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Reading Lucretius made me realize how long the science vs religion debate has been going on. I was introduced to Lucretius through reading George Santayana, the American Philosopher of aesthetics in particular of literature and poetry. I discovered Santayana at about the same time I discovered E.T. Jaynes which is an weird coincidence since they both seem to base their doctrine on untangling the confusion of the mind projection fallacy. They both argue at length that humans attribute too much of what goes in their head to the real world. Santayana used it to argue that religion is poetry and it is an error to believe it speaks of the real universe when it is only meant to metaphorically and poetically represent our internal thoughts about the world.
I find that even if Santayana was no mathematician, his ideas fit very well with Bayesianity. Here are some select quotes from The Life of Reason:
"Science and common sense are themselves in their way poets of no mean order, since they take the material of experience and make out of it a clear, symmetrical, and beautiful world; the very propriety of this art, however, has made it common. Its figures have become mere rhetoric and its metaphors prose. Yet, even as it is, a scientific and mathematical vision has a higher beauty than the irrational poetry of sensation and impulse, which merely tickles the brain, like liquor, and plays upon our random, imaginative lusts. The imagination of a great poet, on the contrary, is as orderly as that of an astronomer, and as large; he has the naturalist's patience, the naturalist's love of detail and eye trained to see fine gradations and essential lines; he knows no hurry; he has no pose, no sense of originality; he finds his effects in his subject, and his subject in his inevitable world."
"Thought, we are told rightly enough, cannot be accounted for by enumerating its conditions. A number of detached sensations, being each its own little word, cannot add themselves together nor conjoin themselves in the void. Again, experiences having an alleged common cause would not have, merely for that reason, a common object. Nor would a series of successive perceptions, no matter how quick, logically involve a sense of time nor a notion of succession. Yet, in point of fact, when such a succession occurs and a living brain is there to acquire some structural modification by virtue of its own passing states, a memory of that succession and its terms may often supervene. It is quite true also that the simultaneous presence or association of images belonging to different senses does not carry with it by intrinsic necessity any fusion of such images nor any notion of an object having them for its qualities. Yet, in point of fact, such a group of sensations does often merge into a complex image; instead of the elements originally perceptible in isolation, there arises a familiar term, a sort of personal presence."
"When this diversity between the truest theory and the simplest fact, between potential generalities and actual particulars, has been thoroughly appreciated, it becomes clear that much of what is valued in science and religion is not lodged in the miscellany underlying these creations of reason, but is lodged rather in the rational activity itself, and in the intrinsic beauty of all symbols bred in a genial mind. Of course, if these symbols had no real point of reference, if they were symbols of nothing, they could have no great claim to consideration and no rational character; at most they would be agreeable sensations. They are, however, at their best good symbols for a diffused order and a tendency in events; they render that reality with a difference, reducing it to a formula or a myth, in which its tortuous length and trivial detail can be surveyed to advantage without undue waste or fatigue. Symbols may thus become eloquent, vivid, important, being endowed with both poetic grandeur and practical truth."
"Science, which thinks to make belief in miracles impossible, is itself belief in miracles–in the miracles best authenticated by history and by daily life"
Lucretius' On the Nature of Things (http://en.wikipedia.org/wiki/On_the_Nature_of_Things) is considered one of the most beautiful epic poem ever written and the subject can be summed up as the rejection of religion in favor of the physical sciences. Written before Christianity even existed, Lucretius describes atoms, the movement of mass, the infinite nature of the universe, and the materialistic nature of the soul. Beautiful indeed.
"It was previously pointed out to me that I might be losing some of my readers with the long essays"
I for one find the long mathematical bayesian proselytizing some of your most fascinating posts. I can't wait for the next ones.
art is a piece of temporary entropic order, the kind that appears extraordinarily from time to time in a universe subject to the second law of thermodynamics.
What's interesting about "Thingspace" (I sometimes call it "orderspace") is that it flattens out all the different combinations of properties into a mutually exclusive space of points. An observable "thing" in the universe can't be classified in two different points in Thingspace. Yes you can have a range in Thingspace representing your uncertainty about the classification (If you're a mere mortal you always have this error bar) but the piece-of-universe-order you are trying to classify is in ideal terms only one point in the space.
