Some mathematical question(not homework)

[becomingagodo]

Okay, I just found out their are objects with infinte volume, but finte surface area.

What the hell?

I know their are objects with infinte surface area and finte volume. I know how to visualize this, however I don't know how to visualize something with a infinte volume but finte surface area.

Without going into higher dimensions I don't see how this is possible. Can someone explain, note the more visual the better.

Man, this has been diving me crazy all day.

Another thing, what is a number?

Now what is a number in it most abstart form i.e. no arguments about real life. I am buying John Conway book on Number and Games, he describes numbers differently(I feel kind of quilty for buying his book, I don't consider Game theory good mathematics). Saying that John Conway work is crazy.

On another note, I can slowly feel infinty grinding away my sanity, hence why I need to understand the top question. Whats that quote that ends in "only a madman can understand infinty". Infinty it could make you cry

[bokonon]

Dunno. Maybe if you consider "the space outside this jar" an object, then the surface area would be the area of the inside of the jar, and the volume would be infinite. Just a guess.

[Fredrik]

I don't know about the first question, but I think the best way to think about numbers (or any other mathematical object) as being defined by their properties. You simply list a bunch of properties that numbers should have (like a+b=b+a), and define the word "number" to be anything that has those properties. This is used e.g. in the usual construction of complex numbers from the real numbers. You start with the real numbers and define complex numbers to be ordered pairs (x,y) of real numbers, with addition and multiplication defined in a certain way. Then you identify the real numbers with pairs of the form (x,0). Now why can you do that? Because those pairs have the same properties as the real numbers we started with.

Another answer is that non-negative integers can be defined using axiomatic set theory. Then you can use that definition to define negative integers and rational numbers. And you can use that to define real numbers, and so on. The set theoretic definition of non-negative integers says that 0 is the empty set {}, 1 is the set {0} that contains the number 0 but nothing else, 2 is the set {0,1} that contains the numbers 0 and 1 but nothing else, and so on...

[andyandy]

you want Gabriel's horn (http://mathworld.wolfram.com/GabrielsHorn.html)

which is formed by considering the surface of revolution of the function y = 1/x about the x-axis for x greater or equal to 1. It has finite volume, but infinite surface area, when you work out the integrals

V = $ \int_1^{\infty} \pi y^2 dx = \pi

S = $ \int_1^{\infty}2\pi y (1+y'^2)^{0.5} dx = \infty

I don't know if this can happen in reality, or if it's a maths anomaly....seems kind of strange to me :)

[bjornart]

you want Gabriel's horn (http://mathworld.wolfram.com/GabrielsHorn.html)

which is formed by considering the surface of revolution of the function y = 1/x about the x-axis for x greater or equal to 1. It has finite volume, but infinite surface area, when you work out the integrals

V = $ \int_1^{\infty} \pi y^2 dx = \pi

S = $ \int_1^{\infty}2\pi y (1+y'^2)^{0.5} dx = \infty

I don't know if this can happen in reality, or if it's a maths anomaly....seems kind of strange to me :)

No he doesn't. Which is why he writes I know their are objects with infinte surface area and finte volume. I know how to visualize this, however I don't know how to visualize something with a infinte volume but finte surface area.

But I'd like to know where he got the idea from. The only hits I get from google are either wrong references to finite volume and infinite surface, or state that it's not possible. And I'm inclined be believe the latter is true. It's provably true for rotational bodies at least.

[Complexity]

Okay, I just found out their are objects with infinte volume, but finte surface area.

What the hell?

I know their are objects with infinte surface area and finte volume. I know how to visualize this, however I don't know how to visualize something with a infinte volume but finte surface area.

Without going into higher dimensions I don't see how this is possible. Can someone explain, note the more visual the better.

Man, this has been diving me crazy all day.


Welcome to math. I don't know the answer to your question.

There is an interesting book by Leonard Wapner, The Pea and the Sun, that will unhinge you. From page xii of the introduction to this book,

"He continued. "So, we have the Banach-Tarski Theorem, or Banach-Tarski Paradox. An equivalent form of this theorem states that a solid of any size, say that of a small pea, can be partitioned into a finite number of pieces, and then reassembled to form another solid of any specific shape and volume, say that of the sun. Consequently, this paradoxical theorem of Stefan Banach and Alfred Tarski is sometimes referred to as the pea and the sun paradox.""

From page xiii of the same:

"As the general population leanred of the theorem, the controversy spread. An irate citizen once demanded of the Illinois legislature that they outlaw the teaching of this result in Illinois schools."

As counterintuitive as this theorem is, it is regarded as valid and as one of the gems of mathematics.


Another thing, what is a number?

