Talk:Mathematical logic/archive 1

Questions and Quibbles

I've got quite a few complaints, and a couple of questions, about this article. I'll start here in the Talk page before upsetting anyone in my edits on the main page.

Although the layperson may think that mathematical logic is the logic of mathematics, the truth is rather that it more closely resembles the mathematics of logic

• Mathematical logic refers to both, and my impression is that overall, in scientific contexts more commonly the former. See, eg. the FOM list.

Mathematical logic was the name given by Peano to what is also known as symbolic logic. In essentials, it is still the logic of Aristotle, but from the point of view of notation it is written as a branch of abstract algebra.

• I haven't heard that the term "mathematical logic" is due to Peano. Do we have a source for this? (I'm not skeptical, just interested)
• The stuff Peano was interested in was FOL, which certainly was not the logic of Aristotle;
• Algebraic logic specifies a particular approach to mathematical logic that definitely is not the whole subject. There are also combinatorial, geometrical and computational approaches to logic, all of which need not be particularly algebraic. This last clause is misleading.

It was George Boole and then Augustus De Morgan, in the middle of the nineteenth century, who presented a systematic mathematical (of course non-quantitative) way of regarding logic.

• Boole's contribution is regarded as brilliantly original, but also widely considered a mess, and De Morgan's attempts to clean it up don't really sort out the mess. Boole did originate the idea of treating logic algebraically, but to call him systematic is not right, IMO, and he doesn't seem to have been a big influence on the real originators of mathematical logic, Frege, Peano, Russell, et al.

Some changes proposed:

• Much material missing from the History section, eg. invention of quantifier and function--argument notation, set theory, unmentioned pioneers (Frege, Cantor, Dedekind, Zermelo, Brouwer, Hilbert, Bernays, Weyl, Ackermann). I'd suggest settling on the time between Frege's Begriffschrift and the publication of Hilbert&Ackermann's work on FOL as the time when the foundations of modern mathematical logic were lain, earlier work is sort of "prehistory", later work, eg. Goedel, Gentzen, Tarski, Turing, is modern.
• Rework "Topics in Logic": separate out applications of ML from the core subject.
• Fundamental results is a nice idea, but it needs better structure. Maybe a timeline is the best idea? If not, by topic might work. The main problem is the huge difference in length of entries, but I think if we had a timeline in the history section, we could expand on them in a "core concepts" section.

Comments? ---- Charles Stewart 21:41, 25 Aug 2004 (UTC)

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Mathematical logic has a link to symbolic logic, which redirects to "Mathematical logic". Change the link or create a new page for symbolic logic? Lebob 00:49, 23 Dec 2004 (UTC)

Technical reference

I would like to move the technical reference somewhere else because:

• This page should be a general introduction to mathematical logic. It should be able to become a featured article. That means it should be accessible to people who don't have the technical background to read the technical reference.
• While first-order logic is central to the field, it is not the only formal logic that is considered. Putting a technical reference here makes it seem like all mathematical logic uses first order logic.

I propose moving the technical reference section to its own page. CMummert 14:46, 14 July 2006 (UTC)

I moved it to Talk:First-order logic/Technical reference for safe keeping. CMummert 12:21, 29 October 2006 (UTC)

Comments and to-do list

This is a High importance article, but it is only at Start class. I read through it and noted the following places where it could be improved. I am noting them here so that I can work on them and others can comment on them. I assume in these comments that this article is, like Geometry, aimed at a reader who may have very little formal training in mathematics; thus formal statements of theorems will have to move elsewhere, and (correct and verifiable) intuition is the key here.

• The lead and history sections have the same issues that Charles Stewart's comment from 2004 (above) discussed.
• The section Fields of mathematical logic needs significant expansion. Each of proof theory, model theory, recursion theory, and set theory could have its own short para. The part on the MSC is probably not interesting to a general reader.
• The relationship between mathematical logic and category theory is probably of interest to many readers.
• A section on Foundations of mathematics is promised by the intro, and needs to be added.
• The Fundamental results section is too terse for an untrained reader.
• The See also list is way too long. Most of those links could be integrated into the article body. CMummert 12:28, 29 October 2006 (UTC)

Math v Logic

Curious, since mathematics already uses logic, what part of logic does mathematics contribute which the art (not science)of logic does not already contain?

