"a Universal Representation Of Rules"

[Doctordick]

Qfwfq, after reading your latest post, I get the impression that you are either simple minded or have utterly no interest in what I say. I don’t really believe you are simple minded so I can only conclude your purpose is no more than trying to stop any serious discussion of my presentation.

 

It is you who keep bringing up Lie algebra, not me.

Really!? Dick, read your very own OP of this very thread. I don't have too much time to waste, not on nonsense at least.

That is a totally thoughtless remark and I think you know it. At least you should if you spent a moment thinking about it. I brought up Lie algebra as a place where the reader might discover and understand some of the issues related to anti-commuting mathematical operators (I would suspect few readers on this forum would even consider such things as possible). I did that for the very simple reason that the only discussions of anti-commuting operators I have ever seen are associated with the presentation of Lie algebras. If you had any familiarity with Lie algebra at all you would be well aware of the fact that there is one hell of a lot more to Lie algebra than simple anti-commuting operators. Since I presume you are familiar with the subject I come to the conclusion that your only goal is here is to create straw man arguments.

 

I use anti-commuting operators for the simple purpose of casting those four basic constraints into what appears to be one equation (it is no more than a notation issue). I use some very specific anti-commuting operators which I explicitly define and nothing else. It is shift symmetry which makes those two representations absolutely and totally equivalent to one another and nothing else. Without the definition of those anti-commuting operators the final equation is “undefined” notation. That is all there is to it and nothing more. No presumption of the validity of Lie algebra has been made!

 

The hypothetical counterexamples in my analogy stand for cases which aren't solutions of your equation, but comply with the initial premises and not with the choice of Lie algebra.

However, I also note that, in those supposed counterexamples, you consistently refuse to use my definitions of those indices I define: that would be “x”, “i”, “tau” and “t”. You invariably presume they represent exactly what you have been taught to believe they represent. Even Rade has managed to see across that trivial cavil. I have come to the conclusion that there is nothing I can do about your jumping from silly cavil to silly cavil as you have no intention of trying to follow my logic anyway.

 

Goodness, it has taken years for your incoherent ramblings to assume any semblance of meaning.

Yeah, I have begun to notice your total inability to comprehend my definitions. I was quite surprised that Anssi picked them up so quickly. And now even Rade seems to have recognized some of the critical issues. For some strange reason, you cannot see anything except my final equation. Reminds me a lot of how I suspect that Egyptian alchemist would probably react to a modern chemistry explanation of some process familiar to him. I should have recognized your problems when you insisted that Newton would not have seen the problem with his definition of time even if the difficulties were pointed out to him.

 

Have fun -- Dick

[Qfwfq]

Qfwfq, after reading your latest post, I get the impression that you are either simple minded or have utterly no interest in what I say.

That's no novelty Dick, You've been under this impression for ages!!!

 

I don’t really believe you are simple minded so I can only conclude your purpose is no more than trying to stop any serious discussion of my presentation.

Actually, my aim has been to make the discussion a bit more serious, but this turns out to be impossible.

 

(I would suspect few readers on this forum would even consider such things as possible).

Why should anybody think this is impossible? Why were you accusing me of this, before I explicitly denied it?

 

If you had any familiarity with Lie algebra at all you would be well aware of the fact that there is one hell of a lot more to Lie algebra than simple anti-commuting operators.

Obviously, Dick. This is the very reason why your choice of operators with those properties winds up leading to what it leads to, when you use it to arrive at the Dirac equation. Which straw man arguments are you accusing me of?

 

It is shift symmetry which makes those two representations absolutely and totally equivalent to one another and nothing else.

This is an unsupported claim. Do you have a logically conclusive argument? Can you prove that no non equivalent choices exist? If you want more folks to take you seriously, be a good mathematician.

 

However, I also note that, in those supposed counterexamples, you consistently refuse to use my definitions of those indices I define: that would be “x”, “i”, “tau” and “t”. You invariably presume they represent exactly what you have been taught to believe they represent.

Er... What have I been taught to believe they represent?

 

Even Rade has managed to see across that trivial cavil.

This reminds me of the time you called in cause your wife having done much better than me, as if to say: "Even a woman beat you!" Dick even recently I've seen Rade querying you about things that were already clear to me, and by his last two posts in this thread he still has some odds with you and shares with me some objections about your Zen Buddhist meditations.