Now what is a number in it most abstart form i.e. no arguments about real life. I am buying John Conway book on Number and Games, he describes numbers differently(I feel kind of quilty for buying his book, I don't consider Game theory good mathematics). Saying that John Conway work is crazy.


Game theory is good mathematics. I haven't read the book you mention, but if it is the same Conway behind the Game of Life, I'd take it seriously.

There are many ways of describing numbers.


On another note, I can slowly feel infinty grinding away my sanity, hence why I need to understand the top question. Whats that quote that ends in "only a madman can understand infinty". Infinty it could make you cry


Don't worry about the 'madman' quote. I don't think we're in any danger of anyone understanding infinity, though some have made good stabs at it.

Infinity is beautiful and very disturbing.

A great book on the subject is Georg Cantor - His Mathematics and Philosophy of the Infinite, by Joseph Warren Dauben. I've read it several times and look forward to reading it again.

I hope you're following through on what we discussed earlier.

[andyandy]

No he doesn't. Which is why he writes

But I'd like to know where he got the idea from. The only hits I get from google are either wrong references to finite volume and infinite surface, or state that it's not possible. And I'm inclined be believe the latter is true. It's provably true for rotational bodies at least.

oops....i should pay more attention :)

...still gabriel's horn is pretty interesting. :D

... the reverse case seems counter-intuitively impossible...even more so than Gabriel and his horn that is....

[Jekyll]

Okay, I just found out their are objects with infinte volume, but finte surface area.

What the hell?

I know their are objects with infinte surface area and finte volume. I know how to visualize this, however I don't know how to visualize something with a infinte volume but finte surface area.

I've no idea what you're on about but here is one such shape (in 4-d).

Take a finite cone and stretch the tip of out to infinity. This cone has infinite surface area and finite edge length. Now embed this in a 4-d space and give the object unit thickness in the new currently empty dimension.

This object has finite 2-d edges and an infinite 3-d surface volume.

I don't consider Game theory good mathematics). Saying that John Conway work is crazy. Conway's work is excellent.

On another note, I can slowly feel infinty grinding away my sanity,
Gosh.

[Math Maniac]

Gagodo, add this: there are degrees of infinity. As an example, both the counting numbers and real numbers are "infinite" but the degree of infiniteness of the counting numbers is less than the infiniteness of the real numbers. Said another way, all infinities are NOT equal.

**Just reminiscing about set theory courses while in college...**

Huh?, you say...so do most people.

[becomingagodo]

Said another way, all infinities are NOT equal.

I know this already. I watched a simple visual proof of it on a program called dangerous knowlege. Simple draw a circle, this is a infintive polygon. Then draw a bigger circle over smaller circle, then expand the points in the smaller circle so they cross the bigger circle. Their would be gaps, which then proves infinities are not equal.
http://video.google.com/videoplay?docid=-3503877302082311448
Skip too 9:25 into this program. It explaines the idea in a minaute.
So you can visualize it.

Okay this is the drinking vessel
or as put in the book
a drinking glass that had small weight, but even the hardiest drinker could not empty.
Quoting from the book
The solid is generated from the cissoid. The canonical curve has equation
y^2=x^3/(1-x),
it has a vertical asymptote at x=1.

The solid concern is contained between the rotation of the upper half of the cissoid and the vertical asymptote about the y axis; it forms a goblet-shaped figure.
It looks like this
http://upload.wikimedia.org/wikipedia/commons/thumb/e/e2/Goblet_Glass_(Banquet).svg/70px-Goblet_Glass_(Banquet).svg.png
Note: without the bottom bit

I can't quote the two pages(this is the only useful piece of information), however the books called Nonplussed! Mathematical proof of implausible ideas.
Christiaan Huygens and Rene Francois de sluze discovered this. Sadly I can't find anything about this in wikipedia.

I don't believe this is true because you can't visualize it. You can visualize higher dimensions, but infinte volume with finite surface area. This is impossible.

[69dodge]

I think you misinterpreted the book's claim. It doesn't appear to deal with surface area at all. The weight of a glass goblet doesn't depend on its surface area, but on the volume of the glass it's made of. So the claim is that a goblet can contain an infinite volume of wine, yet be made of only a finite volume of glass.

You also misinterpreted the video. The part you refer to describes Galileo's confusion about infinity. We understand infinity better now, and know that a small circle and a large circle consist of the same number of points. (But it is true that there are other sizes of infinity, besides the one that is the number of points on a circle.)

