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This question occurred to me after reading the question "Does it make sense to say that one false scientific theory is closer to the truth than another?" From what I gathered, the only way to compare ideas at all would be to compare their 'level of correctness', which seemed far too difficult if lack a complete truth. So, would make more sense to measure what is not true, false, but how would we measure this in a given idea, and how would it be quantified?

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    One does not need "complete truth" to compare how accurately scientific theories reproduce experimental results. Of course, there is more to comparing theories than that, but "complete truth" is not involved in the rest either, see Theory choice and SEP, Bayesian confirmation for some numerical measures.
    – Conifold
    Apr 11 at 20:38
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    It seems that what you are looking for is the (in)accuracy of a scientific theory, not its falsehood. Apr 12 at 13:38
  • Can you make the title of the other question a link? I tried searching for it and couldn't find it.
    – Barmar
    Apr 12 at 15:19
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    If you have a theory that predicts that the speed of light is 1,000 km/sec, it's far more wrong than one that predicts it's 200,000,000 km/sec.
    – Barmar
    Apr 12 at 15:28
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    "accuracy" and "truth" can be orthogonal. I can postulate a theory that gravity happens because an invisible flying spaghetti monster pulls all pairs of masses together with force equal to GMm/r^2. Such theory is clearly accurate (at Newtonian scales) but not true Apr 12 at 21:26

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Fuzzy logic has infinitely many degrees of truth, for any real value from the interval [0, 1]. Truthlikeness (sometimes thought of as "proximity to the truth") is an even more subtle matter, since you can have two entirely true propositions, yet one of them is somehow "more" verisimilar than the other, e.g. "The number of moons of Earth is exactly 1," vs., "The number of moons of Earth is somewhere between 0 and 4."

Per some usual definitions of truth and falsity, truth and falsity "just like that" are too structurally akin to each other for it to seem plausible that ranking theories in terms of proximity to falsity would be logically preferable to ranking them in terms of proximity to truth. However, if one works with some unusual such definitions, perhaps your suggested preference would make sense. See about the pessimistic meta-induction for a reason we might have to look for unusual definitions of truth and falsity in this context.

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  • In mathematics, where you nowadays cannot do without at least some rudimentary form of infinity, this is trickier. Consider Euclid's fifth postulate, or the statement "if you keep adding one, you eventually reach every possible finite quantity". They were introduced as self-evident, to serve as axioms. Nowadays the first has become something local, that can be true somewhere and false somewhere else. While the epistemic status of the second (essentially the induction axiom) is still not clarified well I believe. Apr 13 at 6:00
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All models are wrong, but some are useful (GEP Box)

All scientific theories are models of reality, and they are all going to be wrong or incomplete in some detail. Asking for "truth" is an unreasonable expectation, so science only strives for the best explanation.

Traditionally the value of a scientific theory seems to be based on how much it explains (and/or is consilient with how much other science) for how little theory/assumptions (I'm taking it as read that it isn't obviously falsified by some reliable experiment or observation). I don't think you could make a formal measure from that though (or whether it would have practical value).

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  • My answer explains how to formally measure and compare theories which seek to explain a particular collection of experiments. The paper I linked to, Robinson 2016, is fun; they use computed accuracy of sensors to look for lost aircraft, to model WiFi connections, and more.
    – Corbin
    Apr 12 at 16:34
  • @corbin I don't think that fits the bill - it doesn't include consilience or the range of things explained or the burden of assumptions and theory. It isn't just about the precision of predictions. The paper sounds interesting though (I work in stats/ML). Apr 12 at 16:38
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Predictive statements - of which scientific theories are a subset - have a model, a domain, a precision, and a level of certitude.

Take the statement:

If Alice visits the corner store after midnight, she is more likely than not to meet a tired and disinterested employee.

Model: what is the nature of the prediction? visit therefore meet

Domain: what parts of reality does the model purport to predict? Alice, a corner store after midnight

Precision: with how much detail can the model predict reality? a tired and disinterested employee

Certitude: if we run the experiment many times, how often do we get results that match the prediction? more likely than not, i.e. at least half the time.

A predictive statement is true if the four parts taken together comport with reality, else it is false.