 

Yeah, I have begun to notice your total inability to comprehend my definitions.

Uhm, no, Dick... My inability! :rolleyes: I was talking about your total lack of clarity over the years.

 

I was quite surprised that Anssi picked them up so quickly.

I have often thought of the matters I was discussing with Anssi, ever since childhood. Disagreement with some of you guys' conclusions does not demonstrate lack of comprehension on my part.

 

For some strange reason, you cannot see anything except my final equation.

Dick, I have been through the OP's of these three threads, whether or not you believe it. I waded through the lack of clarity as well as is worth my time (you're lucky that most of my time has been worth very little these days). You keep making these phony accusations, a red herring to distract your disciples or potential ones from taking my points seriously.

 

I should have recognized your problems when you insisted that Newton would not have seen the problem with his definition of time even if the difficulties were pointed out to him.

What are you referring to? As far as I can remember, I might have said Newton could not have suspected the Minkowskian nature of spacetime due to his experience, just like anybody until early 20th century. Newton did have enough sense to avoid being overly bold in making assumptions though, despite what you hear many folks saying nowadays. What grounds do you have about how you suspect an Egyptian alchemist would probably react to a modern chemistry explanations? Are you sure they wouldn't have been eager to learn such a great aid for finding new processes? That's exactly what European alchemists did, over the past few centuries, despite the fact that only after 1905 the "atomic theory" got to be regarded as physical fact rather than a hypothetical model that seemed to work very well.

[Rade]

Doctordick, I have a question that has occurred to me based on one your comments.

 

When you use the word 'explanation', I take it that your definition applies equally to both (1) the person making the explanation and (2) the person(s) to which the explanation is being made ? For example, from your answer to my example about the son explaining to his mother why his nose was broken, you only took the view of the mother, that is, you made the argument that she would have expectations depending on the words the son used. Thus, the mother would have some expectation of understanding (from 0 % to 100%) why the nose was broken from the explanation given by the son (I walked into the garage door). Of course, the son could have offered an infinite number of such explanations, from (2) I have no idea (which would lead to 0% expectation of understanding by the mother), to (2) the door hit me (which would lead to a lower expectation of understanding than the explanation given since there are many doors in the house), etc.

 

On the other hand, the son would likewise have some expectation (from 0% to 100%) that his explanation would be understood by the mother. So, to hide something, the son could provide the explanation (I have no idea why my nose is broken), which would be a single explanation that at the exact same moment in time would result in 100% expectation by the son that the mother would have 0% expectation of understanding. As you see, two completely different sets of expectation, but not by those to which the explanation is given, but between the person explaining and the person hearing the explanation.

 

Thus, in any act of explanation it would appear there are two completely different sets of expectations, those (1) for the person that explains, and (2) for the person(s) that receive the explanation.

 

Now, as I understand, it is the (i) within the (Xi) numerical label that relates to different understanding of explanations by different people that hear an explanation or by different people that provide the same explanation for some circumstance at different time) ? However, it is not clear to me that your (Xi) notation can deal with the situation I have presented, where the expectation differs for (1) the person that explains and (2) the person(s) that receive the explanation ? But I suspect it is just a matter that I do not have clear understanding of the full extent of how the (Xi) notation can be used ? Any help with understanding is appreciated.

[Qfwfq]

Now, as I understand, it is the (i) within the (Xi) numerical label that relates to different understanding of explanations by different people that hear an explanation or by different people that provide the same explanation for some circumstance at different time) ?

That isn't my understanding of it and I think I can answer in the negative without waiting for confirmation from Dick.

 

It seems to me that each [imath]x_i[/imath] is the value of one of the possible attributes of one of the possible entities defined in an explanation. For each [imath]i[/imath] it could indicate how happy or angry one of the people is; it could indicate the colour of something (like, perhaps, one of the three RGB components); it could indicate the material an object is made of; it could be one of the positional coordinates of one of the particles, or one of its momentum components, or a value of angular momentum; it could indicate whether Schrödinger's cat is alive or dead.

 

It seems to me that the [imath]\tau[/imath] index distinguishes different batches of data, so maybe you could consider Mom and son's points of view being different [imath]\tau[/imath] values?

