\section{}
\label{sec:intro}

Even before Hegel's death in 1831, the idea, traditional since the ancient
Greeks, that philosophical speculation could furnish a general model for
scientific thought had been waning in the Germanic world.\footnote{I say
  `Germanic world' rather than Germany so as to include those in surrounding
  nations, in particular Denmark, who were influenced by German thinkers of
  the period; see, for example, \citet[][]{stewart2007}.} What were seen as the
excesses of the romantic and idealist movements, combined with the rapid growth
and manifest success of empirical science, had brought philosophy into
disrepute. Some---the so-called `vulgar materialists'---went so far as to call
for philosophy to essentially identify itself with empirical science
(\citealt[ch. 3]{schnadelbach1984};
\citealt[pp. 182-188]{beiser2014}).\nocite{bacciagaluppi2017}

Mid-nineteenth century neo-Kantianism arose partly as a reaction to these
intellectual currents. Championed, at first, primarily by natural scientists
such as Helmholtz \citep[p. 103]{schnadelbach1984}, the movement did not, as
did speculative idealism, view it as philosophy's task to produce scientific
knowledge by pure thought alone. Nor did these neo-Kantians, with the vulgar
materialists, consider it philosophy's task merely to uncritically catalogue
and systematise the results of empirical science. Rather, with Kant, they saw
it as philosophy's essential, unavoidable, and enduring mission to enquire into
the sources of our knowledge and the degree of its
justification.\footnote{\emph{... sondern sie beabsichtigte nur, die Quellen
    unseres Wissens und den Grad seiner Berechtigung zu untersuchen, ein
    Gesch\"aft, welches immer der Philosophie verbleiben wird, und dem sich
    kein Zeitalter ungestraft wird entziehen k\"onnen}
  \citep[p. 5]{helmholtz1855}.} In other words they, and the philosophers such
as Liebmann, Lange, and Cohen who took up their battle call
\citep[]{zeman1997}, saw the proper task of philosophy as consisting in the
provision of an \emph{epistemological foundation} for science. By the close of
the nineteenth century, neo-Kantianism exerted a powerful influence on Germanic
thought.

It was against this intellectual backdrop that many of the modern era's
spectacular achievements in logic, mathematics, and physical science were
made. Some of these were to eventually deal a heavy blow to the popularity of
Kant's philosophy. Pure logic, Kant had argued, could never provide us with a
genuine expansion of our knowledge; yet Frege's \emph{Begriffsschrift} and the
later systems that drew from it seemed to provide us with tools to do just
this. Euclidean geometry was held by Kant to be both a synthetic and an a
priori science. Yet Hilbert was able to show that it followed analytically from
a set of basic axioms, and the development and physical application (in
relativity theory) of non-Euclidean geometry showed that no one particular
geometry could be regarded as a priori true. Arguably worst of all, the
development of quantum theory seemed to tell against according a fundamental
status to the principle of cause and effect. By the middle of the twentieth
century it was widely held that Kant's philosophy had been definitively
refuted.\footnote{See, for instance, Carnap's summary of the prevailing
  attitude toward Kant in \citet[p. vi]{reichenbach1958}.}

This is the standard historical account of the period after Kant. Yet the truth
is somewhat more subtle and complex than this. For on the one hand, while
neo-Kantianism as a philosophical movement emerged ascendant during the
mid-nineteenth century, its origin in fact predates this period and has its
roots in the thought of earlier thinkers such as Jakob Friedrich Fries, Johann
Friedrich Herbart, and Friedrich Eduard Beneke \citep[]{beiser2014}. These
early nineteenth century thinkers---in a philosophical climate dominated by the
speculative idealism of Fichte, Schelling, and Hegel---attempted to further
develop and clarify the Kantian system understood as a doctrine aiming at
the provision of an epistemological grounding for science and metaphysics. On
the other hand, many of the late-nineteenth and early twentieth century
thinkers whose work had contributed to the demise in popularity of Kantian
philosophy were, despite their divergences from him, substantially influenced
by Kant. Frege, for example, is at pains to call attention to his agreement
with Kant, which he claims far exceeds the extent of his disagreement
\citep[\textsection 89]{frege1980}.\footnote{For more on the parallels
  between Kant and Frege, see \citet[]{cuffaro2012a, merrick2006}.} Hilbert was
also influenced by Kant and in the epigraph to his seminal work on geometry
invokes Kant in support of the spirit of his investigations
\citep[p. 1]{hilbert1902}.\footnote{For more on the Kantian aspects of
  Hilbert's thought, see \citet[]{kitcher1976}.} Reichenbach's conception of
the relativised a priori, the conventionalisms of Poincar\'e, Schlick, and
Carnap, the pragmatisms of C. I. Lewis and others, are best characterised, not
as radical rejections of Kant's philosophical framework, but rather as attempts
to re-explicate the basic Kantian idea that our theoretical frameworks include
an element---what Kant had (mistakenly, according to these thinkers) called the
synthetic a priori---that is conceptually and epistemologically privileged in
some sense.\footnote{I am of course not claiming that any of these thinkers
  would have called themselves Kantians; in particular Schlick and (the
  later) Carnap certainly would not have done so, and nor should we. I
  nevertheless do think that their views as well as the others I have
  mentioned can be seen as continuous with (in the sense of evolving
  continuously out of) a research programme that was begun by Kant.} Viewed
as a research program \citep[cf.][]{bitbol2017}, one may say that Kant's
transcendental approach to philosophy continued, and continued to evolve,
albeit along multiple independent pathways, well into the last century. It is
only with the rise of the Quinean holistic conception of science that these
ideas are rejected in their totality.\footnote{For more on these topics, see
  \citet[]{coffa1991, disalle2002, friedman2009, howard1994, murphey2005}.}

The development of quantum theory in the early part of the last century posed a
particularly strong challenge to the Kantian philosophical viewpoint. Owing to
the indeterminacy intrinsic to the theory, a common opinion expressed at the
time was that this represents a definitive refutation of Kant's philosophy
insofar as it shows that Kant's ascription of a priori status to the principle
of causality cannot be correct. As with other developments in the mathematical
and natural sciences during the period, however, the relationship between
quantum theory and Kant's theoretical philosophy is far more rich and
interesting than this. There has been a flowering of scholarship in recent
years exploring the relationship between Kantian philosophy and quantum
mechanics, and especially between Kant and the influential physicist Niels
Bohr. Commentators such as \citet{kaiser1992}, \citet{chevalley1994},
\citet{cuffaro2010}, \citet[]{bachtold2017}, \citet{bitbol2017},
\citet{kauark-leite2017}, and others have over the years demonstrated that
Bohr's views on quantum mechanics are broadly Kantian in the sense that they
can be motivated from considerations that arise naturally from within Kant's
philosophical framework; further that Bohr's views remain broadly compatible
with---and indeed are very much in the spirit of---a Kantian worldview adapted
to address the situation presented to us by quantum theory.

Most of this literature is aimed at clarifying the philosophical viewpoint of
Bohr, or of related interpretations of quantum mechanics such as that of Werner
Heisenberg \citep[]{camilleri2005}. One of the contributions of this paper, in
contrast, will be to turn an eye back toward Kantian philosophy. In the sequel
it will be my contention that considering Kantian philosophy in the light of
the challenges posed by quantum mechanics illuminates the significance of
several elements of the Kantian philosophical framework, foremost among which
are Kant's synthetic a priori principles (including the principle of causality)
as well as his core doctrine of transcendental idealism.

Transcendental idealism is made up of two claims: first, that space and time
are the necessary subjective forms of all appearances for us, and second, that
they do not attach to things as they are in themselves. With respect to the
second claim in particular, there is dispute within Kant scholarship over
whether to interpret it ontologically (see especially \citealt[]{guyer1987}),
as a claim about how things as they exist independently of us in fact are
(not), or merely epistemically (see especially \citealt[]{allison2004}), as a
claim about what we are licensed to attach to our conception of things as they
are in themselves---in other words what we are in a position to determinately
assert of them---given our particular epistemic limitations.

Allison's interpretation of Kant, at least in this general sense, is in my view
the correct one. However my goal here is not to argue definitively for this
interpretation of Kant, nor is it to somehow refute the ontological
interpretation of Kant's views. My goal, rather, is more cautious and limited
in scope. The epistemic interpretation of Kant's doctrine has been criticised
as amounting to mere stipulation; it does not, it has been argued, sufficiently
capture the real ontological significance that Kant seems to accord to the
transcendental distinction between appearances and things as they are in
themselves, and it does not, it has been argued, sufficiently capture the sense
of `epistemic loss' one feels due to the inability to know things as they are
in themselves. Some stipulations are better than others, however, and in the
sequel I will argue that Kant's doctrine of transcendental idealism, though it
should be understood `merely epistemically' is nevertheless full of
significance for Kant. I will argue that his views in this regard are rooted in
his long-running pre-critical struggle to discover the first principles for
metaphysical cognition, i.e. cognition of things as they actually are in
themselves. Moreover, underlying Kant's doctrine is an ontological posit of a
sort; however this posit on which transcendental idealism rests---the so-called
`discursivity thesis'---is not an ontological posit which pertains to the
mind-independent world but is rather a posit concerning the nature of our
cognition. And far from being, as some have claimed
\citep[cf.][Ch. 1]{strawson1966}, a self-evident truth, discursivity is a posit
which has been denied---and with interesting consequences, as we will see.

The question of the epistemic constraints and preconditions for objective
cognition---the question at the heart of the doctrine of transcendental
idealism interpreted epistemically---is arguably more pressing in quantum
mechanics than in any other area of physics, and it is precisely the question
which occupied physicists, such as Bohr, in their efforts to interpret quantum
mechanics in the initial stages of the theory's development. It is also the
question which during this period occupied Grete Hermann, a
philosopher-mathematician who studied under Emmy Noether as well as the
Friesian philosopher Leonard Nelson. Fries, like Kant, saw transcendental
idealism as an epistemic doctrine. Fries, however, denied the discursivity
thesis. And below we will see how this difference between the orthodox Kantian
and Friesian neo-Kantian understandings of the nature of cognition manifests
itself in a subtle but important difference in the lessons each inevitably
draws for natural philosophy from quantum theory. I will argue that on an
orthodox Kantian viewpoint---for which the view of Niels Bohr may be taken as
representative---the point emphasised is that our fundamental (and necessary)
forms of knowledge are ultimately idealisations and as such are applicable only
within a limited domain. We may nevertheless---because of this very
limitation---continue to employ these idealisations outside of this domain in
order to convey information to one another in a qualified way; i.e. with the
understanding that the information conveyed in this way does not in general
constitute an objective description of the phenomena under investigation. For
Hermann---whom we may take to represent the Friesian viewpoint---there is, in
addition to this, a deeper lesson as well: Quantum mechanics shows us that
physical knowledge is fundamentally `split'. For Hermann, that is, there is a
notion of objectivity that is applicable to quantum phenomena, however it is a
form of objectivity which is valid only from within a particular
perspective. Hermann's Friesian doctrine moreover provides an account of the
relationship between physics and the a priori forms of knowledge which makes
comprehensible how the latter may evolve and undergo revision as a consequence
of developments in the former.

From here on the paper will proceed as follows. In \S\ref{sec:negalt} I will
introduce Kant's doctrine of transcendental idealism and in
\S\ref{sec:firstprinc} I will show how it is motivated, for Kant, by his
struggle to provide first principles for metaphysical cognition in the period
leading up to the publication of the first Critique. In \S\ref{sec:synthap} I
will discuss how Kant's search for the first principles of metaphysical
cognition was transformed, during the critical period, into the search for the
first principles of synthetic a priori cognition, focusing in particular on
Kant's conception of the principle of causality. In \S\ref{sec:quantmech} I
will discuss the challenges faced by Kant's view which arise from the emergence
of quantum theory. Then in \S\S\ref{sec:conceptind}--\ref{sec:schlick} I will
show how the Kantian principles, strictly speaking, remain valid within quantum
mechanics, and that (as Bohr argues) even outside of their domain of
applicability they may be used to convey information regarding the objects of
our investigation in a qualified way. In
\S\S\ref{sec:hermann}--\ref{sec:herm-transcideal} I then consider the views of
the Friesian neo-Kantian philosopher Grete Hermann and discuss her
understanding of the general situation regarding Kantian critical philosophy
\emph{vis \'a vis} quantum theory. I conclude with a brief discussion of the
relative merits of the orthodox Kantian and Friesian viewpoints with respect to
the way they account for the challenges posed to philosophy by quantum theory
and the way they envision critical philosophy's proper response to this
challenge.

