If I observe my mother, is that proof of the existence of my mother?
If I observe an apple on top of a table, is that proof of the existence of that apple and that table?
Someone might claim that they have observed God. Is that proof of the existence of the God they claim to have observed?
More generally, can the observation of anything prove the existence of anything, and if so, under what circumstances?
Welcome to Plato's Cave. 2500 years later and we're none the wiser. :)
If I observe my mother, is that proof of the existence of my mother?
Technically and strictly, this is only proof that whatever it is that is doing the observing believes that it has observed your mother; nothing more or less. From this you can write a 10.000 word pamphlet about what all the words mean. It really depends on your basic beliefs about how the world and yourself works. Nothing of that is "truly known" (but most philosophers or neuroscientists probably have a strong opinion).
A Bayesian would argue that there is no black & white, and every proposition like P1: "I see my mother", P2: "my mother exists" and even P§: "P1 -> P2" has a probability associated with it, but none of them have the probability 0 or 1. So in their worldview, the argument would probably be that the probabilities for all three propositions are pretty high, bordering 1 very closely (but never absolutely).
If I observe an apple on top of a table, is that proof of the existence of that apple and that table?
The same argument.
Someone might claim that they have observed God. Is that proof of the existence of the God they claim to have observed?
Never. This is completely different. In the previous two examples, you did not mention that you would like to convince someone else than yourself from any of the propositions. This new example has P1: "Someone claims they observed God.", P2: "God exists" and P3: "P1 -> P2".
For a Bayesian, the probabilities would be "pretty high" for P1 - there are plenty of precedents of people claiming they observed God in some form or fashion. P2 is completely "free floating" depending on the belief system of the Bayesian at hand. But P3: "P1 -> P2" must be close to zero.
The latter is the simple proposition that solely because someone said something, this must be true. That is a very weak argument. Everybody can say anything whatsoever. Humans are perfectly capable of deceiving, fantasizing, inventing stuff, misinterpreting their observations or plain old lying. Extraordinary claims require extraordinary proofs, so someone simply saying something has no or very little effect on the probabilities of anything.
The more analogous case to your previous two examples would be: "If I observe God, is that proof of the existence of God (to me)." You can do the analysis like before for yourself - the probabilities of this statement (for you) basically reflect whether you already believe or not, or where you are on the spectrum.
More generally, can the observation of anything prove the existence of anything, and if so, under what circumstances?
The term "prove" is difficult here. What does it mean? Experience shows that nothing at all is a "proof" of anything, regarding reality; no matter how well you think you know things, you always have to be open to amending your beliefs. The whole scientific method is built on the concept that we do not even pretend to prove anything at all - we only make theories with a recipe for disproving them; and then work with those theories until disproven. But no scientist ever assumes that any of the theories is fundamentally, objectively "true" forever.
In other areas (philosophy, religion etc.), the concept of "proof" does not even exist, at least not in a meaning one would commonly think it has. If you think of mathematical proofs, they most definitely do not stem from simple observations; they come from creating an abstract, purely self-contained building based on arbitrary axioms and pretty few, strict, logical operations built on top on the axioms. And mathematicians are not even agreeing on which axioms and operations are "correct".
Observation is never theory-free. Even when a person relies on sight, they have a theory of perception that they are implicitly or explicitly relying upon. When we use sensitive scientific instruments to, for instance, observe gravitational waves, that observation itself relies on many theories in order to even say what was observed.
So, no observation is never deductive proof of the existence of a thing: the underlying observational theory may be wrong; an instrument may have failed; our senses may be not be in correspondence with the world.
But an observation is support for the existence of the thing, especially where the theory underlying the observation model is also well-supported and is consistent with other modes of detecting the thing.
See generally Nora Mills Boyd and James Bogen, "Observation and Theory in Science", The Stanford Encyclopedia of Philosophy.
The present formulation of the title question can be improved:
Taken literally your question supposes the existence of “X” when is starts “Under what circumstances is the observation of X ...”.
But when assuming the existence of “X” then it makes no sense to question immediately after the existence of “X”.
A more precise wording seems to me: To observe data and to interpret them as a certain object.
Then your question reads:
Under which circumstances can we derive from observing the data “D” the existence of the object “X”?
Example: Under which circumstances do the observed data from the Large Hadron Collider at Cern prove the existence of the Higgs boson? See Higgs boson.
Apparently the methods are different, depending on the object whose existence is to be proved.
Sometimes it would count already as a big step to reach an agreement about the observed data, e.g., in your third example concerning God.
You ask:
Under what circumstances is the observation of X proof of the existence of X?
It depends on your definition of proof in accordance with your theory of evidence, however, because observations can be flawed in a variety of ways (also depending on your definition of observation), X is construed in scientific thinking as existing if observation is broadly held, peer-reviewed, repeatable empirical evidence open to revision on the introduction of contradictory evidence.
