In an earlier more innocent time, a miracle was something impossible. Everyone knew what was possible and what wasn't possible, and if you saw something impossible then you knew you had seen a miracle, and from that you could deduce the existence of your preferred supernatural entity.
Karl Popper formalised the notion of falsifiability: a scientific theory makes predictions about what will happen in certain situations, and if a prediction turns out wrong, the theory has been falsified.
The concept of impossibility is relevant if a theory makes statements about what cannot happen. For example, the Law of Conservation of Energy tells us that we cannot build a machine which makes energy out of nothing. This can be framed as a positive prediction, i.e. no matter what machine we build the amount of energy that it will create from nothing will be zero, but, because free energy is something we want to achieve, we often prefer to frame this prediction negatively.
The same concepts can be used to formalise the concept of what a "miracle" is: something is impossible according to a scientific theory, if that scientific theory predicts that it will not happen, and if it does happen, then a miracle has occurred, according to that scientific theory.
This notion of "miracle" is scientific, but it is now relative to whatever scientific theory or theories we hold to be true. The scientific notion of miraculousness can be reconciled with our shared existing common-sense notion of miraculousness, if we consider miraculousness relative to those scientific theories that are regarded by most people as being almost certainly true.
(This analysis does leave out another important component of the concept of "miracle", which is that a miracle usually achieves something that someone wants, so the miracle is not only impossible, it is also desirable. There are some things which we consider miracles, more because we know that we want them, and not so much because we can give a precise scientific explanation of why we think they shouldn't happen. A typical example is a "miraculous" cure for cancer, where our skepticism comes from the dubious manner in which such cures are announced, "proved" and promoted, and not because we have a well-defined "theory of cancer" that rules out such cures. The most important consequence of the miracles considered in this article is our own existence as a species, which I guess is something we "wanted", but only after the fact.)
The definition of a miracle in terms of impossibility is useful, but times have changed. Ludwig Boltzmann discovered that the so-called Second "Law" of Thermodynamics is not a statement of prohibition – it is only a statement of improbability. The flow of heat from a cooler body to a hotter isn't impossible, it is just improbable. The probability of it happening to a measurable extent is extremely small, but nevertheless it is a positive number greater than zero (the exact size of which depends on what the temperatures of the "hotter" and "cooler" bodies are and how much heat we want to flow between them).
Quantum mechanics takes this a step further, because in the quantum world almost anything is possible with some non-zero probability. For example, quantum mechanics says that objects can penetrate (classically) impenetrable barriers. You probably won't get through a brick wall just by walking into it, but there is some extremely small but non-zero probability that you will.
Since, in practice, the notion of "impossibility" is a very useful one, we have had to redefine the meaning of "impossible". Where it used to mean a zero probability, now it means a very small probability. Which leads to the question: how small?
This question has no definite answer, and the best answer I can give is: so small that the probability of observing the "impossible" event, on any randomly chosen day, multiplied by the maximum number of days in the forseeable future (like several billion years), multiplied by the number of people who are ever going to exist, should be so small that it will "almost certainly" never be observed at all by anyone. That way, we can safely say that we will "almost certainly" never come to a wrong conclusion by incorrectly rejecting a theory which is used to calculate such a probability. You can then fine-tune the definition of "almost certainly" according to your own personal risk tolerance.
For practical purposes, 10-100 is a small enough probability to be impossibly improbable enough for most people's taste, and it is much larger than the probability of a person quantum tunnelling through a brick wall.
Applying this definition to Popper's concept of falsifiability, we can say that a scientific theory is falsified if it predicts the probability of an event to be lower than our chosen bound, and if the event is observed to occur. So if theory X predicts that event E has a probability of 10-104 of being observed, and event E is observed, then we can deem theory X to be officially falsified.
The so-called Weak Anthropic Principle (which I think used to be just "The" Anthropic Principle, but then various other anthropic principles got invented and so qualifiers had to be applied) can be stated as a principle of necessity:
The critical part of this statement is the "regardless of how improbable". The improbability could be lower than our chosen bound for "impossibility", which means that the Weak Anthropic Principle is telling us that, in certain circumstances, an "impossible" event must have happened.
Thus the Weak Anthropic Principle gives us a scientific theory of "miracles".
One of the biggest mysteries of biology is the origin of life. The combination of Darwin's theory of evolution, the fossil record and modern genetics can tell us a lot about the evolution of life from the earliest single-celled organisms up to the appearance of human beings and other modern life forms. But we remain profoundly ignorant about how the first life form arose from non-living predecessors.
One consequence of this ignorance is that we cannot place very good upper or lower bounds on the probability of life originating. One lower bound is the probability of a fully formed bacterium spontaneously appearing in an environment where it is capable of reproducing itself. The probability of this happening by a fortuitous arrangement of molecules is very tiny, and certainly smaller than our chosen 10-100 "impossibility" limit, but it is still a positive non-zero number. On the other hand, it is quite possible that there is some pathway to the development of an initial life form which does not require anywhere near that amount of improbability, and which falls comfortably within the "possible" range of probabilities. Not knowing how life could have started without a miracle does not imply knowing that life did start with a miracle. And if we think that maybe life was probably going to happen anyway, that is like setting an upper bound of 1 on its probability.