[Rade]

That isn't my understanding of it and I think I can answer in the negative without waiting for confirmation from Dick.It seems to me that each [imath]x_i[/imath] is the value of one of the possible attributes of one of the possible entities defined in an explanation. For each [imath]i[/imath] it could indicate how happy or angry one of the people is; it could indicate the colour of something (like, perhaps, one of the three RGB components); it could indicate the material an object is made of; it could be one of the positional coordinates of one of the particles, or one of its momentum components, or a value of angular momentum; it could indicate whether Schrödinger's cat is alive or dead.

 

It seems to me that the [imath]\tau[/imath] index distinguishes different batches of data, so maybe you could consider Mom and son's points of view being different [imath]\tau[/imath] values?

Thanks, perhaps you are correct ? Whatever the answer, it would help to understand how the different expectations for mother and son are dealt with in the notation. If along the [imath]\tau[/imath] axis that would be fine, but not easy for me to grasp at this point of the presentation.

[LaurieAG]

Hi Doctordick,

 

And what about integration and/or differentiation? Get off your high horse; any defined mathematical operation could be used.

 

Are you exluding calculus from your proof?

 

http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus

 

The fundamental theorem of calculus specifies the relationship between the two central operations of calculus: differentiation and integration.

 

The first part of the theorem, sometimes called the first fundamental theorem of calculus, shows that an indefinite integration[1] can be reversed by a differentiation. The first part is also important because it guarantees the existence of antiderivatives for continuous functions.[2]

 

The second part, sometimes called the second fundamental theorem of calculus, allows one to compute the definite integral of a function by using any one of its infinitely many antiderivatives. This part of the theorem has invaluable practical applications, because it markedly simplifies the computation of definite integrals.

 

The first published statement and proof of a restricted version of the fundamental theorem was by James Gregory (1638–1675).[3] Isaac Barrow (1630–1677) proved the first completely general version of the theorem,[4] while Barrow's student Isaac Newton (1643–1727) completed the development of the surrounding mathematical theory. Gottfried Leibniz (1646–1716) systematized the knowledge into a calculus for infinitesimal quantities.

 

Your work won't stand up without a spine because an undeclared/undefined variable calculus for each point along a continuum is only a continuum where the percentages of base components alone can be compared between readings made at discrete points in time. If the calculus used for each point was defined we could at least be able to normalise the data between points in some way so that all weights and measures remained consistent within the same physical/mathematical universe framework. Variable calculus is just another example of something else that has problems with integration and differentiation.

 

Without definitions all you get is an information gain equivalent to that obtained through selectively examining astronomical observations.

[AnssiH]

Wow, this was confusing... (just read the replies to this thread)

 

Qfwfq, I must say, it is often very hard to interpret your complaints because you tend to only give short and vague comments. I did also interpreted your comments "The argument provided doesn't even attempt to support its necessity" and "what I posted above concerns what you put into the OP of this thread and nothing else" as saying that you did not see an attempt to prove the necessity of FE in the OP. I made that intepretation because personally I would characterize all those other threads leading to this one as a rather meticulous attempt to prove the necessity of the equation.

 

From the consequent posts, I'm guessing you actually meant to refer to the use of anti-commuting operators, and you are concerned that it is because they are used, that some undefendable characteristics are introduced to the equation (i.e. characteristics which are not present in the separated expressions?), and that it is because of those characteristics, that the relationships and definitions of modern physics can be derived later on in the presentation.

 

Would that be a fair assessment of what you are saying?

 

If so, I'm afraid you need to provide more information about why and how do you think this is so. Don't be vague and lay down some details. Surely you understand that even if they do introduce some characteristics that affects the situation in some way, the symmetry arguments probably also play a role in there. Wouldn't that issue be worth investigating quite carefully?

 

Think about that a bit, wouldn't then the real question be "are the additional characteristics such that they require a certain type of information for them to yield (ultimately) modern physics? Or, are the characteristics such that they can always be used as a mechanism to categorize any undefined information?

 

If it's the latter, then yes, it should not be part of the fundamental equation, but it would actually be representative of one of the data-handling mechanisms that we are currently employing in our world-views (i.e. should be part of the later derivations). If it's the former, then it would be representative of actual knowledge about reality in-itself (as in, only valid for our reality).

 

Does the above make sense to you? If not, then, I'm afraid you are not up to speed with the discussion. If so, that's perfectly fine, but you should say what your concern is in as exact mathematical detail as possible. Then at least it might be possible to tell what you are referring to, and judge whether or not the complaint is valid and what its validity would mean.

