Confirmation of Scientific Theories

My primary focus is in scientific confirmation. So I thought I would share several theories about confirmation of scientific theories; they are Bayesian, Likelihoodism, Popperian, and my own Differential Confirmation.

All but the Popperian theory of confirmation revolve around probability, and Karl Popper even started to concede that his view is reducible to some degree of probability. So I’ll start off with a crash course in bayesianism.

Bayesianism gets its name from Bayes’ theorem, a mathematical theorem of probability. To understand this theorem, there are 4 important things to keep track of. First is the posterior probability of a hypothesis given an observation. This is written as P(H|O). Often, in bayesianism, this is what it means to confirm a hypothesis from a given observation. To calculate this, three other values are needed, the likelihood of a given piece of evidence given a hypothesis P(E|H), the prior probability of a hypothesis P(H), and the probability of the evidence P(E). These are related by Bayes’ Theorem: P(H|E)=P(E|H)P(H)/P(E)

To show how these work, let’s have the hypothesis that a particular card drawn randomly from a deck of cards is the ace of spades. So H is “this card is the ace of spades”, and let the observation E, be that the card is black. Now to start entering values.

Prior to the observation, the chance of a card being the ace of spades is 1/52. So P(H)=1/52. The probability of a card being black randomly drawn is 1/2, since half are black and the other half are red. Finally, P(E|H)=1, this is because being an ace of spades entails that it is black, so it gets the maximum probability assignable. According to Bayes’ Theorem, the result is that the probability of this card being the ace of spades is 1/52*1 / 1/2 = 1/26. And this is in line with our intuition that there are 26 possible cards and one of them will be the ace of spades. It works! After all, it’s math, it better work. Additionally, we say that the hypothesis is confirmed because P(H|E)>P(E). Disconfirmation is when the inequality goes the other way, and we have no confirmation when they are equal to one another. In the above card case, we see that 1/26>1/52, so our hypothesis is incrementally confirmed.

So now we come to confirmation of theories. We think of scientific hypotheses as being H, observations as E, and the likelihood as being what we think of as the hypothesis’ prediction. For now, I’ll ignore the problems with filling in some of these values, as it’s very hard in some cases.

Bayesianism councils us to adopt a theory whenever its posterior is above .5. This is because .5 is often the cut off for belief versus agnosticism. Why .5? It’s because P(H) + P(not H) = 1. So in cases of where the value is .5, then P(H) = P(not H), so we cannot decide between the two of them based on probabilities (assuming of course that our degree of belief in a theory should be determined by the probabilities of the theory).

The problems with this are threefold. First is that .5 is arbitrary. There why should we believe a theory to be true just because it’s marginally above .5. Why not some really high probability like .9, so we are really sure that it’s true. Secondly, it does not capture the problems of incremental changes to theories. This is because it’s a law of mathematics that if a new theory, with some additional hypotheses about the way the world works entails an old theory, then P(new theory|E)<=P(old theory|E). So no matter how many novel predictions that the new theory makes, it cannot be more probable than the old theory. Thus, why should we ever believe in an incrementally new theory? We shouldn’t on the bayesian account. Thirdly, what if we only have 2 theories and their posteriors are low? We should still select one or the other, shouldn’t we? What if general relativity’s prior is awfully low, so it’s posterior is never very high, none the less, it makes good predictions, so shouldn’t we still believe it to be true to some extent?

This is where likelihoodism attempts to solve the problems. It’s contrastive, meaning that it only works in comparing two theories. It works by having H1 and H2, as our hypotheses and examines what predictions it makes. In terms of confirmation, all we are interested in are the likelihoods. So all we look at is whether P(E|H1)>P(E|H2). It’s satisfying in that confirmation is only based on predictions. But this suffers from the problem of conspiracies. Conspiracies make the same predictions as a standard scientific hypothesis, so there is no difference between a conspiracy’s confirmation and a real scientific confirmation. Something else needs to be added to make sure that a hypothesis is a good scientific one. And this cannot be based in mathematics.

