Theistic frequentism and evolution

Readers here may be interested in this post of mine on Prosblogion.

Death, suffering, platypuses and echidnas

• It is very surprising (given some appropriate background) that there are platypuses.
• It is very surprising that there are echidnas.
• That there are echidnas is not very surprising given that there are platypuses.

Likewise:

• It is very surprising given theism that persons die.
• It is very surprising given theism that persons suffer.
• That persons suffer is not very surprising given theism and the fact of death.

Therefore, the problem of suffering does not go very far beyond the problem of death.

And of course let me add that it is also very surprising given naturalism that persons die. (For that persons die entails that persons live.)

Magnifying sets of real numbers

If S is a set of real numbers and a is a real number, let aS={ax:x∈S} be the set you get by magnifying S by a factor a.

Here's a funny thing. Some sets get bigger when magnified and some get smaller. For instance, if we take the interval [0,1] and magnify it by a factor of two, we get the interval [0,2], which is intuitively "twice as big". But if we take the natural numbers N and magnify them by a factor of two, 2N will be the even natural numbers, and so 2N will intuitively be "twice as small" as N.

Next observe that if R−[0,1] is the real numbers outside the interval [0,1], then 2(R−[0,1])=R−[0,2] is smaller. Magnifying a set by a factor greater than 1 magnifies both the filled in parts of the set and the holes in the set. The effect of this on the intuitive "size" of the set will depend on the interaction between the holes and the filled in parts.

And if we take the Cantor set C, then magnifying it by a factor of three makes the set be intuitively twice as large. I.e., 3C=C∪(2+C). This makes it very intuitive that the dimension of the set is log 2 / log 3 (which is indeed its Hausdorff dimension). For intuitively if we have an n-dimensional set, and we magnify it by a factor of a, its size is aⁿ. So if n is the dimension of the Cantor set, then 2=3ⁿ, and so n is log 2 / log 3.

Earthly life is very short

We have good a priori reason to think that:

1. If God exists, persons exist forever.

Persons are the sort of non-fungible beings whose cessation of existence would be a cosmic tragedy, and we would not expect that in a world created by God. (Sometimes atheists write as if the hypothesis of an afterlife was an additional posit, brought in ad hoc to counter the problem of evil.)

Our earthly life is very short, then. At most about a hundred years. Which is 0% of eternity. And by (1), we have good reason to think that if God exists, there is infinitely more later.

It is not that surprising that there is evil in a very good existence if the evil occupies only a small portion of the existence. And 0% is a small portion. Moreover, if we were to guess where the evil might be met with, the beginning of existence would be a reasonable guess. For improvement is much better than decay. And while we cannot be self-existent like God, being in some way the co-authors of our goodness is a great value. But that requires the possibility of not being good. And makes plausible the actuality thereof.

Honest manipulation

Suppose I know for sure whether p is true, and you know for sure that I know, and you completely trust me. Moreover, suppose your credence in p is neither 0 nor 1. Then I can slightly manipulate your credence in p, in either direction, with great reliability, and without any dishonesty. Suppose I want to slightly raise your credence. Then I uniformly randomly pick a number N between one and a billion, and inform you whether

1. p is true or N>1

is true. All but certainly, N won't be 1, so I will end up announcing that (1) is true. But then you will ever so slightly raise your credence in p. For if q is the proposition expressed by (1), then P(q|p)=1 and P(q|~p)=1−10⁻⁹, so learning q will slightly raise your credence in p. And if I want to slightly lower your credence, I instead inform you whether

2. p is false or N>1

is true.

The above manipulation observation weakens the manipulation argument I used in this paper, though the manipulation in the paper is much more radical.

An argument from physical disability against naturalism

1. On average, severely disabled people report not being dissatisfied with life.
2. Thus, probably, on average severely disabled people are rightly not dissatisfied with life.

The defeasible inference is due to the presumption in favor of the correctness of people's judgments about their wellbeing.

But (2) is very surprising on naturalism. Given naturalism, one would expect that human wellbeing be extremely tenuous, and that any significant downward push, such as from disability, should push people into the illbeing range. The theist, on the other hand, has a rather better explanation of (2).

