Let's discuss Mathematics

[dutchfire]

@Ninja returns: No, since the analysis exams are oral, they're spread out over de week. (I'd be a bit hard for a single professor to do ~40 exams on a single day )

But to get this a bit more on the mathematics subject:

Let A be a matrix over the GL_n(C) (so an invertible complex matrix of dimension n), and let f' = Af (with f a vector consisting of differentiable functions) be a system of differential equations. Show that the solutionset is a vector space of dimension n.

Well, showing that it's a vector space isn't that hard, you just show closure under addition, scalar multiplication and that 0 is a solution.
However, showing that it has dimension n could be quite tricky. My, slightly sneaky?, solution would be to say that if A is the identity matrix, the solution set obviously has dimension n. Now, all elements of GL can be constructed from the identity matrix by using the elementary operations (switching rows, adding rows to eachother, multiplying a row by a non zero scalar). So showing that doing any of the elementary operations wouldn't change the dimension of the solution set should suffice, shouldn't it?

 

[ParadigmShifter]

1st, don't prove it's a vector space from the axioms, use the subspace criterion (I see you did that by stating the subspace criterion ).

A is invertible, so has linearly independent rows and columns, can't you just use the dimension theory for this? The kernel (null space) has dimension 0.

EDIT: Yep, you did use that. Rank A = number of linearly indep. rows/cols = Dim(Range A). Proof is done via e.r.o.'s.

 

[dutchfire]

Oh, that's right, you can apply the dimension theory for this. A is finite dimensional, even though the basis for your input is something strange like vectors filled with functions.

 

[ParadigmShifter]

Yeah it is Rank A which is important and n is finite, it doesn't matter that the vector space has an infinite dimension. In GLn(F), Rank of any matrix is n.

Hey I could pass the oral exam if I spoke Dutch

Any other questions?

 

[Atticus]

Here's one for y'all math experts. What's wrong with the following argument?

There's some interesting commentary on the linked site, but I think they missed the weakest point in the argument - and I'm curious how many people will pick on the aspect I would pick on.

First of all we must notice that this proof is meant to be fun, so the fact that "special" isn't defined need not count. We could also try to define speciality so that it fits the proof. For example number would be special if it can be uniquely expressed verbally without using numerals.

Or then we can have semi-definition, that we know special number when we meet it. Any way, I suppose that's not what you're after here, Ayatollah, so let's not stick to it.

I suppose the problem you have with it is what ParadigmShifter allready said:

Also, if we say that a number is the smallest non-special number hence making it special, what is special about the next largest non-special number? It can't be the least because we already found it. So if this becomes the new least special number what is special about the number we claimed was the least before?

But this isn't valid critcism. The argument used isn't a process, which eliminates all the nonspecial numbers, it's purpose is to show that the concept of least nonspecial number is contradictory. So it doesn't take away nonspecial numbers one by one and put them in the special numbers basket, but instead it shows that the assumption of nonempty set of nonspecial numbers leads to contradiction.

Spoiler somewhat analoguous situation, which isn't related, but may help someone understand what I meant with the previous paragraph. :

You can compare this with the situation when some people try to refute Canot's diagonal proof: Cantor's idea is to show that the set ]0,1[ of real numbers isn't countable. Cantor makes counterhypothesis, that it is countable, and can be enumerated. Then he chooses a real number, whose n:th decimal is different that the n:th decimal of the n:th number in the list. The number he get's this way isn't on the list, so the counter hypothesis is false. Now I've heard people trying to prove that Cantor was wrong by saying: "so why don't we just put that new number in the begining of the list? We'll have countable amount of numbers then". They don't understand that Cantor's proof is "static", he draws contradiction from the assumption that all the numbers are allready listed, he isn't offering somekind of algorithm to enumerate the numbers.

I don't think there's anything wrong with the argumentation, if not the bad definitions. Trying to produce accurate definition would probably give property which applies to all the natural numbers. If you think a while about the definition candidate I gave before, "uniquely expressible with words without the use of numerals", you'll probably notice soon that it consists of all the natural numbers.

You should notice also how much this resembles Berry's paradox from the early 1900s, which defines n to be "the least natural number expressible with less than hundred letters". It was considered a genuine paradox at it's time. Unfortunately I don't know what logicians of today would say about it.