So what makes a good scientific hypothesis? Karl Popper introduced his own theory which is sticking one’s neck out. A good hypothesis that is confirmed to a greater degree than another is one that makes lots of bold new predictions. General Relativity made for a good hypothesis because it made the bold prediction that if one takes two atomic clocks, and one is flown around the earth at a high speed, then when comparing the two, they will not match up. Additionally, it made the prediction that if I see two lightning bolts striking at the same time, you might not see them striking at the same time. These are bold claims over and above what Newtonian mechanics predicted, so it’s a good hypothesis to be tested. And since these claims came out to be true, we should prefer GR over Newtonian mechanics. However, while this gives some claim to what makes a good hypothesis, Popper only said that one can falsify hypotheses through observation, never confirm. So we never believe a theory to be true, we just say that it isn’t falsified, yet.

My view is that science is concerned with what is confirmed to a greater degree. So rather than focusing on P(H|E), we look at P(E|H)-P(H). We select theories based on which is can be confirmed to a greater degree rather than which has the higher posterior. It’s contrastive so we choose theories over other theories, like likelihoodism. But we also solve the problem of conspiracies and new theories, and arbitrary points of belief. Plus, it’s rather Popperian in nature because a theory that makes a bolder prediction will enjoy a greater degree of confirmation. It takes about another 10 pages to explain all how it solves all of this, but it does work. I’m excited about my project since it captures incremental theory modification. The only concern that I have is that priors show up.

The major problem with priors I’ll leave as a puzzle. How does one figure out the prior probability of general relativity without any evidence? No one’s figured it out, so if you can, publish it. But this is a problem that’s inherited from bayesianism, so at the very least I can push it back on them. But it still concerns me considerably.

The Problem of Grue

What is called the “new riddle of induction” is a problem with grue. What is grue? Why have you never heard of it? Simple, it’s because grue is a made up predicate. A predicate for philosophy is rather simple, take “apples are green”. The predicate of the apples is “x is green”. What Nelson Goodman came up with is that we might also call them “grue”. Grue is a predicate that is split over time, where grue refers to “x is green” if x is observed before the year 2000, and “x is blue” if x is observed after 2000. So then if we observe an emerald before the year 2000, we get to apply two predicates, green and grue. Seem trivial and uninteresting? That’s where philosophy enters!

The problem is a problem of induction. Where induction is that after repeated instances of observations, we gain evidence for the universalization of it. By observing that ravens are black, after observing several million of them, we might be willing to say that all ravens are indeed black. Same goes for emeralds, after observing several million emeralds, it seems that we are on pretty good ground to say that all emeralds are green. But here comes the grue part. If we observed all several million emeralds before the year 2000, then we have equal evidence that after the year 2000, emeralds should be green and emeralds should be grue. That’s just horrifying because that says that all our evidence points to emeralds being green after 2000, and blue after 2000.

This is different from Hume’s problem of induction. Where simply observing repeated instances does not guarantee that any new instance will follow the same pattern. Take white swans. There were plenty of observations that confirmed that all swans were white, all the way up to the point that we observed that there were black ones too. There goes that induction. Hume’s induction works when we are observing instances of the same predicate. In the case of swans, we were observing repetitions that conformed to the hypothesis that “all swans are white”. Goodman’s problem of grue goes a different route.

Goodman points out that we not only have the original induction problem, but also one of not knowing which hypothesis to test in the first place. Selecting the hypothesis that “all emeralds are green” is different from using “all emeralds are grue” they present us with different regularities to observe. But evidence supports both equally, while they diverge in their predictions about the year 2000. Hence, before the year 2000, we don’t know which hypothesis we should even be testing with our data since we can make two different inductions about what happens after 2000.

This is the problem of predicates and language. That our predictions and hypothesis testing doesn’t occur in an objective vacuum of science and measurement, it also occurs in a realm of language. Depending on whether we are talking a grue language or a green language, our data will equally support both languages but will diverge in terms of what sorts of predictions we make. Science, far from being free from language, is ensnared in the language that it needs to make predictions.

Is this a bad thing for science? Yes and no, it depends on how you view science. If you view science as the ultimate arbiter of what is true and false, you might be in trouble. This is because whatever has been confirmed up till now is just like our green predicate, there might also be some other grue-y predicate lurking around that we are confirming as well and that’s the predicate that makes the correct prediction. So our hypotheses about the way the world is might be completely off, not just in an “oops, there’s a black swan” kind of way, but an “oops, our language for even talking about the world is wrong” kind of way.

