Most of us have bought into their mystique and can hardly conceive of a society in which science does not play a leading role, providing the basis for our control over the natural world and with that the key to further improvements in the quality of life.
1. buy into，字面意思为“买进（某公司股份）”，引申意思为“接受，相信（某种观点或抽象概念）”，本句的have bought into 便是引申意思，即“接受/相信自己的神秘观”。该短语动词中buy 为不及物动词，接 into 后才接宾语。所以不可用被动语态 have been brought into。

2. 本句 provide 是及物动词，后面跟了两个并列的宾语 the basis and the key。这两个宾语后面的介词短语 for our control over the natural world，和 to further improvements in the quality of life，分别作定语。
3. with that 这个介词词组，跟 provide 无关。而是一个起连接作用的介词词组，相当于连接副词 then （然后；于是）。that 其实指 the basis，with that （以此；于是）。总之，the key 跟 the basis 是并列成分，作宾语。

4. providing the basis for our control over the natural world and with that the key to further improvements in the quality of life.

为我们控制自然世界提供了基础，以此（于是）也为我们进一步改善生活质量提供了关键。