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If so, does it have a name? I'm especially interested in search terms that I can use for further reading.

In physics, we're familiar with the distinction between "relations of ideas" and "matters of fact." It's especially useful to be clear about what derives from premises and what has been established experimentally, as we usually work in both directions to meet in the middle.

But some arguments seem to straddle both categories and others seem to be neither. For the first type, consider a scientific model, like the Standard Model of particle physics. It is a mathematical model—the first questions we should ask of it is whether it is logically valid. But it is also built to represent facts—the next questions we should ask is whether it agrees with measurements. No one believes the Standard Model as a fact about the world: we expect it to be replaced by successively better matches to empirical fact. A scientific model seems to be a composite thing, judged by the methods of both categories.

But beyond the criteria of logical consistency and agreement with experiment, we also apply criteria that are neither of the above. For lack of better words, these are called "beauty" and "simplicity." Despite the consistency and accuracy of the Standard Model, we distrust it because it has more ad-hoc parameters than we'd expect nature to have.

I can be clearer with an older (more established) example: Ptomley's solar system model of circular orbits with circular epicycles is logically consistent and can be made an arbitrarily good fit by adding epicycles upon epicycles. Kepler's solar system model of elliptical orbits reaches the same precision with fewer parameters. The argument that picks between them doesn't rely on an a priori proof (first prong) or agreement with experiment (second prong) because they both pass both tests. The argument comes from somewhere else. Some people say, "simplicity," but I would call it something more like "insight." Kepler's model makes it easier to discover Newton's, because Newton's gravitation can directly explain ellipses, but has nothing to say about an infinite sequence of epicycles (unless that sequence happens to converge to an ellipse).

"Relations of ideas" and "matters of fact" have different methods for determining truth—mathematical proof and experiments. This "beauty," "simplicity," or "insight" seems to be neither of the above, but it's also not sophistry or nonsense. It seems to belong to a third category that I don't know the name of.

My question is: what's the name of the thing that I'm taking about? Even if Hume didn't discuss it, what's the modern name so I can read up about it?

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    Hume's fork evolved into what is now called the theory/observation distinction. It does not have a third prong, or first and second, it is now recognized that statements lie on a continuum with no "pure" ends. Even the most "observational" statements are said to be "theory laden", see SEP, Theory and Observation in Science.
    – Conifold
    Commented Nov 26, 2020 at 10:25
  • Thanks, I'll read that. The scientific models I described above could be examples of impurity—when a measurement is interpreted in light of the Standard Model, it is not only an observation, but also part deduction—though practicing scientists do take pains to distinguish the theory-derived parts from the direct observations. I'd be willing to believe that there's no pure observation, since all sense experiences are interpreted in terms of some expectations, but mathematics can be "pure," since it can be completely independent of observations, right? Commented Nov 26, 2020 at 15:51
  • And on the other point, there are definitely statements beyond "relations of ideas" and "matters of fact," whose methods of verification are as different as mathematical deduction is from scientific experiment: moral judgements, aesthetic responses, and even metaphysical arguments like Hume's Fork itself are not on a continuum between logic and empiricism. How are those classified? (My example of picking between two models that are both logically consistent and both fit the observations was supposed to be an example of this, not too far removed from science.) Commented Nov 26, 2020 at 16:28
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    The "purity" of mathematics is disputable. While it does not involve literal "testing", its concepts, methods and theories come from and are subject to the same pragmatic considerations of empirical success. They are abandoned when it is lacking. Normative (ethical, aesthetic) judgements are also on a continuum that comes from another Hume's "fork", see the fact–value distinction.
    – Conifold
    Commented Nov 26, 2020 at 23:08
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    Simplicity, elegance, coherence, unification, and other values that influence selection of models are called epistemic values. A classical discussion of their role in science is in Putnam's essay The Collapse of the Fact/Value Dichotomy.
    – Conifold
    Commented Nov 28, 2020 at 9:15

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