Can a new theory be proven just from what it predicts?

Question

If we make predictions with a new theory, but cannot verify its internal mechanisms and causal processes, but find that the predictions turn out to be accurate every single time, is this enough evidence for the theory? Let's assume that the predictions being correct by chance/the null hypothesis are possible but just extremely improbable otherwise.

Answer 1

The ideal standard against which all induction can be evaluated is Solomonoff's theory of inductive inference. This is Bayesian inference with a minimum description length prior.

In Solomonoff's theory, we are first given some observations. We suppose that the observations were generated by some computer program, but we want to know which program generated them.

To do this, we start with a prior distribution over all computer programs, that weights a program exponentially less likely, the longer the program happens to be. Then we rule out the computer programs that do not exactly match all the observations. The posterior distribution assigns nonzero probability to the remaining computer programs (the ones that do exactly match all the observations), with posterior probability exponentially decreasing with program length. This means the shortest program that exactly matches all the data will be assigned the highest probability.

Back to your question. If your theory is accurate every single time, is that enough to prove it?

• No. It must also be short and simple, without shorter competing theories. If there is a shorter theory that is also accurate every single time, the shorter theory will be more likely.
• It also depends on how many observations you test it against. You can never really know when you've done enough testing, although you may decide to stop when the posterior probability of the theory is high enough, such as greater than 95%.
• In practice, it also depends on whether you are testing it against new observations or only fitting it to old data. That a model fits old data is weaker evidence than that the model predicts new data. This is because we imperfect humans use the fact that a model "came to our attention" as a proxy for the model having a high prior probability. This proxy is much closer to the truth if the model came to our attention before having seen the data the model is supposed to predict.

Answer 2

It depends on what you what the theory to do.

Generally scientific theories aren't accepted just because of their predictive power, but for their explanatory power too. This is actually a very complicated question with a lot of disagreements among philosophers of science.

Consider, for example, you went to pre-Newtonian England and said something like "there are these dragons at the center of all planets and stars that pull everything closer to them at rate proportional to this equation : GM/r^2" You would be able to correctly predict the orbits of planets and so forth, but you would also draw some substantial criticism and probably would be ultimately rejected.

So predictive power is not just the sole justification for a theory, it needs to fit in with existing scientific understanding at some level, at least in theory.

Obviously there is a caveat to the need to fit in with existing understanding, (demonstrated by the advent of relativity and QM). It is important to note that the issue isn't entirely to do with QM and relativity being unintuitive or something like that; for example, Newtonian mechanics is a good theory to use when you are calculating the motion of 2 balls or something simple like that, but suppose now you are trying to calculate the motion of an air molecule with the same approach, it would be borderline impossible. Fluid mechanics is a much better theory to use when calculating something like this, does that mean Newton was wrong? Obviously not, because Newtonian mechanics works fine at the appropriate scale.

Here are some papers which explain the issue and give some solutions that you might be interested in :

https://repository.brynmawr.edu/cgi/viewcontent.cgi?article=1016&context=philosophy_pubs

http://www.strevens.org/research/expln/Idealization.pdf

Answer 3

There is a very simple argument that should convince you that the answer should be no. Suppose you have two or more mutually incompatible theories that make the same accurate predictions- they cannot all be true, so the predictions alone must therefore be insufficient to determine the correctness of any individual theory.

Answer 4

I think this question comes from a slight misconception about what constitutes a "theory", and what it means to "prove" it.

A theory's usefulness is in its ability to predict things about the world. A new theory is generally tested by seeing what it predicts about some scenario and finding a place where its predictions differ from some other theory, and then finding a way to observe what happens in reality to compare how well it matches up to the predictions. If observation differs from prediction, the theory is clearly wrong somehow.

If a new theory matches all observed behaviours exactly as well as an existing theory, its only merit is if it can shed new light on the possible causes or mechanisms that give rise to those observed behaviours.

If you have two possible mechanisms that could cause an observed behaviour, by definition you can't declare either to be correct, because the existence of an alternative explanation casts doubt. Whether that is a reasonable doubt depends on how reasonable each theory is, at which point we turn to things like Occam's Razor to decide whether theory A is more probable than theory B - but we can't say that either is "proven".

If your new theory makes accurate prediction about observations that we previously had no theory to explain, then that certainly lends credence to it, but we have to look at what it says about the causes. If your proposed explanation is provably the only thing in existence that could possibly be causing the observed behaviour, then yes, we can probably call that proof, but that's unlikely.

Far more likely is that your theory has two separate parts: a set of rules and logic that predicts the behaviours, and a set of unverifiable speculation about the causes and mechanisms. In this case, we can generally come up with some other set of unverifiable speculations to replace yours with, at which point we're back to the previous paragraph. If the speculative parts can be trimmed out of the theory with no loss of predictive power, then it's less a theory and more a description of observed behaviours. Look at Kepler's laws of planetary motion, which describe with reasonable accuracy the observed behaviour of celestial bodies, but say absolutely nothing about why celestial bodies behave that way.

At this point, "proof" means something slightly different, if it means anything at all - it's perhaps more accurate to say that you can "demonstrate your theory's accuracy" rather than that you can "prove" it.

That's not to say that there's no merit to that speculation. It might be that you were examining a particular field of science and realised that you could use it to construct an explanation for some phenomenon, and even though you couldn't verify the inner workings, you could still see some related effects to provide some evidence for it.

A lot of scientific discoveries started out with someone speculating that some process might be the cause of a phenomenon, and the scientific community spending years (or decades, or longer) coming up with some way to test the previously untestable and finally verify those internal mechanisms. We have theories about what might be happening inside a planet's core, or in the centre of stars, or in the instant of a supernova, but observing those things directly is very difficult so we can't verify with complete certainty that our theories are correct. At this point Occam's Razor comes back into play, or else we get into Bayesian epistemology and a really thorny discussion on what it means to "prove" something, or even to "know" something.