Fun with Math

[scruffy]

Everyone is familiar with the concepts of "point" and "line". We generally consider that there are an infinite number of points in a line.

But here's a mindfuck.

Start with a line (or a line segment). Remove the middle third. For each remaining segment, remove the middle third, and keep doing that to all remaining segments

This construction is called a Cantor Dust, and the interesting thing about it is, the dust can be proven to have THE SAME NUMBER OF POINTS AS THE LINE IT ORIGINALLY CAME FROM.

Makes no sense at all, but the math says it must be true.

Cantor set - Wikipedia

en.wikipedia.org

A further surprise: the Cantor Dust is compact! Totally illogical, but the math says it must be true.

Even weirder still - the Dust can be measured and integrated. It has a Haar measure and a non-zero Hausdorff dimension, even though its Lebesgue measure is 0.

And, when the Haar measure is normalized so the set measures 1, the Cantor Dust becomes formally identical to an infinite series of coin tosses

This remarkable set is self similar (fractal) and has a conservation law associated with it, even though the set is nowhere dense.

 

[gtopa1]

Everyone is familiar with the concepts of "point" and "line". We generally consider that there are an infinite number of points in a line.

But here's a mindfuck.

Start with a line (or a line segment). Remove the middle third. For each remaining segment, remove the middle third, and keep doing that to all remaining segments

This construction is called a Cantor Dust, and the interesting thing about it is, the dust can be proven to have THE SAME NUMBER OF POINTS AS THE LINE IT ORIGINALLY CAME FROM.

Makes no sense at all, but the math says it must be true.

Cantor set - Wikipedia

en.wikipedia.org

A further surprise: the Cantor Dust is compact! Totally illogical, but the math says it must be true.

Even weirder still - the Dust can be measured and integrated. It has a Haar measure and a non-zero Hausdorff dimension, even though its Lebesgue measure is 0.

And, when the Haar measure is normalized so the set measures 1, the Cantor Dust becomes formally identical to an infinite series of coin tosses

This remarkable set is self similar (fractal) and has a conservation law associated with it, even though the set is nowhere dense.

Undefined number imo; how big is infinity or half thereof?? Infinity cannot be "halved".

Greg

 

[scruffy]

Undefined number imo; how big is infinity or half thereof?? Infinity cannot be "halved".

Greg

But it can be doubled, and even multiplied by itself.

One of the key concepts is "countability". Some sets are not countable. Usually it means they have too many members to count, so we call them infinite (in-finite).

Uncountable set - Wikipedia

en.wikipedia.org

Infinities have a lot to do with cardinality. There is an infamous piece of math called the Continuum Hypothesis that says there is no set with cardinality strictly between the integers and the reals.

Aleph number - Wikipedia

en.wikipedia.org

Cardinality is represented by the Aleph number, and N0 is the cardinality of the natural numbers, and N1 is the cardinality of the reals. Alephs are due to Georg Cantor

The interesting result is that 2^N0 = N1

Not every set can be labeled. Normally when we count we just assign labels, 1, 2, 3 ... they're called ordinal numbers. But we could really use "any" meaningful label. This approach is mostly due to Richard Dedekind, the king of the one to one correspondence.

What's an ordinal number? Just a unique label

Ordinal number - Wikipedia

en.wikipedia.org

Infinity - Wikipedia

en.wikipedia.org

 

[scruffy]

Okay, here's an interesting one.

The Partition Function. p(x)

You can look at it like, how many ways can you partition a number.

For example, p(4) = 5, because

1+1+1+1
1+1+2
1+3
2+2
4

Partition function (number theory) - Wikipedia

en.wikipedia.org

This function may look funny, but it has an inverse, and the inverse of its generating function is the Euler function.

Euler function - Wikipedia

en.wikipedia.org

 

[gtopa1]

But it can be doubled, and even multiplied by itself.

One of the key concepts is "countability". Some sets are not countable. Usually it means they have too many members to count, so we call them infinite (in-finite).

Uncountable set - Wikipedia

en.wikipedia.org

Infinities have a lot to do with cardinality. There is an infamous piece of math called the Continuum Hypothesis that says there is no set with cardinality strictly between the integers and the reals.

Aleph number - Wikipedia

en.wikipedia.org

Cardinality is represented by the Aleph number, and N0 is the cardinality of the natural numbers, and N1 is the cardinality of the reals. Alephs are due to Georg Cantor

The interesting result is that 2^N0 = N1

Not every set can be labeled. Normally when we count we just assign labels, 1, 2, 3 ... they're called ordinal numbers. But we could really use "any" meaningful label. This approach is mostly due to Richard Dedekind, the king of the one to one correspondence.

What's an ordinal number? Just a unique label

Ordinal number - Wikipedia

en.wikipedia.org

Infinity - Wikipedia

en.wikipedia.org

Infinity means not to many to count; it means that there are no limits to the "end" of the count. There is NO END. So an infinite number cannot be halved as there isn't ONE NUMBER.



Greg

 

[Wuwei]

This construction is called a Cantor Dust, and the interesting thing about it is, the dust can be proven to have THE SAME NUMBER OF POINTS AS THE LINE IT ORIGINALLY CAME FROM.

It is very easy to show that there are just as many points on a 2 inch line as there are on a three inch line.
It follows from the fact that there is a simple bijective function between each point of the two lines.
That fact can be applied at each level of the construction.

