zen and taoism

Nice to see you again.

You’ve asked a lot of questions, and I’ll try my best.
Ultimately this post becomes another couple lessons. 😉

>>>Countable infinity, or just countable means
>>>that the each number in the set is an integer correct? No decimals?

Actually, no.

Strangely enough, if you consider the set of all fractions between
0 and 1, i.e. not only numbers like 0 and 1, but things like 1/2,
1/3, 2/11, 5/19, 17345/27235, etc., then the total number of all
these things (while “seemingly larger than the set of all integers
due to the infinite density of all the fractions”) is STILL only
the countable infinity. Of course, all the fractions here do
have decimal expansions . . . for example, 1/2=.5 and 1/3=.3333…
etc. The reason why the set of all decimal expansions, i.e.
points between 0 and 1, is the continuum infinity is that there
are a *whole lot* of decimal expansions that are not fractions.
In fact, despite the fact that there are infinitely many fractions,
each of which have decimal expansions (and some infinite decimals),
proportionally speaking it is true that *most* of the decimal
strings you can come up with are *not* fractions (i.e.
representable as a/b).

We saw that the jar of infinitely many blue marbles is
the countable infinity. We also saw in the first three
posts that you can add to the jar a few marbles, and it
is still the countable infinity. We also saw that you
can put “twice as many marbles” in there, and it is still
the countable infinity (i.e. consider {1,2,3,4,…} vs.
{2,4,6,8,…}. In the last example of post #3, we saw
that you can have “ten times as many” and it is still
the countable infinity.

What is a real trip, is say you have an infinite number
of jars of marbles (countably infinite), with each one
of these jars individually having an infinite number
(countably infinite), and yet the total number of marbles
is STILL the countable infinity. This is slightly more
difficult to show, but it is true nonetheless.

So you can’t say that “an infinity of infinities” is a
larger infinity, because in fact, a countable infinity
of countable infinities is still just the countable infinity.
“An infinity of infinities” is not large/strong enough
to launch you out of the countable infinity into a larger
infinity.

What this should hopefully show you, is just how damn big
the continuum infinity is!

I don’t know about you, but I consider this a jaw-dropper.

>>>Continuum infinity, is related to 2 different points
>>>and the space between them?

In all physical examples, yes.

SOME EXAMPLES OF THE TWO INFINITIES:

Countable infinity:
1. All counting numbers (natural numbers), as in
the jars of infinitely many marbles.
2. All integers (counting numbers positive *and* negative)
3. All integers + all fractions as well, any you can think of
4. A countable infinity of collections, each of which
has a countable infinity of items

Continuum infinity:
1. *All* numbers, of any decimal expansion, between 0 and 1
2. *All* numbers, of any decimal expansion, between 0 and 2
3. *All* numbers, of any decimal expansion, without any limits.
4. The number of points on a line segment.
5. The number of points on an infinitely long line.
6. The number of points on any flat surface.
7. The number of points on any imaginary flat surface extending
infinitely in all directions.
8. The number of points in an apple.
9. The number of points in two apples.
10. The number of points in a whole bag of apples.
11. The number of points in infinitely many apples.
12. The number of points in all of 3-D space.

>>>So it’s kind of like listening to a drum LIVE,
>>>where by the sound could be analyzed mathmatically
>>>as being a continuum infinity because each beat of the drum
>>>could be divided into a start and a beginning
>>>(before the silence either side of the beat)
>>>which could then be divided into even more smaller parts
>>>infinatly, kind of like a sine wave and you could zoom
>>>in to any point along that sine wave,
>>>infinate points similar to as you could on you’re ruler example.

Yes.

>>>This would be different to listening to the same
>>>drum beat on a CD or mp3, or anything digital,
>>>where by the sampling rate kind of limits the
>>>different number of points along the sine wave
>>>of the beat on the cd. The sample rate is chosen
>>>so that it is so high that the human ear can’t detect
>>>the difference between the live and the digitally
>>>recorded version. You could however increase the
>>>sample rate so that it were 10x what was required
>>>to be undetectably different for the human ear
>>>to notice between the live and digital drum beat.
>>>You could even increase the sample rate infinately
>>>times for the digital recording but it would always
>>>be a smaller infinity than the number of points
>>>along the LIVE drum beat sine wave.
>>>Am i correct this far?