Different predictive statements may have bigger domains (we could expand from Alice to any person and from the to any corner store), higher certitude, or more precision (we could say that the employee will be wearing a uniform). These make them more predictively useful, but not, strictly speaking, truer.

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  • "Accuracy: with how much detail can the model predict reality?" That's precision. Accuracy is a measure of correctness within a given precision.
    – JimmyJames
    Apr 12 at 16:35
  • @JimmyJames Hard to tell which one uniform-having-ness is, but you're probably right. Will edit.
    – g s
    Apr 12 at 20:34
  • Check my answer. I'll take whatever feedback you have.
    – JimmyJames
    Apr 12 at 20:37
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Truth means correspondence to reality. A scientific theory either corresponds to reality or it doesn't. A scientific theory may solve a problem that another theory doesn't solve while having different unsolved problems. For example, general relativity explains how changes in the gravitational field propagate through space and time. But it fails to solve the problem of providing a quantum mechanical description of gravity. But for each problem a theory either solves it or doesn't so this isn't a measure of truth or goodness or whatever.

For more explanation of how to deal with ideas with binary evaluations, see

https://criticalfallibilism.com/introduction-to-critical-fallibilism/

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  • Since we don't really have any method independent of science for knowing reality, how can we tell whether a theory corresponds to it? This is a fundamental conundrum in philosophy of science.
    – Barmar
    Apr 12 at 15:24
  • @Barmar You work out the consequences of a theory, then do an experiment to test whether those consequences happen in reality. If they don't then there is a problem with the theory. See the work of Karl Popper for more explanation fallibleideas.com/books#popper
    – alanf
    Apr 12 at 19:23
  • That's basically a summary of the scientific method. But it doesn't reveal "truth". And it has limitations, e.g. the "grue" problem in the method of induction.
    – Barmar
    Apr 12 at 19:27
  • Science doesn't work by induction philosophy.stackexchange.com/questions/67110/… The grue problem isn't a real problem philosophy.stackexchange.com/a/21448/5759
    – alanf
    Apr 12 at 20:37
  • That's because we realize we can't really determine "truth", the best we can do is "true enough". Newton's laws of motion are good enough for the vast majority of phenomena people encounter. Einstein's theories of relativity are improvements.
    – Barmar
    Apr 12 at 20:41
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Given some measurement, if it is in reality 5 ft.

And one theory say's it should be 1 ft, and another theory says is should be 4.9 ft, we can compute the error of one theory and says it's +- 4, while the second theory has an error of +- 0.10

Of course, this is the trivial case where falsehood is in regards to measurements.

There are significant complications in practice. Some are:

  1. How could we ever know what the "true" measurement is?

  2. Theories can be more effective in some instances and less in others. So, on the whole- one might need to subjectively weight different contexts depending on what one is trying to do- rather than some generic objective standard.

  3. How do we know that the error is a result of the theory, rather than our measurement techniques? ( This is a related issue to (1))

In the more complex issue kind of falsehood ( 28 Dimensions VS 56 dimensions) It's very hard to quantify degrees of falseness. It's not clear that "more true" is a useful comparison, and instead we turn to utility, simplicity, beauty, ease of use, etc...

As a rule of thumb, theories which assume weaker hypothesis, but still prove the same things - are preferable. This is effectively, the spirit of Occams Razor.

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Don't worry about truth. For empirical science, we can compare the distance between our predictions and our measured values; smaller is better. This lets us iteratively improve. Our distances must be continuous.

The process of computing these distances is called sensor fusion. Summarizing Robinson 2016, one generic way to structure any collection of sensors involves constructing a sheaf (WP, nLab) over the sensors and integrating any sensors which are compatible with each other by overlapping their stalks. Then, a bit of abstract nonsense goes as follows:

  • Every collection of readouts of sensors is a sheaf assignment
  • Every sensor space is e.g. a (real-valued) metric space and has a metric
  • The sheaves over sensors also have a metric space indicating the distance between sheaf assignments

And computer code can be written to compute an assignment which minimizes the error of each individual sensor as well as the entire collection. This not only gives us a way to evaluate each model against empirical measurements, but also to detect whether each sensor is well-calibrated.

