Set List of Theorems

[MolecularMan14]

I believe that there should be a standard list of theorems (#1-#144(?). It is very confusing talking to someone with a different textbook, and referring to theorem #1, when yours is something like "all right angles are congruent", and his is, "If a radius is perpendicular to a chord, then it bisects a chord".

I propose that a universal set of numbered theorems should be established to reduce confusion and help unify mathmatics.

Anyone disagree or, agree?

[ydoaPs]

don't they have names?

[MolecularMan14]

they do, but it's much easier to refer to numbers instead of abbreviations. But i suppose if everything had a short abbreviation like VAT, that could work, but I really cant think of any abbreviations for some! lol

[Dapthar]

I believe that there should be a standard list of theorems (#1-#144(?).

[math joke] I'm sure Mathematicians will get to that right after they standardize notation. [/math joke]

[MolecularMan14]

true.

[MandrakeRoot]

Most important theorems have a name like "Hahn Banach Theorem" or "Banach-Steinhaus" etc...

The other theorems are theorems you will have to cite explicitely in talking with your fellow-mathematicians (In a discussion it is often clear what kind of result you need)

 

Mandrake

[matt grime]

Is it easier to catalogue the millions upon millions of theorems and have a handy reference manual we all can ue (how would we all use it?) than to simply state the required result? Do you know how many theorems there are in the maths world? The important ones have names that are familiar to most, and that is sufficient.

[pi_of_9]

It would be easier if there were standard numbered theorems in plane geometry

[Woxor]

Yeah, you might be able to somewhat standardize each individual field, but even that's kind of asking for a bigger mess than you started with. Voted "no."

[Dave]

Do you know how many theorems there are in the maths world?

 

I don't think anyone would really.

[matt grime]

Some axioms are numbered and reasonably well known. The axioms of (euclidean) geometry for instance. And the seperation properties of topological space (T1, T2, T3 etc). Not to mention that triangulated and other similar categories have reasaonabl well known numberings of axioms (AB1-4).

[MandrakeRoot]

Yeah but it is pretty annoying since you have also T3.5 etc...

Once you have numbered your properties and later you discover another you need to run to 3.5 etc... (I have no idea how that went historically though, but i imagine that people first talked commonly of T3 and T4, before defining T3.5)

 

I would say that numbering is not a good idea, naming could be though

 

Mandrake

[Dave]

Standard numbering of theorems isn't really necessary tbh. The most useful ones usually have names of some sort, and there's just too many to number.

[Ophiolite]

My maths never got past the calculus, but this discussion seems to be more about classification. Molecular Man's proposal seems logical.

 

Surely there would be real benefit in having theorems grouped in some heirarchical structure. Many (most, all?) people learn by consciously or sub-consciously grouping objects, concepts, etc. If there is an already established grouping, then that can facilitate both learning and application. I would have thought that would have been beneficial.

 

Certainly it would be resource intensive to develop and maintain, but the rewards could well justify it.

 

Finally I was perplexed by your remark that there are just to many theorems to number.

A mathematician running out of numbers.

Really, Dave, pull the other one.

[pulkit]

Cataloguing theorems makes little sense to me.

You'll have all sorts of trouble landing at a universle enumeration, and then lots of time wasted to correlate these numbers to theorems (in proofs of other results etc.)

[Dave]

The effort to catalogue all known theorems is a bit of a futile one, because there's just so many of them. You can look at places like Mathworld which is an extremely useful resource, but there's hundreds of thousands of proofs and other obscure theorems which aren't there. Not only that, but I don't think it would be particularly convenient for people to use in their own proofs of other/brand spanking new theorems and lemmas. Referring to them by name is much more convenient because people know the most important ones. You also have to remember that your average mathematician probably has a small library of mathematical textbooks in their office, and usually there'll be some kind of campus library. On top of that, theorems are almost always stated in proofs - I just don't see the point in trying to categorise what must be millions of theorems for no apparent reason.

