm, its state at 2:45 pm, at 2:52.5 pm, and so on, with no “immediate” cause). I defend such a rejection in my infinity book. But the price of taking on board the arguments in my infinity book is that one then has very good reason to accept the Kalaam argument, and hence to deny (1) (since the first physical state will then have a divine, and hence non-microphysical, cause).

We could take care of (5) and (8) by replacing “distinct” with “non-overlapping” in (3). But then the conclusion of the argument becomes much weaker, namely that some mental causes overlap microphysical causes. And that’s something that both the nonreductive physicalist and hylomorphic dualist can accept for different reasons: the nonreductive physicalist may hold that mental causes totally overlap with microphysical causes; the hylomorphist will say that the form is a part of both the mental cause and of the microphysical cause. Maybe we still have an argument against substance dualism, though.

Infinity, Causation, and Paradox

Amazon USA finally has it in stock.

Two kinds of functionalism

There are two kinds of functionalism about the mind.

One kind upholds the thesis that if two systems exhibit the same overall function, i.e., the same overall functional mapping between sequences of system inputs and sequences of system outputs, then they have the same mental states if any. Call this systemic functionalism.

The other kind says that mental properties depend not just on overall system function, but also on the functional properties of the internal states and/or subsystems of the system. Call this subsystemic functionalism. The subsystemic functionalist allows that two systems may have the same overall function, but because the internal architecture (whether software or hardware) that achieve this overall function are different, the mental states of the systems could be different.

Systemic functionalism allows for a greater degree of multiple realizability. If we have subsystemic functionalism, we might meet up with aliens who behave just like we do, but who nonetheless have no mental states or mental states very different from ours, because the algorithms that are used to implement the input-to-output mappings in them are sufficiently different.

If subsystemic functionalism is true, then it seems impossible for us to figure out what functional properties constitute mental states, except via self-experimentation.

For instance, we would want to know whether the functional properties that constitute mental states are neuronal-or-above or subneuronal. If they are neuronal-or-above, then replacing neurons with prostheses that have the same input-to-output mappings will preserve mental states. If they are subneuronal, such replacement will only preserve mental states if the prostheses not only have the same input-to-output mappings, but also are functionally isomorphic at the relevant (and unknown to us) subneuronal level.

But how could we figure out which is the case? Here is the obvious thing to try: Replace neurons with prostheses whose internal architecture does not have much functional resemblance to neurons but which have the same input-to-output mappings. But assuming standard physicalist claims about there not being “swervy” top-down causation (top-down causation that is unpredictable from the microphysical laws), we know ahead of the experiment that the subject will behave exactly as before. Yet if we have rejected systemic functionalism, sameness of behavior does not guarantee sameness of mental states, or any mental states at all. So doing the experiment seems pointless: we already know what we will find (assuming we know there is no swervy top-down causation), and it doesn’t answer our question.

Well, not quite. If I have the experiment done on me, then if I continue to have conscious states after complete neuronal prosthetic replacement, I will know (in a Cartesian way) that I have mental states, and get significant evidence that the relevant system level is neuronal-or-above. But I won’t be able to inform anybody of this. If I tell people: “I am still conscious”, if they have rejected systemic functionalism, they will just say: “Yeah, he/it would say that even if he/it weren’t, because we have preserved the systemic input-to-output mappings.” And there will be significant limits to what even I can know. While I could surely know that I am conscious, I doubt that I would be able to trust my memory to know that my conscious states haven’t changed their qualia.

So with self-experimentation, I could know tht the relevant system level is neuronal-or-above. Could I know even with self-experimentation that the relevant system level is subneuronal. That’s a tough one. At first sight, one might consider this: Replace neurons with prostheses gradually and have me observe whether my conscious experiences start to change. Maybe at some point I stop having smell qualia, because the neurons involved in smell have been replaced with subsystemically functionally non-isomorphic systems. Oddly, though, given the lack of swervy top-down causation, I would still report having smell qualia, and act as if I had them, and maybe even think, albeit mistakenly, that I have them. I am not sure what to make of this possibility. It’s weird indeed.

Moreover, a version of the above argument shows that there is no experiment that we could do that would persons other than at most the subject know whether systemic or subsystemic functionalism is true, assuming there is no swervy top-down causation.

Things become simpler in a way if we adopt systemic functionalism. It becomes easier to know when we have strong AI, when aliens are conscious, whether neural prostheses work or destroy thought, etc. The downside is that systemic functionalism is just behaviorism.