What the Weak Anthropic Principle tells us, is that if there is no way for life to form without some miraculous event E, then, by the principle of necessity, and the observation that we do exist, we can conclude that miracle E did occur.
Most people would feel somewhat uncomfortable with a principle of necessity being provided as an explanation of an impossibly improbable event.
But there is one easy way to make even the most improbable event likely to happen, and that is to provide a sufficient number of opportunities for it to happen. If the probability of event E happening is 10-2000, but there are 103000 opportunities for it to occur, then the nominally "impossible" event actually becomes highly probable. It almost certainly will occur, on some occasion. However, in most cases, we will still almost certainly not observe it, for the original reason that the probability of it occuring on any one occasion is so low, and there is a limit to the number of "occasions" that the human race will ever have to observe anything.
The Weak Anthropic Principle gives us a way out of this difficulty, by providing a selection effect. If the impossible event E is required for the development of intelligent life, then we will observe that it has happened, on that one occasion when it occurred and enabled the development of ourselves. The necessity of our own existence if we are to make the observation has selected in favour of our observing it.
A more precise way to describe the effect of selection is to talk about conditional probability. A conditional probability is the probability of one event E, given the occurrence (or existence) of some other condition X. This is usually written as P(E|X). There is a very simple formula for calculating conditional probabilities, which is that the probability of E given X is equal to the probability of E and X divided by the probability of X. In the case of the Weak Anthropic Principle, the condition X is our existence as an intelligent life form able to make observations, and event E is any historical event which can play a significant role in enabling condition X to come about.
We can restate the Weak Anthropic Principle in terms of conditional probability:
So to summarise so far, sufficient opportunities can make an improbable event likely, and the Weak Anthropic Principle can allow us to observe such an event. But what can be the source of such a large number of opportunities, for example a number large enough to allow a bacterium to be miraculously created from non-living molecules?
There are really only two ways to get a large number of opportunities for unlikely events to happen within a universe like our own:
Either or both of these options will work. As far as I know (but I am not an expert on cosmology), there is currently no known upper bound on the physical size of the universe.
And as for the multiverse, it is not even possible in principle to place an upper bound on the number of distinct "universes", once we admit the possibility of there existing some universe other than our own. If you prefer to actually know that there must exist multiple universes, before you are prepared to believe that there might exist multiple universes, then your best option is probably the Many-Worlds interpretation of quantum mechanics. The Many-Worlds interpretation asserts that the quantum wave-function doesn't "collapse", and this non-collapse forces a reinterpretation of classical approximations to quantum mechanics as ever-splitting classical "universes". (The wavefunction itself doesn't "split", it just carries on being a wavefunction whose evolution is described precisely by the partial differential equations of quantum mechanics.)
The number of "splits" that you can get from the Many-Worlds interpretation is more than enough to create a bacterium-sized miracle.
It might seem that, once we allow for the observation of "miracles", the floodgates are opened, and all possibility of rational science is lost. If miracles are permitted, then surely anything goes.
But this is not the case. Firstly, the Weak Anthropic Principle only allows miracles which are necessary for own existence, and secondly, if there is a choice of different miraculous explanations (of the development of intelligent life), it only permits the least miraculous explanation to be valid.
To show why, I will start by repeating the statement of the Weak Anthropic Principle in terms of conditional probability:
Consider now two possible "miracles", which we can call E and F, where, according to some scientific theory T, at least one of them has to happen if intelligent life is to develop. Suppose that , according to theory T, event E has probability 10-2000 and event F has probability 10-2400. Suppose further that no other miracles are required in either case in order for intelligent life to develop. So the probability of intelligent life developing given either E or F will be some non-miraculous probability. For example P(X|E) might be 0.001 and P(X|F) might be 0.01 (where X equals "the development of intelligent life").
If follows that the probability of X and E = 10-2003, and the probability of X and F = 10-2402. The probability of X occurring by any means is therefore 10-2003 + 10-2402, which is indistinguishable from 10-2003, and we can calculate the conditional properties of E and F given X:
In other words, observing E would not falsify theory T, but observing F would falsify theory T.
We can explain this more simply by saying that E is the least miraculous miracle required to explain X. The Weak Anthropic Principle can explain the observation of the "miracle" E, but it could not explain an observation of the miracle F, because F is more miraculous than is necessary.
In the example I just gave, I was careful to state that once E or F had occurred, no further miracle was required. In considering possible histories with two or more miracles, it is necessary to multiply the individual probabilities together to calculate the total "miraculousness". For example, if miraculous event G1 has probability 10-900 and miraculous event G2 has probability 10-800, then the occurrence of both G1 and G2 has a probability 10-1700, and the occurence of both these events would still be less miraculous than the occurence of one event E with probability 10-2000.
So we can see that is quite possible for a series of "small" miracles to be less miraculous than one "large" miracle. A consequence of the principle of "least miracle" is that if the least miraculous miracle required to explain the development of intelligent life is a series of miracles, then such a series is what we should expect to see.
