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Aspects of the replies to this MO question made me think that a more comprehensive discussion of some related issues might be worthwhile. I remember that this issue also came up very briefly here or on some blog before.
Namely, we have the following sociological phenomenon at the moment:
there are two essentially disjoint research programs active these days, with two different proposals for, roughly, a “new foundation of quantum theory”. Namely
there is a group of researchers who amplify doing quantum mechanics in terms of dagger-compact categories;
there is a group of researchers who amplify looking at quantum systems via their Bohr toposes (or similar toposes).
I think for both these approaches it is true that a) their genuine contribution to new insights in quantum theory has still to be established b) nevertheless the kind of question they address – namely “What is the ’natural’ theoretical formulation of quantum theory?” – and the suggested kind of answer – namely: “It only makes sense using some category theory.” – is found to be of interest more widely.
I think the point is that at the horizon one can see that the puzzlement about the nature of quantum mechanics that its founders still worried about and discussed at lenght does in fact have a useful resolution within a more general abstract perspective.
Whether or not any of the above two proposals do live up to this promise, we can observe some sociological dynamics:
One is this: there is the perception of a certain competion: whose “new foundations of QM” is more foundational? Which one is better?!
One thing I’d like to argue is: this questions is misguided. The two proposals concern roughly disjoint aspects of a bigger story.
Another sociological phenomenon is this: while both approaches are fond of claiming that it is category theory (maybe in its flavor of topos theory) that makes the difference, the reaction to that claim from the pure category theory community often ranged between “not enthusiastic” to “positively displeased”. This could be witnessed on the category theory mailing list, for instance, but also here we have seen comments to this effect.
I have an opinion on that, too, though I am not sure if I can formulate it properly. At least not tonight. Maybe tomorrow. So regard this here as part 1 of a series of little opinion pieces.
I’d certainly like to hear the rest of the series.
I think the point is that at the horizon one can see that the puzzlement about the nature of quantum mechanics that its founders still worried about and discussed at length does in fact have a useful resolution within a more general abstract perspective.
The founders talked about things like the role of consciousness in observation. What kind of resolution might we expect now? How would it be phrased? In terms of information perhaps? Can we take things on from our discussion back here?
> I’d certainly like to hear the rest of the series.
Let me make an attempt. I may want to polish what I say now in a while, but maybe it’s good to start somewhere.
So: what about the reaction of "the pure category theory community" to attempts by physicists to find a category-theoretic re-formulation of foundations of quantum mechanics, or, for that matter, of anything else in physics.
My short answer is: it does not matter.
For the following reasons.
I see little evidence that many people in the community care to understand the issue, the motivation and in fact the need for this aspect of physics or for any other. A notable exception in parts is Lawvere, who apparently built his career out of the motivation to find the category-theoretic foundations of classical mechanics . While his pure-category-theory work is widely appreciated, how many people have ever followed up on his thoughts about the foundations of phyiscs? If he with his sophistication is being ignored on this aspect, no non-pure category-theory outsider can reasonably expect to see any resonance.
With that, what happens next is a classical example in mathematical communication: A approaches B with a half-baked idea that is strongly motivated but not fully worked out yet. B misses the motivation and then dismisses all the rest as what it is without the motivation: trivial to the extent that it does make sense, otherwise incoherent rambling.
Sometimes one also sees this work the other way round: a mathematician sees a construction or theorem of his picked up by phsicists, does not understand the motivation, but gives all the benefit of doubt and simply assumes that it must be all great. (For instance Kostant and -physics.)
Concerning the "trivial to the extent that it does make sense": applications of math to physics is very different to the study of math in itself. Its interest is – at the beginning – not in the strength of the theorems proven from it, if any. It’s a more subtle thing that hasn’t been formalized and is apparently hard to communicate to people who don’t see it by themselves, even though there seems to be a large supply of historical example to draw conclusions from.
Lastly: people who search for the right math to describe their physics typically proceed first by guesswork and then learn the math as they go along. That’s often a pain to watch, much like the process of giving birth. So don’t watch it if you have no family relationship. Wait until the dust has settled.
edit: I should maybe add that where I say “category theory community” above this is a) intentionally vague, exceptions prove the rule, but b) meant to really mean “1-category theory”. In other parts of the field the situation is quite different: I think much of -category theory was all driven from physics (in the form of homological mirror symmetry), for instance. And of course by now the cobordism theorem makes it all clear that this kind of physics is simply a genuine part of -category theory.
