Prospects for a Phenomenological Idealism

[ScottRoberts]

Ok, let's say I was referring to mathematical physicists. But, as you say, seen from the perspective of PoF, the risk for the mathematicians is the same, if they take pride in abstract mathematics, i.e. arbitrarily constructed rules of transformation of arbitrarily defined quantities. The construction is the form of their own thinking life, but in their view they edify or discover an independent, objectified reality. To the extent that the ideal interconnectedness escapes them - blinded by the illusion of purity and detachment from petty worldly concerns, and unaware of the implicit assumptions - they incur the same amorality as the physicists who aim to model the laws of nature as described. Unless any of them starts making use of those same formalized math relations from within a higher cognition of reality, in which case, as said, they realize its intrinsic morality.

Kind of going off-topic here, but I see it differently. In the first place, I would say that mathematical thinking is utterly concrete, not abstract. Indeed, it is almost the only ordinary thinking that is concrete. And that is because a mathematical concept is non-referential, while most thinking is referential. The concept of a triangle is the triangle, while our concept of a lion is not a lion. Also, what they deal with are independent and objective realities. They just don't happen to be very impressive as compared with, say, the thinking that can produce a tree. So to maybe get back on topic, I would say that the problem mathematicians have is, like the rest of us, they have other things to think about, like their academic standing, their relations with other people, etc. Their mathematical thinking in itself is not a problem.

I don't know Tegmark's philosophy. I know very little of all these things, beyond what Cleric and Ashvin have written and linked to in this forum, that I'm trying to apply here.

You're not missing anything. I just like to ponder his idea that all is mathematics, though I would change it to saying that all is fundamentally mathematical thinking, albeit of a hugely higher order of complexity (one that can create worlds), and imbued with feeling (not that our mathematical thinking is utterly void of feeling -- one can experience it as beautiful).

[AshvinP]

I wonder if your project should be reoriented towards summarizing/simplifying or otherwise collating, not necessarily PoF, but Cleric's various metaphorical illustrations on this forum, which themselves explore the PoF ideas but in different ways, from different angles. Very few of them reference Steiner or esoteric scientific concepts, and they provide a very modern language for us to conceptually explore states of higher cognition. For ex., on the question of intuition and 'purely spiritual experience of purely spiritual contents', you could work with the following post. I realize the reference to the first-person perspective of the 'Elohim' in this context is esoteric, but I think it would be relatively easy to work around such references.


viewtopic.php?p=17123#p17123

Now the key to higher cognition is that this structured potential can be known in various degrees. This is evident even from the simple example above - if we recognize our affected stated, we already have grasped some of the curvature of the potential that we're forced to crystalize.
....

I think what you suggest would serve as a Part II to what I am working on. All I would hope to achieve is to show that idealism, and the evidence for consciousness evolution, implies that there is higher cognition, while a Part II would show how to get there. But that's beyond me for the present.

I suppose that I'm not following what you intend for Part 1 when you say, "Now, as I have read some anthroposophy, I think I know what Steiner means by "purely spiritual experience with purely spiritual contents", but the only example I have for this is mathematical thinking. Which I value, but I fail to see how to get from that to moral intuition."

How to get to a somewhat lucid sense of moral intuition without metaphors and examples in our situational life? The problem with mathematical thinking in the ordinary sense is that the mathematical structure explored in thinking is not experienced as symbols for the intentional architecture of agency that weaves our soul-life-physical destiny. The modern intellect simply lacks the depth of experience to use mathematics as an occult tool for connecting with the higher hierarchies like people did right through the Middle Ages. That is, unless it is carried out consciously as a meditative means of attuning to the qualitative spiritual forces that are animating our mathematical thinking. We may indeed experience mathematical thinking as beautiful like we experience a sunset as beautiful, but that in itself does not give us clear intuition of its intentional structure, which I thought is what you were pointing to above as well.