IMO this could explain the way we deal with causality. Why do we say effects have only one cause? Where does the Principle of Sufficient Reason come from? The universe is not actually quantized in pieces that have isolated effects on each other. However, causes and effects are "things", they are points in Thingspace and as "things" they actually represent aggregates, bunches of variable values that when recognized as a whole have, by definition, unique cause-effect relationships with other "things". I see causality as arrows from one area of thing space to another. Some have tried to account for causality with complex Bayesian networks based on graph theory that are hard to compute. But I think applying causality to labeled clusters in Thingspace instead of trying to apply it to entangled real values seems simpler and more accurate. And you can do it at different levels of granularity to account for uncertainty. The space is then most useful classified hierarchically into an ontology. Uncertainty about classification is then represented by using bigger, vaguer, all encompassing clusters or "categories" in the Thingspace and high level of certainty is represented by a specified small area.
I once tried (and pretty much failed) to create a novel machine learning algorithm based on a causality model between hierarchical EM clusters. I'm not sure why it failed. It was simple and beautiful but I had to use greedy approaches to reduce complexity which might have broken my EM-algorithm. Well at least it (just barely) got me a masters degree. I still believe in my approach and I hope someone will figure it out some day. I've been reading and questioning the assumptions underlying all of this lately and specially pondering the link between the physical universe and probability theory and I got stuck at the problem of the arrow of time which seems to be the unifying principle but which also seems not that well understood. A well... maybe in another life.
Well, for example, the fact that two different real represent the same point. 2.00... 1.99... , the fact that they are not computable in a finite amount of time. pi and e are quite representable within a computable number system otherwise we couldn't reliably use pi and e on computers!
Benquo, I see two possible reasons:
1) '2' leads to confusion as to whether we are representing a real or a natural number. That is, whether we are counting discrete items or we are representing a value on a continuum. If we are counting items then '2' is correct.
2) If it is clear that we are representing numbers on a continuum, I could see the number of significant digits used as an indication of the amount of uncertainty in the value. For any real problem there is always uncertainty caused by A) the measuring instrument and B) the representation system itself such as the computable numbers which are limited by a finite amount of digits (although we get to choose the uncertainty here as we choose the number of digits). This is one of the reason the infinite limits don't seem useful to me. They don't correspond to reality. The implicit limits seems to lead to sloppiness in dealing with uncertainty in number representation.
For example I find ambiguity in writing 1/3 = 0.333... However, 1.000/3.000 = 0.333 or even 1.000.../3.000...=0.333... make more sense to me as it is clear where there is uncertainty or where we are taking infinite limits.
James, I share your feelings of uneasiness about infinite digits, as you said, the problem is not that these numbers will not represent the same points at the limit but that they shouldn't be taken to the limit so readily as this doesn't seem to add anything to mathematics but confusion.
Thanks g for the tip about computable numbers, that's pretty much what I had in mind. I didn't quite get from the wikipedia article if these numbers could or could not replace the reals for all of useful mathematics but it's interesting indeed.
I agree that infinity is an abstraction. What I'm trying to say is that this concept is often abused when it is taken as implicit in real numbers.
"We can only "count" because our physical world is a quantum world. We have units because the basic elements are units, like elementary particles. If the real world were a continuum, there would be no arithmetic."
I don't see it that way. In Euclid's book, variables are assigned to segment lengths and other geometries that tie algebra to geometric interpretations. IMO, when mathematics stray away from something that can be interpreted physically it leads to confusion and errors.
What I'd like to see is a definition of real numbers that is closer to reality and that allows us to encode our knowledge of reality more efficiently. A definition that does not allow abstract limits and infinite precision. Using the "significant digits" interpretation seems to be a step in the right direction to me as all of our measurement and knowledge is subject to some kind of error bar.
We could for example, define a set of real numbers such that we always use as many digit needed so that the quantization error from the limited number of digits is under a hundred times smaller than the error in the value we are measuring. This way, the error caused by the use of this real number system would always explain less than a 1% of the variance of our measurements based on it.
