Tuesday, February 19, 2008

Teach This Old Dog a New Trick

I suddenly remembered a quote from a math humor book I read back in junior high. It went something like this:

"The laws of probability are so cleverly formatted that no experiment can refute or prove their validity."

It won't surprise me if there's been a lot of debate about this that may have settled the issue centuries ago, but I haven't been able to come up with a way to falsify the laws of probability from my armchair bit of math. Any particular experimental result is possible under the laws of probability: Some are just more probable than others, and there's no magical cutoff for improbability.

If I can't get anywhere with them, I think I might just have to stick them into the same category of assumptions as "the external universe exists" and "the laws of physics are consistent."


Anonymous said...

Hey now Bronzy, you know this one.

If you wish to test the probability of something, you just have to use a large sampling to see how often it occurs. It won't provide absolute info on the probabilities involved of course. That's no different than observing gravity's consistant operation and not being able to say absolutely that gravity will continue to operate that way.

However, it is sufficient (assuming a large sampling) to conclude at the very least that it would be irrational to operate on the assumption that the probability is something else. That's the most we can hope for and what the universe allows us to do. I believe you've said as much about other experimentation.

So flip that coin 1000 times. If you get roughly 500 heads and 500 tails, it's hardly irrational to conclude 1:1 probability, at least until some hypothetical super large sampling shows tendancies to some other ratio.

Absolute proof isn't science's thing, it's evidence to gain some idea of reality enough that you can base your actions on that, at least tentatively until future evidence emerges.

Oh and, the other two things there are the same way if you ask me. How do you know there's an external reality "for sure"? You don't, but there's enough evidence, that is, the universe operates in a way that suggests there's external reality and there's no evidence to suggest it's all a dream, that you are justified in acting as though it's there.

Does the universe behave in a consistant manner? That's actually one of the first things science, historically, tested for. I mean one of the first trials was to see if the heavens obeyed the same laws of motion as the Earth. And, that works "moment to moment" as well. Just take measurements over time and see if they differ substantially. That is, see if blue is one wavelength today and "chicken blanket mobile" tomorrow. If not, it's safe to conclude physics are consistant. Science depends on laws of physics being at least somewhat consistant, but we'd be able to tell almost immediatly if they different significantly from moment to moment. I doubt life would even be possible if they weren't.

That's all science ever offered to provide, and it's what we get.

Bronze Dog said...

Yeah, I know all that stuff you're saying. I don't know if you missed the question I was trying to ask, or if I wasn't asking it right.

The problem I'm talking about is that probability seems unfalsifiable because any experimental result can be explained by the laws of probability. A theory that can explain anything is unfalsifiable.

I don't see a terribly big problem with that, beyond a bit of mindscrew from thinking about it, just like rejection of solipsism isn't a problem. The laws of probability make the most sense for the observations we've taken so far, and I don't see a better explanation going around.

It's just a little trippy, and I was just wondering if someone thought of a falsification method to resolve the faint philosophical nagging. That's all.

I have no problem accepting any of those assumptions just for practicality's sake, since we can't get anywhere without them.

Anonymous said...

I'd say the way to falsify any question of probability is the same way one would get an initial idea of something's probability.

If you are trying to falsify a coin flip's 1:1 ratio, flip the coin some crazy number of times, the number of times being higher than the number of test flips done across the world or something close to that. If you get some vastly different ratio from that, it's effectively falsified. Again it's not absolute proof of falseness, but it's to a level where thinking it's still true would be unreasonable.

Basically think of it like this. The proposition "the Earth is flat" is falsifiable by finding curvature or going into orbit, but that's not "absolute proof". There's some really small number (dwindling, but still there) who might say either that the curvature of the ocean and watching ships vanish over the horizon is an illusion or all those pictures "from orbit" are just fakes, but that evidence is sufficient that thinking otherwise is unreasonable.

Was that the sort of thing you were asking, or was it something more fundamental like proving math itself? If it comes to that, that's not an issue of testing so much as logic, and there falsifiability would basically be rendering the concept of probability internally contradictory or something like that.

Dikkii said...

Isn't most of what you're asking set by definition?

So if you're looking for a way to falsify a statement like, "probability is the chance that you might select a red apple from a bag of mixed apples," this would be close to completely unfalsifiable, because this is a definition. You would have to redefine probability.

It's like this: can you falsify the statement that, "an apple is the fruit of the malus domestica plant," without redefining an apple?

Anonymous said...

I think you're making a category error.

The laws of probability are mathematical constructs, not experimental hypotheses or fundamental axioms. They are proven (in the mathematical sense) rather than tested (in the scientific sense) or assumed.

The category they belong in is with things like differential calculus. You can't falsify that, either (as far as I can see).

Flavin said...

I agree with Dunc. Mathematical ideas don't need to be, and I'd say shouldn't be, falsifiable. Can you falsify addition?

Bronze Dog said...

Yeah, it's a bit of a difference in category, but I tend to think more in terms of physical world science than math. I think I was asking mostly about the crossover, since sometimes math constructs don't seem relevant to the physical world.

But either way, it's not a problem, just a curiosity I had.

Anonymous said...

I think Einstein said it best:

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

Dikkii said...

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

Or as someone once put it to me, mathematics should really be regarded as a branch
of philosophy (the logic part, at least), rather than being lumped in with the sciences.

Lifewish said...

The problem I'm talking about is that probability seems unfalsifiable because any experimental result can be explained by the laws of probability. A theory that can explain anything is unfalsifiable.

Science is comparative. We don't pick the hypothesis that gets every prediction right every time; we pick the hypothesis that gets more predictions right than any other.

So, for example, people didn't stop using Newton's theory of gravity just because it didn't predict Mercury's precession properly. We waited until we had Einstein's hypothesis, which did get it right.

Similarly, you can't falsify a single probabilistic model of events. But what you can do is compare two models and see which is more predictive, more parsimonious, and more accurate.

The interesting question is the tradeoff you get between these three factors. For example, modelling a process by the normal distribution* N(0,1) would produce stronger, "tighter" predictions than a model of N(0,10). So we should prefer the N(0,1) hypothesis over the N(0,10) hypothesis.

But what if there were more outlying results (suggesting a larger variance)? Then there'd also be a good reason for preferring N(0,10) to N(0,1). What's the crossover point?

And what about parsimony? Say the choice was between N(0,v) and N(0,exp(v^2)/5-1), where v is very small? The former clearly requires fewer magic numbers, but the latter would give more precise predictions. Which do we prefer?

Answers on a postcard, please.

* For those who have no idea what I'm talking about, N(x,y) describes a bell curve centred on x (the mean), with y being a measure of how flat and wide the bell is (the variance)

Anonymous said...

In answer I'd say it depends on how much accuracy you need at the time. In the same way a project like constructing a building tends to use Newtonian physics, as that's all the accuracy one needs. By comparison, building a satellite and plotting it's orbit will require relativity levels of accuracy.