95© The Author(s) 2017 J. Mammen, A New Logical Foundation for Psychology, SpringerBriefs in Psychology, https://doi.org/10.1007/978-3-319-67783-5

Commentary 1: Faith, Hope, and Love

A Commentary to Jens Mammen: A New Logical Foundation for Psychology, Springer, 2017

Niels EngelstedUniversity of Copenhagen, Denmark

Crossing Over

As a young man, fearing to be drawn in completely by the siren song of numbers, sets, theorems, and lemmas, the author of the present book fled his mathematical and physical studies to seek refuge in the faraway land of psychology. He didn’t entirely shed his attire, though; today you still ask him about number theory at your peril.

Cross-dressing—entering one field of science clad in the apparel of another—has been a known way to score points with the natives and gain professional suc-cess. In psychology, mostly frequented by softheads, hardheads from mathematics have had a particularly easy time. Who dare call out the Emperor or question what they don’t understand? This partly explains why the curriculum is stocked with mathematical methodology students are expected to learn and why our official his-tory pivots around mathematical discoveries. Thus G.T. Fechner, a physics profes-sor, is said to have launched modern scientific psychology by subjecting sensory experience to logarithmic scales; George A. Miller is credited with inaugurating the current triumphant line of cognitive psychology with his magical number seven plus minus two; and it was when Daniel Kahneman discovered that Homo economicus has calculating deficiencies that psychology received its first Nobel Prize.

All the cheering notwithstanding, the importance of mathematics in psychology has been exaggerated. You certainly don’t want to belittle achievements like signal

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detection theory, information theory, and game theory, but their virtues untold, such theories at best touch the periphery of the psychological domain, and when the heartland is reached for, usually result in reduction, if not plain distortion. Truth be told, with the proverbial exceptions to prove it—factor analysis, perhaps—it seems to have been the rule that the crossing of psychology and mathematics has contrib-uted little to the first and less to the second.

The present work breaks that rule. Jens Mammen is not a mere cross-dresser; he is a true trans, and his work throws great light upon both psychology and mathemat-ics. That is what makes this work so highly original and extraordinary.

Speaking Prose

What people are most likely to remember from Molière’s comedies is Monsieur Jourdain’s exclamation when to his surprise and delight he is told that he is speaking prose: “My God! Then I’ve been speaking prose all my life without even knowing it!”However risible we think Monsieur Jourdain, speaking prose is what Mammen’s work is all about. It is the theme that clicks together psychology and mathematics. The basic plot is this.

Until Ernst Zermelo in 1904 put forward his axiom of choice, mathematicians had no idea that they were speaking prose. When told, they were surprised and criti-cal; and if eventually prose was accepted, it took a while, and the odd one out still balks. Speaking-prose-neglect is found among psychologists too; only in their case, the cause is not ignorance but partly overfamiliarity, partly dogma. Like the fish is last to discover the water in which it swims, prose comes so natural to psychologists that they rarely give it a thought, and this thoughtlessness is certainly not challenged by the prose-averse Galilean science from which mainstream psychology has taken its cue. Ordinary language is unfit for science, said Galileo; science must be “writ-ten in the language of mathematics … without which it is humanly impossible to understand a single word … [and] one is wandering about in a dark labyrinth.”1

Prose is the flow of ordinary language, structured around the subject-object rela-tion, and the subject-object relation is the soul of psychology. Disbanding prose, Galilean science disbanded the subject-object relation defining of the previous Aristotelian psychology. Galileo was the leading light in Abbé Mersenne’s illustri-ous seventeenth-century circle of mathematical prodigies and pioneering physical experimentalists, and Thomas Hobbes, a member, was first to draw the conclusion, reject the Aristotelian psychology, and turn the human being into a mechanical machine. Upon which, seeing the problem, René Descartes, another member, tried to put the soul back into the machine; upon which, seeing the problem, Baruch Spinoza, a student of his, simply fused machine and soul into panpsychic unity; upon which, seeing the problem, Hobbes’ British compatriots recommended concentrating on subjective experience and leaving aside the material object; upon

1 Galileo (1623), p. 4.

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which, seeing the problem, Immanuel Kant, etc. On and on it went through centu-ries of philosophy and thinking, and it’s still going on.

Into the fray enters Mammen as any serious student of psychology should. In the book, his lifelong campaign against the dragons that keep Lady Psychology captive is both vividly and lucidly narrated. Two things are highly noticeable from his account:

Firstly, that while there seemingly are two—very different—antagonists, suffo-cating mechanical physics at one end and huffing and puffing phenomenology and hermeneutics at the other, they are really the two heads of one and the same monster, a poor beast suffering from an unfortunate disturbance of the subject and object connection.

Secondly, that Saint George has a most unexpected lance to wield: the axiom of choice!

Axiom of Choice

The axiom of choice basically states that if there is food in the fridge, you can get it without specifying it first; you just open and grab. As this seems obvious, why not to mathematicians? The reason was that to mathematicians it was the specification that brought an object into being; without specification, there could be no object; object and rule-bound specification were simply the same. Trying to be helpful, Bertrand Russell explained that prior to the axiom, you could pick a shoe from a closet full of shoes, but not a sock from a drawer full of socks, because the shoe could be specified as a left or right, and the sock could not. After the introduction of the axiom, however, you could get the unspecified sock also.

