Wednesday, July 15, 2009

Borders of the kingdom of science

Sean Carroll wrote an essay about the range of validity of science and the principles that determine how its insights about the "truth" can be extended to new situations. And your humble correspondent actually agrees with all of his main ideas.

Well, more precisely, I no longer think that it is fair to pick religion and religious people as the only "target" in describing some common misconceptions about the "power of science". During the last decade, I learned that who doesn't believe in God is likely to believe in everything else. So he can end - and he often ends - with much more ludicrous ideas than many Christians do. Atheism often becomes a tool to promote more irrational ideas than Christianity.

But if we only talk about religion and science, and if we ignore possible misinterpretations and abuses of Sean's essay, I think he's right. You surely expect me to write a much better text than he did. So here it is. ;-)

Religion: the material vs foggy one

The beginning of Sean's text is dedicated to the definition of the word "religion". I agree with him that definitions can't be right or wrong: they can only be useful or less useful.

In many discussions about religion and science, religion is often claimed to be compatible with science. But when we actually investigate what the word "religion" means in similar contexts, we usually learn that the speakers mean "all parts and weakened statements of religions that are selected in such a way that all opinions that have already been falsified by science are omitted".

Well, if this is what the word "religion" means, then there is no contradiction between science and religion. In fact, this contradiction is a tautology that directly follows from our definition of "religion". This makes the two words consistent but the tautological character of the consistency makes it useless and uninteresting, too.

What's more important is that the "actual religion" that we observe in the real world, the religion as pursued by typical believers, priests, or even theologists, means something else and contains many "material" statements that actually do contradict science. The "religion" in the real world includes a lot of scientifically invalid statements about cosmology, history, and miracles, among other things.

So the champions of religion like to behave as moving targets: they try to defend whatever they seem defensible in a given context. It means that they are modest when they deal with scientists who can rule out many of their claims: in such situations, "religion" only means some very general and abstract things that haven't been falsified yet. But when they return in a more religious community, they happily redefine "religion" so that it includes many things that scientists would not only disagree with but would be able to rationally debunk.

Once again, this dishonest kind of "flexibility" is not inherently connected with religion. Environmentalists and even critics of theoretical physics and many other groups love to act as "moving targets", too. When talking to actual scientists, environmentalists also admit that science makes it virtually impossible to fry the Earth in 100 years or do similar miracles. The term "global warming" only means some unspectacular empirical facts in this context - such as the tautology that the climate is changing. But when these environmentalists get the opportunity to influence more gullible people, they are more than happy to redefine "global warming" and include all kinds of catastrophes.

Not only excessively religious people or environmentalists are demagogues whose opinions resemble a mutating flu virus.

Lee Smolin is perhaps an even better example. Whenever he discusses physics with actual scientists who can easily prove that all his criticisms of physics are scientifically invalid rubbish, he agrees with them. That's not what his "trouble with physics" is supposed to mean in such situations. However, when he talks to the journalists or the laymen, he's more than happy to redefine "trouble with physics". He's very satisfied if they start to believe all kinds of scientific myths and if they extract them from his words about the "trouble with physics". Lee Smolin is as slimy as a snail.

Unfortunately, you won't hear any critical words about far-left atheists such as Lee Smolin or the environmentalists from Sean Carroll. Try to guess why I am talking about this fallacy on both sides of the aisle while he is not.

Empirical character of science vs naturalist religion

Let us return to the positive tone. The title summarizes another topic that Sean Carroll described very well. Science is a method that searches for the best theories of the world by evaluating the empirical evidence with our brains. One must be careful what is really "empirical" about this method. Some people tend to make all kinds of mistakes.

There are people who don't understand that science must actually determine the "truth" by looking at the empirical evidence and by evaluating its consequences with the help of logical arguments. Many people still think that the "truth" should be determined by the degree to which the statements about the world are "philosophical pleasing" according to their pre-existing sentiments. But neither philosophy nor emotions have any power to identify the scientifically true propositions. Clearly, the people who try to decide about the truth by an emotional evaluation of rather vague sequences of words, using their own brains and hormones only, haven't yet undergone the scientific revolution.

