Reality consists of 3 spatial dimensions with time adding a fourth dimension. But what reason could we possibly have for putting such limits on reality? Do higher dimensions have any reality apart from their construction within a mathematical framework?
Plato believed in the reality of higher dimensions, as his allegory of the cave demonstrated. Claude Bragdon considered the fourth dimension to be spatial. He believed our conception of time as the fourth dimension was mistaken. We experience time as a “fourth dimension” because of our lack of ability in sensing this dimension which is spatial. He used an analogy equivalent to Plato’s cave analogy. A flatlander would experience a cube travelling through its plane-wise world as beginning with a point, expanding to a polygon and finally contracting to a point before disappearing from sight. It is obvious to me that this flatlander inhabits a three dimensional world and is itself three dimensional, but can only perceive in two spatial dimensions. It perceives itself and its fellow flatlanders as a polygon which changes over time. It conceives of reality as consisting in two spatial dimensions and one time dimension.
Rudolf Steiner discusses the dimensions of space and beyond in a collection of lectures and discussions collated in the book, “The Fourth Dimension. Sacred Geometry, Alchemy, and Mathematics.”
Here he considers beings consisting of various dimensions. A being of two spatial dimensions would only be capable of perceiving one dimension, a being of three dimensions would only perceive two dimensions and so on. In this resect we are beings of four spatial dimensions but we only perceive three of them. Although each of these three dimensions are unique. They all differ experientially.
Regarding the contemplation of the fourth dimension, Steiner admired the work of Charles Howard Hinton who has been credited with coining the word “tesseract” as the name for the four dimensional equivalent to the cube. As a cube can be represented in two dimensions by the hexagon, so the tesseract can be represented in three dimensions by the rhombic dodecahedron.
In Steiner’s lectures linked to above, he introduces his audience to the mathematical treatment of the higher dimension with accompanying diagrams. He also highlights the differences between mathematical treatment and the reality of further dimensions. He does not view reality as consisting of a series of ever increasing spatial dimensions. The neutralization of polarity in one dimension gives rise to the adjacent dimension. For example when two planes cross there arises a line. The planes give rise to the line but the line has no two-dimensional component.
He considers us to be six-dimensional beings with the three physical dimensions being a reflection of three higher causal, creative dimensions. The plants we perceive are three dimensional images of four dimensional beings. In his “archetypal plant” Goethe caught a glimpse of the reality of plants as four-dimensional beings.
Time is a projection of the fourth dimension into the three spatial dimensions of the physical world. It is a feature of living beings that they change intrinsically over time. Sentient beings encompass five dimensions and self aware beings encompass six dimensions. Thinking is dimensionless.
Now when we try to understand the connection between mind and matter some people regard this as a problem of interaction between an immaterial mind and a material body. This becomes a problem for both materialists and idealists to grapple with. But if we look at this from the point of dimensions we can see a solution. We can take an example of a similar problem in two-dimensional reality. In this two-dimensional world we can imagine a ring with a smaller disc sitting outside it. How can the disc get inside the ring without somehow interfering with the structure of the ring? This would be impossible if reality was limited to two dimensions. But if the disc could be lifted into the third dimension and then moved into the ring, this would seem like a miracle to any being perceiving in just two dimensions. It would be as if the disc disappeared and then reappeared within the ring. It is the same with an act of willing a part of my body to move, i.e. mind affecting matter. The connection does not occur in our familiar three dimensional spatial world but in the higher dimensions in which my inner sentient life belongs. This activity impinges on the three-dimensional world but it is not restrained by it.
I hate to break it to you, but dimensions are not part of reality. They are abstract human constructs. We construct them to help us make sense of the world.
So is that quantities and qualities ruled out? Is there anything you consider to be real?
I regret to inform you that the federation for the advancement of clear communication has decreed that the following words will be off-limits for you the coming twenty years: “field”, “vibration”, “quantum”, “etheric”, “astral” and “dimension”.
Thank you for your understanding
“Real” is a human word. It means what humans want it to mean. And people don’t all agree on what they want it to mean.