This is very important because, if we don't find a way to reintroduce the qualitative moral dimension of a mysterious intentional architecture, then the intellect may try to encompass spiritual reality in the beauty of its mathematical thinking, sort of like this guy:

[ScottRoberts]

I suppose that I'm not following what you intend for Part 1 when you say, "Now, as I have read some anthroposophy, I think I know what Steiner means by "purely spiritual experience with purely spiritual contents", but the only example I have for this is mathematical thinking. Which I value, but I fail to see how to get from that to moral intuition."

How to get to a somewhat lucid sense of moral intuition without metaphors and examples in our situational life? The problem with mathematical thinking in the ordinary sense is that the mathematical structure explored in thinking is not experienced as symbols for the intentional architecture of agency that weaves our soul-life-physical destiny. The modern intellect simply lacks the depth of experience to use mathematics as an occult tool for connecting with the higher hierarchies like people did right through the Middle Ages. That is, unless it is carried out consciously as a meditative means of attuning to the qualitative spiritual forces that are animating our mathematical thinking. We may indeed experience mathematical thinking as beautiful like we experience a sunset as beautiful, but that in itself does not give us clear intuition of its intentional structure, which I thought is what you were pointing to above as well.

This is very important because, if we don't find a way to reintroduce the qualitative moral dimension of a mysterious intentional architecture, then the intellect may try to encompass spiritual reality in the beauty of its mathematical thinking, sort of like this guy:

[video of PI trailer]

I am saying I don't know how to get to a lucid idea of moral intuition without an appeal to the esoteric, which is what I want to avoid in Part I. Which, one might say, is why PoF doesn't quite work as a supposedly non-esoteric work. That is, moral intuition is an esoteric idea. So I recognize it is very important. I'm just not sure, at present, what to do with the idea. Maybe it is possible to show why we need to pursue such an idea in order to overcome our insanity, and Part II will show us how.

[AshvinP]

I suppose that I'm not following what you intend for Part 1 when you say, "Now, as I have read some anthroposophy, I think I know what Steiner means by "purely spiritual experience with purely spiritual contents", but the only example I have for this is mathematical thinking. Which I value, but I fail to see how to get from that to moral intuition."

How to get to a somewhat lucid sense of moral intuition without metaphors and examples in our situational life? The problem with mathematical thinking in the ordinary sense is that the mathematical structure explored in thinking is not experienced as symbols for the intentional architecture of agency that weaves our soul-life-physical destiny. The modern intellect simply lacks the depth of experience to use mathematics as an occult tool for connecting with the higher hierarchies like people did right through the Middle Ages. That is, unless it is carried out consciously as a meditative means of attuning to the qualitative spiritual forces that are animating our mathematical thinking. We may indeed experience mathematical thinking as beautiful like we experience a sunset as beautiful, but that in itself does not give us clear intuition of its intentional structure, which I thought is what you were pointing to above as well.

This is very important because, if we don't find a way to reintroduce the qualitative moral dimension of a mysterious intentional architecture, then the intellect may try to encompass spiritual reality in the beauty of its mathematical thinking, sort of like this guy:

[video of PI trailer]

I am saying I don't know how to get to a lucid idea of moral intuition without an appeal to the esoteric, which is what I want to avoid in Part I. Which, one might say, is why PoF doesn't quite work as a supposedly non-esoteric work. That is, moral intuition is an esoteric idea. So I recognize it is very important. I'm just not sure, at present, what to do with the idea. Maybe it is possible to show why we need to pursue such an idea in order to overcome our insanity, and Part II will show us how.

I see what you're saying. What about the experience of conscience? One could analogize it to 'pangs' of conscience that are, however, experienced with the lucid clarity and precision of mathematical thinking. That is, moral intuition provides a lucid and precise inner orientation for unfolding our personal development in a way that is increasingly harmonious with the development of all other beings.

