Proattitudes, freedom and determinism

On a familiar compatibilistic picture of what things might be like, all our free actions are determined by our proattitudes and beliefs. The proattitudes provide the drive and ends for the action and the beliefs tell us about what does and does not conduce to those ends. On the most traditional version of the story, the proattitudes are noncognitive. I think Warren Quinn's arguments against such a view of proattitudes are sound: noncognitive proattitudes just do not render action rational. I would say they are too much like mere dispositions to act, and dispositions to act, every bit as much as the actions that flow from the dispositions are in need of being made rational. Thus, the proattitudes must have a cognitive component: something like a seeing of an end as good or a judgment of an end as good.

Now consider this dilemma. We either do or do not always act in accordance with the rationally superior attitude. I.e., we either do or do not ever act in accordance with what the attitude presents to us as the rationally called for or the better course of action. If we always do, then we are never blameworthy. For while the judgments embodied in our proattitudes may be wrong, we are not blameworthy for these wrong judgments if we came to them always acting by our better lights.

Blameworthiness requires that at some point we have been responsible for acting against our better lights.

Now, proattitudes are either entirely cognitive or have a cognitive aspect and a conative drive/motativation aspect. If they are entirely cognitive, then when we act against our better lights, then something other then proattitude must be determining our action in cases where we go against the better judgment embodied in these entirely cognitive proattitudes. But on the compatibilist picture, it is being sourced in our proattitudes that makes an action be truly ours. And in the relevant respect, the respect that determines us on the wrong (by our lights) rather than right course of action, the action is not sourced in our proattitudes.[note 1] That makes it very hard to see how we can be responsible.

Next, suppose that the proattitudes have both a cognitive and a conative component. On this picture, the cognitive component is what makes actions rational and the conative is what causally explains the action. On this view, when we act against our better lights, it is because proattitudes with a rationally weaker cognitive component can nonetheless have a causally stronger conative component. But how can we be responsible if that's the ultimate explanation of our wrongdoing? For it is the cognitive component that makes for rational action, for action that is distinctively personal, the sort of thing that is subject to moral evaluation. Imagine taking a brute animal and adding a cognitive component to its noncognitive proattitudes, but keeping the root of the deterministic causal explanation of action on the noncognitive side. That would not make the brute responsible. It would just create a monster.

When one is determined to act in accordance with the rationally weaker but conatively stronger proattitude, one is in the grip of a disorder, a kind of disease of the will (we call it "akrasia" or "weakness of the will"), which causes one to choose the rationally weaker rather than the rationally stronger course of action. But one is not blameworthy for such diseased action unless one is blameworthy for the disease. However, since the story applies all the way back, there is no room for blame left.

This line of thought does not refute the compatibility between responsibility and determinism. For it says nothing against the compatibility between praiseworthiness and determinism. But I think it gives one reason to think that determinism rules out blameworthiness.

Necessary and sufficient conditions

Both philosophers and mathematicians attempt to give nontrivial necessary and sufficient conditions for various properties. But philosophers almost always fail—the Gettier-inspired literature on knowledge is a paradigm case. On the other hand, mathematicians often succeed by the simple strategy of listing one or two necessary conditions and lucking out by finding the conditions are sufficient. And they do this, despite the fact that showing that the conditions are sufficient is often highly nontrivial.

Why do mathematicians luck out so often, while philosophers almost never do? Think how surprising it would be if you wrote down two obvious necessary conditions for an action to be morally wrong, and they turn out to be sufficient. And can philosophers learn from the mathematicians to do better?

1. Subsidiary conditions: Mathematicians sometimes "cheat" by only getting an equivalence given some additional assumption. A polygon has angles add up to 180 degrees if and only if it's a triangle, in a Euclidean setting. And such limited equivalences can still be interesting. While some philosophers accept such limited accounts, I know I often turn up my nose at them. I don't just want an account of knowledge or virtue that works for humans: I want one that works for all possible agents. Perhaps we philosophers should learn to humbly accept such incremental progress.

