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Could you provide a critical assessment of the epistemologies of Positivism and Idealism, specifically strong points and weak points of each?

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To take this step by step, a critical assessment in this case, and in this limited format, would weigh the extent of the truth of a position and provide a detailed examination of the strong and weak points and a statement of how far the writer agrees with the propositions of the epistemologies assessed. Epistemology is defined, briefly, as the study of the origin, source, and scope (structure and limits) of knowledge and of the mode of justification of knowledge, whether it be internal or external to one's mind. Thus when you are asked to critically assess Positivism and Idealism, you are being asked to weigh the truth of the positions and to assess the strong and weak points of each with a view to assessing your level of agreement with these epistemologies. Let's start by briefly defining an overview of each epistemological position.

Positivism is predicated upon the presupposition that (1) a priori metaphysical experience is non-existent and that (2) knowledge comes only through "positive data of experience," with (3) pure logic and pure mathematics existing either as realms beyond data of experience or, as later held, as pure theoretical science. One of the driving forces behind Positivism was a rejection of senseless, abstract philosophical debate that could not be proven or disproven by experience. Positivism joins the three realms of experience, logic and mathematics by holding that all rational assertions can be verified, or proven, by one or the other: by experience, by logic, or by mathematics.

Idealism stands in opposition to Positivism. Whereas Positivism posits knowledge dependent on data of experience and justification of knowledge through that which is external to one's own mental processes, Idealism posits knowledge dependent upon the constructs of the mental processes of one's mind, justifying knowledge through that which is internal to one's own mind. Idealism is predicated upon the presupposition that (1) mind dominates all perceived reality and (2) that knowing and knowledge are mind related and dependent upon mind. In Idealism, mind is conceived as something beyond cogition within one individual's brain; mind is a total universal collective that transcends individuality.

One weak point in idealism is that of the immediacy of experience being in the "now" delimits temporality (time experience) in ways that are contradictory to everyday experience of time. In response, Idealism posits the "thickness" of time and the "extension" of time; it is the extendability of time that permits a correlated thickness. One example given for extension is that of the "tick tock" of a clock during which the tick is not perceived as in the past while the tock does carry a past perception. Thus time is extended through two beats, if you will, of experience and time has become thick.

A common sense strong point of Positivism is that since we all know that our minds did not the originate the plates we wash up and perhaps dropped and broke if clumsy, we recognize that experience, or positive data of experience, is verifiable and reproducible. We also know experientially that the realms of mathematics and logic have verifiable concrete manifestations in experience while mind, if it generates concrete manifestations, does so in such a way that it is not immediately or reliably verifiable.

All that remains for your critical assessment of these epistemologies is for you to decide how far you agree with the propositions put forth by Positivism and Idealism. 

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