[becomingagodo]

"He continued. "So, we have the Banach-Tarski Theorem, or Banach-Tarski Paradox. An equivalent form of this theorem states that a solid of any size, say that of a small pea, can be partitioned into a finite number of pieces, and then reassembled to form another solid of any specific shape and volume, say that of the sun. Consequently, this paradoxical theorem of Stefan Banach and Alfred Tarski is sometimes referred to as the pea and the sun paradox.""
I didn't need to know that. You made everything a million times worse.
http://www.daviddarling.info/images/Banach-Tarski_paradox.png
Don't worry about the 'madman' quote. I don't think we're in any danger of anyone understanding infinity, though some have made good stabs at it.
Isn't mathematics about understanding or insight?
Saying that this does depend on the axiom of choice, which some mathematician don't trust.
The weight of a glass goblet doesn't depend on its surface area, but on the volume of the glass it's made of. So the claim is that a goblet can contain an infinite volume of wine, yet be made of only a finite volume of glass.
Oh yes I can see now.
and know that a small circle and a large circle consist of the same number of points. (But it is true that there are other sizes of infinity, besides the one that is the number of points on a circle.)
I understand that now

[Complexity]

I didn't need to know that. You made everything a million times worse.


:wave1

Evil I am!

I did warn you that it would make you unhinged...

Math and science are full of things like this.

Reality is so much more interesting than woo.


Isn't mathematics about understanding or insight?


Trying to gain understanding and actually gaining understanding are very different things.

I don't think I fully understand a damned thing, and I don't think anyone else does, either.

However, I have a lot of fun trying to understand things, and sometimes I think I'm in the right neighborhood.


Saying that this does depend on the axiom of choice, which some mathematician don't trust.


In some other post, about the loss of certainty in mathematics, I mentioned that sometimes axioms are chosen rather arbitrarily. The Axiom of Choice is one of these axioms, if I remember correctly.

Many mathematical theories have assumed the Axiom of Choice and their theorems may depend upon this axiom.

Other mathematical theories have not assumed the Axiom of Choice and their theorems do not depend upon this axiom.

The Axiom of Choice is neither true or false, and the decision to include it as an axiom of a mathematical theory that you wish to study is fully up to you.

Many may make such a choice out of tradition, philosophical inclination, or pragmatics, but both choices can be interesting.

Math is about the study of consequences.

[Nancarrow]

Hello becomingagodo, I've followed your journey on these boards with some interest. I'm also trying to get to grips with lots of higher maths (whilst actually tutoring 13-18 year olds in what they need to know!). I've no idea about your finite-area-infinite-volume shape, although I share the intuitive sense that it's impossible (but then maths is great for crushing intuitions like the bugs they are). But I thought I'd respond to your other musings.


Another thing, what is a number?


I think Fredrik nailed it. His advice is the same as that given by Tim Gowers (Fields Medalist and fellow of my old college! :Banane01:) in his VSI book[1]. Basically a number is what a number does, and the same goes for other bread and butter concepts of mathematics, such as 'points' and 'lines'. I spent quite a while trying to work out 'what a number is', and the best I can do is 'implicit' definition - a number is an object that obeys axioms x,y and z. Exactly which axioms depends on which kind of number you're talking about and how deep you're going. Here's Stewart and Tall's take on natural numbers from [2] (I think they're in turn riffing off Peano):

A 'natural number' is a member of a set N, upon which a function F:N->N is defined, such that

1) F is not surjective: there is a natural number, let's call it '0', such that F(n)=0 is not true for any n.

2) F is injective: if F(m)=F(n) then m=n. (I believe the converse holds true by the definition of a 'function'... confirmation anyone?)

3) Induction: If 0 is a member of a set S, and whenever n is a member of S, then F(n) is also a member of S, then S=N.

All we need now are a couple of definitions of addition and multiplication:
A) m + 0 = m, and m + F(n) = F(m+n)
M) m * 0 = 0, and m * F(n) = m*n + n

...and, as they say in the book, 'from these slender beginnings we can develop all the usual properties of arithmetic, then later build up the other number systems...'. Pretty cool, huh?

I am buying John Conway book on Number and Games, he describes numbers differently(I feel kind of quilty for buying his book, I don't consider Game theory good mathematics). Saying that John Conway work is crazy.

I've got that book! I got it the same time as Stewart and Tall. I've spent more time with S&T, simply because it seems to me that ONAG is more advanced - Conway likes to think he's making it easy but he really goes very fast! The message I got from it was, Conway was developing a mathematical theory of games (simple kinds, where you have two players who take turns, and one of them loses when they don't have a legal move left), and he found that the structures he created also modelled the real numbers.