\section{}
\label{sec:negalt}

Kant identifies two aspects of objective experience in the \emph{Critique of
  Pure Reason}: \emph{intuition}---the `this', the `that' of experience---and
the \emph{concepts} whereby one synthesises the manifold of intuition. Concepts
belong to the faculty of \emph{understanding}, which we will discuss in \S
\ref{sec:synthap}. Intuition is mediated by the faculty of \emph{sensibility}:
our mind's capacity to be affected by objects (A19/B33).\footnote{References to
  the \emph{Critique of Pure Reason}, or `first Critique', will be to the
  Pluhar translation \citeyearpar{kant1781pluhar}. Page numbers for Kant's
  works cited in this essay (with the exception of the first Critique) will
  be as they are in the standard German edition of Kant's works. In the case
  of the first Critique, page numbers are as in the first (1781) and second
  (1787) edition, where ``A'' denotes the first and ``B'' the second, as
  usual.} The effect on sensibility of some object is called the
\emph{sensation} of it, and with sensation we associate the \emph{empirical}
aspect of our intuition.

We call ``[t]he undetermined object of an empirical intuition'' (A20/B34) an
\emph{appearance}. A shape against the far wall in a dark room, for instance,
which only after some scrutiny is determined to be a chair, is before this
determination merely the appearance of something indeterminate.\footnote{For a
  more thorough discussion of this point, see
  \citet[p. 110-111]{harper1984}.} There are two aspects to an
appearance. First, there is its \emph{matter}; i.e. \emph{what} we sense in
it. Second, there is an appearance's \emph{form}, i.e. that which allows the
manifold corresponding to the appearance to be ordered in certain
relations. Appearance has two characteristic forms: space, associated with
outer appearances, and time, associated with both inner and outer
appearances. As forms of appearances, they are the \emph{formal conditions} for
appearances, in virtue of which they are known a priori as necessary relations
according to which sensations must be ordered for subjects like us (A20/B34,
A26/B42, A33/B49--50). They are also called `pure' in virtue of not in
themselves containing anything belonging to the matter of sensation (A20/B34).

With respect to the intuitions of space and time, Kant's doctrine of
\emph{transcendental idealism} puts forward two theses. The thesis just
asserted, that space and time are necessary subjective conditions for
appearances, is sometimes called the `subjectivity thesis'. Kant also makes the
following claim:

\begin{quote}
Space represents no property whatever of any things in themselves, nor does it
represent things in themselves in their relation to one another. That is, space
represents no determination of such things ... that adheres to objects
themselves and that would remain even if we abstracted from all subjective
conditions of intuition. (A26/B42).  
\end{quote}

This, and the parallel claim for time (A33/B49), are together sometimes
referred to as the `nonspatiotemporality thesis'. Of the subjectivity and
nonspatiotemporality theses, it is nonspatiotemporality which is more
controversial. For on one interpretation of Kant's argument for
nonspatiotemporality, Kant concludes that space and time cannot be
determinations of things in themselves \emph{because} they are necessary
subjective conditions of intuition. Assuming this reading of Kant is correct,
such a conclusion would only follow if subjectivity and spatiotemporality were
mutually exclusive options. But some have argued that there is no reason to
think so. In particular some have pointed out that the very fact that space and
time are necessary forms of appearances may be taken as good evidence that
things in themselves are spatiotemporal.

This objection, that Kant has `neglected an alternative', is most often
associated with Trendelenburg \citep[p. 107]{gardner1999}, but the objection
was also made much earlier by others, for instance by Feder
\citep[p. 140]{sassen2000}. Among more recent commentators, Guyer puts it
particularly forcefully:

\begin{quote}
Transcendental idealism is not a skeptical reminder that we \emph{cannot be
  sure} that things as they are in themselves \emph{are} also as we represent
them to be; it is a harshly dogmatic insistence that we \emph{can be quite
  sure} that things as they are in themselves \emph{cannot be} as we represent
them to be. \citep[p. 333]{guyer1987}.
\end{quote}

Guyer is actually slightly more charitable to Kant than this quote suggests,
for he argues \citeyearpar[p. 366]{guyer1987} that Kant does in fact provide an
(ultimately unsuccessful) argument for the exclusion of Trendelenburg's
alternative. We will pass over the details of Guyer's analysis of the
deficiency in Kant's argument here, except to say that Guyer's criticism---and
the neglected alternative objection more generally---presupposes an ontological
reading of transcendental idealism that was not Kant's.

Kant's transcendental idealism does not in fact amount to a metaphysical claim
regarding the nature of things in themselves as they exist apart from us. It is
best interpreted, rather, as an `epistemic' doctrine
\citep[ch. 1]{allison2004}. That is to say, on Kant's view, implicit in the
concept of objective cognition are the subjective---i.e. epistemic---conditions
under which an object is representable to us. In particular, on Kant's view our
cognition is discursive in the sense that, as we saw earlier, we possess both a
sensible and a conceptual faculty; the former being that through which our
intuitions of external objects are mediated, and the latter being the faculty
through which we spontaneously act upon these intuitions in order to subsume
them under general concepts. Discursivity amounts to the claim that we require
\emph{both} in order for cognition to be possible. As Kant puts it:

\begin{quote}
Neither of these properties is to be preferred to the other. Without
sensibility no object would be given to us; and without understanding no object
would be thought. Thoughts without content are empty; intuitions without
concepts are blind. Hence it is just as necessary that we make our concepts
sensible (i.e. that we add the object to them in intuition) as it is necessary
that we make our intuitions understandable (i.e., that we bring them under
concepts). Moreover, this capacity and this ability cannot exchange their
functions. The understanding cannot intuit anything, and the senses cannot
think anything. Only from their union can cognition arise. (A51/B75--76).
\end{quote}

From the fact that cognition requires the union of sensibility and
understanding, however, it does not follow that we cannot think about objects
apart from the conditions pertaining to our sensibility. We may do so by
abstracting away from these conditions and in this way we may form the concept
of an object `regarded as it is in itself'. But since cognition, given that it
is discursive, requires the contributions of both sensibility and
understanding, it follows that the concept that we form in this way of a thing
regarded as it is in itself is not cognisable. Thus interpreted, transcendental
idealism amounts to a claim about our concept of a thing as it is in itself,
and in that sense we may call this a `conceptual' doctrine: It says simply that
our concept of a thing regarded as it is in itself is not spatiotemporal
because in forming it we abstract away from its spatiotemporal attributes.

Note, however, that it does not follow from this understanding of
transcendental idealism that all talk of things regarded as they are in
themselves should be considered empty or meaningless
\citep[cf.][pp. 28, 32]{ameriks2003}. What transcendental idealism proscribes
is merely the assumption that things which exist independently of us can be
attributed spatiotemporal characteristics either in themselves or in relation
to one another \citep[cf.][p. 24, 36]{allison2004}. But this does not entail
that the concept of a thing regarded as it is in itself is without significance
\citep[cf.][p. 56]{allison2004}, nor that it is ontologically less (or more)
privileged than the concept of a thing regarded under the conditions according
to which it can be cognised by us \citep[p. 47]{allison2004}.

The idea expressed by Kant's discursivity thesis---that cognition requires the
contributions of both sensibility and understanding and that therefore there can
be no standpoint-independent perspective on the objects of cognition
(cf. \citealt[\S 4.3.3]{stang2016})---is central to this understanding of
transcendental idealism. \citet[ch. 2]{allison2004} calls this the
`anthropocentric' paradigm of cognition and as such it may be contrasted with
the `theocentric' paradigm. According to the latter, true cognition is
standpoint-independent in an absolute sense. That is, to know something on the
theocentric paradigm is to know it absolutely independently of the way it is
considered. But according to Kant this is impossible for finite cognisers such
as ourselves.

Understood in this way, both transcendental idealism and the `transcendental
realism' associated with the theocentric paradigm may be thought of as
expressing norms or standards by which to evaluate cognition, and in this sense
can be thought of as `metaphilosophical' doctrines
\citep[p. 35]{allison2004}. Transcendental idealism is a metaphilosophical
doctrine according to which we judge philosophical assertions \emph{vis \'a
  vis} the kind of cognition that is achievable \emph{for us}, and Kant's
critical project more generally constitutes a reconceptualisation of philosophy
in light of this aim.

Of course, this way of interpreting Kant's doctrine is not without its
critics. Scholarly criticisms of the epistemic\footnote{In the sequel I will
  (following Allison) be using the terms `epistemic interpretation',
  `conceptual interpretation', and `metaphilosophical interpretation' as
  synonymous unless I indicate otherwise.} reading include the following: that
it is trivial or tautologous, that despite the name it smuggles in a
substantive ontological position (regarding things in themselves) implicitly,
that the epistemic version of the argument for nonspatiotemporality is no less
fallacious than the ontological version, and that it is merely anodyne (where
this word is used \emph{ad hominem} as a term of abuse) and does not do justice
to the important ontological issues that mattered for Kant. In the next two
sections I will be focusing on the last objection listed. To my mind this is,
despite the unfortunate \emph{ad hominem} character of the arguments often
given in support of it, both the most important of the objections listed as
well as one which has been insufficiently addressed in Kant scholarship to
date.\footnote{Allison's \citeyearpar[pp. 9-11]{allison2004} response to
  Langton, for example, only briefly addresses this `motivational' objection
  in connection with the charge of triviality. \citet[\S 4.4]{stang2016}
  mentions this objection as a way of motivating Langton's own metaphysical
  interpretation of transcendental idealism, but does not consider, as I do
  here, the question of how it might be answered from \emph{within} an
  epistemic interpretation.} (A survey of the other mentioned objections to the
epistemic interpretation, along with some responses, is given in \citealt[\S
  4.3]{stang2016}.) Rae Langton expresses the objection thus:

\begin{quote}
Allison's is an ingenious and attractive solution to an old an ugly
problem. But I would like to suggest that there are reasons for suspicion. I
have an ulterior motive. I have up my sleeve a solution that is, though less
ingenious, more attractive ... Allison's approach makes it analytic that we
have no knowledge of things in themselves. To consider things in themselves is
simply to consider things in abstraction from the conditions of our knowledge:
$K_3$ [that we can have no knowledge of things in themselves] has become the
tautological $A_3$ [that things considered in abstraction from their relation
  to our sensibility are things considered in abstraction from their relation
  to our sensibility]. From one point of view this is an advantage, but from
another it is a grave defect, for it fails to do justice to an aspect of Kant
that ought not to be ignored. What I have in mind is not exactly a Kantian
philosophical thesis ... Rather, it is a Kantian attitude to these
philosophical theses, and in particular [$K_3$]. When Kant tells us that we
have no knowledge of things in themselves, he thinks he is telling us something
new and important. The truth of $K_3$ is a major philosophical
discovery. Moreover, it is not just a discovery with a definite, non-trivial
content. It is a depressing discovery. Kant thinks we are missing out on
something in not knowing things as they are in themselves. Kant speaks of our
yearning for something more, he speaks of doomed aspirations, he speaks of `our
inextinguishable desire to find firm footing somewhere beyond the bounds of
experience' (A796/B824). It is not easy to see how this inextinguishable desire
could be for the falsity of
$A_3$. \citeyearpar[p. 10]{langton1998}.\footnote{Cf. \citet[p. 334]{ameriks1992}.}
\end{quote}

I believe this concern is a legitimate one in the sense that any interpretation
of Kant should account for the attitude of Kant's that is alluded to here. But
in the sequel I will argue that ascribing this attitude to Kant is not at all
inconsistent with an epistemic reading of transcendental idealism. Indeed, for
Kant this metaphilosophical doctrine is motivated by this very attitude. For,
as I will argue below, it was through his long-running pre-critical struggle to
discover the first principles for metaphysical cognition---cognition of things
as they exist in themselves in a mind-independent sense---that his critical
doctrine of transcendental idealism was born, and with it a new set of first
principles pertaining to \emph{synthetic a priori} cognition.\footnote{Kant's
  `pre-critical' period refers to the period of Kant's philosophy up until
  the publication of the \emph{Critique of Pure Reason} in 1781.} We will
also see that Kant's metaphilosophical doctrine does rest on an ontological
posit of a sort; only it---namely the discursivity thesis---is not an
ontological posit which pertains to the mind-independent world but rather to
the nature of our cognition. Later, in \S\ref{sec:fries-nelson}, we will see
that this is not the only ontological posit from which to develop an
anthropocentric standard of cognition.

\section{}
\label{sec:firstprinc}

Kant in fact acknowledged a version of the `neglected alternative' objection
almost a decade before the publication of the first Critique. In a famous
letter to Marcus Herz of February 21, 1772,\footnote{The letter is famous
  because it marks the beginning of Kant's so-called `silent decade' and poses
  many of the questions that he will later take up in the 1781 Critique.}
addressing Lambert's objection to his conception of time,\footnote{Compare also
  the ``Elucidation'' to the Aesthetic, A36-37.} Kant wrote:

\begin{quote}
[Lambert objects that changes] are possible only on the assumption of time;
therefore time is something real ... Then I asked myself: Why does one not
accept the following parallel argument? Bodies are real (according to the
testimony of outer sense). Now, bodies are possible only under the condition of
space; therefore space is something objective and real that inheres in the
things themselves. The reason lies in the fact that it is obvious, in regard to
outer things, that one cannot infer the reality of the object from the reality
of the representation ... \citep[p. 75]{zweig1967}.
\end{quote}

This is the same objection which Guyer puts as follows: ``Why doesn't the
indispensable role of space and time in our experience prove the transcendental
realism rather than idealism of space and time themselves?''
\citep[p. 349]{guyer1987}. Kant's answer, that it is ``obvious'' that such an
inference is invalid, seems unsatisfactory. But before dismissing this as mere
dogmatism we should attempt to understand exactly why Kant thinks it is
obvious. We can understand Kant's position if we place his 1772 letter in its
proper context. Kant's preoccupation, as early as 1755, was with the question
of how one might make metaphysics legitimate.\footnote{In characterising Kant's
  views in this way, I depart from commentators, such as Caranti, who view
  Kant's philosophy (both early and late) as primarily concerned with
  providing an answer to Cartesian skepticism \citep[Ch. 1]{caranti2007}.} For
Kant, although the aim of metaphysics is purportedly rational certainty with
regards to its propositions, in practice its methods are often arbitrary and
hypothetical in character. His (preparatory) project at this stage is thus to
provide a principled methodology or \emph{epistemic grounding} for
metaphysics, now to be thought of as a \emph{science}.