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  • I love sheaf theory but I would like to see more clarity in how you feel sheafs can be applied to epistemology and the philosophy of science in this way. Your post reminds me a bit about Bayesian reasoning which I’ve only just gotten acquainted with, and the idea of “triangulation”. Apr 12 at 17:14
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    Like here’s one idea: you presented sheaf theory in terms of “sensors”. But could you also think about it in terms of probabilistic “hypotheses”? Apr 12 at 17:16
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    How about the dimension curse? When you deal with lots of independent parameters, and you can only approximate the metric in the range which you already have explored well, it can well happen that you are headed towards a particular dead end which is within certain, possibly small, distance from the truth but cannot reach it since it requires assumptions contradicting that truth. While there might be other, trickier routes that in the end do reach the truth but you have been off them all the time. Apr 13 at 5:47
  • @JuliusHamilton:Yes! We can customize the restrictions in each sheaf to reflect our hypotheses; then, the computed distances can be used to perform a comparative analysis over the entire collection of hypotheses. We'd use the same sensors each time.
    – Corbin
    Apr 13 at 20:30
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    @Corbin Many thanks for very interesting references! But are these methods really free from traps? Pattern search can still be attracted to a weak local minimum, while at your link about Nelder-Mead it is explicitly written that the method might converge to non-stationary points Apr 14 at 4:07
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If I understand your question correctly, this is something that has a pretty well-established practice around it. I think a big stumbling block around this is that people tend to want to have a single number, a scalar value that defines correctness. In reality, there's not one single measurement of 'correctness'. There are a number of different ways to define it and they answer different questions. A common example for this is testing for diseases. I think it's a good one because it matters. If someone is tested for an illness, it's important to know not only that the test works, but how well it works. But this is by no means the only context in which such approaches are used. They are ubiquitous in science.

To put this in terms of your question, the 'theory' is that a positive test means that the subject of the test has the disease and that a negative test means the person doesn't have the disease. One way we can refer to these results are 'true-positives' (TP) and 'true-negatives' (TN). A perfect test will have only those results. However, there are other possible results: false-positives (FP) when the test says the person has the disease but does not and false-negatives (FN) when the test says the person doesn't have the disease when they do.

I think it should be obvious that in the real world, there's no such thing as a completely perfect test for any disease (or if there is, it's an extremely rare situation.) So how do we quantify the 'correctness' of the test? There isn't just one way.

The most well-known and common quantification is 'accuracy' which is defined in this context as the number of correct predictions divided by the total number of predictions. Let's say our test is for a rare disease and the test is 99.00% accurate. Meaning that 99 out of 100 tests, the test result was correct. Sounds pretty good right? But, what if this rare disease is only found in 1 out of 10,000 people. We could have a test that does nothing and just gives a negative result every time and it would be more accurate than this test: 99.99%. It should be clear that accuracy is not, by itself, good enough to quantify the correctness of theory.

So, there are other quantifications that we can use which are arguably more useful and informative. Another one commonly used is 'precision' which is defined in this context as the number of correct positive results divided by the total number of positive results. And we also have 'recall' which is defined as the number of correct positive results divided by the total number of people tested who actually have the disease.

There are more such quantifications which you can research if you are interested. The upshot here, though, is that there's no one single number that can fully quantify the correctness of a theory or model.

You might have noticed that this example is very binary: a person has the disease, or they don't. Can we use this kind of thinking when talking about things that are less definite? We can but we have to talk about precision again and define it in a slightly different way. You have probably seen products described as 'precise' e.g., razor blades. In that context, precision means that the blades are milled to a specification that has a very small scale. And 'accuracy' would refer to things such as the percentage of blades a machine produces that meet that specification. To give a silly example, I can claim that the Empire State Building is 3.0854395922356343 meters tall. That's an extremely precise claim that is extremely inaccurate. Or I could say that it's more than 100 feet tall and less than 10,000 feet tall. An extremely imprecise claim which is completely accurate.

So, to wrap up, we can talk about Newton's theories such as universal gravitation and motion. Using it, we can produce highly accurate predictions of where planets will be in a given point of time with a pretty high level of precision (relative to stellar scales, anyway) for large bodies in our solar system. Except for one: Mercury. The model is not accurate for that planet at an equivalent level of precision. A theory which can predict Mercury's position as well as the other planets is 'more correct' and that's exactly what Einstein's theory was able to achieve.

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