Which is perhaps to say that the relationship between physical and mathematical intuition is a subtle one (discussed here), and that people may be strong in one or other, and rarely both. Misunderstanding frequently follows from a lack of appreciation of the other intuition.
I’m also hoping you’ll continue more on the direction I was trying to urge you to follow - the physical intuition in the new approaches to quantum mechanics. Physical intuition is often expressible non-technically. Has something been lost by the disappearance of discussions like the Bohr-Einstein debates?
How might a debating partner respond to
On the philosophical side, I do believe that the question on the existence of something is deeper than the notion of spacetime. What lies at the heart of the former question is the notion of information [Bek], which can tell an observer that there is something. On the one hand, the total information of an observable physical object should be nicely structured. On the other hand, information itself should be some kind of structure that is observable to other structures. Therefore, we can simply define a physical object as a nicely structured stack of information. As a consequence, our physical space is nothing but a network of structured stacks of information, from which spacetime can emerge. (Kong)?
If he with his sophistication is being ignored on this aspect, no non-pure category-theory outsider can reasonably expect to see any resonance.
Well, his sophistication and originality was also an obstacle: very specific and involved terminology, unpublished preprints, and never got involved to cooperate with modern era natural partners. For example the fields of observables on geometric spaces which he calls quantity (for a physicist quantity is usually a thing which gets evaluated to a number with some unit, not a distribution of quantities, but the latter is also sporadically used) are cosheaves, and this has been independently noted also in 60ties in first attempts to do some “spectra” of noncommutative rings, which got culmination in ring theory in 1970s. So the attitude toward that noncommutative spectra community would be to give an example that if the framework is so useful and deep that it works in some examples for that direction, rather than just claiming, of we had it before, why you guys do not use our framework.
On the other hand, why those preprints never get to circulation and posted online. For example, there is a preprint of Lawever from 1960s on probability monad. His student Giry cites it and wrote an article building on it, which is known to have errors. Recently there was some thread on category list that somebody has a scan of the preprint, I asked that person about it (to place it on Lab as a file) and he is uncomfortable to put it online without Lawvere’s consent. In normal circumstances and with commuhnicable people such a consent cold be obtained within a day. I similarly typed hard to find Benabou’s paper and sent him email about posting it to the arXiv, and never got a response about it. It seems it is a feature of category theory of 1960s and 1970s to be self-satisfied and closed community.
Physical intuition is often expressible non-technically. Has something been lost by the disappearance of discussions like the Bohr-Einstein debates?
On this specific point notice that the Bohr topos is named this way for a reason. The idea is that this very topos formalizes the physical intuition that Bohr had been famous for expressing, but didn’t formalize. And how should he have, without a formalism that is able to capture such things?
So this is an example of what I had in mind when I wrote in #1:
at the horizon one can see that the puzzlement about the nature of quantum mechanics that its founders still worried about and discussed at lenght does in fact have a useful resolution within a more general abstract perspective.
The physical intuitions about the foundations of quantum theory have been exchanged at length at the beginning of last century. But the discussion did not lead to genuine progress in the formal theory. Most people in the field therefore became positively uninterested in such exchanges.
What is new now is that the discussion is being revived, but now with a formalism in hand that is in principle capable of capturing the kind of notions of relevance here. It has maybe still not been shown if the Bohr topos genuinely helps with understanding quantum physics. But at least now that question has become a much more formal one that one can study with much more precise tools than those of debate.
But at least now that question has become a much more formal one that one can study with much more precise tools than those of debate.
Birkhoff-von Neumann lattice approach was also perfectly formal and did not lead far. While I like the conceptual novelty in Bohr topos, I disagree that it captures precisely the Copenhagen viewpoint, especially when I see no word mesurement in the terminology. Translating limits of commutative subalgebras into geometric spaces is a novelty but also does not add to the precision, as the data are already precisely in the original noncommutative algebra. I do not see that the abstract limit of subalgebras corresponds to the semantics of measurements, though possibilities to investigate further and coming there potentially are not excluded. Having a commutative subalgebra as an intermediate building block of the story in both cases is parallel just as a building block, not as a story made of building blocks.
Birkhoff-von Neumann lattice approach was also perfectly formal and did not lead far.
But also it didn’t formalize anyone’s intuition, really. What I mean here is that it makes sense to argue that the Bohr topos construction does formalize Bohr’s idea of how to think about a quantum system.
