Carnap's Contributions

There was recently on twitter a good natured thread discussing philosophers of science's contributions to the sciences they study. How often does a card carrying academic philosopher make a direct or first order contribution to the scientific enterprise, it was asked. Naturally I went in to provide some Carnap facts since I thought I could just draw from a previous blog post. But it turned out my main man was already covered so I didn't feel the need to say more. However, on doing a bit of review, it made me realise something that I find a bit interesting about Carnap's contribution to the sciences and how they tended to happen, so a short blog post on that.

(For anyone interested in the more general question concerning the recent influence of philosophy of science upon the sciences they might find these two articles interesting. Somewhat more polemical but with more historical cases this article might also be fun. Note that I am not going to have anything interesting to say on what counts as a science in what follows. If you don't think these are sciences proper then just think of this as Carnap's contribution to these other fields, however they are classified.)

We'll start with the most direct of direct contributions - personally discovering or proving something that is of general scientific interest. As far as I know with Carnap there are not many instances of this. Probably his best case would be the fact that the Logical Syntax of Language contains the first proper statement of the Generalised Diagonalisation Lemma from Gödel's incompleteness theorem. From the outside proving a lemma may not sound too impressive, but in fact if you know the history of logic this aspect of the proof is what allows for the very general self-reference that Gödel brought so vividly to our attention; it's actually conceptually very important. Now Carnap's proof was slightly (but not seriously - see the reply here) incomplete, but none the less I think the norm among historians of logic and mathematics is to credit him, so fair play.

I think the only other contender for such a direct contribution from just Carnap himself would be the discovery of pure inductive logic as a field. This is a very small niche branch of contemporary pure mathematics, not quite engaged in the same project as Carnap but at least interested in the family of formal systems his own work introduced. It does not seem to me to have massively taken off, but they have recently published their own textbook, they have their own set of open problems they turn up at conferences and there seem to be research connections between them and the slightly bigger (and also probably Carnap descended) Objective Bayesian school. So there you go, it's not big but it's not nothing either. 

So re direct contributions to science, therefore, Carnap stated and (almost) proved the GDL and founded a small field of modern mathematics which took his ideas in more abstract and less applied directions. How about slightly less directly?

As far as I can tell they are as follows. First, in linguistics, Carnap's work in Meaning and Necessity forms some of the first steps towards the "possible worlds semantics" which nowadays is so influential. Although even here, Ruth Barcan-Marcus beat him to the punch in actually publishing axioms for quantified modal logic. They seemed to have worked together at Chicago in the years leading up to their first publications on the topic, and Barcan-Marcus work was ultimately not just first but more technically fruitful. (The second chapter of Timothy Williamson's Modal Logic as Metaphysics is quite a nice run down of why Carnap's modal semantics is a sort of close-but-no-cigar sorta thing from the contemporary logician's point of view.) But for all that, there is a path from Carnap's modal semantics to contemporary linguistics -- surprisingly enough not via Kripke but via Montague. This is because Montague semantics builds on some of the technical features of Carnap's work but places them in a generally more productive framework (Montague himself reported that some of these improvements were suggested to him by Carnap in conversation). And Carnap's student David Kaplan did work on demonstratives and indexicals based upon this, and linguists tell me this work is still (somewhat) influential. So not really directly the work Carnap himself did, but the work it inspired, turned out to be productive for linguistics.

Second, there is machine learning and AI. Not only did Carnap secure Walter Pitts a job at a time when he had been unemployed and homeless, but it seems he was very influential on Pitts' (and McCulloch's) thinking about cognition and neural networks. Their seminal paper even uses the notation and systems of Carnap's Logical Syntax to frame its ideas and results. Herb Simon was deeply influenced by Carnap incorporating his ideas into much of his work, even doing scholarship on Carnap. And Ray J Solomonoff directly credits the invention of algorithmic probability to working out how he could build on Carnap's model of inductive logic to just do away with a couple of flaws he saw in it. (Mind you in the same recollection he also mentions how much Freud influenced him and on that basis says not to trust his own memory of what made him do what!) So here the story seems a bit more straightforward. Carnap was directly the teacher of a bunch of the founding figures in machine learning and AI, and they pretty directly credited specific ideas of his for inspiring their work. 

