Wikipedia:Reference desk/Archives/Mathematics/2010 August 13

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is an archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

August 13

Mathematicians and Math (What's the deal?)

OK,So I'm going to ask a few questions about what mathematicians know and don't know.I'd appreciate answers. Thanks guys.Can someone learn math without knowing linear algebra or complex analysis?How about the following. Is it possible to ...

(i)Learn differential geometry without knowing what a top. space is?

(ii)Learn algebraic geometry without knowing what a group is?

(iii)Learn harmonic analysis without knowing what a Hilbert space is?

(iv)Learn fucntional analyss without knowing what a Fourier series is?

(v)Learn ring theory without knowing what a field is?

(vi)Learn set tehory and logic without knowing what anything in math is?

(vii)Learn algebraic number theory without knowing elementary number theory?

(viii)Learn statistics without knowing what a measure space is?

(ix)Learn f.g.t without knowing what a character theory is?

(x(THE MONSTER!)Learn any area out of algebra, analysis, topology, number theory etcetera without knowing any of th eother areas(absolutely NOTHING in fact out of the other areas.)

And lastly,what's the deal with the fake proof of P = NP on the internet?And lastly again,why are mathematicians so uneducated to the point that algebra dudes don't know what a top. space is,top. dudes don't know what a Hilbert space is,analysis dudes don't know what a HOMOTOPY GROUP IS. (I mean how dumb can you get!?)Why do math dudes specialize so much into one thing or the other? I mean,WHY? Why don't they try to do EVERY ARAE? (I couldn't help using "dude" btw ...)I have a typsetting bug btw, hence the punctuation errors. —Preceding unsigned comment added by 114.72.245.4 (talk) 06:42, 13 August 2010 (UTC)

You don't need to know all of maths to get useful work done any more than a person studying insects needs to know all that much about fish, and would they do so much good work if they spent more time studying? And why do you refer to a 'fake proof'? It looks like a quite valiant attempt to me and fake implies wrongdoing. And anyway what gives you the idea a professional mathematician would be completely unfamiliar with something in your list? I'm not a professional mathematician but I've read up on all the things you've mentioned. This sounds like one of those company exhortations that says things like 'we must concentrate on improving on all fronts' ;-) Dmcq (talk) 08:22, 13 August 2010 (UTC)

thanks dmqc but can you give me answers to my questions as well? Thanks. What about other people's opinions? Can others give their opinions as well? I'd like a broad range of opinions from the math community? Thanks.

In the guidance information at the top of this Project Page it states The reference desk does not answer requests for opinions ... Dolphin (t) 07:59, 16 August 2010 (UTC)

Which question did I not answer? Dmcq (talk) 08:44, 13 August 2010 (UTC)

Why do you want to know? I am confused by your questions. Sometimes it is fertile to combine information from seemingly very different areas of science. But your central question "Why don't they try to do EVERY ARAE? " seems strangely naïve to me. Time is limited. There is a time for reading and a time for problem solving and a time for teaching and a time for original research. There is a time for seeking general orientation and a time for seeking knowledge for a specific purpose. As the amount of litterature grows faster than your speed of reading, reading everything is not possible. Bo Jacoby (talk) 09:01, 13 August 2010 (UTC).

At my school every student in the math PhD program had to pass a "qualifying examination" showing they had basic knowledge of all the topics you mention. They would then concentrate more narrowly for their thesis research. There was one loophole: if you wanted to become a logician without studying a lot of math, you could get a logic degree from the philosophy department instead of the math department. In that program you would still have to take a bunch of mathematical logic courses in the math department from math professors, but not much topology or anything like that. 67.122.209.167 (talk) 09:51, 13 August 2010 (UTC)

I think a bit of topology should be a prerequisite nowadays for logic, there's various thinks like the axiom of determinacy or large cardinals which need enough to at least understand things like Baire categories. Dmcq (talk) 12:05, 13 August 2010 (UTC)

Perhaps the above answers need to be "decoded" so that you can understand them. The answer to most of your questions (i)->(x) is "yes". By this I mean, one can learn the subjects mentioned without knowing what you have described. E.g., one can learn what a ring is without knowing what a field is (in fact, I do not know of anyone who learnt this the other way around), one can learn a good deal of statistics without knowing what "measure theory" is, one can learn functional analysis without knowing what a Fourier series is etc.

