OldDog

Yeah, thanks LT for stating that. I had always considered it like and undefined-variable. Maybe that is what has contributed to my confusion. Gonna take a bit of time for that to sink in.

Agreed. Couple of more question before we proceed. Sorry, but this has to do with arrows, again. Is it incorrect to think of Re(z), Im(z) as a point on a plane? It seems that (0,0) as the origin is important to defining an arrow. While it may be a trivial case, it seems without an origin the arrow pointing to z would have no direction or dimension. OMG ... suddenly hit me that to consider some other origin other than (0,0) would mean that such an origin would have to be a point that described by another point, another z with an ordered pair that contained an imaginary component. That is if the plane is consistent.

Ok. So I think where I am at, at this point, I can follow the rules of notation and label a point correctly as Re(z), Im(z). Neatly, this alternate notation avoids the use of i and all the attendant questions about it. Still do not understand the significance/importance of why we are putting an arrow there. Maybe its time to suggest what wecan do with an arrow object. Or, is there more to be discovered about the object?

Yeah, sort of a singer-somgwriter-folk-country-rock band. Glad you enjoyed them.

Ran into this group several years ago. They are a regional band from northern CA. Very good, though. Would love to see them live. Check em out.

Sooo, we are to treat i simply as a variable and defer the notion of imaginary?

That brings up a good point. If it is undefined ... how can we assume it will behave the same way at real numbers? If its undefined I would think we cannot assume anything about it. @wandelaar Hope you will touch on this. You have already suggested in the arrow example that we would be performing some operations on those arrows.

a+bi or ai+b ... it doesn't really matter does it? You are specifying a point on a Cartesian coordinate system (to continue the geometric model) where one dimension id real and the other is not ... contains an imaginary term. The notation is arbitrarty, no? Sorry about the fixation on arrowness. One of the reasons I struggle with math in general ... and imaginary numbers in particular ... is that if I cannot relate it to a real world situation ... a frame of reference for how to apply it ... it seems devoid of meaning. I took a fair amount of math along the way. The only way I could maintain my sanity was to think of it as an arbitrary game with objects and rules for manipulating them. The rules didn't have to make sense, just learned. Kinda like life.

No, I'm stil here .... but struggling still with the idea of arrowness. What is it that give the geometric thing direction ... and for that matter straightness? Why not a curved line ... with or without an arrow on the end?

Agree ... but would add to that. The frame of mind that one approaches any endeavor is important as well as understanding the environment in which one operates. One cannont be a master boatman, swimmer or diver by simply rote learning. One needs to relinquish all preconceptions and be open to the learning experience. Then practice the fundamentals so they can be performed with grace and economy of effort. The reason for pointing to the swimmer and diver is that they have intimate knowledge of the way water behaves through the experience of having been immersed in it. Knowing the environment in which you work will allow you to apply develop and apply skill to best advangage.

OK. So we went down this road because imaginary numbers were mentioned in another post. Wandelaar offered to help with understanding imaginary / complex numbers. Many of us were hoping that he would produce an explanation that could be likened to a concept in taoism. So far the explanations are not capturing my/our imagination. So far, the easiest thing for me to understand has been that the expression i = square root of (-1) is undefined, hence imaginary. This is easy because we all understand that the square of any number, positive or negative is a positive value; that is there can be no number thst when squared produces a -1. At least this is true at the level most of us are thinking. Trying to deal with this geometrically, I think, is a bit confusing. Wandelaar has chosen as object of discussion to use the typical Cartesian corrdinate system as an analogy. I say, analogy because for it to actually apply (as in the example a+bi, the y coordinate must not be real but imaginary. In such a system x coordinate is represented by a and the y coordinate by the imaginary number bi. Where I go of the rails in the geometric example is that the expression a+bi is represented by a straight arrow. Which leads me to the following questions. 1. Is the arrow really straight? 2. If so, is it because the expression a+bi is like the general straight line expression y=mx +b? 3. If a+bi is like y=mx+b ... does that mean i is an imaginary slope? 4. What is it that suggests arrowness? The arrow suggests direction ... or was it really meant to represent a line without arrow? 5. Can it then be said that any expression (straight or otherwise) which contains an imaginary term be considered complex? Maybe the answer to these questions will help understand the nature of a+bi. I think they would for me. Oh, and btw ... 6. The arrow doesn't always have to emanate from (0,0) ... does it?

Soooo ... i is a value (variable?) that we should ignore ... except for its presence. I guess I can suspend my distraction long enogh for the explanation to continue ... but at some point you are going to have to give us a frame of reference for i to go.much further.

I don't think the point is getting across. You have presented two things: a cartesean coordinate system with some arrows drawn and some expressions that (algebraicly) involve a coefficient i. Now, it's easy enough to accept a notation like (a,b) ... but it is failing to relate ... at least in my mind ... to the imaginary number i. Need to get over that hump.

Enjoyed the Feynman lecture ... being a minor aficionado of quantum mechanics ... and will likely visit the imaginary numbers thread ... being a minor aficionado of mathematics. I often sit in wonder of all the fine and detailed knowledge we gather in various technical pursuits ... and at how little it seems to affect our everyday lives.

