Steiner and Schrodinger's Equation

[Federica]

For the cube to become a 'hypercube' in the 4th dimension, it must begin to coincide in some way with other 3-dimensional objects.

By the way, if we are looking for some concrete visual image for how this 4th-dimensional step of 3-dimensional objects 'coinciding' occurs, like we had for the previous steps, I don't think we will really find it. This step can be described by abstract mathematics, since it occurs purely in the stream of (qualitative) time, but I think the only visuals we can get are what Steiner presented in that lecture, or perhaps what was presented in the video series on imaginary numbers.

Thank you! Your illustrations are clear, I can follow the well written connections. I’ve now listened to the lecture again. Steiner also suggest the fourfolded terms: we can’t perceive the fourth dimension because we ourselves are 4d beings. Hence we easily perceive up to 3d, in the same way that a 2d triangle-being, for instance, would only recognize existence of lines, 1d, within its space of perception. Therefore, fourth-dimensional is the nature of our perception-cognition itself, which is why it’s a tough task to wrap our head around it and get a sense of the real nature of the World Content. So the type of effort we make through PoF Part 1 is similar to the attempt to grasp the meaning of 4d. In one case, we work on phenomenological terrain (which is of 3d nature anyway) in the other, we work in symbolic, mathematical terms. In your analogies, we are on yet another possible 'way’, that of the overarching grand architecture of reality that can be initially apprehended indirectly, through clairvoyant accounts, and can illustrate the other two approches. through analogy.

If we epitomize the fundamental unity of reality as: knowing = being = doing, maybe we could associate each of these ‘moods’ with the three approaches. Maybe PoF Part 1 is more like a doing, the symbolic-mathematical approach is knowing, and your illustrations of the higher worlds' organization is like timeless being.

But in the same way that we can strive all the way through the given of experience to touch reality, and that we can develop clairvoyance to directly ascend to the analogies you have traced, I believe the mathematical mode of understanding also can be pursued in itself, with the same intention. That the sphere in the example symbolizes our I-consciousness incorporating itself is insightful, but for me I know an additional step of intuitive mathematical kind is required, otherwise the analogy remains a standalone parallel. I understand that we can't find a visualization in 3d space that would account for the dimensional step up. Still, from my side there is something left to conceptualize-intuit. Clearly that connecting intuition is with you, otherwise you wouldn't have formulated the analogies, and I don’t think there’s more you (or anyone) could say to help. I only feel that I should go further on these mathematical tracks, before switching to the alternative tracks of analogy.

In other words, I want to make sense of the examples in the lecture - why those in particular? Beyond the symbolism of the initial and final states, as you have explained, how are the thought-exercises supposed to symbolize the transformation between states, and what do a point brought within a sphere and a left glove on a right hand have in common? Surely these cues are supposed to elicit recognition of some common feature.

So far, the most insightful visualization for me has been the straight segments of growing lengths, made into bigger and bigger circles, by connecting their edges. The longer the segment, the bigger the circle, and the more imperceptible the curvature applied to the segment, until an infinitely long segment (a line) acquires the 2d quality of circularity, by being locally imagined as portion a circle of infinitely big size. In this way, circularity gives the line a way to ‘come back from its infinity’, and so to encompass ‘direction’, that is, growth, movement. I guess the same applies to a sphere versus a plane. A plane can be locally understood as (a plane is) the surface of an infinitely big sphere. With this knowing, the plane ascends to 3d level, or we could say, it discovers 3d nature within its 2d constitution. This enables the plane, from that higher standpoint, to encompass not only ‘vectors’ of growth/movement, but also self-contained forms and geometrical patterns of any sort - forms with unique character. Similarly, a 3d-shaped being should be able to push its boundaries against its 4d confines and so become fully operational in its 3D environment. For example, a sphere-being could be realized as an infinite sphere volume and its nucleus (potential) at the same time, and so it would become able (from that 4D outlook) to fully apprehend the whole volume of 3D world content and its interconnected dynamics, through the tension between manifestation and potential streaming from within and without the space. And here I should pause for now, before I bring complete chaos to my inner spheres

[AshvinP]

In other words, I want to make sense of the examples in the lecture - why those in particular? Beyond the symbolism of the initial and final states, as you have explained, how are the thought-exercises supposed to symbolize the transformation between states, and what do a point brought within a sphere and a left glove on a right hand have in common? Surely these cues are supposed to elicit recognition of some common feature.