In fact the real numbers were embedded in them - his larger system of 'surreal numbers' has lots of other rather cool things in them. So f'rinstance he defines an 'infinity' which seems to me to be what they usually call 'aleph-null', but then in his system you can have lots of cool numbers like 1/inf, sqrt(inf), 2inf+3 and so on, all of which are distinct 'surreal' numbers. So the first part of his book develops the axiomatic theory of surreal numbers, the second shows what all that has to do with games. (A game is just a surreal number with one key defining restriction lifted)

So technically you could actually use his concept of 'surreal number' to give an axiomatic model of the real numbers. However, as he says in the book, it's probably best to keep teaching real numbers the usual way. In particular, numbers of the form n/2^k ('dyadic numbers', e.g. -1/2, 9/4, 17/64 etc) have a 'special place' within the development of surreal numbers, that they don't really deserve in a tidy number theory (whereas fractions like 1/3 are in a certain sense hideously ugly).

But I ramble...

On another note, I can slowly feel infinty grinding away my sanity, hence why I need to understand the top question. Whats that quote that ends in "only a madman can understand infinty". Infinty it could make you cry

You're not alone. Here's another book for you, Eves[3], late in the book he talks about the three great 'crises of foundational mathematics'. Those are the discovery of the irrationals around the 500 BCs, the increasingly shaky foundations of calculus from 1666 to about the mid 19th c, and all sorts of paradoxes in set theory. The first two crises were resolved. He says the third is still going strong, but the book's a bit old, dunno if that's true. Anyway all those crises have come about as we've been struggling with the concept of infinity.

My advice for brain-mangling ideas such as that is, when you come across a new idea in your studies that has bottomless pits in, start by erecting a big fence around those pits saying 'do not cross'. Get completely comfy with the parts of the ideas that are straightforward. Only once you're very comfy with the 'normal working' of the ideas, then go ahead and take a peek at how the pros have dealt with, or tried to deal with, the bottomless pits.

To take a very simple example, when I'm teaching quadratic equations to GCSE kids, I've got no problem telling them 'if you find yourself taking a square root of a negative number, you can't do this, there's no such thing, it means the quadratic has no solutions'. Usually, if they're fairly bright and I think they can handle it, I'll parenthesise accordingly and tell them to wait for the higher modules in A-level.

Good luck in your journey! Feel free to compare notes!

[1] Tim Gowers, 'Mathematics: A Very Short Introduction', OUP
[2] Ian Stewart and David Tall, 'The Foundations of Mathematics', OUP
[3] Howard Eves, 'Foundations and Fundamental Concepts of Mathematics', Dover

[technoextreme]

Okay, I just found out their are objects with infinte volume, but finte surface area.

What the hell?

I know their are objects with infinte surface area and finte volume. I know how to visualize this, however I don't know how to visualize something with a infinte volume but finte surface area.

Without going into higher dimensions I don't see how this is possible. Can someone explain, note the more visual the better.

Man, this has been diving me crazy all day.


Just ignore it. It happens in two dimensional space also. The integral of the dirac delta function is one even though the width is zero and the height is infinite. Technically speakings it's not a true function but it's extremely useful.

[bjornart]

Just ignore it. It happens in two dimensional space also. The integral of the dirac delta function is one even though the width is zero and the height is infinite. Technically speakings it's not a true function but it's extremely useful.

I think that's different. The integral of the dirac delta function is merely one by definition, isn't it?

[Southwind17]

I vaguely recall being shown, during advanced-level mathematics, how to calculate the time taken for a ball bearing, when dropped on a hard surface, to bounce an infinite number of times. It was, of course, a finite period.

I never could get my head around it, which probably explains why I failed! I doubt I'd understand it any better now. :boggled:

[Jekyll]

I think that's different. The integral of the dirac delta function is merely one by definition, isn't it?

No, you approach it as a limit point of a sequence of increasingly spiky functions of area 1.

[Unalienable]

"The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type."
-- Georg Cantor

"Mathematics is the simple bit. What's complicated are cats. How does one even go about defining what a cat is? I have no idea."
-- John Conway

[Southwind17]

"The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type."
-- Georg Cantor

"Mathematics is the simple bit. What's complicated are cats. How does one even go about defining what a cat is? I have no idea."
-- John Conway

Great, thanks for clearing that up! :confused:

[bjornart]

No, you approach it as a limit point of a sequence of increasingly spiky functions of area 1.

I stand corrected.

[cyborg]

Infinity is beautiful and very disturbing.

No doubt. If you stop anything finitely all the problems melt away.

Here what we are seeing is an interplay between convergence and divergence in two different dimensions - this is quite easy to see in the Gabriel's Horn example: whilst the volume is converging upon a finite value the surface area is constantly diverging away from one. Again if we stop at any point we have a definite and finite answer. If we don't...