On the other hand, there are also many other people who incorrectly think that the empirical pillars of science mean that the conclusions that sound more "naturalistic" are the more correct ones. But once again, as Carroll writes very nicely, this is not what science is actually doing. Science is not a naturalist religion. And even though Carroll wouldn't be too happy to hear it, science is not Marxism, either.

Natural explanations - avoiding supernatural beings and miracles - are brutally preferred by the 21st century science. But this preference is not here with us because of pre-determined philosophical criteria but because the "natural" hypotheses have done a much better job in explaining the phenomena observed during the millenia after the humans became able to ask deep questions.

It was not guaranteed to be the case at the beginning. But it just happened. In some more detailed questions, something else took place. For example, people thought that the world followed deterministic laws, in the sense of the classical determinism. This answer is surely more "materialistic". But science - namely quantum mechanics - has actually shown that the world only admits probabilistic predictions. So the future is never fully determined by the past: only the probabilities are.

Again, this conclusion may contradict some people's philosophy. They may say that it is less "scientific" - but what they mean is that it is less compatible with their "naturalist religion". But what matters is that it is actually consistent with the observations. The best theory describing the microscopic world - quantum mechanics - implies that the world cannot be deterministic. Even though you could think that one cannot "directly" determine whether there are hidden variables etc. that restore determinism in the microscopic world, it doesn't mean that science has nothing to say about it. By a comprehensive evaluation of the evidence combined with clever gedanken experiments, the conclusion is pretty clear: hidden variables can't work and the microscopic world can't follow deterministic laws.

Whether someone finds is philosophically problematic is his psychological problem. It is not a "trouble with physics" or "trouble with quantum mechanics".

Auxiliary, not directly measurable concepts

Another subtle issue that is linked to the "empirical character" of science are concepts that can't be directly perceived by our senses. Are they acceptable in science? Some philosophies - let's call them "extreme positivism" - would argue that such concepts have no room in science and theories using such concepts should be disfavored if not abandoned.

As Sean Carroll says and I confirm, this opinion is utterly unscientific. The real science actually does use many concepts that can't be easily or directly linked to our perceptions. First of all, when we (or you) observe something, we (or you) could say that something is happening with our (or your) souls - and everything else (including other humans) is beyond science. That's the attitude that solipsism takes.

Of course, science wouldn't get too far with this philosophy. Something is happening with our souls because the souls - and brains - are actually receiving the information from our eyes. These eyes do exist: the theories that assume that eyes exist do a better job than the theories that assume that eyes don't exist. The eyes are not the borders of the scientific kingdom, either. It seems obvious that the eyes "see" something because it actually exists, too.

When we use these eyes and brains to observe a particle that is affected by Brownian motion in a bottle of water, the best explanation we can find is that the particle was actually colliding with molecules of water. You may notice that Robert Brown couldn't observe any molecules. But in science, that surely doesn't mean that molecules don't exist! One must carefully distinguish two very different propositions: "molecules haven't yet been seen" and "it has been seen that molecules don't exist". The latter sentence has never been valid.

At some point of the history, the first sentence was true. Could the scientists reasonably conclude that molecules didn't exist? Quite on the contrary. In science, one must be a priori open-minded about propositions such as "molecules exist". When you try to construct hypotheses or theories that have something to do with the microscopic phenomena at this length scale, you will find out that the winning theory that actually accounts for all of these phenomena is a theory that inevitably implies that molecules exist, too.

Many thinkers in the past, including "big shots", have been caught in the trap of the "extreme positivism". They thought that atoms, molecules, quarks, or strings (or fields that exist in the vacuum, without a "material" carrier - which is a kind of opposite example!) couldn't exist - or shouldn't be discussed by scientists - because they couldn't be seen. They were wrong. In some other cases, people with similar opinions could have been right because the objects they criticized were non-existent, indeed. But the they surely didn't give the right explanation why the objects were non-existent if they had only complained that these objects were beyond our perceptions.

Once again, science often critically needs (or predicts) concepts that cannot be directly linked to observations. They're very important in science simply because the most satisfactory theories typically include many concepts of this type. That's why it is very important to talk about quarks and strings in particle physics - and about multiverses in cosmology. And all attempts to prevent scientists from using such concepts - just because they can't be directly "seen" - are simply anti-scientific in character.