Yes. Any digital recording would be a theoretical
countable infinity, and in actual practice is
simply a large finite approximation to it.

>>>So…. The sampling rate, being a countable infinity,
>>>is always playing catchup to the continuum infinity
>>>and can never match it no matter how fast is grows.

As mentioned at the beginning of the post, you really have
to have a powerful and large forcing operation to get
you out of the countable infinity and into the continuum
infinity. If you are “in” a particular infinity, almost
anything that you do will not be strong enough to launch
you into the next one.

>>>But then going back to my apple in a bag idea… [snipped]

See comments under “continuum infinity” above.
Your arguments here don’t work, because being larger in number
by a factor of 10 keeps things larger only when dealing with
finite numbers of things. When you are dealing with infinity
itself, infinity doesn’t care about something “so weak” as
multiplying by 10. You stay right where you are. Behavior
of approaching infinity, while still finite, does not
demonstrate the behavior of infinity itself.

>>>Does the secret of the continuum infinity’s greatness
>>>lay in the fact that there is no counting, no disceting or dividing?

It is certainly one of them.
Continuum infinity is one of the so-called uncountable infinities.
All infinities larger than the countable infinity are uncountable.

>>>Does the “one” experience, experienced in the now
>>>similar to the continuum’s infinity where the
>>>dualistic view is a countable infinity?

I don’t know about this, since there are not just two infinities
(although these are the only ones discussed so far).

>>>Can we actually get a snapshot of infinity in time?
>>>Is it real or is it imaginary?
>>>WHAT IS INFINITY!??!?!? haha 🙂

The more you study it, the better of a picture you can get
certainly, but a complete understanding of all aspects
is akin to a complete understanding of the Tao. You can
get more and more insight, but I’m not sure there is
(in this life) a point where you could not have more
to discover.

Cheers,
Steven

>>>The only thing I can grasp is that
>>>because the blue marbles (1,2,3,4)
>>>are whole and the points are decimals
>>>that there is no fixed relationship
>>>(formula)that can be used to pair
>>>them together? Make them countable?

Check out the two posts I wrote in
response to zoose, it may give you some ideas. 😉

>>>If you break up the marble into
>>>degrees (break up the sphere)
>>>would you be able to pair those
>>>with the points on the ruler?

Exactly. It is very good that you
were able to see this on your own.

Both the number of points in a marble
and the number of points on the ruler
are both the continuum infinity.

Since both infinities are the same
size (i.e. have the same cardinality),
namely that of the continuum infinity,
there is indeed a “pairing function”
that you can come up with to pair them up.

What the actual pairing function is,
is somewhat difficult to construct
and uses technical mathematics, so I’ll
avoid presenting it here, but you are
right nonetheless.

>>>Ahhhhhhh, I don’t get it but am having
>>>lots of fun!!!!!!!!

And that’s the important thing!
Math is supposed to be fun. 😉
(at least I think so anyway)

The important thing is to be able to
observe abstract patterns and have
fun discovering things, and to try
to verify your discoveries.

You get this kind of energy in you,
and it inspires you to investigate
and learn more.

It is this zest that motivated me
to become a mathematician. 😉

I understand it is not for everybody,
but hopefully those that have read these
posts can at least get a taste for
the idea of how it could be fun for someone. 😉

And I had fun giving people a taste of my world. 🙂

>>>By the way, I like myself better as a
>>>brunette!! Lol Steven – you’re the best
>>>Adel

You are pretty cool too.

Keep smiling,
Steven

>>>Hmm, who says this though? How do they know?
>>>I can understand now how you could rate the continuum infinity
>>>as larger than a countable infinity because the time/effort/or
>>>even requirement required to count, divide
>>>or identify each ‘countable’ element means that
>>>the infinities must grow at different rates.

It doesn’t have anything to do with “growing at different rates”.
You have to be careful, because this reasoning is ultimately
the same mistake as with the mistake with the apples that
sort of started this discussion.

It has to do with whether or not you can pair up the items
A for B and show whether or not such a pairing exists such
that nobody from either set is left over.