[becomingagodo]

I think Fredrik nailed it. His advice is the same as that given by Tim Gowers (Fields Medalist and fellow of my old college! ) in his VSI book[1]. Basically a number is what a number does, and the same goes for other bread and butter concepts of mathematics, such as 'points' and 'lines'. I spent quite a while trying to work out 'what a number is', and the best I can do is 'implicit' definition - a number is an object that obeys axioms x,y and z. Exactly which axioms depends on which kind of number you're talking about and how deep you're going. Here's Stewart and Tall's take on natural numbers from [2] (I think they're in turn riffing off Peano):
I don't like that argument.

Basically leave numbers as a set of rules or properties.

Numbers must be something, even if that something is weird. Saying that I haven't got a clue what numbers are.

I've got that book! I got it the same time as Stewart and Tall. I've spent more time with S&T, simply because it seems to me that ONAG is more advanced - Conway likes to think he's making it easy but he really goes very fast! The message I got from it was, Conway was developing a mathematical theory of games (simple kinds, where you have two players who take turns, and one of them loses when they don't have a legal move left), and he found that the structures he created also modelled the real numbers.
I'm don't like real numbers that much, more of a fan of other numbers like hyperreals. I think Conway is going in the right direction or his ideas our right.

[cyborg]

Numbers must be something,

Well if you don't like that numbers are descriptors then that's too bad really - they don't have any physical substance to them. The 'something' that they are is only ever in reference to something else.

It is much like being presented with a 'bunch of apples' and demanding that the 'bunch' has as much substance as the 'apple'. It doesn't. It's just a way of grouping the physical objects - and there are an infinite number of ways of doing that none of which are any more substantial than the other. The only substance you'll get is the substance of the physical computations used to formulate these groups in the first place - namely that spongy organ in your head.

[becomingagodo]

Well if you don't like that numbers are descriptors then that's too bad really - they don't have any physical substance to them. The 'something' that they are is only ever in reference to something else.
I don't believe this to be true. Numbers can be higher things. A example is the order of the prime numbers, this is a deep problem. My intuition tells me that number must be something, even if their just pattarns.

Pattarns are not meaningless and they do show something bigger, Prime number as a example. The pattarn of a Prime number or the Zeta function leads to quantum chaos theory(well thats what I heard). Complex numbers are showing something profond too.
My advice for brain-mangling ideas such as that is, when you come across a new idea in your studies that has bottomless pits in, start by erecting a big fence around those pits saying 'do not cross'.
Isn't that harming creativity.
Gregory Chaitin argues in Metamaths that knowledge is harmful to creativity. What if mathematician are leading you down the wrong road. Mathematics is biased towards things that are considered interesting or improtant. Then what if were asking the wrong questions or that orignallity is lost by reading other works or following them. A good example is Ramanujan who is proberly the most creative mathematician and part off it is proberly due to lack of influence on him by other mathematicians.

A mathematician should drop head first into the bottomless pit, instead of waiting outside.

[cyborg]

I don't believe this to be true.

I understand why you can't accept this.

Numbers can be higher things.

And numbers talking about numbers higher still. Yet this higher order is synthetic.

A example is the order of the prime numbers, this is a deep problem. My intuition tells me that number must be something, even if their just pattarns.

Patterns, numbers etc... are again just in reference to something else. Their substance is dependent on it. They do not stand alone.

Pattarns are not meaningless and they do show something bigger, Prime number as a example. The pattarn of a Prime number or the Zeta function leads to quantum chaos theory(well thats what I heard). Complex numbers are showing something profond too.

Re-read what Complexity said: mathematics is the study of consequences. There is nothing profound here: mathematics is powerful enough to talk about all possible consequences. As such one should not be surprised that it is able to talk about the consequences of the physical universe we happen to find ourselves in.

[Jekyll]

I don't like that argument.

Basically leave numbers as a set of rules or properties.

Numbers must be something, even if that something is weird. Saying that I haven't got a clue what numbers are.

You're looking at this backward.

Numbers are lots of things. Now I can either restrict the set of things that can be numbers, or I can say anything which satisfies my criteria of what a number is is a number. So numbers are anything which follows these number rules and so shares the same patterns.

[Jekyll]

I stand corrected.

To be fair, its a fairly arbitrary method, and you could use the same method to create a delta function that integrates to what ever you like.

[Nancarrow]

I don't like that argument.

Basically leave numbers as a set of rules or properties.

Numbers must be something, even if that something is weird. Saying that I haven't got a clue what numbers are.



Well now hang on a second. Be more careful in your phrasing! I did not leave numbers as a set of rules or properties. I said they were objects that obeyed certain rules and had certain properties. If you insist that numbers must be something, well, that's just what they are!

I'm don't like real numbers that much, more of a fan of other numbers like hyperreals. I think Conway is going in the right direction or his ideas our right.