The history of logic distinguishes reasoning systems from logic, so math might contain a reasoning system independent of logic, but it is not logic! Therefore there is no such thing as mathematical logic. This would be a redundancy, not a tautology.

There are not dozens and dozens of types of logic; Analytic philosophers and mathematicians display an annoying disregard for English grammar and vocabulary, lazily preferring to designate any reasoning system as 'logic', commiting the heresy of dumbing down like any pedestrian.

I have yet to find a nexus between logic and math, so in my humble opinion, there is no overlap between the two.

Mathematical logic, also known as symbolic logic, is a separate field of mathematics, one that analyzes the systems of reasoning behind mathematics using symbols. Like Chalst said below, it is basically a synonym for formal logic. Lebob 00:53, 23 Dec 2004 (UTC)

I would hesitate to call Mathematical Logic a subfield of Mathematics. The logic of George Boole and C.S. Peirce may certainly be considered to be a subfield of mathematics as it is, in essence, Algebra limited to the number 1 and 0 (true and false). Other forms of mathematical logic such as that of Russell or Frege decreed that mathematics could be reduced to logic and not vice versa. So it is a bit of a grey area and maybe not something which should be stated in the first line of the article...? Aindriú Conroy 14:54 01 August 2006

Mathematical logic is generally considered a synonym for formal logic. Formal logic is taught mostly as a part of mathematics, which may answer part of your first question. If you could provide an example from the history of logic, where a reasoning system is distinguished from logic, that the second paragraph may make sense and may have some credibility. Unequivocally there are many types of logics, many different deductive systems, but the main ones used are sentential (sometimes called misleadingly propositional) or predicate logic, with minor differences (like how they treat identity operators). Predicate logic with identity is used in math courses as the basis for axiomatic set theory, for either Zermelo-Frankel or Godel-Von Neumann-Bernays variants, and no more logic is required than that. It would seem from an axiomatic point of view, mathematics is entirely dependent on and presupposes logic, however just like people learn how to speak before they ever take language courses, or play games before learning the rules, people can reason and count, before formally learning either logic or math. Logic is a very useful tool when doing mathematics, indispensible for proving mathematical theorems, doing any interesting work in mathematics. It would seem that set theory is indispensible to model theory, or the semantic part of formal logic. This leads to a circular situation, if you use model theory to prove theorems about logic, and logic to prove set theory theorems, and then set theory to prove model theoretic theorems, where does that leave you?

It's true. This page and several others may more properly be considered as more primarily under Logic rather than mathematics. As was stated earlier. Math is reduced to logic, not the other way around. Since Logic is more fundamental, the organization of the encyclopedia should center around those concepts. At the very least, pages of this sort (both math and logic topics) should have an intro that touches on both the logic and math perspective, and the following sections should generally be organized by opening with the logical applications, and then followed by the math applications. The wikipedia seems to be math-centric in this regard for most of these topics. That's because there are more mathematicians than logicians working on it. I'm not sure this was what people had in mind, but I think we should move in that general direction.

Gregbard 01:23, 3 July 2007 (UTC)

It isn't right that mathematics as a whole is "reduced to logic" - that is the discredited idea of the logicist program. Logic is part of the study of mathematics, just as mathematics is part of the study of physics, but mathematics is not reducible to logic any more than physics is reducible to mathematics. — Carl (CBM · talk) 02:40, 21 July 2007 (UTC)

redundant axioms in the fundations of math...??

Hello.

