Abstract
The duality in humans’ asymmetric subject-object relations to the world can be described as two different ways of selecting, differentiating, and combining categories of objects in the world, sense categories based on objects’ form or features and choice categories based on objects’ existence as matter distributed and moving in space and time. The joint structure of these categories can be described in a simple and elegant axiomatic system using modern mathematical logic and topology, but expressed in terms of ordinary language, and with far-reaching consequences.
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Notes
- 1.
More formally speaking is the axiom of comprehension 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 defined as the members of a having the property P. See, e.g., Crossley et al. (1972, pp. 59–61).
- 2.
In mathematical terms decisions about membership of classes or categories could be seen as continuous mappings of a domain of objects on a discrete set of decisions, ultimately on a “yes/no” set. The extensional method is then, in this frame of reference, the same as a description of the mappings through a description of their inverse images and their structure. In this frame continuity means that inverse images of open sets in a topology are open themselves. The method could also be seen as an ecological generalization of classic experimental psychophysics (Mammen, 2002). Some of these concepts will be explained in later sections.
- 3.
According to the Löwenheim-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.
- 4.
See the later theorem Th. 10 (globality).
- 5.
Remember that this does not necessarily imply that we for any object not belonging to the sense category can decide that it doesn’t.
- 6.
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.
- 7.
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.
- 8.
True duplicates can be reintroduced 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 imply a “softer” or “weaker” and perhaps more realistic interpretation. For simplicity reasons this has been omitted at this place in the presentation.
- 9.
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.
- 10.
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 nonempty 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.
- 11.
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, as discussed above.
- 12.
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 & Mironenko, 2015). You could just as well claim that a book knows what it is about.
- 13.
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 11 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.
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.
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.
-
1.
- 14.
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 of 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 reintroduction (not published) will make no essential difference in the discussion above.
- 15.
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.”
- 16.
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).
- 17.
- 18.
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.
Roger Penrose (1989) has suggested that human perception and thinking may be “non-algorithmic” to explain some phenomena apparently going beyond the capacity of any classical digital computer. This would, e.g., be the case if humans’ set of sense categories has no countable basis. Already the existence of nonempty choice categories is, however, incompatible with digital computers’ discrete/digital and proximal interface with their environment.
- 19.
- 20.
The set of practically available sense categories may change from individual to individual, within individual lifetime and with historical, cultural, and technological conditions.
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Mammen, J. (2017). Sense and Choice Categories. In: A New Logical Foundation for Psychology . SpringerBriefs in Psychology(). Springer, Cham. https://doi.org/10.1007/978-3-319-67783-5_7
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