A posteriori and a priori

Wolff is interested not just of truth and probability, but of different methods for discovering true and probable statements, that is, scientific and philosophical methodology. Now, as we have seen, Wolff has made the bold statement that logic is all that one requires for methodology, and at least when it comes to truth, logic is essentially based on demonstrations. This rather stringent description is easier to understand, once we remember that Wolff would accept also empirical statements as premisses of demonstration.

What I find remarkable is that Wolff mentions two different methodologies: a posteriori and a priori. In all external appearances, we are moving towards Kant, yet, there are still crucial differences. While for Kant a posteriori would mean all knowledge based on experience, Wolff limits the range of a posteriori to mere experiences. Thus, when Kant says that a posteriori knowledge cannot be universal, his statement is far stronger than Wolff's. Indeed, Wolff would verbally accept Kant's statement, but it would mean something less than with Kant – it would be just tautology because individual experiences are always singular.

Now, if a posteriori has no other meaning for Wolff, but individual experiences, a priori must then get everything else, that is, the class of a priori contains all bits of knowledge requiring demonstration. Now, demonstrations as meant by Wolff here can be either direct or indirect, while a priori can be either intuitive or discursive. Intuitive a priori has a nice paradoxical feeling to it, somewhat reminiscent of later and more famous synthetic a priori, yet, is not completely identical with it. Intuitive apriority characterises all those truths that can be directly intuited as true, just by carefully reflecting on the elements of the supposed truth. The class of such truths contains analytic axioms of logic, but also immediate truths about causes and effects, which Kant would have classified as synthetic a priori.

The final class of knowledge statements contains then discursive apriorities, that is, demonstrated truths. As one should remember, these demonstrations could have reliable experiences as premises. Furthermore, Wolff also included inductions as just one modification of syllogistic reasoning. Wolff's class of discursive apriorities contains then a lot that Kant would have classified as a posteriori, and in fact is probably the largest of the three classes.

Within this threefold classification Wolff places his main methodological considerations: when is experience to be relied upon, when can we say that something is a cause, when can we use experiences to draw general conclusions etc. I shall refrain from going into intricate details, and instead, I shall next time look at how Wolff thinks scientific conclusions should be presented.

Rational philosophy or logic (1728)

I recently had the distinct displeasure of reading a rant of a would-be philosopher who disparaged a logician, because modern logical texts are like circuit diagrams – useful perhaps, but meant only for people with no literary taste and ultimately unphilosophical. Personally, I find logical texts of all sorts – whether they be ancient, modern, formal, informal, transcendental or even Hegelian – to be aesthetically pleasing in a way that a beautiful calculation or a brilliant game of chess is also: as delightful in their very existence as brightest of poems, no matter how useful they otherwise might be. And if someone complains about non-existent philosophical import of logic, I am always reminded of Hegel's clever quip about enthusiastic youth who are enamored by Plato's more vivid and lively dialogues and who later become very disappointed when they hit the abstract heights of Plato's Parmenides and its study of dry concepts like one and many. Thus, I am not afraid of the supposed dryness of next book in line, Wolff's Philosophia rationalis sive logica.

Is this all there is to logic?

As someone might remember, Wolff had already published a book on logic, the first in his famous series of reasonable thoughts. The current book, on the other hand, begins Wolff's philosophy anew, except this time in Latin. While the German series was meant mostly for domestic markets and especially his students, the publication of Latin versions of different parts of his philosophy served the purpose of making Wolff's work more known throughout Europe. Because of their more scholarly ambitions, Wolff's Latin books contain also more material than their German equivalents. Thus, while I first thought that Wolff's Latin logic would contain only about 300 pages and not be much longer than its German counterpart, I noticed quickly I had actually picked up a separately published compendium for Latin logic, containing just the table of contents for the actual book, which happened to be over 800 pages long.