Federica,

These are good questions and I don't have any clear answer for them. The most simple thing that comes to mind is that Steiner needs something for our spiritual activity to work through in this domain of the 3rd to 4th dimension transformation, so he isn't simply describing it to us in abstract terms that we passively absorb, but we are actively stirring our inner forces to strive for a conception of the 4th dimension. So he starts drawing on examples of symmetrical mirror images (he speaks later of how the mirror image or polar opposite of a point radiating light is a sphere actively emanating darkness). I think some sort of qualitative differentiation should be utilized to exemplify/symbolize the transformation, rather than only quantitative differentiations that were used for 0-3 dimensions. Steiner later gives the example of cubes with three differentiated colors passing through a fourth color in order to symbolize the transformation (4th lecture).

These four cubes relate to each other in the same way as the three squares in our previous example. We needed six squares to delineate the boundaries of a cube.30 Similarly, we need eight cubes to form the boundaries of the analogous four-dimensional figure, the tessaract.31 In the case of a cube, we needed three accessory squares that simply signified disappearance through the remaining dimension. A tessaract requires a total of twelve cubes, which relate to one another in the same way as the nine squares in a plane. We have now done to a cube what we did with squares in the earlier example. Each time we chose a new color, we added a new dimension. We used colors to represent the four directions incorporated by a four-dimensional figure. Each of the cubes in this figure has three colors and passes through a fourth. The point in replacing dimensions with colors is that three dimensions as such cannot be incorporated into a two-dimensional plane. using three colors makes this possible. We do the same thing with four dimensions when we use four colors to create an image in three-dimensional space. This is one way of introducing this otherwise complicated subject. Hinton used this method to solve the problem of how to represent four-dimensional figures in three dimensions.

Steiner, Rudolf. The Fourth Dimension (p. 38). SteinerBooks, Collected Works. Kindle Edition.

It's interesting because the guy from the video on imaginary numbers and complex functions does a similar thing when speaking of the Reimann surface.





Anyway, Steiner then goes on to say that, if we want to really understand the metaphor of four-dimensional space, we need to do specific inner exercises that give us spiritual perception of the astral plane where 'objective' and 'subjective' qualities coincide.

We have demonstrated at least how we humans can visualize four-dimensional space in spite of being able to perceive only three dimensions. At this point, since you also may wonder how we can gain an idea of real four-dimensional space, I would like to make you aware of the so-called alchemical mystery, because a true view of four-dimensional space is related to what the alchemists called transformation. [First text variant:] if we want to acquire a true view of four-dimensional space, we must do very specific exercises.

Steiner, Rudolf. The Fourth Dimension (p. 39). SteinerBooks, Collected Works. Kindle Edition.

[Federica]

In other words, I want to make sense of the examples in the lecture - why those in particular? Beyond the symbolism of the initial and final states, as you have explained, how are the thought-exercises supposed to symbolize the transformation between states, and what do a point brought within a sphere and a left glove on a right hand have in common? Surely these cues are supposed to elicit recognition of some common feature.

Federica,

These are good questions and I don't have any clear answer for them. The most simple thing that comes to mind is that Steiner needs something for our spiritual activity to work through in this domain of the 3rd to 4th dimension transformation, so he isn't simply describing it to us in abstract terms that we passively absorb, but we are actively stirring our inner forces to strive for a conception of the 4th dimension. So he starts drawing on examples of symmetrical mirror images (he speaks later of how the mirror image or polar opposite of a point radiating light is a sphere actively emanating darkness). I think some sort of qualitative differentiation should be utilized to exemplify/symbolize the transformation, rather than only quantitative differentiations that were used for 0-3 dimensions. Steiner later gives the example of cubes with three differentiated colors passing through a fourth color in order to symbolize the transformation (4th lecture).