Thus, in the \emph{New Elucidation} of 1755, Kant declares that his intention
is to ``establish two new principles of metaphysical cognition''
\citepalias[1:387]{kant1755}. ``By their means'', he tells us, ``you may
acquire no inconsiderable power in the realm of truths.''
\citepalias[1:416]{kant1755}. In his \emph{Only Possible Argument} of 1763,
Kant declares that he will supply---not an actual demonstration---but the only
sure way in which a demonstration of God's existence may proceed: ``What I am
furnishing here is the materials for constructing a building ...''
\citepalias[2:66]{kant1763a}. In his `Prize Essay' of the same year, Kant
outlines the rules by which metaphysics must proceed in its investigations and
``by which alone the highest possible degree of metaphysical certainty can be
attained'' \citepalias[2:285]{kant1763b}. Within metaphysics, Kant tells us,
``One's chief concern will be to arrive only at judgements about the object
which are true and completely certain'' (ibid.). Rationally certain cognition
with regard to the propositions of metaphysics (traditionally construed) is the
goal which by the first Critique Kant will ultimately reject as
unachievable. Yet at this stage, Kant is still hopeful that metaphysically
certain cognition is possible if only one can purify metaphysics' methods.

Kant's solution to the problem was to take its most mature form in his
\emph{Innaugural Dissertation} of 1770. Here we are told that we will achieve
our goal if we first identify and abstract from the form and principles of
\emph{sensible} cognition. In this way we will elucidate the form and
principles of \emph{intellectual} cognition (the proper concern of
metaphysics). Intellectual (or `rational') cognition is cognition which
transcends the limitations imposed by our sensibility. It is subject only to
the laws of the understanding: ``whatever cognition is exempt from such
subjective conditions relates only to the object. It is thus clear that things
which are thought sensitively are representations of things \emph{as they
  appear}, while things which are intellectual are representations of things
\emph{as they are}.'' \citepalias[2:392]{kant1770}.

In order to obtain intellectual cognition, one must abstract from all
conditions related to sensibility. This includes even the form and principles
of sensible cognition. Thus, of space and time, Kant tells us ``that these
notions are not \emph{rational} at all, and that they are not \emph{objective}
ideas of any connection, but that they are appearances, and that, while they
do, indeed, bear witness to some common principle constituting a universal
connection, they do not expose it to view.'' \citepalias[2:391]{kant1770}. Note
that Kant is here using the term `objective' differently from the way he will
later use it in the first Critique. In the Dissertation, `objective' is meant
in the sense of pertaining to the noumenal, or mind-independent, world; in
other words, Kant is operating there with a theocentric standard for objective
cognition. But in the Critique, as we will see shortly, Kant is not hesitant to
grant `objective' status to representations that are dependent on the faculty
of sensibility \emph{as such}, so long as they are not dependent on any
particular sense impressions.

In any case, and again anticipating the neglected alternative objection, Kant
tells us that the forms of sensibility must be excluded even though they seem
to provide good evidence for some underlying analogous connection between
things as they are in themselves: ``the \emph{form} of the same representation
is undoubtedly evidence of a certain reference or relation in what is sensed,
though properly speaking it is not an outline or any kind of schema of the
object ...'' \citepalias[2:393]{kant1770}. ``[E]mpirical concepts do not,''
Kant tells us, ``in virtue of being raised to greater universality, become
intellectual in the \emph{real sense}, nor do they pass beyond the species of
sensitive cognition; no matter how high they ascend by abstracting, they always
remain sensitive.'' \citepalias[2:394]{kant1770}.

To clarify: the fact that spatiotemporal relations necessarily attach to
sensible concepts seems to provide some warrant or evidence for also attaching
these (or analogous) determinations to the concept of the thing in itself
considered apart from the conditions of our sensibility. Nevertheless we
certainly \emph{cannot} say that it follows necessarily from the former that
things as they are in themselves are related to each other in any particular
way, let alone spatiotemporally. For no matter how high we abstract empirical
concepts, they always refer to what is given to us via
sensibility.\footnote{Kant writes, at 2:394, ``a concept of the understanding
  \emph{abstracts} from everything sensitive, but it is \emph{not abstracted}
  from what is sensitive.'' This is explained in the sentence immediately
  preceding: ``The former expression indicates that in a certain concept we
  should not attend to the other things which are connected with it in some
  way or other, while the latter expression indicates that it would be given
  only concretely, and only in such a way that it is separated from the
  things which are joined to it.'' He goes on: ``For this reason, it is more
  advisable to call concepts of the understanding `pure ideas', and concepts
  which are only given empirically `abstract concepts'.''} We are not
warranted, therefore, to attach these attributes to our conception of the thing
in itself.

It may be helpful to consider a mathematical analogy. In mathematics we refrain
from attaching a certain property to a concept even though we know that this
property holds for some particular instances of the concept. Instead, we
include in our concept only that which can determinately be affirmed of the
concept in general. For instance, we say that matrix multiplication is
non-commutative since it is not the case in general that for two matrices, $A$
and $B$, that $AB = BA$, even though, for some particular matrices, $AB$ does
in fact equal $BA$. It is similar with Kant's conception of the thing in
itself, not in the sense that we can attach spatiotemporal attributes to some
but not all particular things in themselves (whatever that would mean), but in
the sense that our concept of the thing in itself, for Kant, can only be
composed of properties we are in a position to determinately affirm of it as
such. But if our concept of the thing in itself as such is composed only of
what we can determinately affirm of it, then spatiotemporal attributes---the
conditions pertaining to our sensibility---certainly may not be attached to
this concept.

In fact the difference between sensible and intelligible concepts is a
difference in \emph{kind}, not in degree \citep[cf.][p. 17]{allison2004}, for
space and time attach necessarily to the objects of sensible cognition, but
must not be attached (in the sense explicated in the previous paragraph) to the
objects of intelligible cognition
\citepalias[2:394]{kant1770}.\footnote{Compare this also with A43/B60 of the
  Critique, where Kant writes: ``Even if we could bring this intuition of ours
  to the highest degree of distinctness, that would still not get us closer to
  the character of objects in themselves. For what we would cognize, and
  cognize completely, would still be only our way of intuiting, i.e., our
  sensibility; and we would always cognize it only under the conditions
  attaching to the subject originally: space and time.''} And since the object
of metaphysics for the Kant of the Dissertation is to attain intelligible
cognition, anything pertaining to sensible cognition---no matter to how high a
degree of abstraction it is raised---must be excluded. Thus in the Dissertation
Kant offers to metaphysicians the ``principle of reduction''
\citepalias[2:413]{kant1770}, which asserts that any concept of the
understanding to which one predicates anything belonging to space and time must
not be asserted objectively (in the sense of the Dissertation), i.e., asserted
as having objective validity independently of all actual---or
possible---experience of it. One might object that the bar that Kant sets for
objective cognition here is high indeed, but one should keep in mind Kant's
project and goals: the attainment of metaphysically certain cognition. Given
these goals, Kant's standards are in my view appropriate, or at any rate not
out of step with his contemporaries.

In any case, once we have purged our concepts of spatiotemporal elements, the
question then becomes one of the formal principle of the intelligible world. In
particular, the question is ``to explain how it is possible \emph{that a
  plurality of substances should be in mutual interaction with each other}, and
in this way belong to the same whole, which is called a world''
\citepalias[2:407]{kant1770}. The principle that is left to elucidate, that is,
is the principle of causality (in a generalised sense) or what Kant elsewhere
calls the principle of the determining
ground.\footnote{Cf. \citetalias[\textsection 2, Proposition IV,
    1:391-393]{kant1755}.} We need not discuss Kant's more detailed comments on
this principle here, but it is worth noting that, just as in his earlier works,
Kant does not actually provide the needed elucidation. Rather, his goal in the
Dissertation is a methodological one; namely to offer this principle and the
principle of reduction as the tools with which one can begin to tackle the
problems of metaphysics.\footnote{Kant does offer a sketch of what one must do
  in order to provide a demonstration of the community of substances:
  ``Granted that the inference from a given world to the unique cause of all
  its parts is valid, then, if, conversely, the argument proceeded in the
  same way from a given cause, which was common to all the parts, to the
  connection between them and, thus, to the form of the world (although I
  confess that this conclusion does not seem as clear to me), then the
  fundamental connection of substances would not be contingent but necessary,
  for all the substances are \emph{sustained by a common principle}.''
  \citepalias[2:409]{kant1770}.}

\section{}
\label{sec:synthap}

By 1781 Kant had, by his own account through reading Hume, been awoken from his
``dogmatic slumbers'' \citepalias[4:260]{kant1783} with regard to the principle
of causality. However the legitimacy of the principle of causality had been
undermined only with respect to metaphysical cognition. Its use had not, in
Kant's mind, been undermined within the domain of natural science. He writes,
in 1783:

\begin{quote}
The question was not whether the concept of cause was right, useful, and even
indispensable for our knowledge of nature, for this Hume had never doubted; but
whether that concept could be thought by reason \emph{a priori}, and
consequently whether it possessed an inner truth, independent of all experience
\citepalias[4:258--259]{kant1783}.
\end{quote}

He continues: ``This was Hume's problem. It was a question concerning the
\emph{origin} of the concept, not concerning its indispensability in use. Were
the former decided, the conditions of its use and the sphere of its valid
application would have been determined as a matter of course'' (ibid.).

Synthetic a priori cognition (a term Kant introduces in the first Critique) is
that cognition in which two (or more) concepts are cognised, in advance of
experience, as necessarily connected in some way to one another. `Ampliative'
cognition of this kind is what metaphysics seeks. But by the time of the
Critique, Kant is convinced that such cognition is impossible without a
reference to the forms of our sensibility, space and time.\footnote{Cf. A5/B9.}
Even the principle of causality has validity only with regard to appearances,
which are always given in space and time. Thus, once we carry out what is
required by the Dissertation's principle of reduction, metaphysical
cognition\textemdash cognition of the intelligible world, i.e., of things as
they are independently of \emph{all} conditions for the (possible or actual)
experience of them\textemdash must be given up forever.

This does not undermine, however, the idea that the concept of cause is
indispensable in some sense, nor does it imply that all synthetic a priori
cognition is impossible. Thus Kant's project now becomes to explicate that
sense and to show \emph{how} synthetic a priori cognition is possible and to
investigate its limits. These limits are \emph{possible} experience, and since
the \emph{form} of possible experience is given a priori, synthetic a priori
knowledge is possible with respect to it. A useful \emph{corollary} to the
results of this investigation is a grounding for the theoretical sciences;
i.e., an answer to the issue that Hume \emph{did not} intend to raise: the
validity and scope of principles, such as that of causality, \emph{within} the
theoretical sciences.\footnote{Kant writes, at B20: ``In solving the above
  problem we solve at the same time another one, concerning the possibility
  of the pure use of reason in establishing and carrying out all sciences
  that contain theoretical a priori cognition of objects''. Cf. also:
  \citetalias[4:280]{kant1783}.} Metaphysics, regarded as a system of synthetic
a priori constitutive principles which \emph{transcend} experience, is declared
impossible. Theoretical metaphysics in the traditional sense is instead
transformed into a system of methodological principles for the investigation of
nature.\footnote{At least part of the reason for the second edition
  emendations of the first Critique seems to have been to clarify this goal
  of delimiting the domain and conditions for the possibility of synthetic a
  priori cognition. See, for example, the Feder-Garve review
  \citep[pp. 53-58]{sassen2000}, and Kant's response
  \citepalias[4:372-380]{kant1783}.}

The `useful corollary' just mentioned---i.e. the grounding for the theoretical
sciences that falls out of an investigation into the scope and limits of
synthetic a priori cognition---is primarily found in the Transcendental
Analytic section of the Critique, and especially in the Analytic of
Principles.\footnote{I do not claim that this is \emph{all} there is to be
  found in this section of the Critique, of course.} Recall from
\S\ref{sec:negalt} that for Kant, possible experience comprises two distinct
aspects: intuition, for which space and time are its pure forms, and
\emph{concepts of the understanding}, which correspond to rules for
synthesising the manifold of intuition. The concept `chessboard', for instance,
corresponds to a rule whereby this particular bit of white, that particular bit
of black, etc. can be associated in one representation. When we synthesise,
i.e. combine, some particular manifold of intuition according to the particular
rule for a concept, we say that the manifold has been subsumed under the
concept and cognised in accordance with it (A68-69/B93-94). A \emph{pure}
concept of the understanding is called a \emph{category} (A80/B106); it is one
of a set of `meta-concepts' that all empirical concepts necessarily
presuppose. Like the pure forms of intuition, the categories are a priori.

