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March 25, 2012 at 8:58 pm #39003StevenModerator
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. ๐>>>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 itemsContinuum 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 ๐The 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,
StevenMarch 25, 2012 at 9:21 pm #39005StevenModerator>>>Also…
>>>Are there any other kinds of infinities apart from countable and continuum?Yes.
Lesson #5:
The size of an infinity is called its cardinality.
The cardinality of the jar of infinitely many marbles
is the countable infinity. The cardinality of the
number of points on a line segment is the continuum infinity.THE POWER SET OPERATION
Suppose you have a set of numbers {1,2,3}
Let’s list all the subsets of this.
By subset, we mean any collection that
contains the same or less than what we have.Subsets of {1,2,3}:
1. Empty set (collection with no objects in it)
2. {1}
3. {2}
4. {3}
5. {1,2}
6. {1,3}
7. {2,3}
8. {1,2,3}Notice that our original set had 3 objects, namely the
numbers 1, 2, and 3.The POWER SET of this is the collection containing
the 8 objects I listed above (the subsets).It is true (maybe it is obvious), that the power
set always has a larger size than the original set.
In the above example, the original set had 3 elements,
and the power set has 8 elements.Now for what is amazing:
This increase in size is true for the infinite realms
just as it is true for the finite realms.If we take our infinite jar of marbles {1,2,3,4,…}
which has the cardinality of the countable infinity,
and we take the power set of this, we get a really
huge set of subsets . . . and this really huge set
has the cardinality of the continuum infinity.Thus, in rough terms from the last post, the power
set operation is a strong enough operation to launch
you out of the countable infinity and take you into
the continuum infinity.But, since I said that one can prove mathematically
that the power set operation always takes you to a
set of larger size, including sets of infinite cardinality,
we can do the same thing with the continuum infinity.Suppose you have a set with cardinality of the
continuum infinity. An example would be the set
of all points on the line segment we discussed earlier.Now take the power set of this.
The resulting really freaking huge set of infinitely
many objects has a cardinality of size LARGER than the
continuum infinity.Thus, we have an ever increasing sequence of larger
and larger infinities, namely:1. Countable infinity
2. Continuum infinity
3. First power of the continuum
4. Second power of the continuum
5. Third power of the continuumAND SO ON.
I think I’ll stop here, as this should be another
“woah dude” moment.Smiles,
StevenMarch 25, 2012 at 9:39 pm #39007StevenModerator>>>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. ๐>>>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. ๐
(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. ๐I 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. ๐And I had fun giving people a taste of my world. ๐
>>>By the way, I like myself better as a
>>>brunette!! Lol Steven – you’re the best
>>>AdelYou are pretty cool too.
Keep smiling,
StevenMarch 26, 2012 at 6:24 am #39009zooseParticipantFirst thanks again for your replies, it’s good to ask questions and talk and learn about it with someone who knows what they’re talking about…
>If we take our infinite jar of marbles {1,2,3,4,…}
>which has the cardinality of the countable infinity,
>and we take the power set of this, we get a really
>huge set of subsets . . . and this really huge set
>has the cardinality of the continuum infinity.
>
>Thus, in rough terms from the last post, the power
>set operation is a strong enough operation to launch
>you out of the countable infinity and take you into
>the continuum infinity.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.
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.
So now from my intuition after what you’ve told me i feel that the continuum infinity is bigger than the countable, but from any test we could apply to *prove* this is the case i think would fail? Is there another way to test if one infinity is larger than the other?
And also this now seems to look like the it’s not that the actual sizes of the infinities are different, just that the rate at which they grow is different.
Then the countable power set infinity that gets launched into continuum infinity… but it still is countable infinity at the heart, so how can it compete with continuum infinity?
I think the countable power set, as it gets higher and higher, the infinity gets more and more complicated and increases at a higher rate for sure, but the growth can be followed on some very high tech computer for some short period of time, but the continuum infinity is instantly infinity. I feel it’s growth can’t be even peeked at, it is beyond being calculated, it just *is*. It doesn’t grow it’s just already there, because the beginning and the end of the continuum are already defined. So now my intuition would tell me the rate of growth of this countable power set infinity would approach the rate of growth of the continuum infinity but could never match it?
And as for the power sets of the continuum infinity, well this seems like it is introducing the countable into the continuum, kind of like tacking on countable power sets to any number of countable points within the continuum. 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? 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.