I'm writting a small paper on the completeness theorem on formal logics. While I was thinking about it, it came to me that while set theory and logics can be formalised in a small group of axioms, they are are also fundamentally interconnected and can not exist one without the other. I mean, check for instance the axiom list avaible on metamath: http://us.metamath.org/mpegif/mmset.html#axioms The fact is that while the axioms for formal set theory can not be written without the axioms of formal logics, for instance: for the axiom of extensionality we of course need the connectives and the formal implication. The opposite is also true and for the axioms of formal logics we need the axioms of set theory because these are in fact set based notions. So as I see it, the complete axiom list like presented in metamath and in many other places is basicly redundant and conceptually paradoxal... its a chicken or egg problem in the fundation of math... i dont think im wrong but please tell if i am or help in any way you can. Im going crazy over this. Uanbiing 13:53, 25 August 2007 (UTC)

Well, you're partly right and partly wrong. Let's start with "wrong": We certainly don't need the full power of set theory just to formalize first-order logic. It can be convenient to do it that way but it's not necessary; a small fragment of arithmetic suffices. (Showing this is the boring part of Gödel's proof of the incompleteness theorems).

But in a larger sense you're right -- foundationalism in general just doesn't work. You always get into an infinite regress. Luckily foundationalism is not the only good reason for studying mathematical logic or "foundations of mathematics". --Trovatore 08:07, 26 August 2007 (UTC)

One common way of looking at it is that when studying mathematical logic you must use some unformalized mathematics to do so, to avoid the infinite regress Trovatore mentioned. This unformalized math is said to be at the "meta" level. Of course, you don't want to assume a lot at the meta level, because the lack of formalization might be a cause for concern. As Trovatore said, it is well known to logicians (but not in elementary texts or the general math community) that it many cases it is possible to use only finitistic methods at the meta level. — Carl (CBM · talk) 12:38, 26 August 2007 (UTC)

Joseph Shoenfield's book "Mathematical logic" (1967) mentions this circle in the following way (footnote on p.9):

"An axiomatic treatment of set theory is given in Chapter 9, but only very elementary results will be needed before then."

However, note that the proof e.g. of the completeness theorem for first-order logic relies on the axiom of choice and some of its implications. Consequently, on p.47, which is still within the part using "unformalized" set theory as a meta-theory for first-order logic, Shoenfield says:

"Next, we need a method for obtaining complete theories. For this, we shall need a result from set theory, which we state without proof. (For an outline of the proof, see Problem 4 of chapter 9)."

But his chapter 9 is exactly about the formalization of set theory within first-order logic! That is, Shoenfield makes a forward reference to a theorem in a specific first-order theory while developing the meta theory of first-order logic itself! OK, admittedly, the completeness theorem is not needed to explain syntax and semantics of first-order logic, but already for explaining the notion of a model, you need some good portion of set theory.--Tillmo 11:38, 6 September 2007 (UTC)

Thanks for your answers, really appreciated (mentioning Godel's proof was particularly helpful, had never thought of it that way). Still, I would like to discuss a bit more this foundationalism issue. My main problem is exactly the use of meta language and informal math to avoid the infinite regress, is there really no way to deal without it? Do you know where can I find more on this subject? Thanks again 89.110.193.137 00:23, 31 August 2007 (UTC)

See e.g. the foundation of mathematics mailing list. It is a closed mailing list, but if you specify your interest, you surely will be let in. --Tillmo 11:38, 6 September 2007 (UTC)

Request for expansion

I would like to suggest that this article be expanded such that members of the general public who take the time to read it would then be equipped to understand the Wikipedia articles on the various aspects of this topic. 68.49.208.76 06:14, 6 September 2007 (UTC)

You can help. It is hard for a professional mathematician to know what needs to be said to help the layperson understand. Point out any aspects of the article that are unclear to you, or that need to be expanded. Rick Norwood 17:39, 6 September 2007 (UTC)

With respect to this article per 68.49's comments, a comparison of the relative treatment of sections suggests that the Formal Logic section may need expansion. While the brief treatment now given is sufficient for persons already familiar with the differences between orders of logic, the vast majority of users may find it difficult to parse and therefore understand the 3-sentence section as too sparse to grasp effectively. Hotfeba 23:06, 8 September 2007 (UTC)