If nobody is left over, the cardinalities are the same.
If you can show that there will always be leftovers, no
matter what the pairing is, then they are not. In this case,
the set with guys always left over has the larger cardinality.

One can show that the power set of the natural numbers
{1,2,3,…} has cardinality of the continuum infinity
by showing that there exists a “pairing function” between
this power set and the set of all real numbers between 0 and 1, i.e. [0,1]
which is already known to have the cardinality of the
continuum infinity by lesson #4.

I have not given you what the actual pairing function is,
because it involves technical mathematics. Therefore,
you will just have to trust me that it can be done.
However, even though I’ll skip a presentation of this,
the truth of the result should not be surprising though,
because after all the set of all points on the line segment
I.E. all points in [0,1], are all infinite decimal strings,
and you know that infinite decimal strings are composed
of nothing other than the numbers {0,1,2,3,4,5,6,7,8,9},
which are the same digits that make up the numbers
in the list {1,2,3,…} (each number here only using
a finite number of such digits).

In a similar fashion, one can show that any attempt
to construct a pairing function between a set X
and its power set P(X) will unavoidably leave leftovers,
similar to the proof I gave in Lesson #4 showing that
the set [0,1] is uncountable and has the cardinality
of the continuum infinity. This again involves
technical mathematics, so you’ll have to trust me.
However, the proof has the same flavor as the proof
I gave in Lesson #4 regarding card([0,1]), so you
are not really missing much.

The beauty of the proof, though, is that it is done
in a completely general way for arbitrary sets X.
Thus, since you can prove that it is always true
that card X < card P(X) for any set X, you know
that every time you "take a power set", you will
end up in a larger infinity. In particular,
card [0,1] >>However now the explaination about the jar of marbles….
>>>You said if for every red marble we could show there
>>>was a blue marble then both infinities were the same size,
>>>but the act of doing this is not practical or applicable
>>>because when you are doing this you are confining the
>>>continuum infinity to the restrains of the countable infinity.
>>>In that way another countable infinity marble can appear
>>>every time you apply the test so that indeed there
>>>is a countable infinity marble for each continuum infinity marble.

As I mentioned to adel in the “extra info for adel” post,
mathematically when you pair things off, you are actually creating
a pairing function that maps everything from the one set onto
the other set in a one-to-one fashion.

Thus for instance, when we had our first example in Lesson #3,
Blue: {1,2,3,4,…}
Red: {5,6,7,8,…}

and we were showing that they both were infinities of the same
size (namely the countable infinity),
we paired them as:
1 5
2 6
3 7
etc.

The mathematical relationship is simply “add 4”.
You can write this as a mathematical formula: f(x)=x+4

There is no difference here between the pairing above
and this formula. They are the same.

NOW YOU ARE RIGHT that if you are comparing things
to infinities larger than the countable infinity, the
idea of making a list of relationships breaks down.
After all, if you can actually list the items, then
you have only made countably infinite number of assignments
(continuum infinity for instance is too large to fit in a
list). HOWEVER, even though you can’t pair the items
off by making a list, you can still pair them mathematically
by creating a pairing formula similar to the f(x)=x+4.

For instance, [0,1] has cardinality equal to the continuum infinity.
We can’t pair things off by making lists, but we can still
use mathematical functions. For instance, if we use
the functional formula f(x)=x+4, we are prescribing the
pairing to all numbers in [0,1] simultaneously!
Under this function, what does .37512226… get paired to?
It gets paired to 4.37512226… etc.
Every number gets assigned simultaneously.

In fact, using the functional formula f(x)=x+4, and the
argument in the previous paragraph, we have just shown
that both [0,1] and [4,5] have the same cardinality.
Thus [4,5] has the cardinality of the continuum infinity also.

It is the mathematical “pairing function” that is the workhorse
here, not the ability to make lists. I just used that as a
basic idea in the first couple of lessons to get the
fundamental ideas across and to avoid getting too deeply
into technical mathematics, especially in the beginning. 😉

SIDE NOTE:
However, you should hopefully start to appreciate how
“technical mathematical formalism” isn’t just a bunch
of confusing, boring nonsense. In fact, although it
takes a fair amount of work to learn such a language
(as it true with any foreign language actually), it
actually simplifies things considerably. When talking
about pairing things with infinitely many items,
even having cardinality of the continuum infinity,
it is difficult for the mind to grasp. Vague and imprecise
ideas arise and creates confusion. But when you simply
say f(x)=x+4 and this maps [0,1] to [4,5] in a
one-to-one and onto fashion, there is no confusion
as to what is occurring. Everything is precisely defined,
and there is no room for possible misinterpretation.

