Scientific Inference and How We Come to Know Stuff. Part 2 - Comments

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mfb
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A nice article!

One mistake I found:
Specifically, it provides the probability that we would observe the data O under the assumption that H0 is correct, p(O|H0)
That value would by tiny nearly everywhere (what is the probability to observe exactly 425435524 radioactive decays within some timespan, for any relevant hypothesis?). You need the probability to observe "O or something that deviates even more from the expectation based on H0" (which you used in the following text) or a ratio of probabilities for different hypotheses.

A nice article!

One mistake I found:
That value would by tiny nearly everywhere (what is the probability to observe exactly 425435524 radioactive decays within some timespan, for any relevant hypothesis?). You need the probability to observe "O or something that deviates even more from the expectation based on H0" (which you used in the following text) or a ratio of probabilities for different hypotheses.
Yeah, thanks. $p(O|H_0)$ is a distribution over $O$. I'll reword that!

In simplified form:

You think something interesting is happening. So you collect data.

Assume that nothing of interest is happening. How likely are you to have gotten such data in this case by pure chance? If it is unlikely, then that gives support to your idea that something interesting is happening.

If someone else gets the same result, then that is further support. And so on. If they get the same result by a different method, that is even better. This is called consilience.

It doesn't mean that your idea is correct, just that it is consistent with experiment. Not the same thing, but it DOES show that your idea has predictive power. In science that matters a lot. I'd say it is the main purpose of scientific theory: to predict what will happen in such-and-such a situation. Explaining why it happens is definitely secondary.

It is quite common for incorrect ideas to give correct results. Indeed, it is very common for theories known incorrect to be used. Incorrect theories may be convenient and good enough for the purpose at hand, especially if they are much simpler than the correct theory.

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It is quite common for incorrect ideas to give correct results.
If they give correct results, why are the ideas incorrect then?

A. Neumaier
2019 Award
The problem with ''while it is impossible to verify a universal statement by observing singular instances, universal statements can be contradicted by individual observations'' is that what is a falsification is not well-defined. If a first year student falsifies Hooke's law through his experimental analysis nobody cares. And falsifications of statistical laws are uncertain by the means with which statistics is created and analyzed. Thus falsification has the same somewhat subjective status as verification: We can never be sure, once the laws are allowed to be imprecise in the slightest - which most modern physics laws are.

stevendaryl
Staff Emeritus
If they give correct results, why are the ideas incorrect then?
Well, in some cases, we know that a model is incorrect (because it ignores relativity, for example), but it still makes predictions that are good enough for practical purposes.

stevendaryl
Staff Emeritus
Just an observation: The type of "universal laws" that are always used in discussing induction or falsifiability almost never become part of a scientific theory. What I mean is that nobody just notices a correlation and proposes it as a universal law: "Hey! Every dachshund I've ever known has a name starting with the letter S. Maybe that's a universal law of physics!" When Newton proposed his law of universal gravitation, he didn't generalize from lots of measurements of the force between objects. Instead, his proposed law of gravitation was a model or hypothesis that allowed him to derive a whole bunch of other facts--namely, the shapes and periods of orbits of planets, as well as the fact that objects drop to the ground when released.

There are certainly cases where people just notice repeated patterns and propose a law that generalizes from those patterns. For example, the wavelengths of light emitted by hydrogen atoms was found to be given by something like $\frac{1}{\lambda} \propto \frac{1}{m^2} - \frac{1}{n^2}$ where $m$ and $n$ are integers, and $n > m$. But it wasn't really taken as a "law of physics", but as a regularity that needed to be explained by physics (and the explanation turned out to be quantum mechanics).

Well, in some cases, we know that a model is incorrect (because it ignores relativity, for example), but it still makes predictions that are good enough for practical purposes.
Does that necessarily mean the model is incorrect? As far as I'm aware, there is no such thing as a perfect model that takes everything into account. And we all know GR is incomplete. So any model is incorrect then?

stevendaryl
Staff Emeritus
Does that necessarily mean the model is incorrect? As far as I'm aware, there is no such thing as a perfect model that takes everything into account. And we all know GR is incomplete. So any model is incorrect then?
I would say that almost any model that we are currently using is not only wrong, but we know that it's wrong. But I suppose there is a distinction between being completely wrong, and being "in the right ballpark". For example, when we compute the energy levels of hydrogen using the nonrelativistic Schrodinger equation, we know the derivation isn't actually correct, because it leaves out relativity, and pair creation, and electron spin, and weak interactions, and GR and so forth. But we expect that taking into account those other things will make a small change, and won't completely overturn how we understand hydrogen atoms.

Just an observation: The type of "universal laws" that are always used in discussing induction or falsifiability almost never become part of a scientific theory. What I mean is that nobody just notices a correlation and proposes it as a universal law: "Hey! Every dachshund I've ever known has a name starting with the letter S. Maybe that's a universal law of physics!"
Agreed. This is why the discussion is focused on hypothesis testing rather than hypothesis formation.