Talk:Mathematical logic

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Archived discussion

[edit] Major reworking

Although this article is Top-priority, it's really barely more than a stub. I'm going to give this article the thorough reworking it needs to get to the quality it should be. Please feel free to help... and don't be surprised at the changes. — Carl (CBM · talk) 15:59, 26 November 2007 (UTC)

Excellent. Unfortunately I am going to be rather busy in real life for a few days, so I can't help much before the end of the week. Just a thought, as I suspect you might be planning to go into rather more detail than there is right now: I don't know if we currently have a definition of what a "logic" actually is. And I am not sure what it is, exactly, as I usually need only first order. But I would imagine that "language = logic + signature", and that deduction rules are related to a logic almost like structures to a signature. I think making clear the modular character of these concepts should really help to get a uniform terminology that makes sense for people from various branches of logic and from universal algebra – necessary for weeding out duplication. (I am not saying this should be part of this article – I haven't thought about it. It's just something I thought I would do some time, and which might be relevant here, perhaps even at an early stage.) In any case, thanks for doing this. I am sure I am going to learn something from the final result as well as from the way you go about it. --Hans Adler (talk) 16:55, 26 November 2007 (UTC)
My first goal is to expand the depth of historical information and to describe the subfields in more detail. I have found it remarkably difficult to find reliable sources that speculate on the nature of logic itself, or define mathematical logic. This is likely because of the culture within math logic of avoiding philosophical rambling. But I have some leads for history books that might prove useful. I expect that once I copy the new version here, other people will round out the coverage.
My goal is to end up with an article that can be put up for A-class review, which includes meeting the scientific citation guideline. — Carl (CBM · talk) 14:51, 29 November 2007 (UTC)
Very good. I just noticed that the French article (fr:Logique mathématique) is largely independent from this one and twice as long. Its introduction makes some interesting points (alas, without footnotes). If you can't easily read French I can put a quick translation here. (And I really like the two footers they are using for the mathematics and logic portals.) --Hans Adler 16:20, 30 November 2007 (UTC)

I copied my working draft here, so that other people can contribute. It is not by any means complete; many paragraphs are just sketches. I plan to add references for all the years in parentheses, just haven't typed them in yet.

The version on French wikipedia isn't bad. You can get google to translate it for you [1]. But I think it spends too much time on symbolic logic, which is only part of mathematical logic. — Carl (CBM · talk) 17:16, 30 November 2007 (UTC)

Hans, thanks for your help this afternoon. The article is, as everyone can see, still very bare-bones with very little exposition. I am adding references, and will eventually convert them to the {{citation}} template. Many of the sections could use rearranging if not complete rewriting. And the history from 1935 to 1950 is almost nonexistent. — Carl (CBM · talk) 21:10, 30 November 2007 (UTC)
Thank you for doing all this work. It's soon midnight for me, so I will probably print the article tomorrow morning to get a better overview. Believe it or not, I learned something very important about model theory from you today. — You noticed that we have contradictory information on the origins of the ε-δ definition of continuity. My impression from what I have seen on the web is that it was first used by Bolzano, then more rigorously by Cauchy (who actually made wrong claims because he didn't think of the problem of uniform convergence), and then rigorously by Weierstraß. The following article should have more precise and reliable information, but as usual I can't read it from home: Walter Felscher, Bolzano, Cauchy, Epsilon, Delta, 2000. --Hans Adler 23:32, 30 November 2007 (UTC)

Regarding the reference to Shoenfield, I think the inline citation should use the year of original publication, because this identifies the era in which the content was written. Later republications are important for purchasing the text but unless the context was changed they aren't going to be accurate about the content. For example, if a book from 1940 was republished in 1990, it's still not going to have information on results proved after 1940. — Carl (CBM · talk) 03:47, 1 December 2007 (UTC)