The application of a category to the manifold of intuition is governed by the
\emph{schema} of that category (A137-147/B176-B187); i.e. the rule by which the
manifold is determined in accordance with the category. For example, the pure
schema of magnitude is number (A142-143/B182). In apprehending the manifold
corresponding to an object, determinate intuitions---`instants'---of time are
produced with each successive act of synthesis. Through these is produced a
\emph{time series} from which one judges as to the extent of the object
apprehended (A145/B184). Through an analysis of the use of the categories in
accordance with their characteristic schema, we are entitled to make a number
of synthetic a priori judgements regarding the objects of cognition in
general. These are called synthetic a priori \emph{principles}
(A159-235/B198-294) by Kant. Of particular importance for our discussion in the
next and subsequent sections, wherein we will be considering the relevance of
quantum theory for Kant's philosophy, is the principle of causality. Kant calls
this a `dynamical principle' since it is a principle governing the connection
of appearances in time. It tells us, according to Kant, that the changes
undergone by an object of cognition are ordered uniquely and objectively
according to a necessary rule (A191/B236). This is in contrast to the series of
subjective perceptions of the object through which we apprehend it.

To illustrate: suppose I lean against a fence at the bank of a river, and watch
a piece of wood as it is carried downstream by the current.\footnote{This is a
  variation on Kant's example of the ship (A191-193/B236-238).} At time $t_1$,
I watch as it comes into view from around the bend in the river some yards
upstream. I then daydream for a while. Eventually, I notice (at $t_2$) that the
log has travelled some distance from the place where I first spotted it. At
$t_3$, I recall to myself the motion of the wood down the river that I
half-consciously observed while daydreaming, after which I continue to watch as
it disappears into the forest ($t_4$). Later that afternoon, I recall that what
aroused me from my daydream was a sparrow alighting on the log (at $t_5$). If
we list these representations in the order in which they are actually
perceived, then this is a \emph{subjective ordering}: $$t_1, t_2, t_3, t_4,
t_5.$$ I can also give them an \emph{objective ordering}, however, according to
which the motion of the log must have actually proceeded in
time: $$t_1,t_3,t_5,t_2,t_4.$$ To determine this objective ordering, I must
discover a rule for the log's motion, for in general ``appearance, as
contrasted with the [representations] of apprehension, can be [represented] as
an object distinct from them only if it is subject to a rule that distinguishes
it from any other apprehension and that makes necessary one kind of combination
of the manifold.'' (B236/A191).\footnote{I have modified Pluhar's
  translation of \emph{Vorstellung} as `presentation' to the more standard
  `representation'. As \citet[p. 348]{allison2001} points out, Kant
  considered this term as equivalent to the Latin \emph{repraesentatio}
  (A320/B376).} The particular rule of succession for the change of state of
the log is something that can only be discovered empirically, for instance by
taking into account the position of the log in the river in my various
perceptions of it, the direction therein in which I perceived the river to be
flowing, etc. However, that objective cognition \emph{presupposes} that there
is some such rule is what the principle of causality tells us. This is \emph{a
  priori}, according to Kant. One notable aspect of such a rule, which we will
return to later, is that Kant seems to require that it be deterministic in the
sense of making possible the perfect \emph{prediction} of an event from what
has immediately preceded it. In Kant's words: ``what precedes an event as such
must contain the condition for a rule whereby this event always and necessarily
follows.'' (B238--239/A193).

The concept of something that is an object \emph{for us}---i.e. of something we
have achieved objective cognition of---presupposes a determination of the
manifold in accordance with the principle of causality and the other synthetic
a priori principles, for Kant (A159/B198). On the other hand, and importantly,
these principles are only regulative with respect to experience `as a
whole'. By this is meant that from a \emph{methodological} point of view we are
required to investigate nature in accordance with them (for otherwise the
objective cognition of anything would be impossible). However it does not
follow from this that the objects of our inquiries are already determined in
accordance with these principles in advance of our investigations, or that we
can know a priori that any particular endeavour to attain objective cognition
will be successful (cf. A509/B537). A particularly striking example of the
failure to obtain objective cognition in this sense is provided by quantum
theory, which we will begin to discuss in the next section.

\section{}
\label{sec:quantmech}

To begin with, consider the following `classical' scenario.\footnote{The
  following example is adapted from \citet[p. 381]{bacciagaluppi2015}, as
  is my discussion of the analogous single-slit example involving quantum
  phenomena which follows. The quantum example is of course originally due to
  \citet[]{bohr1935}. In his paper, Bohr actually discusses (successively)
  multiple variations of the single-slit experiment. The one discussed here
  and in \citet[]{bacciagaluppi2015} is the particular variation discussed on
  pp. 698--699: ``In an arrangement suited for measurements of the momentum of
  the first diaphragm, it is further clear that even if we have measured this
  momentum before the passage of the particle through the slit, we are after
  this passage still left with a \emph{free choice} whether we wish to know the
  momentum of the particle or its initial position relative to the rest of the
  apparatus''.} A medium-sized object is launched towards a diaphragm into
which an opening, or `slit', has been made that is large enough to allow the
object to pass through, but small enough so that it invariably deflects the
object to some degree as it does so. After passing through the slit the object
eventually impacts upon a further screen. Assume that the diaphragm is movable
(e.g. let it be attached to the rest of the apparatus by springs), so that when
the object collides with the edges of the slit as it passes through the
diaphragm, the latter recoils slightly. We would like to describe the state of
the object immediately after its interaction with the diaphragm. Assume that
the (centre-of-mass) positions and momenta of both the object and the diaphragm
just prior to the interaction are known, but that the precise shape and size of
the launched object are unknown, so that we cannot calculate in advance what
the respective positions and momenta of the object and diaphragm will be after
the collision.
Once they do collide, however, we can then measure the momentum of the
diaphragm by observing the amount by which it recoils, which will allow us to
(via the law of conservation of momentum) determine what the momentum of the
launched object is immediately after the collision. Similarly, since the common
centre of mass of the combined system remains at rest, we can determine the
object's position by measuring the position of the slit.

In classical physics, which is adequate to describe the motion of objects such
as the one in the imagined scenario, a precise determination of momentum and
position is sufficient to completely characterise the state of a particle (or
collection of particles) \emph{vis \'a vis} its variable parameters at any one
time. And from the precise characterisation of the state of a particle at any
one time, one can then precisely infer the state of the particle at all other
times, assuming the system is isolated or closed \citep[\S
  2.1]{hughes1989}.\footnote{Note, however, Hughes's caveat in \S 2.4.} Thus
through precisely determining the state of the object immediately after its
impact with the diaphragm, we are able to predict the object's subsequent
trajectory with certainty.

This is in spite of the fact that the diaphragm---our measuring
instrument---disturbs the motion of the object as the latter impacts upon
it. For in classical theory this is in principle unproblematic. In every case
of classical measurement, either the interaction of the measuring instrument
with the object of inquiry can be made negligible for the purposes of the
analysis, or (as above) we can use physical theory to abstract away from this
interaction in such a way as to allow us to precisely determine the object's
various state parameters. Indeed for precisely this reason, after we have
measured the momentum of the \emph{diaphragm}---which will in general disturb
\emph{it}---we can subsequently measure the diaphragm's position, and then use
physical theory to abstract away from the interaction involved in our previous
momentum measurement to determine the position of the object which has just
passed through. In this way we are able to simultaneously ascribe \emph{both}
position and momentum parameters---i.e. a complete state description---to the
launched object \citep[]{bai2017}.

In the words of the previous section, in every case classical theory allows us
to transition from the subjective conditions (represented by the diaphragm in
the above example) under which we perceive an object, to an objective
description of that object wherein these subjective conditions no longer
explicitly appear. They nevertheless remain implicit in the sense that our
description of the object presupposes that it has been determined by us to be
such as we describe it (either directly or perhaps only indirectly) through a
process of measurement in space and time. This process of determination is
characteristic of \emph{all} objective cognition, as we have seen, for Kant.

Things are more interesting when we come to the case of quantum phenomena. Let
the `lanched object', i.e system of interest, now be quantum---a photon, for
example---which passes through the slit in a diaphragm on its way to an
eventual impact with a photographic plate. Similarly as in our previous
example, we would like to objectively describe its state immediately after it
has collided with the slit. In this case, however, quantum mechanics'
well-known `uncertainty'---or as Bohr preferred to call it:
`indeterminacy'---relation for position and momentum precludes us from
ascribing determinate values to these quantities simultaneously. For according
to this relation, as the indeterminacy in the position of an object approaches
zero, the indeterminacy in its momentum approaches infinity, and vice
versa. Expressed in terms of our example, this means that if we choose to
precisely measure the momentum of the diaphragm (so as to ascribe a determinate
momentum to the object that has just passed though it), then the consequent
disturbance of the diaphragm will be such as to make impossible a further
precise determination of the object's position. That is, the diaphragm's
displacement consequent upon the momentum measurement of it will have been, in
contrast to the situation in classical theory, ``uncontrollable''
\citep[p. 698]{bohr1935}---or at any rate not controllable enough---and we will
be unable to account for this displacement and abstract away from it as we did
in the classical case for the purposes of a subsequent position determination
of the object. The situation will be similar if we first choose to measure the
diaphragm's position (and thus the position of the system of interest); through
this choice we will have precluded ourselves from precisely determining the
system's momentum.\footnote{\label{fn:micro1} Note that if we choose not to
  measure the diaphragm at all, then the joint state of the diaphragm and
  photon will be describable as an entangled quantum superposition. \citet[\S
    10]{hermann1935a} makes this point (implicitly) in the context of a
  different thought-experiment (the $\gamma$-ray microscope experiment). The
  role played by the diaphragm in our example is in that context played by a
  photon which is collided with an electron (the system of interest) in order
  to determine the latter's state. In place of a pointer connected to a
  diaphragm, we have in that context a photographic plate which can be set up
  in various ways (or not at all) in order to measure the state of the
  photon. See also fn. \ref{fn:micro2} below.}

\section{}
\label{sec:conceptind}

Bohr expresses the significance of the limitation imposed by the indeterminacy
relations as follows:

\begin{quote}
Indeed we have in each experimental arrangement suited for the study of proper
quantum phenomena not merely to do with an ignorance of the value of certain
physical quantities, but with the impossibility of defining these quantities in
an unambiguous way \citep[p. 699]{bohr1935}.\footnote{An earlier statement
  expressing an essentially identical viewpoint can be found in
  \citet[p. 580]{bohr1928} and is discussed in
  \citet[pp. 311--312]{cuffaro2010}.}
\end{quote}

That is, the significance of the indeterminacy relations is, for Bohr at any
rate, not epistemological in the sense that one presupposes the quantum object
in the above example to be perfectly determinate in itself with respect to all
of its ascribable physical parameters, but yet not completely knowable by
us.\footnote{This is, incidentally, the viewpoint normally associated with (the
  young) Heisenberg, although see the discussion in
  \citet[pp. 92-94]{frappier2017} for a contrary view.} What is being expressed
here, rather, is that their significance is epistemological in a different
sense; a better word would be \emph{conceptual}.

From the point of view of \S\ref{sec:synthap}, we can understand this as
follows.\footnote{I consider Bohr's views on quantum mechanics to have
  substantial elements in common with Kant's philosophy, and I will be
  pointing out some of these elements as we proceed. However my goal in this
  paper is not specifically to present an argument for this interpretation of
  Bohr, as I take that to have already been established by the work of numerous
  others (some of whom have been referred to above). Indeed, even those who
  have identified pragmatism as the more dominant influence on Bohr's thought
  now acknowledge that the influence on him of Kantianism was nevertheless
  substantial \citep[e.g.][]{camilleri2017, faye2017a, folse2017}. In any
  case, as I alluded to in the introduction to this paper, and in agreement
  with \citet[]{folse2017}, I do not see pragmatism as incompatible with
  Kantianism when the latter is construed broadly in the sense of a research
  program \citep[cf.][]{bitbol2017}.} Consider the result of some experiment,
say the mark on a photographic plate, or the particular situation of a pointer
measuring the momentum of the diaphragm in the above example. The pointer and
the mark themselves---considered simply as a mark or as a pointer,
respectively---are describable as classical objects in the sense that they can
each when observed be described in the ordinary way as having definite
spatiotemporal coordinates, and as causally interacting in a definite way with
their surroundings. However our aim (which as we will see cannot be satisfied
in quite the way one would like in the quantum context) is to go beyond the
particular mark and the particular pointer reading and describe these as having
arisen through the interaction of our experimental apparatus with some
independently existing object. Our goal, in other words, is to `get at' this
object as it exists independently of the `subjective conditions' associated
with the particular experiments we subject it to. And the way we do this---or
anyway the way we attempt to do it---is by compensating mathematically for the
interaction between the apparatus and object in our description of the latter.

