A Neo-Kantian View on Causality?

Neither Kant nor any neo-Kantian philosopher before about 1925 knew about quantum mechanics and the new indeterminism of quantum mechanics' description of nature.

Hence it is rather speculative to relate Kant’s conception of freedom to quantum mechanical indeterminism. In addition, Kant strictly separates the causality of nature from the freedom of men to make decisions which start a new chain of appearances. Hence a possible domain of conflict with Kant is the concept of free will - not quantum mechanics.

A key passage from Kant’s “Third antinomy” in the “Critique of Pure Reason” (CPR):

“If, for instance, I at this moment arise from my chair, incomplete freedom, without being necessarily determined thereto by the influence of natural causes, a new series, with all its natural consequences in infinitum, has its absolute beginning in this event, although as regards time this event is only the continuation of a preceding series. For this resolution and act of mine do not form part of the succession of purely natural effects, and are not a mere continuation of them. In respect of its happening, natural causes exercise over it no determining influence whatsoever. It does indeed follow upon them, but without arising out of them; and accordingly, in respect of causality though not of time, must be entitled an absolutely first beginning of a series of appearances.”

(see: I. Kant: CPR, B 479. Translation by Norman Kemp Smith https://web.archive.org/web/20110627054705/http://hermes.arts.cuhk.edu.hk/Philosophy/Kant/cpr/)

Kant states that human decision starts a new chain of appearances, which is not causally determined by nature.

To your first question: We do not have to view the world through the category of causality. The mathematical formalism of quantum mechanics with its indeterminism applies successfully to all experiments in the quantum domain. Nature does not care about determinism or indeterminism. Both are concepts that we humans invented to construct successful models to explain our observations.

To your second question: Which neo-Kantians are competent for a final judgement about the human ability to understand scientific non-deterministic theories?

Aside: Like any other I am always surprised about the remarkable new insights we learn from quantum mechanics. Many of our intuitions and expectations fail in the quantum domain.

This is comparing peaches with apples.

Causality is an ideal manifestation, raising from the macroscopic world phenomena, sensibility and determinism. QM is almost the opposite: it cannot be directly experienced, belongs to a different scale of existence and is non-deterministic (precisely on the Copenhagen interpretation). [1]

Quantum non-determinism seems incompatible with Kant’s defense of 
causality in his Second Analogy."... 

Of course it is incompatible. Just as creationism is incompatible with evolutionary genetics.

According to neo-Kantians will humans never be able to understand
non-causal theories?

A vague question.

If by understanding you mean experience, no; not due to causality, but simply because QM cannot be experienced, exhibiting a behavior incoherent with experience. Feynman addresses such perspective.

If you mean dividing the problem in parts that we can understand (that's what the systemic approach ---part of the systems theory--- allows), as we understand the hardware of a computer (few people understands a CPU, but most people can assemble a PC), yes. We have an excellent formal description of QM, most of its elementary parts being almost simple to understand.

The paper's introduction is quite startling (Eureka! A striking discovery!), but it then follows [1], just to propose a limit between causality and QM which AFAIK follows the traditional approach to discriminate micro/macroscopic phenomena.

In his Critique of Pure Reason (1787), Kant tried to demonstrate that the principle of causality—namely, "everything that happens, that is, begins to be, presupposes something upon which it follows by rule," (1965 ed., p. 218)— is a precondition for the very possibility of objective experience. He took the principle of causality to be required for the mind to make sense of the temporal irreversibility that there is in certain sequences of impressions.

So, whereas one can have the sequence of impressions that correspond to the sides of a house in any order we please, the sequence of impressions that correspond to a ship going downstream cannot be reversed: it exhibits a certain temporal order (or direction).

This temporal order by which certain impressions appear can be taken to constitute an objective happening only if the later event is taken to be necessarily determined by the earlier one (i.e., to follow by rule from its cause). For Kant, objective events are not "given": they are constituted by the organizing activity of the mind and in particular by the imposition of the principle of causality on the phenomena. Consequently, the principle of causality is, for Kant, a synthetic a priori principle.

Ingenious though Kant's answer to Hume was, it was ironic in three respects. Firstly, Kant safeguarded the concept of causality but at the price of making it applicable only to the phenomena and not to the unknowable things-in-themselves (noumena). Secondly, recall that Hume argued that the supposed necessity of causal sequences cannot be observed in the sequences themselves, but is projected by the mind onto the world. Kant agreed with all this, but took this projection by the mind onto the world to be presupposed for the distinction between causal and non-causal sequences. Thirdly, Kant identified causality with the rule of natural law: causal sequences of events are lawful sequences of events. This became the main plank of the Humean philosophical tradition. Stripped from objective necessity, natural laws boil down to worldly regularities.

QM is widely thought to be a strongly non-deterministic theory. The theory does not say what happens in a given case, but only says what the probabilities of various results are. So, for example, according to QM the fullest description possible of a radium atom (or a chunk of radium, for that matter), does not suffice to determine when a given atom will decay, nor how many atoms in the chunk will have decayed at any given time. The theory gives only the probabilities for a decay (or a number of decays) to happen within a given span of time. Einstein and others perhaps thought that this was a defect of the theory that should eventually be removed, by a supplemental hidden variable theory[6] that restores determinism; but subsequent work showed that no such hidden variables account could exist. At the microscopic level the world is ultimately mysterious and chancy.

So goes the story; but like much popular wisdom, it is partly mistaken and/or misleading. Ironically, quantum mechanics is one of the best prospects for a genuinely deterministic theory in modern times!

The fundamental law at the heart of non-relativistic QM is the Schrödinger equation. The evolution of a wavefunction describing a physical system under this equation is normally taken to be perfectly deterministic.[7] If one adopts an interpretation of QM according to which that's it—i.e., nothing ever interrupts Schrödinger evolution, and the wavefunctions governed by the equation tell the complete physical story—then quantum mechanics is a perfectly deterministic theory

More commonly—and this is part of the basis for the popular wisdom—physicists have resolved the quantum measurement problem by postulating that some process of “collapse of the wavefunction” occurs during measurements or observations that interrupts Schrödinger evolution. The reason is simple: the collapse process is not physically well-defined, is characterised in terms of an anthropomorphic notion (measurement)and feels too ad hoc to be a fundamental part of nature's laws.[8]

This small survey of determinism's status in some prominent physical theories, as indicated above, does not really tell us anything about whether determinism is true of our world. Instead, it raises a couple of further disturbing possibilities for the time when we do have the Final Theory before us (if such time ever comes): first, we may have difficulty establishing whether the Final Theory is deterministic or not—depending on whether the theory comes loaded with unsolved interpretational or mathematical puzzles.

Moreover, the non-causal world concept is not a regular one and one wonders what does it represent!

ref.

http://science.jrank.org/pages/8539/Causality-Kant.html http://aporia.byu.edu/pdfs/pederson-causality_and_objectivity.pdf https://plato.stanford.edu/entries/determinism-causal/