In fact, in many cases, scientific theories are actually most beautifully described in formalisms that contains objects that are unobservable even in principle. For example, gauge theories are most conveniently written in terms of a gauge potential. But the Yang-Mills potential is not a physical observable because it is not gauge-invariant. If you change it by a gauge transformation, you obtain a physically indistinguishable situation (even in principle!). Nevertheless, the perturbative description of gauge theory is prettiest if you use this ambiguous, uniquely unobservable quantity at each point! It's just what the mathematical analysis of the equations implies.

The opinion that such concepts shouldn't be a part of science is a philosophy, and science has actually proven that this philosophy is incorrect. The gauge field is just one example out of many.

How do we generalize observations in science?

Once we accept that the observations force scientists to incorporate seemingly "unobservable" concepts to the very hearts of their theories, we should ask the question how it is possible that science can actually predict anything. Why is it a mystery?

Well, we have observed some phenomena. But one can always say that any event that will take place in the future (or that has taken place in a different place) is uncertain because we haven't observed it yet. Strictly speaking, it is a different event than all the events we have witnessed in the past. So science can't say anything about it, can it?

Yes, it can. In fact, it is the very main purpose of science to say such things - to make predictions. How is it possible that science can predict new things?

There exist "primitive" examples of a similar phenomenon. Because the sunrise occurred on every day in your life so far - thousands or tens of thousands of days - it is sensible to assume that the Sun will rise tomorrow, too. Now, this is not an argument based on the cutting-edge science. And in fact, we know that this argument will fail on a sunny day sometime in a few billion years from now.

But nevertheless, this example is enough to show how the scientific reasoning works, even at a very primitive level. Let's inflate the numbers a little bit. Imagine that you believe me that we have evidence that the sunrise has occurred on the most recent 1.7 trillion days in the past. The question is whether the Sun will rise tomorrow.

An obvious hypothesis is that the Sun will rise on every day in the future because it has done so for 1.7 trillion days in the past. This hypothesis is compatible with all the "simple" or "binary" data about the sunrise we have.

You can create a lot of hypotheses that imply that the sunrise only occurs on some days but not others. For example, on days number "p" where "p" is a prime integer, there is no sunrise. A vast majority of these hypotheses will be instantly ruled out because the sunrise did occur every day, as far as we can say.

So the simplest alternative hypothesis you can make that is not immediately ruled out will imply that the sunrise takes place on every day for the first N days but this process fails on the day N+1. This is a one-parameter class of hypotheses. (Each hypothesis, labeled by N, is actually a union of many hypotheses that differ in the predictions of the "days after tomorrow", greater than N+1.) And such hypotheses are only compatible with the observations if N is greater than 1.7 trillion - because we have always seen the sunrise so far.

You must divide the prior probability in between the different hypotheses with different values of N. It is very sensible (but not quite rigorously proven!) to assume that the hypothesis with N (the lifetime of the Sun) equal to 1.7 trillion is comparably likely to the hypothesis where N is equal to 1.7 trillion plus one. So the probability that the minimum N is actually equal to a very specific integer close to 1.7 trillion can't be bigger than one trillionth or so. It is therefore negligible. It means that the probability that tomorrow will be the first day without the sunrise is negligible, too.

(Various believers in catastrophic theories should study this simple argument and learn something from it.)

As I have mentioned, science has collected much more detailed data than the simple binary (Yes/No) observations of the sunrises every day. The theories must therefore be compatible with many more things. It follows that the viable scientific theories in 2009 are much more sophisticated than the theories that only tell you how many sunrises will occur before the Sun fails for the first time. The logical reasoning that is needed to deduce such conclusions is much more complex, too. In the case of the Sun, we actually know that the Sun will die sometimes in a few billion years.

But what I want to emphasize is that the basic logic of science is actually very similar to the logic I used to determine that the Sun will rise tomorrow (using a primitive approximate theory that will break down in a few billion years). The only difference is that in real science, our theories must be compatible with much more detailed and extensive empirical datasets than the observed fact that the Sun has risen for quite some time. I am not going to discuss nuclear physics and the burning Hydrogen because this text is about the philosophy of science, not about astrophysics.

