This post is really about a question regarding confirmation theory but the main question arises in a way that may interest some folks here. I’ve been spending a few hours each evening learning some math I’ve always wanted to know (pretty exciting, huh?) and I’ve been very impressed by certain mathematical structures. For instance, projective geometry is beautiful. A fundamental principle of projective geometry is the principle of duality, which says, basically, that interchanging point and line throughout any theorem of projective geometry results in another theorem. For instance, the dual of Pascal’s theorem is Brianchon’s theorem. I found this principle of duality surprising and impressive. At least some mathematicians are likewise impressed. The author of the book I’m currently reading notes that mathematicians working in projective geometry are mainly attracted by its aesthetic qualities. It struck me that someone might view the beauty of projective geometry as some evidence for theism. I don’t think a person who thought this would be terribly mistaken but then again I find it hard to accommodate this intuition in standard confirmation theory. The beauty of projective geometry is that it has a certain structure. Presumably, though, that structure exists in every world. So it’s not the case that theism predicts this structure anymore than any other theory (assuming that universal possibilism is false). So, how should one try to accommodate this intuition—if at all? The general issue is how to account for confirmation by necessary truths.
The beauty of projective geometry is that it has a certain structure. Presumably, though, that structure exists in every world. So it’s not the case that theism predicts this structure anymore than any other theory
Hi Ted,
I'm not sure theism does not predict the structure. Suppose God exists only if God necessarly exists and suppose further that there is a priori reason to believe that perfect beings would actualize aesthetically pleasing worlds (I actually don't believe this, but just for the sake of argument). In that case we might expect aesthetically pleasing structures in every world in which God exists. Failing to find these structures in some worlds would be evidence that God exists in none. Finding them in every world would be evidence that he does exist.
Barring a convincing naturalistic explanation, couldn't the intuition also be supported by the plausibility that God would imbue his creatures with cognitive faculties designed to produce pleasure when observing the fundamental structures of creation? Of course, this approach would have the consequence that the nature of the structures themselves would be irrelevant to the argument. But it defends the intuition.
Hi Mike,
That’s interesting! The oddity, though, is that if necessities are outside God’s control the mere existence of necessities doesn’t indicate providence.
I’m tempted to think that if the existence of necessities doesn’t indicate providence then a significant group of theistic arguments is potentially bad. Take the moral arguments. Suppose moral truths are necessities then those facts don’t indicate providence. More significantly, take the aesthetic arguments for theism. (As an aside I think these arguments are some of the most significant reasons most people actually use to support theism.) Suppose, as is plausible, that beauty is grounded in symmetry properties and that symmetry properties are either necessary or consequences of simpler properties (a number of naturalists hold that latter claim). The principle of duality, for instance, is just a symmetry property on the space of projective geometry. In this case the existence of beautiful things doesn’t indicate providence. I think theists should be worried about these things but I haven’t run across any discussion of this (maybe, though, I just don’t read enough). Anyway--on the idea that one man’s modus ponens is another man’s modus tollens—this leads me to think that there very well may be something about the actual structure of necessity that indicates providence. But I’m not at all sure how to accommodate that without universal possibilism.
Joshua,
Yes, that’s the most straightforward way of defending the intuition. Theism predicts and naturalism doesn’t that we have faculties able to discover and delight in principles of projective geometry. Of course, as you note, that changes the confirming fact, but the resulting argument isn’t implausible.
The oddity, though, is that if necessities are outside God’s control the mere existence of necessities doesn’t indicate providence.
That's certainly right. I thought you had in mind that these appealing geometrical properties are instantiated in every world. It might be that, though instantiated in every world, such an instantiation nonetheless depends on the existence of a perfect being. So they are both necessary and dependent on God's existence. Compare the property of omniscience. That property is instantiated in every world iff. God exists in every world. So it might be true that necessarily something is omniscient and also true that that necessary truth confirms the existence of God.
Ted,
It seems that you need to distinguish the fact that the theorems of projective geometry are true from the fact that they are beautiful. My initial intuition is that the first is necessary but that the second isn't. If that's true, then you don't actually have a case of confirmation by necessary truths--which was the larger issue you were interested in.
It seems to be a contingent fact that we are creatures such that we find certain truths of mathematics beautiful. It's not a contingent fact that these theorems have the symmetry properties they do. But that we find those properties beautiful, however, is contingent.
The route from the contingency of our appreciation of beauty to the contingency of beauty will not be a simple one--if indeed there is such a route. The mere fact that it is contingent that we can comprehend certain necessary truths doesn't undermine the necessity of those truths. Something like this will or may be true in the case of beauty as well.
