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Using a Topological Model in Psychology: Developing Sense and Choice Categories


A duality of sense categories and choice categories is introduced to map two distinct but co-operating ways in which we as humans are relating actively to the world. We are sensing similarities and differences in our world of objects and persons, but we are also as bodies moving around in this world encountering, selecting, and attaching to objects beyond our sensory interactions and in this way also relating to the individual objects’ history. This duality is necessary if we shall understand man as relating to the historical depth of our natural and cultural world, and to understand our cognitions and affections. Our personal affections and attachments, as well as our shared cultural values are centered around objects and persons chosen as reference points and landmarks in our lives, uniting and separating, not to be understood only in terms of sensory selections. The ambition is to bridge the gap between psychology as part of Naturwissenschaft and of Geisteswissenschaft, and at the same time establish a common frame for understanding cognition and affection, and our practical and cultural life (Mammen and Mironenko 2015). The duality of sense and choice categories can be described formally using concepts from modern mathematics, primarily topology, surmounting the reductions rooted in the mechanistic concepts from Renaissance science and mathematics. The formal description is based on 11 short and simple axioms held in ordinary language and visualized with instructive figures. The axioms are bridging psychology and mathematics and not only enriching psychology but also opening for a new interpretation of parts of the foundation of mathematics and logic.

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  1. In the present context with a special focus on the modern development in foundational mathematics, mathematical logic, theory of sets and general topology. See e.g. Kuratowski (1961), Crossley et al. (1972), Moore (1982).

  2. The terms “numerical” and “qualitative” identity can be a little misleading because of their linguistic similarity with the modern distinction between “quantitative” and “qualitative” methods, etc., to which there are no simple relations, however. In contrast, the distinction is rather close to Aristotle’s classical distinction between objects’ matter and form, respectively. “Matter” is referring to objects’ individual existence and their continuing identity in time. “Form” is referring to their belonging to classes defined by similarities and differences. These distinctions will be elaborated further in the paper.

  3. More formally speaking the Axiom of Comprehension is a rule for defining sets and therefore also sometimes just called the Comprehension Schema. It says that to define a set b it is not sufficient that members of b have some property P. You have to have another set a and then define b as the subset of a having the property P. See e.g. Crossley et al. (1972, pp. 59–61).

  4. These proximal stimulations are of course carrying information of their distal sources (objects, etc.), and we are accordingly interpreting them as or allocating them to these distal sources. The question of how this localization is accomplished, how some spatial frame or “coordinate-system” for the sensory information is established, has a long history in psychology, including how it is related to the organization of our actions or “the motor system” (e.g. H. Lotze, I.M. Sechenov, E. von Holst, J.J. Gibson). However this will not be further discussed here. For the only known Danish experimental investigation of spatio-temporal integration in the sensory-motor system, see Mammen (1969).

  5. Gibson is one of the few 20th century investigators of sensing and perception who stresses the practical activity of the individual as fundamental for understanding our knowledge of the world. Another tradition, building on the “romantic” view of Fichte, Schelling and Hegel has via Marx had great influence on Activity Theory (Mammen and Mironenko 2015) and has a much more general concept of practical activity transcending Gibson’s which is restricted to immediate physical interactions with the environment. Unfortunately this tradition has had very little influence on modern mainstream psychology.

  6. If we think of measurement as an act of comparison of a given object x with some norm, or point of reference, it will be close to practical reality to say that a difference can always be detected within finite time, however small, but that identity is undecidable in the sense that we can never know if an undetected difference is due to identity or due to that we have not yet reached a level of precision or a step in the comparison procedure where a difference would be detected. There is an asymmetry between detecting difference and identity. Difference can be detected with certainty within finite time, identity can never. This is parallel to the example with the Mandelbrot Set given above.

  7. The “naked” senses are restricted in their discriminative capacity by sensory thresholds, “just noticeable differences”. Equipped with measuring instruments, microscopes, etc., in principle yielding digital output, these thresholds are irrelevant and are replaced with uncertainty of measurements defined by the instruments’ interaction with the objects of measurement, ultimately restricted by quantum uncertainties.

  8. We have already, before selecting by our senses, chosen a “domain” to partition by selection, cf. the Axiom of Comprehension mentioned above.

