Wikipedia:Reference desk/Archives/Mathematics/2006 September 9

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September 9

Most complex strand of Math

generally speaking, what is the most complex and hardest to master strand of mathematics in the world? Jamesino 17:15, 9 September 2006 (UTC)

That is a fairly meaningless question, as 'hardest' is impossible to define objectively. If you want something that is pretty tough to get to grips with, I would recommend trying to understand Mumford's book "Abelian Varieties". Madmath789 18:28, 9 September 2006 (UTC)

The most complex is complex analysis and the hardest is solid geometry. Fredrik Johansson 19:16, 9 September 2006 (UTC)

Lol. – b_jonas 19:24, 9 September 2006 (UTC)

Well, there is the list of Unsolved problems in mathematics, if you want problems that are so hard that nobody in the world knows how to solve them (yet). They're by no means the only ones, but they are some of the most famous because of their importance to other aspects of mathematics and, in some cases, because they have been unsolved for many years after they were first proposed. --Robert Merkel 02:27, 10 September 2006 (UTC)

Opinions are all mine:

Category theory is difficult only because it has abstracted away its subject metter. However, there are mathematicians who understand this material very well.

Number theory is difficult because most of your physical intuition about continuously changing variables is completely unhelpful in that field. However, there are mathematicians who understand this material very well.

So, what are my fields of interest? Category theory and number theory. WHat was I thinking? -- 66.103.112.140 01:22, 12 September 2006 (UTC)

The Langlands program, as I understand it, unifies a number of disparate fields of mathematics, and thus requires an especially wide background to tackle. It also has resulted in a number of awards for those who have made progress in it, which could be taken as a metric of difficulty (similar to the Fields Medals awarded to Smale, Freedman, and Perelman). Tesseran 22:43, 13 September 2006 (UTC)

prime interest

how can i find we the prime interest rate gose up and down in the federal reserve bank and why the banks and the lenders and the investors look at this closely —The preceding unsigned comment was added by 64.107.153.70 (talk • contribs) .

I would start at the Wikipedia article Prime rate, which includes descriptions of indicies, and links to data on the prime rate. --TeaDrinker 19:26, 9 September 2006 (UTC)

Aliquot Sequences

I just read in a book that

No more chains were discovered until 1969 when Henri Cohen checked all aliquot sequences starting under 60,000,000 and found seven chains of four links each. No chain of three links--nicknamed a "crowd"!--has ever been found, though no one has a reason why they should not exist.

It occurred to me - if it's true they don't exist, does there have to be a good reason for it? Is it possible that for some things that happen in number theory, the best explanation is "it just turned out that way"? Or does it seem more reasonable to think that anything can be explained in an interesting way from the basic characteristics of the numbers involved? Black Carrot 20:38, 9 September 2006 (UTC)

I am not familiar with the terminology in your question. So, I looked up Aliquot sequence in Wikipedia and gained some knowledge. But I found no reference to chain, nor in Wiktionary. Perhaps you could fill out the Aliquot sequence article. --MathMan64 21:22, 9 September 2006 (UTC)

"It just turned out that way" is not an explanation. But, maybe there is no explanation. It also depemds on what you consider an explanation. For Goldbach's conjecture and the Collatz problem there are "explanations" based on treating numbers as if they are random variables. A true explanation implies a proof. But is a proof, conversely, always an explanation? --Lambiam^Talk 22:04, 9 September 2006 (UTC)

A "chain" (his word, not mine) is in our article a set of sociable numbers. There are no known sets of three sociable numbers, apparently. That doesn't matter though, it was just an attempt to give context. Black Carrot 05:11, 10 September 2006 (UTC)

I suppose I should have fleshed my question out more in the first place. Here's the thing. I've been putting a lot of work into studying number theory (from books, so it's a bit broken up), and I've been among other things trying to get an idea how the discoveries themselves are structured. Get a feel for how they work, if you see what I'm saying. Since someone got famous a century ago for proving some things can never even be proven, it doesn't seem like a wholly fruitless pursuit. Unfortunately, I don't know of any standard code words for the distinction I'm trying to get across. I'm describing it mostly in terms of the words that come to mind when I think about it. So, it's more like defining continuity by "it's smooth and doesn't have an holes" than by "The limit of f(x) as x approaches c must exist and be equal to f(c). (If the point c in the domain of f is not an accumulation point of the domain, then this condition is vacuously true, since x cannot approach c.)" (from the relevant article). So, bear with me. When I see most explanations for why things work, I think, that makes sense. That's a good reason. I see a strong justification for no number one less than a square being prime: x²-1=(x 1)(x-1). It didn't just happen to turn out that way, things didn't just fall together like that. It's that way because that's how the numbers work. Looking at the numbers in a table, it might seem like there was no good reason for it, it might even seem like it wasn't going to keep going all the way up. It might seem like dozens of real patterns slammed together and made something coincidental. But, after seeing that description of it, it all makes sense. So, that's what I'm asking. Does it seem like anything could come up that, even once there was no doubt it was true(ie, it was proven, by whatever method presented itself), had no more compelling description than "that's the way it turned out"? I brought up the 3-amicable numbers because that's what suggested it to me, which happened because it seems a likely enough candidate for this. Even if it's possible to prove it, perhaps by computing all possibilities (unlikely, given there are infinite choices, but with quantum computing, who knows?) or perhaps by a more practical method, does it seem possible that it would have no actual justification? That there being no 3-long strings of an already rare structure would strike people as an infinite (but verifiable) stretch of "that's the way it happened"? Or, and this is the important part, does it seem like by the nature of number theory, anything that made it to a full proof would have that quality of sense and revelation when put next to a significant experience of the numbers involved? Black Carrot 05:11, 10 September 2006 (UTC)