 

If you don't want to put in the time to do this, then I guess all that should be said is your concern is noted, and thank you for your time.

 

Rade, about your questions regarding prediction and explanation. It's very simple; being able to draw predictions based on undefined information, always implies "an understanding" of some sort exists. I.e, if you have "expectations", then a transformation from "undefined" to "defined" must exist. If you don't have definitions, what would you have expectations of? :)

 

So, having an explanation entails having definitions, which means you are thinking in terms of those definitions. That contains everything that you would call "understanding". (Your description of "understanding" just categorizes and breaks it down further, but that categorization is not necessary or useful in DD's analysis)

 

Thus, in any act of explanation it would appear there are two completely different sets of expectations, those (1) for the person that explains, and (2) for the person(s) that receive the explanation.

 

...

 

Now, as I understand, it is the (i) within the (Xi) numerical label that relates to different understanding of explanations by different people that hear an explanation or by different people that provide the same explanation for some circumstance at different time) ? However, it is not clear to me that your (Xi) notation can deal with the situation I have presented, where the expectation differs for (1) the person that explains and (2) the person(s) that receive the explanation ? But I suspect it is just a matter that I do not have clear understanding of the full extent of how the (Xi) notation can be used ? Any help with understanding is appreciated.

 

The "i" in [imath]x_i[/imath] relates the definitions of an explanation with the underlying information, in that the value of the "i" tells you what defined entity it is, and "x" represents the underlying information. Different explanations can make those assignments differently. (Qfwfq's examples are not exactly correct, btw)

 

But first things first, about your example, you are just getting confused by semantics again. "I explained my night out to my mother" is not an example of "explanation" as DD defined it, even if it's a "procedure" and even if it yields an expectation that my mother gets upset.

 

You should understand by now, that by explanation he is referring to generating an understanding or conceptualization of some "noumena". So to speak...

 

So when you make another example, check first if you can replace the word "explanation" with "world-view" and "explaining" with "forming a world view". If you can't, you are just confusing the semantics of the word "explanation".

 

E.g; "...I take it that your definition applies equally to both (1) the person forming the world view and (2) the person(s) to which the world view is being formed?" <- see, that's off the mark.

 

-Anssi

[LaurieAG]

Hi AnssiH,

 

E.g; "...I take it that your definition applies equally to both (1) the person forming the world view and (2) the person(s) to which the world view is being formed?" <- see, that's off the mark.

 

I agree because the standard communication process requires the sender/receiver to communicate via a communication context that includes the knowledge from their own individual zones of experience. If there is no overlap in the respective zones of experience then what follows can be more like programming rather than communication unless some sort of bridge can be built via mutual consent.

 

Just because committing a communication to a receivers own zone of experience might change a receivers perception it will not change the reality of any situation if the information communicated is false. The Dali Lama had some interesting thoughts on what understanding the difference between perception and reality was but you should be able to find this out for yourself.

[Qfwfq]

Wow, this was confusing... (just read the replies to this thread)

True, look at the last new input.

 

Qfwfq, I must say, it is often very hard to interpret your complaints because you tend to only give short and vague comments.

When I do this, you always fail to appreciate my reason for it.

 

Would that be a fair assessment of what you are saying?

You could put it into those words if you like.

 

If so, I'm afraid you need to provide more information about why and how do you think this is so. Don't be vague and lay down some details.

Anssi, a well known tenet in debate is that the onus of proof rests upon the one who makes a claim. I said pretty much where I found non sequitur and what is ad hoc, there's no point asking me for greater detail.

 

Surely you understand that even if they do introduce some characteristics that affects the situation in some way, the symmetry arguments probably also play a role in there. Wouldn't that issue be worth investigating quite carefully?

Probably? On what grounds? That's what Dick answered and I told him to prove it, if he wants any serious philosopher to take him seriously.

 

Think about that a bit

Your two questions are a bit vague so I think we should avoid the possibility of misunderstandings. But in any case, instead of asking me about those questions, you could study the relation between Lie Groups and Lie algebras, possibly even their representations and then see the specific case of the Lorentz group and Dirac algebra (or Weyl algebra for weak interactions). The entirety of Group Theory is a vast thing to study but fortunately this isn't quite necessary; I only ever did what is essential to theoretical physics, with the groups of main concern to it.