[Federica]

Ok, let's say I was referring to mathematical physicists. But, as you say, seen from the perspective of PoF, the risk for the mathematicians is the same, if they take pride in abstract mathematics, i.e. arbitrarily constructed rules of transformation of arbitrarily defined quantities. The construction is the form of their own thinking life, but in their view they edify or discover an independent, objectified reality. To the extent that the ideal interconnectedness escapes them - blinded by the illusion of purity and detachment from petty worldly concerns, and unaware of the implicit assumptions - they incur the same amorality as the physicists who aim to model the laws of nature as described. Unless any of them starts making use of those same formalized math relations from within a higher cognition of reality, in which case, as said, they realize its intrinsic morality.

Kind of going off-topic here, but I see it differently. In the first place, I would say that mathematical thinking is utterly concrete, not abstract. Indeed, it is almost the only ordinary thinking that is concrete. And that is because a mathematical concept is non-referential, while most thinking is referential. The concept of a triangle is the triangle, while our concept of a lion is not a lion. Also, what they deal with are independent and objective realities. They just don't happen to be very impressive as compared with, say, the thinking that can produce a tree. So to maybe get back on topic, I would say that the problem mathematicians have is, like the rest of us, they have other things to think about, like their academic standing, their relations with other people, etc. Their mathematical thinking in itself is not a problem.

Yes, let’s look at that for a minute, I don’t think it’s really off-topic. I understand your use of “concrete” and “non-referential” here. Cleric and Ashvin would say “sense-free thinking” and I understand why you call it "utterly concrete". Now I’ll try to point out the PoF perspective, I trust Ashvin will correct if necessary. The point is, there is still a way to think the concrete thought of a triangle - or the concrete thought of a much more complex objective mathematical reality (not independent though, independent of what?) - abstractly. It all depends on the quality of the mathematician's thinking activity.

Let's say there is a certain reality whose dynamism is expressed in form of mathematical relations. The mathematician’s thoughts condense into that reality. In this way, the reality becomes perceptible in thought-images of mathematical relations. If the mathematician understands those relations appearing in his thought-perceptions as an independent reality he just discovered, in the same way Joao da Nova discovered the island of Saint Helena but simply not referred to the world of senses, then his “concrete” thinking remain collapsed into abstract thought, since thinking doesn’t recognize itself as awakening in the living interconnectedness with the beings of the hierarchies who are holding on those relations, infusing them with their own being, so that the mathematician can fill them with self consciousness. In which case, his mathematical thinking is definitely abstract, and therefore, it is a problem.

Their mathematical thinking in itself is not a problem.

But there is no mathematical thinking in itself. There is only conscious mathematical thinking flowing within certain concrete reality, or abstract mathematical thinking of those same relations.

I don't know Tegmark's philosophy. I know very little of all these things, beyond what Cleric and Ashvin have written and linked to in this forum, that I'm trying to apply here.

You're not missing anything. I just like to ponder his idea that all is mathematics, though I would change it to saying that all is fundamentally mathematical thinking, albeit of a hugely higher order of complexity (one that can create worlds), and imbued with feeling (not that our mathematical thinking is utterly void of feeling -- one can experience it as beautiful).

Just as it can't discover independent worlds, so mathematical thinking can't create them either, in PoF perspective, no matter the order of complexity. It can only become conscious of these worlds, conscious of their shared reality, by growing into it from within. If we imagine that our non-referential, thought-out mathematical concepts create worlds - only because these thoughts have no reference in the world of matter, like a lion has - then we are falling in the trap of abstractness, when we remain unaware of the concentric nature of our thinking with respect to the rest of reality, that is, with respect to the complex interconnectedness of the spiritual beings the concepts are made of. We don't experience the true nature of that reality, we are only spellbound to its precipitation, that we read as an independent new world. But maybe you meant it in a different way that I haven't understood?