For a Kantian, it is of course never possible to \emph{completely} eliminate in
this way the subjective conditions pertaining to our observation of an object
from our objective description of it, since any description of an object that
is cognisable for us must be determinable with respect to space and time. This
is perfectly acceptable, however, since our standard for objective cognition is
anthropocentric rather than theocentric. And since space and time are the pure
forms attaching to any possible experience, we can in principle achieve a kind
of objectivity through the attainment of a description of the object which will
be valid for all discursive cognisers such as ourselves.

Now in order to describe something objectively in this sense---i.e. as an
object of possible experience that exists independently of us in the sense that
its description does not depend on the particular way that we \emph{actually}
come to know it---it must be determinable in space and time in accordance with
the synthetic a priori principles. Earlier I mentioned causality as an example
of a `dynamical' principle of this kind. In addition to the dynamical
principles there are also what Kant calls \emph{mathematical} principles
(B198--B294).\footnote{The mathematical principles are the \emph{Axioms of
  Intuition} and \emph{Anticipations of Perception}; the dynamical
  principles are the \emph{Analogies of experience} and the
  \emph{Postulates of empirical thought as such}.} According to the latter,
anything that appears to us must be apprehended as having, determinately, both
an extensive (length, breadth, etc.) and an intensive magnitude (i.e. a
degree). The dynamical principles, in contrast, are not principles for the
apprehension but for the connection of appearances in time. They state, first,
that all change presupposes something permanent; second, that all change must
occur according to the law of cause and effect; third, that all substances that
are perceived as simultaneous are in mutual interaction.\footnote{Here, I only
  consider the \emph{Analogies}, as the \emph{Postulates} are not directly
  relevant for our discussion.}

An objective description in the above sense is one that is determined in
accordance with \emph{both} sets of principles. Together, they assert that the
determination of any appearance as an object of possible experience must be
such that at a determinate instant in time, it has a determinate extent
(constrained by the mathematical principles) and hence a determinate position
in space, and that there is a law (subject to the dynamical principles) by
which it dynamically interacts with its surroundings in and through time. In
the context of our example of the slit, one can interpret this as signifying
that any description of a quantum object that purports to pick out an object of
possible experience for us must be such as to ascribe to that object both a
determinate position and a determinate momentum parameter. But according to the
indeterminacy relations, it is impossible in principle to describe the
particle's momentum with any degree of precision without a corresponding loss
of precision with regards to its spatial coordinates. The upshot of all of this
is that we cannot complete our description of the object according to the
Kantian criteria for objects of possible experience. And yet these are
\emph{necessary} criteria, for Kant, in the sense that, as we have seen,
objective knowledge is impossible for us without them. We may get around this
in some sense by ascribing only indeterminate values of position, momentum,
etc. to the `object' of our investigations, but the resultant `unsharply
defined'\footnote{\citet[Cf.][p. 582]{bohr1928}.} description can as a result
never be an object \emph{for us}---i.e. it can never be an object which we can
have possible experience of and thus never be real for us in that sense. We can
consider it merely as a noumenon, or abstract object.\footnote{See
  \citet[p. 313]{cuffaro2010}.}

For a Kantian, the situation thus seems hopeless. An objective description just
is one in which we determine something according to both spatiotemporal and
dynamical criteria---we simply have no other choice. And yet these criteria
cannot fulfil their intended function in the quantum domain, for a
determination of one necessarily excludes a determination of the other in the
sense of the indeterminacy relations. Further, there is a different (though
related) sense in which they mutually exclude one another as well, which stems
from the so-called `wave-particle duality' of quantum phenomena. As Bohr points
out \citep[p. 581]{bohr1928}, in the equation expressing this duality: $E\tau =
I\lambda = h$, Planck's constant ($h$) relates quantities that are incompatible
from a classical point of view. That is, in the first relation, $E$ (energy) is
associated with the concept of a particle given with definite spatiotemporal
coordinates, while $\tau$ (the period of vibration) is associated with a
wave-train ``of unlimited extent'', not conceptualisable with respect to
definite space-time coordinates. Likewise, respectively, for $I$ (momentum) and
$\lambda$ (wavelength). Bohr's point is that it is inconsistent from a
classical point of view to describe an object as being, in accordance with the
above relations, both given at some definite spatiotemporal location and of
unlimited extent in space and time. However a violation of the indeterminacy
relations in the description of a quantum object would imply
\citep[p. 311]{cuffaro2010} a precise determination of that object with respect
to both of the above parameters, which would entail the simultaneous
applicability of these mutually exclusive conceptualisations of the object,
which cannot be (i.e. on pain of contradiction). Thus not only do the
classical---or Kantian for our intents and purposes---criteria mutually exclude
one another \emph{vis \'a vis} their determinability. Even if we could
determinately ascribe both spatiotemporal and dynamical attributes to the
object, the resulting object would be self-contradictory.

Ironically it is the uncertainty relations which save us, at least to some
extent. They guarantee that we can nevertheless achieve a unified---albeit
abstract---description of quantum phenomena by `patching together' the mutually
exclusive dynamical and spatiotemporal descriptions that result from our
various experiments. As Bohr puts it:

\begin{quote}
The apparently incompatible sorts of information about the behaviour of the
object under examination which we get by different experimental arrangements
can clearly not be brought into connection with each other in the usual way,
but may, as equally essential for an exhaustive account of all experience, be
regarded as ``complementary'' to each other \citep[p. 291]{bohr1937}.
\end{quote}

The uncertainty relations guarantee that a dynamical description can never
contradict a spatiotemporal description\textemdash that the two can be used in
a complementary way in our description of an abstract quantum object\textemdash
for any experiment intended to \emph{determinately} establish the object's
spatiotemporal coordinates \emph{can tell us nothing} about its dynamical
parameters, and vice versa.

\begin{quote}
the proper r\^ole of the indeterminacy relations consists in assuring
quantitatively the logical compatibility of apparently contradictory laws which
appear when we use two different experimental arrangements, of which only one
permits an unambiguous use of the concept of position, while only the other
permits the application of the concept of momentum
... \citep[p. 293]{bohr1937}.
\end{quote}

We are not licensed, however, to take the next step and ascribe physical
reality to this `patched together' object of our descriptions, for the object
is not real but abstract, and its classical spatiotemporal and dynamical
attributes are idealisations.

\section{}
\label{sec:schlick}

For one seeking to defend the Kantian philosophical framework, which includes
the Kantian synthetic a priori principles---and in particular the principle of
causality---this is good news, at least in one sense. For despite the fact that
the indeterminacy relations imply that in general it is impossible to predict
with certainty the future behaviour of a quantum object from a complete
characterisation of its present state, i.e that the latter in general does not
``contain the condition for a rule whereby [a subsequent] event always and
necessarily follows'' (B238--239/A193--194), Kant's synthetic a priori
principles are not thereby invalidated. For quantum state descriptions are
merely abstract objects from a Kantian point of view and thus are not objects
of possible experience for us. It is only the concept of the latter which
presupposes that it be determinable in accordance with the principle of
causality and other synthetic a priori principles. And it is only for such
objects of possible experience that the perfect prediction of its subsequent
states from a determinate description of the object is implied.

There is, nevertheless, a potential problem, if not for Kant's framework as a
whole then at least for the principle of causality. For as we saw, the
principle of causality is, in addition to being presupposed by the concept of
objective cognition, also (arguably for precisely this reason) a regulative
principle for the purposes of the investigation of nature generally
speaking. But if objective cognition in the Kantian sense is typically excluded
in quantum mechanics, and if moreover the doctrine of complementarity allows us
to continue to do physics in spite of this, then it seems that we can do away
with causality as a regulative principle even if, strictly speaking, it is not
contradicted by quantum mechanics. The logical empiricist Moritz Schlick
expressed this anti-Kantian objection in 1931 as follows:

\begin{quote}
The principle of causality does not directly express a fact to us, say, about
the regularity of the world, but it constitutes an imperative, a precept to
seek regularity, to describe events by laws. Such a direction is not true or
false but is good or bad, useful or useless. And what quantum physics teaches
us is just this: that the principle is \emph{bad}, useless, impracticable
within the limits precisely laid down by the principle of indeterminacy. Within
those limits it is impossible to seek for causes. Quantum mechanics actually
teaches us this, and thus gives a guiding thread to the activity that is called
investigation of nature, an opposing rule against the causal
principle. \citeyearpar[p. 285]{schlick1962}.
\end{quote}

This objection strikes deep---indeed to the very heart of Kant's philosophical
project---which as we saw was, at least as early as 1755, the methodological
one of providing a principled means for acquiring objective cognition---of
providing the ``materials for constructing a building'' as he put it in 1763
(2:66). For if the Kantian principles are inapplicable within a particular
domain of physics, it can be but small comfort to know that they are (for that
reason) still strictly speaking valid in general. Kant's epistemological
grounding, this objection asserts, is irrelevant within this specific domain
and of no use to us.

By way of response, one might note that Schlick's construal of the situation
regarding the principle of causality in quantum mechanics only applies to the
quantum state description itself, i.e. it shows us that our `patched together'
description of a quantum object is not the description of an object of possible
experience and thus not subject to the principle of causality in that sense (as
we saw in the previous section). But while it is true that one cannot
simultaneously attach determinate spatiotemporal and dynamical attributes to
this description, it remains the case on the viewpoint expressed here that we
anyhow require the Kantian categories and principles in order to `patch
together' this very description. That is, they are necessary for the very
interpretation of the results of the measurements involving mutually exclusive
experimental arrangements from which we construct our description of the
quantum object. As Bohr puts it:

\begin{quote}
Here, it must above all be recognized that, however far quantum effects
transcend the scope of classical physical analysis, the account of the
experimental arrangement and the record of the observations must always be
expressed in common language supplemented with the terminology of classical
physics. \citep[p. 313]{bohr1948}.

\vspace{12pt}

The main point here is the distinction between the \emph{objects} under
investigation and the \emph{measuring instruments} which serve to define, in
classical terms, the conditions under which the phenomena
appear. \citep[pp. 221-222]{bohr1949}.
\end{quote}

We require the classical concepts, not only to observe, but also to communicate
experimental results:

\begin{quote}
... the requirement of communicability of the circumstances and results of
experiments implies that we can speak of well defined experiences only within
the framework of ordinary concepts \citep[p. 293]{bohr1937}.
\end{quote}

In this sense the `classical concepts' do in fact continue to function at least
as methodological \emph{tools} for the investigation of nature. The
epistemological lesson driven home by quantum mechanics, one might say, is that
epistemic primacy---the conditions under which we can assert that we have
cognised some object---does not imply ontological primacy. In other words the
necessary principles according to which we cognise something as an object of
possible experience cannot be construed as necessary conditions for the
possibility of objects in themselves. But this is something we (as Kantians)
should already know, for it is what is asserted under an epistemic
interpretation of Kant's transcendental idealism. That is, it \emph{does not
  follow} from the fact that we necessarily apprehend the world in this way
that it must conform to our way of apprehending it. In the context of Kant's
synthetic a priori principles this means that it does not follow from the fact
that we are required to investigate nature in accordance with these principles
that the objects of our inquiries are already determined in accordance with
them in advance of our investigations, nor that we can know a priori that any
particular endeavour to attain objective cognition will be successful
(cf. A509/B537). These principles are constitutive for objects, but only
regulative for nature as a whole.

But how, one might ask, is this `patching together' of the information
regarding the object which we glean from our various experiments in accordance
with the classical concepts actually accomplished? The indeterminacy relations
show us only that (as emphasised by Bohr) this patching together cannot lead to
contradictions, but it remains to be clarified just how the classical concepts
can be used in this way. This is, in part, the contribution of Grete
Hermann. And as we will see in the next section, in addition to this she lays
down an alternative---but still broadly Kantian---understanding of the lesson
provided to philosophy by the discoveries of quantum theory, which can be seen
as stemming from a Friesian-Nelsonian conception of the fundamental nature of
cognition.

