Marxists seem to stress the importance of analysis based on facts which are obtained from our sense organs and claim to follow the "scientific" method with everything from the study of history, the study of society to theorising about revolution. Anarchist authors do not seem to have the same tendency to proclaim everything they say to be scientific.
Marxists' insistence of claiming their theory to be scientific however is in contradiction with Marx's analysis of history and the economy as being based on social relations, which, by definition, do not conform to any scientific patterns. From my understanding, it is the positivistic influence on the the largely reformist and evolutionary Second International, which existed in the period of the ascendant bourgeoisie, which is to blame for this kind of theorising on the parts of Marxists.
Also, from wikipedia:
Comte offered an account of social evolution, proposing that society undergoes three phases in its quest for the truth according to a general 'law of three stages'. The idea bears some similarity to Marx's view that human society would progress toward a communist peak. This is perhaps unsurprising as both were profoundly influenced by the early Utopian socialist, Henri de Saint-Simon, who was at one time Comte's mentor. Both Comte and Marx intended to develop secular-scientific ideologies in the wake of European secularisation.
Is my understanding correct? What is the real extent of positivist influence on anarchism/marxism? Are there any cogent critiques of positivism from anarchists or Marxists?
working class, I'll leave the Anarchists on this site to answer for themselves, but as a Marxist I can give you a partial answer.
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I think a real problem here, working class, is that you seem yourself to see 'science' as something positivistic. I think philosophers of science now accept that 'science' isn't an "analysis based on facts which are obtained from our sense organs", and that "social relations" are an integral part of 'science'.
So, it is possible to see "Marx's analysis of history and the economy as being based on social relations" as entirely 'scientific'.
In some sense, the real question now is not 'is Marxism a science' but rather 'is science a Marxism'?
For example, Einstein famously said, "it's the theory that determines what you can observe" [think 'Higgs Boson'], and as 'theories' are human, and humans are social, and current society is a class society, it seems reasonable to conclude that 'science' is not 'science in the 19th century positivistic sense', but that 'science' is a social activity, containing real social divisions.
So it is your statement that "social relations, which, by definition, do not conform to any scientific patterns" which is scientifically incorrect. It's your 'definition' which is at question. 'Social relations' can be examined scientifically.
Well, almost every philosopher of science in the 20th century criticised 'positivism', so you don't even have to be a Marxist to provide a 'cogent critique'. Popper, Kuhn and Lakatos spring to mind.
FWIW, if you regard 'Marxism' as a Lakatosian 'research programme', you won't go far wrong.
Perhaps a starting point would be to define who you mean by 'Marxists'. Are you asking questions about the thinking of 'late 19th century Marxists' (more a historical issue) or 'Marxists in early 21st century' (more a political issue)? I would regard these as separate 'research programmes'.