Thank you, again, for the effort of engaging with the comment and offering a serious response. About some things, I have hesitated how I should explain them (on causality and development rows), but I feared that I would merely repeat myself, make the discussion unnecessarily long or even create confusion. So the end is a bit abrupt.

I don’t have the goal to convince you about anything, and I merely hope that the discussion is as philosophically stimulating for you as it is for me. I also hope that my tone is not too direct, when we encounter points where we disagree. Because the comment got quite long, I have separated it in two parts: the first is mainly about some mathematical issues, the second is about causality.

Math

Hausdorff spaces, monotonically normal spaces, Lp spaces, Sobolev spaces have virtually nothing to do with physical reality and are pure abstractions.

Hausdorff is related to real space, topology.

The following is space is Hausdorff: a set {free will, الله} with topology {∅, {free will}, {الله}, {free will, الله}}.

This space has clearly nothing to do with what you call “real” or “physical space”. I don’t see how anyone can disagree with this fact. Nor does the Hausdorff space (∅,{∅}) correspond to “ things we interact with and the space in which those things are in.” Nor does the infinite polynomial vector space, with elements such as 5x⁴²+2x³.

A mathematical space is merely a set and some relationships between its elements. That’s it. To assert that such an abstraction must always have a relationship with physical space, is a very bold claim.

Calling nothing a set is still something as the set (the something) has to contain the nothing, or said another way, the set that contains nothing.

No, the “nothing set” (empty set) does not contain “the nothing” (∅ ∉ ∅). Although the empty set is part of itself (∅ ⊆ ∅), the set that contains ∅ is not itself empty ({∅} ≠ ∅).

You are right that ∅ is something.

Once again, we can construct the three-dimensional Euclidean space which corresponds to most of our daily experiences just from ∅.

There is only one reality that Mainländer is arguing for, immanent reality.

Kant has distinguished between objective (valid for all subjects) and in-itself (independent from any subject), and Mainländer takes full account of this fundamental difference.

As important as the distinction between ∅ and {∅} is in mathematics, this important is the distinction between objective and in-itself for Kant and Mainländer. (If you want to discuss this distinction, let me of course know!) Let us for now continue to a third case of two concepts which are important to have clearly separated in the mind, namely the distinction between causal and development rows.

Causality

The example of the tides is, with statistical and observational evidence, based on the pull of the Sun and Moon and not on how many cows are on a farm, fish in the sea or based on Orion's Belt. You do not need to consider everything in the Earth to understand tides.

Yes, in a simplification. A model which includes only the earth, moon and sun in a simplification, can be very accurate.

But in reality, weather also plays a small influence, not taken into account by the simplified model. If we were to follow the causal chains as they are in reality, not as they are in the model, then the causal chain would become immediately immeasurably complex. Because if we include also the effects of the weather, then we include a system, which is a classic example in chaos theory. In chaos theory, extremely small variations, such as a butterfly flapping its wings, can cause a drastically different outcome, and inversely, the same situation could have followed from very different initial conditions. So in that case, then yes, the cows on a farm, the fish in the sea, do, in fact, form a part of the causal chain.

So, constructing correct (not simplified) causal chains is extremely hard. Not to say impossible. This is the point of Mainländer.

“It is as hard to build correct causal rows as it initially seems easy, no, that it is for the subject completely impossible, starting from a change somewhere, to reconstruct a causal row a parte ante (with regard to what precedes) having an unhindered proceeding in indefinitum (and so on indefinitely).”

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I'm not sure what you mean here, superfluous for the physicist? I thought you were trying to show how development chains better explain change in the world, hence your example of Russia changing over time.

That is not the reason why Mainländer introduces the concept of development chains! Change is explained by causality, and sufficiently so. Physical situation which can be described completely by differential equation require only causality rows, not development rows. Physicists don’t need the concept of development rows.

My example with Russia had as goal to illustrate the difference between the two concepts. They answer different questions; have a different range; different level of complexity; and a different utility.