\section{}
\label{sec:hermann}

Hermann's reflections on quantum mechanics and its significance for the
`critical philosophy'---the philosophical tradition begun by Kant and continued
by Fries as well as by her own teacher, Nelson---are for the most part
contained in a long essay that she published in 1935
\citep{hermann1935a}.\footnote{For more on Grete Hermann's life and
  intellectual background see, for instance, \citet[]{hansenSchaberg2017},
  \citet[]{leal2017}, and \citet[]{paparo2017}.} She begins this essay with the
admission that the emergence of quantum theory has ``shaken'' (p. 239) this
tradition and in particular has shaken the idea that a priori principles
discoverable through critical philosophical analysis lie at the foundations of
natural science. Ultimately she will argue that the challenge posed by quantum
theory to the critical philosophy can be met. However it cannot be met, she
argues, simply by recapitulating the critical deduction that had previously been
made for these principles.\footnote{As we will see in more detail later,
  `deduction' is being used here in a specific technical sense that is
  relevant to Friesian critical philosophy. Throughout her essay, Hermann
  generally presupposes her readers to be familiar with Friesian doctrine
  including its characteristic jargon (her essay was originally published in
  the Friesian journal \emph{Abhandlugen der Fries'schen Schule}).} She writes:

\begin{quote}
For, even if the physical development of the theory is not sufficient to put
the foundations of the thus achieved knowledge of nature into the sharp light
of awareness, still the scientific progress that has been obtained in these
theories precisely through the willingness to abandon or revise old familiar
concepts provides the guarantee that new and fruitful points of view have been
introduced here into research. Only their philosophical interpretation and
elaboration will produce clarity concerning both the philosophical arguments
for the a-prioricity of natural-philosophical principles and the objections to
them arising from the side of physics. \citep[p. 240]{hermann1935a}.
\end{quote}

In looking to quantum theory to illuminate the critical philosophy in this way,
Hermann's attitude differs starkly from that of her teacher Nelson. Nelson, as
we will see later, was far from denying the general idea that philosophical
insight is attainable through an analysis of scientific knowledge. However
Nelson held that the physics of the early twentieth century (not just quantum
theory) was still too immature and full of contradictions to provide the kind
of insight required to illuminate the critical philosophy. He rather
recommended that---for the time being---critical philosophers adopt ``the
self-denying stance of Conventionalism'' \citep[p. 253]{nelson1971} toward
physics and consider it to be ``of purely heuristic significance'' (ibid.) for
the purposes of critical philosophy.

But while Hermann maintained that real insight into the a priori philosophical
principles of natural science could be gained through a critical philosophical
analysis of the challenge posed to them by quantum theory, she noted that this
could not take the form suggested by physicists such as Born, i.e. of
attempting to disprove quantum theory by empirical means so as to restore these
principles to their former unexceptioned status \citep[]{born1929}. This is
because the question raised by quantum theory is whether such principles are a
priori precisely in the sense of being amenable to a purely philosophical,
albeit in some sense empirically informed, analysis
\citep[pp. 240--241]{hermann1935a}.

With respect to the principle of causality in particular, Hermann also argues
(in agreement with Schlick\footnote{Cf. \citet[\S 6]{schlick1961}.}) that it
would be pointless, in the face of the challenge posed by the indeterminacy
relations, to abandon the criterion through which causal connections in nature
can be known; i.e. the criterion of prediction. She writes:

\begin{quote}
One who wished to brush this off with the excuse that, while the
\emph{knowledge} of the causes determining the processes is limited, the
\emph{existence} of such causes is not put in doubt, removes the law of
causality from the realm of the principles governing natural knowledge into
that of mysticism. Where it is impossible in principle to decide what falls
under a given concept in nature, the statement \emph{that} anything falls under
it also loses its meaning. \citep[p. 242]{hermann1935a}.
\end{quote}

Hermann nevertheless cautions against the use of the criterion of prediction in
the ``positivistically distorted form'' (ibid.) employed by certain
philosophers of her day. Presumably she has in mind those, such as Schlick, who
simply identify causality with prediction. For Hermann, apparently, being able
to satisfy the criterion of (perfect) prediction is necessary, in some sense,
in order to have a meaningful notion of cause. But the concept of causal
connection is for Hermann nevertheless quite distinct from the concept of
prediction. The latter functions as the criterion by which we identify causes,
but does not in itself express the concept of causal connection.

Now our application of the criterion of prediction is limited within quantum
mechanics by the indeterminacy relations. But for Hermann the significance of
this limitation is not that quantum objects always possess joint determinate
values of, say, position and momentum that we must necessarily remain ignorant
of to some extent. Rather, for Hermann their significance consists in the fact
that the simultaneous subsumption of our description of a quantum object under
the wave and particle conceptualisations of phenomena is only possible via
limited applications of these two pictures to the phenomena
\citep[p. 246]{hermann1935a}. Hermann's view is in this way similar to
Bohr's. However, as we will see shortly, Hermann's understanding of the wider
significance of this circumstance marks a subtle but important difference with
Bohr that is relevant to our earlier discussion of transcendental idealism.

In any case, precisely because a quantum object cannot, according to these
relations, be simultaneously described as having a determinate position and
momentum, it follows that it cannot be the case that its future motion is in
principle determined by (and hence perfectly predictable from) these state
parameters as it would be for a classical object. But this raises the question
of whether there may be further (testable) parameters, not described by quantum
theory, which in fact precisely determine the quantum object's motion. This
question, Hermann maintains, is a meaningful and legitimate one to which a
positive answer is conceivable despite the various so-called proofs (which she
argues are unsound) purporting to show the impossibility of such parameters in
general \citeyearpar[p. 254]{hermann1935a}.\footnote{Notable is her analysis
  \citeyearpar[pp. 251-253]{hermann1935a} of von Neumann's impossibility
  proof which anticipates the famous objections of Bell
  \citep[see][]{seevinck2017}. Note that the argument presented in her 1935
  paper is mostly a reproduction of the arguably clearer version from her
  earlier unpublished paper \citep[]{hermann1933}.} Indeed, for Hermann ``there
can be only one sufficient reason for abandoning as fundamentally useless the
further search for the causes of an observed process: \emph{that one already
  knows these causes}.'' \citeyearpar[p. 254]{hermann1935a}.

Nevertheless (and perhaps surprisingly) Hermann maintains that quantum
mechanics gives us no reason to continue the search for additional causes of
the processes it describes, since quantum mechanics itself already provides the
resources with which to identify them. In particular, Hermann notes that if one
interprets the particular situation of a measurement pointer as a statement
about the current state of a quantum system of interest, this interpretation
presupposes a (classical) theory of the interaction that has taken place
between the quantum system and the measuring apparatus through which the
particular situation of the pointer has arisen.\footnote{Compare this with the
  statements attributed to Einstein in \citet[pp. 63-64]{heisenberg1971}.}
For instance, in the above example of the diaphragm and slit one appeals to the
conservation law for momentum in order to interpret the reading of the pointer
after one has measured the diaphragm as asserting a fact about the quantum
object which has just passed through it. But in making such an inference,
Hermann argues, the reading of the pointer is thereby ``explained as the
necessary effect that the system to be measured has imposed on the instrument
in the process of measurement'' \citep[p. 255]{hermann1935a}. In other words we
posit a fact about the quantum system of interest---that it has a particular
momentum---as an explanation which, in light of the classical conservation
theorems, accounts for the fact that the pointer reading has a particular
value. Moreover such a posit is testable, for quantum theory predicts that,
given this posit regarding the system of interest's momentum, a subsequent
measurement of that parameter will (experimental error aside) yield the same
value again with certainty.\footnote{This is just the statement that quantum
  measurements are repeatable.}

Because of this there is no need, Hermann claims, to seek for the physical
features overlooked by quantum mechanics which would make possible a causal
explanation of the measurement pointer's particular reading. For a (classical)
causal explanation for this particular measurement result is already provided
by quantum mechanics in just the way mentioned. Likewise, the classical theory
of interaction appealed to in this case also provides a classical causal
explanation for the particular value we ascribe to the system of interest
itself subsequent to its interaction with the diaphragm. Nothing about this
analysis changes essentially, moreover, if it is a microscopic object that
plays the role for us of a measuring instrument \citep[p. 255]{hermann1935a},
for example when we use a microscope to collide a photon with an electron in
order to measure the state of the latter.\footnote{\label{fn:micro2} In \S 10
  of her exposition, Hermann explores the microscope example alluded to here
  at length. However Hermann's essential conceptual point, at least for our
  purposes, is made in the section we have been focusing on: \S 9. Besides a
  concrete illustration of the discussion of \S 9, \S 10 additionally
  contains further contributions to quantum foundations. It contains, for
  example, one of the earliest discussions of quantum entanglement in either
  the physical or philosophical literature. A detailed analysis of this and of
  the particulars of Hermann's microscope example is inessential for our own
  discussion, however, thus we will not consider them here, and will for the
  most part continue our discussion in the context of Bohr's example of the
  diaphragm and slit. For more on the particulars of Hermann's microscope
  example, see \citet[]{filk2017, frappier2017}. A short summary of the
  relation between the two examples is given above in
  fn. \ref{fn:micro1}. For a more detailed discussion and comparison of the
  two examples, see \citet[]{bacciagaluppi2017a}.}

Returning to the example of the diaphragm and slit: Recall that due to the
uncontrollable displacement of the diaphragm consequent upon a measurement of
its momentum, we are (as we have already seen) unable to form a posit regarding
the system of interest's \emph{position} immediately after having interacted
with the slit. This results in a kind of `one-sidedness' in our
characterisation of the system of interest. As Hermann puts it:

\begin{quote}
the quantum mechanical description by which, on the basis of some observation,
a physicist determines his system, does not characterise this system completely
and absolutely, but (so to speak) reveals only one aspect of it---precisely the
aspect that presents itself to the researcher on the basis of the observation
made here. \citeyearpar[p. 256]{hermann1935a}.
\end{quote}

To put it a different way: For any experiment performed on a quantum system of
this kind one can, in the way described above, appeal to a classical law by
which the reading of our measurement apparatus emerges as the necessary
consequence of a posited feature of the quantum system measured. In that sense
we classically explain the reading of the pointer via this posit and the
relevant classical conservation law. However at least from a classical point of
view such a description of the quantum system of interest is not a fully
objective one, in the sense that it leaves out information which we require for
a classically complete characterisation of the system's state. But as Soler and
others have correctly pointed out,\footnote{See, especially, p. 65 of
  \citet[]{soler2017}. \citet[p. 257]{banks2017} makes a similar point, and
  so does \citet[p. 140]{bacciagaluppi2017a}, although in the latter case
  \citeauthor[]{bacciagaluppi2017a}'s focus is on this difference between
  Hermann's view and Bohr's rather than Kant's.} this presents a problem for
one seeking to defend a strictly Kantian conception of causality. Hermann seems
to depart substantially from Kant's conception, in fact. For recall that for
Kant a causal process is one in which the sequence of appearances of an object
can be connected together in time (see \S \ref{sec:synthap} above). But the
appearance of an object is an appearance which can be given an objective
description. And an objective description is one in which \emph{both} dynamical
and spatiotemporal parameters can be ascribed to the object. Thus the causal
explanations that Hermann claims can be reconstructed from our various
observations of phenomena do not seem to be causal in a Kantian sense. One may
doubt, therefore, whether Hermann's analysis of the situation in quantum
mechanics, even if one deems it to be a success otherwise, can be understood as
a vindication of Kant's critical philosophy.

To be fair to Hermann, however, she nowhere claims to adhere completely
strictly to the critical philosophy as it was expounded by Kant
himself. Indeed, as we will see further below, she is in fact critical of Kant
in her essay. We will also see how the conception of causality that results
from Hermann's analysis of quantum theory can actually be seen as a natural
consequence of applying the critical method as it was elaborated by Fries and
Nelson---as a natural extension of Friesian neo-Kantianism, if you will.

\section{}
\label{sec:fries-nelson}

Fries, like Kant, saw the goal of critical philosophy to be primarily one of
explicating the a priori in cognition, where the norm or standard for cognition
is understood (to use Allison's terminology) not in a theocentric but in an
anthropocentric sense. Recall that for Kant, the nature of our cognition is
discursive. By this it is meant that cognition for us is composed of both a
sensible and a conceptual aspect---the faculties of sensibility and
understanding, respectively. Kant also refers to the faculty of \emph{reason},
and indeed dedicates the lion's share of the Critique (the Transcendental
Dialectic) to this faculty. But by reason he means, according to Fries, merely
dialectical reason, i.e. the ability to make inferences. For Fries dialectical
reason is not essentially different from understanding; for understanding is
the faculty of judgement, and dialectical reason is merely a kind of
judgement. For Fries, understanding and reason in this sense should rather be
thought of as together comprising a single faculty, namely the faculty of
mediative or reflective reasoning. But in addition to this faculty and the
faculty of sensibility, Fries holds that there must be a third autonomous
faculty involved in our cognition. This is pure reason properly speaking, which
operates spontaneously and involuntarily and is, for Fries, the proper source
of the a priori principles of our cognition \citep[pp. 227--228]{leary1982}.

