Lukacs argued that there is dialectic in history but not in nature, he subsumed the catergory of nature under history and criticized the extension of the dialectic to nature (I think Lukacs's target was not Engels because Dialectics of Nature was published after History & Class Consciousness, if my memory was correct). ... The 3nd point was then and now an interesting question/puzzle to me: does Marxism need a dialectic of nature to justify the validity of historical materialism or to make Marxism 'Scientific'? My knowledge in science did not go beyond high school elementary level so I'm not in a position to say yes or no on Engels's position, not to mention whether modern science has confirmed a dialectic in nature or not. But it seems that if the three laws of dialectic: quantity to quality, negation of the negation etc. were laws in the sense of, say, Newton's universal gravitation or Law of Motion, then the implications are determinism as well as the 'algebra of revolution'.
Furthermore, how could these laws enable us to understand concrete history event: e.g. European workers and their Social Democratic Parties, contrary to all expectations, voted in support to go for war in 1914. Commentators used to say a dialectic of nature is just another version of Schelling and Hegel's futile philosophy of nature. In this respect I think Lukacs is correct in insisting on a dialectic of revolution and rejecting a dialectic of nature.
But you have a much stronger scientific background, so if you have the time perhaps you could give an explication on the dialectic of nature and its relation to Marxism and some related issues (of course, in layman's terms), other members may be also be interested.
Julio Huato: Is math another CONCRETE science, in the sense of being concerned with some aspects of REALITY (space forms and quantity relations) to which it constantly has to refer for verification? Or, is it, as most mathematicians would have it, a FORMAL science, in the sense of being EXCLUSIVELY concerned with its own internal logical consistency once a few universal definitions and axioms are laid out and accepted as "valid" (not necessarily as "objective")? What's the essence of mathematics, the objectivity of its definitions, axioms, and theorems or its mode of reasoning whatever the relation of such definitions, axioms, and theorems to the objective world? Why?
... how do you test a mathematical theorem directly against empirical reality? Moreover, how do you test axioms and definitions? What kind of practice validates a mathematical definition, axiom, or theorem?
Flora Tristan: the dialectic does not exist outside human consciousness i.e. it is a method we humans can use to analyze our condition and the condition our condition is within. Thus, we can observe that after a certain quantity of heat is applied to water, it changes in quality from liquid to steam.
Dialectics of Nature
A response to Alex, Julio and "Flora"
It seems to me that anyone who recognises that dialectical forms are manifested in history and asserts that such laws are not manifested in other aspects of nature, has the onus of showing why this is so. It is easy to see why the principle of division of powers between executive and legislative bodies would be manifested only in human society, or the form of value of commodities, or Romanticism - but dialectics constitutes a category of logic, an extremely general concept, whose specificity to social relations totally escapes me.
Let us leave dialectical logic aside for the moment, and just consider formal logic. If we can prove the case for formal logic, then hopefully nothing more need be said for the more general, more adequate logic of dialectics. Further, this may help us in answering Julio's question, since there is a very close relationship between formal logic and mathematics.
Lev Vygotsky and Jean Piaget have both shown, on the basis of extensive observation and experiment, that babies are born without any concept of logic or anything else, or any intuition, or capacity to coordinate their actions other than a capacity to accomodate their actions to the world acting on their senses and assimilate that action by extending it beyond the act of accomodation (which is initially confined to grasping and sucking reflexes). Both agree that primitive intuitions are formed through the interiorisation of the coordination of practical actions, i.e. actions which are already accomodating objective (natural or social) movements. Vygotsky, in extending the range of actual observation and experiment up to young adulthood was able to further show that concepts reach the young person through the action of society around them, are interiorised in an abstract way (accomodation). Theoretical thought becomes possible only when intuitions which exist only at the level of coordination of actions come into contact with socially acquired concepts which can then be assimilated.
Now Piaget makes the mistake of making no distinction between primitive intuitions and concepts but while that error leads him wildly astray when it comes to developing an epistemology, it does not detract from out point here.
Before we have the concept of logic, we have already learnt to coordinate our practical activity, such as movements of our hands and eyes, so as to accomodate ourselves to the objective world - whether that objective world be one dominated by human beings or lions and tigers or vegetables is neithe rhere nor there. The only point at issue is that it is an objective world, outside the child's mind (such as it can be said to exist at any particular stage in development).
Thus this faculty of accomodation to the objective world and extend this accomodation unilaterally so to speak, assimilate the properties of the objective world, (which is not yet consciousness) is a reflection of the objective world, of a specific animal character, which will develop further in a specifically human character.
For example, we learn to reverse a movement from left-to-right by means of an action of our arms from right-to-left, to put something back where it was. Now, we do not yet have the concept of the reversibility of displacements, but we have learn to reverse them of out own volition. Once we have learn to reverse lots and lots of different movements of the world and our own bodies, that is, we have interiorised this law of nature in the form of facets of the coordination of our sensori-motor activity (which is called "intuition"), when we much, much later come into contact with the concept of reversibility, and learn to operate with it, we can recognise it in our own action. (This example is of a very early intuition which appears in the development of mathematics very late as it consitutes a high level of abstraction - a fact which puzzles Piaget who is very proficient with "modern" mathematical logic, but ignorant of dialectics.)