I have a guess at what is in the next lesson. continuum powerset of the continuum? Just a guess then that would be like a full continuum infinity on both an x and y axis? And then if you had a continuum powerset of the continuum powerset of a continuum powerset it would start spreading in the 3rd dimension? What would happen to the next powerset added? where would it start going then?
… I think i’m getting ahead of myself, i’ll wait patiently for the next lesson ๐
March 26, 2012 at 1:31 pm #39011adelParticipantIt’s like having different dimensions that live
side by side, but you can cross a bridge from
one to another thru the power sets.Not sci fi, but spiritual math.
AdelMarch 26, 2012 at 1:44 pm #39013adelParticipantWow,
Our bodies must contain all of these
infinities, just aligning in one way
will pair you with one type but you are
still able to bridge to others!!!!!And
I’m sure, all simultaneously.Sorry, the language eludes me but it’s
HUGE! Your world is magical. Wish that
we could all have had a taste of that
magic in school, glad that there are
teachers like you.Adel
March 27, 2012 at 5:44 am #39015StevenModerator>>>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+4There 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. ๐SIDE 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. ๐Once 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. ๐
>>>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,
StevenP.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. ๐
March 27, 2012 at 5:46 am #39017StevenModeratorMarch 27, 2012 at 5:47 am #39019StevenModeratorMarch 28, 2012 at 8:43 am #39021zooseParticipantSteven you are undoubtedly very smart. 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. If they had have given good scenarios in their lessons like you have provided here i’m sure i would have liked to continue my study of it. It seemed to me at the time that it became more complex without any real enjoyment and so i didn’t go any further.
What you present is mind bending no doubt and it’s going to take me a bit to digest it.I understand why you enjoy it and i certainly would of if i had have gone a little further. It’s like giving yourself concrete evidence of what you think of is correct. Thankyou very much for the time you have spent giving me your answers but i’m sure i will still have some annoying questions in the days to come…
I’m still waiting for a gold star of my very own so give me a few days and i will be back ๐
However i have one last thing that will help my digestion… The original countable infinity, if it starts from 1 and goes forever…. FOREVER… then any infinity, no-matter how great, seems just as big as the original infinity. For any new number created in any other super infinity there could always be a new number in the countable infinity to match it. If the rate of growth of infinity is not an issue, then we have unlimited time to create a new number in countable infinity to match any super infinity. I know you must have complex calculations to justify what you say but surely it should be able to be explained in laymen’s terms also? Nomatter how complex a calculation or finding may be it still has to be bought back to the normal world for any real benefit.
I have one more query…. i remember you saying in an earlier post you sit in wu wei while you are at work, but to me it is impossible to do mathematical calculations or complex thought of any kind (especially mathematics) while staying in wu wei. How do you possibly manage it. How do you watch yourself doing complex math without any self involvement at all? I know it’s not explainable but having worked as a software developer i find it impossible this type of thought. I can speak, listen, hear and read in wu wei, but when i need to apply myself, understand, or develop something new i need to think and wu wei goes out the window.
… a mathematician operating in wu wei…. i can’t fathom that!
Thanks again!
March 28, 2012 at 4:21 pm #39023StevenModerator>>>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? ๐Realistically, 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 ๐
S
March 28, 2012 at 5:07 pm #39025StevenModerator>>>However i have one last thing that will help my digestion…
>>>The original countable infinity, if it starts from 1 and
>>>goes forever…. FOREVER… then any infinity,
>>>>no-matter how great, seems just as big as the original infinity.
>>>For any new number created in any other super infinity
>>>there could always be a new number in the countable infinity to match it.
>>>If the rate of growth of infinity is not an issue,
>>>then we have unlimited time to create a new number
>>>in countable infinity to match any super infinity.
>>>I know you must have complex calculations to justify
>>>what you say but surely it should be able to be explained
>>>in laymen’s terms also? Nomatter how complex a calculation
>>>or finding may be it still has to be bought back to the
>>>normal world for any real benefit.No, this is not true actually.