The answers to some of your other questions is, to some extent, "no". E.g., learning a little algebraic number theory without knowing elementary number theory is possible if you stick to the abstract concepts such as Dedekind domains, DVR's etc. and know in your own mind what a prime ideal is without needing to motivate it from elementary number theory, but there will come a time when intuitions from elementary number theory play a role. Similarly, one can learn basic Fourier analysis without knowing, at least abstractly, what a Hilbert space is, but concrete Hilbert spaces occur throughout Fourier analysis and so one needs to, at least at some point, learn how to identify an abstract Hilbert space. Also, it is a must to know what a ring is before one does algebraic geometry, and I assume, at least I hope that I do not need to "assume" this, that all students who can identify rings can also identify groups. Even in a more concrete sense, group schemes and algebraic groups are central to arithmetic geometry.

On the other hand, "learning" and "doing research" are different things to a certain extent. One can learn finite group theory (I presume that is what you mean by "f.g.t") very deeply without knowing much about characters. But seeing as character theory has provided efficient solutions to problems across finite group theory for which no purely group-theoretic proofs are known, "avoiding" character theory would significantly narrow your research interests. Furthermore, even though it is possible to do elementary differential geometry without having familiarity with the abstract definition of a topological space, many important more advanced techniques in differential geometry rely on other fields such as algebraic geometry where abstract concepts such as topological spaces play a role.

I agree with Bo Jacoby and Dmcq here. Although it is extremely important for a research mathematician to broaden his interests as much as possible, the amount of mathematics in existence is currently too great for a single individual to learn in a lifetime. I suppose you can at least "see" most of the central areas of mathematics to some extent, and indeed, many first-rate mathematicians have worked on several areas in their lifetime (e.g., Michael Atiyah, Jean-Pierre Serre and Alexander Grothendieck). But just because someone specializes in an area does not mean that that person cannot do good work; nearly every mathematician specializes in some area or the other!

Finally, I would like to draw an analogy with the "world map". Supose we take an (immortal) individual with no knowledge of the "global structure" of the Earth: where other lands are located, what the shape of the continents are, succinctly, absolutely no knowledge whatsoever of the world map. Then we put this individual somewhere randomly in the world with no tools (perhaps except for ample amount of food and water, and a ship to travel between lands) and asked him/her to draw the entire world map as accurately as possible after travelling across the world. (And by this I mean, nearly as accurately as people have depicted the world map today using advanced technology.) The amount of time this would take probably does not even compare to how much time it would take to learn all of the mathematics currently in existence. But collectively, all individuals on Earth do know the structure of the world map upon communication with each other. The situation is identical with mathematics; no-one knows every area of mathematics at a research level. That is why people collaborate with each other. I think that is the answer to your question: not every mathematician has the time or energy to learn many branches of mathematics to some extent; perhaps these are examples of the mathematicians you have seen. But together, mathematicians have been able to do remarkable things. PST 10:24, 13 August 2010 (UTC)

The user 119.72.245.4 should be directed to the summary recently written up by Charles Matthews at the conclusion of a similar discussion at WPM. Tkuvho (talk) 11:04, 13 August 2010 (UTC)

114.72.245.4 (aka 110.20.24.219), is there any particular reason that you have twice modified the IP address in SineBot's sig of your initial post to read 119.72.245.4? Do you wish you were in Tokyo? -- 1.46.68.59 (talk) 13:08, 13 August 2010 (UTC)

permutations and combination

There are four sections in a test each carrying 45 marks.In how many ways a student can pass if the pass or cut-off mark is 90 marks? —Preceding unsigned comment added by Imteyazmohsin (talk • contribs) 11:12, 13 August 2010 (UTC)

Basically, how many ways can you select at least n items from 2n? The number of ways to select any number of items would be 2²ⁿ. Half of that would be the number of ways to select more than n plus half of the number of ways to select exactly n.--RDBury (talk) 12:46, 13 August 2010 (UTC)

If I am reading this correctly, it is asking how many ways can you get >=90 when adding up four integers each with a value 0 to 45 inclusive. 2,271,181. -- kainaw™ 13:37, 13 August 2010 (UTC)

That's certainly quicker than counting up to 811,753,894,769,360,571,756,961,179,474,567,246,637,861,540,684,562,488. -- 117.47.211.213 (talk) 14:29, 13 August 2010 (UTC)

Two ways that I can see. Either do the work learning and revising or cheat. I'd advise the former. :) Dmcq (talk) 18:57, 13 August 2010 (UTC)

