Friedrich Eduard Beneke: New foundation for metaphysics (1822)

Beneke’s Neue Grundlegung zur Metaphysik is a programmatic declaration of the possibility of metaphysics. Beneke does not give a definite explanation of metaphysics, but he evidently contrasts it with a mere study of human consciousness - metaphysics, in comparison, deals with being and not mere thinking.

Problem of justifying the possibility metaphysics means then for Beneke a problem for showing that we can think of something we know to exist. Beneke’s not that original solution is to note that when we think of our very activities of thinking, we necessarily think of something that exists, because self-evidently our own activities of thinking must exist, whenever we happen to think them.

Beneke’s aim in pointing out this rather clear-looking fact is to argue against a position he calls strong idealism.This strong idealism is apparently supposed to endorse the notion that we can never connect thinking with being, but are always closed within the circle of mere representations of being. It is not at first clear why Beneke would call such a position idealism, but he is at least thinking about Kant’s notion of inner sense, which he explicitly denies.

It might be that Beneke and Kant are speaking of two different things. When talking about inner sense, Kant wants to emphasise that we cannot know even ourselves from a perspective reaching over our experience and that we therefore cannot know e.g. whether we are immortal and immaterial entities or material objects. Beneke accepts this, but insists that we still can know something about our mental life, namely, its existence as such, and even more, what this mental life feels to us. Of course, Kant might object that none of this is enough for knowledge, but we might well ask whether Kant just has a too tight definition of knowledge.

Although Beneke speaks against idealism, he is also willing to speak against a crude sort of realism that would state that we have as immediate connection with other things as with ourselves. Beneke still admits, against what he calls weak idealism, that we can know the existence of other things, although only mediately. The link between our immediate knowledge of ourselves and mediate knowledge of other things is, according to Beneke, our knowledge of our own bodies - we know our bodies just as we know our mental life, but our bodies are also causally connected to other bodies (Beneke deals here briefly with the Humean problem of how we recognise causality, and like in his earlier work, he insists that we can through an indefinite number of repeats become more and more certain of the existence of such a causal connection).

Although the lens of the body allows us to recognise the existence of other things, Beneke says, it also restricts our knowledge to a certain perspective. In other words, Beneke is of the opinion that we know other things more completely, if they happen to resemble ourselves. Even things quite removed from us, such as mere material objects with no life or consciousness, can be known only through the lens of our own bodily feelings - for instance, we can understand the physical movement of objects through our own experiences of moving through space.

Augustin-Louis Cauchy: Course of analysis (1821)

1789-1857

Analyzing a rich text like Cauchy’s Cours d'Analyse is almost impossible. Thus, I shall instead concentrate, firstly, on general features of the book, and secondly, on some interesting peculiarities of the work.

The title of the book will probably not say much to anyone not experienced with what is nowadays called mathematical analysis - and indeed, it is not clear whether Cauchy’s intentions coincide exactly with this modern notion. Hence, instead of trying to define analysis at this context, I shall merely point out what appears to be the topic of this book, namely, functions.

When Cauchy speaks of functions, he seems to be generalising from individual mathematical operations, such as addition, multiplication, logarithm and sine - in a quite general manner we could say functions involve taking number or numbers and using them to calculate or determine other numbers. Unlike in the current notion of function, Cauchy allows that functions might have several alternative results. For instance, Cauchy notes that although we usually restrict the notion of square root to positive square roots, we might as well take also a negative square root as being one possible result of applying the function of square root.

We can point out two major questions involving functions that interest Cauchy. Firstly, Cauchy, like other mathematicians of the time, is interested of the general question of finding what could be called roots of a function. The function in question will take one number and produce another number as its result, and the root of this function is then a number that when applied to the function will produce zero as a result.

Secondly, Cauchy is interested of what happens when functions are applied to infinitely large or infinitely small quantities. These terms are a remnant from an earlier period of mathematics, when the unclear notion of an infinitely small number was used in making sense of results in differential calculus. With Cauchy, these notions are inevitably connected with the concept of a variable: when we speak of an infinitely great number, we mean a quantity that is meant to change its value by becoming larger and larger and eventually exceeding every finite number. Similarly, an infinitely small number is a quantity that is meant to become smaller and smaller, eventually diminishing beyond all positive numbers, without ever reaching zero.

Now, with the aid of the notion of infinitely small or infinitesimal number, Cauchy defines the concept of a continuity of function. His notion of continuity has something to do with a formula f(x + a) - f(x), where f refers to a function of some type, what is in () is the number applied to the function f, and a is an infinitesimal number. In effect, Cauchy wants us to think of the area where the results of a function lie, when the numbers applied to the function become less and less varied. If the function is continuous around certain number, Cauchy says, this means that the area of results will also be infinitesimal - in other words, the less variation we have with numbers applied to function, the less variation we have with the results.

Although Cauchy’s definition can be used with all numbers applied to functions, he is especially interested of cases where the numbers themselves are either infinitely small or infinitely great, that is, when we think of them as variables decreasing toward zero or increasing without any limit. This operation of finding a limit for a function appears to be yet another function, and just like with all the other functions, Cauchy accepts the possibility of a function having more than one result. For instance, the values of sine function vary constantly between -1 and 1, thus, when the numbers applied to the function increase indefinitely large, the limit of this function, Cauchy says, consists of all the numbers between -1 and 1.

While differential calculus had been the original spur for mathematician’s developing the notions of continuity and limit of a function, for Cauchy this is already just a one possible application of these notions, while other applications, such as the question of an unending series of sums of numbers, might be even more important.

An important application of the notions concerns the other interest mentioned earlier, that is, the question of roots of a function. Cauchy notes that if a function is continuous, whenever numbers are taken from some continuous part of number line, and the same function has a positive result with some number from that part and a negative result with another number from the same part, then the function has a root somewhere between those numbers. This sounds evident, but it requires some careful thinking to actually prove it. In effect, Cauchy notes that due to the continuity of the function in this area, we can find between the numbers described both numbers giving negative and numbers giving positive results, as close to one another as we like. These two series of numbers must have the same limit, which then can have neither positive nor negative result, that is, it must be the root of the function.

Another major theme Cauchy deals with in his book is imaginary - or as we would nowadays say, complex - numbers. Discussions thus far have been grounded in actual numeric operations that make some concrete sense. Thus, negative numbers can be understood as a simple way to speak about operations of subtraction (i.e. - a is just a summarised form of saying “subtract a”), raising number to a fractional expression 1/a is another way of saying that we are taking the ath root of that number, and any difficult calculation involving numbers expressible only as limits of certain number series (such as raising a number to pith power) refer to limits of functions, when applying them to numbers from that series.

Now, when Cauchy starts to speak of square roots of negative numbers, he doesn't really mention any means to make such a mathematical formula sensible (he has the means by which it could be done in his use, but that’s another story). Cauchy then accepts that what mathematicians are speaking about when dealing with square roots of negative numbers is just imaginary of fictional - the signs apparently referring to such roots mean nothing. Still, majority of the operations used in the context of real numbers work as well in the context of these imaginary numbers. Since these imaginary numbers can be used as tools for finding meaningful results to questions involving just real numbers, the interest for this part of the mathematical analysis is then also justified.