I'll respond to Tobias in a post rather than a blog reply because he raises several points, and I want to include a picture or two.
1. TS: "When you cross the real-world boundary, i.e. when real-world items (such as wave lengths) are mapped onto cognitive categories (colors perceived), you are talking about something else since the real world is not a module."
The arguments I was making hold equally well for representations within the central nervous system (CNS), for example between the retina, the lateral geniculate nucleus and V1. Real-world spatial relations are mapped partially veridically onto the retina (due to the laws of optics). The spatial organization of the retina is (partially) maintained in the mapping to cortex; that is, LGN and V1 are retinotopic. So the modules here are the retina, LGN and V1, which are certainly modules within the CNS.
The same sort of relationship is true for acoustic frequency, the cochlea, the medial geniculate nucleus (MGN), and A1. Acoustic frequencies are mapped partially veridically onto the coiled line of hair cells in the cochlea (due to laws of acoustics). That is, frequency is mapped into a spatial (place) code at the cochlea (this is not the only mechanism for low frequencies). And the cochlear organization is partially preserved in the mappings to MGN and A1, they are cochleotopic (= tonotopic). There is an "arbitrary" aspect here: frequency is represented with a spatial code. But the spatial code is not completely arbitrary or random, but organized and ordinal, such that frequency increases monotonically from the apex to the base in the cochlea, as shown in the diagram from Wikipedia, and is preserved in tonotopic gradients in A1. That is, the mappings between the modules are quasimorphisms.
2. TS: "when I use the word "arbitrary" I only mean the above: the fact that any item of list A may be associated with any item of list B."
Then I think you should find a different term. I also think there has been far too much focus on the items. As I have tried to explain, items enter into relationships with other items, and we need to consider the preservation of these relationships across the interface or the lack thereof; we need to keep track of the quasimorphisms. So it is not the case for many of the intermodular interfaces in sensation and perception that any item on one side of the interface can be mapped to any item on the other side of the interface. Spatial and temporal and other ordering relationships tend to be preserved across the interfaces, and this strongly constrains the mapping of individual items. Remarkably, this is true even in synesthesia, see Plate 9 from Cytowic 2018.
3. TS: "That's all fine, but I am talking about the mind, not about the brain. Whatever the wirings in the brain, they won't tell us anything about how cognitive items of two distinct vocabularies are related (Vocabulary Insertion), or how a real-world item is associated to a cognitive category (wave length - color)."
I am not a dualist, and I doubt that this blog is a good forum for a discussion of the merits of mind/body dualism. Here is a quote from Chomsky 1983 on the mind/brain, he reiterates this in Chomsky 2005:257 and in many other places.
"Now, I think that there is every reason to suppose that the same kind of “modular” approach is appropriate for the study of the mind — which I understand to be the study, at an appropriate level of abstraction, of properties of the brain ..."
Just to be clear, I am not saying that cognitive scientists should defer to neuroscientists, but they should talk to them. The idea that we have learned nothing about color perception and cognition from the study of the human visual pathway is simply false.
4. TS: "is there evidence for interfaces that are not list-based?"
Yes, almost any (non-linguistic) set of items with an ordering relation. When aspects of the ordering relation are preserved across the interface the mapping will be a quasimorphism, and thus the item-to-item mappings will be strongly constrained by this, that is, if a < b then f(a) <f f(b). What's unusual about the lexicon is that small changes in pronunciation can lead to enormous changes in meaning. In many of the other cases we instead end up with a very small, almost trivial look-up table, something like the sets of basis vectors for the two spaces, as with homomorphisms between groups in algebra.
5. TS: "is there evidence for associations that correspond to partial veridicality, i.e. where the to-be-related items are commensurable, i.e. allow for the assessment of similarity?" ...
"The same goes for the association of real-world items with cognitive categories: trying to assess the (dis)similarity of "450-485 nm" and "blue", as opposed to, say, "450-485 nm" and "red" (or any other perceived color for that matter) is pointless. Wave lengths and perceived colors are incommensurable and you won't be able to tell whether the match is veridical, non-veridical or partially veridical."
This isn't pointless at all. In fact, remarkable progress has been made in this area. See, for example, Hardin 1988, Hardin & Maffi 1997, Palmer 1999 and Bird et al 2014. The match is partially veridical in a variety of ways. Small changes in spectral composition generally lead to small changes in perceived hue; the mapping is a quasimorphism. Importantly, the topology of the representation changes -- and thus is a non-veridical aspect of the mapping, from a linear relation to a circular one in the cone cells of the retina to an opponent process representation in LGN.
6. TS: "The secret key is the look-up table that matches items of the two modules."
I agree with this, except that I want the look-up table to be as small as possible, the "basis vectors" for the spaces. In my opinion, the best way to accomplish this is with innate initial look-up tables for the features, giving the learner the initial conditions for the Memory-Action and Perception-Memory mappings. The feature-learning approaches, including Mielke 2008, Dresher 2014 and Odden 2019, start with an ability to perceive IPA-like phonetic representations. I simply don't believe that this is a plausible idea, given how difficult even simple cases are for such an approach, as explained in Dillon, Dunbar & Idsardi 2013.
Bird CM, Berens SC, Horner AJ & Franklin A. 2014. Categorical encoding of color in the brain. Proceedings of the National Academy of Sciences, 111(12), 4590–4595.
Chomsky N. 1983. The Psychology of Language and Thought: Noam Chomsky interviewed by Robert W. Rieber. In RW Rieber (ed) Dialogues on the Psychology of Language and Thought. Plenum.
Chomsky N. 2005. Reply to Lycan. In LM Antony & N Hornstein (eds) Chomsky and his Critics. Blackwell.
Cytowic RE. 2018. Synesthesia. MIT Press.
Dillon B, Dunbar E & Idsardi WJ. 2013. A single-stage approach to learning phonological categories: insights from Inuktitut. Cognitive Science, 37(2), 344–377.
Hardin CL. 1988. Color for Philosophers: Unweaving the Rainbow. Hackett.
Hardin CL & Maffi L. 1997. Color Categories in Thought and Language. Cambridge University Press.
Palmer SE. 1999. Vision Science: Photons to Phenomenology. MIT Press.