Er, ok, except that earlier you said you didn't think game theory was 'real maths' and you thought Conway was 'crazy'. :boggled:

Isn't that harming creativity.
Gregory Chaitin argues in Metamaths that knowledge is harmful to creativity. What if mathematician are leading you down the wrong road. Mathematics is biased towards things that are considered interesting or improtant. Then what if were asking the wrong questions or that orignallity is lost by reading other works or following them. A good example is Ramanujan who is proberly the most creative mathematician and part off it is proberly due to lack of influence on him by other mathematicians.

A mathematician should drop head first into the bottomless pit, instead of waiting outside.

Well I didn't say you should fence off the bottomless pits indefinitely, only until you're comfortable with the easier stuff around them. IOW if you can't do simple A-level questions you're probably not going to have much luck with topology or category theory!

I certainly agree with you that the maths-community-at-large selects certain problems as more interesting than others, there's no reason you can't go off and study your own questions. But first you ought to know what sort of questions can be asked. And you ought to know what questions have already been answered, if you don't want to waste your life reinventing the wheel!

[becomingagodo]

Er, ok, except that earlier you said you didn't think game theory was 'real maths' and you thought Conway was 'crazy'
We need more crazy people in mathematics. At this moment I am so confused I don't even know what mathematics is.
I said they were objects that obeyed certain rules and had certain properties. If you insist that numbers must be something, well, that's just what they are!
Numbers must be something more then this.
Well I didn't say you should fence off the bottomless pits indefinitely, only until you're comfortable with the easier stuff around them.
But then you lose originallity, you become a good mathematician. The best method would to go creatively crazy and the only way to do that is to jump into the bottemless pit.
But first you ought to know what sort of questions can be asked. And you ought to know what questions have already been answered, if you don't want to waste your life reinventing the wheel!
Thats a good point.

[cyborg]

Numbers must be something more then this.

Why must they?

[Complexity]

becomingagodo - Becoming a mathematician is a lot like becoming a doctor.

There is a great deal to be learned, some of which is best learned in a certain order.

There are ways of being exposed to things, ways of teaching, metaphors, and experiences that have been demonstrated to be of value.

Those things that excite interest in the field and motivate one's involvement in the field should not be allowed to distract one from learning the basics thoroughly.

Learning some things out of order can make going back and learning the early things well and thoroughly, exercising them enough so that become part of you, seem to be unbearably tedius. You may skimp on acquiring these skills and this knowledge, to your permanent injury.

Mentoring is good.

There are many fascinating specialties, but one must not specialize too early.

Becoming competent will take a long time.

Few will become great.

Please don't be so impatient that you mess things up for yourself.

Think of those fenced-off areas as the barricades parents put in place to keep their kids from falling down stairs, eating the garbage, or getting into medicines or poisonous cleaning supplies.

The barriers are put there to keep kids out of trouble, but they aren't usually there for life.

You are in danger of maiming your ability to learn mathematics by constantly darting off to look at crashes or oooo and ahhh at sparkly things. You are very distractable right now. It would be far better for you to spend a lot of time on the basics and keep a notebook of things that you'd like to learn about or look into later.

The madness / crazy thing isn't a useful line of thought.

I truly wish you well in your studies. I wouldn't be posting all of this if I didn't.

[becomingagodo]

becomingagodo - Becoming a mathematician is a lot like becoming a doctor.
I don't see the comparison.
There is a great deal to be learned, some of which is best learned in a certain order.
Most mathematical concepts get used to death. Mathematics is more then just learning.
Those things that excite interest in the field and motivate one's involvement in the field should not be allowed to distract one from learning the basics thoroughly.
But, thats boring. Plus if you don't know the concepts behind it then how can you understand something throughly. I wanted to understand calculus and infinty is their, how can one sleep at night when his knowledge depends on being a robot? Nobody understand infinty and at best they become good at calculus.
You may skimp on acquiring these skills and this knowledge, to your permanent injury.
Or permanent freedom.

[becomingagodo]

There are many fascinating specialties, but one must not specialize too early.
I don't plan to specialize, it keeps you back. I still can't see why you just don't learn complex analysis? all mathematician think the same, I don't see how you can be weak at something.
Think of those fenced-off areas as the barricades parents put in place to keep their kids from falling down stairs, eating the garbage, or getting into medicines or poisonous cleaning supplies.

The barriers are put there to keep kids out of trouble, but they aren't usually there for life.
What trouble can you get by thinking about higher mathematics.
darting off to look at crashes or oooo and ahhh at sparkly things.
No, I wouldn't describe it like that. It more like someone pointing a gun at you. Would you look away and do something else. Again calculus is a good example, do you look at infinty pointing a gun at you or just ignore it.
It would be far better for you to spend a lot of time on the basics
The basics, and what about creativity and originallity?
It would slowly melt away.