2. Different tasks: Philosophers often don't just ask for necessary and sufficient conditions. We want conditions that are prior, more fundamental, more explanatory. It may be true that a necessary and sufficient condition for an action to be wrong is that it is disapproved of by God, but that doesn't explain what makes the action wrong (assuming that the Divine Command theory is false). Moreover, sometimes we even want our necessary and sufficient conditions to work in impossible scenarios: we admit that God has to disapprove of cruelty, but we argue that if per impossibile he didn't disapprove of it, it would still be wrong (I criticize an argument like that here). This would be an absurd requirement in mathematics. "Granted, being a Euclidean polygon whose angles add up to 180 degrees is a necessary and sufficient for being a Euclidean triangle, but what if the Euclidean plane figure were a triangular circle?" The mathematician isn't looking to explain what a triangle is, but just to give necessary and sufficient conditions.

It is no surprise that if philosophers require more of their conditions, these conditions are harder to find. Again, I think we philosophers should be willing to accept as useful intellectual progress cases where we have necessary and sufficient conditions even when these do not satisfy the stronger conditions we may wish to impose on them, though I also think these stronger conditions are important.

3. Ordinary language is rich and poor: There are very few perfect synonyms within an ordinary language. There are subtle variations between the properties being picked out. Terms vary slightly in their meaning over time. But now necessary and sufficient conditions are very sensitive to this. Suppose that it were in fact true that x knows p if and only if x has a justified true belief that p. But now reflect on how many concepts there are in the vicinity of justification and in the vicinity of belief. Most of these concepts we have no vocabulary for. Some of these concepts were indicated by the words "justification" and "belief" in other centuries, or are indicated by near-synonyms in other other languages. If the English word "belief" were slightly shifted in meaning, we would most likely have no way of expressing the concept we now express with that word, and we would be unlikely to be able to give an account of knowledge. It can take great linguistic luck for us to have necessary and sufficient conditions statable in our natural language. Only a small minority of possible concepts can be described in English. (There are uncountably many possible concepts, but only countably many phrases in English.) What amazing luck if a concept can be described twice in different words!

I may be overstating the difficulty here. For sometimes the meanings of terms are correlated, in the way that vaguenesses can be correlated. Thus, "know" and "belief" may be vague, but the vaguenesses may neatly covary. And likewise, perhaps, "know" and "belief" can shift in meaning, but their shifts might be correlated.

Final remarks: The point here isn't that giving explanatory necessary and sufficient conditions won't happen, but just that it is not something we should expect to be able to do. And I should be more willing to accept as intellectual progress when we can do partial things:

1. give conditions that are necessary and sufficient but not explanatory
2. give conditions that are necessary and sufficient in some limited setting
3. give necessary but not sufficient conditions, or vice versa.

Uncountable continuum?

Suppose space and time are non-discrete. Do we have good reason to think that they form an uncountable continuum of the real-number sort? One might first speculate: Perhaps points in space have coordinates that are triples of rational numbers (in some coordinate system)? That would, however, make it impossible to rotate an object by 45 degrees: the coordinates after such a rotation would no longer be rational numbers. And that's implausible. But there are bigger countable sets than that of rational numbers that one might invoke that would get out of problems like that. So why suppose our space and time have the structure of the real numbers?

Impossible worlds

Suppose w is an impossible world. Then impossible things may be possible at w. For instance, w might be a world where square circles are possible. But an impossible world need not be such that impossible things are possible at it. After all, an impossible world w might have the same modal truths as our world does and violations of them. Thus, there will be two impossible worlds: One where there are square circles and square circles are possible, and one where there are square circles despite their impossibility. Moreover, there will be an impossible world that is just like ours except for some or all modal truths. Imagine a world just like ours except that every proposition is possible and another just like ours except that no proposition is possible.

