> what if some outside force were causing the 111 elements transitions to speed up, but that outside force had no effect on the 1 who was immune to that cause? How can we ever be sure of anything?

If two models generate the same predictions, they are the same model, and the simpler (in the Kolgomorov complexity sense) is best understood as the truth (in so far as that concept is meaningful at all). A force acting on everything in the universe except one atom, or a (equal and opposite) force acting on that one atom, is the same thing[1], just as a universe where all distances are multiplied by two is indistinguishable from the current universe. Again, there's no way to make an observation from outside the universe; we can only make relative measurements, observe patterns, and use them to make predictions.

[1] ultimately, this comes down to: they generate the same sense-impressions that I subjectively experience - but for simplicity let's assume an objective universe exists, other humans exist, and we can observe the same things - I believe this assumption is justified as the most parsimonious explanation of the patterns I perceive in the sense-impressions I experience

> is best understood as the truth

No! It's preferred as an explanation because it is more elegant and convenient, NOT because it is more true. It is the same true!

That's just getting into semantics, which is pointless. What is "truth"? Ultimately, that depends on your point of view. If you base it off of experience (which is how you learn about everything in the first place), then the experience is "truth". If you base it off of something "outside" of yourself, that reference point itself becomes "truth". But of course, that reference point - be it real or not - is still something you imagine is there, and therefore might not be real, and the burden of referencing everything to that "source" of truth still falls on that one thing you are trying to avoid: experience. Sorry for being overly philosophical. For a continuation of this thought, try web searchs for "solipsism" and "world of mind".

Elsewhere in the thread someone gave the example of heliocentrism versus geocentrism. You can form a heliocentric model that makes accurate predictions, you just have to add epicycles to everything. But I would say that heliocentrism is not merely more elegant and more convenient, but also more true. Wouldn't you? For a more extreme example, in the story from http://lesswrong.com/lw/i4/belief_in_belief/ I think we would say that it isn't true that there's a dragon in the garage.

In that case however we have other data than just the model to suggest that geocentrism is false, whereas the topic of this thread is models in physics mathematics that merely differ in complexity, with no apparent difference in "truth".

> Just as a universe where all distances are multiplied by two is indistinguishable from the current universe.

Uh, wouldn't you have a speed of light problem? I think most of the fundamental forces are all inverse square laws, which would also make things weird.

I'm sure you have a much better understanding that my casual interest. But, uh, simple linear scaling seems detectable.

I mean a universe in which the speed of light was multiplied by two, and so was every linear distance measure.

so the earth is twice as big, but the same mass? seems like there are some detectable changes there. but maybe you're increasing mass in a really tricky way. it'll need to be more than double to have the same effect. at a given distance.

The other one that seems like it would be tough is strength of a magnetic field. inverse cube law, rather than inverse square. seems really hard to do something and get the same answers.

i'm an idiot. by distance you mean something more like a pure scalar value. if you double all the secret constants in the universe, we can't tell the difference.

You double k1 and k2. Now something involving k1*k1/k2 is twice it was before.

Why would it matter though? I read a analogy somewhere (in a Carl Sagan's book, I think) about how a goldfish in a convex fish bowl might create a set of rules for motion of objects that will work fine in predicting motion of objects as seen from inside the fish bowl but will be totally different from the true laws that people outside that bowl will formulate. As long as the rules keep working for the fish, why would the fish care if they are wrong? More importantly, how would you know that you are living in the outermost layer of reality? (not sure what the technical term should be)

As Vladimir Bartol famously opined: "Nothing is true."

How do we know for sure that we're not in the Matrix? Can we be sure? Does the very fact that we question our reality confirm our sentience or are our thoughts merely the output of algorithms we've been programmed with from the outset?

> How do we know for sure that we're not in the Matrix? Can we be sure?

If we observe no effects then the most parsimonious explanation for our experiences is that we're not. We can't ever be sure, because someone could be in the Matrix and have absolutely identical experiences to our own. Occam's Razor tells us what's most likely (and you can get more specific than that, putting numbers on it - see Kolgomorov complexity), but it's not 100% reliable.

> Does the very fact that we question our reality confirm our sentience or are our thoughts merely the output of algorithms we've been programmed with from the outset?

Try reexpressing that question in terms of something you could empirically observe. If you can't, the question probably isn't meaningful.

That concept of surety we cannot grasp with the observation tools we have available, but it's also not necessary to grasp it; the assumption that the 1 element was changing rather than the 111 only becomes functionally "incorrect" if it contradicts some other observation we make.

Beyond that, the grand truth of what's actually happening becomes a bit uselessly murky. The real function of Occam's Razor is to allow us to simplify models for our own benefit (because simpler models are easier to interpret and use), not necessarily because the simple answer is the Actual Truth(TM), which we can't know outside of observation.

There's a thought experiment in college philosophy: how do you know the universe wasn't created from nothing precisely six seconds ago, with all matter and energy in the universe arranged with the precise concentrations and momenta they would need to become the present universe in precisely six seconds?

> The real function of Occam's Razor is to allow us to simplify models for our own benefit (because simpler models are easier to interpret and use), not necessarily because the simple answer is the Actual Truth(TM), which we can't know outside of observation.

What about assigning greater prior probability to hypotheses that are simpler in some sense? Is there perhaps something to that too?

I'd say it's legitimate, just because simpler models have turned out to be true more often in the past, as best as we can tell.

Maybe even _that_ reasoning relies upon simple hypotheses being more likely. In this case, the simple, and therefore favoured, hypothesis would be that "models with a simplicity that is similar to that of other successful models are more likely to be true".

Assuming that other successful models are also true and not just a wolf in sheep's clothing.

Let's forget that I said "true", and switch it to "successful" instead.

That sounds like a more complex hypothesis to me.

Maybe it's simpler than "models with a simplicity that is dissimilar to that of other successful models are more likely to be true", though it's not obvious what should be considered simple and not.

Kolgomorov complexity seems like the obviously correct measure, though AIUI it's only defined relative to some fixed Turing machine.

I have an idea of doing a PageRank-style weighting of Turing machines, based on how simple/complex simulating one under another is. That would align with my intuition that most basic universal Turing machines are equally valid, but one that has (say) a short instruction that outputs one particular long sequence of symbols is probably not a valid machine to use.

Surely this is the same paradigm that would have opined Copernicus's heliocentric incorrect and the geocentric model accepted by the masses the correct model(?)

Is it not scientific method to prove correctness, not simpleness?

The geocentric model isn't simpler; that was the problem underpinning the shift to a heliocentric model.

The geocentric model required the addition of several epicycle adjustments with no observable cause. Re-framing the local system with the sun at the center eliminated most of the epicycles.

(... and the remaining errors in the heliocentric model relative to the observed motion led to the discovery of the moons and, eventually, Newton's gravitational theory. The driving function, overall, is simplification of the model by removing special cases and the need for acausal "I don't know why; this is what we observe" rules).