Kant on the Peculiarity of the Human Understanding and the Antinomy of the Teleological Power of Judgment

Kant argues in the Critique of the Teleological Power of Judgment that the first stage in resolving the problem of teleology is conceiving it correctly. He explains that the conflict between mechanism and teleology, properly conceived, is an antinomy of the power of judgment in its reflective use regarding regulative maxims, and not an antinomy of the power of judgment in its determining use regarding constitutive principles. The matter in hand does not concern objective propositions regarding the possibility of objects or actual features of certain objects, namely, organisms. It is rather a methodological issue as to the appropriate way to explain the generation, development, and function of organisms. Taken in this manner as subjective maxims guiding the explanation and inquiry of organisms, the principles of mechanism and teleology need not necessarily be seen as contradictorily opposed but instead can be combined in the study of organisms. This, however, is not enough to complete the analysis of the antinomy of the teleological power of judgment. In order to show that there is an antinomy in this case, Kant has to establish that both seemingly conflicting maxims are necessary and natural to the human mind. He does it by grounding them in the ‘special character’ or peculiarity (Eigentümlichkeit) of the human understanding. However, it is not entirely clear just what exactly this peculiarity of the human understanding is. Paul Guyer argues that Kant suggests two different accounts of the peculiarity of the human intellect. According to one account, this peculiarity consists in the fact that our understanding forms general concepts and according to another, in its propensity to proceed from the parts to the whole. I will argue in this paper that Kant puts forward a single account, in which the combination of these two features demonstrate the peculiarity of the human understanding manifested in the encounter with organisms. This account explains the necessity of the regulative maxims of mechanism and teleology, and thus completes Kant’s analysis of the antinomy of the teleological power of judgment

Harmonic functions on locally compact groups of polynomial growth

We extend a theorem by Kleiner, stating that on a group with polynomial growth, the space of harmonic functions of polynomial of at most $k$ is finite dimensional, to the settings of locally compact groups equipped with measures with non-compact support. This has implications to the structure of the space of polynomially growing harmonic functions.Comment: 20 page

Leibniz, the Young Kant, and Boscovich on the Relationality of Space

Leibniz’s main thesis regarding the nature of space is that space is relational. This means that space is not an independent object or existent in itself, but rather a set of relations between objects existing at the same time. The reality of space, therefore, is derived from objects and their relations. For Leibniz and his successors, this view of space was intimately connected with the understanding of the composite nature of material objects. The nature of the relation between space and matter was crucial to the conceptualization of both space and matter. In this paper, I discuss Leibniz’s account of relational space and examine its novel elaborations by two of his successors, namely, the young Immanuel Kant and the Croat natural philosopher Roger Boscovich. Kant’s and Boscovich’s studies of Leibniz’s account lead them to original versions of the relational view of space. Thus, Leibniz’s relational space proved to be a philosophically fruitful notion, as it yielded bold and intriguing attempts to decipher the nature of space and was a key part in innovative scientific ideas

On the Cryptographic Hardness of Local Search

We show new hardness results for the class of Polynomial Local Search problems (PLS): - Hardness of PLS based on a falsifiable assumption on bilinear groups introduced by Kalai, Paneth, and Yang (STOC 2019), and the Exponential Time Hypothesis for randomized algorithms. Previous standard model constructions relied on non-falsifiable and non-standard assumptions. - Hardness of PLS relative to random oracles. The construction is essentially different than previous constructions, and in particular is unconditionally secure. The construction also demonstrates the hardness of parallelizing local search. The core observation behind the results is that the unique proofs property of incrementally-verifiable computations previously used to demonstrate hardness in PLS can be traded with a simple incremental completeness property