Nevertheless, I seem to have the intuition that there must be some such route. It is hard for me to believe that, even if certain theorems are true in all possible worlds, that they are beautiful in all possible worlds. There seems to be a tighter connection between being beautiful and being taken to be beautiful than there is between being true and being taken to be true.
James
Ted,
The relation between beauty and appreciated beauty just struck me as similar to that between truth and knowable truth. Just as some have defended the principle that all truths are knowable, I wonder whether it might be possible to defend the principle that the beauty of all beautiful theorems is appreciable. Being appreciable is something that beautiful theorems could possess in all possible worlds--even in those worlds where there is no one to appreciate their beauty.
Some such principle seems to be the right way to go for someone who wants to defend the possibility that beautiful theorems are beautiful in all possible worlds and yet concede that beauty has some important connection to appreciators of beauty.
Also, I wonder whether it might also be possible to construct a paradox analogous to Fitch's paradox of knowability. (Here I'm just brainstorming.) If one accepts the principle I suggested above about the appreciability of beauty, I wonder whether one is also thereby committed (in Fitch-like-paradoxical fashion) to the thesis that there is no unappreciated beauty.
Since Ted's question occurs in the context of evidence for the existence of God, this paradox may not be as paradoxical as it would be in other contexts. If there is no unappreciated beauty and if (a la Fitch's paradox) there are no unknown truths, perhaps this is reason to believe in a knower and appreciator who knows all truth and appreciates all beauty. Such a being cannot be merely a contingent being.
Caveat: I don't know the literature on Fitch's paradox. So, I may be making some blunders here.
James
It is hard for me to believe that, even if certain theorems are true in all possible worlds, that they are beautiful in all possible worlds.
James,
That's an interesting suggestion. But it's pretty much a platitude I think--at least, so says Michael Smith, Frank Jackson, etc.--that moral and aesthetic properties supervene on descriptive properties. If you have the same subvening properties in every world--in this case, I take it the subvening properties would be the geometrical properties of symmetry, etc.--then you will have the same aesthetic properties in every world. More generally, any two worlds that are descriptively indiscernable are aesthetically indiscernable. This is consistent with your other claim, of ocurse, that the supervening properties are not appreciated in every world.
I can't see any way to get the confirmation unless we are talking about universes that contingently instantiate the relevant geometrical properties. Of course, maybe there is some other way I just don't see. Or maybe you have some argument that aesthetic properties do not supervene in the allegedly platitudinous way.
A theme appears to be developing: to get confirmation the focus needs to be on the appreciation of beauty rather than on beauty itself. In this case theism doesn’t explain beauty but it explains why we find so much beauty in the world. The issues here are related to the debate over R in the EAAN but that’s for another day.
I like Mike’s idea that we can get confirm for theism by some necessary truths—like omniscience—but I think those examples are instances in which the confirming fact entails the hypothesis. I was curious as to whether there were cases in which the confirming fact is necessary but doesn’t entail the hypothesis. Mathematicians, at least some, recognize cases of this sort, when, for example a computer program tests a potential theorem by proving it for case after case after case, in search for a counterexample. After the program runs for say a year without turning up a counterexample, some mathematicians hold that that confirms the theorem. This is just a case of instance confirmation in which the instances are necessary truths. There are actual cases of this but my memory fails me right now. It’s discussed in some recent books of number theory, though.
BTW James, I think you can generate an analogue to the knowability paradox. There are many ways of actually generating such a paradox as T. Williamson has demonstrated over and over. But I think you just need the operator to be factive and to distribute over conjunctions (plus some simple logical rules). It seems to me that “appreciated beauty” is factive and distributes.
Brainstorming (hopefully this sort of thing is permitted in a blog on Saturday afternoons…): I wonder if there’s any interesting connection here with Plato’s theism and the appreciation of the forms. Plato thought that the forms were the true objects of beauty and there was some being that was the embodiment of nous. Perhaps there’s some connection in the idea that appreciating the forms is participating in divine activity.... Crazy idea: doing mathematics is participating in divine activity.
There may be another way to explain confirmation by necessary truths in a case like this without appealing to anything like Plato’s Pythagoreanism. The trick is to focus on the accessibility relationship for evidence. I’ll assume that evidence must be known or believed with justification. When one learns the principle of duality that principle becomes accessible to serve as evidence. One might hold in this case that necessary truths in one’s evidence can raise the probability of a hypothesis because other hypotheses don’t predict that those truths would be part of one’s evidence. The situation is like this. The proportion of theistic worlds to non-theistic worlds in which the principle of duality is evidence heavily favors theistic worlds. When one learns the principle of duality one can learn something about the class of worlds one is in, the worlds in which the principle is accessible.