I do not recognize them as a governing body (3-D or otherwise), so I’m pleased to tell them that they can (to put it plainly) F.A.C.C. off 🙂
So you are only speaking for yourself when you say that dimensions are not part of reality. Would you say that in your opinion dimensions are not part of reality, but it’s not something you can state as a fact?
How real are the dimensions used when surgeons are preparing for and performing keyhole surgery?
Yes, I was expressing my opinion. I assume that many mathematicians and physicists would agree. But there would also be those who disagree.
I already have a post here, where I argue that facts themselves are just human artifacts.
Plato would never have said that the Forms are a different or higher dimension than sensible particulars. That’s a really bad misreading of the Allegory of the Cave.
Neil Rickert,
Thanks for that link, Neil. Good times… 😏
LOL
Seriously, though. I do not see how you get from
to
My body moves because of muscles contracting, which soundly reside in the three dimensions visible to us. Nothing disappears or reappears. In short, I do not see what the word “dimension” did to clarify matters.
Thanks for the link. You wrote, “The traditional view is that we pick up facts, and most of cognition has to do with reasoning about these facts”.
Well I would not put it like that. In my opinion we first apprehend the world as fragmented, and the process of learning is a striving for unification. Reality becomes less fragmented as we learn to make connections. Rather than picking up facts we discover relationships.
In your first example of a “fact” you give the coordinates of Chicago. Obviously these are a human artefact. In isolation, 41.8819° N, 87.6278° W is a group of meaningless symbols. To treat this as factual we would need to know what the facts are. And the fact is that this is a system that humans invented in order to have an agreed upon standard with which to locate any point on the surface of the earth. It is a fact that those coordinates are a human artefact, but does this mean that all facts are human artefacts?
What about the angle subtended by an arc formed between the centre of mass of a grain of sand on Lake Michigan and the centre of mass of a pebble on the lower reaches of the River Thames (the location of the ray’s origin being the centre of the earth) is a precise fraction of one full revolution of the ray from the centre which passes through both? Is this fact a human artefact?
I didn’t say that Plato would have said this.
The allegory of the cave has many sides to it. He explores the relationship between images and reality. We can understand that those confined in the cave can only see two-dimensional images and the world from which these images come has three dimensions. That gives us something to think about.
Through the physical eyes we apprehend physical dimensions. But we also possess the mind’s eye with which we gives us the opportunity “see” into other dimensions which are not spatial at all. In relation to these dimensions the spatial dimensions are but images; very useful images, but images nonetheless.
The problem does not lie within the brain, nerves and muscles, it lies between the thought and the action. I think about performing an action and I will it to happen. What initiates this process?
My view is pretty much the opposite. We divide the world up into parts, and then we name the parts. This dividing process is what creates the fragments.
Yes, we then try to put it back together. That’s how we learn — by dividing the world into parts and then describing how to put them back together.
Are our views really so far apart? Can we not agree that the fragmented world is of our own making.
Although it would be fatal to physically divide, say, a mouse into its component parts and then put it back together into its former state.
I did not suggest anything about physical dividing.
My point, though, would be that the mouse has no component parts, until we choose to divide it into parts. Again, that is not about physical dividing. And I won’t call it “logical dividing” because that is not a good description. The dividing is perceptual, part of perceptual discrimination.
I don’t think the difference in dimensionality is what interests Plato. He is fairly explicit that the distinction is one of degrees of variability. The shadows are constantly changing, and that is why it is foolish for the prisoners to claim they have any real knowledge of them at all.
Plato follows Heraclitus in holding that the world of the senses is a world of constant change, and he also follow Parmenides in holding that only that which is truly Is can be truly thought. His solution is to treat the world of becoming as a second-rate kind of reality, whereas only those things that are grasped wholly by the intellect are fully and completely real.
Interestingly, he positions geometric figures as in between sense-perception and the intellect — geometry deals with things that are more stable and unchanging than the world of the senses, but still falling short of what is grasped solely by the intellect alone.
Because what is beheld by the intellect alone is that which stands to mathematics as mathematics stands to perceptible objects (as he puts it in the Divided Line), I think it is a mistake to use the geometric concept of dimensionality to understand what he means by the Ideas. Ancient Greek mathematicians had the concept of dimensionality, and if that’s what Plato had in mind, he would have said so (probably).