We learn to sort and categorise things, to form sets and classes of things. We do this in the first place with our hands and eyes - sensori-motor activity. As we become proficient in doing this, which is an essential skill for the survival of a young human, and we develop what Piaget calls "sensori-motor intelligence". So long as we clearly recognise this as a unity of opposites (sensori-motor is NOT intelligence), this is a useful notion. Formal logic is nothing more nor less than the behaviour of sets of things.
The only qualification is that "formal logic" refers to a thought-object, a phenomenon of the subject, and obviously cannot exist other than in the mind of a thinking organism. However, the question is not whether logic can exist before or other than in the minds of thinking organisms, it is a question as to whether this thought object is true, or has truth.
Now I hope you can see the purpose of the psychological diversion from the epistemological question. People may have all sorts of ideas and concepts in their heads (the subject matter of psychology), but the essence of those thoughts, which we study in logic, is precisely that which gives us humans an adequate reflection of the objective world, which our practice proves to be an adequate reflection of the objective world.
Dialectical Logic is a more general, comprehensive development of logic than formal logic. Formal logic is based on the relation of identity and simplicity of concatenation. Piaget has shown quite materially how the intuition of identity develops oout of the flow of "one damn thing after another" in the mind of the child and upon that basis the intuitions which are reflected in formal logic.
Dialectical logic reflects relationships / movements at a deeper level, inasmuch as identity is but a relative concept; things do not remain the same, and this transformation of a thing into other than itself, out of itself, is the object of dialectics. Consequently, the grasping of dialectics in the form of concepts arises later in historical and cognitive development than formal logic. It describes how the formal logic which is adequate at one moment breaks down and re-emerges in the form of a formal logic reflecting different things at another.
This is somewhat like the more general fields of mathematics such as topology which are capable of representing whole areas of mathematics, different algebras, as values within its higher categories and comprehend the transformation of one into another - simpler, more comprehensive, but from the point of view of notion, more developed, higher.
We act dialectically all the time. It could not be otherwise. To be conscious of that, to have a concept of the dialectics of our own action, or any other action, is something else. The concepts of dialectics are very general concepts, just as rigorous as those of mathematics, but much more general; concepts which can be understood on the basis of practical, theoretical activity. It is enough to be able to follow a chain of reasoning, a chain in which every concept is well-defined and finite, to have the basis for learning formal logic. To understand dialectical logic, one must have experience in the transformation of one "paradigm" (to use the very apt term of Thomas Kuhn) into another, to have seen and participated in a "logic of events" and felt its power and necessity, and done so repeatedly in different circumstances, etc., etc., so as to be able to interiorise the concepts of dialectics and give meaning to the words one learns, if one is lucky enough to have read the right books, of dialectics.
Without the benefit of book-learning, such experience may lead to intuitive understanding, successful coordination of actions, but still lacking the capacity to reflect in conceptual form and therefore to consciously assimilate them and extend the coordination of actions beyond the moment. With book-learning, but without the necessary breadth of experience, knowledge of dialectics will be formal and immature and unnconnected with practical activity, not useful for the coordination of practical activity - unless you are a Hegel, and are privileged with being the one to elaborate those laws for the first time.
Just like formal logic.
It is a matter of unceasing wonder to me that at the time Hegel wrote the Science of Logic, The Origin of Species was still 50 years from publication and Lyell's science of the genesis of the Earth's crust 30 years away. At the state of knowledge at his time, the only thing which had a genesis was human history and individual organisms. I say "wonder", but of course, this is a phenomenon often repeated in history.
According to the unemployed civil engineer, Louis Althusser, the dialectic of Hegel described the Idea and that of Marx the "real world" - what a load of old cobblers, as the English would say. For Marx or Hegel, the concepts of dialectics are thought-forms which are adequate to thought-forms; for Hegel, nature (and society - of course!) express the dialectic, express the Idea, and have to be discovered within history; for Marx, the concepts of dialectics are very good, very concrete abstractions from the material world, knowledge of which is a step towards understanding the world more and more deeply. But the material world is inexhaustible, and Julio's points on that score are absolutely pertinent.
The development of mathematics as a science, as an activity of human beings, is continually driven by the whole development of the material world. The development of mathematics could not be adequately comprehended solely on the basis of its own concepts. As a whole, it does receive both stimulus and checking from social practice.
But of course, its individual proofs, theorems, concepts, theories etc., can only beproved on the basis of its own methods and concepts not any other. (By the by, we must now exclude "and spatial relations" when talking of the subject matter of mathematics, just "quantity", and quantity in the strict (very broad) scientfic definition given by Hegel).
Mathematics is a science. That is to say it reflects an aspect of the truth of the objective world, that is, of the world outside the subject's consciousness. When a mathematician tests the validity of a new theorem she/he has thought of, it is NOT a matter of his/her Will whether that theorem turns out to be provable or not. It is an objective fact, and the mathematician seeks to establish that truth or not, objectively. His/her mathematical activity is the form of practice adequate to such testing. I do not think the word "empirical" is appropriate, just "objective" - the two words certainly do NOT have the same meaning.