That’s why the abstract mathematical machinery is so useful
for analyzing mathematical ideas. It simplifies
difficult ideas and removes all imprecision in your
arguments. All the mathematical symbols have very precise
meanings, so the notation saves your brain the work of
having to keep things in your head while simultaneously
eliminating subtle variance in meaning. Unfortunately, the
way mathematics is presented in school is to just to drill
the mathematical machinery down the throats of uninterested
students who have no idea why it is useful and therefore
are not motivated to learn it. It also creates the mistaken
belief that the TOOLS for doing mathematics is mathematics,
when in fact mathematics is the investigation of abstract
ideas, observation of patterns, and trying to verify/prove
these discoveries. The abstract mathematical machinery
is simply a set of very useful tools for doing this, but
it is not math itself really. It is my belief that all
people that say that they don’t like math are people that
have never actually seen what math is. 😉

Once a person can get a taste for what math is, a person
can start to see that there is an actual useful need
for the abstract mathematical machinery, and that this
stuff actually makes things more simple rather than the
other way around.

I’ll stop waxing philosophical here. 😉

>>>And as for the power sets of the continuum infinity,
>>>well . . . [snipped]
>>>My first thought was that it couldn’t be bigger than
>>>a continuum infinity because any power sets tacked on
>>>would already be contained somewhere along the continuum?
>>>It’s only a guess but is that right?

Nope. It actually is bigger, believe it or not.

>>>But now i am also thinking maybe by doing this
>>>the infinity jumps out of the continuum (say the 1 meter ruler)
>>>and starts growing in the other direction..
>>>like in a graph, the infinity now not only runs along
>>>the x axis it is now along the y axis too.
>>>It has escaped the original continuum?
>>>haha it’s only a guess too but this is starting
>>>to sound like sci-fi stuff now.

Actually, the power set operation is even more powerful than just
splitting into another direction. Recall from “Lesson 5 part 1” post,
actually all the points in 2-D space, or even 3-D space, still
only combine to be the continuum infinity. For a given infinity,
two infinities, three infinities, etc. all do not make things
any larger–it is still the same infinity. In fact, the most
surprising thing (I’ll not show this, just trust me) is that
an infinity of infinities is no larger than the original infinity.
You actually need an operation MUCH STRONGER THAN THAT.
The power set operation is such an operation.

It is only sci-fi stuff if you start talking about the Q Continuum, lol.
Bonus points if you get the reference.

Smiles,
Steven

P.S. So, in any case, we now have more than two infinities.
We have the following:
1. Countable infinity
2. Continuum infinity
3. First power of the continuum
4. Second power of the continuum
5. Third power of the continuum
Etc.

This is quite amazing and mind-bending, if you ask me. 😉

>>>I used to love maths but when i did my final year of high school
>>>and did specialist maths, even though i got a good score
>>>i started disliking it because it started to get
>>>into the realm of unusable.

Unusable is sort of a relative idea.
For instance, when–in everyday life–are you going to directly “use”
the ideas of different infinities? 😉

Realistically, you aren’t; but at the same time, looked at in the
right light, it can simply be fun to investigate and it provides an
opportunity to expand the awareness of the mind.

Too often through elementary school, high school, etc., teachers
promote the idea that math is “useful” because it helps aid
your understanding of science, engineering, etc. While this
is sometimes true, it is like saying that it is NOT useful
unless it is good for something else. This is saying it is NOT useful
in its own right. Then once this “lesson” has been taught, and
the students feel that they can’t “apply” this stuff to regular
life, they feel that learning it is pointless. This is the tragedy.

This is something that doesn’t happen in art.
Nobody is in an art class and is being taught how to create a painting,
and then says “when are we going to use this?”.

This is something that doesn’t happen with people who plant flowers.
Nobody asks them, “how is this useful?”