These four cubes relate to each other in the same way as the three squares in our previous example. We needed six squares to delineate the boundaries of a cube.30 Similarly, we need eight cubes to form the boundaries of the analogous four-dimensional figure, the tessaract.31 In the case of a cube, we needed three accessory squares that simply signified disappearance through the remaining dimension. A tessaract requires a total of twelve cubes, which relate to one another in the same way as the nine squares in a plane. We have now done to a cube what we did with squares in the earlier example. Each time we chose a new color, we added a new dimension. We used colors to represent the four directions incorporated by a four-dimensional figure. Each of the cubes in this figure has three colors and passes through a fourth. The point in replacing dimensions with colors is that three dimensions as such cannot be incorporated into a two-dimensional plane. using three colors makes this possible. We do the same thing with four dimensions when we use four colors to create an image in three-dimensional space. This is one way of introducing this otherwise complicated subject. Hinton used this method to solve the problem of how to represent four-dimensional figures in three dimensions.

Steiner, Rudolf. The Fourth Dimension (p. 38). SteinerBooks, Collected Works. Kindle Edition.

It's interesting because the guy from the video on imaginary numbers and complex functions does a similar thing when speaking of the Reimann surface.





Anyway, Steiner then goes on to say that, if we want to really understand the metaphor of four-dimensional space, we need to do specific inner exercises that give us spiritual perception of the astral plane where 'objective' and 'subjective' qualities coincide.

We have demonstrated at least how we humans can visualize four-dimensional space in spite of being able to perceive only three dimensions. At this point, since you also may wonder how we can gain an idea of real four-dimensional space, I would like to make you aware of the so-called alchemical mystery, because a true view of four-dimensional space is related to what the alchemists called transformation. [First text variant:] if we want to acquire a true view of four-dimensional space, we must do very specific exercises.

Steiner, Rudolf. The Fourth Dimension (p. 39). SteinerBooks, Collected Works. Kindle Edition.

Ashvin,

I am not very good at jumping onto a part number 13 on a series on imaginary numbers, so I have given up on the video for the moment since I don't know what a Riemann surface is, and I doubt it would be practical to figure that out now, plus the entertaining style of the video is probably not my favorite for this type of content. But I will continue with Steiner's fourth dimension lectures, that's sure.

[AshvinP]

Ashvin,

I am not very good at jumping onto a part number 13 on a series on imaginary numbers, so I have given up on the video for the moment since I don't know what a Riemann surface is, and I doubt it would be practical to figure that out now, plus the entertaining style of the video is probably not my favorite for this type of content. But I will continue with Steiner's fourth dimension lectures, that's sure.

Right, I wasn't sure if you had already worked through those videos or not. I really only wanted to call attention to one specific part of the video that is comparable to what Steiner (via Hinton) does in those lectures for the 3- to 4-dimensional transformation.

There is another variable, the imaginative part of z (function), we call this y, that is not included in our visualization... this is the danger of visualizing high-dimensional mathematical concepts. What we are looking at here is just a projection, a shadow of our full 4-dimensional surface... there are inherent limitations to the type of structures we can visualize in the 3 dimensions... one approach is to expand our visualization to include another dimension of human perception, such as color... our colors now give us a nice idea of what's happening in our missing 4th dimension, y.

It's interesting how our mathematical thinking in this respect naturally leads us to reintroduce the qualitative dimension of our perception (in this case color), no longer relying on only 'number, weight, and measure' as we can for the lower dimensional transformations. But that is only if we are trying to work with concrete images of the transformations, i.e. we remain somewhat connected to the first-person thinking perspective, instead of only abstract equations that we can keep as independent realities to think about from the side.

[Federica]

Thanks Ashvin, all your points make sure sense, except the impossible meat, to which one can only say: bleah...
By the way, it reminds me that few days ago you made a questionable statement in that area, that all the most pleasurable foods are also the most unhealthy ones - highly questionable, if you ask me

This makes me curious, do you disagree and take great sensuous pleasure in eating healthy foods??