When I say these things, I seem to be near the boundary of coherence—and maybe on the wrong side of it. But one can give precise descriptions of such worlds by saying precisely which propositions are true at them. For instance, consider a world w₁ such that a proposition p is true at w₁ if and only if p is actually true or p is a proposition expressing the possibility of a proposition q (for any q), and all other propositions are false.

Natural mathematical structures

This post is inspired by Heath White's comment here.

There are lots and lots of different kinds of mathematical structures. Here's an operation on the real numbers: a#b = a^3b+7. You can study this operation heavily, but chances are that you won't get anything very interesting (but maybe you will!).

But on the other hand, take something like addition or multiplication (or both). These have many beautiful properties, and lend themselves to many kinds of abstraction: groups, fields, rings, monoids, etc. When this happens, it is evidence that the structure one was studying is somehow natural. While in some way any coherent set of coherent axioms might be fruitfully studied, there both seem to be particularly natural axioms for a structure--like, commutativity--and particularly natural clusters of axioms--like those defining a group or a ring--that seem worthy of study. Anyway, around a particularly natural structure there springs up a wealth of mathematics.

Some of the most creative mathematics seems to be the identification and introduction of natural structures. For instance, one of the things I learned in my recent work in formal epistemology is that classical probability is a very natural structure. On the other hand, hyperreal-valued probabilities of the sort that some philosophers like seem to be quite an unnatural structure--one doesn't get the same wealth of neat results. The more one plays with hyperreal probabilities, the more they look like Frankenstein's monster. (On the other hand, the R(I) monoid I discuss in a recent post is rather more natural, though it may not seem that way initially.)

What is this naturalness of structure? David Lewis took natural properties to be more basic, and unnatural ones to be constructions from the more basic ones. That is not the case for natural mathematical properties. If we consider mathematics set-theoretically, all the properties--both the natural and the unnatural ones--we are studying are constructions out of set-theoretic properties. A natural cluster of axioms might be no simpler than an unnatural cluster of axioms. Moreover, the naturalness seems independent of the foundational grounding. Suppose one day we have a better foundation for mathematics than set theory. (Not unlikely!) Group theory and probability theory will still be studying something natural.

What, then, makes a mathematical structure natural? Is it purely extrinsic, with the natural properties and clusters of axioms being those that are mathematically fruitful? Or maybe there is no distinction: Maybe if the amount of effort that has gone into analyzing addition were put into analyzing the # operation I gave at the beginning of this post, we would find just as beautiful mathematics? Maybe such deflationary stories are the whole story about mathematical naturalness. But maybe there something deeper about the natural properties and clusters of axioms. Aquinas thinks all creation in some way reflects God. Perhaps the more natural properties--whether empirical or mathematical--are those that somehow more deeply reflect God's mind?

This post comes after spending over a week on some mathematical issues only to find today that I committed a subtle (perhaps only to me!) error at the beginning of the investigation, and almost all of the work has come to naught. This reminds me of the famous joke about dean talking to the physicist: "You always want money for more equipment! Why can't you be like the mathematicians? All they need are paper, pencils and garbage cans. Or better yet, why can't you be like the philosophers? They don't even need the garbage cans."

Mathematical beauty

I keep on going back and forth on the question whether the beauty of mathematics is something surprising and metaphysically significant. I find myself going between two views.

Deflation: Mathematical beauty is just a matter of selection. There are many beautiful theorems. But there are many, many more ugly theorems. It's just that the ugly theorems don't get published, unless they are of practical importance or are appropriately connected with beautiful mathematics. Imagine that we got a book of all the theorems of arithmetic. There would be many beautiful things in the book. But intuitively a large part of the book (if that makes sense to say: it's an infinite book after all!) will just be boring theorems like "18883 x 77891 = 1470815753" or "The equation x²+9873773873+8383883=0 has no solutions."

