# Gödel's Incompleteness Theorems and Implications for Science

A few days ago, I heard a biologist mention that one implication from Gödel's Incompleteness Theorems is that an unlimited number of general statements can account for a given set of observations. Therefore, confirmation of observational consequences of a given general statement does not only confirm that statement, but also all the others and thus nothing is learned about reality. Initially, I doubted the validity of the claim, but after a while, I could see that this could be a reasonable implication. I'm wondering if somebody could confirm why this claim is true, or explain why it is false.

• "an unlimited number of general statements can account for a given set of observations" - what's the connection with Gödel?
– user20153
Oct 22, 2016 at 21:14
• "confirmation of observational consequences of a given general statement" - can you explain what you mean by this?
– user20153
Oct 22, 2016 at 22:44
• @mobileink 1. Given any set of observations O, there exist an infinite number of statements that account for the facts. 2. Suppose we have some statement S that accounts for an observation set O, and S implies implication I. Then if we confirm that I is in fact true, that gives a confirmation of S, as well as other statements that account for the observations O. A quick, finite example of (1) is how Brahe's geocentric and Galileo's heliocentric theories could explain the phases of Venus and retrograde motion of the planets, viewed from Earth. For (1), we have infinitely many of these claims. Oct 23, 2016 at 3:01
• Gödel's Incompleteness Theorems has NO implications whatever regarding "observational statements", period. Oct 23, 2016 at 12:51
• Okay, now I have enough rep to comment here. You're talking about underdetermination, which is an interesting topic but has basically nothing to do with Gödel. Oct 23, 2016 at 13:38

I'd offer this as a comment, but for some reason I can't comment until I have 50 reputation, and right now I only have 41.

There might be a relevant implication of the Incompletness Theorems that I'm forgetting, but this sounds to me more like a version of the problem of induction than incompleteness. Spurious references to incompleteness are pretty common; for examples, see the Wikipedia and the Stanford Encyclopedia entries on the Incompleteness Theorems.

@jobermark mentioned Popper, but I think neglected to explicitly mention his major contribution, falsifiablilty, relevant to your question (though I think he hinted at it). What's not relevant is that observations may confirm one or more theories. Conversely, it's the sine qua non of a (scientific) theory that it must be falsifiable. Then any observation contrary to a theory's predictions disproves it.

You're kind of trying to say that observations confirm theories, and in fact may confirm many different theories. But no number of observations confirm any theory; they're merely consistent with one (or more) theories. And there's no problem about the consistency of a set of observations with several different theories. Popper's point is that as you make more and more observations, you'll (hopefully) begin to come across inconsistent observations that eventually falsify all-but-one of your theories. And even then, the remaining theory hasn't been absolutely confirmed, per se. Theories aren't confirmable, they're falsifiable.

This is just another way of putting Popper's objection to verification in science in general.

Statistically, and in other objective ways, only the experimental failure of a statement can really mean something, in the sense of clearly causing other statements to be more or less likely to be true, and even then only to a very limited degree. Since failure to fail can be taken as a sort of meta-statement, verification has an effect, but is an almost meaninglessly weak force, operating very indirectly upon our warranted expectations. (Since it has significant psychological force, that indicates that verification acts primarily upon our unwarranted expectations.)

It is not really a consequence Goedel, but of the overall structure of model theory and mathematical models in general. Whether or not a scientific theory is completable in principle, it is never complete in practice, and what applies to its truth is more of a matter of informal statistics than formal logic.

If you wanted to put a basis under this idea from formal logic, it would not be Goedel, but something like the Lowenheim-Skolem theorem, that every axiomatization with any infinite model has more than infinitely many different models -- at least one of every cardinality. And any individual fact from one of those models can be 'observed' and added and still leave more than infinitely many options. But that would still be overkill.

It sounds like you're asking whether the underdetermination of scientific theory by evidence, is legitimate. You also seem to be asking whether this problem is implied by Gödel's incompleteness theorems, which I don't quite have the background to answer.

It's easy to show that underdetermination is true in a trivial sense by constructing an infinite set of theories that could each explain all possible observations. Consider the proposition that all of our experiences (including our scientific observations) have been forged by an omnipotent trickster demon, a la Descartes, and call this proposition D. By stipulation, this theory could explain anything we observed, even though it lacks predictive power or falsifiability. We can then construct infinite propositions of the form "D ∧ P", where P is any proposition that's logically consistent with D. To show that there are infinite possible values of P, and not just very many, consider the following infinite series of propositions that are consistent with D: P1 = "there is exactly 1 planet in the universe", P2 = "there are exactly 2 planets in the universe", ..., Pn = "there are exactly n planets in the universe", as n approaches infinity. This is hardly an exhaustive list of possible propositions, and the content I chose for them was mostly arbitrary, but we know that there infinite values of n, so there must be infinite theories "D ∧ Pn" that can explain any given observation. The theories constructed this way aren't empirically useful, however, because they don't imply anything we can predict or test.

It's harder to know whether this is true in a non-trivial sense, i.e. that there are always infinite (or even multiple) "empirically equivalent" theories that explain the evidence, according to the link I posted in the beginning. However, even if we assume that the scientific consensus on something is the best theory we've ever considered, "Stanford claims that in the past we have repeatedly failed to exhaust the space of fundamentally distinct theoretical possibilities that were well confirmed by the existing evidence, and that we have every reason to believe that we are probably also failing to exhaust the space of such alternatives that are well confirmed by the evidence we have at present." So even if it's not true that there are infinite falsifiable theories that are equally explanatory (which may or may not be the case), there's a strong argument that there are some such theories.