Well, it also depends on how we define healthy foods. I can say that my perception of food has changed since I stopped eating animal corpses and drinking alcohol, and I appreciate food differently today, though it has taken some time to adjust. For a while I’ve had the habit of indulging in some so-called comfort foods, with the special intention to compensate for the ‘deprivation’, like chips or ice-cream. It didn’t work. Leaving aside the health element, these foods were not improving but actually worsening the situation. They were charged with unreasonable expectations, and eating them was just a prolonged negative experience, from overly anticipating, to agitated, to then disappointed. So I’ve learned to call forth this sense of ‘wrong’ invariably felt with those food compensations when grocery shopping, or, if I happen to be out, just before the moment of choice. And by paying more attention to the question, I have noticed that what makes food really pleasurable, comforting and satisfying are the care and preparation we put in and around it, both in thought and in practice. The more a meal comes about through careful, creative choices and positive intentions towards oneself and/or others, the more it’s satisfying, no matter how basic or healthy it is, and when this is not possible I try not to expect much. Not saying that I achieve this every time, not at all, but at least I now know how to create a positive experience around food. It literally can’t work if the primary focus is compensating pleasure. And I don’t regret what I no longer eat/drink, since it was basically unconscious behavior (what I sometimes do regret is to be lazy and not careful enough).

[AshvinP]

Thanks Ashvin, all your points make sure sense, except the impossible meat, to which one can only say: bleah...
By the way, it reminds me that few days ago you made a questionable statement in that area, that all the most pleasurable foods are also the most unhealthy ones - highly questionable, if you ask me

This makes me curious, do you disagree and take great sensuous pleasure in eating healthy foods??

Well, it also depends on how we define healthy foods. I can say that my perception of food has changed since I stopped eating animal corpses and drinking alcohol, and I appreciate food differently today, though it has taken some time to adjust. For a while I’ve had the habit of indulging in some so-called comfort foods, with the special intention to compensate for the ‘deprivation’, like chips or ice-cream. It didn’t work. Leaving aside the health element, these foods were not improving but actually worsening the situation. They were charged with unreasonable expectations, and eating them was just a prolonged negative experience, from overly anticipating, to agitated, to then disappointed. So I’ve learned to call forth this sense of ‘wrong’ invariably felt with those food compensations when grocery shopping, or, if I happen to be out, just before the moment of choice. And by paying more attention to the question, I have noticed that what makes food really pleasurable, comforting and satisfying are the care and preparation we put in and around it, both in thought and in practice. The more a meal comes about through careful, creative choices and positive intentions towards oneself and/or others, the more it’s satisfying, no matter how basic or healthy it is, and when this is not possible I try not to expect much. Not saying that I achieve this every time, not at all, but at least I now know how to create a positive experience around food. It literally can’t work if the primary focus is compensating pleasure. And I don’t regret what I no longer eat/drink, since it was basically unconscious behavior (what I sometimes do regret is to be lazy and not careful enough).

Yes, I agree and I am also trying to cultivate that more intentional, idealized expectation and satisfaction with healthy food, which is another way of saying more realistic. That is why I would emphasize it's only the sensuous pleasure that comes from the most unhealthy foods. Here I would define 'sensuous pleasure' as that which always comes at the expense of deliberately conscious awareness of and attunement to the deeper layers of experience, instead sucking us into subconscious feeling experience asymptotically approaching unconscious will experience. That was the great risk after the Fall - that the enthusiastic will of humanity would turn itself more and more towards immediate gratification of desires for sensuous pleasure and thereby snuff its own consciousness out. The situation was inverted by the capacity to become consciously aware of the psycho-physical risks associated with such pursuits and thus freely pursue the ideal-spiritual source of satisfaction more directly.

[Federica]

This makes me curious, do you disagree and take great sensuous pleasure in eating healthy foods??