Of course the naturalist can talk about evolutionary and social mechanisms that make severely disabled people report satisfaction even though their lives are on balance unsatisfactory. But that isn't a naturalistic explanaton of (2). It is an explanation of (1) coupled with a patronizing denial of (2).

The theist, like the naturalist, might question people's self-reports. But she could also accept (2). Of course, the theist will then have a serious problem of evil in the case of the large numbers of severely disabled people whose life satisfaction self-report is negative. But since the theist has very good reason to think that life continues infinitely beyond death, a dissatisfaction—even a rightful one—over a finite initial period is not perhaps quite that surprising.

Conditional analysis of ability

It's sometimes fun to beat a dead horse. Consider the classic compatibilist conditional analysis of ability:

• x can do A if and only if were x to choose to do A, x would do A.

A student in class today pointed out that then the indecisive Kirk in The Enemy Within can save the crew, since if he were to choose to do so, he would. But of course the point is that he can't (or at least wouldn't be able to if the condition progressed).

It gets worse:

• My pen can make a choice. For if it were to choose to make a choice, it would make a choice.
• A shark can entertain thoughts about moral duties. For if it were to choose to entertain thoughts about moral duties, it would entertain thoughts about moral duties.

An argument against the compatibility of freedom and determinism with lots of auxiliary assumptions

Let L be the laws of nature. Suppose:

1. Everything in the causal history of the coming into existence of an entity is essential to the entity—it is impossible that that very entity exist without it. (Essentiality of origins)
2. The complete causal history H of the coming into existence of Bob together with L entails every action of Bob's. (Determinism-plus)
3. Bob cannot exist unless L is true. (Strong nomic boundededness)

It follows from these that it is metaphysically impossible for Bob ever to do otherwise. For Bob cannot exist if H or L is not true, and H and L entail every action of Bob's. But:

4. If it is metaphysically impossible for Bob ever to do otherwise, Bob is not free.

So:

5. Bob is not free.

Essentiality of origins is very controversial, but I think there are very good theoretical reasons to posit it. It yields, for instance, a very neat account of transworld identity.

Determinism-plus isn't just determinism. For suppose we live in an infinite deterministic universe and relativity theory holds. Then the backwards light cone of every action of yours is wider than the backwards light cone of your initial coming into existence, and there may be causal influences on your actions that did not influence your coming into existence. In such a setting, determinism-plus is false. On the other hand, determinism-plus is going to be true in a more Newtonian universe, where everything affects everything else instantaneously (say, via gravity), and so the causal history of your coming into existence contains all of the universe's state prior to your coming into existence. Also, in a some small finite relativistic universes, we might have determinism-plus.

Furthermore, determinism-plus's understanding of laws of nature rules out miracles. That's a serious problem. It may render determinism-plus incompatible with theism. Still someone, especially a non-theist, might think:

6. If freedom is compatible with determinism, freedom is compatible with determinism-plus.

For it really shouldn't matter for freedom whether we live in a small finite or infinite universe or whether miracles are possible. Still, determinism-plus is a definite weakness of this argument. It's false, I think.

And I know there are compatibilists who are willing to deny (4).

This is a pretty weak argument. Still, it's worth thinking about. For the problems with the premises don't seem to be such as to be deeply relevant to whether there is free will. Maybe there is some way of building a better argument out of these options. I leave that for the reader.

So close to a good ordering of all subsets of reals...

Let X be the set of integers, or a circle, or the real line, or Euclidean n dimensional space. Imagine a point is "uniformly" randomly chosen in X. For any two subsets A and B of X, we would like to be able to say if one of the subsets is more probable as the location of the point. Here are some conditions we want to impose on the ≤ comparison:

1. ≤ is a total preorder: for any A, B and C, we have A≤B or B≤A; A≤B and B≤C implies A≤C; and A≤A.
2. For any translation t of X (where we deem rotations on the circle to count as "circular translations"), we have tA≤tB if and only if A≤B. (≤ is translation-invariant)
3. If A is a proper subset of B, then A<B (i.e., A≤B but not B≤A).
4. If m(A)<m(B) for d-dimensional Hausdorff measure (including of course Lebesgue measure), for any d between 0 and the dimension of the space (inclusive), then A<B.

