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Falsifiability has a troubled history, but fundamentally it's not a bad idea. Put simply, an idea is falsifiable if it is capable of being disproven. It is considered a desirable quality in meaningful statements, especially in the sciences. Some have even used it as the essential property of scientific statements that separates science from non-science. I personally wouldn't go that far - I think that there's good thinking and bad thinking, and that scientific thinking is just a particular blend of good thinking. But any idea or hypothesis that claims to describe the real world should be "falsifiable", in some sense.

Why am I hedging? Why not simply say "Falsifiability is good, and any unfalsifiable statement are meaningless"? Because in reality, falsifiability is just a rule of thumb - a simplified version of a deeper truth. This deeper truth is that which is expressed in Bayes' theorem: hypothesis are judged by the probabilities with which they predict the data: that is, by their likelihoods. Effectively, you can think of a hypothesis as being defined by their likelihoods. Now, it may be that a hypothesis is "unfalsifiable" in the sense that none of its likelihoods are zero, and therefore there would be no observation which could absolutely disprove it. But even in such cases, as long as you don't cheat by breaking the rules of probability in assigning these likelihoods (say, by setting all likelihoods to 1), an observation will judge the hypotheses that made predictions about it.

It's also worth noting that observations are never absolutely certain. All of science is based on experiments, and all experiments have uncertainties. These uncertainties mean that a hypothesis could never be "disproven" to the point that enough positive evidence in its favor couldn't make up for it. Consider the case of the neutrinos reported to be travelling faster than the speed of light. An observation seemed to "disprove" relativity, but because of the strength of relativity as a theory, many people stood by it despite the observation. And they turned out to be right.

It may also be that, depending on how the hypothesis is "disproven", it can be modified to easily accommodate the new data. Before the nineties the Big Bang theory was understood to predict a slowing expansion for the universe. When this was observationally "disproven" - with an accelerating expansion, of all things - we did not reject the Big Bang theory, but made small adjustments to it instead. We in fact invented something completely unknown - dark energy - to make the theory fit the data.

These cases are a complete travesty if you only have "falsifiability" as the defining criterion for science. "We're making up stuff to make the theory fit the data? How wrong is that?" But we do not doubt that these are in fact the proper steps to scientifically proceed, even if they violate the principles of "falsifiability".

So, "falsifiability" is false - at least when it is understood strictly. But when understood as a rule of thumb meant as a simplified expression of some deeper truth in Bayesian reasoning, it's useful. The full application of Bayes' theorem easily makes sense of the above examples where "falsifiability" fails. (Exercise for the reader - try it. If you understood my series on Bayesian reasoning, this should not be too difficult.) And when you look at the equations - especially Bayes' theorem in odds form - many ideas clustered around "falsifiability" can be salvaged. These are ideas like "a hypothesis should make a real prediction, instead of saying that anything is possible", "a hypothesis that assigns more extreme likelihoods is a 'better' hypothesis", "you should readjust your beliefs according to the evidence", and "a nearly-zero likelihood has a good chance of dramatically changing your beliefs, because it may create enormous likelihood ratios". Falsifiability, understood in terms of these ideas, is an important quality to have in a hypothesis.

So, how is this related to Christianity?

Christianity is the most falsifiable worldview I know. Right there in the Bible itself, it tells you exactly how to falsify it: 1 Corinthians 15:14 basically says that if Christ did not rise from the dead, my faith is in vain. Come up with a definitive, physical proof that Jesus didn't rise, and you've successfully falsified my hypothesis.

Am I saying that such physical evidence is possible? Absolutely. The following are some of the things that Christianity assigns nearly zero likelihoods to - that is, things that would "falsify" Christianity by disproving the resurrection. This is what would cause me to make a post saying "Hey guys, it turns out I was wrong about this whole Christianity business".

A hidden cave is discovered in Israel, containing the mummified body of a man whose wounds are consistent with execution by crucifixion. The man was about 30 years of age at the time of his death, and his wounds include all the small details which are atypical of a crucifixion, such as puncture marks by a crown of thorns and a fatal spear thrust to the side. All dating techniques - radiometric, dating small coins found on the body, the style of burial cloths used, a tree growing over the coffin whose age is known, etc - conclusively date the body to 30 AD.

Within a cavity of the body itself, in a protective container, is found a number of manuscripts which detail a massive conspiracy to deceive the world by starting a new religion. In addition to the disciples and Jesus himself, the members of the conspiracy include Pontius Pilate, Herod Antipater, and Emperor Tiberius. Their membership is unquestionably verified through their official imperial Roman regalia (seals, etc) present in the coffin, which checks out with known history. These artifacts and the manuscripts are also all definitively dated to 30 AD. The conspiracy is to have Jesus (which these documents clearly identify as the mummified body) crucified then pretend to have risen from the dead, by having the disciples steal the body. These manuscripts furthermore contain text describing the future doctrines of this new religion. This section of the document contains much of the text of the New Testament, decades before they were supposed to have been written and thereby unquestionably proving that the writers of this document were also the authors of the New Testament. 

In a completely separate development, an entirely different group of document is found that details that Jesus had small, unobtrusive physical defect - say that there was a small but distinct imprint of a cross on his skull above his hairline. These new documents are unquestionably written by the Pharisees, who are enemies of the conspiracy and have no reason to collaborate with them. These documents passes all the usual tests for authenticity. And sure enough - our mummified body has this defect. 

Lastly, far in the future humans invent time travel - the kind that allows for watching, but not changing the past. When we travel back in to the time of Jesus' death and burial, you see some disciples scurrying about the tomb, and everything stated in the above conspiracy theory scenario is verified.

You may object that this is an unlikely discovery: that doesn't matter. Falsifiability attacks those theories that are IMPOSSIBLE to disprove, not unlikely. The theory of gravity is falsifiable because if I drop a brick it may fall upwards. The fact that this is unlikely actually means that gravity is a strong theory. All strong scientific theories are like that - possible to be falsified, but unlikely to be.

You may also object that this is a weakness in my belief, that leaving this door open may cause me to lose faith someday and is therefore antithetical to believing in God. On the contrary, this is a mark of a healthy faith. Faith without works, or actions, is dead. Those whom Jesus commended for their faith were those who acted upon it. Leaving this possibility open is me stepping out in faith, for I have faith that God really did raise Jesus from the dead. It is like Peter's faith in stepping out out to walk on water. Imagine if Peter instead had said, "Um, actually Lord, I'll stay in this boat here. I'm sure that you can have me walk on water if you wanted to. I believe you. So I don't actually need to go on the water". What would Jesus have said to him?

Peter stepped out in faith, and he walked on water - if only for a short while. My faith is like that in that it is falsifiable, just as Peter had the possibility of sinking into the water.

But wait - Peter became afraid, and began to doubt and sink into the water. So was Peter's faith falsified? Only in the strict, unproductive sense discussed at the beginning of this post. This is like the case of the Big Bang theory only needing minor adjustments after the discovery of the accelerating expansion of the universe. Likewise, if I have made some mistake in my understanding of the resurrection, or my construction of the conspiracy theory scenario above, I still have faith that Jesus will reach out and catch me, as he did with Peter.