we see the picture of an endless maze of connections and interactions, in which nothing remains what, where and as it was, but everything moves, changes, comes into being and passes away. ([#!Eng76!#] page 24)In order to try to make sense of this, the first task is that of analysis -- the separation of the different components, their isolation in the laboratory, and the detailed study of one phenomenon while ignoring all others. That is the great achievement of natural science. Mathematics and formal logic are the highest abstractions of this process.
So, in the foundations of mathematics the basic elements reflect this analytic approach. In classical logic a proposition p possesses a definite truth value -- true or false, there is never any doubt about it. And once p is made true (or false) it is fixed for all time. Similarly in set theory, if we have a set S, we can always tell whether an object x belongs to S or not and the membership of S is fixed.
But consider (as Engels does) the set of all living things. For everyday purposes (leaving aside the house of Lords) it is perfectly clear what this means, but there is a problem in accurately defining the boundary. At exactly what stage can we say that a complex protein is not just a chemical but alive? When do we say that a person is dead -- when the heart stops, when there is brain-death, when rigor mortis sets in or what? Of course we can make a definition of `being alive' but the definition will be arbitrary to some extent. Secondly, every living thing embodies a struggle for existence. A living organism may be in transition to death. The true proposition p is becoming false. So the set of living things is not fixed, its membership is constantly changing. Furthermore, as medical science progresses even the definitions and boundaries will evolve and get extended.
Set theory and classical logic thus fail to capture this process of transformation. So, it must be borne in mind that the properties and laws discovered by this analytic method are not absolute and final, but contingent on the very narrow conditions that were imposed on the experiment. The second task, therefore, is that of synthesis. Here it is the relations between things which concern us, the way in which a particular event fits into the totality. A property is no longer considered as definitely true or false, but dependent on other factors and therefore in a state of change. The abstraction of this wider view is dialectics.
It would be a mistake to oversimplify this point. It is not true that mathematics can only deal with static or deterministic events. The differential calculus, for example, is devoted to the study of change, while probability theory allows the handling of indeterminacy. Mathematics starts off as a rigid, static discipline but finds that this is inadequate to deal with a range of phenomena. The subject is forced to expand in order to deal with this, but it is not achieved without difficulty -- it took 200 years before the methods used in the calculus were given a satisfactory justification.
But even at its most advanced level, the understanding of the world through mathematics involves the reduction of some process to a formal string of symbols (a formula). This might be a differential equation (for processes which change in time) or a stochastic matrix (for indeterminate processes) but the history, or the probability distribution, is fixed for all time by a single formula. This inevitably leads to problems when a qualitatively new process evolves. So in economics, for example, quite sophisticated models using differential equations and probabilities can take into account as many factors as you like and are run through a computer. The economists are always surprised when their predictions fail to materialise as the system moves into a new phase, governed by new laws. No doubt mathematics will evolve to take more of this type of development into account, but it always involves the reduction from the living world to a dead formula.
Dialectics provides the philosophical framework for reasoning about change and interaction and the process of synthesis.