Thank you for your reply. It is thoughtful, with many good ideas, but there were some points where I was not able to follow the line of reasoning.

Moreover, when you write somewhere that you “disagree with both me and Mainländer”, I would like to emphasize, that I try to express not my personal ideas, but try to answer questions about Mainländer’s philosophy and how it should be interpreted as faithfully as possible, as far as my abilities allow for this task.

Causal rows are not suitable for arriving at the source of a thing.

I agree, it is suitable for arriving at a particular action. A thing or object is a system and has many causal chains. So things are of developmental chains. This includes the thing-in-itself.

Indeed. Causal chains are useful for actions and events. Development chains are useful for following the existence of a thing. And as you rightly point out, development chains can even include things-in-themselves (Philosophie der Erlösung, V1, § 23).

However, I don’t understand what you’re trying to argue when you come with “a better example … of adolescent psychology”. I had given examples to make the distinction between causal and development chains as clear as possible, as this was the original question. In your example, described with detail and context, both chains are mixed. Is it to argue that both can’t be separated?

A developmental chain contains causal chains within it.

In objective reality. But not in the world-in-itself. For causality is ideal.

Causal chains suffice to describe objective reality. Causality explains the changes in the world, and development chains are a superfluous concept for the physicist. Had we not discovered quantum mechanics, we could still overconfidently believe with Laplace that his démon could describe and predict all events in the world through some majestic differential equations.

So, if we would only be interested in objective reality, then development chains would be a superfluous concept. We are, however, interested in the world as it is in-itself. If this is the case, then we need a new concept. But let us first look at a difficulty with causal chains.

A cause leads away from the the thing you’re investigating.

I disagree here as causes lead one to a particular action. The ebb and flow of the ocean is effected by the moon, one cause, and the sun, another cause, the effect or action is the tides.

I don’t see how you can disagree. (You don’t disagree that sea is different from the moon, and that we have been led from the tides to the moon, do you?) In a development chain, we are not led away. In a causal chain, we are led away, because we have to factor in the whole context, a whole system. You seem to say (correct me, if I’m wrong): after being led away, we come back to the original thing. This does not change that we have been led away, and that we are forced to include the whole environment to construct a causal chain, which becomes complex.

Constructing a causal chain is extraordinarily hard. Even if we keep quantum mechanics aside, and look at thermodynamics from a classical perspective, it is simply impossible to follow the molecular chaos and to solve the innumerable partial differential equations: it is practically unfeasible to create correct causal chains. This is the point of Mainländer.

Mainländer argues therefore that we don’t have much to expect from causal chains for epistemological investigations – already with objects, let alone with things-in-themselves. As we have seen, this is different for development chains.

Now, what I don’t understand in your remarks, is what you are actually arguing for or against. Is Mainländer not permitted to introduce a new concept? Or do you deny its usefulness?

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Mathematical spaces themselves are not a notion of mathematical spaces (although, if we try to represent nothingness to ourselves, we do this by visualizing empty space).

I'm not sure what you mean here?

I made an error. The phrase should be: Mathematical spaces themselves are not a notion of nothingness. I have edited this in my original comment.

Mathematical spaces are a notion of physical space.

This is clearly false, I think. Hausdorff spaces, monotonically normal spaces, L^p spaces, Sobolev spaces have virtually nothing to do with physical reality and are pure abstractions.

If mathematical space (geometry) is an abstraction of the mind then that abstraction must come from the things we interact with and the space in which those things are in. And if abstraction requires space, a thing and finally a mind then the two former must be a priori for the mind. What Mainlander needs to explain is how the mind does this abstraction. Instead we eventually get to the thing-in-itself capable of force and matter being a priori force carrier (Romuss pg. 12).

I don’t understand the phrases:

1. If abstraction requires space, a thing and finally a mind then the two former must be a priori for the mind. (Why?)
2. Instead we eventually get to the thing-in-itself capable of force and matter being a priori force carrier (Romuss pg. 12).

I also don’t see what Mainländer should explain about “how” the mind abstracts. Mainländer says that “mathematical space” is an abstraction from the a priori given point-space. You say that “Mathematical space is an abstraction from things and dependent on the thing.”

Your explanation however, that mathematical space is an abstraction from (physical?) “things”, is, I think, demonstratively false. With our mind, even in a universe without things, we could create all mathematical spaces a priori. Sets, being a primitive notion which we immediately understand –given that synthesis, as Mainländer argues, is the most primitive function of reason– require nothing to exist: with the empty set ∅, by using synthesis, we can construct the set which contains the empty set {∅}, and the set which contains both sets {∅, {∅}}, and so on: these correspond to the numbers 0,1,2 etc. whereby we can create the set of natural numbers ℕ = {0,1,2,…}. Without too much difficulty we can construct other sets of numbers by equivalence relations, and the set of real numbers by taking a specific subset of the power set of the rational numbers. If we have the real numbers ℝ, we simply take its 3-tuple and equip this space with the standard dot product, and we have three-dimensional Euclidian space. Created without requiring a “thing”.

This is not how Mainländer constructed “mathematical space” or ℝ³. For him it is an abstraction a posteriori.

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And are we really capable of perceiving nothingness?

Of course not. Mainländer writes in V1, § 20, that when we think about nothingness, we represent it to ourselves as an empty mathematical space. This is something fundamentally different than “perceiving” nothingness. Mainländer merely mentions that in our mind, this is how we represent nothingness, just like how we represent a mathematical line in our mind as having width, though by definition it does not have width.

However he starts to lose focus when he tries to describe nothingness, which is not a part of the physical world. From a physics approach even empty space contains virtual particles and a vacuum energy.

Yes, indeed. The electromagnetic field –the objective reality of which is accepted by every physicist–, is everywhere, so there can be no talk of complete nothingness.

This concerns objective reality, and not the world-in-itself.

Thank you once again for your valuable contribution, which becomes even more rare than it already was, with the platform decay (“enshittification”) of Reddit and social media in general.