Just like its German counterpart, Wolff's Latin logic contains much that would not be dealt in a logic course these days: it is more of a book of methodology. Thus, it is also meant to be the first book of Wolff's Latin philosophical works in the sense that reader should first grasp how philosophy works before actually reading some philosophy: the true first philosophy is then ontology, because all the other parts of the philosophy depend on it.

As starting points of series, both books begin with an account of what philosophy is all about. But the inflatedness of the Latin logic shows itself in the very start, with Wolff's novel discussion of three forms of cognition – well, it is actually novel only from the perspective of Wolff, because it is quite reminiscent of Bilfinger's disputation with this very topic. What is important in this beginning, is Wolff's clear commitment on empiricism: all cognition begins with a historical phase, where one can just learn facts through observation. The cognition could then develop into mathematical cognition, by quantifying the results of observation, or it could turn philosophical by attempting to find explanations for the facts (note that nothing speaks against cognition that is both philosophical and mathematical, especially if the quantification helps us to discern causal relations).

If philosophical cognition means finding explanations for observed facts, philosopher is then a person who can give such explanations – that is, an expert on some topic. Philosophy, on the other hand, is for Wolff not just any expertise. Just like in his German logic, Wolff defines philosophy as a science of what is possible. I already noted that this definition means actually just what science does: capacity to demonstrate assertions from indubitable premisses.

Whereas German logic left a rather rationalistic impression, in Latin logic Wolff admits that experiences and experiments can well give science its required premises, provided that they just are reliable. Indeed, although Wolff does equate philosophical and mathematical method, he does accept also the construction of hypotheses or reasonable, but unproved assumptions as an incentive to scientific development. Thus, completely axiomatic-deductive system is admitted to be a mere ideal that we can perhaps approach, but never completely satisfy. The ideal also instigates philosophers to remain moderately skeptical in dilemmas where none of the options can be proven indubitably.

Wolff also notes that philosophy might be cognized only historically, that is, we could just e.g. read Wolffian system and learn all its propositions. Such a historical knowledge of philosophy might be useful, but true philosophical cognition of philosophy is achieved only when we try to understand what philosophers say, for instance, by repeating the experiments described in a text book.

Just like in German logic, in Latin logic Wolff also presents a general division of philosophy. What is remarkable is that the new division is more detailed, especially as it comes to more empirical parts of Wolffian system. This no doubt reflects the fact that Wolff has now actually worked out his system in more detail and has especially realized how important empirical observations are to the development of science. In addition, Wolff also helpfully indicates how each part of his system depends on some parts and serves as a foundation for others.

I’ll be continuing for a while with my account of Latin logic, and next time I shall take a look at the difference between natural and artificial logic.

Christian Wolff: Remarks on Reasonable thoughts on God, the world and the human soul, also on all things in general (1724)

There are two ways to deal with additions, remarks and clarifications meant for explaining one's own philosophical text. Firstly, it is possible to incorporate such additional material to the old text and sell it as a new edition – this is what philosophers such as Kant and Hegel will do. Then again, one can also create a completely new book meant to elucidate the first. This second strategy was used by Schopenhauer and before him Wolff in the commentary of his Magnum opus on metaphysics: Anmerckungen über Die vernünfftige Gedancken von Gott, der Welt und der Seele des Menschen, auch allen Dingen überhaupt. When Wolff dealt with other sciences or branches of philosophy, he often made references to his earlier works and especially to German metaphysics. Wolff thus had a reason for choosing this manner of publication – incorporating additions to the original would have meant changes in the paragraph numbers used for reference purposes. This seems different from Schopenhauer, who probably was just too lazy to edit the first part of his masterpiece.

The motivation behind Wolff's commentary is naturally the need to clarify some points that had not been understood properly. As we have seen, Wolff was especially criticized in the pietist circles of German academic life, who regarded Wolff as a atheist in disguise continuing the work of Spinoza. It is then no wonder that the longest comments Wolff makes are aimed at Lange and his compatriots.