Proposition 1. Given the Axiom of Choice, there is an ordering ≤ satisfying 1-4.

Proof: Start with an ordering such that A is less than B if and only if A=B, or m(A)<m(B) for any d-dimensional Hausdorff measure, or A is a proper subset of B. This ordering is translation-invariant, and it extends to a preorder satisfying 1-4 by the main theorem of Section 2 here.

That sounds great! We can finally compare probabilities of landing in arbitrary sets, it seems. Well, almost. Given a uniform distribution, we would at least want the invariance also to hold for coordinate reflections (where we reflect the kth coordinate, for any k).

Proposition 2. There is no ordering ≤ satisfying 1-3 and the coordinate reflection condition.

That's a consequence of the final proposition in the paper I linked to above.

What a surprising difference these reflections make! With just translations, we have a lovely invariant order (though presumably not unique) respecting strict inclusions of sets. When we add coordinate reflections, we don't. Technically, the difference is that once we have reflections and translations, our symmetry group is no longer commutative. And of course, in the Euclidean space case if we add rotations, all is lost, too (that, too, is easy to show).

Philosophical corollary. There can be incommensurably probable events, and hence incommensurably valuable events (since two chances at the same good will be incommensurably good if the chances are incommensurably probable).

"Quotation marks"

I don't ever remember seeing any misuse of "quotation marks".

Two thoughts on theologians who say "God does not exist"

Some theologians like to say that God does not exist. They say this to mark the radical difference between God and creatures.

1. If one is going to say such things, a more helpful way to speak would be: "God exists but we don't." For that would still get across the radical difference between God and creatures, but get right the fact that God is the one who is the more real. Compared to God's reality, we are but shadows. It is said that God said to St Catherine of Siena: "I am who I am, and you are she who is not." This poetically conveys a deep truth. We are but shadows, and "shadow" is often an overstatement.

2. There are many metaphysicians who like to say that complex artifacts like tables, chairs and blowguns don't exist. But many of them say this only in philosophical contexts and not in "ordinary" contexts, or they qualify the "don't exist" with a "really". They may or may not be misguided in the form of their odd denial, but what they (we!) are getting at is plausible: There is a deep difference between the kind of being that a table, chair or blowgun has, and the kind of being that a horse or a photon have (some of these philosophers will class the horse with the chair; that's mistaken, but the basic point I am making isn't affected). The ordinary language sentences "The pig exists" and "The car exists" have very different (nonpropositional) grounds: the former is grounded in a single thing while the latter is grounded in the arrangement of many things. Well, these theologians, like these metaphysicians, are also impressed by a deep ontological difference (a deeper one, perhaps). But like the metaphysician who is willing to speak with nonphilosophers in ordinary ways, these theologians should be willing to say "God exists" in contexts of ordinary worship. Or like the metaphysician who says that computers don't really exist, she could simply make a qualification: "God doesn't exist in the shadowy way." Or, more perspicuously?, she could say: "We don't really exist, but God does." (Though I think that if one does that, one should also distinguish us from artifacts. Perhaps the distinction could be marked with "really" and "really really"!)

An argument against Christian materialism on a pro-life view

1. No one is saved who does not have a love for God in this life.
2. If materialism is true, early human embryos do not have a love for God.
3. At least some, perhaps all, people who die as early embryos are saved.
4. So, materialism is false.

One might think that in premise (2), the antecedent isn't doing any work: that it is simply true that early embryos do not have a love for God. Not so. For if love for God is a matter of a certain orientation of the soul towards God, then people who do not have a brain might nonetheless have a love for God.

One might even try to run the argument with young infants instead of embryos. But there, I think, the argument could run into difficulty. For it may be that a young infant's brain hardware is sufficiently developed to love God, but simply does not have the software for it. And God could miraculously give the infant the software. I suppose the Christian materialist could think that God could miraculously join the embryo with a brain, perhaps a brain in another dimension. But it is not clear that that both such an embryo would then be one of us humans and that brain would be its brain.

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.