> We study logic in the light of the Kantian distinction between analytic (untyped, meaningless, locative) answers and synthetic (typed, meaningful, spiritual) questions. Which is specially relevant to proof-theory: in a proof-net, the upper part is locative, whereas the lower part is spititual: a posteriori (explicit) as far as correctness is concerned, a priori (implicit) for questions dealing with consequence, typically cut-elimination. The divides locative/spiritual and explicit/implicit give rise to four blocks which are enough to explain the whole logical activity.
Honestly, this and the rest of the paper read like a caricature of (a particularly grumpy) continental philosopher. I suspect many mathematicians don't engage with this because they are drawn to precision. From what I read (and from your comment), I have no idea what this program really is about. Maybe there's something of value in there but it seems really hard to tell.
Somebody who worked through the two volumes of "Proof Theory and Logical Complexity", and worked through the later chapters of "The Blind Spot" will already be used to the style, so won't find it an obstacle when reading the Transcendental Syntax papers. And those who didn't read these works? Well, they don't know the prerequisites anyway (the author assumes), so making the style more palatable to them is not a priority.
The aforementioned works are better starting points: they cover classical proof theory, so there are many other textbooks that can be consulted while the reader develops familiarity with Girard's style. And one needs to know the material to understand Transcendental Syntax anyway.
It always bothers me when that is apparently not the case.
His main contribution, through linear logic, are ways to separate 'elementary' operations of logic into simpler ones. Linear logic decomposes the classical "and" and "or" into multiplicative and additive ones, and some modalities. This gives a more precise encoding of operations that can directly link logic to complexity theory, and have interesting applications to compiler design.
Note that academia has been very receptive of his technical achievement, the proof of consistency of System F through reducibility candidates, but much less welcoming of some of his other papers that he deemed way more interesting (as we saw recently with Terence Tao "I got a paper rejected today"). I think him writing this way is a reaction to that; rather than try to court people who might reject his ideas, he already filters people who are looking for the good stuff in them.
To confirm, s/continuous hypothesis/continuum hypothesis/ ?
Also, for the curious, here's a paper pushing back on the idea that the continuum hypothesis et al. even "need" to be resolved in the first place: https://arxiv.org/abs/1108.4223 (The set-theoretic multiverse, JD Hamkins, 2011). (I don't know anything about set theory myself, so I can't personally comment on what it says—but AFAICT Hamkins is respected in the area.)
Girard has been a vocal critic of foundational problems, labeling them as "hell levels", with the typical approach of set theory and tarskian semantics as the lowest one and category theory as a "less worse" one (at least one level above). One issue with his program is that he mixes abstract, philosophical ideas with technical ones. So even if some things have interesting technical applications, they may be different when seen from a more philosophical point of view. For instance, set theory as the foundations of mathematics is a pretty solid model but it is seen as fundamentally unsatisfying for many reasons -- most famously the continuous hypothesis. Gödel and other very high-profile mathematicians thought it was a very unsatisfying issue even though from a mathematical model theory point of view, it's not even a paradox. So the new foundational approaches tend to have maybe deeper philosophical problems about them; for example see Jacob Lurie's critic of the Univalent foundations program (after the "No comment" meme he expressed a long list of issues with it).
The other issue with this particular work is that it uses new vocabulary for everything to avoid the bagage of usual mathematical logic, but it kind of give a weird vibe to the work and make it hard to get into without dedicating much work.
The result is therefore something that is supposed to solve many longstanding problems in philosophical and technical approaches to the foundations of mathematics but has not had a big impact on the community. This is not too surprising either because the lambda calculus or other logical works were seen as trivial mathematical games. We'll see if it's a case of it being too novel to be appreciated fully, and this work seems to try to explore it in a technical way to answer this question.
https://girard.perso.math.cnrs.fr/Logique.html (in French, it gives an overview of the program)
https://girard.perso.math.cnrs.fr/Archives.html (transcendental syntax papers are there in English)