I understand what you are trying to say; The problem is that it is entirely subjective. Given some proven statement, one might say that the proof is beautiful and it explains why the statement must be true, while another might say that the proof is completely technical and it just "turned out that" the proof exists. As far as I can tell, the only "objective" approaches are the two polar opposites, "every proof just happens to exist, and every correct statement just happens to be so", and "every proof is a reason why the statement must be true". There are other problems with roots deep in the theory of mathematical logic, in which I don't have enough proficiency to portray them accurately. I'll just say that Gödel's incompleteness theorems suggest that the device of "proof" is flawed; In every interesting system (number theory included), there are statement which follow from the axioms but cannot be proven from the axioms. So, you might say that there are truths which "just happen to be", in the sense that you cannot prove them, thereby you are unable to explain their reason to be true (of course, since they are unprovable, you will never know that they are actually truths, so no concrete examples of this can be given). -- Meni Rosenfeld (talk) 07:15, 10 September 2006 (UTC)

On a completely unrelated note, 3 is a prime one less than a square. It is the only one, of course - but be careful when saying there are "no" such numbers. -- Meni Rosenfeld (talk) 07:19, 10 September 2006 (UTC)

A (very) slightly more complete way to describe the incompleteness theorem is this: In a nontrivial system of axioms, there are statements that are (1) well formed (a technical distinction that means "not gibberish"), (2) not provable by a finite sequence of inferences starting at the axioms, nor is its converse, and (3) not disprovable by a finite sequence of inferences starting at the axioms, nor is its converse. The slightly less pedantic way to asy it relies on a mental image. Imagine a plane. There are points on the plane colored bright white. These are the axioms. They are true (by definition/assumption/supposition/...). There are points on the plane colored deep black. These are the negations of the axioms. They are false (again by definition/...). There are rules of inference that allow one to conclude that a certain pattern of white points can be added to a known constellation of white points. Similarly, the rules of inference allow us to infer more black points as well. We may imagine the process of applying all (finite) sequences of inferences to all constellations of white and black points. This is equivalent to applying all finite proofs. In this process, we may imagine that the frontiers of uncolored points are continuously pushed back as more and more statements are resolved white or black ("follow from the axioms" and "do not follow from the axioms"). Now we have a couple of things to notice. There are points that would be colored white or black if infinitely long chains of inference were allowed. These points are typically not classified. Godel's proof (and a few other results) showed something stronger. There are points that would never be classified even if infinite proofs were allowed. These points are statements that cannot be classified by the given axioms and rules of inference. If more axioms were allowed, then some of these points would be classified (but not all). If more rules of inference were allowed, then some of these points would be classified (but not all). These uncolored points are called independent of the axioms.

There are "boring" ways to add new axioms: (1) add a point that is already colored white (then no new points are classifiable since all deductions from this statement were already colored), (2) add a point that is already colored block (making all points be colored both black and white -- i.e. there is a contradiction in the axioms and the system is vacuous in that it no longer separates the plane into two sets), (3) add a point that is not already colored at all. This third choice, to add an independent statement, is similar to (in linear algebra) adding another basis element to a partial basis so that the span of the basis is increased. We can repeat this construction and continue to classify points, selecting unclassified statements at each step.

One problem is that at any step, if a statement is uncolored, then so is its negation. So, we can add any axiom we like. An example is the parallel postulate of Euclidean geometry. It is an unclassified point based on the other Euclidean axioms. So we may add it or any variant of it that is unclassified. This freedom allows not only Euclidean geometry, but also non-Euclidean geometries. Each choice is a valid extension of the "simpler" Euclidean axioms. So, we may add any unclassified statement, even the negation of the one that we might intuitively pick based on our experiential background.