[Complexity]

I don't see the comparison.


Well, work harder at understanding it. That's rather a glib dismissal.


Most mathematical concepts get used to death. Mathematics is more then just learning.


You've got some wrong ideas that are holding you back and you need to dump them, the sooner the better.

If you don't, you are not going to become even a competent mathematician.

I am unaware of any mathematical concepts that 'get used to death'. Give some examples.

Mathematics is more than just learning. However, the learning has to come first, and it will take many years.


But, thats boring. Plus if you don't know the concepts behind it then how can you understand something throughly.


I'm sorry that you find learning the basics boring. That is a fairly common reaction, I'm afraid.

The life work of many mathematicians have gone into creating and refining the basics, however, and they found it well worth their lives to do so.

The basics that you so cavalierly dismiss are essential and must be mastered with sufficient enthusiasm to carry you through the rough or tedius parts.

You're not going to get a concept all at once. You'll encounter concept X several times during your mathematical education and later research. Each time you encounter concept X, you'll have more knowledge, more experience, and more maturity - you'll be able to understand more about X, be able to use it more effectively, and richer questions emerge in your mind.

For example, consider how we teach children about atoms. We give them a sequence of metaphors that are known to be inaccurate but which become increasingly closer to how we understand the world.

The world is made up of atoms that are can't be divided and that bounce around in a void.

An atom is like a little solar system, mostly empty space, that has a nucleus for a 'sun' and electrons as 'planets'.

The nucleus of an atom is made up of protons, which carry a positive charge, and enough neutrons, which carry no charge, to somehow keep the protons from flying apart.

Electrons shouldn't be thought of as little points of negative charge - each electron is spread out in some way.

The 'shapes' that electrons occupy/are aren't at all like planetary orbits but rather can be spherical or even a donut shape with lobes.

Quarks.

Quantum mechanics.

...

The model that is presented is carefully tailored to the knowledge and experience of the child in question.

The same applies to the learning of mathematics. You will be introduced to concepts in a way that experience has shown works well. You will revisit concepts many times at different points in your education, each time being given a richer version of the concept.

Will you ever fully understand a concept? I say 'no'. This is not a reflection on you. I don't think anyone fully understands anything. This simply means that we must do the best with the understanding that we have of things and try to improve our understanding of things.


I wanted to understand calculus and infinty is their, how can one sleep at night when his knowledge depends on being a robot?


You've brought up this 'robot' business before. You seem to think that learning 'tedious' things, memorization, and working a large number of problems are things that a robot can do / should be doing, and have nothing to do with fun and creative mathematics.

You're wrong. You may find learning many basics boring, but they aren't, really - they'll be as boring as you make them. Work on them with enthusiasm and dedication and you'll be much better off.


Nobody understand infinty and at best they become good at calculus.


As I said, no one fully understands infinity (or anthing else), but our collective understanding of infinity continues to grow.

You will come to learn much more about infinity as you learn more mathematics. You'll become able to appreciate richer treatments of infinity as you progress through mathematics.


Or permanent freedom.


Wrong. This is the 'mad genius' trap.

Quit trying to rush things. Learn the basics really well and in a reasonable order.

[jsfisher]

all mathematician think the same, I don't see how you can be weak at something.


That is just not true at all. I don't mean this as any sort of put-down, becomingagodo, but you are still young and your exposure to mathematics barely scratches the surface of arithmetic with a dash of geometry. Your rejections of Turing and Conway as real mathematicians is a symptom of this. There is so much more out there to appreciate; take the time to explore and discover.

Complexity, for example, has already told you he has a side interest in the RH, but his passion is for graph theory. Graph theoretic concepts, however, aren't anything like what is needed to explore non-Euclidean geometries.

I, on the other hand, like automata theory and I can hold my own in some parts of modern algebra (which is nothing like high-school algebra). I don't cope as well with topology, and differential equations are best ignored by me.

The point is different branches of mathematics -- just like different branches of science -- require different interests, talents, and mindsets. Mathematicians do not all think alike.

[Complexity]

I don't plan to specialize, it keeps you back. I still can't see why you just don't learn complex analysis? all mathematician think the same, I don't see how you can be weak at something.


You don't yet have any idea of how huge mathematics has become.

There was a time, perhaps as recently as 200 years ago, when there were still a few mathematicians who were knowledgeable about all fields of mathematics, but the sheer amount and diversity and richness of mathematics has grown so much in that time that no one is aware of all of the different fields of mathematics, let alone being knowledgeable about them.