 

As far as I can construe your meaning:

If it's the latter, then yes, it should not be part of the fundamental equation, but it would actually be representative of one of the data-handling mechanisms that we are currently employing in our world-views (i.e. should be part of the later derivations). If it's the former, then it would be representative of actual knowledge about reality in-itself (as in, only valid for our reality).

Uhm, sure you haven't run into some confusion between them? If so then, yes, maybe I've guessed your meaning and your two questions above do make sense to me but these conclusions seem flawed. It seems to me the first case would be more like my opinion. I'm afraid it is you that is not up to speed with the discussion.

 

If you don't want to put in the time to bridge your gap and see how Dick's presentation relates to the standard, then I guess all that should be said is that you are free to choose what to believe.

 

Rade, about your questions regarding prediction and explanation. It's very simple; being able to draw predictions based on undefined information, always implies "an understanding" of some sort exists.

I've never gone to bed to sleep the night without being sure the next morning would come and the sun would rise, because it has always been so, ever since I can remember, my father can remember, my grandfather could remember, his father and grandfather could remember and so on. What better understanding could one go looking for?

 

I.e, if you have "expectations", then a transformation from "undefined" to "defined" must exist. If you don't have definitions, what would you have expectations of?

Oh, so it isn't the "undefined" poltergeist we must apply the notation to and have expectations about, after all? What a relief! Actually, this was already clarified in a recent post by Dick, with his analogy of three cases about some chemical process, which makes it clear to me that considerations of information theory do apply.

 

That contains everything that you would call "understanding".

So there's no chance astronomers would come along and tell me they have a better understanding of day and night than the one I expressed above.

 

(Your description of "understanding" just categorizes and breaks it down further, but that categorization is not necessary or useful in DD's analysis)

Are you sure it doesn't mean Dick's analysis isn't very useful?

 

(Qfwfq's examples are not exactly correct, btw)

If I simply hadn't placed sufficient hype on the arbitrarities, I humbly beg your kind Lordship forgiveness for the frailty of my effort, otherwise tell me exactly where I have erred. Edited January 7, 2011 by Qfwfq
slight revisions

[Rade]

I have another question for AnssiH and/or Doctrodick. I am having problem seeing 'exactly' how the {X1,X2,X3..X3, t index} approach makes a distinction between explanation of chronology and history of the past.

 

What I mean is that, to say you can put events for some circumstance into a set of Xi notation labeled with a t index, is nothing more than a type of chronology, a sequence of events indexed by t with no reference to how the various Xi may or may not be related to each other. It would be like the individual files of a file catalog of books by different authors with different copyright dates, one after another, indexed by the date of copyright, not when the book is discovered to exist.

 

Now the historian is not so much interested in the sequence of the Xi, as they would be to uncover some veiled patterns that may exist within the sequence that may have relevance to the present.

 

So my first question. Where exactly within the presentation does the Xi notation change from being a set of chronology of Xi, to a set of historical value ?

 

And second, in what sense does any explanation provided by a historian of a chronology as given in my example reach the conclusion that the explanation relates to "noumena", I mean, where is the thing-in-itself that underlays any explanation put forward by the historian concerning the sequence of files in the catalog ? Would the noumena be the "words within each book", clearly information that is both undefined and unknown by reference to the Xi notation found on the files in the catalog ?

[Rade]

AnssiH, on 06 January 2011 - 05:39 PM, said:

I.e' date=' if you have "expectations", then a transformation from "undefined" to "defined" must exist. If you don't have definitions, what would you have expectations of?[/quote']I do not understand what you mean. You would have expectation (i.e., some probability from 0 to 1) that some undefined information exists that allows definitions to be formed, First comes the undefined information --->, then the concept of it --->, then the definition of the concept. See how the first arrow allows for "expectation" prior to any definitions. So, sorry, I really have no idea what you are trying to say.

[LaurieAG]

Hi Rade,

 

AnssiH, on 06 January 2011 - 05:39 PM, said: I do not understand what you mean. You would have expectation (i.e., some probability from 0 to 1) that some undefined information exists that allows definitions to be formed, First comes the undefined information --->, then the concept of it --->, then the definition of the concept. See how the first arrow allows for "expectation" prior to any definitions.

 

Its a pity that Hypography doesn't have more applied scientists because the transformation from philosophical theory to real experimental science involves resolving the undefined so that it can become defined and measured in a consistent manner.

