I would like to invite everyone to discuss another text that has been submitted to the international workshop “Computational Modeling II” (organized in Cracow, on 11.03.2019, at the UPJPII University).

The whole text is available HERE.

The text, enriched with some new elements by Paula Quinon, is an extract from two articles by Paweł Stacewicz: 1) “On different meanings of analogicity in computer science” (already published in Polish, in the journal “Semina Scientiarium”), 2) “Analogicity in Computer Science. A Methodological Analysis” (submitted and currently reviewed in the journal “Studies in Logic, Grammar and Rhetoric”).

Therefore it is not a fully original work.

Nevertheless, we decided to submit it for discussion, because we are now working on a new publication devoted to **analog/continuous computations**, and all additional critical input, and each additional discussion will be for us very precious.

Thus, we will be grateful for any comments that may contribute both: the improvement of the text no. 2 (which still is in the reviewing process), and the **development of our new ideas**.

To encourage you to read the whole text, we put two representative passages below:

**Two basic (general) meanings of analogicity**

Regardless of the (technical) aspect that is considered in contemporary computer science there exist two different (yet not necessarily separate) ways of understanding analogicity.

The first meaning, we shall call it AN-A, refers to the concept of *analogy*. It acknowledges that analog computations are based on natural analogies and consist in the realisation of natural processes which, in the light of defined natural theory (for example physical or biological), correspond to some mathematical operations. Metaphorically speaking, if we want to perform a mathematical operation with the use of a computational system, we should find in nature its *natural analogon*. It is assumed that such an analogon simply exists in nature and provides the high effectiveness of computations. The initial examples of AN-A techniques (that will be developed later) are: the calculation of quotient using the Ohm’s law (an illustrative example) or the integration of functions using physical integrators (a realistic example).

The second meaning, we shall call it AN-C, refers to the concept of *continuity*. Its essence is the generalisation (broadening) of digital methods in order to make not only discrete (especially binary) but also continuous data processing possible. On a mathematical level, these data correspond to real numbers from a certain continuum (for example, an interval of a form [0,1]), yet on a physical level – certain continuous measurable variables (for example, voltage or electric potentials).

In a short comment to this distinction, we would like to add that the meaning of AN-A has, on the one hand, a historical character because the techniques, called *analog*, which consisted in the use of specific physical processes to specific computations, were applied mainly until the 1960s. On the other hand, it looks ahead to the future – towards computations of a new type that are more and more often called *natural* (for example, quantum or computations that use DNA). The meaning of AN-C, by contrast, is more related to mathematical theories of data processing (the theoretical aspect of computations) than to their physical realisations. Perhaps, it is solely a theoretical meaning that, in practice, is reduced to discreteness/digitality (wedevelop this subject in section 2.2) due to physical features of data carriers.

Additionally, it is important to note that analogousness does not exclude continuity. This means that both continuous and discrete signals can be processed as analogons. Therefore, the above-differentiated meanings are not completely opposed.

**The physical realisation of continuous (hyper)computations**

Another methodological issue is related to analog computations in the sense of AN-C, that is continuous. Theoretical analyses indicate that computations of this type – described, for example, with the use of a model of recursive real-valued functions – have the status of *hypercomputations*. This means that they allow solving problems that are out of reach for digital techniques which are formally expressed by the model of universal Turing machine. One of such problems is the issue of solvability of diophantine equations.

Although the theory of continuous computations does predict that they have *higher computational power* than digital techniques, the important question about practical *implementability* of continuous computations arises. That is to say: if the physical world, the source of real data carriers and processes to process data, was discrete (quantised), we would never be able to perform any analog-continuous computations.

The question about the separateness of the mind (or even the mind-brain understood as a biological system) from the physical world, to which real digital automaton belong, is related to this issue. Perhaps the fact that the mind’s computational power is higher than the power of digital machines – which, according to some people, is proven by the observed ability of the mind to solve intuitively difficult mathematical problems – can be justified with the *continuity* of mental sphere (or even the continuity of nervous system).