You do.
I know you didn’t. But I thought it relevant and worth considering. It is a major difference between organisms and machines.
My previous remark that “we first apprehend the world as fragmented” might have been better written, “our sense experiences give us a disjointed view of the world”. And our thought processes give us the ability to recombine and thus give meaning to the separate sensations.
Seeing the mouse as an object in our vision is but a small fragment of the reality that is the mouse. The more we understand about its life processes, its habits, its internal structures and organs, its relationships, the less fragmented our view becomes.
And I would say that perceptual discrimination is not an act of fragmentation. Discriminating involves relationships. It is an act of gaining an understanding of how the various pieces of the jigsaw puzzle fit together. The picture emerges slowly and can involve many false placement of the pieces.
In what dimension do the original thoughts of inventors of mechanical devices exist prior to the production of an actual machine?
If they can be said to have any location at all, I’d hazard a guess that they live in precisely the same dimensions as the inventors themselves.
But even when a prisoner is freed and ventures out of the cave, when his eyes become accustomed to the light, still only sees a world that is constantly changing. The difference is that he has turned from looking downwards towards the images, to looking upwards to that which allows us to see images. That is the sun. But although he is gazing on the highest, it is limited to the world of the senses. There is a still higher realm (dimension) and that is to be seen by the mind’s eye. This is what the analogy of the divided line is all about, and the analogy of the cave is told to help clarify the divided line analogy.
And according to Socrates the freed prisoner has a duty to return to the shadowy world from which he came in order to help his fellow prisoners to know the truth.
I wouldn’t say he treats the world of becoming as a second-rate kind of reality. It is more like the basics which have to be learned if anyone is to reach an intellectual understanding of “the good”. Who would say that the times tables are a second-rate form of mathematics? Socrates gives a specific order in which learning should proceed. Arithmetic, plane geometry, solid geometry, on to the movement of solid bodies, and onwards.
As Socrates says in The Republic, “knowledge at which geometry aims is knowledge of the eternal, and not of aught perishing and transient”. The Platonic solids and his ideal tringles are the ideal forms out of which any material body is but an image. The shadows on the cave wall are two-dimensional images and likewise the three-dimensional figures outside the cave are also images.
In The Republic Socrates tells Glaucon that they had proceeded in the wrong order:
When we practice geometry we use our physical senses to observe and manipulate these geometrical figures, but these are just images in the three-dimensional sense world. Plato considers the reality of these figures to exist in the ideal world. The physical dimensions are a mirror image of the spiritual dimension in which the ideal forms reside. This can be accessed by the intellect, the mind’s eye, just as physical figures can be accessed by the senses. That is how Plato divides the line.
That’s right, “I” do. The “I” being the highest of the four elements that makes up the physical human being. The next lower principle, the sentient “body”, the seat of my emotions, might give me the urge to act and I can either carry out the action or resist my urges.
Uhm, no I don’t think that is entirely correct. You might want to check a periodic table to look up the correct number of elements.
Anyway, my question was: what does the word “dimension” do to clarify matters? And believe you me, you desperately need to clarify matters.
The divided line makes pretty clear that the intelligible stands to the visible as the things perceived stand to images in reproductive art.
More importantly for our conversation, the divided line has four segments arranged in precise order. The intelligible and the visible are each subdivided. The shortest (and thus least important) is artistic images as a sub-section of the visible, Then comes sense-perceptions as the next sub-section of the visible.
But the intelligible is also sub-divided into geometric figures and arithmetic units, followed by the intelligible all by itself. The key difference is that the former (mathematics) begins with axioms and deduces conclusions from them, whereas the latter is the search for ultimate first principles themselves.
Yes, though interestingly the prisoner had to be forced to leave the cave and is then forced back down into it.
Yes, that’s the order in which geometric concepts are mastered, but that’s not relevant to the point I’m trying to make.
That’s nonsense. Material bodies aren’t “images” of geometrical objects. And while yes, geometry aims at knowledge of the eternal, it falls short of true knowledge of the forms.