Well, it also depends on how we define healthy foods. I can say that my perception of food has changed since I stopped eating animal corpses and drinking alcohol, and I appreciate food differently today, though it has taken some time to adjust. For a while I’ve had the habit of indulging in some so-called comfort foods, with the special intention to compensate for the ‘deprivation’, like chips or ice-cream. It didn’t work. Leaving aside the health element, these foods were not improving but actually worsening the situation. They were charged with unreasonable expectations, and eating them was just a prolonged negative experience, from overly anticipating, to agitated, to then disappointed. So I’ve learned to call forth this sense of ‘wrong’ invariably felt with those food compensations when grocery shopping, or, if I happen to be out, just before the moment of choice. And by paying more attention to the question, I have noticed that what makes food really pleasurable, comforting and satisfying are the care and preparation we put in and around it, both in thought and in practice. The more a meal comes about through careful, creative choices and positive intentions towards oneself and/or others, the more it’s satisfying, no matter how basic or healthy it is, and when this is not possible I try not to expect much. Not saying that I achieve this every time, not at all, but at least I now know how to create a positive experience around food. It literally can’t work if the primary focus is compensating pleasure. And I don’t regret what I no longer eat/drink, since it was basically unconscious behavior (what I sometimes do regret is to be lazy and not careful enough).

Yes, I agree and I am also trying to cultivate that more intentional, idealized expectation and satisfaction with healthy food, which is another way of saying more realistic. That is why I would emphasize it's only the sensuous pleasure that comes from the most unhealthy foods. Here I would define 'sensuous pleasure' as that which always comes at the expense of deliberately conscious awareness of and attunement to the deeper layers of experience, instead sucking us into subconscious feeling experience asymptotically approaching unconscious will experience. That was the great risk after the Fall - that the enthusiastic will of humanity would turn itself more and more towards immediate gratification of desires for sensuous pleasure and thereby snuff its own consciousness out. The situation was inverted by the capacity to become consciously aware of the psycho-physical risks associated with such pursuits and thus freely pursue the ideal-spiritual source of satisfaction more directly.

I agree with the definition, but I doubt it’s useful to classify foods as objectively healthy or unhealthy, since we mainly make them more or less healthy-unhealthy through our activity around them. Even water can become unhealthy if we drink it in unreasonable quantities, for example. Therefore, I would turn your statement around: “it's only the sensuous pleasure that comes from the most unhealthy foods” and I’d rather say: it’s our quest for sensuous pleasure first, that makes the food unhealthy, and the quest ultimately unsatisfied. Which is why I dislike impossible meat: its reason to exist, and to be consumed, is precisely to prioritize sensuous pleasure. As you say, to the extent that we can sacrifice its pursuit and use food as an opportunity to expand our state of consciousness of our flow (for example with foods that require some action long before they are consumed) food can become a source of more elevated satisfaction too.

[AshvinP]

I want to share two resources that are relevant to our prior discussion here on mathematical reasoning and the etheric formative forces. The first is a video by Linnell which is mostly rooted in Steiner's lectures, The Fourth Dimension. I found it really helpful to visualize the various forms and transformations involved. Apparently, it was recorded about 10 days ago, which is right about the same time Federica mentioned those same lectures!





The second is a great book on using the visual language of projective geometry to explore the phenomena of life, which manifests as a polar interplay of central (physical) forces working from the Earth's center and peripheral (etheric) forces working from the Cosmic expanses. I haven't made it too far into the book yet, but the first few chapters have been very informative and illuminating. I may have some more thoughts to share on it later and would be interested to hear anyone else's as well.


https://www.aetherforce.energy/the-plan ... chapter-1/

The fundamental notions of projective geometry are not really difficult of attainment; they are less difficult and certainly far less abstruse than many of the mathematical ideas applied, for example, in the physics of our time, in which large numbers of people take an intelligent interest. These fundamental notions are, however, comparatively little known. To enter into them requires only a certain effort in active and imaginative thinking, such as does not always come easily to people today.