At the very beginning of the commentary Wolff notes that his criticizers had mistakenly thought that he had denied some doctrine, because he had not wanted at that stage to commit himself to any position concerning that doctrine: for instance, he had not at first wanted to say anything about the possible independence of the world, because he was not yet in a position to disprove it, and some reader (clearly Lange) had concluded that Wolff actually believed in the eternity of the world. Wolff is clearly dedicated to the way of presenting theorems that occurred in the mathematical works and especially in Euclid's Elements: one should not use premises one has not yet proven to be correct.

One aim of the commentary is then to emphasize the various interconnections between the different parts of German metaphysics and even different parts of Wolff's whole philosophy. The strict Euclidean method of presentation often prevents such discussion: you cannot say that proposition proven here will help to prove another proposition there, because we are not yet in a position to do the actual proving. The more relaxed form of commentary allows this, and thus Wolff can justifiably note in it that e.g. proposotions of psychology will be used as premises of morality.

Despite the task of showing interconnections, Wolff's commentary is still rather fragmentary: only some paragraphs require comments and of these only few require a lengthier discussion. Thus, it is no wonder that my texts about the commentary will also be fragmentary in the sense that they are rather short and form no coherent whole.

I shall begin unraveling this confusing mishmash by studying the notion of ground or reason. Until next time, then!

Ludwig Philipp Thümmig: Institutions of the Wolffian philosophy provided for the use of academics (1723)

In the development of a theory there becomes a time, when the ambiguities of academic research become distilled in the succinct form of a text book. In the development of Wolffian philosophy this distillation occurred with Thümmig's Institutiones philosophiae Wolfianae in usus academicos adornatae. The book appeared in two parts, firs of which dealt with the theoretical part of Wolff's philosophy – it covers issues dealt in Wolff's logical, metaphysical and physical works.

Summarising an intricate philosophical work is undoubtedly an achievement in itself, but one might wonder how original it can be. Then again, Thümmig's work was not completely without its novelties. While Wolff himself had written his main works thus far in German, Thümmig wrote in Latin, making Wolffian philosophy so available for an international audience. Indeed, many of the Latin terms used for concepts of Wolffian philosophy – e.g. ontologia – are fixed for the first time in Thümmig's work.

An interesting example of a terminological novelty is the notion of infinite judgements. In Wolff's logic judgements are divided into affirmative and negative judgements (respectively, ”A is B” and ”A isn't B”). Now, Thümmig mentions also a third possibility, where the form of the judgement is affirmative, but the predicate is negative (i.e. ”A is not-B”). The notion of infinite judgement was to be important later on, because it allowed Kant to classify judgements in triplets according to their quality (more of this when we reach Critique of pure reason.)

Now, it is undoubtedly questionable whether these terminological novelties were truly Thümmig's own inventions: the notion of ontology had appeared even before Wolffians used it and I suspect that same is true with the idea of an infinite judgement. Furthermore, it is doubtful whether Thümmig was really the first Wolffian to use these terms. A year later Wolff noted in a preface to a work on teleology that Thümmig's works were essentially faithful representations of Wolff's own doctrine. This makes one suspect that Wolff himself had already used the terminology in his lectures and private correspondences and Thümmig had merely wrote down what Wolff had said.

Whomever the real innovator is, Thümmig's book does contain in addition to terminological novelties also some substantial additions to and reworkings of Wolff's original writings. I shall discuss few of them in later blog texts, but for now I shall concentrate on the question of what was the ideal of science in Wolffian school.

Ever since Leibniz the field of truths had been divided into truths based on the laws of logic and truths based on empirical facts. Following this division, Thümmig speaks of a priori and a posteriori cognitions. The terminology is interesting. At least since Kant, philosophers have been accustomed to speak of a posterior cognition, based on experience, and a priori cognition, not based on experience. Now, this hasn't been the case always. Originally, a priori referred to reasoning that derived effects from their causes, while a posteriori reasoning referred to the opposite method of deriving causes from effects. I am sure that someone has already investigated the topic, but it would be interesting to know when exactly the two terms changed their meaning – certainly it happened then before Kant.