Once again, **we invite** everyone to discuss our text — Paula Quinon & Paweł Stacewicz.

In order to grasp the meanings of analogicity under investigation in Paula and Paweł’s paper (henceforth P&P), a reasonably straightforward exercise might be to try and find the contrasting terms and synonyms for each meaning.

For analogy understood as AN-C, it will be discreteness or digitality – i.e. the discreteness of the operations of a digital computer – whereas the key synonym of analogy will be continuity, as P&P already highlighted. For example, and for the purposes of the present discussion, scientific models may be analog, material or digital, computer-based in their mode of realisation.

For analogy understood as AN-A, the contrasting terms and synonyms might be a bit harder to pin down. One antonym might be identity, the matching synonyms being correspondence or, more specifically, isomorphism. Compare: If “The world is its own best model” (Rodney Brooks), it will not be a model but simply the world. It neither corresponds nor is isomorphic to itself. (Note that Brooks coined this phrase as argument against the use of explicit models in Artificial Intelligence.) Conversely, if one is developing models (in AI or elsewhere), specific sets of correspondence relations between selected properties of the model (M) and selected properties of its object or “target system” (S) will have to be established. Paradigmatically, these correspondence relations are formally expressed as isomorphisms, understood as a one-to-one bijective mapping between the set of selected properties of M and the set of selected properties of S.

However, a second, probably more obvious antonym of analogy is its straightforward negation, that is, disanalogy (note that this transformation does not work with “analogicity”), with similarity being the corresponding synonym – which also figures in P&P’s text, as the “second level” on which the quality of analogueness manifests (p. [4]). Here, the relation in question is of a different kind. Similarity is a phenomenal, partly observer-dependent quality, not a formally defined relation. One might seek to establish formal criteria of similarity, but this is something that follows a, perhaps intuitive and probably subjective, perception of similarity. In contrast, isomorphisms do neither rely on nor necessitate similarities.

If this reconstruction is to the point, the question will be whether and to what extent the example chosen by P&P – the calculation of quotient using Ohm’s law – meets either or both ways of parsing the AN-A concept of analogy.

I will not enter deep into a discussion of the primary distinction in P&P’s paper – between AN-A and AN-C – because I would be writing a lengthy treatise in response, which will have to wait for another time. For now, I will leave it at raising the suspicion that many critics of the epistemic value of computer models tend to conflate AN-A and AN-C, by discrediting the analogies these models produce, in terms of both isomorphism and similarity, on the grounds of their discreteness and digitality: as a well-known objection goes, they can only deliver discrete approximations to continuous functions and will therefore never fully capture the nature of those functions. The value of P&P’s distinction in this context is this: We may grant to the critic that computer models can indeed only produce discrete approximations, but this does not per se disqualify AN-A relations, as these are always and by definition partial, either as partial isomorphisms or as partial similarities. This plainly is what models are, independent of the question whether they are computational and digital (AN-C) or analog and material (non-AN-C) in their realisation of AN-A relations.

Thank you very much for your commentary.

My first observation stems, perhaps, from my imperfect familiarity with English, especially when it comes to recognizing certain terminological nuances. The term “analogicity” (in Polish: analogowość) I chose for the general designation of methods/computations, which in computer science are called analog (called ambiguously).

The problem is that among these methods/computations there are both continuous computations (enabling to operate on continuous signals) and empirical computations (consisting in realization of dedicated physical processes, corresponding to mathematical operations).

In my view, therefore, the term “analogicity” covers both the feature of continuity of computations and the feature of their empiricality (which, perhaps erroneously, I called analogousness). Depending on the context, therefore, “analogicity” is either “continuity” (related to AN-C) or “analogousness” (related to AN-A).Therefore, it is not in line with my intention when Hajo writes “AN-C analogy” and “AN-A analogy”. Because only computations of the AN-A type refer to the concept of analogy. From my point of view it should rather be: “AN-C analogicity” and “AN-A analogicity”.