Third and finally, Carnap seems to have had a lot of indirect influence on what we might nowadays call information theory. And curiously here it almost all seems to have been via his philosophical work rather than his directly technical work. For it seems one of his students, Richard Jeffreys, took Carnap's logical empiricism as the starting point for his development of radical probabilism in Bayesian epistemology and decision theory. Jeffrey's apparent interpretation of his relationship to Carnap, at least, was that radical probabilism kept what was correct in Carnap's logicist and tolerant approach to the world, but dropped the erroneous dogmas of empiricism. Abner Shimony, who played a vital role in operationalising Bell's inequalities, reported being inspired by Carnap's model of clarity and work on testability and meaning (along with his thinking on induction) - although he noted that at no point did he ever really agree with Carnap. Shimony later edited and published Carnap's essays on Entropy - which as far as I can tell were substantially correct. Carnap himself developed and published work on semantic information with Bar-Hillel, and David Lewis developed the first signalling game as a way of vindicating Carnap's conventionalism against Quine re philosophical foundation of logic and language. 

So there you have it there's my gathered case for Carnap's influences on the sciences. When I find out about more I will add to the post. (Edit: one already! Thanks to Richard Zach for pointing out that Carnap's early work on axiomatics was a big influence on Gödel and Tarski; also in the vein of productive but not ideal, nicely fitting our theme.) But for now I just want to highlight something: how often all this goes via Carnap's mistakes.

I recall Eric Schliesser saying somewhere (but cannot now find! Will update with links if I get them) that Carnap is interesting because for all his skill he was a habitual bungler. In another circumstance I remember Glymour saying that Carnap was ever a day late and a dollar short. Looking at these lists it's impossible not to agree, right? Genuinely the scientific legacy just described is one that any of us in academia would kill to have... yet it is basically all just a result of him being wrong, and inspiring someone else to do better. Even the GDL, probably his best single claim to real high level scientific credit, was kinda mistakenly done in his hands and had to be fixed up by later workers. 

Now neither Schliesser nor Glymour intended what they said to be entirely dismissive of Carnap. I think they both have some affection for him and his work. And obviously I do too! So I think it is worth reflecting on how it can be that such an impressive legacy can be composed near entirely of errors.

To me the lesson I take away concerns the place of academic philosophy in the present academy. Namely I think this: if one aims to be a kind of naturalistic or science-facing-interdisciplinary scholar, then one will usually do better to be clear and bold and thus clearly wrong, than one will do to be intricate and careful and likely correct. Of course I cannot really draw a comparative claim like that from just the information about one person, so implicitly I am here drawing upon my background knowledge about lots of other philosophers and their legacies and how they might compare to Carnap's. So I think that Carnap's legacy is just a very dramatic illustration of the conditional claim.

The reason I think the conditional holds is to do with the comparative advantage of the academic philosopher, and is perhaps also related to Frieidman's work on the history of paradigm changes. I do not think we are especially good at establishing claims as true. What we instead have is wide latitude to explore ideas that would be ruled out by the stricter paradigms of other fields. We have tools and clarity norms that can make us unusually good at being explicit about what elements of our ideas are driving what consequences, making it easier to see what ought be retained or discarded. And, if only because what we do is so cheap, we have unusually wide remit to make use of academic norms permitting riskier publications. So if anyone is going to do the work of "explore the consequences of far reaching but speculative or commonly presupposed ideas and see how they play out" we are very well positioned to be that person, and we are likely to do so in a way that facilitates others realising what they like/do not like in the results of this, and modifying their behaviour accordingly. 

Carnap's legacy is a dramatic illustration that done well this can be very fruitful despite its high propensity to generate error. I do not think it is the only way to be a good philosopher, indeed I do not think we should presume that all the ways of being a good philosopher are consistent with being a good academic, but it is certainly one way to do things of real benefit to science and society. And whatever we are trying to do in our work, not many of us can honestly boast of in fact having done better than Carnap in our own spheres.