There are some basics that are common to the education of most mathematicians, but each field that is touched on more enormous than you can imagine, and most fields aren't touched on.

I've taking courses in real and complex analysis. I'm not ignorant of these fields, but I do recognize that I don't feel a strong affinity to them. I find them fascinating, but I haven't found reason to believe that I'm going to ever become good enough at them to do creative work in those fields.

I keep learning about many fields of mathematics (e.g. knot theory) regardless of whether I think I'll ever learn more than introductory aspects of the field.

However, since I want to do some creative work in at least one area of mathematics, it makes sense for me to focus much of my time and energy on field(s) that I feel an affinity for and that fascinate me.

If you don't specialize and attempt to learn a little bit of everything, you'll be incredibly frustrated, you'll fail in an impossible quest, and you will be very unlikely to do any significant mathematics.

Mathematicians do not all 'think the same'.

I will never learn of the existence of most of mathematics. I will be incredibly weak in most of the fields that I become aware of. I may become a bit competent in some aspects of mathematics. I hope that I become the first to gain a better understanding of some sliver of mathematics.

There are many much better mathematicians than I am. In fact, I'm self-conscious about applying the term to myself, even though it accurately describes some of what I do.

I think that the description of my mathematical possibilities (two paragraphs ago) applies to all mathematicians and would-be mathematicians, including yourself.


What trouble can you get by thinking about higher mathematics.


Before you have mastered the basics? You can get so distracted and discouraged and confused that you may abandon mathematics, never really having given yourself a chance to learn it properly and with respect.

By all means, learn about what you can of mathematics while you are learning the basics with care. I have found that reading the history of mathematics is a great way of doing this. It is fascinating and a great motivator when one is beginning a problem set of 60 calculus problems.


No, I wouldn't describe it like that. It more like someone pointing a gun at you. Would you look away and do something else. Again calculus is a good example, do you look at infinty pointing a gun at you or just ignore it.


You learn enough to get by for now, and trust educators and mentors about how much is enough.


The basics, and what about creativity and originallity?
It would slowly melt away.


No, it doesn't. Keep a notebook, write down your ideas, explore them given what you understand at the time. Do this while plugging away at what you need to learn.

[Southwind17]

I still can't see why you just don't learn complex analysis? all mathematician think the same, I don't see how you can be weak at something.

What trouble can you get by thinking about higher mathematics.

The basics, and what about creativity and originallity?
It would slowly melt away.

Complexity is right. I assume you've never learnt to play a musical instrument. It takes time and patience, and whilst it's tempting to try to learn the tricky stuff before the basics it will lead to bad habits that are difficult to correct later, that's assuming your interest doesn't wane through frustration first.

[fuelair]

And what are those abstart numbers anyway?

[Soapy Sam]

I'll believe in any infinite surfaced (or volumed) object when I see one painted blue and filled with beer.

Call me a sceptic.

As for numbers- they're whatever you want them to be .

If you have three bricks and you take away the bricks - what do you have now?

[Mashuna]

I'll believe in any infinite surfaced (or volumed) object when I see one painted blue and filled with beer.

Call me a sceptic.

As for numbers- they're whatever you want them to be .

If you have three bricks and you take away the bricks - what do you have now?

Hmm, if you have three bricks and I take away the bricks, I've got three bricks. :p

[cyborg]

I'll believe in any infinite surfaced (or volumed) object when I see one painted blue and filled with beer.

You couldn't see it. You are too small.

Call me a sceptic.

You're a sceptic.

If you have three bricks and you take away the bricks - what do you have now?

Omega.

[Stir]

I'll believe in any infinite surfaced (or volumed) object when I see one painted blue and filled with beer.


OK -- refer all the way back to post #2, which looks a bit like a joke, but is in fact completely correct. First fill an infinite universe with beer, then pick one tiny bubble and paint its inside surface blue ... and Soapy has his infinite volume of beer with a finite blue surface.

The important thing is that mathematically a surface does not define an 'inside' and an 'outside'. Think about a small circle on the surface of the earth. That circle divides thearth into 2 regions, and most people would considet the smaller of the 2 to be the interior. Let that circle get larger (i.e. increase its radius) until it's a "great circle" i.e. roughly the size of the equator ... and keep going. As you increase the radius beyond that point, suddenly the region considered the interior is larger than the exterior ... and you can continue increasing the radius until the area of the 'exterior' is very small and the interior approaches the area of the globe. Did the interior and exterior suddenly switch? or was the original perception of interior and exterior mistaken?

With that in mind: an infinite volume can be divided by a simple finite surface into two pieces, one of which could be finite, and at least one must be infinite. And that finite surface bounds an infinite volume.