Gödel, Marx, Godel, Hegel, Goedel, foundations, mathematics, incompleteness theorem, philosophy, theorem, incompleteness

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Marx, Hegel and Goedel's theorem

By K. Podnieks

Creative Commons License This work is licensed under a Creative Commons License and is copyrighted © 1999 by  me, Karlis Podnieks.

Hegel died in 1831, Marx - in 1883. Would they be pleased reading Goedel's paper in 1931?

In the former Soviet Union, all students and doctorands were forced to study Marxist philosophy (the disciplines were called "Dialectical materialism" - the general philosophy, and "Historical materialism" - the social philosophy). So, I had the "opportunity" of studying Marxism for quite a long time - from 1968 to 1975.

Of course, as a politicized discipline taught under a totalitarian system, you cannot take Marxist philosophy 100% seriously. It contains interesting ideas mixed up with opinions of people who were simply defined as experts (or even "classics") by the system.

But let us drop the later add-ons due to Lenin and stalinists, and let us drop some naive opinions of Engels, then, what kind of interesting ideas will remain of the Marxist philosophy?

The "dialectical materialism" is based on two main ideas - materialism and dialectics.

Materialism

Materialism as an idea (not as a system!) simply recommends to think that I am living in an environment that exists independently of me and my ideas about it. Let us call this environment "reality". This is a very useful and very practical hypothesis that allows to bring into order my mental world. Do you know a more useful and more practical idea?

If you were programming a robot, you would be forced to think about your robot and its environment. The environment exists "as it is", it does not depend on the robot. Hence, you will be forced to store some model (or theory) of this environment in the robot's memory. I am a kind of robot (a very good one!).

But, of course, materialism is not the only useful idea. Surprisingly, even the directly opposite idea - "doubting reality" (Marxists call it "subjective idealism") appears to be very fruitful. For example, let us recall the exciting history of Hume & Kant -> Mach -> Einstein that resulted in the relativity theory. Dialectics?

Dialectics

The second idea - dialectics - Marx adopted from Hegel's thesis, antithesis and synthesis. Marxist dialectics is a kind of heuristics. It recommends a specific (or universal?) way of solving problems: always search for contradictions and opposing forces involved. Marxists derive this heuristics from the following "ontological" hypothesis: no fixed complicated schema or system can exist unchanged for a long time. Someday, inevitably, it will be blown up by its inherent opposing forces.

Goedel's incompleteness theorem seems to be an evidence in favor of this hypothesis. Indeed, as a pure mathematical result, this theorem says that no formal theory can be simultaneously powerful, consistent, and complete. Are formal theories an adequate model of fixed, self-contained systems of reasoning? If yes, we arrive to the following "dialectical" conclusion: no fixed, self-contained system of reasoning can be 100% perfect. Such systems are inevitably either very restricted in power (i.e. they cannot express the notion of natural numbers with induction principle), or they are powerful enough, but then they lead inevitably either to contradictions, or to undecidable propositions.

Hegel died in 1831, Marx - in 1883. They would be pleased reading Goedel's paper in 1931, oder?

April 18, 1999

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Gödel, Marx, Godel, Hegel, Goedel, foundations, mathematics, incompleteness theorem, philosophy, theorem, incompleteness