But if you’re a Popperian, and you think that science maybe gets close to the truth, but really proceeds in a negative fashion, then grue doesn’t pose nearly as much of a threat. After all, both predicates are falsifiable, we just need to get around to falsifying one of them to see which one survives. We withhold the judgment that the other is true, but we at least know what languages do not work. It’s a relatively negative view of science, but one that a lot of scientists have also latched on to. That science proceeds not through confirming hypotheses, but through falsifying bad ones. And in a sort of survival of the fittest, we get to hypotheses that are closer to the truth (verisimilitude for those jargon junkies out there).

And that’s the problem of grue, in a nutshell of course. But for those of you who think that maybe there’s something fishy about grue as a predicate. Michael Titelbaum (in Not Enough There There) has recently published a proof that this problem holds for all first order logic. So take that you doubters. Personally, I find the problem of grue intractable, and it forces me into a popperian outlook even more. That science is a collection of hypotheses not yet falsified, but may be in the future. There are problems with this of course, like the duhem-quine problem of auxiliary hypotheses, but that’s better than the conjunction fallacy in my opinion. But that’s enough jargon speak for me, maybe d will understand what I’m babbling about.

Raven’s Paradox and Hypothetical-Deductivism

I’m on a science kick lately, if you can’t tell by the last few posts. I’ve gotten my phil science batteries recharged with my current course and I feel like sharing more of what I do with the world. So today I’m going to talk about the hypothetical-deductivism model of science and the raven’s paradox. Technical words, but not that scary. What hypothetical-deductivism (from now on HD since I’m too lazy to write it out again) is, is a model for how science confirms and disconfirms hypotheses. It’s an old model, but it’s been updated to what is now called bayesianism, which I find suffers from similar problems.

So what is HD? It’s a logical approach that has two parts, confirmation and disconfirmation. Confirmation happens when we have an hypothesis that logically implies some observation. The hypothesis that all ravens are black entails that if x is a raven, then x is black. If we find something that is a raven and is black, then this confirms that all ravens are black. It can be much more interesting than this, but it’s mainly a common sensical approach to confirmation. If an hypothesis logically entails a prediction, and that prediction is correct, then the hypothesis is confirmed. Hence the name Hypothetical-Deductivism, one deduces observations from a hypothesis and if those observations are correct, confirmation happens, yay!

Disconfirmation happens in just the opposite way, that if it entails that an observation will not happen, but it does, then the theory is disconfirmed. So when Newtonian mechanics predicted that light from stars will not bend around the gravitational field of the sun (gravitational lensing), and then we observed that it did, it disconfirmed Newtonian mechanics while confirming general relativity.

It’s a straight forward approach that came out of a time when we thought that logic could solve all of our problems. It hasn’t, and it won’t. Why is this? Two reasons: white shoes and bumblebees.

The white shoes problem, usually known as the raven’s paradox, is a problem that arises from a fact in logic. It is a logical truism that if a hypothesis implies some observation, then the negation of that observation implies the negation of the hypothesis. So in the case of ravens, the HD model of confirmation, if x is a raven then x is black, is logically equivalent to if x is not black then x is not a raven. But as before we saw that if we find something that is black and a raven, then it confirms the first implication, so a white shoe that is neither black nor a raven will confirm the second implication. So white shoes confirm that all ravens are black. This is a problem, white shoes shouldn’t confirm anything along those lines.

Second is the irrelevant conjunction problem. Again, it follows from logic. You’ll have to take my word for it, but it is a logical fact that if A implies B, then (A and C) implies B. So we can take some hypothesis about bumblebees dancing to direct others toward honey and attach it to general relativity. General relativity implies that starlight will bend, and logically so will General relativity and bumblebees dancing imply that starlight will bend. The meat of this problem is that if we observe starlight bending, then we confirm both the hypothesis of General relativity, and bumblebees dancing. Hence the name, the problem of irrelevant conjunctions. We could even take this a step further and append bumble bees to the ravens implication such that white shoes confirm that all ravens are black and that bumblebees dance to show where the honey is.