"It’s so much darker when a light goes out than it would have been if it had never shone." - John Steinbeck

Looking upward, we contemplate with its help the signs in the heavens; looking down, we examine the lines of the earth. Thus we come to know the circumstances of the dark and the light. Going back to the beginnings of things and pursuing them to the end, we come to know the lessons of birth and of death. - Ta Chuan: The Great Commentary

Kinda figured this would be a controversial concept. I really believe that there is a bit of truth in all these points of view ... because my experiece and reason compels me to a holistic view of Taoism ... one that embraces the I Ching, early proto-daoist ideas, Laotzi, Zhuangzi, Guanzi and other sources on into the early centuries. The idea that Heaven, Earth and Man form a trinity comes to me from the I Ching, specifically from Wilhelm/Baynes. The eight trigrams are images not so much of objects as of states of change. This view is associated with the concept expressed in the teachings of Lao-tse, as also in those of Confucius, that every event in the visible world is the effect of an “image,” that is, of an idea in the unseen world. Accordingly, everything that happens on earth is only a reproduction, as it were, of an event in a world beyond our sense perception; as regards its occurrence in time, it is later than the suprasensible event. The holy men and sages, who are in contact with those higher spheres, have access to these ideas through direct intuition and are therefore able to intervene decisively in events in the world. Thus man is linked with heaven, the suprasensible world of ideas, and with earth, the material world of visible things, to form with these a trinity of the primal powers. (p lvii) ... and then, later ... The six lines of each hexagram are divided among the three primal powers, heaven, earth, and man. (p 289) In the Ho-Shang Kung Commentary, Dan Reid translates .. 'Dao gave birth to the One' Dao sent forth life, and there was Oneness 'The one gave birth to Two' The One gave birth to Yin and Yang 'Two gave birth to Three' Yin and Yang gave birth to harmony between clear and opague. This separated into three ebergy-breaths, creating Heaven, Earth and Humanity Reid later makes a compelling case for the prevalence if early Taoist ideas being in common circulation prior to Laotse and shows how these went on to find expression in other viens of Taoism, including Nieye. So I think the varied interpretations of what in Ch 42 is presented as ... One emerging, One > Two, Two > Three ... may not necessarily be describing a strict generation sequence but rather a relationship of things among the Ten-Thousand things. This is about as far as my understanding has developed thus far. As I do further research no doubt these notions will change.

Some explain it as ....Heaven, Earth and Man.

The line following the three comparisons seems to be key to this chapter. Lin Yutang's interpretation seems close. Therefore, he who loves most spends most, He who hoards much loses much. But I think there is even more to it. It seems that in the world, the things we hold on to too tightly ... be it money, goods, ideas, people ... are damaged or destroyed. Only by letting go of them are they allowed to work their own way in the world ... find their best truest expression. One who truly has high regard ... love ... things will let go of them and remember them fondly.

I guess there are just some authors/translators whose writings I don't get. Chad Hansen is one of these. His choice of words and manner of expression I find difficult. Not that that's bad ... it stops me in my tracks and forces me to think hard about what he is trying to convey. For instance, what does ... the advantage of lacking deem-acting ... really mean. Such an odd turn of a phrase. I think there is more to this line than inaction or non-doing , as suggeted by others. I think Lok Sang Ho expresses it best with the benefits of not contriving.

Ths first few lines ... Out of Tao, One is born; Out of One, Two; Out of Two, Three; Out of Three, the created universe. The created universe carries the yin at its back and the yang in front; Through the union of the pervading principles it reaches harmony. ... along with Ch 40 ... Reversion is the action of Tao. Gentleness is the function of Tao. The things of this world come from Being, And Being (comes) from Non-being. have always demonstrated to me a direct connection to the Yijing.

As pointed out by Wandelaar in his quote, the last four lines of this chapter don't seem to follow ... kinda non sequitur. After reading several translations I sort of settled in on a meaning of ... those that force things do not come to a natural end.

I am struggling a bit with this chapter. As a whole, it talks about what the manifest world (heaven, earth and the myriad things) is like when with Oneness and what happens without Oneness. It speaks (most typically translated) of the myriad things attaining or acquiring Oneness, as if they came into existence without oneness. How do the myriad things attain and lose oneness? Lin Yutang only refers to the myriad things being in possession of oneness ... as if that were the natural state upon coming into being ... before going on to decribing things without oneness. How do the myriad things lose the oneness they were born with? Flowing Hands, significantly and perhaps more importantly, speaks of the myriad things coming from Oneness, does not really speak of attainment and only speaks of loss of oneness in relation to man. Is man really a distinct case? How does man attain and lose ... or lose and attain? Thoughts?

I noticed that too ... and while I admire David Hinton's work, I wonder why he decided to not use the definite article. Is it his means of drawing attention to the ineffable ... indefinite ... nature of Dao? Seems a bit contrived ... possibly to the point of detracting from the translation. Lofty? But that's the idea behind offering multiple translations ... to see how bbc.co others have understood the text and would communicate it to others.