Theology: Mathematics seems to be have more in the way of surprising beauty than we would expect from the selection hypothesis. It happens not infrequently that as a working mathematician one writes down some obvious necessary conditions for something to happen, and then one proves—often in a highly nontrivial fashion—that these necessary conditions are also sufficient. Or maybe there is just a little bit to add, and then they become sufficient. Of course, often no such thing happens—we're just stuck with necessary conditions. But the number of times that the necessary conditions are also sufficient is surprisingly large, large enough to call out for an explanation.

And that need for explanation pulls me in one of two theological directions. First, there is Augustine's idea that mathematical objects are in the mind of God, and so we would expect to find beauty in them, since God is supremely beautiful. Second, one might have the thought that we are divinely designed, among many other things, for the kind of reasoning found in mathematics. Of course, one might also offer a naturalistic evolutionary explanation. But I am not sure that will be satisfactory: finding utterly exceptionless necessary and sufficient conditions is just not something that happens much in the practical life that our evolutionary development is driven by.

Something positive about Bayesian regularity

The brunt of a lot of my recent posts has been that there is no hope for Bayesian regularity if one requires natural invariance conditions. But here is a positive result. For this result, we will need the values of the probabilities to be taken in a very special space which is a variant of a space defined by Dos Santos. We now define this space. Let I be a totally ordered set under ≤. Let R(I) be the set of monotone non-increasing functions f from A to [0,∞] with the property that either f(x)=0 for all x or there is a unique (!) member i of I such that 0<f(i)<∞. Note that R(I) is itself a totally ordered set under pointwise comparison and it has a natural pointwise addition operation that respects the ordering. You can think of R(I) as very much like a set of non-negative hyperreals where numbers whose ratio is infinitesimally close to 1 are identified.

It is fairly easy to see that it follows from Proposition 1.7 of Armstrong that if G is a supramenable group and X is any space acted on by G, then there is an I and a finitely additive measure P on all subsets of X with values in A that is strictly positive in the sense that P(B)=0 if and only if B is the empty set. Moreover, we can normalize P into something like a probability by supposing that I has a final element, call it 1, and P(X) is the member f of R(I) such that f(1)=1.

In particular, there will be a strictly positive R(I)-valued finitely additive measure on the circle and the line, invariant under isometries. But not in dimensions greater than one due to Banach-Tarski related stuff.

Fact: There is a natural correspondence between real-valued Popper functions on X that make every non-empty subset normal and strictly-positive finitely-additive R(I)-valued measures. It's easy to see how this correspondence goes in one direction. Suppose we have such a strictly positive measure P. We want to define P(A|B) for some non-empty B. Choose the unique i in I such that P(B)(i) is in (0,∞) and then define P(A|B)=P(A∩B)(i)/P(B)(i). Moreover, the Popper function will be strongly-invariant (P(gA|B)=P(A|B) if gA and A are subsets of B)) if and only if the corresponding R(I)-valued measure is invariant.

For epistemological purposes, this is a move in the happy direction, but the fact that nothing like this can work in Euclidean settings in higher dimensions is a problem.

Note that P as above will be regular in the weak sense that 0<P(A) if A is non-empty but typically not in the strong sense that if A is a proper subset of B, then P(A)<P(B).

Art as discovery and mathematics as art

There is a very large but probably finite number of possible images that the human eye can distinguish. Among these possible images, it seems that a relatively small subset is very beautiful (or has some other aesthetic quality to a high degree—I'll just stick to beauty for now). One way to see that visual artist is as a discoverer and communicator of beautiful images: in that very large finite space of possible images, she discovers a beautiful one, and then realizes it. The realization makes it possible for her to communicate her discovery to others. (Of course, the tools of discovery will often not be entirely mental—paintbrushes, texture of canvas, and the like all are tools of discovery, like a scientist's instruments or a mathematician's calculator or scrap paper.) Likewise, the musician searches the very large but probably finite number of possible sequences of sounds that the human ear can distinguish for that small minority that are very beautiful, and realizing the possible sequence communicates her discovery to others.

