Miriam Yevick was a mathematician who corresponded with physicist David Bohm in the 1950s and went on to publish a very interesting article on the formal structure of perception and cognition: Holographic or fourier logic [1]. The abstract:

A tentative model of a system whose objects are patterns on transparencies and whose primitive operations are those of holography is presented. A formalism is developed in which a variety of operations is expressed in terms of two primitives: recording the hologram and filtering. Some elements of a holographic algebra of sets are given. Some distinctive concepts of a holographic logic are examined, such as holographic identity, equality, contaminent and “association”. It is argued that a logic in which objects are defined by their “associations” is more akin to visual apprehension than description in terms of sequential strings of symbols.

In 1978 she commented on an article by John Haugeland, The nature and plausibility of Cognitivism, and spells out the implications of her idea for cognition ([2] p. 253):

The author here points out a distinction between two modes of understanding our environment: the first identifies objects by quasi-linguistic representations; the other apprehends objects by means of nonarticulate skills. This dichotomy; which is undoubtedly related to the complexity of the concrete objects to be recognized or manipulated, was projected as follows by von Neumann (1966, pp. 51-54): “certain objects are such that their description is more complex than the object itself.”

We can explicate this proposition on a theoretical level in the domain of optical patterns. (See Yevick, 1975op. cit.). Such patterns or objects are thin, white regions on a black background. These can be simple (regular), like the outlines of rectangles; or complex, like the outlines of Chinese characters or random-like motions. The following holds true: a complex object requires a long (sequential, quasi-linguistic) description but yields a sharp recognition (auto-correlation) spot under holographic filtering; hence it is identified most readily by holographic recognition, or holistically. A simple object requires a short (quasi-linguistic) description but yields a diffuse recognition spot; hence it is identified most readily by quasi-linguistic representation or description.

Description and holographic recognition thus appear as two (comple- mentary) modes of identifying an object: the more complex the object, the longer its description and the sharper its auto-correlation spot, and vice versa. The more complex they physiognomy of a person, the more unique, and hence sharper, its identity and ease of recall; the more simple, the more common and hence “unidentifiable.” Perfect holographic recognition ob- tains for a totally “random object”, that is, one with an infinitely long description; for a perfectly sharp point the opposite is true.

Suppose that one is given a store of objects with which one is familiar, a holographic recognition device, and a quasi-linguistic mode of representa- tion; one is then presented with an arbitrary object to be “identified.” An approximate match is obtained either by producing a description of acceptable length or by holographic recognition of a subset of similar (associated) objects from the store. The mode of identification that will be more appropriate then depends on the complexity of the unknown object. If it is simple, we "know" it by a short linguistic description; if it is complex, by the "associations” it evokes.

If we consider that both of these modes of identification enter into our mental processes, we might speculate that there is a constant movement (a shifting across boundaries) from one mode to the other: the compacting into one unit of the description of a scene, event, and so forth that has become familiar to us, and the analysis of such into its parts by description. Mastery, skill and holistic grasp of some aspect of the world are attained when this object becomes identifiable as one whole complex unit; new rational knowledge is derived when the arbitrary complex object apprehended is analytically described.

She goes on to point out that the same distinction holds in the domain of abstract objects (254):

A careful scrutiny of the various presentations leading to Godel’s result reveals that the “abstract objects” that are the entities under discussion in a formal system actually occur in two modes: as objects identified by bold- faced pictures or shapes or marks on paper, and as objects generated recur- sively from certain zero-entities, recognized in some way by their rank, that is, the first of a certain list (Quine, 1950); an expression of length one (Shoenfield, 1967); a sequence of one symbol (Godel, Pred. 15 in van Heijenoort, 1970; Mendelson, 1964); entities generated by a successor operation on a pair of arguments (Kleene, 1970, pp. 247, 251-252; Pred. Dn 1 should readry s”0). We recognize abstract objects of the first kind byosten- sion (holistically); those of the second kind are recognized by sequential generation or description. But whereas in the case of concrete objects discussed above, it is possible to assert that these two modes of (approximate) identification refer to the same object, the abstract objects have no identity recognizable beyond their formal mode of representation or generation. Thus, going beyond Haugeland’s remark and using the word “mode” for his “dimension,” we note that the mixing of modes is already present in the argu- ment that yields Godel’s undecidability result: it rests essentially on the identification of abstract objects (“formal numerals”) recognized in two dis- connected modes. The well-known confusion between Mention and Use reappears here as a confusion between showing and telling or display and enumeration, that is, as a mixing of dimensions.

The following quotation from Freudenthal (1960), who attempted to construct a language, “Lincos,” aimed at cosmic communication, clearly projects the irreducible duality: “We have agreed to abstain as much as possible from showing (concrete things or images of concrete things) but we cannot entirely abstain from it. Our first message will show numerals as an in– troduction to mathematics. Such an ostensive numeral, meaning the natural number n, consists of n peeps with regular intervals; from the context the reader will conclude that it aims aishowing just the natural number n.”

For minds to communicate or to do formal mathematics, they must possess both a quasi-linguistic (sequential, rational) and a holistic (ostensive, associative) dimension.

This has an obvious bearing on the current controversy in artificial intelligence between partisans of a pure neural network approach and those who argue that symbols are necessary as well. Neural networks exemplify Holographic or fourier logic, literally so in the case of convolutional neural networks. But symbolic reasoning is necessary as well. They apply to different classes of objects.

David Hays and I gave Yevick’s work an important place in our papers on the natural intelligence [3] and metaphor [4].

References

[1] Yevick, Miriam Lipschitz, Holographic or fourier logic, Pattern Recognition, Volume 7, Issue 4, December 1975, Pages 197-213, https://doi.org/10.1016/0031-3203(75)90005-9

[2] Yevick, Miriam L., The two modes of identifying objects: descriptive and holistic for concrete objects; recursive and ostensive for abstract objects. Brain and Behavioral Sciences, 1(2), 253-254. 1968, doi:10.1017/s0140525x00074458

[3] William Benzon and David Hays, Principles and Development of Natural Intelligence, Journal of Social and Biological Structures, Vol. 11, No. 8, July 1988, 293-322, https://www.academia.edu/235116/Principles_and_Development_of_Natural_Intelligence

[4] William Benzon and David Hays, Metaphor, Recognition, and Neural Process, The American Journal of Semiotics, Vol. 5, No. 1 (1987), 59-80, https://www.academia.edu/238608/Metaphor_Recognition_and_Neural_Process.