Talk:Independence (mathematical logic)

merge

When I wrote this article, I was unaware of the existence of the logical independence article (I think I had seen it before; I just forgot). That article is more complete, but this one has the better name, so I put the merge tags from logical independence to independence (mathematical logic). The usage notes in independence (mathematical logic) about the word "undecidable" and about the sense of "independent" meaning simply "not proved" (rather than "not proved nor refuted") should in any case be maintained. --Trovatore 05:15, 24 January 2006 (UTC)

Instead of a merge, I had what I had written at independence (mathematical logic) deleted (after saving a copy locally), and logical independence was then moved here. Then I selectively reincorporated my text from the previous independence (mathematical logic) into the current article. --Trovatore 18:01, 16 February 2006 (UTC)

latest edits

Here are my objections to DesolateReality's latest edits:

1. "Independent of a maximally consistent body of propositions". The only thing I can understand by "maximally consistent" is that any proper extension is inconsistent. But then nothing can be independent of that! (In the sense of "independent of and consistent with"; see next objection.)
2. The latest edits have as the primary meaning of "independent" the sense of "cannot be proved" rather than "can neither be proved nor refuted", and refers to the second sense as "informal". I kind of doubt that this really reflects general usage.
3. The claim The existence of independent statements is of philosophical interest. It puts into question Hilbert's program, casting doubt as to whether a complete formalism of mathematics is possible. is not really accurate; the fact that a theory fails to prove or refute something may just mean that you haven't made the theory strong enough. The argument against Hilbert's program has to do with the necessary incompleteness of any theory satisfying certain hypotheses. Without quantifying over theories, the claim is severely misleading. (A related problem is that no sentence is "independent" full stop; it can only be independent of some specified theory.)
4. The "standard technique" section is mostly accurate but does not strike me as being written in encyclopedic style. --Trovatore 21:45, 10 June 2007 (UTC)

Thank you, Trovatore, for the revert. I agree with you generally about your objections. Here are my specific replies:

1. I wanted some way to refer to the term "theory" for a general audience. I agree that "maximally consistent" doesn't make sense.
2. I agree.
3. My intention here is to bring out why logicians are interested in independence proofs. Until I find a better way to phrase this, I agree with the revert.
4. I think later editions of this article should try to incorporate the observation that independence of σ from T is usually proven by exhibiting a model of T + ¬σ. Such a method of proof is usually the first thing taught to logic students immediately after the notion of independence is explained.--DesolateReality 04:43, 11 June 2007 (UTC)

publications?

Are there any publications that elaborate on this topic further? Can they be listed? --Farleyknight (talk) 02:31, 25 December 2008 (UTC)

"Undecidable" in the sense of decision problems

I undid edit, which had the following edit summary:

This is in fact a specific application of the meaning of "decidable" as applied in a decision problem, as a sentence is independent if and only if its truth value can be decided by an algorithm that enumerates all proofs.

That's not the same thing at all. The IP is arguing that a statement independent of a formal theory is one that is not decided by a particular program, and he/she is right about that. But an undecidable problem in the sense of decision problems is one that cannot be decided by any program whatsoever. There is no such thing as a problem with only one (or finitely many) instances that is undecidable in this sense. You can always write a program that will just say "yes" or "no" unconditionally, and that "decides" the problem for that single instance. --Trovatore (talk) 19:46, 9 January 2015 (UTC)

Please revert to old title "logical Independence"

In the last ten years Logical Independence has become understood as having crucial significance and affect in Physics

References

In Physics and in other areas of Mathematics, the term Independence has its own meaning other than the logical one. Physicists cannot use the term proposed by this article's title because in Physics, that reference would be at least ambiguous, but in fact misleading.

Please revert tto the old title. Then I can place a link from the article at Quantum indeterminacy, to this page. — Preceding unsigned comment added by User:Stephiefaulkner (talk • contribs)

You can link to any page. In the wikitext editor, you could use this: [[Independence (mathematical logic)|logical independence]] to produce logical independence. However, since a WP:Redirect has been created, you only need to type [[logical independence]] and let the software figure it out.

In the visual editor, type and select the text logical independence on the page, click the link button in the toolbar, and search for the page you want to link to. WhatamIdoing (talk) 21:04, 17 June 2016 (UTC)

In 2006, I appear to have thought that independence (mathematical logic) was the better title. In 2016, I don't see any huge advantage for either over the other.