Why is this interesting? Because HD is rather much like a common sense approach to science. If an observation is derived from a hypothesis, then either it is confirmed or disconfirmed by the observation. But common sense is wrong on this point. And that’s what philosophy likes to do. It likes to start from a common sense beginning, and then take it further and see how it fares. Often, it does not do so well. which is why philosophy is interesting, it questions what we take for granted as true and often finds that it doesn’t work. That doesn’t mean it has all the answers, believe me, you don’t go into philosophy for answers, but in some sense, like science, at least we know what doesn’t work.

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For those familiar with bayesianism, the problem of irrelevant conjunctions plagues it as well. First is just that since an irrelevant hypothesis will not impact the probability of an observation, it too will be confirmed by observation. Or, in a more technical perspective, the HD model is a subset of bayesianism, HD is just when the likelihood of Pr(observation|hypothesis & irrelevant hypothesis)=1, which then means that the probability of Pr(hypothesis & irrelevant hypothesis | observation) = Pr(hypothesis & irrelevant hypothesis) / Pr (observation) under Bayes’ theorem. And this is always greater than than Pr(hypothesis & irrelevant hypothesis) because Pr(observation) < 1.

On the Mathematization of Nature

Around the time of Galileo, a major advancement occurred in science, the mathematization of phenomenon. This is not to say that mathematics had not been employed before hand, but Galileo produced a systematic approach to it. It was a major leap forward since it produced a mechanism of prediction that was robust and exact. This continued on through the work of Kepler, Newton, Einstein, and Feynman to the point that now we do science almost in a vacuum of mathematics. It was then the advent of bizarre phenomena at the level of quantum mechanics that we began to run into some interesting problems (though, it also existed before when atomic theory was developed). The problem that developed was the predictions of mathematics in describing and predicting the existence of things that were in a sense unobservable to humans. What could legitimize our inference that such entities existed?

To understand this question, we should look at competing theories, particularly one produced by Bas van Fraassen. The overview of this is simple. When one conducts scientific experiments, one observes regularities in nature at the macroscopic level. In fluid dynamics, one observes motion called brownian motion that is detectable. It was a step forward in science that the mathematics predicted the existence of atomic particles that produced such motion. The position that atomic particles exist as a result of such observations is termed realism; van Fraassen says that all we observe and thus all we can confirm is the observational level of brownian motion as a regularity in our observations, thus he remains agnostic to the existence of atomic particles. It seems peculiar to take such a stance. We’re pretty sure that atoms exist in everyday life, but van Fraassen questions just how much evidence we have for atomic particles over and above what confirms regularities at the macroscopic level.

There is a relatively good basis for this. In the contrastive empiricism developed in brief by Elliot Sober, we see a simple understanding of this in bayesian terms. The idea is that all the theories that assert the existence of atomic particles imply the existence of macroscopic theories like brownian motion. Thus, Realism implies Macroscopic Skepticism. Given that this is an implication, under rules of bayesianism, Realism cannot be more probable than Skepticism. Think of it this way, the antecedent in an implication cannot be more sure than the consequent, otherwise there will be times when the antecedent is true and the consequent is false. If rain is the only cause of humidity, then a 50% chance of rain cannot make the chance of humidity greater than 50%, since there is only one cause of humidity, rain (this is simplified, but it’s a way of understanding the rule). Thus, whatever is confirmed through observation at the macroscopic level cannot make the existence of atomic particles more likely than the macroscopic theories. So why should we adopt a theory that is less probable? The bayesian answer is that we should not.

And there is the conundrum. In terms of mathematical logic and probabilistic mathematics, one encounters the counter intuitive idea that while mathematics predicts the existence of unobservable particles, one cannot confirm them to a greater degree than macroscopic principles. By the logical choice that one should believe what is most probable, one should believe the macroscopic principles to be true, as in the mathematics of the macroscopic theories to be representative of the regularities in nature; and one should at the very least remain agnostic toward the existence of atomic particles.

Furthermore, this is all thanks to the mathematization of nature. Where one can convert microscopic theories into macroscopic regularities, thus establishing the implication that leads to a problematic conclusion. Personally, I resist this idea. My thesis is focusing on the failure of the implication in the direction that Sober asserts. Rather, I believe that it is justified that the implication runs the opposite direction, that macroscopic principles imply the existence of microscopic principles through the conversion of macroscopic principles into idealized microscopic particles, such as the Hamiltonian model that is used in quantum mechanics. Under my view, one observes macroscopic principles, idealizes them, and then develops the microscopic principles that follow from the idealization. This is just an overview of what I’m working on, but it’s a fascinating twist to the history of science that I hope to help resolve in part.