[AshvinP]

Ok, let's say I was referring to mathematical physicists. But, as you say, seen from the perspective of PoF, the risk for the mathematicians is the same, if they take pride in abstract mathematics, i.e. arbitrarily constructed rules of transformation of arbitrarily defined quantities. The construction is the form of their own thinking life, but in their view they edify or discover an independent, objectified reality. To the extent that the ideal interconnectedness escapes them - blinded by the illusion of purity and detachment from petty worldly concerns, and unaware of the implicit assumptions - they incur the same amorality as the physicists who aim to model the laws of nature as described. Unless any of them starts making use of those same formalized math relations from within a higher cognition of reality, in which case, as said, they realize its intrinsic morality.

Kind of going off-topic here, but I see it differently. In the first place, I would say that mathematical thinking is utterly concrete, not abstract. Indeed, it is almost the only ordinary thinking that is concrete. And that is because a mathematical concept is non-referential, while most thinking is referential. The concept of a triangle is the triangle, while our concept of a lion is not a lion. Also, what they deal with are independent and objective realities. They just don't happen to be very impressive as compared with, say, the thinking that can produce a tree. So to maybe get back on topic, I would say that the problem mathematicians have is, like the rest of us, they have other things to think about, like their academic standing, their relations with other people, etc. Their mathematical thinking in itself is not a problem.

Yes, let’s look at that for a minute, I don’t think it’s really off-topic. I understand your use of “concrete” and “non-referential” here. Cleric and Ashvin would say “sense-free thinking” and I understand why you call it "utterly concrete". Now I’ll try to point out the PoF perspective, I trust Ashvin will correct if necessary. The point is, there is still a way to think the concrete thought of a triangle - or the concrete thought of a much more complex objective mathematical reality (not independent though, independent of what?) - abstractly. It all depends on the quality of the mathematician's thinking activity.

Let's say there is a certain reality whose dynamism is expressed in form of mathematical relations. The mathematician’s thoughts condense into that reality. In this way, the reality becomes perceptible in thought-images of mathematical relations. If the mathematician understands those relations appearing in his thought-perceptions as an independent reality he just discovered, in the same way Joao da Nova discovered the island of Saint Helena but simply not referred to the world of senses, then his “concrete” thinking remain collapsed into abstract thought, since thinking doesn’t recognize itself as awakening in the living interconnectedness with the beings of the hierarchies who are holding on those relations, infusing them with their own being, so that the mathematician can fill them with self consciousness. In which case, his mathematical thinking is definitely abstract, and therefore, it is a problem.

Their mathematical thinking in itself is not a problem.

But there is no mathematical thinking in itself. There is only conscious mathematical thinking flowing within certain concrete reality, or abstract mathematical thinking of those same relations.

This lecture on Imaginative cognition should provide some additional clarity to this issue. In a certain sense, mathematical thinking only becomes 'concrete' when it raises out of the sensory spectrum, weaves thinking freely in its own sense-free element (and thereby deconditions from sensory stimulation and purifies the soul-structure), and then returns to the living processes of Nature in this more conscious, purified mode of activity. We aim with this strengthened and purified thinking to unveil the hidden etheric and higher processes of Nature - which are not only lawfully ordered, but also aesthetic and moral processes - normally missed by our aliased attentional capacity. We could say that we are discovering the lawful processes that structure everything we already experience and therefore helps us make better sense of that experience, and also fashioning new worlds in that we are seeding the imaginative, inspired, and intuitive foundations for metamorphosed natural kingdoms to precipitate through future rounds of Earthly evolution. In the process of gaining new concrete insights into the World Content, we are indeed fashioning new organs of thinking-perception, and in the process of translating those insights into redemptive deeds, we are also fashioning a new organism as individuals and for the Earth as a whole.


https://rsarchive.org/Lectures/GA324/En ... 18p01.html

Most important is the fact that in the process of mathematical thinking, one is assured of continually following everything one does with full, clear consciousness. I believe I am not exaggerating when I say that clarity of consciousness can be measured against mathematical thought, its highest standard. In fact, when we engage in mathematical thinking, there is no possibility to doubt that each single manipulation we perform is accompanied by our inner conscious activity — for each is inwardly visible. We have ourselves in complete control, so to speak, when we think mathematically.