At any rate, when you try to create the most sensible or most likely hypothesis that is compatible with everything we have observed (up to a few exceptions that you can identify as errors or manifestations of new physics that probably has a negligible effect on all other phenomena you want to predict), you are doing something very similar as the "naive" scientist who has correctly predicted that the Sun will rise tomorrow.

If you do such things right, you will find out that pretty much everything we have ever seen is a manifestation of the Standard Model, General Relativity, a few cutting-edge additional phenomena, and some sensible assumptions about the initial conditions. Moreover, if you analyze how these a priori separated portions of your theory may fit together, you will determine that the world around us is almost certainly an excited vacuum described by string/M-theory. Most laymen misunderstand how ambitious the claims of the current science actually are.

Science is not just a tool to understand a few very isolated phenomena. In principle, the current scientific picture of the world claims to describe pretty much all phenomena we have ever observed. Equally importantly, it also claims that there can't really be any other phenomena in the regimes that science claims to have mastered. There are no dragons or witches in these realms.

Of course, in reality, the contemporary science has some limits but they're already pretty far. The likely location of these "borders of science" is a question that every scientist must care about; it is a part of science that can be studied and is being studied by the scientific method, too. The borders exist in an abstract multi-dimensional parameter space of our knowledge.

Borders of current science

Let me describe some of the limitations of science as of 2009. Current science can't unambiguously tell us everything that happens at distances shorter than 10^{-18} meters - because it's hard to determine what's happening in this extremely small empire. It can't tell us what are the detailed properties of dark matter - because it's hard to measure (or indirectly determine) the properties of particles that don't interact with us too much or too frequently.

It can't tell us what happened at the very beginning of the Universe or the beginning of life on Earth - because it's hard to extract accurate information from some very old "fossils" or their cosmological counterparts - or even to find such "fossils". (But we can go very far, and in cosmology, we can surely go to the time when the Universe was just a few seconds old.) And science can't tell us what is exactly driving many complex phenomena - because they're complex and we're often unable to arrange all the necessary bits and pieces in our hands (or computers), to do the required difficult calculations, and to be sure that we have the right answers and explanations.

But on the other hand, science can tell us many things - and understand at least most of the isolated events that appear at the ordinary length scales and energy scales. If we assume that Jesus Christ existed and was crucified at some point, the first droplet of His blood that fell from the cross followed Newton's mechanical laws of gravity - at least with a good enough accuracy.

How do I know it? How do I dare to speak about His blood? First, I have had the moral right to think about such questions since the wars between science and religion a few centuries ago: science has won, if you have forgotten. Second, what are my rational reasons to be so certain? Well, it's basically the same kind of extrapolation I used for the sunrise. In the case of His blood, it is slightly more sophisticated than the case of the sunrises (because it uses some continuous functions and elementary differential equations) but it is much less sophisticated than the predictions that the cutting-edge physics is actually doing today.

You can invent an alternative theory where His blood followed a very different path. But in order to be consistent with other observations of mechanical phenomena that seem to be compatible with Newton's theory, His blood will have to be a kind of exception. A theory where His blood is an exception from the equivalence principle and other laws is awkward.

It is not just an emotional, aesthetic argument. Very sane refined, quantitative versions of this argument imply that such an awkward hypothesis must be given a much smaller prior probability than the theory which includes no exceptions for Christ's blood. The reason is similar as the reason why the very long but finite "life expectancy of the Sun" was unlikely to be exactly as long as needed for the Sun to die tomorrow. Why? Because there are many possible theories where Christ's blood - or something else - gets some special "miraculous" treatment that differs from the otherwise known laws of physics.

All these inequivalent "miraculous mutations" of the conventional laws of physics are mutually exclusive. So the prior probability that any hypothesis is valid - that must sum to one - must be divided between this astronomical number of "mutated theories". That makes the probability that each of them is valid very tiny. In fact, not only the hypothesis that the first Christ's droplet of blood followed a trajectory exactly copying Moses' face is unlikely. But the total hypothesis that any droplet on Earth gets such an exception is extremely unlikely because I can invent many more methods how to "add exceptions" to the laws of physics and all these mutated classes of hypotheses must share the limited prior probability.