For Thümmig, a posteriori cognition was based in experience, while a priori cognition was based something called pure reasoning. Experience was the epitome of intuitive cognition that required a direct intuition of things. Judgements based immediately on intuitions concerned always individual things, and experience was a sort of generalization from intuitive judgements. The transition was possible, because at least the predicates of intuitive judgments were general and therefore even they had something to do with generalities. Thus, by knowing properties shared by many individuals we could discover empirical laws connecting certain general properties.

Pure reasoning, on the other hand, was the high point of symbolic cognition, which used words or other symbols to stand for things themselves. Reasoning in general had to do with making discursive judgements, that is, judgements deduced from other judgements by means of syllogisms. Reasoning was pure, when among the starting points of deduction there was no intuitive judgement, but everything was based on mere definitions and self-evident axioms.

As it was common at the time, Thümmig characterized mathematics as the primary example of a priori cognition – both Hume and Leibniz would have agreed that mathematics was based on self-evident axioms. We have seen that Rüdiger had criticized such an idea, because at least geometry appeared to have an intuitive aspect. Kant in a sense struck a compromise between the two positions, because on his opinion mathematics is both a priori and intuitive – here Kant had obviously changed the meaning of a priori and intuitive.

A primary example of a posteriori science is for Thümmig physics. Although Wolff and Wolffians were mistakenly thought to disparage empirical matters, we can immediately see that over half of Thümmig's book is dedicated to physical and hence empirical questions.

A more intriguing problem is where in the classification metaphysics should be situated. We have seen that Wolff at least apparently tried to axiomatize at least a major portion of metaphysics: everything begins from the self-evident principle of non-contradiction, while even the crucial principle of sufficient reason is supposedly deduced from it.

Thümmig, on the other hand, does not even mention this deduction. Instead, he emphasizes the justification that Wolff had barely mentioned – the principle of sufficient reason is required so that we can distinguish between a dream and reality. Thümmig thus apparently bases the main principle of metaphysics on an empirical proposition.

Does that make Thümmig's version of Wolffian metaphysics then a posteriori? Not necessarily. The possibility to distinguish dreams and reality is in a sense a necessary presupposition of even having experiences. We might hence interpret the justification as transcendental – metaphysics would then be synthetic a priori in the Kantian sense.

Next time I'll be looking at Thümmig's metaphysics in a more detail.

Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Levels of certainty

The Platonic analogy of a broken line is probably familiar to all philosophy enthusiasts. Just like the knowledge of shadows and pictures is to the knowledge of real things, so is the knowledge of the whole world of sense experience to the knowledge of the world of ideas, and how the knowledge of the sense experience is to the knowledge of the ideas, so within the knowledge of the ideas is the doxa to the episteme. I am not doing Plato-study here, so I won't consider more closely how e.g. the doxa is to be differentiated from the episteme. What is interesting here is the idea that certainty of knowledge comes in grades: for instance, that knowledge of sense experiences is unreliable compared to our knowledge of mathematical issues.

This idea was inherited by later philosophers and eventually also reached Germany, where Wolff finally translated the Platonian classification of the levels of certainty to German, although Wolff apparently left out the lowest rang of the Platonic ladder. Comparing to the sense experience Wolff speaks of Glauben, Platonic doxa is replaced by Wolff's Meinung and the highest level of episteme has been transformed into Wissenschaft. The three terms play an important role in the later German philosophy, so let's have a look at them in more detail.

The term Wissenschaft or science is already familiar to us. One might wonder why Greek episteme corresponds with science, when words like ”epistemology” suggest that the Greek original has something to do with knowledge in general. Yet, if we look at how Plato and Aristotle used the word, science or Latin scientia is a very apt translation. For instance, in Aristotle's Posterior analytics episteme refers to a deductive system of knowledge based on indubitable axioms and definitions. As we saw in the previous text, in Wolff this mathematical ideal of science has already been replaced by a more modern notion of science as based on both axioms (in mathematics) and reliable experiences (in experimental sciences).