There are also examples of axioms that appear to be limit points of theorems. The axiom of choice (AC) appears to be one of these. It's relatively straightforward in (normal) axiomatizations of set theory to show that the finite AC is true ("colored white"). However, AC is known to be independent of Zermelo–Fraenkel set theory (ZF) (q.v. the Axiom of Choice article). So there are "standard" examples of the accumulation point of a sequence of theorems being an independent statement.

Also, there are statements that are suspected independent that are not known to be independent. (And note that "independence" is only with respect to a specific set of axioms. Add another axiom and that (previously) independent statement may sudden become colored black or white.) The Continuum hypothesis (CH and "generalized continuum hypothesis" = GCH) is independent of both ZF and ZF AC, but AC is a white point in ZF GCH.

Now which axioms we pick as our starting points for coloring the plane of statements is our choice. Non-Euclidean geometries weren't studied for most of the history of the parallel postulate because non-Euclidean geomatries were perceived to be non-physical and therefore useless. Euclidean geometry told you how the world worked; non-Euclidean was therefore defective. But since we have a choice, every classified theorem "just turns out that way" because of our choice. It is our assignment of external value to specific sets of axioms that makes the result either "meaningful" or "a pedagogical exercise". Even worse, when working in a set of axioms after adding an independent statement, neither the "usefulness" nor "uselessness" of which is a priori clear, one may say that a newly provable result "just happens" because of the adoption of the additional axiom. An example is the Generalized Riemann hypothesis (GRH). The GRH is irritating because we currently don't know if it's white, black, or uncolored. There are proofs under the assumption that GRH is either white or uncolored (and a couple of these are in the article). I know independently of at least one result predicated on the GRH being black or uncolored (which is interesting, because most mathematicians intuit that the GRH is true). IF GRH is uncolored, then we may say that either set of dependent theorems "turns out" to be true based on whether you extend your axioms with GRN or its negation. If GRH is colored, then one set of theorems will "turn out" to be true only because GRH turned out to be true or false.

To come back to your example... That 3 is the only prime that is one less than a square depends on your axioms. How do you define "number" and "prime"? There are generalizations to these ideas where there may be more than just one (or perhaps none). The difference is the choice of axioms. -- Fuzzyeric 03:10, 12 September 2006 (UTC)

Then there's the even more restrictive version of "it just turned out that way". One may take that statement to apply to *every* colored point in the plane of statements. "It just turned out that there are Pythagorean triples in the set of integers described by our axioms." This is a perfectly valid statement, if one assumes that every set of axioms is equally important. However, mathematics is actually more practical than that. We: (1) don't get excited about contradictory axioms, and try to not work with contradictory sets of axioms (although Godel has shown that showing that axioms are non-contradictory may be infeasible), (2) tend to work with the smallest set of axioms that is (a) able to prove all the things we can prove with other presumably equivelent axioms and (b) able to prove "easy" things "easily", and (3) tend to work on axiom systems that have applications in the physical sciences and to other branches of mathematics. So rather than mathematicians being machines for turning arbitrary axiom sets into theorems, mathematicians tend to pursue theorems in axiom systems that are applicable to some problem. From this point of view, nothing "just turns out to be true". It was true a priori in the application domain and the choice of axioms and the proof provide a way to lift that intrinsic truth out of the murky realm of intuition and into the Platonic ideal of "true". They are the ladder that the statement climbs to go from "just turns out to be true" to "is true and here's why". -- Fuzzyeric 03:10, 12 September 2006 (UTC)

Some good points, but keep in mind, I mean what I mean by "it just turns out that way", not what you mean by it. As far as I can tell, what you've said doesn't change that. And the specific set of rules I'm operating under are those of, unsurprisingly, arithmetic. BTW, it might help to point out that proof itself isn't as inherently pure and perfect as all that. It's based on the same subjective thing I'm basing my question on. In this case, though, it's the sense that "it just has to be that way", a feeling that many people don't share. Have you ever tried to argue proof with an intensely religious (or agnostic or athiest for that matter) person? It can be pretty much impossible, because given any statement in the proof, all they have do is say "That's not true." Sure, under a system of logic you've made up that statment can be rendered meaningless, but how do you know that system is right? The proofs aren't "true", they're convincing and they fail to lead us astray most of the time. I don't see how the statement "it is or it isn't" can be false. Until, that is, I realize that the excluded middle isn't always justified. So I adjust it to be convincing again, etc. So, the axioms people have retroactively fitted counting and addition with are very important, since they're the main way we study it, and since I'm asking about number theory the philosophy of it is important, but they're only tangentially related to my question. What I'm asking about is an evaluation (equally subjective, but newer and less polished) of something's inherency to the numbers themselves. Black Carrot 15:47, 15 September 2006 (UTC)