  9. An intensional definition of something (e.g. a set of objects) is a definition by properties, and accordingly is an intensional definition of a concept a description of the meaning of the concept. An extensional definition of something (e.g. a set of objects) is a listing of all the members of the set, and accordingly is an extensional definition of a concept a reference to all examples falling under the concept. In a broader sense intensionality refers to the way you define something, and extensionality to the resulting class. The definition of gold from the periodic system is intensional, and statements about its geographic distribution and rarity are extensional. There are some parallels to the distinction between connotation and denotation, respectively, as used by e.g. Bertrand Russell, and to the type-token distinction, as used e.g. by C. S. Peirce. “Intensional” should not be confused with “intentional” which refers to having a purpose or goal or philosophically to be directed towards or referring to something, as e.g. used by Franz Brentano for whom this directedness is a defining property of consciousness. But already Aristotle used equivalent concepts.

  10. Formal logic and theory of sets also share the so-called duality rules. In formal logic they say that the negation of a conjunction (“and”) is the disjunction (“and-or”) of the negations, and vice versa that the negation of a disjunction is the conjunction of the negations. In theory of sets the duality rules say that the complement of an intersection is the union of the complements, and vice versa that the complement of a union is the intersection of the complements.

  11. According to the Löwensheim-Skolem-Tarski Theorem (Mammen et al. 2000, pp. 81–84) this makes the statements “immune” in relation to cardinality of infinite sets, which is a precondition for some later conclusions in the paper, but which will not be discussed explicitly in the present context.

  12. See the later theorem Th. 10 (globality).

  13. Remember that this does not necessarily imply that we for any object not belonging to the sense category can decide that it doesn’t.

  14. This presupposes that there is some “order” in the set of sense categories so that one of them can indeed be “found” to which the object belongs. This is a reasonable assumption if the set of sense categories is finite. But if the set is infinite this is not trivial, cf. the above discussion of the mathematical Axiom of Choice. However, at this stage of analysis we are as said the senses’ advocate and accept the statement in Ax. 3.

  15. As with Ax. 3 the claim that such a difference can be found, if the potential set of differences is infinite, is not trivial, cf. also here the above discussion of the mathematical Axiom of Choice.

  16. True duplicates can be re-introduced in an expanded analysis. In this paper they will, however, be excluded. In a later section of this paper there is a short discussion of the introduction of time and the consequences for the below axiom Ax. 4 (Hausdorff), which implies a “softer” or “weaker” and perhaps more realistic interpretation. For simplicity reasons this has been omitted at this place in the presentation.

  17. See earlier footnote.

  18. In geometrical topology an axiom is added which defines the topological space as connected, i.e. that it can’t be divided in two disjunct non-empty open sets. It can’t be “cut in two”. This axiom makes our Ax. 5 to a theorem as any connected Hausdorff-space is also perfect, while a perfect Hausdorff-space is a more general structure which need not be connected. Usually one further axiom is added in geometrical topology which defines the space as compact, i.e. not “open-ended” as the real axis or real plane but rather “limited” as a circle, a sphere or a torus. Geometrical topology is thus about compact, connected Hausdorff-spaces, and primarily about their systematic classification.

  19. This is true in any case when talking of classical physics within the mechanistic frames rooted in Renaissance physics. In modern physics, including quantum physics, there may be a need to introduce something like choice categories also. This will not be treated in detail here.

  20. A manifestation of that is of course the use of computer models for human cognition. The computers’ only relation to the world is through their input received as physical signals. All ascriptions of symbols to the computer and all references to phenomena in the world, which tacitly presuppose choice categories, are nothing but projections from naïve cognitivist researchers (Mammen and Mironenko 2015). You could just as well claim that a book knows what it is about.

  21. Formally any object in Ù is defining a continuous function of time in “geographical” space, a connected trajectory, and also a continuous function of time in a “featural” space. The objects in Ù are thus linking or “pairing” functions in the two spaces. If now choice categories are interpreted on the set of trajectories and sense categories are interpreted on the set of featural functions, using the compact-open topology for function spaces (Fox 1945), all the eleven axioms will again be satisfied. There is even a correspondence between this more general dynamic “time dependent” case and the static case which could be considered a “snapshot”, or a cross-section with time in the dynamic case. In fact the “snapshot” case is a limiting case of the more general “time dependent” case, which on the other hand can be seen as a generalization of the static case. This means that the discussions so far are equally valid for the two cases.