The meaning of Wolff's Meinung or opinion is also easy to understand, although unlike the Greek original, Wolff appears to evalue opinion as the lowest cognitive state. Opinion is essentially a weaker version of science: ”If we assume definitions that appear to be possible and in inferences assume some axioms, which appear to be correct, although we have not yet demonstrated them, and which we cannot corrobarate through indubitable truths – then we arrive to opinions”. That is, opinions might be argued for, but they still lack the ultimate certainty of science based on incontrovertible truths: if I have an opinion of something, things might still be different than I think. Furthermore, opinions are more subjective than science, because my opinions might known to be false by another person. One might even think that one's opionions are scientifically certain, if one is not aware of how things are demonstrated in science.

It is Glauben that is the most distant from its Platonic predecessor, pistis, but this just reflects the development of the Greek word. While for Plato pistis referred simply to sense experiences, even in Aristotle's Rhetoric pistis meant conviction and trust invoked by a good speaker, while in Pauline letters pistis refers to the first member of the triad ”faith, hope and love”. German Glauben means similarly both belief and faith.

Wolff's use of Glauben reflects Aristotle's rhetorical use of pistis: Wolff understands by Glauben the approval that is given to a statement because of a testimony of someone else. Yet, Wolff extends the role of such a conviction on a testimony from mere judicial matters. While opinion is only a sort of diluted version of science, Glauben is the counterpart of science. Remember that for Wolff science, at least in humans, deals only with possibilities, for instance, with what can be done: these are the things that can be demonstrated. What has actually happened, instead, is beyond scientific proof and we just have to believe the testimony of our own senses and of others, when it comes to such historical questions.

In addition to methodology of mathematican and experimental sciences, Wolff's logical work then also contains the rudiments of a methodology for history. An important element in these rudiments is to recognise how reliable a person describing some events is. Wolff suggests several rules of thumbs how one could decide e.g. whether a witness would have some reasons for lying about what has happened, but does not move further beyond such rules of thumbs.

Although Wolff appears not to use Glauben in the sense of religious faith, we might apply his definition also to faith. Then religion and faith would become intersubjective, communal issues. Having faith on certain religious dogmas would mean being convinced that the people ascribing to those dogmas are reliable witnesses who have no reason for lying on such matters and who are linked through a chain of equally reliable persons to an original witness who was there to actually see what the holy books describe.

Christian Wolff: Reasonable thoughts on the capacities of the human understanding and their correct use in knowing truth - Rationalism vs. empiricism

A common trope in philosophy text books is the supposed battle between rationalists and empiricists, in which the first wanted to base all knowledge on reason and the latter on experience and which was finally solved by Kant who discovered that knowledge was based on both reason and experience. It takes no great historian to discover that this simple tale of two battling schools with three great names on both sides is largely fictitious, not least because e.g. Leibniz did not form a common school with Descartes and Spinoza, but opposed the two in some issues even more than he opposed Locke, the only empiricist of note to have written at the time.

I am not sure who actually invented the fable of the two schools of philosophy, but the first signs of it is the already familiar Kantian tale of Locke as the intuitionist and Leibniz as the intellectualist and Kant himself as the necessary symbiosis of the two. But even after Kant this paradigm was not a given when interpreting the history of philosophy. For instance, Hegel distinguished empiricism from metaphysical school, which apparently included, in addition to the traditional rationalists, ancient philosophers like Plato and Aristotle. Even the separating principle of the two schools is not the same as in the separation of rationalists and empiricists. Metaphysical school, says Hegel, based science on common experiences and analysis of these experiences, while empiricists tried to base it on individual perceptions and then noted that no science of necessities and universalities could be based on them – this description fits not Locke, the paradigm empiricist, but characterises at most a caricature of Humean philosophy. Furthermore, Kantian philosophy is for Hegel not a symbiotic combination of the two schools, but more like a modification of empiricism in the sense that both disagree with metaphysical school about the possibility of certain kinds of knowledge.