The simple mechanical hypothesis without exceptions is qualitatively different from its "mutations" because of its uniqueness, so it belongs to a class of its own. Its prior probability is not diluted by a huge factor.

That's the "Bayesian reasoning" that actually explains Occam's razor - i.e. the fact that scientists don't want to add awkward exceptions unless it is necessary. As I have written above, similar things are sometimes necessary. A new kind of a phenomenon (or a new particle such as the muon) is sometimes observed and our theories have to become more complex, non-minimal, and so on. But scientists don't add such "arbitrary hair" for no good reason, and the previous paragraph gives you a logical justification why they behave in this way.

Such an approach is not based on 100% certainty: nothing in science is really 100% certain. But the actual certainty about many propositions can be and often is extremely close to 100% because the logic is often rooted in extensive and accurate evidence. If you analyze the detailed logic behind my conclusions that the Sun will rise tomorrow and that the droplet of Christ's blood followed the Newtonian path, you can see how robust the logic is and what the possible loopholes might be. (If you want to be unbiased, you should also try to make all the improvements to my arguments even if I didn't write them explicitly.) I think that if you do it right, you will conclude that it is extremely unlikely that there are any loopholes that will change the conclusions.

Science as a battling kingdom

To summarize, science is trying to describe an increasing spectrum of phenomena by an increasingly "canonical", "unified", and "beautiful" - and therefore "a priori" more likely - theories. It has done an amazing job in this program. While it couldn't have been obvious to the ancient Greek philosophers that all the observed phenomena and all their detectable details could have been described with a few pretty equations (up to uncertainties and errors that are stunningly small in all real applications), the research has actually shown that it is the case.

Scientists working on scientific theories try to extend their range of validity. They always try to determine where the borders lie. The champions of particular theories try to prove that their kingdoms are very large. They try to prove that no other theory can describe the same class of phenomena: they try to kill the foreign soldiers on the battleground, if you wish.

On the other hand, there may also exist other theories or "philosophies" - that try to expand and take over a class of phenomena - that are generalizations of theories that have been successful elsewhere (or that were invented from the scratch). These theories often predict new phenomena in a particular regime. A key question is whether such new phenomena exist - and more generally, whether the predictions are valid.

In this battle of nations, comparisons between the observations and the predictions (that often need complicated calculations to be made) are the judges that determine which army wins and which theory covers the given province. The judges are not perfect (because imperfect people are needed for the tests of the theories) but they're good enough to make progress and to choose the better army in most cases.

Real scientists typically don't support just one army: they neutrally play with all of them and try to figure out which army is winning. Individual scientists are often packages that contain competing generals as well as the judges. But even when the individual scientists only do some of these jobs - if they're biased, in this sense - the whole scientific community is expected to cover all the jobs in a balanced way.

In many cases, the battle remains open for a very long time. In many other cases, the winner is completely clear and the losing army is eliminated up to its last man. And in most cases, the winning army actually catches some memes from the losers. So the true winners are not exactly the same soldiers who entered the bloody fight but somewhat wiser and more skillful soldiers - a refined, more accurate theory.

Science is a gigantic battlefield covered by kingdoms (scientific theories) that often follow a very different logic. Each kingdom tries to use its own territory in the optimum way (to increase the accuracy of its predictions and to refine all the details) and to expand into other kingdoms (to generalize its propositions). Whether such excursions are successful ultimately depends on the empirical criteria (observations). The kingdoms often change their internal system. Sometimes, two kingdoms get unified (and such a unification makes Einsteinian theoretical physicists really happy and more certain). Sometimes, a new detail is revealed (an unanswered detailed question) and it tears a kingdom into two parts that continue the fight as two entities.

Quite often, the kingdoms take over regions that haven't been governed by any king so far. In such cases, science may try to attack questions that used to belong to philosophy or religion. But whether science is ultimately successful in extending its kingdom into the hostile territory of philosophy also depends on the empirical criteria - and the ability to tightly connect the old kingdom with the new colonies. If the transportation of products, commodities, and soldiers is efficient enough (i.e. if the new application or conclusion of a theory seems to be a pretty reasonable or inevitable generalization of the insights made in the "normal conditions" where a theory is known to work), the kingdom may keep the colony.

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