Status Updates From Reconsidering Logical Posit...

Accordingly, Carnap there adopts a strategy of "purely structural definite descriptions" that aims to individuate all objects of science solely on the basis of their formal or structural properties within the logic of Principia Mathematical and it is in this way, finally, that Carnap ultimately breaks decisively with both neo-Kantianism and Husserlian phenomenology.

Indeed, Carnap pushes such logical-conceptual objectification far beyond anything envisioned by Weyl; for Carnap (1928a/1967) explicitly asserts that the objectivity of science requires that all purely intuitive or ostensive elements must be completely and definitively expunged

Physical space, in particular, becomes a purely abstract mathematical object (the set of quadrupes of real numbers which is distinguished from other isomorphic relational structures only by the circumstance that our previously constituted epistemic subject is perceptually embedded within it - at a definite "point of view," as it were.

Now Carnap, in his later writings, soon leaves the intuitive space of his
dissertation completely behind. Indeed, Carnap (1928a/1967) adopts a
stepwise constitution of objectivity via purely conceptual or logical means that is parallel to both the neo-Kantianism of Cassirer and the symbolic construction of Weyl.

That this intuitively spatial structure then necessarily lacks what Weyl
calls a determinate orientation of the purely infinitesimal metrics can only mean, for Carnap, that no such orientation inheres in physically spatial reality at all - in other words, that the full metrical structure of physical space can only be conventional.

On Carnap's account, by contrast, the a priori infinitesimally Euclidean nature of space is conceived of as a direct reflection of the necessary structure of our spatial intuition. And it follows, for Carnap, that specifically spatial structure (as opposed to purely formal logical-mathematical structure) can only be intuitively spatial structure

For Weyl himself rejects the idea that physical metrical structure is conventionally stipulated, and instead sees in the general theory of relativity the culmination of Riemann's suggestive remarks according to which the metric of physical space is empirically determined by "binding forces" acting on the underlying metrical continuum.

It is against this backdrop that we should understand the
striking disagreement between Weyl and Carnap over the conventionality of physical metrical structure.

Weyl comes more and more to defend the view that, although our
mathematical-physical understanding of space must indeed begin with immediately intuitive acts (by which the subject locates and orients itself in the "here and now"), the outcome of this procedure is a wholly nonintuitive "symbolic" construction by which we represent the physical-spatial world by abstract mathematical symbols having no intuitive content.

Both Carnap and Weyl thus react to the new situation created by the general theory of relativity, not by adopting a straightforwardly empiricist conception of the foundations of geometry, but rather by generalizing the Kantian notion of the synthetic a priori to the infinitesimally Euclidean character of space

We should also note, finally, that Schlick's "causal realist" theory of
perception and knowledge still harbors significant strains and tensions... For the heart of this theory is the idea that private and subjective, intuitively given spatiotemporal relations constitute the basis for our construction of the objective and "transcendent" realm of nonintuitive spatiotemporal realities.

...in virtue of which, at the same time, our abstract conceptual systems obtain a relation of designation or coordination to the "transcendent" realities in objective space and time outside of our consciousness.

The latter are not given to our consciousness; the former are not (at least not yet) objects of conceptualization and knowledge. We can, nevertheless, set up a relation of coordination or correspondence between the two sides via the method of coincidences,...

Schlick's understanding of general relativity is thus intimately intertwined with a version - a sophisticated and abstract version, to be sure - of a classical "causal realist" theory of perception and knowledge. On one side are the intuitive realities of acquaintance directly given to our consciousness; on the other side are the "transcendent" realities in objective space and time...

For Schlick, therefore, it is not only the case that Einstein's theory destroys the Kantian bridge between thought and reality-namely, pure intuition but it also, via the method of coincidences, shows us how to restore such a bridge in a radically new form.

Reality, for Schlick, includes, paradigmatically, the domain of our private,
immediately given data of consciousness, which constitute the totality of
intuitive objects of acquaintance. But this realm of intuitively given data is not the only reality,... On the contrary, there is also an objective spatiotemporal reality, described, paradigmatically, by modern mathematical physics,

... His explanation, appropriately enough, is inspired by the second great advance in the foundations of geometry of his time, the application of non-Euclidean geometry to nature effected by Einstein's general theory of relativity.

To now explain how knowledge of reality is possible - that is, in his own terms, how we can nevertheless set up a relation of designation or coordination between a Hilbert-style uninterpreted axiom system and some domain of real objects - is thus the sine qua non of Schlick's early scientific epistemology...

The problem is that a Hilbert-style axiomatic system,
precisely in its purely formal-logical, essentially uninterpreted character, is deliberately and self-consciously divorced from all contact with reality: Implicit definition, by contrast [with concrete or ostensive definition], never stands in community or connection with reality, it denies this intentionally and in principle, it remains in the realm of concepts.

Second, Schlick has abandoned the realism of General Theory of Knowledge. In fact, he articulates the characteristic claim of later logical positivism that the difference between realism and strict empiricism cannot itself be expressed or stated. It makes no difference at all whether the objects of physics are viewed as logical constructions from experience or as "independent realities."

Yet the philosophical pressures leading toward a "conventionalist" misunderstanding of general relativity were quite irresistible (even to Einstein himself), and the notion of covariance is remarkably subtle and elusive. As a result, verificationist misinterpretations of general relativity have persisted almost to the present day.