 

[ParadigmShifter]

Well it's not a set it's a fuzzy set. The least non-special number has partial membership in the fuzzy sets of both special and non-special numbers

EDIT: The remaining members of the non-special numbers aren't as special as the least either

 

[Atticus]

Ok, it's a little fuzzy area of mathematics for me, so I won't comment that.

Perhaps this is a fuzzy paradox, and is contradictory only to a degree x, where x is in [0,1].

 

[ParadigmShifter]

Also, the line "So S has a least member, that's pretty special!" says nothing about the other members of S.

It's mainly contradicting the notion of S being a set I think.

I'm not too familiar with fuzzy logic either but the vagueness of the definitions makes me think in terms of these objects.

EDIT: Interesting topic though, I'd like to see what the original poster thinks is the false part of the argument.

 

[dutchfire]

Amazed you didn't spot this obvious error ParadigmShifter:

Clearly zero is a very special number, being the first natural number.

Since I need to know some set theory tomorrow (), I'm curious if this set can be defined using the ZF(C)-axioms. You can't use the axiom of specification without properly defining special for every natural number. So I think you can't define this set in terms of the Zermelo Fraenkel axioms without rigorously defining what you mean by special.

 

[ParadigmShifter]

I let the zero thing slide since it's controversial EDIT: And { 0, 1, 2, ... } is isomorphic to { 1, 2, 3, ... } anyway so they have the same set-theoretic properties

Yeah, you'll need a rigorous definition for ZFC set theory.

It also talks in terms of degrees of specialness making it a non-predicate.

 

[dutchfire]

Especially the fact that being "non-special" can make something "special" (as the lowest non-special number), meaning you can't properly define the the special criterion.

 

[Atticus]

I'm waiting him to tell that too.

Since all nonempty sets of N have the least member, it's enough to show that there's no least element. You can conclude that the set (that is: speciality) is badly defined, or that the set of nonspecial numbers is empty.

So I think only valid critcism is what you already said: the definition. However, I think we have some intuitive notion of what it means to be a special number, and we can use it to define "speciality".

You could try to define it with degrees: The only even prime is obviously special, the least perfect number little less obviously, the least odd prime and the least square of a prime less obviously.

You could take some basic properties, and issue the degree of speciality by the number of those properties needed to specify the number. These would lead to fuzzy sets.

Or you could say that number is special if and only if it can be expressed with those properties. Or to use the definition I already gave as an example: a number is special if it can be uniquely expressed with words without using numerals.

In order that the proof will work with the definition, the definition must satisfy at least these criteria:
1. Least number satisfying some property of specified list must laways be special.
2. The property of nonspeciality must be in the list mentioned in 1.

This is of course very similar to numerous paradoxes, which use a "circular" way of definitions. I'm not sure what is the consensus of such definitions, but as far as I know, some of them have been considered acceptable.

EDIT: x-post, aimed at PS.

EDIT2: The two criteria I give above also show clearly why the proof is kind of trivial. To see it, let's spell out definition that satisfies rules 1&2:

Let B={A_i} be a family of properties of natural numbers (i.e. sets of N). The property S is special with respect to B, if the least number of every set in BU{N\S} is in S (when the set has least number, i.e. is nonempty).

You could also of course add that the least numbers of intersections of sets in B should also be in S, but in order for this to work, N\S has to be a set whose least number is included in S. But since S and N\S are disjoint, this is contradiction unless one of them is empty..

So the proof relies on the intuitive understanding of speciality, that the property N\S is one property which "clearly" counts. However if that would be spelled out the contradiction would be trivial. (Basically what ParadigmShifter and Dutchfire have already said, but with other words).

 

[ParadigmShifter]

Yep, but you got to use BUNS are in S, well played

 

[Ayatollah So]

@ Paradigm and Atticus,
I'm waiting for more responses. Preferably some by people who aren't so math-whiz. You've both hit on my concern, though not in the same words I would use. The reason I'd use different words probably comes down to not sufficiently understanding yours

I do like what Atticus said about using an intuitive definition of "special", though. I can't see anything wrong with using an intuitive definition. But doing so might highlight the problem I have in mind...