Perhaps Paula will speak here, who has a greater intuition in translating from Polish into English.

And another issue. Hajo correctly noticed that AN-A computations are more difficult to describe methodologically than AN-C. I have to think about his remarks.

As a synonym reflecting the essence of AN-A computations I propose to consider “empirical”. So analog computations of AN-A type would be as much analogous as empirical. In my other works I even use the acronym AN-E. Perhaps it is more appropriate. The essence of the analog computations of type AN-A (or AN-E) is their empiricality; to put it differently: dependence on the existence of appropriate, dedicated physical processes in nature.

And the last issue. Hajo writes about models and modelling. However, I would like to pay more attention to the context of the implementation of a certain type of computations (and problem solving) than to the context of modelling one or other phenomena. Nevertheless, the comments about modelling are interesting and I have to think about them.

I see that there is a lexical issue with the similar-sounding, related-but-different terms “analogy”, “analogicity”, “anlogueness”, which, if and when taken to mean roughly the same, will lead precisely to the conceptual ambiguities we are trying to resolve here. So I am happy to concede the terminological point to Paweł. (And I have a slight suspicion that even if we ask native English speakers for advice, their opinions on the semantics of those terms will not be unanimous.)

As to my emphasis versus P&P’s non-emphasis on analogies in the context of modelling, I sense there might be a somewhat different understanding of models between us. When Paweł says that he sometimes uses “empirical analogicity” instead of analogousness, it appears to me though that we might be after the same thing after all. My usage of the terms “model” and “analogy” is strongly informed by Mary Hesse (1966) and Max Black (1962), two mid-20th century philosopher of science who highlighted the importance of the use of models and analogies in the development of the sciences, and who largely equated the terms “model” and “analogy” in this context. So the scholarly tradition might be a different one, but the concern with relations in nature that are to be captured in a certain form, computational or other, does look similar to me. Perhaps my discussion of isomorphisms, as introduced by Black (1962), might provide the best link to Paweł’s empirical analogicity?

References:

Black, Max. Models and Metaphors. Cornell University Press, Ithaca, 1962.

Hesse, Mary B. Models and Analogies in Science. University of Notre Dame Press, Notre Dame, 1966.

Yes, Hajo is right when he writes about analogy and modeling.

Thanks!

In my previous comment I hurried a bit, writing that AN-A (or AN-E) computations have little to do with the modelling process. Of course they do…

If models are considered as a starting point of a theory (when a theory is being developed), or as a key element of a theory (when models are being developed within an already existing theory), then they are crucial for empirical computations.

I wrote this myself :)

I quote:

[

The first issue is directly related to the first way of understanding analogicity (AN-A) and concerns the reliability (in a narrow sense: accuracy) of computations based on the principle of natural analogy. As already indicated in point 1.2., the mathematical reliability of procedures of this type (that is efficiently using them to perform some mathematical operations) must depend on the level of adequacy of a theory that connects formulas and results of computations with physical reality (more accurately: processes that perform these computations).

The mentioned theory – being the result of idealisation procedure, typical of empirical sciences, which consists in examining phenomena ignoring factors that are recognised as unimportant – is never hundred percent appropriate. Thus, if the results of mathematical operations are sought directly in the reality that is described by the theory (for example, through experiment, measurement, etc.), they must be distorted by the very same factors that have been omitted during idealisation. Metaphorically speaking: the procedure of idealisation works both ways. It allows to create a cognitively effective theory but trying to realise theoretical computations by referring to (not idealised) reality, it must cause mistakes.

]

If we use the word “model” instead of “theory” in the above passage, we obtain an extremely strong connection between computations (empirical, natural, based on natural analogies…) and modelling procedure.

As I wrote above, this relationship both gives the basis for AN-A/E computations and causes some problems…

Thank you Hajo for pointing out interesting sources. I am also curious to comment on the above-mentioned problems…