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Freud broaches the idea of the “death instinct” in the following quotation from section V: “At this point we cannot escape a suspicion that we may have come upon the track of a universal attribute that has not hitherto been clearly recognized or at least not explicitly stressed. It seems, then, that an instinct is an urge inherent in organic life to restore an earlier state of things which the living entity has been obliged to abandon under the pressure of external disturbing forces; that is, it is a kind of organic elasticity, or, to put it another way, the expression of the inertia inherent in organic life” (36). The idea here is that organic life seeks conservation and/or to avoid stimulus and change. This goes counter to most theory that states organic entities seek evolution, development, and change. This idea of conservation, argues Freud, is what leads to repetition.
Section VI of Beyond the Pleasure Principle opens with the following line: “A sharp distinction between the ‘ego-instincts’ and the sexual instincts, and the view that the former exercise pressure towards death and the latter towards a prolongation of life” (44). Freud spends much time trying to explore his point using biology, germ cells, and other such things (I have read that much of the "science" Freud employs in this part of his text has since been disproven). Another fundamental point of BTPP is that Freud works to set himself apart from Jung by claiming that his theory of the pleasure principe in conflict with the death instinct is dualistic and Jung’s idea of libido is a monistic theory (I guess the point here is that a two tiered Freud trumps a single tiered Jung in explaining the workings of the human unconscious, or to put it another way, "mine's bigger than yours").
Freud concludes Beyond the Pleasure Principle by saying that he is not sure if he believes the conclusions he has arrived at in the text. He says he feels more comfortable with the first two steps of his major theories (1. the idea of sexuality and 2. the development of narcissism) than with the third, the death instinct that is discussed in the essay.