It’s not very deep, actually a very simple idea. And it is about measurement: Bohr’s point was that whatever one can say about a quantum system must be expressible in terms of a bunch of experiments in classical physics. Now “classical context” translates into “commutative subalgebra of observables” and so we want to be looking at things that can be “probed” by commutative subalgebras of a given quantum system. This are presheaves on these.
That alone is not a strong argument for anything, but it deserves to be looked into. I think it becomes a bit stronger if one notices that several of the foundational theorems that characterize quantum physics are naturally formulated in terms of presheaves on commutative subalgebras, too. Here people always mention Kochen-Specker, but maybe Gleason’s theorem is more relevant, even.
I similarly typed hard to find Benabou’s paper and sent him email about posting it to the arXiv, and never got a response about it.
which paper? In emails to him he has told me he has probably hundreds of pages of notes, which he airs in seminars and talks. And as far as I can tell (I may have the wrong end of the stick), he really really doesn’t want to let the ideas out, fearful they will be ’stolen’, only trusting them to people close to him. I offered to host scans of his notes on the nLab, to disastrous effect.
Well, his sophistication and originality was also an obstacle: very specific and involved terminology, unpublished preprints, and never got involved to cooperate with modern era natural partners.
To those people that he should have communicated with, yes. But it is precisely the behaviour that otherwise the cat-theory community appreciates, as you note. So therefore my point: if this community does not even react usefully to physics presented in this style, then it is unlikely that it will react usefully to it presented in any style.
@David re Bénabou: It sounds like he doesn’t understand how time-stamped copies of his notes on the Internet would help him to establish priority. As it is, he goes on the catlist to say «I remember this lecture 40 years ago, which establishes priority.», and others are sceptical. It’s too late for those, but if Johnstone writes another book, then he could have all of the evidence on the Web ahead of time.
Not that he should trust us; he should host them himself.
@Urs #6 I can see how formalisms can cut through misunderstandings in verbal debates, but something it seems that has also gone from the Bohr-Einstein debates is the thought experiment. Do we have anything from the past few decades to rival these? The Eppley and Hannah thought experiment?
Maybe that’s wrong:
Thought experiments continue to play an important guiding role in theoretical physics: the ideal experiment conceived by J. Bekenstein [2] in order to elucidate black hole entropy or the thought experiment imagined by S. Hawking to tackle the problem of unitarity violation in black hole space- times are some examples. (A Gedankenexperiment in Gravitation)
something it seems that has also gone from the Bohr-Einstein debates is the thought experiment.
I may be wrong, but my perception is that the thought experiments that drove the debate over the foundations of quantum mechanics essentially left everyone puzzled and have as yet to be usefully resolved. The reason why these debates are no longer had is because, to put it bluntly, people got fed up with this state of affairs.
That’s where the phrase “shut up and calculuate” originates from: the discussion was going in circles leading nowhere. Even worse, maybe, those claiming to still seriously consider these debates have, to my mind, wandered off into territory that one should not enter.
Not sure if you have seen the recent “discussion” about the “many worlds interpretation of quantum mechanics and the multiverse”. This makes me shudder. Maybe the appeal to topos theory in the “new foundations” makes topos theorists cringe, but compared to this state of affairs it is a vast improvement to have at least a proposal for a formalization of the question.
Bayesian interpretation FTW! (^_^)
Hi Toby,
since your message has no verb, and since I can’t figure out what "FTW" stands for (reminds me of "RTFM", but going in this direction leads me into territory which I won’t explore here…) I have to guess a bit. I am guessing maybe you mean to imply something like:
Adopting an Bayesian attitude towards probability theory makes all open issues of foundations of quantum mechanics disappear.
Is that what you mean?
I wouldn’t quite agree with that. I am happy with the Bayesian attitude towards probability theory, I find it obvious. But I don’t think that this makes the foundational issues of QM go away.
And it seems to me that the authors of the article behind the "FTW"-link
Carlton M. Caves, Christopher A. Fuchs, Ruediger Schack, Quantum probabilities as Bayesian probabilities (arXiv:quant-ph/0106133)
would agree with me on this:
in the intro they say
In the classical world, maximal information about a physical system is complete in the sense of providing definite answers for all possible questions that can be asked of the system. In the quantum world, maximal information is not complete and cannot be completed.