This also seem to require that we distinguish mathematics on natural numbers which represent countable whole items, and mathematics that represent continuous scales which would be best represented by the real numbers system with the limited significant digits.
Now this is just an idea, I'm just an amateur mathematician but I think it could resolve a lot of issues and paradoxes in mathematics.
Nominull, I kind of agree that they are the same at the limit of infinite digits (assuming by 2 you mean 2.000...). It just seems to me that working with numbers that are subject to this kind of limit is the wrong approach to mathematics if we want maths to be tied to something real in this universe, especially when the limit is implicit and hidden in the notation.
I think that one of the difficulties inherent in monotonous logics comes from the fact that real numbers are not very good a representing things continuous. In order to define a single point, an infinite number of digits are needed and thus an infinite amount of information. Often mathematicians ignore this. To them, using the symbol 2 to represent a continuous quantity is the same as the symbol 2.000... which seem to make for all kinds of weird paradoxes caused by the use of, often implied, infinite digits. For example, logicians seem to be unable to make a distinction between 1.999... and 2 (where they take two as meaning 2.000...) thus two different definable real numbers represent the same point.
When using real numbers that represent continuous value, I often wonder if we shouldn't always be using the number of digits to represent some kind of uncertainty. Using significant digits, is one of the first thing students learn in university, they are crucial for experiments of the real world, they allow us to quantify the uncertainty in the digits we write down. Yet mathematicians and logicians seem to ignore them in favor of paradoxical infinities. I wonder if by using uncertainty in this way, we might not do away with Godel's theorem and define arithmetics within a certain amount of relative uncertainty inherent to our measuring instruments and reasoning machinery.
Oh and Stephan, why not have instead something like the Church or Reality an open source reason based religion, or even an atheistic compassion based religion like buddhism? Instead often violent divide and conquer based religions such as the abrahamic religions you mentioned. These religions are very immoral if you ask me.
Stefan Pernar, you are right, christianity is fitter than atheism in an evolutionary kind of way. It's members reproduce, spread, divide and conquer like cancer. That's why they exist. But is that such a good thing? Utility wize cancer's strategy is widely unoptimal imo.
I believe that ideally speaking the best choice is the torture, but pragmatically, I think the dust speck answer can make more sense. Of course it is more intuitive morally, but I would go as far as saying that the utility can be higher for the dust specks situation (and thus our intuition is right). How? the problem is in this sentence: "If neither event is going to happen to you personally," the truth is that in the real world, we can't rely on this statement. Even if it is promised to us or made into a law, this type of statements often won't hold up very long. Precedents have to be taken into account when we make a decision based on utility. If we let someone be tortured now, we are building a precedent, a tradition of letting people being tortured. This has a very low utility for people living in the affected society. This is well summarized in the saying "What goes around comes around".
If you take the strict idealistic situation described, the torture is the best choice. But if you instead deem the situation to be completely unrealistic and you pick a similar one by simply not giving a 100% reliability on the sentence: "If neither event is going to happen to you personally," the best choice can become the dust specks, depending on how much you believe the risk of a tradition of torture will be established. (and IMO traditions of torture and violence is the kind of thing that spreads easily as it stimulates resentment and hatred in the groups that are more affected.) The torture situation has much risk of getting worst but not the dust speck situation.
The scenario might have been different if torture was replaced by a kind of suffering that is not induced by humans. Say... an incredibly painful and long (but not contagious) illness.
Is it better to have the dust specks everywhere all the time or to have the existence of this illness once in history?
But Eliezer, Wikipedia says about the Copenhagen interpretation:
Aage Petersen paraphrasing Niels Bohr: "There is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature."here is no quantum world. There is only an abstract physical description. It is wrong to think that the task of physics is to find out how nature is. Physics concerns what we can say about nature."
Doesn't this imply that Bohr didn't believe in inherent randomness but in randomness in the "description"? This seems like the same position as Jaynes and Einstein to me. Is Wikipedia wrong here? What am I missing???