    In fact the correspondence between the cases is so strong that the satisfaction of the axioms in one of the cases implies the satisfaction in the other case with two important exceptions:

    1. 1)

      The satisfaction of axiom Ax. 4 in the static “snapshot” case implies the satisfaction of the axiom also in the “time dependent” case but not necessarily vice versa

    2. 2)

      The satisfaction of the axioms Ax. 5 and Ax. 6 in the “time dependent” case implies the satisfaction of the two axioms in the “snapshot” case, but not necessarily vice versa.

    The interpretative consequences of this are supportive for the axiomatic system but too comprehensive for the present context. For technical details see the more elaborated discussion in Mammen et al. (2000, pp. 224–250). Here shall just be mentioned one consequence: If the axioms from the beginning had been defined on the general dynamic, “time-dependent” case, with the static “snapshot” case as only a limiting case, all axioms would also be satisfied in this limiting case except Ax. 4 (Hausdorff) which would not necessarily be satisfied. If two objects have different “feature functions” in some interval of time it can’t be excluded that they momentarily have the same features. This could open for a little weaker and perhaps more realistic interpretation of Ax. 4.

  22. That the 11 axioms are really logically independent has been proven in (Mammen 1996, pp. i–xiii). An updated shorter version in English can be found in the Appendix to this paper.

    Sense categories being “true unica” and choice categories being “true duplicates” have, as already noted, been excluded in this version of the axiomatic system for the sake of simplicity. Their re-introduction (not published) will make no essential difference in the discussion above.

  23. In this case is used a version of the Axiom of Choice (see Moore 1982) known as Zorn’s Lemma, close to Hausdorff’s “maximal” principle, mentioned above. Hoffmann-Jørgensen’s proof refers to a mathematical structure called maximal perfect topologies (Hewitt 1943; van Douwen 1993; Mammen et al. 2000, pp. 165–193). The reader of the proof is warned, that the section “4. Example” on the last pages is an independent demonstration of a constructive example of a perfect Hausdorff-space on a countable point-set but with a non-countable basis, and not part of the proof itself. What I here have called “decidable categories” is by Hoffmann-Jørgensen called “resolvable categories” which corresponds to subsets in what Hewitt calls an “irresolvable” topology, which again corresponds to what I have called “complete”.

  24. As a potential (cf. “Hoffmann’s Conjecture”) version of the Axiom of Choice the completeness of categories in the axiomatic system expressed in theorem Th. 11 has the virtue of being first-order-language equivalent and hence with no reference to cardinality which many other versions of the axiom have (Moore 1982, pp. 330–333).

  25. See discussions of maximal perfect topologies in Hewitt (1943), van Douwen (1993); Hoffmann-Jørgensen (2000), and Mammen et al. (2000).

  26. This will be the case if sense categories in some way can be represented in an “alphabet” of symbols. The equivalent case has a central role in mathematical proofs of incompleteness, e.g. Kurt Gödel’s famous proofs. In Mammen (1996, pp. xiv–xviii), it has been proven that if the set of sense categories has a countable basis the space of decidable categories can’t be complete.

  27. These mathematical concepts, defining degrees of “order” in organized sets will not be described further in the present context. About “countable basis” see Mammen (1996) and Mammen et al. (2000).

  28. The set of practically available sense categories may change from individual to individual, within individual lifetime, and with historical, cultural and technological conditions.

  29. The intersection of two disjunct open intervals is Ø which is not in S.

  30. The union of two disjunct open intervals is not an open interval and hence not in S.

  31. Irrational numbers are not member of a subset in S.

  32. {2} is the union of Ø in S and {2} and hence itself in S.

  33. No subset in C is finite.

  34. The intersection of a subset in C defined by −∞ < q ≤ 0 and a subset defined by 0 ≤ q < ∞ is {0} which is not in C. On the other hand no intersection with a subset in S can “isolate” {0} as member of C using Ax. 11.

  35. {1,2}, the union of {1} and {2}, is not a choice category.

  36. Ø is in S and its intersection with {1} is Ø which is not in C. This example is a correction to the example in Mammen (1996) due to an adjustment of Ax. 8 (see Mammen et al. 2000, pp. 260–261).