This is the Kochen-Specker theorem, characterizing one of the foundational aspects of QM. And this is an invariant fact that cannot go away by re-gauging our concepts of probability. And the authors don’t claim it does, they instead claim that
Using this distinction, we show that any Bayesian probability assignment in quantum mechanics must have the form of the quantum probability rule,
When I look through the text to find the point where they fulfill this promise, it seems to be happening around equation (4). As they amplify themselves, their conclusion is nothing but Gleason’s theorem, another of those characterizations of the nature of QM.
They give its statement some Bayesian sugar, and I think that’s actually a nice way of speaking about it, but I don’t see that this answers what I would think are the foundational questions that the ancients (Bohr, Einstein et al.) struggled with, and which the "new foundations" (just so that we have a term for it) want to address.
So I find it noteworthy that the substance of their discussion is a) Kochen-Specker’s theorem and b) Gleason’s theorem. This are precisely the two theorems that one can argue motivate looking at the Bohr-topos: because both make statements about presheaves on commutative subalgebras of observables.
To summarize:
I agree that the Bayesian attitude is the correct way to think about what probability means when applied to physics (obviously! I would say, wasn’t that implicitly clear to everyone even before Bayes?);
assuming this, then we don’t have to worry about what it means to have a probabilistic theory of physics (such as statistical classical mechanics or quantum mechanics or, for instance, astrology );
but that’s not the worry anyway, I think: the worry is that even as a probabilistic theory, QM is fundamentally different from classical mechanics, and the feeling is that this fundamental difference deserves a better conceptual understanding.
Notice that for instance in all the Bohr topos-discussion the probabilistic nature is taken for granted anyway: we just speak about states as functionals on the observable algebra, which is the context of quantum statistical mechanics anyway. The passage via the GNS construction to a space of pure states is not at the center of attention at all.
But now that I said so much in reply to you just throwing out a keyword, maybe you can expand on what you mean?
FTW means (I believe) ’for the win’. I’m still not sure I understand what this means, though. It is an Americanism probably derived from sport.
(edit: wiktionary definition)
(edit 2: not sport-related:
Originated from the game show Hollywood Squares where the result of the player’s response is expected to win the game.
from urban dictionary)
I can’t figure out what “FTW” stands for
If it meant “RTFM”, I would suggest that you’re on the Internet, so one web search away from knowing what out means. But actually, “FTW” is not that condescending; it just means “for the win”, an expression of enthusiastic approval.
You hypothetically attribute to me:
Adopting a Bayesian attitude towards probability theory makes all open issues of foundations of quantum mechanics disappear.
No, you also have to apply this attitude to quantum theory itself; merely applying it to probabilities is not enough. Quantum theory is a noncommutative generalisation of probability theory, so this is not trivial. But the same philosophical attitudes that lead one to adopt a Bayesian interpretation of probability theory should also lead one to adopt a Bayesian interpretation of quantum theory.
However, this leaves open the possibility that there remain open issues in the foundations of probability theory, and these will necessarily also be open issues in the foundations of quantum theory. And even if the commutative case is fully satisfactory, there may still be issues in the noncommutative case.
All the same, I find the Bayesian interpretation almost fully satisfactory, so announcements of new foundations to solve open problems seem fairly empty to me. Doubtless the partisans of many worlds or Bohm feel the same way.
As they amplify themselves, their conclusion is nothing but Gleason’s theorem, another of those characterizations of the nature of QM.
Actually, it’s Gleason’s Theorem for POVMs, which is so different from the classical Gleason’s Theorem that I don’t think of them as the same thing. (This is a minor point.)
There is no deep mathematics in this paper, because the whole point is that no new mathematical ideas are necessary, only an appropriate interpretation of the usual ideas. I suspect that they just put some theorems in there to make people feel like the paper is not too trivial; but the idea is philosophical, not mathematical.
That said, there is one step of mathematical sophistication that helps me. The category of commutative von Neumann algebras is dual to the category of localisable measurable spaces, and probability measures correspond to strongly continuous states. Quantum theory may be founded on arbitrary von Neumann algebras and so literally becomes noncommutative probability theory. This is not to denigrate other mathematical foundations of quantum theory, of course; but this is one that I use to settle philosophical issues (by seeing what the issue becomes in probability theory and considering how to resolve that).
There remain, of course, philosophical issues that apply only to the noncommutative case, but these aren’t the ones that most people seem to worry about.