On the other hand, if there is swervy top-down causation, and this causation meshes in the right way with mental functioning, then we are once again in the experimental philosophy of mind business. For then neurons might function differently when in a living brain than what the microphysical laws predict. And we could put in prostheses that function outside the body just like neurons, and see if those also function in vivo just like neurons. If so, then the relevant functional level is probably neuronal-or-above; if not, it's probably subneuronal.

The centrality of the natural numbers

The more I think about the foundations of mathematics, the more wisdom I see in Kronecker’s famous saying: “God made the natural numbers; all else is the work of man.” There is something foundationally deep about the natural numbers. We see this in the way theories of natural numbers is equivalent (e.g., via Goedel encoding) to the theories of strings of symbols that are central to logic, and in the way that when we fix our model of natural numbers, we fix the foundational notion of provability.

The context problem for Lewisian functionalism

One problem for functionalism is the problem of defect. David Lewis, for instance, talks of a madman for whom pain is triggered by something other than damage and whose pain triggers something other than avoidance. Lewis’s functionalist solution is to define the function of a mental state in terms of the role it normally plays in the species.

Here is a problem with this. Suppose that in mammals pains is realized by C-fiber firing. But now take the C-fibers inside a living mammalian skull, disconnect their outputs and connect external electrodes to their inputs. Make the C-fibers fire. Since the C-fiber outputs are disconnected, causing them to fire does not cause any of the usual pain behaviors, the formation of memories of pain, etc. In fact, it seems very plausible that there is no pain at all. Yet according to Lewisian functionalism, there is pain, because it is the normal connections of the C-fibers that define their functional role.

This thought experiment shows that the physical realizers of mental states need to occur in their proper context. But this bumps up against Lewis’s madman, in whom the pain states, and presumably their physical realizers, do not occur in their proper context.

It seems that what the functionalist needs to say is that in order to realize a mental state, a physical state must occur in a sufficient approximation to its proper context. If it’s too far, as in the case of the C-fibers with severed outputs, there is no mental state. If it’s close enough, as in a moderate version of the madman case (I don’t know what to say about Lewis’s more extreme one), the mental state occurs.

But how is the line to be drawn?

Perhaps there is no problem. Pain is not in fact C-fiber firing. Perhaps enough of the brain needs to be involved in conscious states that one cannot plausibly remove the states from their normal functional context? Still, this is worth thinking about.

M-rays

Let me tell a story. Some neuroscientists detected a physically novel form of radiation, M-rays, that is emitted by brains of subjects who are thinking consciously. They discovered this much like X-rays were discovered, namely by finding something impinging on equipment in the lab in a way that could be explained by conventional physics. Further experiment showed that M-rays are not emitted by anything that clearly isn’t conscious. Moreover, the line between animals that emitted M-rays and animals that didn’t seemed to correspond to a noticeable difference in cognitive sophistication. Finally, in humans the M-rays turned out to be modulated in a way that has a natural one-to-one correspondence with the phenomenal states reported by the conscious subjects, so that the scientists eventually learned to discern from the M-rays what the subject’s conscious state is. (The CIA was very interested.)

In this case, it would be eminently reasonable for a physicalist to conclude that consciousness is the emission of M-rays.

This thought experiment shows that mysterians like Colin McGinn are mistaken in holding that no discovery we could make would solve the hard problem of consciousness.

But of course few physicalists actually expect to find a physically novel phenomenon in the brain.

Independence of FOL-validity

A sentence ϕ of a dialect of First Order Logic is FOL-valid if and only if ϕ is true in every non-empty model under every interpretation. By the Goedel Completeness Theorem, ϕ is valid if and only if ϕ is a theorem of FOL (i.e., has a proof from no axioms beyond any axioms of FOL). (Note: This does not use the Axiom of Choice since we are dealing with a single sentence.)

Here is a meta-logic fact that I think is not as widely known as it should be.

Proposition: Let T be any consistent recursive theory extending Zermelo-Fraenkel set theory. Then there is a sentence ϕ of a dialect of First Order Logic such that according to some models of T, ϕ is FOL-valid (and hence a theorem of FOL) and according to other models of T, ϕ is not FOL-valid (and hence not a theorem of FOL).

Note: The claim that ϕ is FOL-valid according to a model M is shorthand for the claim that a certain complex arithmetical claim involving the Goedel encoding of ϕ is true according to M.