I see, that's is not how I had understood it. I guess I should just leave this stuff to physicists.
I dunno Nick, your link implies the 'multiple universes' interpretation of quantum theory, and like Jaynes and Einstein, I tend to disagree with this interpretation. But yeah, I'm sure there exists some kind of physical explanation that when written down is more similar to a scientific article than a religious text. We just don't know it yet.
I call myself an atheist. However, I actually think believing in a vague god is based on probabilisticly rational and bayesian kind of thinking, at least for the limited context humans live in.
I say 'vague god' because I believe most people who believe there is a god and have somewhat solid arguments supporting this fact often use fallaciously the wrong level of conceptual abstraction to support their own specific god. The word god is not very well defined and there is quite a large margin around the definition to play with. I find the best arguments, like the prime-mover or entropy argument, are bayesian in a certain context but even where they make sense, they prove nothing but a very vague god. Theists have a very annoying tendency to use these arguments, which in reality, only support the fact that there is 'something' that somewhat fits the definition of "god" (in that it is a creator) that is complex enough to have 'created' the universe (assuming the concept of 'creation' makes sense outside the universe), or at least something which created the thermodynamic order found in the universe. There is never any good evidence for the specific gods, only for some vague god that is probably more similar to a physical phenomenon like the big bang than to the gods of religious literature.
Now why do I think the vague gods are, in some sense, rational ? It came to me while I was thinking about bayesian probabilities, while reading Jaynes book. In most problems, propability is conditioned on some variable I, representing general contextual knowledge. The equations often take the form of P(H|O,I) which represents the probability of an hypothesis H knowing some observations O and other more general facts 'I'. Jaynes never said much about 'I' except that it is whatever else we know about the problem. I like to think of 'I' as a sort of low enthropy bounded context. I sometimes call it the 'contextual urn' because probability texts often idealise this information into an urn. The contextual urn need not have a hard boundary like a real urn, its bounds can be empty space as distance itself or even time can isolate things in the universe. (As an aside, I think studying how we recognise these contexts and their bounds could explain a lot about how we reason and how to make predictions about the universe. It is a hole in probability theory which needs to be understood before we can build Jaynes rational robot) 'I' is some recognisable context that allows us to make predictions. The fact that it is recognisable means it has properties that we have seen before. The contextual urn defines a sitation, a spacio-temporal region, that is low entropy enough to be recognisable and that repeats itself often enough that we can learn things about it.
The next thing I noticed about the relationship between 'I', 'O' and H is that we can kind of view 'I' and O as a cause of H and effects seem never to be more complex than their causes. This is particularly true about creation as far as we can take a creator and his creation to be a cause and effect (Which philosophers like Hume accepted). Taking an information theoretic perspective, if something can create someting else, it contains all the information to create it and probably more. It is at least as complex entropically as the thing it creates. Humans have always lived in a world where this was true almost all the time and hence it is perfectly reasonable for them to deduce using bayesian reasoning that's how things pretty much always work. It is not hard to see then that living organisma, humans or even the universe in general contain a great amount of complexity and there has to be something even more complex which created them. e.g. god.
If we look further than our immediate existence, we find out that it is not always true that a cause is more complex than its effect. Because of random variations, an effect is not very probable to be more complex than its cause but it CAN happen sometimes. And as a result of natural selection, it is possible for the complexity of populations of effects to increase given a bias which makes the more complex survive more than the less.
Evolution is not something that happens in the time-scale of a human life therefore it is not very useful to us. We thus have evolved and rationally learned during our lifetime that effects are probably always less complex than their causes. And in the context of a relatively short life span this is right!
We have to look at a wider timespan to see that there is actually another way for complexity to arise and that it explains the complexity we observe much better than the gods of religions. This is of course the theory of evolution.
I think this explains why theists feel so threatened by evolution. It's because it is the only good alternative for the creation of complexity. And although most people don't understand the principles of entropy and thermodynamics, most people's innate Bayesian reasoning leads them to the right conclusions: When they see the alternative explaining the creation of complexity and when they see how well the theory of evolution fits historical evidence, their last argument for the belief in god vaporises.