The divided line makes it tolerably clear that geometry stands to the Forms as reproductive art stands to sense-perception. So trying to understand the Forms as geometric figures is not consistent with what Plato is actually saying.
But the dimensions have a polarity. In this materialistic age any point in three-dimensional space is measured from the point where the x, y and z axes meet in a space in which the infinitely distant plane is regarded as fixed. But if thoughts are not regarded from the point as is physical substance, but have their origin from the polar opposite plane-wise direction, then they are beyond being precisely measured in space. In my opinion reality consists of the physical, measurable, pointwise, radial forces in polarity with the spiritual, immeasurable, planer, encircling forces.
There is a polarity between space and counter-space which is most easily observed when we examine plant life following the lead of Goethe. Here we see phases of expansion and contraction, radial forces of the rising shoots and planar nature of unfolding leaves. Plants force their way up from below but they are also drawn upwards through heavenly forces.
I think there is a bit of a language problem here. I used the word “element” as to mean a principal component. I prefer to use language flexibly and not be too rigid.
Sometimes we have to put in a bit of effort to understand each other when we are discussing complex and obscure topics. It’s not always easy to make oneself understood or to understand the point someone else is trying to make.
“Dimension” can refer to space, it can refer to time, we can think of dimensions of the mind. Plato’s divided line is a good example of where a physically measureable entity can be used as an analogy to gain an understanding of that which is not amenable to this type of measurement.
But if thoughts are regarded as a physical phenomenon spatially restricted to brains, then we don’t need any extra dimensions. I think you were begging the question when you asked where decisions initiate.
That’s up to you of course, but I reserve the right to reject arguments that are unclear to me.
Well, you could resist talking about polarities and pointwise and planar forces and other stuff you know I do not accept. For my part, I think I can see what the analogy is supposed to show. As hinted above, I think your analogy fails because you are smuggling in the very thing you are trying to demonstrate: that minds and thoughts are non-physical phenomena. People who have no problem with minds as physical phenomena associated with brains will not see the point of adding bonus dimensions.
Representational art or reproductive art is not what any of the segments of the divided line represent. I’ve attached an image from an interesting lecture on the divided line. Here he has divided the line into segments AG, GC, CH, and HB.
AG represents images (eikasia). Works of representational art would only belong in this section if they were being confused with what was actually represented, otherwise they would belong in segment GC, seen as an actual object of art. This section concerns images such as shadows or mirror reflections which are taken as real through a lack of knowledge of the objects casting the shadows or reflections.
Socrates was less concerned with the relative lengths of each segment and more concerned with proportions between segments. The whole reflected in the parts. Using the notation of the attached image we get; BC:BA = BH:BC = CG:CA. He was asking Glaucon to consider a line segment which was arbitrarily divided into two unequal segments. If he was concerned with relative lengths don’t you thing he would have stipulated where along the segment the dividing point should lie?
The latter, BH (noesis) is not a search. Searching belongs in the former, CH (dianoia). Noesis is the direct apprehension of reality. As I like to point out, an example would be the perception of the ideal triangle in the mind’s eye. All physical triangles are representations of this triangle.
He would willingly have returned before his eyes had become accustomed to the new conditions. Acclimatizing is a slow process but when it does occur he returns to the cave with knowledge that his fellow prisoners can know nothing of and so they cannot make any sense of what he is telling them. Is it any wonder he would be reluctant to go back down?
But it is interesting that it is a logical sequence through the physical dimensions.
Material bodies are geometrical objects, which are imperfect representations of geometrical forms.
Do you not think that people can distinguish between reproductive art and the object which they are supposed to represent? Notwithstanding the work of Magritte the average person would be able to distinguish a painting of a pipe with a real pipe.
In order to understand the ideal triangle of Plato it is not enough to have a mental picture of a triangle. The mental picture is also a representation which would require movement to better represent the ideal.
Surely that is what’s under debate. Are thoughts physical phenomena?
And I endorse your right.
I’m not asking you to accept anything I say. I am happy that you are at least giving my posts a little thought.