These a priori principles, for Fries, are immediate, though they are not
sensations. They are cognitive, and yet they are not a species of reflective
cognition. They thus constitute a kind of \emph{immediate cognition}, which
when applied in particular instances manifests itself in what Fries calls a
`feeling-for-truth' \citep[p. 178--179]{nelson1971}. However because the
cognition represented by these a priori forms is not reflective cognition, it
cannot be demonstrated via deductive proofs in the way that Kant had, according
to Fries, aimed to do in the Critique. Rather, these forms of cognition may
only be `deduced'---i.e. discovered and described---through an analysis of the
subjective contents of our cognition. In this sense the method of the critical
philosophy is, according to Fries, \emph{empirical}. Through a philosophical
analysis of the `data' that is our subjective consciousness, we `deduce' the a
priori forms of knowledge that are implicitly relied upon therein
\citep[pp. 228--229]{leary1982}.\footnote{For a detailed account of Fries's
  method of `deduction' see \citealt[pp. 164--196]{nelson1971}. We will be
  considering part of Nelson's exposition in more detail below.} In this sense
Fries's conception of critical philosophy can be thought of as, as Nelson puts
it, a ``combination of Kant's notion of the Critique of Reason with Plato's
idea that philosophical knowledge is fundamentally obscure''
\citep[p. 160]{nelson1971}.

The reader will at this point be forgiven for thinking that Fries's critical
philosophy is itself obscure, however. The idea of a `feeling-for-truth' is
unclear and seems to amount to a bald `appeal to intuition' characteristic of
pre-Kantian philosophy or even of contemporary analytic metaphysics. In this
sense it does not seem to have much in common with Kantian philosophy. More
importantly the idea of an immediate cognition which, despite its immediacy, is
only known obscurely seems incoherent \citep[cf.][p. 78]{beiser2014}. And it
seems problematic (to put it charitably) to hold that our knowledge of the a
priori can be grounded empirically. It is thus quite fortunate that one of
Nelson's most important contributions to Friesian critical philosophy was to
clarify these and other aspects of Fries's views.

The distinction between the a priori forms of knowledge (what, in Nelson's
terminology, is called the `system of metaphysics', by which he does \emph{not}
mean metaphysical cognition in the sense of Kant's \emph{Innaugural
  Dissertation}) and the Friesian critical philosophy, Nelson explains, can be
understood in terms of what we would today call the distinction between
object-language and meta-language \citep[pp. 184--190]{nelson1971}. Nelson does
not use exactly this terminology, though he comes close, and in any case
expresses the essential ideas:

\begin{quote}
The peculiarity of the Critique of Reason, as practised by Fries, lies in this:
the knowledge contained in the system of metaphysics forms the object of the
knowledge contained in the Critique of Reason. \citep[p. 184]{nelson1971}.
\end{quote}

This is interesting and noteworthy, since the lecture notes from which these
words are quoted (published posthumously in German in 1962 and then in English
in 1970-71) date from the years 1919-1926---significantly earlier than Tarski's
and Carnap's introduction and formalisation of these concepts in 1933 and 1934,
respectively.\nocite{tarski1956, carnap1937} In any case Nelson illustrates the
distinction via a discussion of the axiomatic method in mathematics. In the
system of mathematical knowledge that we call geometry, Nelson explains, one
often needs to answer questions about the various statements in the system and
their relations to one another. To do so one uses what we would now call a
meta-language, within which one is able to formulate assertions regarding what
we would now call the object-language (in this case the system of geometrical
statements). In this way one can express propositions about the object-language
in the meta-language such as: `$\phi$ is unprovable within the system
$\mathfrak{G}$' (where $\phi$ is a statement in the object language
$\mathfrak{G}$), `The minimal set of axioms for $\mathfrak{G}$ is
$\{\phi,\psi,\chi\}$', and so on \citep[p. 185]{nelson1971}.

To illustrate how the knowledge contained in the meta-language can be acquired
empirically, suppose that one has not been given a complete specification of
the system of interest (the object-language) at the beginning of one's
investigation into that system.\footnote{This illustration of how one may
  learn about an axiomatic system `via observation' is not actually a part of
  Nelson's discussion, but it is fully in the spirit of that discussion.}
Suppose instead that one is successively presented a series of theorems in the
object language. In such a case it is evident that as each new theorem of the
object language is exhibited, one's understanding of the object
language---which one encodes within one's meta-language---will grow; we learn
empirically, so to speak, about the characteristics of our object-language
(e.g. its axioms).

This relationship between `meta-language' and `object-language' is entirely
analogous, Nelson maintains \citeyearpar[p. 185]{nelson1971}, with the
relationship between Fries's critical philosophy and the `system of
metaphysics' (which in the present context is, as we noted above, just the
system comprising the a priori principles of cognition). But in place of
mathematical statements, our object-language will in this case contain
statements such as `$B$: Every change has a cause'. Within the meta-language,
on the other hand, analogously to the mathematical meta-language which contains
statements such as `$\phi'$: $\phi$ is unprovable', the critical philosophy
will contain a statement asserting: `$B'$: $B$ recapitulates some item of
immediate knowledge', where $B'$ is `deduced' in the Friesian sense by way of
the empirical knowledge we have acquired thus far.

Fries's tripartite conception of cognition amounts, as does Kant's bipartite
discursive conception, to a kind of ontological posit. However in both cases
these are posits regarding the concept of cognition. They do not aim to make
substantive assertions about the properties of objects in the mind-independent
world (cf. \citealt[][p. 166]{nelson1971}, \citealt[\S 12]{beiser2014}). As for
the actual a priori principles of cognition which are discovered through the
Friesian critical process of `deduction', these in the end turn out to be
identical to those arrived at by Kant. In this sense one may say that Fries's
contribution to critical philosophy pertains not to the elaboration of the end
result of critical investigation but rather to the method that is to be
followed so as to arrive at that result. Moreover since this critical method
is, for Fries, an empirical one, it seems that it cannot be ruled out that
further investigation will uncover new a priori forms or even modifications in
our understanding of the a priori forms we have already uncovered. \emph{The
  critical method may in principle yield surprises.} And this, Grete Hermann
maintains, is precisely what occurs when we re-examine the critical philosophy
in the light of the discoveries of quantum mechanics.

\section{}
\label{sec:herm-transcideal}

Hermann begins her discussion of the deeper consequences for the critical
philosophy of the discoveries of quantum mechanics in \S 16 of her 1935
essay. She begins with the observation that the nature of the Kantian system of
natural philosophical principles (i.e. his synthetic a priori principles) is
intimately bound up with his doctrine of transcendental idealism, which she
characterises as follows: ``according to [transcendental idealism, the
  synthetic a priori principles] cannot provide adequate knowledge of reality
`in itself' but only a limited knowledge of nature that stops at the conceiving
of `phenomena'.'' \citep[p. 271]{hermann1935a}. Hermann then considers two
Kantian arguments for transcendental idealism. The first begins from what we
referred to above as the subjectivity thesis: that space and time are the
formal and necessary subjective conditions for appearances. She takes Kant's
argument for transcendental idealism from subjectivity to be unsuccessful,
interestingly, for the same reason that most ontological interpreters of
transcendental idealism do, namely because it does not follow from the fact
that space and time are the necessary subjective conditions of appearances for
us that they necessarily do not attach to things in themselves as they exist in
the mind-independent world apart from us. In other words his argument depends,
according to Hermann, on ``the erroneous assumption that already the a priori
character of these notions robs them of the objective significance of
determining reality in itself.'' \citep[p. 271]{hermann1935a}.

Hermann is in fact echoing similar statements by both
\citet[pp. 190--197]{nelson1970} and
\citet[pp. XXIV--XXV]{fries1828}. Friesians as a rule criticised Kant's
arguments from `formal idealism'---a moniker which appears only in the second
edition of the Critique---as well as the second edition emendations to the
Critique more generally. In particular they objected to Kant's substitution of
the subjective deduction of the first edition for the transcendental or
`objective' deduction of the second edition, despite Kant's assertion that
these are merely different ways of presenting one and the same doctrine
(BXXXVIII). In light of our discussion in \S\S
\ref{sec:negalt}--\ref{sec:firstprinc} it should be clear, however, that these
thinkers have misinterpreted Kant's doctrine of formal idealism and the notion
of objectivity appealed to therein. It should also be clear, though, that these
thinkers did not interpret Kantian-Friesian doctrine ontologically
themselves. They rather saw Kant as confused. Nelson writes:

\begin{quote}
[Kant's question of] how items of knowledge can relate to objects is wrongly
put. The relationship of knowledge to its object is not a topic for scientific
investigation at all, for it is impossible to compare knowledge with its
object. In order to check on the objective validity of our knowledge, its
agreement with the object, we would need to know the object independently of
our knowledge. But we know it only by the knowledge which we have of it. So we
should have to take this knowledge as valid, in order to compare it with the
object. We should be moving in a circle. In another context Kant realized that
knowledge cannot be shown to agree with its object, but in his attempt at an
Objective Deduction he got involved in this fallacy. What is correct in this
Deduction---and it does contain many correct and important points---is really
concerned with a totally different question, not the objective validity, but
the mutual relationships of items of knowledge, taken subjectively; the
agreement of \emph{a priori} knowledge with the experience of which it is a
condition. Take it in this way, as a question about the mutual relationship of
items of knowledge, and Kant's answer is entirely
correct. \citep[p. 191]{nelson1970}.
\end{quote}

Part of the misunderstanding here seems to be that Nelson (and Hermann) take
Kant's use of the term `object' in a stronger sense than Kant actually
intends. As we have seen, an `object' for Kant is nothing other than an
\emph{objective determination} of the manifold of appearances in accordance
with the synthetic a priori principles for the application of the categories,
where these latter are rules for the synthesis of sensible intuition as such,
whose pure forms are space and time. There is no further ontological inference
here (either positive or negative) to the nature of things in themselves
existing apart from us. None of this is in conflict with anything asserted by
Nelson in the just quoted passage.

Regarding the `other context' mentioned by Nelson: This is most likely Kant's
discussion of the antinomies. These arise, Nelson explains, ``through our
silently assuming that Space and Time could contain a Whole of all existing
things'' \citeyearpar[p. 170]{nelson1970}, and are resolved via the doctrine of
transcendental idealism properly construed:

\begin{quote}
We can only describe the phenomena perceived as far as our experience goes,
though no limits are set to the progress of that experience. So there is indeed
one Nature, in space and time, one network of phenomena, bound by necessary
laws. But it is not a Universe, not an absolute Whole of all existing
things. Knowledge of nature is knowledge only of
phenomena. \citep[p. 171]{nelson1970}.
\end{quote}

In her own essay, Hermann presents a variation of this argument that is set in
a modern physical context. She first asks us to consider the principle of cause
and effect, and notes that it is not really possible, strictly speaking, to
apply it even within classical physics, for the relation of cause and effect
presupposes that we can distinguish between two contiguous states of an object,
an earlier and a later state. However the temporal evolution of classical
physical states---which in the formalism of classical physics is expressed by
the fact that its laws are given in the form of differential equations---is
\emph{continuous}. This means, however, that for every state of the system
$\omega'$ that comes earlier than $\omega$, there is actually a state
$\omega''$ in between $\omega'$ and $\omega$, and thus properly speaking
``there are no temporally \emph{contiguous} states and hence for no state of a
system can one specify another that has directly brought it about or has been
caused by it.'' \citep[p. 272, emphasis added]{hermann1935a}.

To illustrate this, Hermann invites us to consider the `acceleration field'
which surrounds the Earth in the context of Newtonian gravitational
theory. That is, Newtonian theory specifies a law that associates, with a given
point in the space surrounding the Earth, a particular acceleration that will
be felt by any body situated at that point \citep[cf.][p. 261]{stein1967}. Thus
one might say that this law causally links the presence of a body at that point
(the `cause') with the acceleration imparted to it there (the `effect'). This
acceleration, however, is defined via a differential equation, i.e. as the
change in velocity experienced at the given point. `Change in velocity',
however, is not an independent physical process but rather a ``\emph{relation}
that obtains between the instantaneous velocity and the subsequent course of
nature'' \citep[p. 272]{hermann1935a}, i.e. the further points on the object's
trajectory.

In fact it is impossible, Hermann argues, to determine the acceleration
imparted to an object existing \emph{at} a specific point in space \emph{at}
the specific instant that it is there, since by considering only this single
(idealised) spatiotemporal point one can in no way determine the object's
velocity. Rather one must consider a wider space-time region and then compute
the change in velocity of the object within that region. In considering a wider
region, however, we thereby go beyond the particular point at which the
instantaneous acceleration imparted to the object has been attributed. Further,
within this wider region the object will experience additional changes in
acceleration as it moves about. So we see that in this case it is not really
possible, strictly speaking, to isolate the effect (the acceleration) at a
specific point for the purposes of attributing it to a cause (i.e. that it was
present at that point).