This is something that doesn’t happen when people exercise to become fit.
They don’t look at the treadmill, realize that they are merely going
nowhere, so that it “accomplishes nothing” and then says “when
are we going to use this [skill] in everyday life?”

Yet, this happens in math.
And the reason is because students since the time of being really little,
are indoctrinated with the idea that the reason “why you want
to do it, is because it helps science, engineering, and everyday life”,
rather than it is interesting in its own right (like art), is
intrinsically fascinating and beautiful (like flowers) and is a
great training tool for expanding the mind (like a treadmill is
for expanding your physical fitness). And most importantly, it is fun.

Yet because people are trained to believe
the “useful for daily living” idea, and when people see
that outside of simple arithmetic, you don’t “need it”
for everyday living, they “correctly deduce” it is a waste of time.

Based on the false indoctrination,
I understand WHY people would come to this conclusion.
It’s the only reasonable conclusion that one would come to.
However, it is a tragedy.

If the real reason behind learning it were presented
(namely as I said above:
it is interesting in its own right (like art), is
intrinsically fascinating and beautiful (like flowers) and is a
great training tool for expanding the mind (like a treadmill is
for expanding your physical fitness). And most importantly, it is fun.),
then math would be viewed through a completely different lens.

AND as to the FUN aspect, the way in which people can have fun
is if they are given the opportunity to explore, investigate,
question, and are led to discover the mathematical discoveries
themselves–rather than to be simply told a bunch of recipes
to memorize (because it’s good for ya).

It’s just bothersome to me, because I see the beauty and interest
in it, and yet whenever I teach a class, on the first day of
that class . . . 98% of the incoming students don’t want to be
there. They are taking it because of some requirement. They
don’t *want* to learn it, and they project all their negative
energies on you. Similarly, whenever in daily life, I meet
someone and I tell them I study math . . . usually the first
response I get is either “I hate math” or they tell me how
bad they are in it.

At least this isn’t so much a problem in Healing Tao classes.
If someone is taking a Healing Tao class, they are not
going into the class “hating qigong” . . .

END RANT 😉

S

Do you teach qigong? Where do you teach? Where did you find your students?

I feel i am ready to teach, i’ve practiced countless hours over 6-7 years (for 1 year i did 4-7 hrs a day and felt i’ve worked it all out myself and worked myself though all of my mistakes. and just put up a sign out the front of my house a couple of weeks ago but so far nobody has come. Hope i can help somebody in the not too distant future!

Outside of doing some privates when people contact me,
I am not actively teaching qigong right now.
Mainly my energies are focused on getting my PhD.
I just want to get it out of the way, and not have
any side projects going on . . . plus, since I may
end up going to a completely new area in a year or so,
I likewise don’t feel it to be fair to any students.
Instead I use any free time I do have, to bolster my skills
and qigong resume to be more effective when I do begin
an active teaching role again.

Once I am no longer in graduate school, hopefully in
around a year, and both a job and location I expect
to not be moving from for awhile, I will begin with a
more active qigong teaching role.

If you want to do qigong teaching, I would recommend
that you look into having an outside location. For
most people, qigong is a new thing, and when you couple
that with the idea of going to someone’s house that you
don’t know, it is pretty intimidating. If you have
an outside location, it is more neutral and more inviting.

Prior to my years in the PhD program, I taught tai chi
and always did it in an outside location. I was always
able to find students . . . sometimes only a few, and
sometimes up to 10-15, but averaging 5 regulars . . .
and this was during a time when I really only had 3 years
of tai chi experience, no HT background, and was in my
mid-to-late twenties! So I think you can get students,
you just have to do it in the right way.

I would look into sharing some space with a yoga studio
or dance studio, as usually they don’t occupy their space
all day long, and would appreciate the money from you renting
their space for a few hours during their off-time. Other
options (if getting your own space is intimidating) would
be to look into teaching qigong at the Y or a community center.
This option is a little easier to advertise, but you’d have
to expect the place to get a bigger cut and you probably have
to be a little more careful about mentioning too much that
could be viewed as spiritual or religious.

But this is the route I’d recommend, and I’d bet you find
it easier to get people . . . because most are not going
to go to someone’s house they don’t know–no matter how nice
they appeared–for anything. There are too many weirdos
out there, and it becomes a safety issue.

Steven