This model of the artist as discoverer and communicator makes the artist not that different from the pure mathematician, who also searches a large space of abstracta—say, the space of proofs or the space of theorems—for the few that exhibit some property, often an aesthetic one such as beauty (mathematicians also talk of "interest", but when the mathematics is pure, that "interest" is a kind of aesthetic quality, and for simplicity I'll stick to beauty) and then communicates these to others.

How exactly the analogy between the artist and the mathematician works out depends on whether Platonism about propositions (and similar objects) is true. The musician and painter in producing sounds and paintings do not merely represent the beauty of the possible sound or image: they make the possible sound or image actual. If such Platonism is true, then the mathematician does not realize possibilia in presenting a proof or a theorem, but only represents them. In this way, the mathematician is more like a composer or a novelist whose product is also a representation of a thing of beauty, rather than the thing of beauty itself. (Of course, the inscription of a theorem or a musical composition can be beautiful—the the quality of the calligraphy, say, but this is not mathematical or musical artistry per se.) On the other hand, if Platonism is false, then we might think of the very token inscriptions of a theorem or a proof as realizations of the possibilia that the mathematician has discovered: the mathematician searches the space of possible theorem inscriptions and finds beautiful ones.

Of course the discovery model of the artist's work isn't the only model of the artist's work. I think a creation model is more common. This model lays an emphasis on producing a thing of beauty (or other aesthetic qualities, of course). But I think that the discovery model works particularly well for a composer, who can be a great composer upon composing a beautiful work even if no one performs it.

The creation model makes the artist more like God. Is that a merit or demerit of the model?

But remember I am no philosopher of art.

Merging Lewisian worlds

According to Lewis, any pair (or, more generally, plurality) of concrete (he doesn't even restrict it this way) of objects has a mereological sum. Now, suppose that x and y are concrete objects in worlds w₁ and w₂ respectively. Let z be the mereological sum of x and y. According to Lewis, worlds are maximal spatiotemporally connected sums of objects. Now, here are some plausible principles:

1. Spatiotemporal connection is transitive and symmetric.
2. If a is spatiotemporally connected to a part of b, then a is spatiotemporally connected to b.

Consider any concrete objects a and b in w₁ and w₂, respectively. Then a is connected with x, since all objects in a world are connected. And y is connected with b. Moreover, by 2, a is connected with z since x is a part of z. And by 2, b is connected with z. Thus, by 1, a is connected with b. Thus, all objects in w₁ and w₂ are mutually connected, and so by Lewis's account of worlds, there is only one world. Which is absurd.

Explaining the simplicity of theories

The following is a basic presupposition of science:

1. If two scientific theories equally well fit our observations, and one of them is by far simpler than the other, then the simpler theory is more likely to be true.

Granted, we don't have a good account of "far simpler" or even of "equally well fit", but nonetheless something like (1) is surely true. And that is an amazing fact about the world. What explains that fact?

Note that we cannot really explain (1) simply by citing the fundamental physical laws of nature. For (1) is true in reality as discovered across the disciplines, not just in fundamental physics. It is surely true of biological, geological, astronomical and sociological theories.

Fact (1) suggests that the laws and other structure of our world are generated in a way that tends towards simplicity given the same empirical outcomes. Why? Well, I see three stories.

Theism: Simplicity is good, either intrinsically for aesthetic reasons or instrumentally because it helps agents like us get the good of empirical knowledge, and so a perfect being will prefer simpler structures when they can produce the same empirical outcomes.

Axiarchism: Simplicity is good, as above, and there is a fundamental law of metaphysics that all must be for the best.

Logocentrism: The world is generated by something like a random process that randomly generates a complete coherent descriptive sentence, in a non-gerrymandered language, with longer sentences having lower probability.

I find Logocentrism incredible: Why should the length of a linguistic expression matter except where there is a mind?