I find the stuff at Bohr topos perfectly compatible, by the way; it just doesn’t answer any questions of interpretation for me. And I have trouble seeing how it would help anybody else in this way. When you started this thread, it didn’t seem to me to be about that at all.
Hi Toby,
thanks, that helps. So we have found out that there are two different issues here and that we have partially been talking cross-purpose.
indeed, I am not concerned with "interpretation of QM" here. This is not what I mean by "foundations of QM". I am not concerned with the philosophical questions here, but only with structural questions. What these approaches that I termed "new foundations" propose to offer (whether or not they succeed) is (luckily!) not another item in the list that contains the words "Bohm", "many worlds" etc. I think the discussion behind these words is, in my above words, "territory that one should not enter". This is fruitless and misguided (I think).
Instead those “new foundations” are proposals for how the mathematical structures are usefully understood.
Maybe to give an example to illustrate this: this is a bit like the discussion I have with Zoran here every now and then, about what the "correct" mathematical way is of describing noncommutative geometry. Is it just the study of the category of non-commutative algebras with some extra information (Connes) or is it rather the study of the sheaf topos over it, or its it in fact that study of commutative 2-algebras (categories of quasicoherent sheaves) or actually that of commutative -algebras (-categories of quasicoherent sheaves). And given any of these answers, how is ncg thereby an example of a general notion of "geometry" that also includes the traditional commutative geometry, or it is something different? How are we to think of ncg?
Of course this example is closely related to the issue of QM, which introduced the fact that phase spaces are non-commutative spaces.
Or are they? That’s another part of the question here. Instead of saying that a non-commutative algebra of quantum observables is a non-commutative phase space, one can just as well – and even more naturally – argue that it is in fact a commutative but non-associateve Poisson manifold: take the commutative non-associative product to be the Jordan algebra product and take the Lie bracket to be the commutator . Inspection of the formalism – for instance of the definition of states show: the second point of view is just as consistent, and maybe even more natural . (It’s strange that this is not said more often, as it is such an obvious observation that should make one think at least once if all the ncg-activity has been going in the wrong direction from the get-go. But for instance Weinstein highlights it briefly on page 80 of his lectures on quantization.)
I find it interesting that for instance the Bohr topos-formalization already allows to have some theorems that shed light on this. As discussed there, Andreas Doering has shown that equivalences of Bohr toposes correspond to _Jordan-_isomorphisms of the algebras that they are built from. So that’s now an interesting fact about QM that we wouldn’t have learned without this new formalization. It indicates that the standard idea "quantum phase space is a non-commutative space" may not actually be useful, and that "quantum phase space is a certain commutatively-ringed topos" is actually more true to what’s going on.
Maybe I am distorting the history now (am I?) but it would seem to me that this type of question is actually what drove the founders of QM to those fanous debates. What they were puzzled by is: how can it be that the formalism of QM looks so radically different from the formalism of physics known before? Is there a way to reconcile this? Is QM a weird piece of structure that we have to accept to live with, or can we see that it is secretly not all that weird, really.
And here by "weird" I don’t mean what Bohm et al apparently find weird ("Oh, it doesn’t look like the universe is a clockwork of point pieces as Laplace imagined. But wait, if we just decompose the simple formulas in a funny way then it does look so just a little bit again – what a relief.")
No, by "weird piece of structure" I mean: compared to the mathematical structure that we do know controls classical physics, classical geometry, etc.
[continued in the next comment…]
[…continued from the previous comment]
For instance the Coecke-school of "new foundations" is all about observing: look, the entanglement and what goes with it may look weird from the outside, but structurally what is going on is simply that we pass from cartesian monoidal categories to more general monoidal products. This allows to realize that much of quantum structure is actually very natural, it is the structure of monoidal (dagger) categories. As John Baez once nicely observed in his Quantum quandaries : this simple observation (way too simple to make any pure monoidal category theorist take notice!) goes a long way towards making one of the weirdest aspects of QM become entirely natural: it is entirely clear that it makes sense to think of QM in terms of cobordism representations, and also the product on is not cartesian, and in a natural way so. The non-cartesianness of the codomain is a direct reflection of this.