Here Steiner gives answers in a Q & A session in 1922 . Steiner argues that the reality of the fourth dimension and beyond does not continue ad infinitum. This continuation of dimensions can be dealt with mathematically, but it is an abstraction. He explains that the fourth dimension actually negates the third spatial dimension. So if we designate the increase in the standard dimensions as +a for the first, +b for the second, and +c for the third; the continuation results in -c, -b, and -a. We begin with a dimensionless point and end with a dimensionless point.
Putting this in the context of “The Divided Line” as in the image above, The lower end “A” is dimensionless and the upper end “B” is also dimensionless.
So maybe Steiner did not understand relativity.
I am not aware of any information to the contrary.
CharlieM,
At Republic 509d-510a, Plato writes, “In terms of relative clarity and opacity, one subsection of the visible consists of images. And by images I mean, first, shadows, then reflections in water and in all close-packed, smooth, and shiny materials, and everything of that sort”.
It seems tolerably evident that “close-packed, smooth, and shiny materials” refers to pigments, marble, and clay — the materials used in Greek ornamental art, especially pottery, painting, and sculpture. (It may be helpful to notice that a common theme of Greek ornamental art is tales from Greek mythology, and we know that Plato would prohibit the poets from being part of the ideal city because poison the souls of the people by telling lies about the gods. So here Plato is saying that the visual artists are those who know the least about reality because they are furthest removed from it.)
We also find in Republic 510b, “In one subsection [of the intelligible] the soul, using as images the things that we imitated before, is forced to investigate from hypotheses, proceeding not to a first principle but to a conclusion. In the other subsection, however, it makes its way to a first principle that is not a hypothesis, proceeding using forms themselves and making its way through investigation of them.”
In the highest part of the divided line, “reason itself grasps, by the power of dialectic. It does not consider these hypotheses as first principles but truly as hypotheses — but as stepping stones to take off from, enabling it to reach the unhypothetical first principle of everything. Having reached this principle, it reverses itself and, keeping hold of what follows from it, comes down to a conclusion without making any use of anything visible at all, but only of forms themselves, moving on from forms to forms, and ending in forms” (Republic 511b-511c).
Plato is quite clear that in geometry, we begin with images and proceed from those images to conclusions — whereas in dialectics, we travel up to first principles themselves and investigate the forms without using any images at all. This is consistent with what we know about how the ancient Greeks practiced geometry. For them, doing geometry meant drawing figures in the sand. A geometric proof was not a set of symbolic notations to be manipulated on paper, but a concrete sequence of physical movements to be enacted by tracing out lines in the sand.
I know you like to say that geometric objects are examples of the Forms. I am pointing out to you that Plato explicitly denies that. If you want to disagree with Plato, that’s your prerogative. But then stop invoking the authority of his name for advertising your own beliefs.
Well, thoughts don’t seem to be physical. But the question is whether that licenses the inference of immaterialism or dualism. After all, material bodies don’t seem to be mostly empty space, even though physics tells us that they are. So there’s actually a nice question here whether the seeming non-materiality of thoughts is any different from the seeming non-emptiness of material bodies.
The going view amongst cognitive scientists is that thoughts are really material in roughly the same sense that material bodies are really mostly empty space. But it’s never a bad idea to examine the philosophical assumptions that this view relies on, especially because it’s true.
Nobody really understands what conscious thought is or how it originates so why shouldn’t it spring forth from some realm unknown to contemporary physics? I think the main argument for anchoring it in everyday physics is that, to the best of our knowledge, conscious thought requires electrochemical activity in nervous systems. In previous discussions, Charlie’s arguments focused on severing that link by talking about ghosts and near-death experiences. Not the most convincing approach IMNSHO. But IF the postulate of some mind realm could serve as a plausible explanation for as yet unexplained phenomena, that would be a powerful argument. Not seen any, so far.
Steiner from the book, “The Fourth Dimension,…”
Living bodies are built up by a physical, radial, expansion of substance into space. But that is only one half of the reality. If this one sided view was all that’s involved the expansion would be without form. The complimentary pole in the formation of organisms consists of centripetal inwardly directed forces which originate not from individual points but from the the whole of the surrounding sphere.