More fun with infinity

Here is a cute proof that 0.999999.... = 1 exactly.
Let x = .99999....
Multiply both sides by 10:
10x = 9.9999.....
10x = 9 + .99999....
10x = 9 + x
9x = 9
x = 1
.9999 ... = 1

More fun: Look up The Infinite Hotel Paradox.
.

 

[scruffy]

Infinity means not to many to count; it means that there are no limits to the "end" of the count. There is NO END. So an infinite number cannot be halved as there isn't ONE NUMBER.



Greg

Not so. Many infinite series converge to a single well defined number.

And, while one can't exactly say where the halfway point is, one can take "this much of it".

 

[scruffy]

It is very easy to show that there are just as many points on a 2 inch line as there are on a three inch line.
It follows from the fact that there is a simple bijective function between each point of the two lines.
That fact can be applied at each level of the construction.

More fun with infinity

Here is a cute proof that 0.999999.... = 1 exactly.
Let x = .99999....
Multiply both sides by 10:
10x = 9.9999.....
10x = 9 + .99999....
10x = 9 + x
9x = 9
x = 1
.9999 ... = 1

More fun: Look up The Infinite Hotel Paradox.
.

So... partitions of zero.

Work has been done on rings like Z/nZ, and also zero frequency spin with phase partitions.

Zero is usually by definition "a" point or the empty set. Yet it can be partitioned if constructed more carefully.

 

[scruffy]

Partitions of unity:

Partition of unity - Wikipedia

en.wikipedia.org

 

[scruffy]

So, "partitions" in the traditional sense have to do with the relationships between numbers. (Number theory).

But partitions in the more general sense, are anything that adds up to the number. Hence partitions of 0, and partitions of unity. There are plenty of functions and sets of functions that have this behavior.

 

[zaangalewa]

It is very easy to show that there are just as many points on a 2 inch line as there are on a three inch line.
It follows from the fact that there is a simple bijective function between each point of the two lines.
That fact can be applied at each level of the construction.

More fun with infinity

Here is a cute proof that 0.999999.... = 1 exactly.
Let x = .99999....
Multiply both sides by 10:
10x = 9.9999.....
10x = 9 + .99999....
10x = 9 + x
9x = 9
x = 1
.9999 ... = 1

More fun: Look up The Infinite Hotel Paradox.
.

But what when on the 3rd quadrillionst position after the dot is a water drop and no digit?

 

[Wuwei]

But what when on the 3rd quadrillionst position after the dot is a water drop and no digit?

The basic assumption is that all integers following the decimal point are nines. If not then the proof does not apply to that case.

 

[zaangalewa]

The basic assumption is that all integers following the decimal point are nines. If not then the proof does not apply to that case.

Is this the real world? Or is this "only" a pure spiritual construction which never anyone will be able to prove? I ask this because this is politically or ideologically relevant today: "Materialism vs spirituality". Many scientists seem to be today extremely bad philosophers although "natural philosophy" and "physics" are the same. In this context I personally call mathematics "the spirituality of physics" - but mathematics is much more than to be only a servant of natural science. The question is in this context : Is mathematics a pure human invention? If in such a case something exists in the universe (in the nature) what is not mathematics - are we able to understand it in this case? Concrete here: Where's the place for this never ending row of number nines? Only in our brains? Between our brains? In the nature? In heaven? ... Are we able to imagine a universe without mathematics? ... How do we call this? ... ¿"Helll"?

 

[Wuwei]

Is this the real world? Or is this "only" a pure spiritual construction which never anyone will be able to prove? I ask this because this is politically or ideologically relevant today: "Materialism vs spirituality". Many scientists seem to be today extremely bad philosophers although "natural philosophy" and "physics" are the same. In this context I personally call mathematics "the spirituality of physics" - but mathematics is much more than to be only a servant of natural science. The question is in this context : Is mathematics a pure human invention? If in such a case something exists in the universe (in the nature) what is not mathematics - are we able to understand it in this case? Concrete here: Where's the place for this never ending row of number nines? Only in our brains? Between our brains? In the nature? In heaven? ... Are we able to imagine a universe without mathematics? ... How do we call this? ... ¿"Helll"?

I always wondered if an infinite decimal of 9's was infinitesimally smaller than 1 or exactly 1. The proof I gave shows it is exactly 1. It is just a curiosity.

Physics is useful for discovering the behavior of nature and modeling it with math as a precise description. For example physics does not know what a photon is, but knows exactly how it behaves under various conditions. Understanding what a photon is would be under metaphysics.

My feeling is that math is not a human construct, but a discovery of something that already exists. I am astounded by the fact that the universe can be modeled so exactly by math. It's something that many take for granted.

 

[zaangalewa]

I always wondered if an infinite decimal of 9's was infinitesimally smaller than 1 or exactly 1. The proof I gave shows it is exactly 1. It is just a curiosity.

Physics is useful for discovering the behavior of nature and modeling it with math as a precise description. For example physics does not know what a photon is, but knows exactly how it behaves under various conditions. Understanding what a photon is would be under metaphysics.

My feeling is that math is not a human construct,

Are we a construct of mathematics? Or of a pure spirit?

but a discovery of something that already exists. I am astounded by the fact that the universe can be modeled so exactly by math. It's something that many take for granted.

That's every day astonishing me again. Mathematics says something should be like that - and reality answers this is correct. That's unbelievable - but I never saw any exception. The world is reasonable - perhaps not perfect but reasonable. Are we reasonable?