Learn English – “By which time” meaning

It's something of an internet meme. There's a picture that's popular of a skeleton playing a trumpet. Because trumpets (in a very vague sense) make something of a 'doot' noise, people started referring to the meme by using variations on 'doot'. Some of these are 'updoot' (a variation of 'upvote') or 'doot doot mr. skeltal'.

This sentence is so horribly written it's difficult to know where to begin. Few sentence elements appear as single nouns, adjectives, or even participles. The overall effect is bloat and confusion.

If the constitutive element of the modern world is, as the parenthetical phrase defines it, man (sic) as the producer of rational hypotheses, then it is difficult to understand how "man" can be articulated with the ideology of progress and the machine. This ideology will emerge triumphantly paraphrased at the end of the sentence, now logically a product of itself.

The machine is defined with a restrictive relative clause as the machine that transforms the methods and theories of [the] natural sciences, apparently distinct from some machine that doesn't. Or does the author mean the machine as such? In which case, the relative clause should be non-restrictive.

So what is doing the arming of the tools and methods and renovating their research/experimental capacities? The ideology of progress and the development of the machine, or the machine that transforms? So let's take the nearest available: machine. The machine transforms by arming and renovating.

It seems to me that despite the author's failed attempt at describing a complex process, you have understood its contours. It's a pity that self-criticism did not extend to a sentence that should never have been generated in the first place.