We shall introduce them here descriptively, as simply as possible, with the help of illustrations, it being of paramount importance to activate one’s picture-thinking in a mobile and qualitative way, thus evoking a healthy feeling of form. The relationships and metamorphoses are related to the living rhythms of our life, which penetrate not only our abstract thinking, but our whole being. The meaning and content of the geometrical truths will become apparent through description and illustration, but we shall leave aside proofs. Reference can be made by those requiring a systematic treatment of the geometry to the relevant publications to which reference is made in the Notes and References.16

Historically, projective geometry, the modern form of geometry, which transcends Euclid’s ancient form, arose out of the the transformation of geometrical figures. Their transformation to begin with, as the name implies, is by perspective, where a plane figure, for example, is “projected” from one plane into another. “Transformation” means “metamorphosis”; it is not surprising that this kind of geometry is akin to the Goethean morphology.

[Federica]

I want to share two resources that are relevant to our prior discussion here on mathematical reasoning and the etheric formative forces. The first is a video by Linnell which is mostly rooted in Steiner's lectures, The Fourth Dimension. I found it really helpful to visualize the various forms and transformations involved. Apparently, it was recorded about 10 days ago, which is right about the same time Federica mentioned those same lectures!





The second is a great book on using the visual language of projective geometry to explore the phenomena of life, which manifests as a polar interplay of central (physical) forces working from the Earth's center and peripheral (etheric) forces working from the Cosmic expanses. I haven't made it too far into the book yet, but the first few chapters have been very informative and illuminating. I may have some more thoughts to share on it later and would be interested to hear anyone else's as well.


https://www.aetherforce.energy/the-plan ... chapter-1/

The fundamental notions of projective geometry are not really difficult of attainment; they are less difficult and certainly far less abstruse than many of the mathematical ideas applied, for example, in the physics of our time, in which large numbers of people take an intelligent interest. These fundamental notions are, however, comparatively little known. To enter into them requires only a certain effort in active and imaginative thinking, such as does not always come easily to people today.

We shall introduce them here descriptively, as simply as possible, with the help of illustrations, it being of paramount importance to activate one’s picture-thinking in a mobile and qualitative way, thus evoking a healthy feeling of form. The relationships and metamorphoses are related to the living rhythms of our life, which penetrate not only our abstract thinking, but our whole being. The meaning and content of the geometrical truths will become apparent through description and illustration, but we shall leave aside proofs. Reference can be made by those requiring a systematic treatment of the geometry to the relevant publications to which reference is made in the Notes and References.16

Historically, projective geometry, the modern form of geometry, which transcends Euclid’s ancient form, arose out of the the transformation of geometrical figures. Their transformation to begin with, as the name implies, is by perspective, where a plane figure, for example, is “projected” from one plane into another. “Transformation” means “metamorphosis”; it is not surprising that this kind of geometry is akin to the Goethean morphology.

Ether Force - what a site! Thanks for sharing. Is the book you are reading by Olive Whicher the linked one on plants?
I wonder, since the same author has this one titled Projective Geometry:
https://icedrive.net/s/ZS5PCYhyXuYA3YVB4FTBwXj2PbTz

I am listening to Linnell's presentation as an exercise. Surely a great exercise for me, because it's proving very tough to keep going after 30 minutes, and it's more than 2 hours in total! But I'm motivated And regarding the coincidence you mention, I don't even want to start thinking about it

[AshvinP]

Ether Force - what a site! Thanks for sharing. Is the book you are reading by Olive Whicher the linked one on plants?
I wonder, since the same author has this one titled Projective Geometry:
https://icedrive.net/s/ZS5PCYhyXuYA3YVB4FTBwXj2PbTz

I am listening to Linnell's presentation as an exercise. Surely a great exercise for me, because it's proving very tough to keep going after 30 minutes, and it's more than 2 hours in total! But I'm motivated And regarding the coincidence you mention, I don't even want to start thinking about it

You know by now there are no coincidences, Federica, your soul-entanglement with Linnell has led you back to him

Yes, the book is the one linked there and available for download - the Plant Between Sun and Earth. I also ordered the Projective Geometry book a while back, but that's the one I stopped and will return to.