We encounter similar difficulties, according to Hermann, when we try to
characterise the state description of a (classical) physical system as a fully
objective description (in the strong sense) of the properties possessed by that
system. For the physical quantities attributed to the system via its state
description (mass density, velocity, etc.) must be specified for every point
comprising its extension as well as for every point along its trajectory. But
these specifications make use of differential equations which, as we just saw,
actually require extended spatiotemporal regions for their determination. So we
cannot say, according to Hermann, that the system `really bears' these
properties which we ascribe to it via its state description:

\begin{quote}
One can thus say nothing as to the specification of the properties of physical
objects or events that determine these as they are constituted in themselves;
rather, the alleged properties of physical systems in truth only specify
certain relations between the parts of the system, without these parts being
themselves unambiguously specifiable. \citep[p. 273]{hermann1935a}.
\end{quote}

We see, therefore, that even in classical physics the conception of things
which exist in space and time and which stand in causal relationships to one
another is in fact of only limited applicability; we can in principle
successfully describe phenomena in accordance with this conception, but only if
we do so `in the limit'. That is,

\begin{quote}
As long as one is dealing only with finite, spatially and temporally extended
physical systems whose own inner structure need not be considered beyond a
certain limit, one escapes the difficulties of the limiting process that make
impossible the full application of the concepts of cause and of
substance. \citep[p. 273]{hermann1935a}.
\end{quote}

In this sense one can employ the fundamental concepts of `substance',
`causality', etc. within physical theory only if one understands one's use of
these concepts in this context in terms of an `analogy'
\citep[p. 273]{hermann1935a}. For a complete specification of the object in
spatiotemporal terms is on the basis of the above considerations
impossible. But as analogies, i.e. in these concepts' application to finitely
specified objects---objects for which everything beyond a certain specified
bound has been `cut' away from our description of them---these fundamental
concepts can be used fruitfully, and indeed are indispensable. For physical
investigation always begins with the observation of a finite extended system
which we conceive of as an incompletely specified thing existing in space and
time in interaction with its surroundings. And it is the very fact that this
specification is incomplete that presents us with the occasion for better
determining the object of our investigations beyond our arbitrary initial `cut'
\citep[p. 274]{hermann1935a}.\footnote{Cf. Harper's
  \citeyearpar[]{harper2011} account of `theory-mediated measurements'.} And
indeed we do so. And yet although we can move this cut further and further
inwards we can never do away with it entirely, for it alone guarantees that we
can fruitfully describe the object of our investigation as a thing, albeit only
by way of analogy; i.e. not as a thing considered in itself apart from the
conditions under which it is represented by us, but always only in relation to
the spatiotemporal limitation imposed upon our representation of it by the
forms of our sensibility.

One might worry that this doctrine seems to depart substantially from
Kant's. For Kant, recall, it is constitutive of what it means to be an object
that it is fully determined in space and time in accordance with the
mathematical and dynamical synthetic a priori principles. Hermann seems to be
arguing, however, that in fact a full determination of this kind is
impossible. But from that it would seem to follow that Kantian objects are
strictly speaking impossible as well.

Now it is true, as we have seen, that for Kant the concept of an object of
cognition presupposes the ascription to it of determinate spatiotemporal
parameters. And it is also true that for Kant, our representations of space and
time are representations of continuous, infinitely divisible, quantities (B211,
B255). Yet it is not a part of Kant's doctrine that an object must be
determinable beyond limit. This would only follow (as Hermann is of course
aware) if we took our spatiotemporal characterisation of the object to pertain
to it as it is in itself. In that case a truly objective characterisation of
the object would indeed require us to describe it exhaustively via a completed
infinity of spatiotemporal divisions. Kant's synthetic a priori principles,
however, are only applicable to appearances, and completed infinities of this
kind are not to be found in experience. To assume so can result only in
antinomies (A523--527/B551--555).\footnote{See also \citep[\S 4]{bell2017}.} In
this sense we can see how Hermann's discussion constitutes, exactly as she
maintains, an argument for transcendental idealism motivated by a consideration
of the antinomies.

Now Hermann points out that one might respond to this line of argument as
follows \citep[p. 275]{hermann1935a}: One might concede that the conception of
things existing objectively (in the strong sense) in space and time must
dissolve, if pushed to its inevitable logical conclusion, into a network of
spatiotemporal relations which \emph{cannot} be characterised in terms of such
things, and therefore that such a conception is impossible. Nevertheless one
might maintain that it is possible that further investigation into the natural
world will reveal that this relational network itself is objectively
determinable via a consideration of things (in the weaker sense) describable in
space and time. In other words it could be that the spatiotemporal relational
network is itself objective even if it cannot literally be construed as
pertaining to objects (thought of in the strong sense).\footnote{It is possible
  that the view represented by the hypothetical objector here may actually be
  that of Nelson. In the context of practical philosophy, for instance,
  Nelson maintained that the moral law, although only discoverable
  empirically and `subjectively', was nevertheless objective. This was a
  conclusion with which Hermann disagreed \citep{leal2017}.}

Far from disagreeing, Hermann maintains that this is a question which can only
be decided by empirical research; and it has in fact, she argues, actually been
answered in the negative by quantum theory. Quantum theory shows us that this
relational network is not objective but rather `split'.\footnote{See also
  \citet[]{crull2017}.} \citet[p. 275]{hermann1935a} writes:

\begin{quote}
[Quantum mechanics] does away with the notion that these relational networks
should be determined at any rate through objective circumstances of things in
space and time, and shows them in turn to depend on the manner in which the
observer obtains knowledge of the system. \citep[p. 275]{hermann1935a}.
\end{quote}

This is so because, as we have seen, our ascription of a causal history to a
quantum object depends on the experimental context under which that object is
considered. In the experiment (described earlier) which we set up to determine
the momentum of a given quantum system of interest, for example, we explain the
fact that the pointer took on a particular reading by positing that the quantum
system had a particular momentum at the time of its interaction with the
diaphragm. That is, this posit, in conjunction with the classical conservation
theorems, accounts for the fact that the pointer reading took on a particular
value in the context of this measurement. In performing such a measurement,
however, we cut ourselves off from ascribing a determinate value of position to
the quantum system of interest at the time of its interaction with the
diaphragm, for the determinateness of our attribution of position to the
quantum system varies inversely with the determinateness of our attribution of
momentum. Further, not only does what we learn about the quantum system depend
on our observational context, it is also the case that (see also \S
\ref{sec:conceptind} above) these observational contexts are in general
incompatible. Yet as Hermann points out, ``the limit up to which one or the
other model finds application is \emph{not itself an objective property of the
  object}'' \citep[p. 270, emphasis added]{hermann1935a}.

Within quantum mechanics there is no viewpoint, therefore, from which one can
give the quantum object a complete characterisation in the classical sense of
determining both its position and its momentum; it is therefore not an object
in that sense. Nevertheless it is possible to speak objectively about the
quantum object \emph{from within} a particular observational context. We can
say, that is, that (experimental error notwithstanding) the fact that the
quantum object had a particular momentum was \emph{the cause} of the pointer's
having indicated a particular value, because there is no information which we
lack in order to complete our characterisation of the object from the
perspective of that experimental context---in other words we have a complete
characterisation of `the object' that our experiment was set up to ascertain:
the photon's momentum.

To come back now to the question we first considered in \S \ref{sec:hermann}:
Is Hermann's view actually Kantian? In fact there is a sense in which it is
Kantian and a sense in which it is not. Let us begin with the latter: Kant has
developed his system a priori, beginning from a posit regarding the nature of
our theoretical cognition as involving the contributions of both the faculties
of sensibility and understanding, along with an account of the a priori forms
of these faculties (space and time and the categories). From this he has
deduced a number of synthetic a priori principles which are presupposed by the
concept of objective determination as such. In the context of physical theory
these principles imply that the objective determination of something must be
such as to ascribe to it values for both its position and its momentum. The
`objects' described in the previous paragraph, however, do not satisfy this
criterion and in this sense they are not objects of theoretical cognition; the
principle of causality is in this sense not applicable to them. We may
nevertheless, as emphasised by Bohr, achieve a unified---albeit
abstract---description of quantum phenomena by `patching together' the mutually
exclusive dynamical and spatiotemporal descriptions that result from our
various experiments on quantum systems. But we must not call this patched
together description an object---at any rate not an object of possible cognition.

In a different sense, however, Hermann's viewpoint is indeed quite Kantian. For
after all, Kantian philosophy teaches us that there are multiple ways of
considering phenomena. We may consider ourselves as either objects of
theoretical or practical (i.e. ethical) inquiry, for instance, and each of
these domains of inquiry comes with its own criteria for cognition. Moreover
since our cognition of objects is limited to appearances it does not follow
that any one domain can be thought of as either ontologically superior or
reducible to the other.\footnote{Cf. \citet[p. 48]{allison2004},
  \citet[p. 247ff]{nelson1971}.} Further, within each domain distinct notions
of justification are operative, along with distinct notions of causality---most
generally thought of as change in accordance with law
\citepalias[4:446]{kant1785}---and the other fundamental concepts of the
Kantian system. ``The will is a kind of causality'', Kant writes, ``belonging
to living beings insofar as they are rational'' \citepalias[4:446]{kant1785}.

What quantum mechanics teaches us, according to Hermann, is that even
\emph{within} a single domain of inquiry, such as physical science, one can be
confronted with a splitting of perspectives in the above sense:

\begin{quote}
the splitting of truth goes deeper than philosophy and natural science had
previously assumed. It penetrates into the physical knowledge of nature itself;
instead of merely delimiting its scope against other possibilities for grasping
reality, it separates various equally legitimate representations within the
physical description that cannot be unified into a single picture of nature
\citep[p. 277]{hermann1935a}.
\end{quote}

Beginning from such motivations, therefore, it seems natural for a Kantian to
want to relativise the concept of causality to different perspectives in this
way within the domain of physical science. And yet for an orthodox Kantian this
way is barred for the reasons which we discussed above. Hermann, however, is
not an orthodox Kantian but a Friesian neo-Kantian. And as we saw in
\S\ref{sec:fries-nelson}, for Friesians the method of critical philosophy is
not a priori but \emph{empirical}. Friesians are allowed to learn and adapt the
critical philosophy in the face of the challenges posed to it by quantum
mechanics in a way that orthodox Kantians seemingly cannot. The way is open,
therefore, for Hermann's relativisation of the concept of causality (and along
with it, objectivity) within physical science. And she need not deny the
central tenets of critical philosophy as it was understood by the Friesian
school of neo-Kantianism---namely its core conception of the critical
\emph{method}---in order to do so.

\section{}
\label{sec:conc}

Let us now summarise our discussion and conclude. I introduced and motivated
Kant's critical philosophy, including his doctrine of transcendental idealism,
in \S\S\ref{sec:negalt}--\S\ref{sec:firstprinc}, and then his principles for
synthetic a priori cognition in \S\ref{sec:synthap}. In \S\ref{sec:quantmech} I
considered the challenges for Kant's view which stem from the emergence of
quantum theory and in \S\S\ref{sec:conceptind}--\ref{sec:schlick} I showed how
these principles may be maintained on an orthodox interpretation of Kantian
philosophy along Bohrian lines. In
\S\S\ref{sec:hermann}--\ref{sec:herm-transcideal} I then introduced the views
of the Friesian neo-Kantian philosopher Grete Hermann and in particular her
understanding of the general situation regarding Kantian critical philosophy
\emph{vis \'a vis} quantum theory.

I argued that on an orthodox Kantian viewpoint---for which the view of Niels
Bohr may be taken as representative (though he never called himself such)---the
point emphasised by quantum mechanics is that our fundamental (and necessary)
forms of knowledge are ultimately idealisations and as such are applicable only
within a limited domain. And yet we may nevertheless---because of this very
limitation---continue to employ these idealisations outside of this domain in
order to convey information to one another in a qualified way; i.e. with the
understanding that the information conveyed in this way does not in general
constitute an objective description of the phenomenon under investigation. As
for Hermann---whom we may take to represent the Friesian viewpoint---I showed
that there is, in addition to this, a deeper lesson as well: Quantum mechanics
teaches us that physical knowledge is fundamentally `split'; for Hermann, that
is, there is a notion of objectivity that is applicable to quantum phenomena,
however it is a form of objectivity that is intrinsically
perspective-dependent.

Which of these positions is to be preferred? I will not attempt to answer this
question definitively here. There were other neo-Kantian thinkers---to name
one: Ernst Cassirer \citeyearpar[]{cassirer1936}---who commented on quantum
mechanics, and any judgement regarding the correct path for a Kantian to take
in the face of quantum theory should more properly attend upon a detailed
consideration of these others in addition to the two thinkers which we have
considered here. But perhaps we may make one or two tentative statements in the
meantime. A merit of Bohr's view is that it emphasises the lesson of
transcendental idealism, i.e. that epistemic primacy---the conditions under
which we can assert that we have cognised some object---does not imply the
ontological primacy of those conditions. Quantum mechanics comes out as a
striking example of this on the Bohrian view. For Hermann, on the other hand,
the lesson is that in general objectivity (in the weak sense) must be thought
of as relativised within physical science to a particular perspective---i.e. a
particular measurement context---within which we may continue to meaningfully
employ principles such as that of causality to the representations of phenomena
that are confined to that perspective. Quantum mechanics teaches us this, on
Hermann's view. And it is a merit of her viewpoint that it makes sense of the
fact that quantum mechanics can teach us \emph{anything at all} about the a
priori forms of our cognition. Moreover it hints, perhaps, at a more general
account of the way that the a priori forms of our cognition---what in another
context have been called `framework assumptions'---may change their scope and
meaning over time.

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