I don't want to bias the answers (not like I have a "better answer" but still), so ... wait a bit.

 

[dutchfire]

Exams went pretty well

Interesting thing I had to proof during the analysis exam:
A set X is infinite if and only if there exists a proper subset Y with the same cardinality as X.

 

[ParadigmShifter]

Interesting, I'll have a go

I'll number the statements

1) X is infinite
2) There exists a proper subset Y of X s.t. |Y| = |X|

I think proof by contradiction is going to be useful here, so I'll try proving ¬(1) -> ¬(2)

Suppose X is finite with cardinality n. Then a proper subset Y of X can't have cardinality |X| since if it did then Y would have n members too and hence be equal to X.

¬(2) -> ¬(1)

I'll have to think about this one

 

[dutchfire]

Well, I tried proving 1 -> 2, so all you have to do is explicitly giving such a subset Y. If X is countable, it's pretty easy, but if it isn't countable, you've got a bit of a problem.

The proper way to solve it involves taking a sequence from X (using the axiom of choice)

 

[Atticus]

Suppose X is countably infinite, so X={x_1,x_2,....}.

Now define Y=X\{x_1}, and f:X->Y: f(x_k)=x_{k+1} is bijection. So X and Y have the same cardinality.

Now suppose X is uncountably infinite. It therefore has countable subset {x_1,x_2,...}=:A. Now define Y=X\{x_1}, and
f: A->Y : f(x_k) = x_{k+1}. Now the function

g:X->Y: g(x) = f(x), when x\in A, and id(x) otherwise*

is bijection, and thus X and Y have the same cardinality. QED

X finite => no equinumerous genuine subset
is trivial, and therefore
equinumerous genuine subset => X infinite
is also trivial.

(If they aren't trivialities, you'll have to specify what infromation are we allowed to use).

*) id= identic funtion.

 

[Ayatollah So]

he seems to argue degrees of membership in the set of special numbers which is in fuzzy logic territory which isn't classical set theory.

Well it's not a set it's a fuzzy set. The least non-special number has partial membership in the fuzzy sets of both special and non-special numbers

Perhaps this is a fuzzy paradox, and is contradictory only to a degree x, where x is in [0,1].

Well you guys smelled the rat - the fuzzy membership of the "special" category - but I guess you're leaving it up to me to kill it.

Or then we can have semi-definition, that we know special number when we meet it.

Absolutely. Always take an argument and each of its statements at face value, unless forced to do otherwise. Specialness is intuitive, and also fuzzy. So consider each nonzero level of specialness, from 1 down to infinitesimal. Degree 1 specialness is what you would grant to be special even at your worst mood. Degree 0.01 specialness covers what you'd grant to be special only on your most generous 1% of days. Notice that I'm making specialness pretty darn subjective - I think that's within the spirit of the whole thing.

Suppose that the next highest rating you give to any but your most specialest numbers - the next highest after 1.0 - is 0.98. So consider the smallest number that fails to rate a 1.0 degree of specialness. Does that very fact - being the smallest number that fails to rate a 1.0 degree of specialness - make it special? Well yeah, a little. But not special to degree 1.0, or even 0.98, more like, oh, about 0.05 degree. So, if we consider "being special to the utmost degree" versus "not special to the utmost degree", we can see that "being the least number in the latter category" is not a contradiction. Also we can see that the same applies to "being special to a very high degree", where we arbitrarily define very high degree as, say, >= 0.95.

But what about "special to at least a modest degree", where a modest degree is defined as 0.05 or greater? Does the assumption of a nonempty set of not-even-modestly-special numbers, lead to a contradiction? Only if being the least number to fall short of the 0.05 mark were special to degree 0.05. But, that's not plausible. There are plenty of numbers just over, and just under, the 0.05 mark, which makes the 0.05 mark a really boring neighborhood. Being the first guy to fail to be in that boring neighborhood, is an even-more-boring fact.

In summary, there may well be no level of specialness for which the argument works. The "paradox" is probably contradictory to degree 0.

 

[Harbringer]

Ive always wanted to get into mathematics heavily, but I just dead end so much on the understanding level and was wondering if any of you would happen to know a good place to start in getting into math. Im still at pre-algebra level math