Of Alterations: Leibniz, Frege, Wittgenstein

“The cottage fades before his sight,
The garden and its lovely charms;
The guests are scattered through the land
—For the eye altering, alters all—;
The senses roll themselves in fear,
And the flat earth becomes a ball.”

~William Blake “The Mental Traveller”

The similarities between Liebniz, Frege, and Wittgenstein really are uncanny. It should, of course, come as no surprise to anyone that Frege knew some of Leibniz since he mentions his philosophical ancestor multiple times over his career, but the relationship goes much deeper, I think, than the offhand quotation. It is also clear that Wittgenstein had access to Leibniz’s work in the form of Russell’s 1900 publication of A Critical Exposition of the Philosophy of Leibniz, as well as the likely exposure to Louis Couturat’s 1901 publication La Logique de Leibniz. (Whitehead mentions in his autobiographical notes that all of his own interaction with Leibniz came from Couturat, and that Russell knew of Couturat as well.) Even the discursive reader can see that the projects are all linguistic at their cores, that each of the projects is an exercise in logic, and that all of them stand firmly against psychologistic conceptions of logic. While all of this is true, I certainly don’t believe its the end of the story.

This post has two purposes. First, I want to show that Frege and Leibniz and Wittgenstein all have the same worry with clarity in the language amongst the three of them. Second, I want to show how they each stand differently oriented on that issue. I will not go into any real depth. Depth is for another time.

In his Dialogue on the Connection Between Things and Words, Leibniz states the following:

For even though characters [words] are as such arbitrary, there is still in their application and connection something valid which is not arbitrary; namely a relationship which exists between them and things, and consequently, definite relations among all the different characters used to express the same things. And this relationship, this connection is the foundation of truth.

Later,

[...] truth is not based on what is arbitrary in characters but on what is permanent in them: namely, the relationship which marks among themselves have to things.

Don’t jump ahead. Ill mention Ludwig in a moment. Leibniz turns at this point to mathematics, and in his Preface to the General Science points out that in mathematics all signs are clearly determined and that it is this clarity which allows for methodological demonstration of our conclusions. He then states, turning towards the language and reason, that

Whence it is made manifest that if we could find characters or signs appropriate for expressing all our thoughts as definitely and as exactly as arithmetic expresses numbers or geometric analysis expresses lines, we could in all subjects in so far as they are amenable to reasoning accomplish what is done in Arithmetic and Geometry.

From this, it seems clear that Leibniz has come across the problems of clarity and definition in the language which results from the improper or unclear use of signs. His method of clarification, however, is to replace the common signs with new ones which are not arbitrary and are always clear. This would be the “alphabet of thought,” a set of basic concepts which would allow for rational calculus.

Compare this to Frege’s introduction to Begriffsschrift. Frege begins with a picture of common scientific deliberation and comes to the point that “the most reliable way of carrying out a proof, obviously, is to follow pure logic.” He then wonders whether or not the classification of Arithmetic falls under the purely logical, or if it admits of experiential justification. At this point Frege tells the following story:

My initial step was to attempt to reduce the concept of ordering in a sequence to that of logical consequence, so as to proceed from there to the concept of number. To prevent anything intuitive from penetrating here unnoticed, I had to bend every effort to keep the chain of inferences free of gaps. In attempting to comply with this requirement in the strictest possible way I found the inadequacy of language to be an obstacle; no matter how unwieldy the expressions I was ready to accept, I was less and less able, as the relations became more and more complex, to attain the precision that my purpose required. This deficiency led me to the idea of the present ideography.

(Frege maintains that he is working solely on the level of conceptual content soon after this.) What is striking in the comparison between Frege and Leibniz here is the lack of clarity, the way that language can trip up our efforts if we don’t pay attention to it. Both Frege and Leibniz want to address this problem with their scripts —systems of logical symbolism. Frege’s discussion of Leibniz in the introduction is also helpful. “Leibniz, too,” he says, “recognized —and perhaps overrated— the advantages of an adequate system of notation.” He later states that “It is possible to view the signs of arithmetic, geometry, and chemistry as realizations, for specific fields, of Leibniz’s idea. The Ideography here [Frege's] adds a new one to these fields indeed the central one, which borders on all the others.” Frege, too, sees that he is bound up with the Leibnizian problematic.