“Seek and ye shall find”. 🙂
Any person who claims to be an artist of any sort but who does nothing but produce imitations or copies of the reality around them produces work that does belong along with the shadows and reflections of the first line segment. And that is where Plato quite rightly places them.
But “all close-packed, smooth, and shiny materials” includes any reflective surface. “The ancient Egyptians, Romans and Greeks were quite fond of mirrors and often manufactured mirrors from polished copper and bronze.”
The images displayed in these mirrors belong in the first segment.
I absolutely do not say that geometric objects are examples of the Forms. Geometric objects are imperfect imitations of the Forms.
Socrates states that geometricians:
The generals need to have a practical knowledge of geometry in order to defend the state but this concerns becoming and is looking back to the lower half of the divided line. The philosopher is concerned with the “being” of geometry. And for Plato knowledge of mathematics is not something that one newly acquires, it is a process of recalling. Learning geometry for its own sake is a process of drawing out the student’s knowledge of being.
From your quote, it is now clear that Steiner was confused about relativity.
And are confused by woo.
When did I talk about ghosts? Can you provide a link?
Can you give us more to go on than an assertion?
During a previous discussion you posted a link to this guy (among several. It was over two hours of footage in total):
I can’t be bothered to find the exact time, but IIRC the patient claimed to have interacted with someone who was deceased. This was supposed to support your claim that thinking is possible without brain activity.
To be able to find something, it needs to exist. Otherwise you are wasting your time.
Steiner is looking at relativity as three dimensional. But the time dimension is fully entwined in relativity. You cannot understand relativity as only three dimensional.
It’s just models Charlie. Einstein’s model used time as a fourth dimension, and helped better match observations, and give an explanation to gravitation, and provided some predictions later confirmed. But it’s a model. Other models in physics, I don’t remember if in string theory, seem to converge to 11 dimensions, if I remember the number correctly. That doesn’t really mean there’s 11 dimensions, it means that 11 dimensions seem to explain/predict some observations/phenomena. “Derived” woo-woo, aka mental masturbations about dimensions, is just overenthusiastic fantasy, inspired by mistaken notions of what the models might mean.
So I didn’t actually talk about ghosts. I referenced someone who may have spoken about ghosts among other things.
No, Steiner discusses Einstein’s theory of relativity in relation to three spatial dimensions and a time dimension. He regards Einstein’s theory to be correct as far as it goes, but it leads to abstractions.
Steiner also gives his own account of dimensionality.
Regarding the addition of dimensions, a further dimension is generated through the movement of the previous dimension. Move a point and a line is produced. Move a line segment sideways and the plane figure of a rectangle is produced. Move a square sideways the same distance as the length of its side and a cube is produced. Similarly moving the cube will produce a tesseract.
Mathematically this can be continued to produce dimensions of continuously greater numbers. But these are abstractions.
According to Steiner in reality we live in a three dimensional world in which each of these dimensions have their own characteristics. He talks about the way the mental/spiritual sphere connects to the material/bodily sphere. Our bodily nature is spatial and we live and have an interaction of our thinking, feeling and willing with our bodies. Our willing has a relationship with our bodies in three dimensions. Our feeling life is active in two dimensions, and our thinking in one dimension. Our egos are dimensionless.
So rather than dimensions continuing on endlessly there is a polarity in which the fourth dimension becomes a negative third dimension, the fifth dimension becomes a negative second dimension and the sixth dimension becomes a negative single dimension. (I use the term “negative” purely as a contrast to the physical dimensions).
And this is where I deviated from Steiner’s scheme in the op. I said, “thinking is dimensionless”. After meditating on this I now realize that his scheme makes more sense. If I could rewrite it I would say the ego is dimensionless.
Plato and Socrates were wise men. Models belong with the images of reality in in lower segments of the divided line. The upper segment foregoes models and images and apprehends reality directly.
Perceiving the ideal triangle is a case in point.
Space and time are abstract, even without relativity.
Is “the ideal triangle” isosceles, equilateral, or scalene? How do you know which one is more ideal than the others?