Whence the idea that proof of nonexistence is impossible?

from this thread.

Didn’t Euclid prove that there does not exist a largest prime number? Didn’t the Pythagoreans prove that there does not exist a rational number x such that x[sup]2[/sup] = 2? Didn’t Cantor prove that there does not exist a bijection between the integers and the reals?

Thus my question: whence came the idea that proof of nonexistence is impossible?

Mathematical proofs notwithstanding, this guy gives a pretty good explanation (or at least as good as I can find at midnight).

This statement,

from this thread in GD, seems to be at odds with

from your cite.

Michael Roy seems to be saying that an entity’s nonexistence prohibits us from knowing its properties, while ultrafilter’s argument allows us to postulate the hypothetical entity’s properties first, and then do one of two things:

[li]Find them contradictory and conclude that the hypothetical entity cannot exist.[/li][li]Find no contradictions among them and conclude that the hypothetical entity might exist.[/li][/ul]

I find ultrafilter’s argument more persuasive.

What if a hypothetical entity has a property which contradicts something which is known to be true? Like a country larger than Asia? Can’t you then conclude that the hypothetical entity cannot exist?

This criterion might be a useful addition to ultrafilter’s argument. These two criteria will catch not only internal contradictions, but also external contradictions, that is, properties of the hypothetical entity that disagree with facts that are known to be true. Some atheists make use of the latter when claiming that the existence of evil proves the impossibility of an omniscient, omnipotent, benevolent god.

Good question, and good examples.

I was recently struck by a corresponding conumdrum (that might be hijack, but is so close in spirit it seems fair to raise).

Whence the idea that circularity in definitions is necessarily a bad thing?



Seems to me a perfectly good definition of X is given, in terms of X.

Is the x in your definition really well-defined? It seems that there are two possible real values of x that would satisfy your equation. Perhaps you meant to define x as sqrt( x + 1 ) instead.

However, defining a variable x in terms of x is often done in programming, especially loops. For example, the thread about calculating square roots by hand suggested this algorithm. Given b > 0, to find x such that x[sup]2[/sup] = b, first make a guess for x and then make subsequent refinements of the guess by redefining

x = 0.5 (x + b/x)

This algorithm is a perfectly valid use of circular definitions. An invalid use of circular definitions is the dictionary that has “large” as the definition of “big” and “big” as the definition of “large”. Without prior knowledge of the meanings of the two words, the reader can only infer that they are synonymous. Such a dictionary thus fails in its task of conveying the meaning of each word.

Dictionaries! They keep me awake at night, worrying about them.

First an anecdote, browsing a dictionary once (sad, I know) I stumbled upon the word narwhal, not a word I knew so I looked eagerly to the definition, a form of monodon, it said, not as helpfully as I expected.

I skimmed to monodon, fingers-crossed for a more forthcoming definition. And my hopes were well-founded, it told me a monodon was a horned whale, and I was satisfied.

BUT, what if I hadn’t known what a whale was, I would have had to look that up too, it would have told me that the whale was a large equatic mammal, which is fine if you know what a mammal is, but if you don’t, you start the journey again.

And so on, but not ad infinitum – because the dictionary is finite, one must inevitably come across a self-defining cycle of words (I hope this is clear).

So, in what way do dictionaries “define” anything? Don’t they ultimately define everything circularly, and hence, invalidly?

The answer to these questions is what irks my insomniac mind.

An essay by Richard Carrier, “Proving a Negative”, which some of you may find interesting:

As for the OP, my guess is that the “proof of nonexistence is impossible” myth came from selective use of the fact that nothing can be proven with complete certainty. The reason it’s so popular is that it gives people an easy way to avoid difficult issues.

Possibly along similar lines is something I remember from reading Carl Sagan’s works. Whether it originated with Sagan, I won’t speculate, but he said:

Absence of evidence is not evidence of absence

and left it pretty much as a dictum.

The context was UFO’s and other phenomena of that ilk.

(other similar statements from Sagan can be found at this link and a discussion of same at this thread.)

This reminds me of something I heard on the radio a few weeks ago . Somebody had orderd some goods via mail order and they had not arrived . When he complained to the company he was asked to prove to them that he had not received them . How on earth can you prove a negative such as this ?.

Easy, just return to them all the packages they sent to you (i.e. none), and then send them a follow up letter to confirm that they received nothing.


Think about it. If you claim to have a big green Martian sitting in your bathtub, then a total absence of evidence (on close inspection of your bathtub) for the presence of a big green Martian is proof that you don’t have a big green Martian sitting in your bathtub.

(Example taken from Richard Carrier’s essay, above.)

Staale Nordlie, I think your example of the Martian in the tub is one variation of this dictum. But other variations might include something like:

I didn’t know there was a law against killing puppies on this street, therefore you can’t convict me on it. (Ignorance of the law is no excuse, sort of thing)

Where it gets tricky is when your argument for the existence of something cannot be backed up with some reasonable evidence, it is not therefore a certainty that the thing whose existence you’re arguing for does not in fact exist. This falls into the atheist/believer argument’s realm, I suspect.

Fair enough, but if I claim to have an invisible pink unicorn sitting in my bathtub, you might search my bathtub carefully and find zero evidence of its existence (absence of evidence), but also zero evidence of its nonexistence (evidence of absence). You cannot conclude that an IPU is not sitting in my bathtub.

I have to nitpick this. It’s a perfectly valid use of circular redefinitions. As you note, you start by making an initial guess for x. Something like x = 10, I suppose. Something necessarily not circular. However, if you use x before giving it a value, even if it’s in the line where you give it a value, a good compiler will give you a warning.

Now, a similar thing in mathematics is recursive definitions. It’s a little hard to explain why these are not circular, because they look like circular definitions, but it’s a completely different thing:

As I pointed out earlier, you cannot be absolutely certain about anything. All you have to go by is probabilities.

In order to “know” something you have to reject an infinite number of potential claims with no evidence to support them. If there is no evidence to support a claim then the only reasonable “action” (“inaction”?) is to treat that claim as false.

(Depending on how you define “evidence”.)

In this case I don’t have to search your bathtub at all to cunclude that an IPU is not sitting in your bathtub. A unicorn can’t be both “pink” and “invisible” at the same time. It is absurd. It could be that I am wrong - maybe the reasoning which leads me to reject the IPU is faulty or irrelevant. But then we’re back to the same fact - ultimately you cannot prove anything in the absolute sense.

It seems to me that this proof of nonexistence only works because you’re trying to disprove something that, if it existed, would leave recognizable evidence.
If, on the other hand, you’re trying to disprove (or prove) the existence of something like a spiritual being that set the universe in motion and then left it alone, or a spiritual being that only intervenes in the world by inspiring/motivating humans to act in various ways, or free will, or other things the evidence of which would be nonexistent or difficult to pin down, then you’re out of luck.

In fact, there might be a Martian in my bathtub that is hiding itself by the means of sufficiently advanced technology/magic. Alien mind control rays. Gotta wear a tin foil hat :D.

Here’s how jjimm explains the IPU:

See the OP for examples of facts that were proved in the absolute sense.