Faculty opening at Baylor Philosophy

BAYLOR UNIVERSITY, Waco, TX announces a tenure-track Assistant Professor position in the Department of Philosophy beginning in the fall of 2014. AOS and AOC:  Open. Salary is competitive. Teaching load and scholarly expectations are consistent with those of a research university. Review of applications will begin immediately and will continue until the position is filled. To ensure full consideration, the completed application should be received by November 1, 2013.

Baylor, the world's largest Baptist University, holds a Carnegie classification as a "high-research" institution.  Baylor's mission is to educate men and women for worldwide leadership and service by integrating academic excellence and Christian commitment within a caring community.  Because Baylor aspires to become a top tier research university while reaffirming and deepening its distinctive Christian mission, Baylor is actively recruiting new faculty with a strong commitment to scholarly activity and an equally strong commitment to teaching.

The letter of application should respond to Baylor's most recent mission statement Pro Futuris (available on the web at http://www.baylor.edu/vision) and include an account of the applicant's own religious views. In addition to a letter of application, the candidate should submit a CV, a professional writing sample, three letters of recommendation, and official transcripts.  Send applications to Dr. C. Stephen Evans, Chair, Search Committee, Department of Philosophy, Baylor University, One Bear Place #97273, Waco, Texas, 76798-7273.Baylor is affiliated with the Baptist General Convention of Texas and as an AA/EEO employer; Baylor encourages minorities, women, veterans, and persons with disabilities to apply.

Progress report: Positive results for invariant Popper functions

This is a very technical note. I've spent a fair amount of time this week thinking about invariant Popper functions. Say that a group G is neatly supramenable if there is a Popper function P on G with every non-empty subset normal and satisfying the strong invariance condition P(gA|B)=P(A|B) whenever gA∪A⊆B. Neatly supramenable groups are supramenable: every non-empty subset A has an invariant finitely additive measure m such that m(A)=1. Anyway, I think I can prove—I now have two proofs drafted, so that makes me more confident—that every exponentially bounded group is neatly supramenable. Thus, every elementary supramenable group is neatly supramenable.

One philosophically interesting upshot of all this is that n-dimensional Euclidean space supports a Popper function with all non-empty subsets normal that is invariant under all translations as well as under single-coordinate reflections ((x₁,...,x_i,...,x_n) going to (x₁,...,−x_i,...,x_n)). But when one adds rotations into the mix, this is false for n≥2. So there is something philosophically problematic about rotations for the notion of uniform probability.

Don't quote the result yet as the proofs use mathematics that I am not very familiar with (ultrafilters, non-standard analysis, etc.).

Oh, and all of this uses the Axiom of Choice.

Soul-body interaction

I have a post on soul-body interaction on Biola's Center for Christian Thought blog that may be of interest to my readers here.

Trust and lies

You promise to meet me for dinner at 7. We say that the promise normally makes it appropriate for me to trust you'll show up at 7. But that's not quite right. What is more appropriate to trust is that you'll meet me for dinner at 7 or have good moral reason not to be there. This point applies even if I know that you won't have such good moral reason. For that you won't isn't s matter of trust of you, but of prediction.
By the same token, if it can ever be permissible to lie, and you assert something, I never ought to trust you that you are being truthful. Instead at most I ought to trust that you either are being truthful or have good moral reason to lie.
So if it is ever appropriate to take it on trust alone that you are being truthful, lying is always wrong.

An argument for incommensurable goods

One of the upshots of a number of my posts on the limitations of probability theory is that there are events that are probabilistically incomparable—neither can be said to be more likely than the other. (For instance, this post.) But an objective chance at a good is good, and better the greater the chance and worse the lower the chance. Chances p and q at the same good G will, then, be incommensurably good when the chances p and q are incomparable. Hence, if there are incomparable objective chances, there can be incommensurable goods. But it's plausible that there can be incomparable chances (see the post I linked to above, for instance). So there can be incommensurable goods.