The Proposition is yet another nail in the coffins of formalism and positivism. It tells us that the mere notion of FOL-theoremhood has Platonic commitments, in that it is only relative to a fixed family of universes of sets (or at least a fixed model of the natural numbers or a fixed non-recursive axiomatization) does it make unambiguous sense to predicate FOL-theoremhood and its lack. Likewise, the very notion of valid consequence, even of a finite axiom set, carries such Platonic commitments.

Proof of Proposition: Let G be a Rosser-tweaked Goedel sentence for T with G being Σ₁ (cf. remarks in Section 51.3 here). Then G is independent of T. In ZF, and hence in T, we can prove that there is a Turing machine Q that halts if and only if G holds. (Just make Q iterate over all natural numbers, halting if the number witnesses the existential quantifier at the front of the Σ₁ sentence G.) But one can construct an FOL-sentence ϕ such that one can prove in ZF that ϕ is FOL-valid if and only if Q halts (one can do this for any Turing machine Q, not just the one above). Hence, one can prove in T that ϕ is FOL-valid if and only if I holds.

Thus, in T it is provable that ϕ is FOL-valid if and only if G holds. But T is a consistent theory (otherwise one could formalize in T the proof of its inconsistency). Since G is independent of T, it follows that the FOL-validity of ϕ is as well.

Preventing suffering

Theodicies according to which sufferings make possible greater moral goods are often subjected to this objection: If so, why should we prevent sufferings?

I am not near to having a full answer to the question. But I think this is related to a question everyone, and not just the theist, needs to face up to. For everyone should accept Socrates’ great insight that moral excellence is much more important than avoiding suffering, and yet we should often prevent suffering that we think is apt to lead to the more important goods. I don’t know why. That’s right now one of the mysteries of the moral life for me. But it is as it is.

Famously, persons with disabilities tend to report higher life satisfaction than persons without disabilities. But we all know that accepting this data should not keep us from working to prevent disability-causing car accidents. While higher life satisfaction is not the same as moral excellence, the example is still instructive. Our reasons to prevent disability-causing car accidents do not require us to refute the empirical data suggesting that persons with disabilities lead more satisfying lives. I do not know why exactly we still have on balance reason to prevent such accidents, but it is clear to me that we do.

Mother Teresa thought that the West is suffering from a deep poverty of relationships, with both God and neighbor. Plausibly she was right. We probably are not in a position to know that affluence is a significant cause of this deep poverty, but we can be open to the real epistemic possibility that it is, and we can acknowledge the deep truth that the riches of relationship are far more important than physical goods, without this sapping our efforts to improve the material lot of the needy.

Or suppose you are witnessing Alice torturing Bob, and an oracle informed you that in ten years they will be reconciled, with Bob beautifully forgiving Alice and Alice deeply repenting, with the goods of the reconciliation being greater than the bads in the torture. I think I should still stop Alice.

A quick corollary of the above cases is that consequentialism is false. But there is a deep paradox here that cuts more deeply than consequentialism. I do not know how to resolve it.

Here are some stories, none of which are fully satisfying to me in their present state of development.

Perhaps it is better if humans have a special focus on the relief of suffering and improvement of material well-being of the patient. An opposite focus might lead to an unhealthy condescension.

Perhaps it has something to do with our embodied natures that a special focus on the bodily good of the other is a particularly fitting way for humans to express love for one another. While letting another suffer in the hope of greater on-balance happiness might be better for the patient, it could well be worse for the agent and the relationship. Maybe we should think of what Catholics call the “corporal works of mercy” as a kind of kiss, or maybe even something like a sacrament.

Perhaps there is something about respect for the autonomy of the other. Maybe others’ physical good is also our business while moral development is more their own business.

I think there is more. But the point I want to make is just that this is not a special question for theism and theodicy. It is a paradox that all morally sensitive people should see both sides of.

Coming back to theodicy, note that the above speculative considerations may not apply to God as the agent. (God cannot but condescend, being infinitely above us. God is not embodied, except in respect of the Incarnation. And we have no autonomy rights against God, as God is closer to us than we are to ourselves.)

Minecraft cake cake

Our youngest wanted a Minecraft cake for her birthday. We settled on a Minecraft cake block cake using this clever method. (But we froze the frosting overnight on parchment paper—maybe wax paper would have been better—and instead of candy melts just used more frozen colored frosting; we also used only two layers.) The brown bottom is cinnamon—we had three people apply it simultaneously.

