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#1: Socialism and National socialism:Since national socialism is a fascist ideology and marxism is the polar opposite of fascism then why do they both share the name 'socialism'?
In the interwar years in Germany, capitalism was unsurprisingly seen in a bad light. For what it's worth, the NSDAP did begin as a scatterbrained radical-populist group that felt that their brand of socialism was better suited to countering what they saw as "capitalism" as well Marxist socialism (Communists) and Reformist socialism (ie social democrats of that time).
The term socialism and its use by the NSDAP during their campaigning was their way of trying to get working class support and taking the support of their main opponents away. In some ways one could see this is a way of the NSDAP shoving it in
Ideologically there was a lot different. On a surface level some will point out welfare measures and economic regulation, which to some people is sufficient enough for "socialism" to be applied.
"National Socialism" took cues from corporatist approaches that Italy championed and it was incorporated into their system. There was a concept of class collaboration built along nationalism.
Social relations were not changed much from a capitalist society. Really an American or someone else from another developed nation probably would have not felt too alienated or out of place in Germany at the time. Some of the economic proposals were not too different from social democratic proposals, but they were directed for a different purpose than those people were. But in the end it was a capitalist society. Indeed along with Italy a number of Americans, ranging from big businessmen (Ford) down to small time demagogue tended to express admiration for these countries for being stable, ending labor danger, etc... I don't need to tell you that Marxism is built on a concept of the proletariat, working class, assuming control of the state through a revolution and working towards communism. National "socialism" is not that.
In public speeches early on they made use of it to get on the anti-capitalist mood, but in private Hitler expressed issues with the use of "socialism" in their party and for the most part they began phasing it out later on.
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#2: Left communism: What exactly are the beliefs of left communists and why do they oppose PSL members ( uhhh...marxist-leninist would be the correct term right?)?
Zan answered this well enough.
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#3: Utopian Socialism: What is this creation and does it factor into the modern world anymore?
Utopian Socialism, as the name implies, refers to forms of socialism which were not feasible. In the years before Marx and Engels formulated their critiques of Capitalism and their development of socialism, what we knew as socialism was typically cult-like in nature. Some early examples of so-called utopianists:
-Charles Fourier
-Robert Owens
-Saint-Simon
These men and their followers attempted to make communities founded on their principles. Taking a look at them they tended to appeal a lot to religious sensibilities of people. What they shared in common was almost immediately failing. The duo would also see it as socialism that was developed out of a crude interpretation of class.
Taking a look at some selections from Engels' Socialism: Utopian and Scientific...
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These were theoretical enunciations, corresponding with these revolutionary uprisings of a class not yet developed; in the 16th and 17th centuries, Utopian pictures of ideal social conditions; in the 18th century, actual communistic theories (Morelly and Mably)[2]. The demand for equality was no longer limited to political rights; it was extended also to the social conditions of individuals. It was not simply class privileges that were to be abolished, but class distinctions themselves. A Communism, ascetic, denouncing all the pleasures of life, Spartan, was the first form of the new teaching. Then came the three great Utopians: Saint-Simon, to whom the middle-class movement, side by side with the proletarian, still had a certain significance; Fourier; and Owen, who in the country where capitalist production was most developed, and under the influence of the antagonisms begotten of this, worked out his proposals for the removal of class distinction systematically and in direct relation to French materialism.
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This historical situation also dominated the founders of Socialism. To the crude conditions of capitalistic production and the crude class conditions correspond crude theories. The solution of the social problems, which as yet lay hidden in undeveloped economic conditions, the Utopians attempted to evolve out of the human brain. Society presented nothing but wrongs; to remove these was the task of reason. It was necessary, then, to discover a new and more perfect system of social order and to impose this upon society from without by propaganda, and, wherever it was possible, by the example of model experiments. These new social systems were foredoomed as Utopian; the more completely they were worked out in detail, the more they could not avoid drifting off into pure phantasies.
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The Utopians’ mode of thought has for a long time governed the Socialist ideas of the 19th century, and still governs some of them. Until very recently, all French and English Socialists did homage to it. The earlier German Communism, including that of Weitling, was of the same school. To all these, Socialism is the expression of absolute truth, reason and justice, and has only to be discovered to conquer all the world by virtue of its own power. And as an absolute truth is independent of time, space, and of the historical development of man, it is a mere accident when and where it is discovered. With all this, absolute truth, reason, and justice are different with the founder of each different school. And as each one’s special kind of absolute truth, reason, and justice is again conditioned by his subjective understanding, his conditions of existence, the measure of his knowledge and his intellectual training, there is no other ending possible in this conflict of absolute truths than that they shall be mutually exclusive of one another. Hence, from this nothing could come but a kind of eclectic, average Socialism, which, as a matter of fact, has up to the present time dominated the minds of most of the socialist workers in France and England. Hence, a mish-mash allowing of the most manifold shades of opinion: a mish-mash of such critical statements, economic theories, pictures of future society by the founders of different sects, as excite a minimum of opposition; a mish-mash which is the more easily brewed the more definite sharp edges of the individual constituents are rubbed down in the stream of debate, like rounded pebbles in a brook.
To make a science of Socialism, it had first to be placed upon a real basis.
If you read through the rest of it, they go into depth about Fourier, Saint-Simon, and Owen in particular, and admit they had a good idea of the ills of society. However what they proposed was a naive and often wildly imaginative ideas of socialism which Marx and Engels felt that socialists in their day should move beyond.
There were often communities set up like I mentioned earlier. The United States had their share of them too, all of them with the same fate. Nowadays, like it was in the past, it is employed as a derogatory term towards people who hold a crude understanding of socialism, or at least one that isn't really rooted in an analysis of the world than it is in more abstract elements.
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#4: Statism: This is what again? The belief that a socialist state should be run by a single supreme leader, or something else?
This is more of a derogatory term to be honest. In the most general term it means the use of a state to solve matters in society, but in a more negative connotation would refer to a dictatorial use of the state, which anarchists would say is inherent in anything which tries to use the state (and to that end they would say a true revolution would take down both the state and capitalism)