Image via WikipediaI recently posted a report on the conference I attended on Salomon Maimon. However, running out of steam at the end, I gave the gist of Gideon Freudenthal's talk rather briefly and he has sent to me both an abstract of the talk and a discussion of the overall argument. I reproduce these here:
Maimon develops his philosophy in the analysis of mathematics which is the best of human knowledge in his view. However, even mathematics does not really satisfy his criterion of rationality which depends on intelligibility in terms of the insights of human understanding rather than intuition and imagination. The specification of intelligibility is through analytic and synthetic criteria. Maimon tries to reduce mathematics to analytic propositions and, when this proves impossible, aims to make it conform to his synthetic criterion, the latter consisting in a principle of determinability, which distinguishes meaningful predications from category mistakes. Both programs fail to fully achieve their goal but, rather than give up on his criterion of intelligibility, Maimon envisages an infinite progress to his goal.
Maimon's philosophy is imbued with mathematics. To him, only mathematics is knowledge proper, and he develops his theses in discussion of mathematical examples. However, also mathematics does not satisfy his criteria of rationality.[see note 1 below]
Maimon's main concern is intelligibility. His criterion of rationality is insight of the understanding as opposed to intuition and imagination. Intelligibility is specified to an analytic and a synthetic criterion on which two comprehensive philosophical programs are based, one more demanding than the other. The first criterion is logical truth, and the program consists in the reduction of all synthetic propositions to analytical ones, concepts of substance to concepts of function. The less demanding criterion was Maimon's Principle (or Law) of Determinability. The principle formulates the conditions of a "real synthesis". In “real synthesis” a new object is produced from which new consequences that follow neither from the original subject nor from the predicate concepts alone, but only from their synthesis. Thus a triangle has certain “consequences” (e.g., that the sum of its internal angles equals two right angles), whereas the Pythagorean theorem is a consequence of the synthesis of “triangle” and “right angle.” This criterion does not dispense with intuition, nor substitutes analytic for synthetic judgments, but accepts the (temporary) reality of synthetic judgments (a priori). Both criteria presuppose one supreme concept from which they proceed either analytically or synthetically.
The central motif of Maimon's philosophy is hence that proper knowledge must be based on the understanding. Intuition is not only opaque to reason but may also deceives us. Maimon learned this lesson in The Guide of the Perplexed of Maimonides, his early source of philosophical education. Maimonides discusses asymptotes. The imagination (or intuition) shows that these two lines must intersect, the understanding proves that this is false. What source of knowledge do we trust, the imagination (or intuition) or the understanding? In his commentary, Maimon emphasizes the prerogative of reason over imagination which alone establishes the preeminence of man over beasts, and supplies a three pages discussion with a simplified version of Apollonius' proof accompanied by a diagram. He was evidently very proud of this proof and mentions it also in his Lebensgeschichte.[ see note 2 below]
Maimon's philosophical program was not successful in either version. Mathematics depends on axioms, postulates, and natural numbers which are not the product of the understanding but imposed on us in intuition. He therefore concludes that even mathematics is only subjectively necessary and not objectively and apodeictic. We thus receive the following hierarchy: Pure logic is objective and apodeictic, arithmetic and even more so geometry are subjectively necessary, mathematical physics is contingent, propositions dependent on perceptions ("The square is red") are not yet knowledge in this form.
The uniqueness of Maimon's philosophy consists in upholding these criteria of rationality on the one hand, and claiming that they have not been met even by the best of human knowledge, on the other. The gap between actual and ideal knowledge is a permanent challenge and - because it diminishes by the progress of knowledge - it is also a motivation to further efforts. The full fulfillment of the program is the prerogative of the "infinite intellect". The present state of mathematics - and its gradual transcendence towards the ideal of the infinite intellect! - are the share of the finite intellect. Insisting that our knowledge is not based on firm ultimate foundations and does not conform to the criteria of proper knowledge, that it rather begins and stops in the "middle" in a mixture of logic and intuition, and that also philosophy - his own included! - is merely hypothetical, is the core of Maimon's anti-Kantian philosophy of human finitude. The optimistic counterpart is the claim that we proceed towards ever more objective knowledge. If indeed an isomorphism obtains between the knowledge of the finite and the infinite intellect, then we may hope that we progress in the right direction. But because we cannot know this, our progress may be an aberration. This is Maimon's radical skepticism. The hope that from the "middle" we progress towards "proper knowledge", and the skeptical fear that this might be an illusion, designate the opposite poles of Maimon's "Rational Dogmatism" and Skepticism.
 There have been a number of cursory discussions of Maimon's philosophy of mathematics, but the only serious analysis is Lachterman's 1992.
 Maimonides' example is the leg of an hyperbola and its asymptote which is the outline of the cone itself. See The Guide of the Perplexed I, 73. See Apollonius Conica II, 1, 2, 14. See Maimon's commentary in GM 142 - 149, esp. 146-148; Lebensgeschichte I, 381 see also GW III, 232.