Groups and roles

I’ve had a grad student, Nathan Mueller, do an independent study in social epistemology in the hope of learning from him about the area (and indeed, I have learned much from him), so I’ve been thinking about group stuff once a week (at least). Here’s something that hit me today during our meeting. There is an interesting disanalogy between individuals and groups. Each group is partly but centrally defined by a role, with different groups often having different defining roles. The American Philosophical Association has a role defined by joint philosophical engagement, while the Huaco Bowmen have a role defined by joint archery. But this is not the case for individuals. While individuals have roles, the only roles that it is very plausible to say that they are partly and centrally defined by are general roles that all human beings have, roles like human being or child of God.

This means that if we try to draw analogies between group and individual concepts such as belief or intention, we should be careful to draw the analogy between the group concept and the concept as it applies not just to an individual but to an individual-in-a-role. Thus, the analogy is not between, say, the APA believing some proposition and my believing some proposition, but between the APA believing some proposition and my believing that proposition qua father (or qua philosopher or qua mathematician).

If this is right, then it suggests an interesting research program: Study the attribution of mental properties to individuals-in-roles as a way of making progress on the attribution of analogous properties to groups. For instance, there are well-founded worries in the social epistemology literature about simple ways of moving from the belief of the members of the group to the belief of the group (e.g., attributing to the group any belief held by the majority of the members). These might be seen to parallel the obvious fact that one cannot move from my believing p to my believing p qua father (or qua mathematician). And perhaps if we better understand what one needs to add to my believing p to get that I believe p qua father, this addition will help us understand the group case.

(I should say, for completeness, that my claim that the only roles that human beings are partly and centrally defined by are general roles like human being is controversial. Our recent graduate Mengyao Yan in her very interesting dissertation argues that we are centrally defined by token roles like child of x. She may even be right about the specific case of descent-based roles like child of x, given essentiality of origins, but I do not think it is helpful to analyze the attribution of mental properties to us in general in terms of us having these roles.)

My experience of temporality

This morning I find myself feeling the force of presentism. I am finding it hard to see my four-dimensional worm theory as adequately explaining why my experience only includes what I am experiencing now, instead of the whole richness of my four-dimensional life. I am also finding it difficult to satisfactorily explain the sequentiality of my experiences: that I will have different experiences from those that I have now, some of which I dread and some of which I anticipate eagerly.

When I try to write down the thoughts that make me feel the force of presentism, the force of the thoughts is largely drained. After all, to be fair, when I wrote that I have am having trouble “explaining why my experience only includes what I am experiencing now”, shouldn’t I have written: “explaining why my present experience only includes what I am experiencing now”, a triviality? And that mysterious sequentiality, is that anything beyond the fact that some of my experiences are in the future of my present experience?

The first part of the mystery is due to the chopped up nature of my consciousness on a four-dimensional view. Instead of seeing my life as a whole, as God sees it, I see it in very short (but probably not instantaneous) pieces. It is puzzling how my consciousness can be so chopped up, and yet be all mine. But we have good reason to think that this phenomenon occurs outside of temporality. Split brain patients seem to have such chopped up consciousnesses. And if consciousness is an operation of the mind’s, then on orthodox Christology, the incarnate Christ, while one person, had (and still has) two consciousnesses.

Unfortunately, both the split brains and the Incarnation are mysterious phenomena, so they don’t do much to take away the feeling of mystery about the temporal chopping up of the consciousness of my four-dimensional life. But they do make me feel that there is no good argument for presentism here.

The second part of the mystery is due to the sequentiality of the experiences. As the split brain and Incarnation cases show, the sequentiality of experiences in different spheres of consciousness is not universal. The split brain patient has two non-sequential, simultaneous spheres of consciousness. Christ has his temporal sphere (or spheres, if we take the four-dimensional view) of consciousness and his divine atemporal sphere of consciousness. But seeing the contingency of the sequentiality does not remove the mystery in the sequentiality.

It makes me feel a little better when I recall that the presentist story about the sequentiality has its own problems. If my future experiences aren’t real—on presentism they are nothing but stuff in the scope of a modal “will” operator that doesn’t satisfy the T axiom—then what am I anticipating or dreading? It seems I am just here in the present, and when I think about this, it feels just as mysterious as on four-dimensionalism what makes the future impend. Of course, the presentist can give a reductive or non-reductive account of the asymmetry between past and future, but so can the four-dimensionalist.

So what remains of this morning’s presentist feelings? Mostly this worry: Time is mysterious and our theories of time—whether eternalist or presentist—do not do justice to its mysteriousness. This is like the thought that qualia are mysterious, but when we give particular theories of them—whether materialist or dualist—it feels like something is left out.