This is why I think Coecke-school "new foundations" is a genuine improvement over what the standard textbooks have and what the ancients had, simple as the basic idea may be. On the other hand – which is why I started this thread – one should not make the mistake (as sometime I see people make, such as in the MO comment that I mentioned in #1) to think that this alone is now already the complete foundations of QM. No, it’s just one small aspect of it, just about the monoidal structure. In fact, the most immediate restriction it has is that it currently concentrates on the fully topological case: finite dimensional (fully dualizable) Hilbert spaces. If I were in Bob Coecke’s group, I would try to push the whole undertaking to non-finite spaces (almost fully dualizable or "Calabi-Yau objects") to get to the physically interesting meat.
Other aspects completely left out by Coecke-foundations is the relation to phase spaces, observables, etc, the whole step from Schrödinger picture to Heisenberg picture. This is where the Isham-initiated ideas and then the Bohr topos comes in, I think. Together it may give a complete picture, in the end.
These last two comments ought to appear as a post at the Cafe, if that has any purpose still, unless it’s the lack of public audience in the forum which allows the bolder expression of opinions. I’d like to hear what Bob et al. would say to it.
unless it’s the lack of public audience in the forum which allows the bolder expression of opinions.
Funny that you say this, to me it feels the other way around. There may be more readers in total at the Café than here, but I don’t see there the audience that I am looking for. I have given up on it.
But in the last two days with http://theoreticalphysics.stackexchange.com/ in its beta phase, I am growing hope that that’s going to be a good place to hang around. When it becomes public in a few days, maybe we can move patrts of this discussion here to there. Hm, only that of course that place is not really for discussion . Sigh :-)
Maybe you have a point - the post Bohr Toposes - didn’t exactly provoke a flurry of discussion. But if the theoretical physics stackexchange is like MO, then discussion won’t be allowed there. I’m not completely convinced the Cafe is dead for the kind of discussion necessary. Perhaps Bohr Toposes didn’t spark discussion because it was written as exposition, and people are nervous responding to factual assertion by someone who obviously knows much more than them. What you’ve written above seems to me more likely to provoke a response (asking questions “Maybe I am distorting the history now (am I?)”, making suggestions “If I were in Bob Coecke’s group, I would try to push the whole undertaking to non-finite spaces “, expressing opinions “I think Coecke-school “new foundations” is a genuine improvement “, etc.). We could ask Bob and Andreas directly for their views.
Surely the question would come up as to how, if
Together it may give a complete picture, in the end.
is right, could this togetherness be captured mathematically.
Okay, maybe my way of making posts to the Café has been all wrong (though this is the first time that I hear the suggestion that it is too expositional, which confuses me now ;-). On the other hand, Zoran and Toby who usefully commented here almost never comment there. Also, I seem to remember the two or three times that I had made a comment in reaction to something Bob Coecke wrote there, I didn’t get a reaction.
Anyway, I realized I feel much better since I decided to ignore the Café since a few weeks back, which is a sign of something. I’ll leave it that way. Let’s come back to the topic:
could this togetherness be captured mathematically.
Yes, certainly. The idea would be to consistently replace algebras of observables with Bohr toposes and spaces of states with the corresponding internal construciton of classical states. For instance by vdBergh-Heunen the relevant category of (partial) -algebras has a tensor-structure, and by Nuiten we know how to transport this to the Bohr toposes.
Lots of things to play around with here. Hopefully people will investigate it. I am not investing a huge amount of energy into it, myself, though. I have started thinking about it at all only because I happen to be surrounded by people who do. Also I feel a bit nervous: the topic is so young, it is hard to see what the fruit to pick really is. That was a risk with Nuiten’s thesis. But I think it worked out well.
Zoran and Toby who usefully commented here almost never comment there.
It’s been a long time since I’ve regularly read the Café. I couldn’t keep up with it. I don’t read everything in the Forum either.
So there’s a distancing from the Cafe, but it would nice if someone competent, along with David Roberts, could say something sensible about John Huerta’s first post there (even if it’s to tell him to stop wasting his time there and join the forum community:) ).
I’ll post a reply. Only just arrived in my hotel after a transatlantic flight…
What you should do, Ben, if you think Urs should read your paper, is give a link to it.
I tried to perform a service and provide a link to the paper (it sounded interesting to me anyway), but I didn’t have luck finding the one article in particular.
Here is Ben’s web page.
The search led me to several interesting papers by Prakash Panangaden:
These are all subjects close to my heart. I’ve been having a bit of fun lately reviewing one of my old papers (with Urs) whose goal was not completely dissimilar:
Googling for keywords in Ben’s last message produces this pdf.
But, Ben, this is not really the kind of stuff that we were discussing here, I think.
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