[Woxor]

This runs into the same sort of problems I had when I tried to figure out a better way to express an organism's taxonomic position:

 

The problem with the current system is that it's non-cladistic (i.e. just because X and Y share a common ancestor Z doesn't mean that the smallest taxon containg X and Y will contain Z, and a taxon containing Z may not contain X or Y), and that it uses very few heirarchical levels of classification (species, genus, order, etc.). Ideally, there would be the same number of levels as there have been speciations in a given species' geneology.

 

Since speciation is never really thought of as happening between more than two species at the same time, you can look at it in a binary way: let each speciation that ever happened be represented by a single binary digit, and put all the digits in chronological order. Thus, 11...01 would be a species that diverged from species 01...00 at the very beginning, but 101...100 and 101...101 would have diverged the most recently. Thus, we have a perfect system for classifying organisms according to strict cladistics and without arbitrary heirarchical levels.

 

EXCEPT: The system is obviously ridiculously impractical. There would be millions if not billions of digits for each species, we would have no idea what most of the digit values should be, there's no account for weird bacterial rejoining and such, and there is little allowance for alteration of the system in light of new evidence -- say you find out that X speciated from Y rather than Z -- then every subsequent digit has to be rethought, there might be an additional number of digits ... you get the idea. The thing doesn't work, and I obviously came up with it only as a thought experiment and without anything like the kind of education I would need to come up with a real, practical system.

 

You run into the same kind of problems here. What if we want to add a new theorem into the system, as we would need to do very often? How do you number the theorems?

 

In chronological order according to when they were proved? That wouldn't be terribly useful, and we would have trouble verifying each historical case anyway.

 

In order of "importance"? How do you decide what's important, and how do you compare the negligible importance of the thousands and thousands of minor lemmas to one another?

 

In groups according to subject? This makes the most sense, but it pretty much eliminates the need for a universal numbering system, leaving each subject to be numbered independently, which is almost what we have now. But you're still left with the above problems on a smaller scale.

 

Furthermore, what about systems with different axioms? Do you really want Euclidean theorems mixed in with those from hyperbolic geometry? If not, how do you determine a way to keep them distinct? How do you denote that each theorem is only true under certain axioms?

 

Well, that's my take on it. I just see too many impracticalities.

[MandrakeRoot]

I clearly agree with you. Surely since lemmas are often only there too make the proof of a theorem more readable. So when two mathematicians would proof the same ("big") theorem they might end up with a different number of lemmas.

 

Mandrake

[Dave]

I like the idea of things like MathWorld, Wikipedia etc, since they allow people to maintain the list of theorems etc themselves and hence it's fairly easy to keep them up to date, and because of the entire peer review thing it's all pretty accurate.

[MolecularMan14]

How exactly do mathmeticians refer to theorems now? As abbreviations?

[matt grime]

By names if they have them. Examples were given:

 

Hahn-Banach

Fundamental Theorem of Calculus/Algebra/Arithmetic

3-epsilon or epsilon over 3

Morera's Theorem (for triangles)

Cauchy's Theorem (there are many of these)

Green's Theorem

Stoke's Theorem

 

and the list goes on.

 

Those unnamed are described, even if you're writing a thesis/paper and you number something as theorem 3.3.1 it is *BAD* mathematical presentation to, at a later point, simply say "this follows from 3.3.1", and infinitely preferable to say "this follows from 3.3.1 which showed that ...." Not everyone has a photographic memory.

[MandrakeRoot]

I agree that it is rather tiring to look back all the time what was again theorem 3.3.1 when reading a book.

 

Mandrake

[matt grime]

I once had to read a book where typically "the proof of Proposition 4.5.6" was:

"this follows from 2.1.2 and 2.4.5 and definition 1.2.1 and corollary 1.6.7"

 

some proofs even required external references to be checked, which is acceptable in papers, but unnecessary in textbooks if the proof is essential.

[Dave]

Most of my Algebra I lectures are like that - it makes it impossible to understand how he's proving things when you can't remember what he's already said.

[MandrakeRoot]

That is so annoying. You are right indeed that that is unacceptable from a textbook !

Also that is not really a proof, since you might as well say it before stating the theorem. Something like : The following result follows easily from theorem 2.1.2, ....

Stating the result as a pseudo proof doesnt really add anything to the book.

 

Books should be largely self contained in my opinion

 

Mandrake