And, dear friends, the condition of consciousness present in mathematical thinking is in fact what a person strives for who strives toward what I call imaginative knowledge. When we think mathematically, what is really the content of our soul? It is the numerical world, the spatial world, and so on. I will speak of this later. Thus we have in our soul the content of a particular field with a certain pictorial representation. To work in a similar condition of soul but toward another pictorial content, is what constitutes the development of imaginative cognition. And this brings me to the following.

When we apply mathematics to outer nature (at first we can hardly do otherwise if we are accustomed to this approach), we apply it to only one part of nature, which we call the mineral world. In the mineral world we are presented with something that in a certain way is fully suited to a pure mathematical approach. But the moment we rise from the merely mineral to the plant or other kingdoms of nature, then the mathematical approach to which we are accustomed is of no use to us. A person who strives to rise to the imaginative level of knowledge desires to gain something more in his soul life than geometrical constructs or numerical relationships. He would like to gain forms that will live in his soul in exactly the same way as these mathematical forms, but which go beyond the mathematical in their content. He would like to gain forms that he can apply in the same way to the plant kingdom as he applies purely mathematical forms to the mineral kingdom. I will speak later concerning exact methods which lead in the direction of imaginative forms. Our first concern must be that everything that leads to an imaginative level of knowledge shall take place in a condition of soul that is absolutely equivalent to mathematical cognition. Actually, the best preparation for the development of imaginative cognition is to have dealt as much as possible with mathematics — not so much in order to reach particular mathematical insights as to be able to experience clearly what the human soul does when it moves in the realm of mathematical structures. This activity of the human soul, this fully conscious activity, is now to be applied to another area. It is to be applied in such a way that out of our inner constructs — if I may use the expression in a wider sense — we form further constructs which enable us to penetrate plant life in the same way that we penetrate mineral nature, chemical-physical nature with mathematical constructs.

[ScottRoberts]

I am saying I don't know how to get to a lucid idea of moral intuition without an appeal to the esoteric, which is what I want to avoid in Part I. Which, one might say, is why PoF doesn't quite work as a supposedly non-esoteric work. That is, moral intuition is an esoteric idea. So I recognize it is very important. I'm just not sure, at present, what to do with the idea. Maybe it is possible to show why we need to pursue such an idea in order to overcome our insanity, and Part II will show us how.

I see what you're saying. What about the experience of conscience? One could analogize it to 'pangs' of conscience that are, however, experienced with the lucid clarity and precision of mathematical thinking. That is, moral intuition provides a lucid and precise inner orientation for unfolding our personal development in a way that is increasingly harmonious with the development of all other beings.

I agree, conscience is the ordinary phenomenal experience that suggests a moral reality.

[ScottRoberts]

But maybe you meant it in a different way that I haven't understood?

I think so, and I think I should apologize for trying to express things without also providing sufficient background for why I express things in the way I do. The following may help to provide that background:

https://sites.google.com/site/nondualistlogic/thinking-and-feeling-language-and-perception

So, roughly, I define mathematical thinking as non-referential thinking, and I am saying that the thinking of beings of the spiritual hierarchy that creates worlds is non-referential. But the same is happening in the worlds that our mathematical thinking creates. It's just that they are tiny and don't have the strength to think an atom into existence.

[ScottRoberts]

This lecture on Imaginative cognition should provide some additional clarity to this issue. In a certain sense, mathematical thinking only becomes 'concrete' when it raises out of the sensory spectrum, weaves thinking freely in its own sense-free element (and thereby deconditions from sensory stimulation and purifies the soul-structure), and then returns to the living processes of Nature in this more conscious, purified mode of activity.

I disagree with the need to return to the living processes of nature for mathematical thinking to be concrete. But I agree that one needs to return to nature with this sort of thinking to find concreteness there. It will be a different mathematics, though.

