Category talk:Mathematical logic stubs

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See Wikipedia:Stub types for deletion/Log/2007/July regarding an archived discussion on a closed recommendation to delete or rename this stub category, resulting in a decision to keep this stub and also have a general logic stub.

A member of the WikiProject Logic has raised significant points on the use of terminology by philisophers and non-mathematical logicians compared the usage of the same terms by mathematical logicians. In Gregbard's analysis, the sentence arose: "The notion expressed more than once here is that 'mathematical logic' isn't logic."

• I will assume ordinarily reasonable people will generally agree that the above statement implies "logic isn't mathematical logic." In some cases it does have this meaning, but in some cases it does not. Acknowledging this point should be no problem for most if not all philosophical and mathematical logicians.

• I will further assume that there are two points of view here; a mathematical point of view that uses the terms "logic" and "mathematical logic" as equivalent out of our implicit desire to represent concepts as the shortest written form possible to still maintain universal recognition among mathematicians, and a separate point of view that uses "logic" as equivalent to the phrase "all instances of forms of logic" or "that which is always and generally applicable as logic."

• By the precise term "point of view", I refer to that which is acceptable in given socially recognized and socially valid spheres and NOT in any way to be construed in a class that would be a violation of WP:NPOV; otherwise there would exist a glaring contradiction of the concept that the Wikipedia was involved in the censorship of ideas at such a fundamental level as to render the Wikipedia worthless for open discussion. As it is not our purpose to render the Wikipedia as worthless, the view of the Wikipedia must allow for the existence of those two legitimate points of view as implicit in the wiki structure.

• The points of view appear to be separate by their separate graphical representations as sets. Mathematics has a restrictive view on what constitutes mathematical logic, and with apologies to everyone else, when a mathematician uses the term "logic", he or she is referring to "that which is mathematical logic", and because mathematicians speak in those terms, we should be forgiven for stating anything like "philosophical logic is not logical" when we really mean that "philosophical logic permits arguments to be constructed in ways that have been deprecated in mathematical logic" as a statement of fact between mathematicians. These mathematicians nearly all accept this statement of fact as true and therefore openly communicate to and with other mathematicians in this way. This is both sociologically real and valid in its reality. Computer science can be and is mathematically seen as an extension of mathematical logic because first-order formal logics can be used to conveniently express concepts of computability, complexity, and other measures of algorithmic behavior and performance, and so here we are on the Internet. For computer science types who approach that science from a mathematical standpoint, philosophical logic is more art than science in a fuzzy logic sort of way, and it is algorithmically difficult and many times impossible to encode philosophically valid arguments into fragments of functional programs. This statement means that mathematicians either know, are aware of, or are capable of discovering in the literature, proofs of that difficulty or impossibility where the proofs have mathematical rigor and precision in logic. A value judgement that some things just are not worth even contemplating arises when the impossibility of some entire class of concepts is a fact known with mathematical certainty.

None of this detracts from the philosophical validity of arguments allowed in philosophical logic, which is sociologically valid for many more people than there are of mathematical logicians, but hopefully it will explain the entrenchment of sides in the above-linked discussion in a place, the Wikipedia, where more of those mathematical logicians might tend to congregate and edit. (This is WP:OR and should be deleted as such except for the facts that (1) it is documented in "stable" do-not-edit form available by hypertext transfer protocol (HTTP) at the Wikipedia, at the above link, (2) it is true, and (3) it is an attempt to defuse a situation wherein the presence of two valid viewpoints has a potential for personally divisive conduct among editors. If this is deleted, it is already available elsewhere at academia.wikia.com in the Wiki Journal or elsewhere else. The necessity for deletion points to a Wikipedia policy paradox wherein it must be true that WP:FAIL for a living system with inter-communicating (Internet) components.) Hotfeba 02:28, 25 July 2007 (UTC)

I've tried to read the above a couple of times, and to be honest, I'm still not sure where you're coming from. Let me address a couple of points, though:

• The term mathematical logic, as used by mathematicians, means the union of proof theory, recursion theory, model theory, and set theory (including descriptive set theory), and arguably also includes type theory, category theory, universal algebra, and topos theory. These fields have some hard-to-describe quality in common, along with common historical roots -- but for the most part, with the possible exception of proof theory, they are not "logic" in the sense of "the science of making valid inferences".
• When mathematical logicians use the term "logic" unmodified, they may or may not mean "mathematical logic". It depends on context. I once heard a great set theorist say he "wasn't good at logic". The context was a colloquium where I asked him if it wasn't obvious that something he had spent some effort explaining was true, just on the grounds of some variable appearing free in some statement (or something like that).
• When mathematical logicians want to be unambiguous about "logic" in the sense of "the science of making valid inferences", they call it "philosophical logic".