In place of a strict division of two schools, various premodern philosophers form then more of a continuum of different standpoints from, say, Spinozan axiomatics as the ultimate in rationalism to Humean bundle of impressions without any necessary connection as the ultimate in empiricism. What I would now like to do is to see where in this continuum Wolff's philosophy fits in. One would expect that Wolff as the supposed follower of Leibniz would be closer to the rationalist end of the line. Yet, Wolff is distinctly aware that many sciences can be based only on experiences. Indeed, in addition to the method of syllogistic reasoning, Wolff tries to describe, however crudely, a method of experimentation, by which basic propositions could be discovered.

Wolff defines experience as something that can be known through perception. Note that he does not identify experience with perceptions. Instead, experience is in a sense more stable than a perception: while a perception might vary from one person to another, experiences are only such perceptions that we know to be capable of being at least in principle communicable to other persons. Thus, experiences are essentially intersubjective.

Despite this stability, experiences deal still only with individual things and might even be deceptional, because human perceptions are not always reliable. Yet, Wolff admits that true universal propositions could be based on experiences. The method for this universalisation is careful experimentation: one varies the situation and so tries to determine the conditions in which the experienced phenomena appears.

Wolff's methodoloy of experiences appears surprisingly empiricist. Still, he is not a pure-bred Lockean, although he does mention latter's work favourably at the beginning of his logic. Wolff does accept also the possibility of substantial knowledge being based on self-evident or analytic axioms, as we already saw in his treatment of mathematics. In other words, analytical propositions are not empty or tautologies according to Wolff. Wolff's justification of the substantiality of such propositions is characteristically pragmatic and even pragmatist. An axiom or a definition can be informative, because it might help us to determine postulates, that is, self-evident ways to affect things. Thus, because we know what a circle is, we know also how to produce one, if suitable materials are given. Even logic is not for Wolff a mere formal system but a helpful tool as a methodology of science.

Christian Wolff: Elements of all mathematical sciences - The axiomatic method in Wolff

We have seen how Wolff differentiates between nominal definitions, which merely explain what a word means, and real definitions which tell how to generate a thing corresponding to a definition. In the actual text of the Anfangs-Gründe, all the Erklärungen are mere nominal definitions, while no explicit real definition is mentioned. We shall see in a while that despite this the work does contain many real definitions.

From mere nominal definitions one cannot really deduce anything substantial and no philosopher has ever denied this. Well, Aristotle appears to say in his Posterior analytics that one could literally deduce all theorems from mere definitions, but Aristotelian definitions are not supposed to be nominal, but reveal the essence of something, so this might not actually be an exception. In any case, we do need some basic propositions (Grundsätze, as Wolff says in German) and from these basic propositions together with nominal definitions we can then derive other propositions, which Wolff calls Lehrsätze.

In addition to this division of basic and derived propositions, Wolff dstinguishes propositions merely describing a state of affairs from propositions stating a method or a need of a method for doing something. This division resembles the division of definitions to nominal definitions describing how to use a word and real definitions telling how to manufacture a thing of a certain kind. Similarly, basic propositions can be divided into axioms describing certain fundamental states of affairs from which proofs can begin and into postulates describing certain fundamental methods for doing things. Finally, derived propositions can be divided into theorems that need to be proved and into problems that need first to be solved and then these proposed solutions need to be proved to solve the problem. We might thus classify the constituents of the axiomatic method in a following form:



Unfortunately, this classification oversimplifies Wolff's thought. Firstly, while nominal definitions can be separated from axioms and theorems, real definitions are not a type of statements different from postulates and problems. Indeed, real definitions tell us how to find or manufacture certain things and thus are included either in postulates or in problems.

Secondly, Wolff notes that we can actually do without the problems altogether, because they could be turned into theorems. Theorems, Wolff says, are always of the form ”if p, then q”, whereas problems could always be translated into this form in the following manner: if the solution is followed, the problem is solved.

