Andreas Rüdiger: True and false sense - Sensational mathematics

I have often wondered where Kant got the idea of dividing judgments into analytic and synthetic, analytic referring to judgments where the content of the predicate was included in the content of the subject and synthetic referring then obviously to judgements where this inclusion did not hold. It's not any difficulty in the definitions I am speaking of, but of the nomenclature that would have in Kant's days reminded the reader of two different methods of reasoning, analysis and synthesis.

Originally analysis and synthesis were used by Greek geometers as referring to processes that mirrored one another. In analysis, one assumed that the required conclusion – proposition to be proved or a figure to be constructed – was already known or in existence. One then had to go through the conditions of this conclusion in order to find self-evident principles on which the conclusion could be based.

If analysis moved from conclusions to premisses, the synthesis moved the other way. One began from some principles already assumed or demonstrated to be true and from methods that one already knew how to use, and from these principles and methods set out to prove some new theorem or to draw a new sort of figure.

A certain step in the evolution of the mathematical methods into Kantian judgement types is symbolised by Rüdiger's notion of analysis and synthesis. Just like in the tradition, Rüdiger uses analysis for a method moving from consequences to principles behind them. Yet, he also calls such method judicial and separates it from synthetic method, which he describes as invention. That is, analysis does not produce any new information, just like in Kant's analytic judgement predicate does not reveal anything that wouldn't already be in the subject. Instead, analysis merely determines whether a given proposition is clearly true or at least probable.

Rüdiger's account of synthesis or invention of new and informative truths includes even more aberrations from the traditional account. For Rüdiger, synthesis might involve also mere probable conclusions that are based on the correspondence of various sensations – for instance, by seeing that a certain effect follows always from certain conditions, we may conclude that a new occurence of similar conditions would probably lead to similar effects. Here we see Rüdiger's empiricist leanings, but he does not restrict synthesis to mere empirical generalisations – in addition he also accepts necessary demonstrations.

Rüdiger divides demonstrations into three classes, according to three components required for thinking. One type is based on the verbal form of thinking and grammar: for instance, we deduce from the statement that Jane hit Mary the related statement that Mary was hit by Jane. The second type contains various forms of reasoning, such as traditional syllogisms, but the common element Rüdiger suggests is that all of them are based on the relations of ideas – we might name these forms logical.

By far the most interesting is the third type of reasoning, the mathematical. Leibniz and Wolff had thought that mathematics was based on inevitable axioms and even empiricists like Hume grouped mathematics with logical reasoning. Rüdiger, on the other hand, clearly separates logic and mathematics. Logical reasoning is based on the relations of ideas, while mathematics is based on the sensuous element of thinking.

Rüdiger's position shares some interesting similarities with Kant's ideas on mathematics. Both Kant and Rüdiger are convinced that mathematics are not mere logic, but synthetic or inventive. True, Kant speaks of mathematics as based on intuitions, while Rüdiger speaks of sensations, but this might not be as great a difference as it first seems. Rüdigerian concept of sensation is clearly more extensive than Kant's and would probably include also what Kant called pure intuitions. Indeed, Rüdiger also separates mathematical reasoning from mere empirical generalisations – mathematical truths are not mere probabilities.

The reason behind Rüdiger's desire to separate mathematics from logic is also of interest. Once again Spinoza is the devil that one wants to excommunicate. Spinoza's Ethics is supposedly philosophy in a mathematical form, but Rüdiger notes that this is intrinsically impossible. Mathematics can rely on certain sensations, when it constructs its definitions and divisions – it can tell that triangle is a meaningful concept, because it can draw triangles. Philosophy, on other hand, does not have a similar possibility for infallibly finding sensations for its concepts – a very Kantian thought.

So much for Rüdiger this time. Next time we are back with Wolffian philosophy.