We aim with this strengthened and purified thinking to unveil the hidden etheric and higher processes of Nature - which are not only lawfully ordered, but also aesthetic and moral processes - normally missed by our aliased attentional capacity. We could say that we are discovering the lawful processes that structure everything we already experience and therefore helps us make better sense of that experience, and also fashioning new worlds in that we are seeding the imaginative, inspired, and intuitive foundations for metamorphosed natural kingdoms to precipitate through future rounds of Earthly evolution. In the process of gaining new concrete insights into the World Content, we are indeed fashioning new organs of thinking-perception, and in the process of translating those insights into redemptive deeds, we are also fashioning a new organism as individuals and for the Earth as a whole.


https://rsarchive.org/Lectures/GA324/En ... 18p01.html

Most important is the fact that in the process of mathematical thinking, one is assured of continually following everything one does with full, clear consciousness. I believe I am not exaggerating when I say that clarity of consciousness can be measured against mathematical thought, its highest standard. In fact, when we engage in mathematical thinking, there is no possibility to doubt that each single manipulation we perform is accompanied by our inner conscious activity — for each is inwardly visible. We have ourselves in complete control, so to speak, when we think mathematically.

And, dear friends, the condition of consciousness present in mathematical thinking is in fact what a person strives for who strives toward what I call imaginative knowledge. When we think mathematically, what is really the content of our soul? It is the numerical world, the spatial world, and so on. I will speak of this later. Thus we have in our soul the content of a particular field with a certain pictorial representation. To work in a similar condition of soul but toward another pictorial content, is what constitutes the development of imaginative cognition. And this brings me to the following.

When we apply mathematics to outer nature (at first we can hardly do otherwise if we are accustomed to this approach), we apply it to only one part of nature, which we call the mineral world. In the mineral world we are presented with something that in a certain way is fully suited to a pure mathematical approach. But the moment we rise from the merely mineral to the plant or other kingdoms of nature, then the mathematical approach to which we are accustomed is of no use to us. A person who strives to rise to the imaginative level of knowledge desires to gain something more in his soul life than geometrical constructs or numerical relationships. He would like to gain forms that will live in his soul in exactly the same way as these mathematical forms, but which go beyond the mathematical in their content. He would like to gain forms that he can apply in the same way to the plant kingdom as he applies purely mathematical forms to the mineral kingdom. I will speak later concerning exact methods which lead in the direction of imaginative forms. Our first concern must be that everything that leads to an imaginative level of knowledge shall take place in a condition of soul that is absolutely equivalent to mathematical cognition. Actually, the best preparation for the development of imaginative cognition is to have dealt as much as possible with mathematics — not so much in order to reach particular mathematical insights as to be able to experience clearly what the human soul does when it moves in the realm of mathematical structures. This activity of the human soul, this fully conscious activity, is now to be applied to another area. It is to be applied in such a way that out of our inner constructs — if I may use the expression in a wider sense — we form further constructs which enable us to penetrate plant life in the same way that we penetrate mineral nature, chemical-physical nature with mathematical constructs.

I think any difference in outlook depends on whether we want to define mathematics to keep it as "which we are accustomed to", or to extend it to non-referential thought in general, which I am doing.

By the way, I am amazed that you are able to find these apropos lectures.

[AshvinP]

This lecture on Imaginative cognition should provide some additional clarity to this issue. In a certain sense, mathematical thinking only becomes 'concrete' when it raises out of the sensory spectrum, weaves thinking freely in its own sense-free element (and thereby deconditions from sensory stimulation and purifies the soul-structure), and then returns to the living processes of Nature in this more conscious, purified mode of activity.

I disagree with the need to return to the living processes of nature for mathematical thinking to be concrete. But I agree that one needs to return to nature with this sort of thinking to find concreteness there. It will be a different mathematics, though.

