Iconic virtues of diagrams In his Critique of Pure Reason, Immanuel Kant claimed that, being grounded on the forms of sense intuition, arithmetic and geometric propositions are both synthetic (i.e. informative) and a priori true. Bernard Bolzano, followed by the logicist movement (from Gottlob Frege to Rudolf Carnap), answered that the generality and necessity of mathematical propositions and proofs can only be grounded on conceptual analysis.Even though, like Frege, Charles Sanders Peirce is one of the fathers of formal logic, he provides some semiotic reasons to think that Kant was right: diagrams do convey general meanings and provide some knowledge which is necessary and not trivial. Unlike logical analysis, visual presentation of concepts in schemas or diagrams helps to explore concepts by stressing some of their “side” features in such a way that enables new knowledge to be acquired: “diagrams evolve what was involved”.

Immanuel Kant (born Emanuel Kant; 22 April 1724 – 12 February 1804) was a German philosopher. Born in Königsberg in the Kingdom of Prussia, he is considered one of the central thinkers of the Enlightenment. His comprehensive and systematic works in epistemology, metaphysics, logic, ethics, aesthetics, political theory, and the philosophy of religion have made him one of the most influential and hi Immanuel Kant (born Emanuel Kant; 22 April 1724 – 12 February 1804) was a German philosopher. Born in Königsberg in the Kingdom of Prussia, he is considered one of the central thinkers of the Enlightenment. His comprehensive and systematic works in epistemology, metaphysics, logic, ethics, aesthetics, political theory, and the philosophy of religion have made him one of the most influential and highly discussed figures in modern Western philosophy. Kant's philosophy is centered on the human subject and motivated by the desire to secure the possibility of both knowledge and morality against the threats of skepticism and determinism. In the Critique of Pure Reason (1781/1787), Kant argues for transcendental idealism, the doctrine that space and time are mere "forms of intuition" (German: Anschauung) that structure all experience and that we have knowledge only of "appearances" and not of the nature of things in themselves. Kant drew a parallel to the Copernican Revolution in his proposal to think of the objects of experience as conforming to people's spatial and temporal forms of intuition and the categories of the understanding, instead of the traditional method of showing how the mind might conform to its objects. Kant believed that reason is the source of morality and that the categorical imperative binds all rational agents. He believed that aesthetics arises from a faculty of disinterested judgment. Kant hoped that perpetual peace could be secured through an international federation of republican states and international cooperation. Kant believed that true religion is grounded on morality. The exact nature of his religious views is a matter of dispute. At age 46, Kant was an established scholar and an increasingly influential philosopher, and much was expected of him. In correspondence with his ex-student and friend Markus Herz, Kant admitted that, in the inaugural dissertation, he had failed to account for the relation between our sensible and intellectual faculties. He needed to explain how we combine what is known as sensory knowledge with the other type of knowledge—that is, reasoned knowledge—these two being related but having very different processes.…

arrived at from arbitrary constructions in mathematical matter have applicability to objects of experience? Might not mathematics be a purely imaginary science

Kant's arithmetical judgements do not meet his standard for mathematical construction. In his view, mathematics is synthetic a priori because it constructs mathematical objects a priori. The method of constructing is not one that philosophy can mimic but the resulting intuitive objects are an indication of how ampliative judgments can have a secure ground and do serve as guide to philosophy. While it can be separately argued that arithmetic is synthetic a priori, construction requires the exhibition of an object in intuition and mathematical construction in particular produces a continuous magnitude. In geometry, geometrical concepts have non-conceptual content via elementary constructions that correspond to a priori intuitive activities. A priori intuition conditions the successive nature of considering arithmetical judgements but it does not provide arithmetic with non-conceptual content. Absent arithmetical intuitive objects, we cannot characterize arithmetic as constructive, something Kant views as endemic to the mathematical method. To deny the in concreto constructive method of arithmetic is to deny its status as an organon of mathematics. While J. Michael Young's tracing of the arithmetical process closely aligns with the framework of construction it falls short of mathematical construction in subtle but significant ways. Arithmetic remains synthetic a priori but achieves syntheticity because arithmetical concepts cannot be exhaustively and exclusively contained within