Perhaps the intended problem was: how many different combinations of scores are there (from the four sections) that constitute a passing grade? In other words, how many combinations of integers a, b, c, and d are there such that each is in the range [0,45] and a + b + c + d >= 90? -- Tom N (tcncv) talk/contrib 00:04, 19 August 2010 (UTC)

Unsolved problems

How much math would you have to learn before you could reasonably begin trying to solve the famous unsolved problems in math? I mean this both in terms of years of education and level of math (i.e., calculus, analysis, number theory, etc.) In theory you could try without knowing much beyond high school maths but you probably wouldn't succeed that way. 99.137.221.46 (talk) 14:49, 13 August 2010 (UTC)

A somewhat related question was asked a short while ago, and can be seen here. -- Meni Rosenfeld (talk) 15:29, 13 August 2010 (UTC)

Going by the history of famous problems that have been solved up to now, a lot, think post-doctorate. It would also seem to take a good deal of inborn talent and a certain amount of luck. Usually famous unsolved problems are famous because the greatest mathematical minds of the day have tried to solved them and failed.--RDBury (talk) 17:47, 13 August 2010 (UTC)

You may be interested in the (oft-repeated) story of George Dantzig, who, as a graduate student, solved two famously unsolved problems when he came into class late and mistook them for homework. -- 140.142.20.229 (talk) 01:52, 14 August 2010 (UTC)

Topological 'closeness'

I've learned an embarrassing amount of topology without really ever getting clear in my head some of the foundational concepts, so I thought it was time to (at least partially) remedy that. I'm hoping if I get these points cleared up, something will finally click, and I'll have a complete idea of how the subject works.

I understand the progression from metric spaces to topological spaces, since a lot of the proofs for metric spaces just generalise when you talk about 'open sets' which satisfy some of the same properties, but as soon as we drop the notion of distance and start dealing with abstract topological spaces, I sort of lose focus. I've heard the notion of 'closeness' talked about with topological spaces - obviously, with metric spaces, the distance function allows for a clear visualisation of this, but how do you talk about points x and y of a topological space as being 'close' (obviously relative to some standard of 'closeness'). Is it to do with the number of open sets in which both points appear together? Thanks, Icthyos (talk) 15:10, 13 August 2010 (UTC)

The basic relation of a topological space can indeed be seen as a closeness relation, but between points and sets, not between points and points: a point x is close to a set A if x is in the closure of A. Algebraist 15:14, 13 August 2010 (UTC)

(edit conflict)When you generalise to topology, you inter alia drop scale factors. A "very small" metric space (say, with a diametre of 0.001), and a "large" space (say, with the diametre 1000) certainly are different as metric spaces, but may be essentially equivalent (i. e., homeomorphic) as topological spaces. Say, that the first and the second space actually have the same underlying point set X, but that all distances are a million times larger with the metric (distance function) in the second space. In other words, say that the relation between the metrics only is a matter of a rescaling:

$d_{2}(x,y)=1000000d_{1}(x,y)$ for all x,y ∈ X.

Note, that the same subsets of X are open, if you choose the one or the other metric. Thus, indeed, the topology is the same in both cases.

However, assume that there are two points x and y in X, such that $d_{1}(x,y)=0.0001$. Are they "close" or not in the topology? 0.0001 is not much, but if we rescale, then we find that $d_{2}(x,y)=100$, which is a good bit more. And this was just using one scale factor; we could rescale by means of any positive real number, and get a different metric on X, with the same topology. Thus, the question whether just two points x and y are "close" or not is meaningless in an (ordinary) topological space.

As Algebraist just explained, there are a lot of other questions about closeness which are meaningful. If x is a point in X, and S is an (infinite) subset of X, then x is close to S, if there are elements in S arbitrarily close to x, with respect to any one of the aforementioned metrics. Meditate a bit over this; you ought to see that the scale factor doesn't matter for x being close to S in this sense. Arbitrarily close means that for any ε > 0 there should be a y ∈ S with $d_{*}(x,y)<\epsilon$; and you should see that you may counter a rescaling by a judicious choice of an "auxiliary" ε.

I hope this is to some help with your trouble with the intuition. Since this seems to have been your problem, I limited my arguments to such topologies as may be derived from metrics; in particular, these are Hausdorff. The nice thing with reformulation in terms of open sets (only), is that the results work not only independently of the metric, but also for topological spaces where no compatible metric at all exists. However, that may be a later step in abstraction... JoergenB (talk) 20:21, 13 August 2010 (UTC)

To elaborate slightly on this notion: a point x is close to a set A when there are points in the set arbitrarily close to x. What does arbitrarily close mean? In a metric space, it means we can get within any ε of x. In a topology, it means we can get within any open set containing x. Open sets determine what "arbitrarily close" means in a topology.