I’m inclined to reject S5 for some tracts of language, then it’s contingent whether certain necessary truths obtain. However, I think some of the necessities are “conceptual” necessities and do satisfy S5 (different fragments of language are governed by different modal structures says I), and math very plausibly goes here.
Still, necessary truths can raise subjective probability. Suppose I’m a nihilist and think we live in a fundamentally irrational, chaotic universe. Then I learn about the beauty of mathematics and I think “Wow, maybe reality is rational and structured. Maybe there’s even a necessary person at the top of it all.” I think that’s quite rational. If your worry is just the ideal evidence problem, then I don’t think that’s much of a worry, just a technical quirk.
Your last comment reminds me of Garber's approach to the problem of old evidence (special case of ideal evidence). I like his approach a lot and so I like yours too. I think ykou ought to flesh out that last thought, it seems worth pursuing.
Ted, I agree with you that this is something that theists should worry about. In fact, I've mentioned this worry in a (so far unpublished) article.
(1) A theist, I think, should say that God is the source of all beauty.
(2) But mathematical propositions are true whether or not God exists, and
(3) the truths they express are beautiful.
(4) Therefore there is beauty that does not have its source in God, but this contradicts (1), therefore...
Well, obviously, the theist could just abandon (1), and some people might say that it isn't integral to theism. But I want to assert (1) as a theist: I certainly don't want to abandon it without a fight.
As for (2), there's already a large literature on this subject, to which my unpublished paper is intended as a small contribution. So for the moment, let's just pretend that I have discovered a truly marvellous demonstration of this thing, but the margin of this blog is not sufficient to contain it. In other words, we'll take a rain-check on (2).
On (3),James raises interesting points about whether beauty must be appreciable, and it's being a contingent fact that we are able to appreciate mathematical beauty. It is a remarkable fact, so it seems to me, that appreciation of beauty is so important in the quest for mathematical truth.
(I remember, incidentally, attending a seminar in which Roger Penrose spoke at length about how great mathematicians always seek theorems that are beautiful rather than provable, because they know that eventually, a proof will emerge. Richard Dawkins, who was in the audience, said that he had been disturbed the suggestion that great mathematicians seek theorems that are beautiful, but reassured by the fact that these theorems are believed, ultimately, on the basis of proof.)
Here is what I think the atheist could say:
We have evolved so as to find certain shapes to be more beautiful than others, and this helps us in such activities as finding a mate. For example, we find people whose features are symmetrical more attractive than those whose features are non-symmetrical. With our capacity for abstract thought, we have learned to apply concepts such as symmetry, first learned in the world of everyday life, to abstractions such as mathematical truths. One could say that we then make an unnecessary fuss about those mathematical truths that happen to have features we find pleasing. But, if Penrose is correct, the connections are deeper: if we search for a mate with symmetrical features, we are likely to find a mate who will produce healthy off-spring. If we search for a theorem that is aesthetically pleasing, we are likely to find a theorem that will produce further theorems, opening up new realms of mathematical research. Why should this be?
The answer could be that factors such as symmetry are important in defining what can and cannot be the case mathematically, and we live in a world of structures that conform to mathematical laws, because there is no other way for them to be, so, if symmetry is important mathematically, it is no surprise that it should also be important physically, and given its physical importance to us, it is no surprise that we should have evolved to respond positively to its presence. Of course, what I'm doing here on behalf of the atheist is not so much advancing an argument, as proposing a research program which, if successful, might result in an argument.
The mathematically significant is bound to be physically significant, because physical reality is constrained by mathematics.
We, as physical beings, respond to the physically significant.
So, when we study mathematics, although our sense of what is significant was formed by responding to physical structures, it still guides us.
Saying 'that is beauty' is our way of responding to perceived significance.
Thanks for the helpful discussion. The upshot appears to be that to get confirmation in a case like this needs to find some contingency (or absence of conceptual necessity), whether it’s that there are persons that appreciate the theorems of mathematics, whether it’s that our evidence includes necessary truths, or whether the theorems themselves are subject to divine control. This claim, though, conflicts with the practice of some mathematicians that there are cases of incremental confirmation for necessary truths from necessary truths. The deeper concern, though (as I now realize), is about the evidential import of the structure of necessity. If moral and aesthetic properties supervene of descriptive properties which are necessary then on the standard approach such properties are evidentially irrelevant.
Ben, can you send me that paper you mentioned?