But I don't really understand the complaint. Is it thought that, because it's mathematical logic, it can't be relevant to physics? That doesn't seem to follow. I haven't looked into the new material, but I haven't seen anything that contradicts the claim that the physics papers are using "independent" in a sense that comes from mathematical logic. --Trovatore (talk) 21:15, 17 June 2016 (UTC)

Will anyone object if I change the title of this page back to "Logical Independence"? 2nd December 2020

I come back to the same issue as before; the question of the name of this subject; and therefore the title of this page. Where mathematical Logic is being applied to Physical Theory, this is known as Logical Independence, not Independence (mathematical logic). Are researchers in this area, talking about this, supposed to tag the paretheses "(mathematical logic)" continually? When I began to research and write about this I followed up on what had gone before; and that was "Logical Independence". Go through the litrerature and you will see. In the world of Physics, the term "Independence" does not carry the same meaning. You may argue that it does have the same meaning, but it certainly does not invoke the the weight carried by the discipline of Mathematical Logic. In the past you say:

In 2006, I appear to have thought that independence (mathematical logic) was the better title. In 2016, I don't see any huge advantage for either over the other. But I don't really understand the complaint. Is it thought that, because it's mathematical logic, it can't be relevant to physics? That doesn't seem to follow. I haven't looked into the new material, but I haven't seen anything that contradicts the claim that the physics papers are using "independent" in a sense that comes from mathematical logic. --Trovatore (talk) 21:15, 17 June 2016 (UTC)

  In effect you say you see no evidence of an advantage in what I propose, after having not looked for any such evidence. Try doing a google search for the term "independence" and then another for the term "logical independence".  Take a look at this paper: Logical independence in quantum logic https://link.springer.com/article/10.1007/BF02059228

I think I would somewhat object, yes. This article is about a specific formal concept from mathematical logic. "Logical independence" is much vaguer. I am also concerned about the reasons for the proposed change; it seems to relate to a line of inquiry in physics in which one of the key investigators has a name similar to your user name. It is also not clear to me that the notion in that physics paper is exactly the same as absence of a formal proof in first-order logic. --Trovatore (talk) 18:41, 2 December 2020 (UTC)

I agree with Trovatore. From the abstract at Paterek.Kofler.Prevedel.2010 it seems they describe a kind of quantum computing mechanism that implements a theorem prover that is able to check independence in the original mathematical sense. So I don't see a deviating meaning in physics. - Jochen Burghardt (talk) 12:04, 3 December 2020 (UTC)

OK. Would you object to the unlinking of the term logical independence from being re-directed to this page? It was originally redirected here because the term logical independence was said to be synonymous with independence (mathematical logic). But now it turns out these are not synonymous. I quote: "This article is about a specific formal concept from mathematical logic. "Logical independence" is much vaguer."

The two terms cannot be both synonymous AND much vaguer. Furthermore; if logical independence is not what this page is talking about, then since the section on physical theory IS talking about "Logical independence" in physics; IE the Natural Universe; then if logical independence is not the same as independence, then that section should be removed from here, along with the references it cites. And then the section on physical theory can be placed where it belongs --- in a page about logical independence. By the way; If you read the Paterek et al paper, titled Logical Independence and Quantum Indeterminacy, you will find that its formalism is about functions that can neither prove nor disprove one another. And further; the terminology, logical independence, in that article, was adopted to replace the term mathematical undecidability used in an earlier draft, after discussions with CJ Chaitin, according to the acknowledgements. If you read the paper by Szekely, you will find it is a rigorous piece of first-order logic, and you will find he uses the term logical independence. What he is doing is logic; not physics. But he's talking about physics.

So in short; what I'm asking is: undo the redirection to this page, so that the application of mathematical logic, to the physical sciences, can be properly recorded on the Wikipedia AND understood in the language used by physicists working in this area. — Preceding unsigned comment added by Stephiefaulkner (talk • contribs) 13:20, 3 December 2020 (UTC)

It seems to me there are three possible cases:

• You think the quantum indeterminacy article, or perhaps other related articles, are using "logical independence" in some natural-language sense distinct from the sense of this article, but that this natural-language sense is not appropriate for its own article (or you just don't want to bother right now). In that case your course is clear. Just find the link [[logical independence]] in whatever article you're worried about, and unlink it, leaving the text as simply logical independence without the [[...]].
• You think there's a distinct sense of "logical independence" that you're talking about, and you can find reliable sources that discuss it and establish its notability. In that case, go ahead and write the logical independence (physics) article, or whatever it should be called, and link to that one.
• You think that the sense being used in "quantum indeterminacy" is in fact the one used in this article. In that case you don't have to do anything at all. Just leave the link as [[logical independence]], it redirects here, and everything is fine. In this third case I don't see what the objection to the current state of affairs is in the first place.

Any possibility I've missed? --Trovatore (talk) 18:44, 3 December 2020 (UTC)

I thank you for those options. My view is the third one: The term logical independence, used by physicists, is identical in meaning to the meaning in this page. However, just leaving the link in tact, redirecting here, is not "everything is fine"; because, in that state the page is not optimally helpful; and it should be. — Preceding unsigned comment added by 84.167.223.148 (talk) 19:23, 3 December 2020 (UTC)

In what way is it "not optimally helpful"? --Trovatore (talk) 19:28, 3 December 2020 (UTC)