I agree that that's a problem. For me the balance was only slightly in favor of 2001, so I am not surprised you prefer 1967. I have changed the footnote to "A classic graduate text is the book by Shoenfield (2001), which first appeared in 1967."
An observation: The Citation tag has the year of the first edition as "origyear=1967", although I don't know if that's the correct use (since the current edition seems to be a reprint of the second edition from 1973, and I found no documentation on the intended purpose of origyear). When I started using these tags a few days ago, this would have resulted in something like [1967](2001), but that's no longer the case. --Hans Adler 09:56, 1 December 2007 (UTC)
I'm not sure about the citation tags. Mentioning the original year in prose is fine with me. One area where I am not strong is the early history of model theory, which is why it is currently just a single sentence about Tarski. You thoughts above about the nature of logic are relevant to the section on formal logic. — Carl (CBM · talk) 16:35, 1 December 2007 (UTC)
Well, I hoped that somebody had given a reasonable mathematical definition for what I would call a "logic", but I am beginning to suspect that I was wrong. "Logical systems" in Lindström's theorem come close to what I mean, and so do institutions. But they also include the model relation, which may not be needed for proof theory, and no inference rules. I was looking for a word just for a functor from signatures to languages, which could then be equipped with functorial model relations and functorial inference rules. — Yes, I imagined that you left the model theory bit for me. I am thinking about this.
I have done all the obvious or trivial changes immediately after proof-reading. Now I will soon start with a few things where I am not entirely sure what to do. You might want to have a look at them afterwards. --Hans Adler 18:31, 1 December 2007 (UTC)
I agree the easy changes are getting harder to find, which is grear. There are still a few gaps in the coverage, and I made a list tonight of several more primary sources to cite, but I think the coverage is filling out well. This is good, because the article is approaching the recommended maximum length.
As it stands, the article now has a lot of information about historical developments, and some information about milestones in particular fields. I think it's weak on analysis, criticism, and other "secondary source" material. So my next goal is to try to add a little more criticism (preferably with sources). I'm not used to writing "nontechnical" articles like this, so it's an experiment for me to find an acceptable presentation. — Carl (CBM · talk) 03:57, 3 December 2007 (UTC)
Yes, the article seems to be converging very well. I am glad you have embarked on this experiment, which really looks like it's going to be a great success. I am not sure that I want to know how many hours you spent on it. Did you have time for eating during the weekend? --Hans Adler 11:37, 3 December 2007 (UTC)

[edit] Cantor

I'm delighted to see the improvement in the article, and have just one suggestion.

Cantor first appears in this article in the discussion of the well-ordering principle. Shouldn't his contribution be mentioned earlier in the history section?

Rick Norwood 14:25, 3 December 2007 (UTC)

Yes, certainly, a synopsis of Cantor's work in set theory needs to be added to the 19th century section before the article is complete. Also Hermann Weyl's Das Kontinuum needs to be mentioned. Thanks for reading through the article and giving other suggestions, or editing it. — Carl (CBM · talk) 14:38, 3 December 2007 (UTC)

[edit] Vaught's Conjecture

I noticed that in the section "Model theory" it is written that Robin Knight refuted Vaught's Conjecture. However there was an error in Knight's 2002 construction and circa 2003-2004 there were attempts to patch it, but not everyone was satisfied with his arguments. Unless I've missed some new development here in the past few months or so, there's no consensus yet in the model theory community about the status of Vaught's Conjecture.

Skolemizer (talk) 06:41, 5 January 2008 (UTC)

I had never heard about this result before I read it in this article, but since it fell into a period in which I was not very active in mathematics I assumed I had just missed it and was going to read it. Now based on your warning I have asked an expert, who told me that there is in fact no consensus that the counterexample is correct. I think the proposed example should not be mentioned in this article, and I will remove it. --Hans Adler (talk) 00:18, 9 January 2008 (UTC)

This is very interesting. I'm no model theorist, and people I respect referred to Knight's result as a proof at some point, so I assumed they were correct. Wasn't Knight's result published? Thanks for correcting my error, in any case. I'm glad other people are watching these pages. Are there other things that could be added to the model theory sections? — Carl (CBM · talk) 00:15, 11 January 2008 (UTC)
So far as I know it was never published. There was a special session on this at the British Logic Colloquium 2002, so I would assume that's how it became well known to people outside stability theory. The example is extremely complicated, and so it probably took some time for people to make up their minds. His home page has a second draft from January 2003 and more corrections from November 2003. He also says there that he is working on a simplified example which he hopes to have complete "in June" (presumably June 2005, since the page was last changed in May 2005). So unfortunately it looks like this example is dead.
I am feeling a bit guilty that I haven't revised the section on model theory otherwise, as I meant (and promised) to do. I find it very hard to describe what I think is a large, heterogeneous subject in just a few words. --Hans Adler (talk) 11:00, 11 January 2008 (UTC)