Philosophy and the Journey to Atheism

I guess I’ll share a little bit of how philosophy has influenced my life.

For background purposes, I go to a college that stresses analytic philosophy. This is a school of philosophy that grew out of mathematicians and logicians and stresses logical coherence and clarification of concepts, very much along the lines of Wittgenstein. There are, of course, other forms of analytic philosophy and there is no general form of it beyond the fact that it’s generally the philosophy of english speaking countries as opposed to continental philosophy which includes existentialism, deconstructionism, and others along those lines.

This stress on consistency and evidence in the use of clarification is what brought me to atheism. Above all else was the striving for consistency in the way that I view evidence and infer from it. By consistency I mean that I treat all similar bodies of evidence in a similar manner. In the case of other religions that I rejected, I noticed that I demanded a large body of evidence from them in order for me to ever believe them. Miracles could be explained in alternate naturalistic ways. Claims to prophethood is something that seems to be relatively common, so it’s not a claim that genuinely comes from actually being a prophet (since I thought there was only one prophet, Jesus). Quite simply, I rejected other faiths because I required a significant body of evidence to believe their claims, but then I noticed that I did not require the same amount from my own beliefs, despite the similar bodies of evidence.

I also noticed that it didn’t approach my common sense epistemology. If someone were to approach me with a book, in which it claims to be the truth in terms of claims about the afterlife, miracles, fantastic claims about history, I would require quite a bit of archeological evidence and some pretty powerful arguments within it about the truth of its claims. Just as I learned to do with any philosophical text. But for some reason, at the time, I noticed that I didn’t apply this scrutiny to the Bible. I treated it as exceptional, but I didn’t have an argument for why it is exceptional. I allowed for the possibility that it was, but I couldn’t find any arguments for why it was special.

Some of the special arguments were appeals to moral authorship. But philosophy generally rejects this with many good reasons. Ones that I couldn’t argue against. Another was biblical foreknowledge, but learning about history showed me that many ancient civilizations were very advanced, so it wasn’t unique. And over time, I began to believe that it wasn’t unique. Thus I became a relatively weak form of atheist, which is that I cannot decide between religions, so I would choose none of them.

And that was enough at the time, to simply find it nonunique so it isn’t a positive option above all others. In bayesian terms, the probability that it is right is no higher than any others, so no choice can be made.

I later came to apply this same bayesian analysis to the concept of gods in general. Where the characteristics and the number of gods could also not be shown to be more or less probable than alternative hypotheses. And then, faced with an alternative hypothesis of naturalism that was confirmed in a way that I could see, while the number of gods I could conceive of I could not see how they were directly supported over and above naturalism; I adopted a parsimony argument, that gods overcomplicated the evidence, that I should only support what is directly supported by observational evidence, which are naturalistic probabilities.

Philosophy influenced me at every step of the way by pointing out how to argue, different arguments, what constituted an argument, and a logical background for claims. The biggest contributor to my understanding of the world is definitely owed to David Hume, whose arguments against the existence of God are generally used even today by pop atheist books. Philosophy also stressed consistency as a primary goal as opposed to feelings. Combined with being bipolar, I now deemphasize feelings about things to a great degree since I can become delusional.

Now, working in philosophy of science, I realize that evidential support is incredibly complex and it’s reasonable to even doubt scientific accomplishments. Works by people like Nancy Cartwright and Bas van Fraassen, showed me how it’s difficult to even demonstrate the existence of hard to observe phenomena like atoms. So now I’m even more skeptical of the existence of a God since I have enough problems trying to argue for things.

And that’s my basic transition to atheism through philosophy’s influence. It’s relatively mild, without a lot of pomp and rejection. Just slow sober reflection and an emphasis on consistency along with reading philosophy. The combination of these made it very difficult to believe, and so I stopped. I guess you could call me more of an agnostic than an atheist, but I don’t care for labels. I behave as though there is no god, and that is enough. Beyond that is quibbling about semantics about my beliefs that I’d divine no discernible difference in my life. I guess I’m a pragmatist in that way, and a Wittgensteinian.

