Interesting undertaking to analyze the history of the way memory has been understood and studied historically, going back to the ancients. Unfortunately, as with many attempts to critique scientific undertakings, this work makes liberal use of terms by intentionally re-defining them to make a point, such as referring to things like file cabinets as "external memory." This has the effect of obfuscation with regards to the language used in writing about the concepts being presented, thereby giving some level of false credence to the idea that a file cabinet is somehow "memory" just as whatever happens in the mind/brain is also memory. This allows for seemingly legitimate attacks on the way so-called reductive and "positivist" science is utilized in the study of memory. Aside from fascinating recounting of the historical contexts of how memory was regarded, I found the point being made to be tired and outdated and ultimately unsupported.
I also took issue with statements like "It certainly does not look as though the organization of external memory required only the projection of an organization already established in the human brain. If that had been the case, one would have expected far more rapid advances in the organization of external memory than are observed in human history. The slow rate of progress suggests rather a co-evolution of external memory and the corresponding cognitive functions." (Location 104 of 5004 in the Kindle edition.) This is essentially a meaningless statement. Here's why: How does the author know that the development of this "organization of external memory" wasn't rapid? Maybe it happened at the most "rapid" rate possible because it WAS a projection of an organization already established in the human brain.
By the way, I'm not saying that "reductive" and "positivist" science are the ultimate way to study memory. Indeed, when one considers recent ideas in complexity as applied to the brain (and, hence, the mind), it cannot be denied the this approach has its limitations, but it is also an indispensable process in the science of psychology. Danziger doesn't deny that, but I think he is overemphasizing those limitations.