The interesting question undoubtedly is how Wolff suggests the axioms to be justified, that is, what is the ultimate basis of his demonstrations. Now, the Anfangs-Gründe investigates also handicrafts like architecture or artillery and Wolff admits that he must use information from experience as a justification of some theorems in these sciences. Yet, in case of pure mathematics, Wolff suggests that just by looking at the (nominal) definitions we can immediately see that certain basic propositions are valid. In Kantian terms, Wolff thnks that axioms are analytic.

Wolff's suggestion appears rather outdated for anyone who is acquainted with how axiomatics are dealt in modern mathematics: axioms are chosen not because we would be somehow immediately certain of their truth, but because we are interested in finding out what propositions follow from these axioms. Indeed, when Wolff suggests as an axiom that between two points we can draw only one straight line, it is difficult to see how this could be decided merely from Wolff's definition of straight line: straight line is a line where a part of the line is similar to the whole.

Actually one might criticise even Wolff's definition: how can he be sure that only the supposed straight lines are such that their parts are similar to the whole? Indeed, we might well imagine a sort of fractal line where the details of the whole would be repeated ad infinitum in smaller and smaller pieces of the whole. Yet, we should not be too harsh on Wolff, who surely could not have been aware of the exotic geometric shapes in modern mathematics. Furthermore, Wolff's definition is probably derived from tradition, because it appears to be an improved modification of Euclid's definition of straight line as a line which lies evenly with the points on itself.

Even if we accepted Wolff's definition, it would appear that his axiom presupposes characteristics of space that are not included in the definition of straight line. In Kantian terms we would say that Wolff's axiom is actually synthetic, and indeed, this is what Kant said of a related proposition that a straight line is the shortest line connecting two points. In other words, geometry requires according to Kant something more than mere conceptual analysis.

Although Kant's viewpoint on the axioms of geometry seems a clear improvement on Wolff's, it is interesting to note that on Hegel's opinion the very axiom on straight lines that Kant had considered is actually analytic. This does not have to mean that Hegel would have returned to Wolffian idea of axiomatics. To see this, we must first consider why Kant would have called the axiom synthetic. He probably had an independent notion of a straight line, perhaps something similar to Wolff's definition.Then Kant just needed to notice that this definition did not imply the axiom, and the syntheticity of the axiom became evident.

Hegel, on the other hand, apparently does not begin from an independent definition of a straight line. True, he begins his justification of Kant's axiom from the idea that a straight line is the simplest of all lines, but here this simplicity indicates not so much a characteristic, but a search for a characteristic. The most basic type of line is to be called a straight line and we need to choose what sort of line are we to take as the most basic. Kant's supposed axiom is then just the required definition of a straight line: the minimality of the length of a line is chosen as a criterion for its simplicity.

Understood as a definition, Kant's supposed axiom is undoubtedly analytic: being a shortest line between two points is just what being a straight line means. The analyticity of the axiom would still not undermine the syntheticity of geometry. We would still have to take as an axiom, firstly, that there is something corresponding to this definition – that there is no infinite series of shorter and shorter lines connecting two points – and secondly, that a line determined by two points is unique – that there are not more than one lines between two points equalling the lower limit of all lines connecting the two points (note that elliptical geometry does not satisfy the latter condition: for instance, there are infinite numbers of equally short routes from the North to South Pole on the surface of the Earth).

Interestingly, if we accept Kant's axiom as the definition of straight line, as Hegel suggests, Wolff's axiom becomes in a sense also analytic: indeed, there can be only one unique shortest line connecting two points. That is, if there just is any shortest line: as we have seen, this is not guaranteed by the definition.

We are still not completely through comparing Wolff's idea of mathematics with the ideas of later philosophers. Next time, we shall see whether Hegel's criticism of Wolff's notion of differential calculus was justified.

Christian Wolff: Elements of all mathematical sciences (1710)

The mathematical magnum opus of Christian Wolff is undoubtedly his Der Anfangs-Gründe aller mathemathischen Wissenschaften, which covers in four volumes areas such as arithmetics, astronomy, hydraulics and algebra. The book was apparently heavily used in teaching, because several editions were printed: the one I read hails from 1738. In addition to the book itself, Wolff also wrote a supplement with trigonometric tables, a shortened version and a Latin translation. Apparently he had already realised that the true source of wealth for academics are good text books.