A beef I have with the new atheist movement

I have a beef with the new atheist movement. This is not to say that I’m on the side of people like Chris Mooney, who I think is just pandering away his beliefs to remain popular. And I’m not going to say that it’s the tone either. There are plenty of atheists who have a perfectly respectable tone, and to be stick up for one’s beliefs is something that all religions have done, so it’s perfectly reasonable that atheists should also stand up.

My beef with them is that their arguments can’t be taken seriously. I can barely read Richard Dawkins because he’s really just rehashing arguments made by David Hume, only in a less rigorous manner. Same with Christopher Hitchens. I usually put down their books after they essentially turn into a literary rant about either the problem of evil or that religion often makes some pretty far out claims.

Then there’s Sam Harris. Who thinks that science can pretty much provide a foundation for morality. Forget the derivation of is from ought that he neglects to argue against, all that his neural scans show is a reflection of current morals and current moral reasoning. Then he adds in a dash of utilitarianism and voila, he’s supposed to be a genius.

PZ Myers is also pretty bad in his arguments. Again, it’s the bad god + weird claims argument. He has some more, but it never goes anywhere further than simply ridicule.

The difficulty that I have with reading these books and authors is that they don’t take religious claims seriously and that there are serious arguments out there that are beyond the sophistication of the usual new atheist crowd’s knowledge base. Take for instance the fine tuning argument. It says that there are an infinite number (or at least a great number) of combinations that physical constants could take, but in our case the physical constants appear to fit just right for life.

Most arguments against this is that of the anthropic principle. That it this cannot be used as evidence since if they were not right, then we wouldn’t be around to observe these other constants. But this merely dismisses the point by saying “I don’t know, and neither can you” so the event doesn’t require any explanation. This is one way to go about it, but I’ll turn to Elliot Sober for another scenario similar to the fine tuning argument.

Enter the Firing Squad Scenario:

A prisoner is lined up in front of 12 marksmen to be shot. After firing 12 shots apiece, the prisoner is still alive and all 144 shots missed him. Now he remarks that this event does not require any explanation because if he was killed, then he wouldn’t be around to observe it. However, there is still something to be explained, why is it that he is still alive, standing in front of 12 marksmen. Two explanations can be given: 1) there was collusion and the marksmen all conspired to miss, 2) it was an accident that they all missed.

The explanation for this in probabilities is that the probability of a firing squad sparing a victim, given that the victim survives, is greater than the bare probability that the firing squad spares its victims. Thus, in the above case, the prisoner can indeed make the inference that there was collusion among the firing squad members to spare him. Looks like the anthropic principle does not work here! In fact, it appears to be the opposite since the victim can make the inference. So a simple dismissal using the anthropic principle is not fine grained enough to work in all instances. Hence it is not really the best argument out there. In arguing with someone, one must make sure that the principle argument that one uses can be generalized to other cases and still work.

However, using the probability form that we just saw up above; can we say that the probability of an intelligent designer existing given that the constants are right is greater than the bare probability of an intelligent designer existing? We cannot. The reason being is that Pr(intelligent designer exists) doesn’t have any data to provide a value for it. So the inequality that would allow us to make a choice: Pr(Intelligent designer exists | constants are right) > Pr(Intelligent designer exists), cannot be established. Through simple probabilities we can see that in the case of the fine tuning argument, there cannot be an inference made, while in the case of firing squad inference, we can say that there was collusion since we have data to back up Pr(Firing squads collude).

The argument is a little more involved, requires a little knowledge of probabilities, but it gets the inferences right, which is more than can be said by the bare anthropic principle. And this summarizes my beef with the new atheist movement: they ignore better arguments. This doesn’t necessarily mean that atheists win hands down, but if you’re going to make claims against the vast majority of people, at least get some better arguments in hand or else you’ll be dismissed by more intelligent people. This is especially important since most new atheists pride themselves on being more rational. If you’re going to be more rational, read some philosophy, not a biologist’s rehash of enlightenment materials.

 

References:

Sober, Eilliot http://philosophy.wisc.edu/sober/design%20argument%2011%202004.pdf