Guide (in French): https://tsguide.refl.fr/ts.html

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

This is so French that a baguette appeared in my hand after I clicked the theory link. Probably a cognitive hazard to anyone who is really into generics.

I haven’t been keeping up with Girard as much as I would like, but am I correct in intuiting that this is the next step past Ludics?

Update: I don’t see a citation, so I guess this is an exploration in a different direction.

This is a further step in the same programme. The programme itself is somewhat agnostic as to the underlying dynamics; the ‘stellar resolution’ mechanism (Eng's terminology, not Girard's, who AFAIR doesn't name the system) is a better-behaved replacement for the ‘designs’ of _Locus Solum_, which IMO remains the best introduction for the bigger ideas of the programme.

I've not been following since around 1985 when I realized there's no such thing as semantics. Is this mainstream now?

Of course. Pretty hard to deny it when transformers learmed to produce useful human language not by figuring out how it relates to some Tarskian "true reality" but by looking at a lot of text and figuring out its internal use, exactly as Girard (and implicitly Kant) predicted it would happen.

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).

Good to hear. I need to look up my Philosophy major friend who disagreed with me in 1985 and issue a formal "I told you so" notice ;)

>but the mainstream has caught up since then.

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".

For me, particular syntax of a particular representation and implementation is one thing, the isomorphic implementation-free math describing what it represents is 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.

On the topic of Girard, I've been very attracted to his ideas about the "dynamics" of logic and "communication without understanding". It always appears to me that such ideas and things like the "execution formula" should have profound implications on static analysis of algorithms and things like abstract interpretation and collapsing stacks of interpreters.

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)?

Thank you for pointing out these distinctions which I had not grasped, and the relevant literature! Time for me to dig back into Girardesque pages.

Is "Logique 2.0" this paper https://girard.perso.math.cnrs.fr/logique2.0.pdf? Is there an English translation available?

Claude translation of the intro:

> 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.

It may not help with clarity, but I do love dearly how, so refreshingly, he allows me to feel how he himself feels of the world and its significance with his work, and in that I do feel I understand his point of view better.

Yes, it's this paper but there's no English translation. Girard mostly (only?) write in French now. I believe LLM like ChatGPT or Claude do a great job nowadays?

This is an implementation of Girard's "transcendental syntax" program which aims to give foundations to logic that do not rely on axiomatics and a form of tarskian semantics (tarskian semantics is the idea that "A & B" is true means that "A" and "B" is true; you've simply changed the and to a "meta" one rather than the logical one). This program is more than 10 years old, with first written versions appearing around 2016, and the ideas appearing in his talks before that.

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)

I tried looking into the Transcendental Syntax I paper. It starts with

> 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.

Girard is one of the top logicians of his time. He can write standard mathematical prose when he wants to. In turn, when he chooses to use the continental philosophy style, it's a deliberate choice: people who can't read it are not in the target audience anyway.

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.

Yeah, after doing some research I realised that he isn't some crank and actually has done a lot of relevant work in logic. I still don't think that justifies writing in a deliberately obtuse style, but I guess he can do whatever he wants.

Agreed! I would think the target audience might include people who would like to be in the target audience.

It always bothers me when that is apparently not the case.

His books "the blind spot - a course in logic" is despite its title a bad offender in mixing fairly outrageous claims and philosophical rants with technical statements; and it is one of his most accessible texts. His writing is deliberately off-putting and hard to read; you actually need to already understand his ideas to understand his writing.

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.

> 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.

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.)

So he's not crazy? After a period of my life where I had just read tons of type theory and logic and found his "Locus solum" by accident but couldn't understand a damned thing. It feels like an insane asylum. I could never figure out if he was a crank or a genius.

http://girard.perso.math.cnrs.fr/0.pdf

He's indeed very special and looks like he's writing to himself (especially in his last works). However, after trying to understand his last works for 3-4 years in full-time, I can't say he's "crazy". I would rather say that he gave up trying to make his texts understandable, and he probably has fun writing like that. I once met him and he said jokingly that he was a poet.

I like this guy. Saving for later. Thanks!

I’m sympathetic towards Girard’s dissatisfaction with the current state of model theory, but I don’t see how his project may in any practical sense fix it.

Why are all comments in this thread by engboris (the author of the repo) dead? They appear really relevant.

pinging dang in case he reads it

Seems to be solved now!

I'm an admin here. You just got caught in a (totally erroneous) spam filter. Sorry! It's hard to make these things work properly.

Fortunately other HN users vouched for your comments, and emailed us so we could mark your account legit. Everything's fine now and I hope you'll continue to participate in the community!

Thank you very much!

Jean-Yves Girard really let it go.