 

When you start this applied transformation you note the points where important variables become masked by theoretical variables that are then manipulated without any concern for the impact on the underlying variables that they mask (the o--o--o furphy being a good example).

[Rade]

the transformation from philosophical theory to real experimental science involves resolving the undefined so that it can become defined and measured in a consistent manner.

Thank you for the comment. I note that Aristotle never made this mental transformation, from philosophy theory to real experiment. I am only aware of a single real experiment suggested by Aristotle, yet he remains one of the most comprehensive thinking people that ever lived. I often wonder why this is, why did he never come to realize the explanatory power of experimentation ? So, rather than drop a ball from a height and make measurements, he developed a philosophic theory to explain the observed path the ball took (which was found to be incorrect once the experiment was performed). Perhaps, as you say, he never made the applied transformation, he never made note of the points where important "real" variables become masked by theoretical variables ?

[LaurieAG]

Hi Rade,

 

Perhaps, as you say, he never made the applied transformation, he never made note of the points where important "real" variables become masked by theoretical variables ?

 

Yes exactly, I respect Aristotle for what he did successfully transform not what he didn't. Euclid was successful almost a century later.

 

http://en.wikipedia.org/wiki/Euclid

 

Although many of the results in Elements originated with earlier mathematicians, one of Euclid's accomplishments was to present them in a single, logically coherent framework, making it easy to use and easy to reference, including a system of rigorous mathematical proofs that remains the basis of mathematics 23 centuries later.

 

This is just like using A--B--C to model a degenerate triangle. Point D, the real variable that is actually connected to points A, B and C alone, is omitted (undefined and not even considered) and the real structure is masked by a perception that is wrong in both 2D and 3D.

[Qfwfq]

Aristotle never made this mental transformation, from philosophy theory to real experiment.

Actually, he believed very much in the importance of observation but complexity got to him. He stressed that it is pointless to hold an opinion which plainly contradicts observation, but he repudiated quantitave measurement and analysis because the inevitable imprecision and uncertainty really got to him; he lacked modern technique of data analysis which overcomes these problems. Complexity got to him so much that there's a lore, according to which he anded his life by throwing himself off the coastal cliffs after watching the motion of the Aegean sea waters at great length; he came to the conclusion he would never understand them.

 

So, he usually preferred to match his experience up with considerations on qualities and attributes.

[LaurieAG]

Hi Qfwfq,

 

Actually, he believed very much in the importance of observation but complexity got to him. He stressed that it is pointless to hold an opinion which plainly contradicts observation, but he repudiated quantitave measurement and analysis because the inevitable imprecision and uncertainty really got to him; he lacked modern technique of data analysis which overcomes these problems. Complexity got to him so much that there's a lore, according to which he anded his life by throwing himself off the coastal cliffs after watching the motion of the Aegean sea waters at great length; he came to the conclusion he would never understand them.

 

So, he usually preferred to match his experience up with considerations on qualities and attributes.

.

 

Good point considering what Aristotles real field of expertise was. He was the expert at observing human behaviour/opinion and the drivers for modifying these behaviours/opinions through sets of laws and instructions despite mutual social agreements that provided otherwise.

 

Unfortunately the universal calculus* of human behaviour/opinion is as fluid and everchanging as the ocean is when compared to the relative immutability of the laws that govern these same human beings. * calculation; estimation or computation: the calculus of political appeal.

[Rade]

Hi Rade, This is just like using A--B--C to model a degenerate triangle. Point D, the real variable that is actually connected to points A, B and C alone, is omitted (undefined and not even considered) and the real structure is masked by a perception that is wrong in both 2D and 3D.

Thank you, yes, this make perfect sense, when I hold a 2D triangle in such a way that all I see before me is a straight line, and for a 3D triangle all I see is a plane (it looks like my 2D triangle). Perhaps the reality of the degenerate triangle becomes unveiled in the 4D representation used by Doctordick, when the tau dimension is added to the three x,y,z spatial dimensions ? Seems to me this should hold true, but perhaps I error, because I cannot visualize the outcome of perception of a degenerate triangle in 4D. If this is true, then deep reality (the triangle as triangle) may be sensible to humans only as a 4D thing, yet such representation may be completely outside knowledge of the thing it itself. Edited January 11, 2011 by Rade