The book is also famous as the target of Hegel's criticism in his Wissenschaft der Logik, where Hegel makes fun of how Wolff tries to formalise mere handicrafts like constructing a window. Indeed, it is rather peculiar to find in the section on architecture definitions of windows and doors (window, for instance, is an opening in the wall from which light can fall into the building). Even more disturbing is the section on artillery, where Wolff defines such things as bombs, hand grenades and mines (e.g. hand grenade is a bomb small enough to be thrown).

As is already apparent, the book would nowadays be read by future engineers and not by future mathematicians. Even in the case of arithmetics and geometry Wolff is more interested of the practical capacities implicit in these sciences: arithmetics, for instance, Wolff says to be of use in counting money. Indeed, the sciences dealt with are as mathematical as the method of Spinoza in Ethics is geometrical. Nowadays we would say that both Spinoza and Wolff applied or at least attempted to apply the so-called axiomatic method. Wolff's book even contains a description of mathematical or axiomatic method, which is undoubtedly the most interesting part of the book for a modern philosopher.

The basic idea of the axiomatic method is that certain propositions are to be developed from other propositions of a more foundational level. Some of these propositions merely tell us what things are or define them, while others or axioms spell out some further characteristics of things. I shall now concentrate on what Wolff has to say about definitions and leave the axioms to my next writing.

It has been customary to separate mere verbal or nominal definitions, which merely explain how certain words are to be used, from real definitions describing the essence of the defined thing. Although the distinction seems clear, it is somewhat difficult to determine when we have truly defined the essence of a thing or even whether thungs have any essences.

Wolff accepts the traditional notion of a nominal definition, but his idea of a real definition seems more creative. Real definition, for Wolff, tells how a thing is generated. Although in one example – that of a solar eclipse – Wolff clearly refers to general causes of a phenomenon, in most cases the generation refers to an actual production of a thing. Thus, Wolff suggests that a real definition of a circle describes how a circle is drawn.

Wolff has replaced the question about essences of things with a more pragmatic aim of extending our abilities to do things. This is clearly in line with Wolff's rather pragmatic attitude towards sciences. Indeed, many of the theorems in Anfangs-Gründe are just such real definitions: we are first told the nominal definition of a window, but only later on do we learn how to build windows. Although German idealists work on a more refined level than at mere handicrafts, one might think that Wolff's idea is a sort of precursor of the later idea of the primacy of practical reason.

According to Wolff, sciences are then more about how to do things than about what things are, that is, sciences are intrinsically connected or even identifiable with technologies. The most interesting consequences of this connection arise in the realm of pure mathematics. Notably, Wolff defines arithmetics not as a science of numerical relationships, but of methods for discovering numbers: 5 + 7 = 12 does not represent a relation between certain objects or sets of objects, but tells that by adding five to seven one gets a new number, that is, twelve.

The difference between the two interpretations of arithmetics is not so obvious when we remain at the level of finite set of numbers, because within a finite set we could at least in principle check whether the set contains a number having a desired property. But in the case of infinite sets the difference becomes more obvious. For instance, it is not at all obvious that we can find solutions to some difficult equations and we cannot even go through all natural numbers to find out if they do have.

This difference actually is what separates the so-called Platonist and constructivist interpretations of mathematics: the former focuses on the mathematical relations and supposes that these relations do exist, even if we cannot know anything about them, while the latter focuses on the actual methods of solving mathematical problems and regards it meaningless to discuss whether these problems would have solutions without any method of solving them. Wolff is unconsciously heading towards the constructivist option, which is interesting, because I think that some of the German idealists have affinities with a constructivist reading of mathematics.

So much for definitions,next time I shall be looking at what Wolff says about axioms, postulates, theorems and problems.