The Uncertainty Principle of language

In quantum mechanics, if two operators do not commute, there exists an uncertainty relation between the physical quantities represented by the two operators. Consequently, the uncertainty principle in language can be expressed formally as follows.

Corollary 4   If $S_1$ and $S_2$ are two linguistic formulation operators which do not commute, we have

$$\left[ {S_1,S_2} \right]\equiv S_1S_2-S_2S_1=i\hbar I$$

where $I$ is the identical operator. As a consequence, we have

$$\Delta S_1\Delta S_2\ge {\hbar \over 2}$$

The non-commuteness of two linguistic formulation operators can be easily checked by the eigenbasis of the operators: If $S_1$ and $S_2$ have different eigenstates, they do not commute. Thus, the non-commuteness can be established on account of counter-examples. For instance, ``Taube'' in German and ``dove'' or ``pigeon'' in English indicates that there is an uncertainty relation between German and English^4.5.

There is another aspect of Uncertainty Principle as far as language understanding and language formulation is concerned. This can be derived directly from Postulate 5.

Corollary 5   If $U$ and $F$ are the two operators associated with language understanding and language formulation, we have

$$\Delta U\Delta F\ge {\hbar \over 2}$$

The implication of Corollary 5 is twofold. First, it is an Uncertainty Principle of interpersonal communication. It undermines the possibility of perfect communication. But this is not to say that there is no effective communication^4.6. In fact, the effectiveness can be understood in a stochastic sense and has to be established through engaged ``tuning up'' of the common language used by the participating parties. They can achieve this by negotiating the definitions in their vocabulary, for example. Secondly, Corollary 5 applies to the ``private'' language and thought of an individual as well. It therefore implies that the reality envisaged by an individual is intrinsically not perfect, for an individual has to memorize his/her eidetic experiences -- either by keeping them in the head or writing them down on paper -- for later perusal. In this sense, Corollary 5 can be interpreted as an Uncertainty Principle of memory and of the eidetic experience the memory is supposed to represent.

Understanding the Uncertainty Principle in language does not imply that the apparent logic and stability of language will fall apart. Instead, it tells us in which situation a property is stable and can be counted on. For one thing, it implies that aggregate properties of language are quite predictable, albeit stochastic. Moreover, since the Uncertainty Principle is effective only if the eigenbases of two operators commute, one can ``tune'' the language so that the symbols in the language remain eigenstates for a long period of time (technically speaking, that is keeping the system in coherent). The language used in mathematics may be an example.

Indeed, mathematics is perhaps the best example in which the symbols of the language are well tuned. The eigenstates of a mathematical discourse are kept by writing them down on a blackboard or paper constantly. This process is the ``classicization of symbols.'' For the purpose of mathematical inference, these symbols are then constructed to form pure states and are subject to reversible logical inferences, after which the pure states remain pure. (This may be where an impression of mathematical objectivity can be built). According to quantum mechanics, the pure states of mathematical language are coherent states in the underlying representational language. So they can survive a long time without spreading out.