This is kind of the whole point to the miraculous proof
in Lesson #4 that the number of points in the line segment [0,1]
is larger than the countable infinity.I understand that it is hard for the mind to accept, and
the mind wants to rebel against it, but this is precisely the
reason that the first mathematician to carefully study and analyze
infinity ended up going mad and had to be institutionalized.The only thing I can say here is to go back and reread
Lesson #4 again and take some time to really meditate and
absorb the argument used to show that the number of points
in [0,1] is larger than the countable infinity.The crucial idea, whose foundation is laid in the first 3 lessons,
and then reaches its major mathematical orgasm in Lesson #4, is one in
which you really can’t just look at it once and feel confident with.
You sort of need to just shake your head, clear your mind,
and start reading it over from the beginning. It is one
that takes repeated passes before complete understanding
can be found. This was true for me as well
when first being exposed to it.To avoid creating a situation where you become supersaturated,
assuming you are not already, I think saying less
is the better answer than for me to say more. ๐>>>I have one more query…. i remember you saying in an
>>>earlier post you sit in wu wei while you are at work,
>>>but to me it is impossible to do mathematical calculations
>>>or complex thought of any kind (especially mathematics)
>>>while staying in wu wei. How do you possibly manage it.
>>>How do you watch yourself doing complex math without
>>>any self involvement at all? I know it’s not explainable
>>>but having worked as a software developer i find it
>>>impossible this type of thought. I can speak, listen,
>>>hear and read in wu wei, but when i need to apply myself,
>>>understand, or develop something new i need to think
>>>and wu wei goes out the window.Wu wei to my understanding means “effortless action”;
it does mean “mindless action”.It simply means living your life, as it is, in a natural way,
without needing to add any effort to what you are doing.If you are having a pleasant conversation with somebody,
you don’t need to put forth any effort to do so. It is
natural. You don’t need to put effort into coming up with
words, constructing sentences, and the like. You don’t
even think about such things, you are just present with
other person engaged in an exchange.Yet, if you were to start learning Mandarin Chinese,
while a native speaker could carry on such “wu wei conversations”,
you could not. It would require effort on your part
to even form simple sentences. But if we switch to
English, it becomes effortless . . . as effortless as breathing.The point is, is that the longer you do an activity, the
more integrated it becomes into your infrastructure, and
it starts becoming natural and effortless. That’s the
best way I can explain it.If I am doing my PhD work, which is at the threshold of
my understanding, it requires a lot of effort. If, on the
other hand, I were doing calculus, it is simply effortless
at this point–in fact, I have had dreams where I’ve given
whole calculus lectures. At this point, anything at the
level of college Calculus II or lower, I can do while
being in a coma. It has become as easy as breathing.Sometimes I’ll slip into these modes when doing my PhD work,
if I see how the argument is going to go and it is a matter
of spending the hour or so to fill in the details. Then
doing mathematics becomes like a form of high-level
meditation.But then there are other times when I am stuck and
the ideas are not coming easily. Then I am using a lot of
effort, and after maybe a couple of hours, I start feeling
tension arise and I need to stop and take a break.I don’t know if that helps. I don’t really know how to
explain it any better. Probably in retrospect, the
best way to understand it is the idea of a foreign language . . .
in that it requires effort to use, but the more you use it,
the more natural it becomes, until eventually you don’t
need to use any effort whatsoever to engage with it.Steven
March 29, 2012 at 5:27 am #39027zooseParticipantDo 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!
March 29, 2012 at 5:47 am #39029zooseParticipantYeah i understand you fully with your wu-wei explaination…. Just taking a break has always been my weak part. Knowing when to take a break is now easy because i do tasks, watching myself and i know when i need a break but i really enjoy the stress, just not what it does to me.
My most recent finding in my meditation and daily life, which has been with me for a while but is the most important finding of all for me, is that i always need to be watching myself, the moment i don’t i find myself doing silly things like tensing up in my normal areas. My body is really like a child, can’t leave it unattended for a moment or it starts being silly! …and then the problems start.
I don’t know what the difference is between mindless and effortless action but for me it’s best when my body and mind just does as it does, effortlessly, and i just watch it. My mind is empty and aware of everything that is going on, and whenever it wants to it can just offer a suggestion, and because it is the only thought it is so clear. I feel this is best because the power of the mind is still available without totally letting go and watching my life unfold, but it is also not muddled with conflict where it fights against it’s self or even worse is so noisy that there is no power behind any thought.
I will go back through your lessons and revise. Might not get a chance to reply for a couple of days though because i’m going away for a kickstart to my 2 week holidays from work! ๐
March 29, 2012 at 7:46 am #39031StevenModeratorOutside 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
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