We aim with this strengthened and purified thinking to unveil the hidden etheric and higher processes of Nature - which are not only lawfully ordered, but also aesthetic and moral processes - normally missed by our aliased attentional capacity. We could say that we are discovering the lawful processes that structure everything we already experience and therefore helps us make better sense of that experience, and also fashioning new worlds in that we are seeding the imaginative, inspired, and intuitive foundations for metamorphosed natural kingdoms to precipitate through future rounds of Earthly evolution. In the process of gaining new concrete insights into the World Content, we are indeed fashioning new organs of thinking-perception, and in the process of translating those insights into redemptive deeds, we are also fashioning a new organism as individuals and for the Earth as a whole.


https://rsarchive.org/Lectures/GA324/En ... 18p01.html

Most important is the fact that in the process of mathematical thinking, one is assured of continually following everything one does with full, clear consciousness. I believe I am not exaggerating when I say that clarity of consciousness can be measured against mathematical thought, its highest standard. In fact, when we engage in mathematical thinking, there is no possibility to doubt that each single manipulation we perform is accompanied by our inner conscious activity — for each is inwardly visible. We have ourselves in complete control, so to speak, when we think mathematically.

And, dear friends, the condition of consciousness present in mathematical thinking is in fact what a person strives for who strives toward what I call imaginative knowledge. When we think mathematically, what is really the content of our soul? It is the numerical world, the spatial world, and so on. I will speak of this later. Thus we have in our soul the content of a particular field with a certain pictorial representation. To work in a similar condition of soul but toward another pictorial content, is what constitutes the development of imaginative cognition. And this brings me to the following.

When we apply mathematics to outer nature (at first we can hardly do otherwise if we are accustomed to this approach), we apply it to only one part of nature, which we call the mineral world. In the mineral world we are presented with something that in a certain way is fully suited to a pure mathematical approach. But the moment we rise from the merely mineral to the plant or other kingdoms of nature, then the mathematical approach to which we are accustomed is of no use to us. A person who strives to rise to the imaginative level of knowledge desires to gain something more in his soul life than geometrical constructs or numerical relationships. He would like to gain forms that will live in his soul in exactly the same way as these mathematical forms, but which go beyond the mathematical in their content. He would like to gain forms that he can apply in the same way to the plant kingdom as he applies purely mathematical forms to the mineral kingdom. I will speak later concerning exact methods which lead in the direction of imaginative forms. Our first concern must be that everything that leads to an imaginative level of knowledge shall take place in a condition of soul that is absolutely equivalent to mathematical cognition. Actually, the best preparation for the development of imaginative cognition is to have dealt as much as possible with mathematics — not so much in order to reach particular mathematical insights as to be able to experience clearly what the human soul does when it moves in the realm of mathematical structures. This activity of the human soul, this fully conscious activity, is now to be applied to another area. It is to be applied in such a way that out of our inner constructs — if I may use the expression in a wider sense — we form further constructs which enable us to penetrate plant life in the same way that we penetrate mineral nature, chemical-physical nature with mathematical constructs.

I think any difference in outlook depends on whether we want to define mathematics to keep it as "which we are accustomed to", or to extend it to non-referential thought in general, which I am doing.

By the way, I am amazed that you are able to find these apropos lectures.

If we are extending mathematical thinking to necessarily include higher cognition, then I suppose you are correct. I try to stay away from the term 'non-referential', because that can be misleading. Most mathematical thought doesn't refer to sensory (spatial) forms, but it does refer to spiritual (temporal or even timeless) forms-relations that are not perceived. I don't think we should imagine that, by thinking mathematically, we are necessarily experiencing the 'things-themselves'. In fact, we are not even doing so with Imaginative cognition.

By 'concrete', I basically mean 'practical' or 'pragmatic'. If we weave our thinking in the beautiful symmetries and relations of mathematics but can't find an avenue in which this higher insight can be applied to more effectively steer our stream of becoming towards higher ideals in the manifest world, then I think that would make it more abstract than concrete.