It is also important to note that Frege sees himself as limiting Leibniz’s endeavor. His script is like Leibniz’s in that it is a “device invented for scientific purposes,” and as such is an attempt to improve the methods of truth valuation. This is where he says Leibniz also saw the need for “adequate notation.” But in Leibniz’s case,

His idea of a universal characteristic, of a calculus philosophicus or raciocinator, was so gigantic that the attempt to realize it could not go beyond the bare preliminaries. The enthusiasm that seized its originator when he contemplated the immense increase in the intellectual power of mankind that a system of notation directly appropriate to objects themselves would bring about led him to underestimate the difficulties that stand in the way of such an enterprise.

It was for this reason, Frege says, that his script is a way of temporarily restricting the problem in order to deal with it piece by piece. Frege’s notation is only supposed to differ in scope from Leibniz’s idea.

Now, lets turn to Wittgenstein. In reference to Leibniz’s Dialogue on the Connection Between Things and Words, we can turn to a number of moments in the Tractatus, the Blue and Brown Books, and PI:

Tractatus:

2.2: The picture has the logical form of representation in common with what it pictures.

3.322: It can never indicate the common characteristic of two objects that we symbolize them with the same signs but by different methods of symbolizing. For the sign is arbitrary. We could therefore equally well choose two different signs and where then would be what was common in the symbolization.

3.323: In the language of everyday life it very often happens that the same word signifies in two different ways —and therefore belongs to two different symbols— or that two words, which signify in different ways, are apparently applied the same way in the proposition. [...]

In 3.324 and 3.325 Wittgenstein says that errors of this kind are the source of fundamental confusions, and that Frege and Russell didn’t expel all the possibilities of such error in their symbolism. (Note that his example of just such a confusion is Russell’s Paradox.)

Here’s the Blue Book:

Without a sense, or without the thought, a proposition would be an utterly dead and trivial thing. And further it seems clear that no adding of inorganic signs can make the proposition live. And the conclusion which one draws from this is that what must be added to the dead signs in order to make a live proposition is something immaterial, with properties different from all mere signs.

But if we had to name anything which is the life of the sign, we should have to say that it was its use.

For the sake of time, I will mention one in Investigations as well:

(From 38) This odd conception springs from a tendency to sublimate the logic of our language —as one might put it. the proper answer to it is: we call very different things “names”; the word “name” serves to characterize many different, variously related, kinds of use of a word —but the kind of use that the word “this” has is not among them.

Of course, we should call up PI 43 here as well, in light of what was said in the Blue book. All together now:

For a large class of cases—though not for all—in which we employ the word ‘meaning’ it can be defined thus: the meaning of a word is its use in the language.

Good. That was fun.

Here’s the relation of all of these quotations between Leibniz, Frege, and Wittgenstein: The signs alone are arbitrary, limp, dead, unused. Together, in a system, they are meaningful and can express truth value. Moreover, meaning is not psychological, all three philosophers deny any such claim. However, if proper attention is not payed to how the signs relate to each other in a system, a lack of clarity will arise. That the signs remain arbitrary is one such area of possible confusion, and each philosopher treats this last issue in a different way. Leibniz wanted universality by means of the signs never being arbitrary but purely representational, like numbers in mathematics. Frege employed quantification theory in order to remove the lack of clarity unconnected signs may have, thus making his system articulate. Wittgenstein shows that the signs never stand alone since they are always and everywhere used, and since this is the case the meanings are logical ones even in the ordinary language. (Frege’s context principle hovers over everything in Wittgenstein.)

Here is a final thought on the difference between these three. Frege says in Begriffsschrift that he “can best make the relation of [his] ideography to ordinary language clear if I compare it to that which the microscope has to the eye.” His system looks down to the language. Leibniz says in 1677 that his language would be an “instrument which will exalt reason no less than the telescope does to perfect our vision.” His system creates a new perfect language and looks upward from that creation. Wittgenstein gives us the picture theory of language, he doesn’t give us a camera. The picture is already there, its the ordinary language.

Keep in mind this post is not exhaustive and I do not mean it to be.

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