But what if I forget about standard four-dimensionalism and presentism, and just try to see what theory of time fits with my experiences? I then find myself pulled towards a view of time I had when I was around ten years old. Reality is four-dimensional, but we travel through it. Future sufferings I dread are there, ahead of me. But I am not just a temporal part among many: there is no future self suffering future pains and enjoying future pleasures. The past and future have physical reality but it’s all zombies. As for me, I am wholly here and now. And you are wholly here and now. We travel together through the four-dimensional reality.

But these future pains and pleasures, how can they be if they are not had by me or anyone else? They are like the persisting smile of the Cheshire cat. (I wasn’t worried about this when I was ten, because I was mainly imagining myself as traveling through events, and not philosophically thinking about my changing mental states. It wasn’t a theory, but a way of thinking.) Put that way, maybe it’s not so crazy. After all, the standard Catholic view of the Eucharist is that the accidents of bread and wine exist without anything having them. So perhaps my future and past pains and pleasures exist without anyone having them—but one day I will have them.

Even this strange theory, though, does not do justice to sequentiality. What makes it be the case that I am traveling towards the future rather than towards the past?

And what about Relativity Theory? Why don’t we get out of sync with one another if we travel fast enough relative to one another? Perhaps the twin who travels at near light speed comes back to earth and meets only zombies, not real selves? That seems absurd. Maybe though the internal flow of time doesn’t work like that.

I do not think this is an attractive theory. It is the theory that best fits most of my experience of temporality, and that is a real consideration in favor of it. But it doesn’t solve the puzzle of sequentiality. I think I will stick with four-dimensionalism. For now. (!)

Extended simples

There are at least two reasons to think we are simple:

1. It is difficult to explain how a non-simple thing can have a unity of consciousness.

2. There is David Barnett’s “pairs” argument.

But we are clearly extended.

So, we are extended simples.

So, there are extended simples.

(That said, while I am happy with the idea that we are extended simples, I am suspicious of both 1 and 2.)

Ownership and ontology

We can own dogs, trees, forests, cars, chairs, computers and cupcakes, but of these examples, only dogs and trees really exist. Many of the things we own do not really exist. This makes me sceptical of the idea that there are strong property rights independent of positive law.

You might stop me by saying that my ontology is simply too restrictive. Maybe forests, cars, chairs, computers and cupcakes all really exist. I doubt it, but the examples of non-existent things we can in principle own can be multiplied. It is just as reasonable to talk of owning the vacuum inside a flask as it is to talk of owning the cocoa inside a cup. In both cases, labor was needed to generate the “thing” owned, and there is a reasonable moral expectation of non-interference with respect to it. (I would be destroying your property if I beamed a gas into your vacuum flask.)

What does this have to do with scepticism of strong property rights independent of positive law? First, it becomes very difficult to draw a principled line between ownables and non-ownables. Second, once we recognize that we can own things that don’t exist, such as vacua, it becomes difficult to distinguish “things” we have created and own from other kinds of outcomes of our activity. It then becomes plausible that the relevant right is one that should apply to outcomes of activity without much regard for whether that outcome is a thing that exists, a “thing” that doesn’t exist, or some other kind of outcome, such as a mountain’s being enchanted. There seems to be some kind of a right not to have the intended outcome of one’s virtuous activity destroyed without good reason. But how good the reason has to be will vary widely from case to case, so it is unlikely that this kind of a right will ground a strong view of property rights independent of positive law.

But the difficult is not the impossible. For it may be that although it would be difficult to make the needed distinctions, these distinctions could be grounded in highly detailed facts encoded in our natures.

The unity of consciousness

Here’s a familiar kind of argument:

1. A spatial arrangement of ingredients of a mental life would not yield a unity of consciousness.

2. We have a unity of consciousness.

3. Our mental life is not constituted by a spatial arrangement of ingredients.

4. So, our minds are not spatially extended entities (and in particular they are not brains).

(The last step requires some additional premises about extension and mereology.)

But our unity of consciousness also includes ingredients that take time. We are aware of motion, and motion takes time. We consciously think temporally extended thoughts. If we take the argument (1)-(4) seriously, it looks like we should similarly conclude that our souls are not temporally extended entities.

This might be a reductio ad absurdum of the line of argument (1)-(4). For it seems that even the dualist will recognize the essentially temporally extended nature of many of our conscious states.

Or maybe it’s an argument for a Kantian view on which we have a noumenal self that is beyond space and time as the physicists conceive of them.