Hope this clarifies how I see the issues. I think this is a bit different from the way you thought I thought about them, but frankly it's a bit hard to tell. --Trovatore 21:45, 25 July 2007 (UTC)

I agree with everything you pointed out. A formal system is defined with formal terms, a list of symbols, another list of axioms, some rules for making deductions in the system that yield theorems, and extensions are made to that formal system that eventually provide us with some model that has an interpretation that we can agree is a set theory, or a theory of recursion, or some other particular theory that has some qualities approaching consistency and completeness, and we may describe this as a union of theories when taken as a whole, even if that union is arrived at by some process of logical construction in which some ideas are more fundamental than (and deductively must be arrived at before) others. Generally, when informed mathematicians communicate with each other about those matters, there seems to be no problem except where a new, not-yet-accepted theory needs to be reviewed for any errors that may nullify its anticipated or intuitive results, another deductively logical exercise. Other problems arise when a theory is not complete or has some aspect of inconsistency; that is when mathematical minds go to work to find some similar theory that is "better" (more likely to be proved complete or consistent or both). Of course, these words suffer from me not regressing to offer fully-explicated precise definitions such as in the manner of a Russell, but then my words are intended for a more general audience than than that limited only to mathematicians (I would not expect the casual reader of Wikipedia articles or a writer interested only in pop fancruft to be a graduate student or professor of mathematics, which is why I pointed out earlier that there are too few Wikipedia editors willing to edit or merely review all of the stubs in the category), or I would have just copied some mathematical logic text verbatim from first page to last to be thorough. The curious without a math degree can spend a year or more with Mendelson's introductory work (a source that was used in an earlier edition to write the Encyclopaedia Britannica article) if one wishes to see a demonstration of a first-order formal system for logical deduction that is developed and extended into number theory, set theory, recursion and the like; even as an introduction, this is a book that does not "read" like A Series of Unfortunate Events or some other work of fiction.

There is a mathematical point of view of deductive logic which as we may say is not unambiguous unless (1) mathematical logicians want to be unambiguous and (2) we make the effort to fully explain things in unambiguous terms, so that modestly-educated people or even a philosopher can understand what is being discussed. A potential for endless argument that will not advance the goal of completing articles for the Wikipedia occurs when mathematical logicians are confronted with philosophical arguments which the mathematical mind refuses to buy into, a matter of misunderstanding. So, of course everything depends on context, and the context of the Wikipedia determines that 98%+ of readers simply won't care if there is or isn't a mathematical logic stub type, or that to some philosophers in the context, it was silly and territorial to have created the stub category in the first place. But of the ones to whom it does matter -- and it does matter to me -- there will be "no changing of religion" just because a philosopher does not see the validity of mathematically precise formal systems with mathematically precise definitions that mathematical logicians, computer scientists, and other readers of the literature in those fields feel comfortable with for proving theorems and exploring the completeness and consistency of theories. Both points of view are sociologically valid, and there is much conflict to be avoided when each side recognizes the validity of the other in its own sphere of communication where they happen to intersect here on the Internet. Hotfeba 03:57, 26 July 2007 (UTC)

Hotfeba, please, sometimes less is more, OK? If you'd state your claims more concisely I might have some idea what you're talking about.

The idea vaguely trickles through to me that you think my objection, and that of my party, to considering "mathematical logic" to be part of "logic", is that "logic" is unformalized. As Prufrock might have said, that is not what I meant at all; that is not it, at all. The objection is rather that mathematical logic simply is not logic; it's mathematics. It uses logic, of course, but so does almost all intellectual endeavor; if you cast that wide a net you might as well say that biology is logic.

And by the way, the sort of logic that we use, in mathematical logic, is mostly unformalized. We study formalized logic, but we don't really use it; it's just not practical nor adapted to human thought. --Trovatore 04:18, 26 July 2007 (UTC)

And that perhaps is the great inside joke of mathematics. We want others to believe that we are logical and reasoned, but behind all of their awe, conjecture, and puzzlement regarding our unified theories and theorems... behind all of that, we're just guessing, and our guesses stand until a philosopher has the simple courage to challenge and defeat them. In the context of the vast majority of people who use the Wikipedia, this is surely concise enough for them in reaching a concensus on the sociology of an in-group that uses specialized vocabulary as a gatekeeper to admission.

My apologies to you, but as a former mensan, I happen to see mathematical logic constructs everywhere I look every single day... paraphrasing Newton, everything is a sum. In my philosophical ignorance, I imagine that the Wikipedia could be the sum of those sums...

By the way, did you ever notice how many of our informalized guesses in mathematics never see the light of day until somebody else decades later does the heavy lifting and provides a proof that invariably involves the usage of mathematical logic? As for your example that biology is logic, I concur and have already made a WikiProject Logic proposal to categorize most if not all areas of study as applications of logic. Hotfeba 17:33, 26 July 2007 (UTC)

If you think you're agreeing with me, then one of us is missing what the other one is saying (or maybe both of us are). The biology thing was a reductio ad absurdum, not a proposal, for Pete's sake. As for your point about "informalized guesses", you're going to have to give me an example there. --Trovatore 20:32, 26 July 2007 (UTC)