Note that the notion of relative closeness does not exist in Topology; one point cannot usually be said to be closer or further from x than another point (I may be wrong about this, but I think the only time you can really argue such a statement is if the space is disconnected, then points in a different connected component might be considered to be further than points in the same connected component, but even that isn't really true). Rather, we have a more global notion of points that are arbitrarily close to x. --COVIZAPIBETEFOKY (talk) 19:49, 13 August 2010 (UTC)

I think you could justifiable describe y as closer to x than z is close to x if any open set containing x and z also contains y and there is an open set containing y and x but not z. I can't think of any space where that holds, other than ones constructed specially for the purpose, though. --Tango (talk) 20:20, 13 August 2010 (UTC)

This is a nice idea from Tango. I can't think of a counterexample. Moreover, I can think of an example that goes against standard metric topology. Think of the real line, and the points x = 0, z = 1, and y = 2. With the metric topology, x is closer to z than it is to y, i.e. | x − z | < | x − y |. With the particular point topology, where the open sets (along with R and ∅) are the sets containing y. Every open set containing x and z must also contain y. But there is an open set containing x and y, namely {x,y}, that doesn't contain z. So in this topology, in Tango's sense, x is closer to y. — Fly by Night (talk) 00:20, 14 August 2010 (UTC)

Well, yes, that just proves that you can have completely different topologies on the same underlying set. The only thing the real line with the standard topology and your space have in common, really, is the cardinality of the underlying set. --Tango (talk) 00:44, 14 August 2010 (UTC)

It's even deeper than that. The underlying set, probably, has little to do with anything. Any set can be given many, many different topologies. All of which give that set very, very different properties. The question that we ought to ask ourselves is this: what is the cardinality of the topologies on a given set? — Fly by Night (talk) 01:07, 14 August 2010 (UTC)

Well, as far as topologies are concerned, a set is completely defined by its cardinality. That cardinality is very important, but other than that sets only differ by the names we give the elements. --Tango (talk) 02:11, 14 August 2010 (UTC)

Not quite. The topology of a set depends exactly upon its topolgy (I would worry if the topology of a set did not depend upon the set's topology). Take as a counterexample the real line with the trivial topology and the real line with the discrete topology. The real line and the real line have the same cardinality, but they are not homeomorphic when they carry these two topolgies. (Although, two sets with the trivial topology are homeomorphic if and only if they have the same cardinality.) — Fly by Night (talk) 03:20, 14 August 2010 (UTC)

I wasn't talking about topological spaces; I was talking about sets. You can define exactly the same topology on the real line as on the Euclidean place as on the complex plane as on the real projective line, etc., and get exactly the same topological space. All those sets are essentially the same because they all have the same cardinality. --Tango (talk) 09:27, 14 August 2010 (UTC)

OP, from my limited experience the whole point of using open sets instead of distance function is for the ease of it. It's much more time consuming to define metrics on sets, when all we do is talk about the open sets. So I think of R^n as a product space and not a metric space. Furthermore, some topologies are not metrizable (there's no distance function that give precisely the same open sets), for example the long line and the Zariski topology. Also, when you said "I've learned an embarrassing amount of topology without really ever getting clear in my head some of the foundational concepts", this is why you don't rush math. Money is tight (talk) 04:33, 14 August 2010 (UTC)

The original question by Icthyos seemingly concerned such topologies as may be derived from metric spaces. There are numerous other, and rather interesting ones, too. Consider the spectrum of a (commutative and unitary) ring R,

X = $\mathrm {Spec} \;R:=\{{\text{prime ideals in }}R\}$,

together with its Zariski topology. Note, that if p and q are elements in X, i. e., prime ideals in R, such that p is a proper subset of q, then every open set that contains q also will contain p, but not vice versa. This indicates that one should be a little careful in defining "closeness between two points" in a general topological space; beware of coming up with a definition, such that "p is close to q, but q is not close to p".

However, my advice to Icthyos is: Ignore non-metrisable topologies for the time being (but remember that the general topology you try to learn will cover more than the examples you choose to consider right now)! JoergenB (talk) 13:23, 14 August 2010 (UTC)

Thank you all for the detailed responses. I believe something has clicked! Icthyos (talk) 00:04, 15 August 2010 (UTC)