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I thank the organizers of the workshop July 5–8, 2015, in Remich, Luxembourg: “Structuring of dynamic borders: Topological models for life-course transitions” Jaan Valsiner and Dieter Ferring for the invitation and the participants for useful discussions. I want to thank Jens Kvorning for help with the figures and for constructive suggestions on how to visualize the ideas at the workshop and in the paper. I further want to thank Jaan Valsiner, Jens Kvorning, Niels Engelsted and Jørgen Hoffmann-Jørgensen for useful commentaries to a draft and Ib Madsen for useful suggestions for improving the precision and clarity of the mathematical presentation in the paper.

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Proof of Independence of Axioms for Sense and Choice Categories

This proof of independence of axioms is a free translation from Mammen (1996, pp. vi-xiii), with some omissions and with changes in terminology and the naming of the axioms. The correspondence between the naming of the axioms in Mammen (1996) and in the present paper is listed in Mammen et al. (2000, p. 259–261). There are 11 axioms:

Ax. 1: There is more than one object in Ù

Ax. 2: The intersection of two sense categories is a sense category

Ax. 3: The union of any set of sense categories is a sense category

Ax. 4 (Hausdorff): For any two objects in Ù there are two disjunct sense categories so that one object is in the one and the other object in the other one

Ax. 5 (perfectness): No sense category contains just one object

Ax. 6: No non-empty choice category is a sense category

Ax. 7: There exists a non-empty choice category

Ax. 8: Any non-empty choice category contains a choice category containing only one object

Ax. 9: The intersection of two choice categories is a choice category

Ax. 10: The union of two choice categories is a choice category

Ax. 11: The intersection of a choice category and a sense category is a choice category

The proof uses the method of “models”, i.e. referring to examples of mathematical “spaces” on a point-set or “universe” Ù of points or objects where some of Ù’s subsets are appointed sense categories and other subsets are appointed choice categories. In each of the examples S denotes the set of all sense categories and C the set of all choice categories. S is not necessarily defining a topology in Ù.

The logic of the proof is that consistency of a set of axioms is proven if there exists an example of a space where they are all valid. Given this proof it is further proven that an axiom is independent of the other axioms if there exists an example of a space where all axioms are valid except the one in question. So we need 12 examples of spaces for the proof, one for the entire set, and one for each of the axioms.

The examples are all referring to subsets of the real axis, here denoted R. The subset of all rational numbers in R is denoted Q. The empty set is denoted Ø. The subset of all unions of open intervals in R including Ø is denoted O. The examples have no interpretative relation to the set of objects treated in this paper which are neither points in nor parts of R. The examples are chosen only for the purpose of technical proof, and an infinity of other examples could have served the same purpose.

Here are the 12 examples:

All axioms valid: Ù = R; S = O; C = all finite subsets in R.

All except Ax. 1: Ù = {1}; S = Ø; C = Ø and {1}.

All except Ax. 2: Ù = R; S = O except Ø; C = Ø and {1}.Footnote 29

All except Ax. 3: Ù = R; S = all open intervals in R and Ø; C = Ø and {1}.Footnote 30

All except Ax. 4: Ù = R; S = all intersections of Q and O; C = Ø and {1}.Footnote 31

All except Ax. 5: Ù = R; S = O and all unions of O and {2}; C = Ø and {1}.Footnote 32

All except Ax. 6: Ù = R; S = O; C = all subsets in R.

All except Ax. 7: Ù = R; S = O; C = Ø.

All except Ax. 8: Ù = R; S = O; C = all intersections of O and Q.Footnote 33

All except Ax. 9: Ù = R; S = O; C = Ø and all unions of a finite non-empty subset in R not including {0} and an intersection of O and subsets of q ϵ Q defined as −∞ < q ≤ 0 and/or 0 ≤ q < ∞.Footnote 34

All except Ax. 10: Ù = R; S = O; C = Ø, {1} and {2}.Footnote 35

All except Ax. 11: Ù = R; S = O; C = {1}.Footnote 36

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Mammen, J. Using a Topological Model in Psychology: Developing Sense and Choice Categories. Integr. psych. behav. 50, 196–233 (2016).

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  • Sense categories and choice categories
  • Mathematics, Formal Logic, Theory of Sets, Topology, Decidability, Axiomatic method, Axiom of Choice
  • German Philosophy
  • Aristotle
  • Naturwissenschaft vs. Geisteswissenschaft
  • Affection and attachment
  • Standards and reference points