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Imagine scientists observe a linear correlation between two measurable quantities x and y in some type of phenomenon. They induce an experimental law: y=kx. However, the degree of precision of their measurements was finite. To what extent can we expect that the correlation will still be linear if, say, we double the level of precision of measurements? I'm not talking about expectations with regards to the value of k, which I assume will be updated, but about the linear form of the law. What kind of inference (inductive, Bayesian, inference to the best explanation) is required to justify such expectation?

My first thought was that the linearity of a correlation could be justified by induction: given that the correlation was linear so far, it is rational to expect that it will remain linear for higher degrees of precision (note that induction is faillible, but it's not the problem I'm interested in here). But this approach is problematic, because there are infinitely many functions that approximate a linear correlation (y=k•sin(x) for example), and all of them would be as much justified by induction, including the ones that no more approximate linearity at higher levels of precision.

We have a similar problem using Bayesian inference: there are infinitely many possible functions that could correspond to our observations, and it's not clear why the ones that will continue to approximate linearity with higher precision would have more weight than the others.

The only inference that seems to justify that the correlation will still be linear is an inference to the best explanation: a linear correlation is more simple, more explanatory, hence more likely than another type of correlation. But this type of inference can be criticised (why would nature be simple?).

So it's not clear why we should expect the form of experimental laws to be preserved with higher precision, and yet this kind of expectation seems to me implicit in scientific reasoning: scientists are not expecting experimental laws to break down every time they increase the precision of their measurement (to some extent at least, I don't mean indefinitely). But maybe I'm wrong? Am I missing something and are there ways of justifying this type of expectation in a Bayesian framework for example? Are there resources in the scientific or philosophical literature on this topic?

  • But the form of experimental laws is not preserved with higher precision, and linear rules are expected to be replaced by more elaborate ones, as in Hooke's, Ohm's or other laws. The practice of running high precision tests even for established theories like SR is quite widespread, which would not make sense if such inferences were made. It is not that preservation of linearity is inferred, but rather that having it to some precision is often good enough, and one pragmatically assumes it further absent positive evidence to the contrary. – Conifold Apr 19 at 7:34
  • 1) Just getting a correlation does not create a Law. Theories need a supporting mechanism or argument. It is the theory that supports the observation that has expectations laid upon it. That something appears to take a given form, entirely from data, is not enough. 2) Science does not have meta-expectations like this in general. If a theory comes along, that either better integrates with other accepted theories or better reflects data, or that is equally good but simpler, it will be adopted. -- The contest of theories is the kind of inference you need, and you don't have a theory here. – hide_in_plain_sight Apr 19 at 10:04
  • @hide_in_plain_sight I'm talking about experimental/phenomenological laws here, I have in mind something directly associated with data models, not theoretical laws. – Quentin Ruyant Apr 19 at 10:20
  • You are talking about justification in science. There are not different parts of these disciplines that work by different rules. Data does not have a logic of its own. – hide_in_plain_sight Apr 19 at 10:22
  • @hide_in_plain_sight well the distinction exists. Usually, theories are taken to explain experimental laws, which are more connected to observations. Duhem already mentioned the distinction. Data models are constructed by means of statistical technics from brute data. It involves a bit of theory (models of experiments etc) but a data model does not derive from a theory in the same way a theoretical model does. This is well documented – Quentin Ruyant Apr 19 at 10:31
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It may help to consider the end-user of the experimental law, as follows.

Engineers use predictive models all the time that are based on empirical results from experiments where, for example, getting three-place accuracy out of a linear fit to the data over a specified range of values gets the job done almost all of the time. And the engineer using that model doesn't necessarily care why the relationship was linear in the first place, nor why the fit isn't linear for certain values.

Physicists, on the other hand, generally develop models from first principles which may or may not be linear, and then compare the model to reality in the interests of determining whether or not there exist physics that the model is missing.

Nature is under no obligation to present us with relationships that are perfectly or even approximately linear.

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