A standard view of psychologists on the mind mechanisms was based on logic.  Symbols and abstract concepts in the brain were considered similar to logical symbols. During the last decade, a predominant view became that symbols and abstract concepts are represented in the brain by symbols grounded in perception and action. Among leading authors developing theories of grounded symbols are Barsalou, Cangelosi, Talmy, Tomasello. However, their psychologically-based theories have not been mathematically formalized.

In this paper the authors developed a mathematical model for Barsalou’s Perceptual Symbol Systems. This development required overcoming computational complexity, which have caused impasse during decades of developing computational linguistics, artificial intelligence, and other methods of modeling the mind mechanisms. The authors developed an original mathematical method, which seems to be a breakthrough, overcoming decades of difficulties.

Mathematical formalization in this paper has led to novel understanding of cognitive processes. Whereas Barsalou developed a powerful idea of distributed representations of mental concepts in the brain, and Bar demonstrated in neuroimaging experiments that mental representations are vague and not concrete, their ideas of distributed mechanisms and vague representations seems to require significant corrections in view of the mathematical models developed in this paper. Mathematical models, however powerful, cannot be considered a final scientific truth, until they are validated by experimental data. A number of experimentally verifiable predictions are made in this paper, so soon we might know the validity of the developed theories.

I taught a course on a similar topic

This paper offers a most interesting and  unconventional theory about how knowledge is represented in/by the mind. It relies on the  marriage  of  the Neural Modeling Fields – Dynamic Logic (NMF-DL) framework put forward by Perlovsky in the late 1990s  and the Perceptual  Symbol System (PSS) proposed by Barsalou at more or less the same  time. The PSS rejects the standard view that amodal symbols represent knowledge in semantic memory and in doing so it introduces  the elusive notion of perceptual (modal) symbols and simulators.

I found the paper particularly useful because it highlights my difficulty to understand the usefulness (I’m a pragmatic person) of PSS. In fact, NMF-DL is in its essence an unsupervised (clustering) algorithm, and the application illustrated in the paper – identifying situations defined by the appearance of  specific groups of objects  – is indeed typical of such algorithms. Once an unsupervised learning algorithm has finished its task of categorizing the objects into classes, the interpretation and further manipulation of those classes are left to "a higher order level in the hierarchy"  which  invariably means… a human! No wonder the authors have chosen to refer to  the object categories  as  "situations". I see the same type  of difficulty in the wet version  - PSS - of the unsupervised learning algorithms: amodal symbols must enter the game at some point.

Of course, nobody knows where the amodal symbols are in the brain or even if this concept has any meaning at the neuronal level, but since most of our thinking is done in linguistic terms and language is an amodal symbolic system I would rather stick to the devil I know.

I found particularly interesting, even premonitory, the assessment by the authors of the present status of standard amodal symbol systems:

 "… they promised to provide elegant and powerful formalisms for representing knowledge, because they captured important intuitions about the symbolic character of cognition, and because they could be implemented in artificial intelligence. As we discuss in the next section, these promises were unfulfilled due to fundamental mathematical difficulties.

As  Google has proved unquestionably, sheer computer power can work miracles and so it is not too farfetched to expect most of the mathematical difficulties pointed by the authors will be reduced to minor nuisances in a few decades.   In fact, many promising approaches to produce human level understanding, such as the Cogent  Confabulation theory of Hecht-Nielsen, rely heavily  on that bet.

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