For instance, the ODNI acknowledged the existence of UAPs (UFOs) in the "UFO Report" in 2021. That is, the ODNI has made the claim that UAPs exist and routinely violate US protected airspace based on at least 144 credible observations. From the WP article:
On June 25, 2021, a nine-page preliminary assessment was issued.2 It states that the UAPTF focused on 144 observations of "unidentified aerial phenomena" by the U.S Armed Forces, mostly from U.S. Navy personnel, from 2004 to 2021... There were "11 reports of documented instances in which pilots reported near misses with a UAP."2 The report established five potential explanatory categories: airborne clutter, natural atmospheric phenomena, U.S. government or American industry developmental programs, foreign adversary systems and a catch-all "other" category.
So, UFOs exist according to the ODNI. Now, are they drones? Weather balloons? Swamp gas? Little green men from outer space? Among these questions, no ontological commitment of any sort was made by the report beyond establishing their existence. Thus in this way, wide-scale, credible observations form the backbone of a phenomena long before some phenomenon is complemented by a complex body of theory to explain it.
An observation is evidence but not proof, nor definition. There can be all kinds of errors involved (illusions), and even if not, the thing that was observed may not be the thing we think it is (e.g. an apple on a table might actually be a fake made of plastic).
So even if a thing had been observed, many more observations of that thing would be required to know it's attributes.
Arguably any god is too great, complex, whatever to ever be observed enough to know them and their properties. This apart from the problem that gods don't typically show their drivers license for us to know it's the same god as we observed last week.
In general, observation cannot be used to prove existence. There's a few related reasons for this:
The observation of X "is simply to be accepted as it gives itself out to be" according to Husserl's principle of all principles, as described by Heidegger in On Time and Being (1964), pages 62-3:
For Husserl, the Cartesian Meditations were not only the topic of the Parisian lectures in February, 1920. Rather, since the time following the Logical Investigations, their spirit accompanied the impassioned course of his philosophical investigations to the end. In its negative and also in its positive sense, the call "to the thing itself" determines the securing and development of method. It also determines the procedure of philosophy by means of which the matter itself can be demonstrated as a datum. For Husserl, "the principle of all principles" is first of all not a principle of content, but one of method. In his work published in 1913 [Ideas, tr. 1962], Husserl devoted a special section (section 24) to the determination of "the principle of all principles." "No conceivable theory can upset this principle," says Husserl (ibid., p. 44).
"The principle of all principles" reads:
that very primordial dator Intuition is a source of authority (Rechtsquelle) for knowledge, that whatever presents itself in "Intuition" in primordial form (as it were in its bodily reality), is simply to be accepted as it gives itself out to be, though only within the limits in which it then presents itself.
"The principle of all principles" contains the thesis of the precedence of method. This principle decides what matter alone can suffice for the method. "The principle of principles" requires reduction to absolute subjectivity as the matter of philosophy. The transcendental reduction to absolute subjectivity gives and secures the possibility of grounding the objectivity of all objects (the Being of this being) in its valid structure and consistency, that is, in its constitution, in and through subjectivity. Thus transcendental subjectivity proves to be "the sole absolute being" (Formal and Transcendental Logic, 1929, p. 240). At the same time, transcendental reduction as the method of "universal science" of the constitution of the Being of beings has the same mode of being as this absolute being, that is, the manner of the matter most native to philosophy. The method is not only directed toward the matter of philosophy. It does not just belong to the matter as a key belongs to a lock. Rather, it belongs to the matter because it is "the matter itself." If one wanted to ask: Where does "the principle of all principles" get its unshakable right, the answer would have to be: from transcendental subjectivity which is already presupposed as the matter of philosophy.
While what we think of the world is clearly informed by our observations, single "normal" observations by individuals without technical equipment like cameras etc. are not proof at all. There are just too many possibilities to err or be deceived, even if we assume a fairly straight-forward relationship between reality and our observations (which, of course, is not a given, if you have read Plato or Descartes or Philp K. Dick, or have seen the Matrix or Carpenter's They Live or Fight Club).
Even single carefully controlled experiments are not proof; usually, replication is required. It is, unfortunately, and for a multitude of reasons, not all of which are malevolent, not uncommon that replication attempts fail and the theories based on the first observation must be retracted. Successful reproduction of an experiment, by contrast, strengthens our confidence in the underlying theory.
This process leads in the direction of a Bayesian epistemology which is the formalization of something we all do unconsciously anyway: We have a lot of confidence in theories supported by widely reproduced experiments; observations which contradict widely accepted theories, by contrast, are met with an initial skepticism. That is why we would not think that the laws of gravity are wrong, even if a trustworthy friend tells us of their credible observation of, say, a levitating rock. We would, instead, ask them whether they were high or possibly dreaming or observed a mirage. A special detail of Bayesiansm is the Sagan standard: "Extraordinary claims need extraordinary evidence." It is, of course, possible that even well-established theories (like Newtonian physics) are utterly wrong; but it is rare. Additionally, as in this case, the old theory can be proven wrong only under conditions that exceed the ones during the original, widely reproduced observations.
Therefore, in the realm of science, the answer to your question "under what circumstances is the observation of X proof of the existence of X" is: When the observation is intersubjective (i.e., can be made by more than one person) and reproducible.
In other realms, such as religion, this is not a condition: Miracles are highly subjective and about the opposite of reproducible.
