Transcendental Syntax
github.com
github.com
Girard's ideas are indeed cryptic and may sound like a caricature of (a particularly grumpy) continental philosopher. One of my goals is to make these ideas more understandable and more down-to-earth, to show that they can be illustrated in a toy programming language. But there's a still a lot of work to be done (way more than what you could imagine just from reading Girard's papers). Although I studied Girard's last works for 4 years, I'm still far from understanding all its consequences. That's because it is related to pretty much everything in logic and computation (what is a type/formula/specification, how do you know whether an object is of some type, what is a proof, what is meaning, what is a system, what are logical rules, what all models of computation have in common, what all logical systems have in common, what is a program, what is an algorithm).
The idea is to go beyond the current proof-as-program correspondence (on which proof assistants like Coq are based). After its analysis of the notion of proof, Girard wanted a computational space in which elements would be able to "test themselves", without relying on some "hard-coded" semantics. In programming terms, it would correspond to the ability to build types from programs. There is no primitive types, not even the arrow type of functional programming. We just have Prolog-like bricks of terms which can interact with each other. They can express programs or tests for programs. You build everything with that like how you do chemical experiments. To give a probably familiar illustration, I'm often asked whether we could "do transcendental syntax" with the untyped lambda-calculus. It would be possible if for any type T you could find a finite set of lambda-terms acting as "tests" such that interaction with it ensures being of type T. The space of lambda-terms is not "large enough / flexible" for that. You need to introduce exotic objects like how you extend the set of rationals Q to the real numbers R to solve more equations. You can also think of how to tell whether a finite automaton recognizes some (infinite) rational language without relying on an external semantics. You only have input words and infinite testing by feeding it all words of the language. But yet, we are able to do it with an external semantics. So the question is how you internalize semantics in the object language.
If there's anyone who has any inspirations or insights, I'm open and interested. The right path to follow is still undefined. I just have some tools coming from my understanding of Girard's works.
Theory: https://ncatlab.org/nlab/show/transcendental+syntax
More Theory: https://ncatlab.org/nlab/show/Geometry+of+Interaction
Some sort of explanation: https://x.com/noncanonicAleae/status/1874893997791170719
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)
> 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.)
Has there been any literature on concrete steps in this direction, or is there anything holding back Girard's theories from practical application (I know that Girard's Geometry of Interaction was supposed to have problems with additives, of which exact implications I do not quite comprehend, which may or may not be relevant)?
I don't think there is any concrete steps in this direction. However, those ideas still live in the transcendental syntax since it is the successor of his geometry of interaction programme.
> or is there anything holding back Girard's theories from practical application
In theory, I don't think so. In practice, it's very difficult to approach his ideas. Not only because he's hard to read but also because you have to read and understand a lot of his previous works but also of other people's work to put everything into context. Once you do it, you then need enough practical knowledge to have an idea of what applications can come out of it.
> I know that Girard's Geometry of Interaction was supposed to have problems with additives, of which exact implications I do not quite comprehend, which may or may not be relevant
It's not very clear. From what I understand, he now considers he was "doing it wrong" and then the problem disappeared because he now distinguishes between "local" (asystemic/particular) and "global" (systemic/generic) mechanisms which are apparently and mistakenly mixed in logic. Full additives are global and live in a "system" (which defines contraints over a "free" computational space -- think of "complex systems") although we can define weaker "local" additives. This difference between local (he calls it first-order but it has nothing to do with FOL/predicate calculus) and global (he calls it "second-order") is mentionned in his "Logique 2.0" paper.
Is "Logique 2.0" this paper https://girard.perso.math.cnrs.fr/logique2.0.pdf? Is there an English translation available?
> Abstract:
> In this tract, I lay the foundations for a radical re-reading of logic. I illustrate this with technical developments: in particular, a notion of truth based on the Euler-Poincaré invariant.
> Introduction: The Return of Philosophy
> At the end of the 19th century, logic experienced a spectacular renaissance. But, like a snake that grew too quickly while forgetting to shed its skin, logic remains prisoner today of a Nessus tunic - the scientistic format concocted by the founding fathers that has become obsolete. This obsolescence is made manifest by the proof networks derived from linear logic [2]. It is high time to change our reading grid and carry out a "Copernican revolution": the transition to logic 2.0.
> The first point of resistance is the dominant philosophy - analytical, to simplify - whose main thesis is that... philosophy serves no purpose: a simple translation would allow bypassing it by reducing it to predicate calculus. This thesis, due roughly to Russell, places logic in the position of an irrefutable arbiter. And thus, how can we judge it if it is its own jury? To top it off, modern "analytics" reduce philosophy to logic... from Russell's time, the only one they know. This outdated logic thus dictates its law under the cover of scientistic philosophy.
I am not sure the poetic flourishes are helping with clarity and formalization... but I would dearly love to watch a video of him shouting in this style from upon a box in a town square.
Update: I don’t see a citation, so I guess this is an exploration in a different direction.
There might be links or transcendental syntax may subsume both geometry of interaction and ludics but nobody studied the relationship with ludics as far as I know.
My current understanding is that we should distinguish "proof-objects" (a sort of certificate) and what I call "proof-process" (a recipe, a method for building proof-objects). GoI/TS mostly speak about proof-objects. If you know about proof-nets, I don't see them as the essence of proofs behind sequent calculus proofs but as formal certificate constructed by sequent calculus proofs (seen as recipes). The link between ludics and GoI may be there but I don't know. There's nothing written about this.
Girard even noted that we'd know progress was happening as soon as early models started making certain specific kinds of mistakes children also tend to make, such as the answer to "which is heavier, 5kg of bricks or 5kg of feathers".
So yes, Kant, Girard and you got this right early on, but the mainstream has caught up since then.
Of course, semantics still works well as a technical tool in formal logic, though it has no link to its philosophical counterpart (not that this prevents B-grade philosophers from abusing the unfortunately chosen terminology to equivocate them).
I'm not sure which mainstream you're talking about, but I'm relatively interested in mathematics (and a bit in formal logic), so I should be way ahead of the mainstream (generally uninterested in math), and I have no idea what you mean. I mean you may be right, but I don't don't even understand a hypothesis "there's no such thing as semantics".
I.e. A natural number is a list of successor relations. It directly maps to the reality of collecting a series of units, by actually being a series/list of units.
Whereas, a natural number represented in binary doesn't look at all like what it represents, but is an isomorphic representation. It behaves the same way.
And can be interpreted the same way, when it is being used for that purpose.
So: Series/list of steps = Semantics, for a syntactic implementation of binary natural numbers. Only in the right context, where the latter is mean to represent the former, does the isomorphism relate the syntax to the semantics.
In another context, a series of binary values might just be a random pattern we used to mark a bunch of things we think are related. In that case, the semantics would be completely different, even though the syntax is identical.
But I have no idea how other's think about these things.
pinging dang in case he reads it
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