In many cases, if we succeed in constructing an explanation of the development of intelligent life which involves a miracle, we may later discover an alternative story which "splits" the miracle into two lesser sub-miracles, with a certain amount of non-miraculous development occurring between the two sub-miracles.
For example, consider the previously mentioned miraculous bacterium. The miraculous bacterium is supposed to have arrived on the scene with a complete and complex interdependent set of systems for reproducing DNA, translating DNA to RNA, and translating RNA to proteins (where some of the RNA molecules and the proteins take part in the DNA reproduction and translation systems).
But the roles played by RNA in this system strongly suggest the existence of an earlier RNA-only life form. This is the RNA theory of the origin of life. According to this theory, in the earliest life forms RNA played the roles of both gene and enzyme, roles which are now mostly taken by DNA and protein respectively.
From the point of view of someone skeptical about unexplained evolutionary histories, this theory does not completely eliminate the need for a miracle, it just replaces the single bacterium miracle by two lesser miracles: firstly an "RNA miracle", which resulted in the appearance of a self-reproducing RNA life form, and then a "DNA/Protein" miracle, which resulted in the appearance of the DNA/protein mechanisms added to the base RNA system.
Replacing one miracle with two may not seem like an improvement, but what matters is that the total miraculousness of the two miracles is probably less than the miraculousness of the original one miracle. The principle of the "least miracle" tells us that any miracle story justified by the Weak Anthropic Principle is only plausible as long as we are not aware of any alternative miracle which is less miraculous. So the possibility of the less miraculous history with two miracles renders the more miraculous single-miracle history too miraculous to be plausible.
One thing that makes the split RNA and DNA/Protein miracles less miraculous is that some of the development can be explained by a certain amount of ordinary non-miraculous evolution occuring between the initial appearance of the RNA lifeform and the second step of developing a DNA/protein system.
It may turn out that this miracle "splitting" can be carried out repeatedly, with the final result being a long sequence of very small miracles. The miraculousness may never be completely eliminated, but probably there will cease to be any really large miracles left in the explanation.
The development of industrial scale gene-sequencing has resulted in an overwhelming amount of genetic information which can be applied to reconstruct the detailed history of genetic evolution. In the future the amount of sequencing data is going to continue increasing, and the reconstructed histories are going to become even more detailed. This will give us a lot of data which can be searched for evidence of "miraculous" evolution.
One of the supposed "fallacies" of evolution is the idea that there is something special about the sequence of evolutionary steps leading to our own existence. Evolution is the result of the natural phenomena of mutation, recombination and natural selection acting on our ancestors in the same way as on the ancestors of all other living creatures. To think otherwise is to give ourselves an unjustified special place in the scheme of things, as if we were the very thing for which the Universe had been constructed (for example, by God).
It's all very humbling to accept that we are not special, except that the Weak Anthropic Principle suggests that maybe we are still special. The conditional probabilities are conditioned by the principle on our existence, and not on the existence of any other species of living thing. If there are any miracles in the path of evolutionary development, those miracles will only be observed on that part of the evolutionary pathway which leads directly to ourselves.
As I suggested above, these miracles may consist of a large number of small miracles, rather than one or a few large miracles. As the reconstructions of genetic evolution become more detailed, it may turn out to be possible to spot these miracles, or, at least, to spot characteristics of evolution on the "main line" (i.e. the line leading to us) which are statistically differentiated from evolution on all the other lines (i.e. after branching off the line leading to us).
What would this miraculousness look like? One possible symptom would be too much evolution occuring within a given time frame between two points on the "main line" where branch lines are known to have branched off. This would suggest the occurrence of a small miraculous "jump" from one genome to another, which involved too many simultaneous "lucky" mutations to be attributed to non-miraculous "chance". To confirm that such a jump was a "miracle" explained by the Weak Anthropic Principle, it would be necessary to verify that no similar jump could be observed on non-"main-line" evolutionary paths.
Until we find hard evidence that some miracle has occurred in the origin and evolution of life, it might seem that the Weak Anthropic Principle is a theory looking for something to explain. But there is one mystery that it can help clear up, which is the mystery of the missing aliens.
We would expect that if an alien civilisation developed to the stage of current human civilisation, then soon afterwards it would expand into space and have a visible effect on its surroundings, and one way or another we would be able to observe this effect ourselves.
But there is no visible sign of any life in the universe other than our own, leading to speculation about what "happened" to the aliens. Perhaps it is impossible for an alien civilisation to avoid some kind of self-destruction? But it seems unlikely that all technological civilisations are doomed. A simpler explanation of why we can't see other civilisations is that they don't exist – there are no aliens, and we are all alone.
If the development of intelligent life requires one or more miracles, then we will never see the development of any intelligent life independent of our own origins. Of course if there's a multiverse of universes, then there will be aliens living in other universes, but almost every intelligent species will be utterly alone in the universe that it inhabits (and we must presume that there is no way to travel between universes, otherwise we would be back to having to explain why we can't see any aliens ourselves).