Still those socks created some anxiety with the mathematicians. “It is not alto-gether uncontroversial that the axiom of choice should be accepted as something that is universally valid,” wrote the renowned physicist Roger Penrose. “The trouble with this axiom is that it is a pure ‘existence’ assertion, without any hint of a rule.”2 Penrose’s statement brings us right to the heart of the matter: rule or existence?

Rule in our context is synonymous with specification and description, and this again is the same as appearance or phenomenon. In other words, mathematicians originally shared the conviction of the subjective idealist philosophers in Britain, who held—in the words of George Berkeley—that esse est percipi, that being is appearance.3 If we here mention that the land of psychology, in which our fugitive author was seeking asylum, was ruled by a rigid school of phenomenology led by a professor, who took George Berkeley as his model; that the first string of theoretical papers Mammen wrote were directed against this professor and the narrow subjec-tivism of that school; and that this criticism went to the barricades in a 1968 student

2 Penrose (2004), p. 366.3 Not that strange since mathematical objects, like circles and triangles, are best thought of as ideal Platonic forms without material substance.

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rebellion of which Mammen was an organizer, then this obviously goes a long way to explain his alertness to the heterodox and heretical mathematical axiom that claims that existence matters and appearance is not everything.

A dictionary tells us that “phenomenal” means cognizable by the senses, whereas “existential” means having being in time and space, and with this pair we have reached the duality that runs like a red thread through Mammen’s work. Sense categories and choice categories, he calls them, the first referring to sensory appear-ance and the latter to the axiom of choice with its existence claim.

The distinction between sensory appearance (aka qualitative identity) and existence as temporal-spatial being (aka numerical identity) is—like speaking prose—not something to which we usually pay attention. Those familiar with detective series will be well aware, however, that identification is not grabbing, nor grabbing iden-tification; a culprit on a police file and a culprit in custody are two quite different things; you can have the one without the other.

With the axiom of choice, grabbing is introduced in mathematics. As Mammen perceptively notes, it was always there through the backdoor—take a circle, said the teacher from the blackboard to the class—but until the axiom of choice, never through the front door. Grabbing is a subject-object relation, which means that with the axiom, psychology enters mathematics in a formal and axiomatic way. If, how-ever, the axiom of choice is the ingress of psychology in mathematics, it is also the ingress of mathematics in psychology. There was never before a gathering like this, and Mammen’s claim to have introduced a new logic is completely justified. If there be reason and justice in the world, its future impact should at least equal Fechner’s psychophysics a century and a half ago.

In the book, playing both fields, Mammen elaborates and develops both what the new logic means for mathematics, and what it means for psychology. I shall leave the first for the hardheads to savor and, in this commentary, keep to psychological issues we have both shared.

Beginning with the Beginning

Having first met when he was external examiner on my PhD thesis on human evolu-tion and human exceptionality,4 my work dovetailed with Mammen’s, when, read-ing his doctoral dissertation5, I became convinced that his category of choice fitted the idea I was working on, namely, how the psyche—the subject-object relation—came into the material world in the first place.6 Years of friendship and cooperation followed,7 and the above idea—to link intentionality with thermodynamics via the living being’s locomotion through time and space in quest of food—has been

4 Engelsted (1984).5 Mammen (1983).6 Engelsted (1989).7 Mammen , Engelsted, & et al. (2000).

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included in the present work, where Mammen very aptly calls it the symmetry breach that saves psychology and the world from panpsychism. The threat of pan-psychism must be averted, of course, when Mammen, in another of the book’s important contributions, lectures us that the notion of physics habitually entertained by psychologists is grossly antiquated, and that modern physics in its portfolio has long had a full range of phenomena traditionally—and falsely—reserved only to life and psyche, spontaneity, active selection, and discontinuity among them.

The primordial subject-object sequence mentioned is the easiest entrance to some of the intricacies and challenges contained in Mammen’s work. The sequence, valid from amoebas to human shoppers, has—as seen in Fig. 1—five steps, the last of which has two options.

First the subject sets out into the blue in search of its object, food in the case of amoebas, merchandise in the case of shoppers. Next, with luck, informative traces are picked up to guide the subject toward the object. When tangible contact is reached, the subject tries to grab and hold on to the object. Finally, the object is either consumed, always with amoebas, or safekept, with humans also a possibility.

If you have grown tired of mind and behavior and want another short definition of what psychology is about, faith, hope, and love would not be the worst choice, which justifies this commentary’s title. More to the point, the three concepts are also highly pertinent to the issues at hand, which explain their reappearance in the figure. Mammen practically makes love the defining subject matter of human psychology. Comparing humanism and psychology to Goethe’s Doctor Faustus and his deplor-ing fate of “eternal emptiness,” the rescue is also the same, says Mammen, namely, “love, or a hitherto ignored structure in the world, and in human existence, with love as a paradigmatic example.” To explain that is—besides to present the new mathematics—the aim of his book, he says.8 Before we turn to love, however, we shall begin with faith and hope.

Long before faith and hope became the mental concepts we today think them to be, they were operationally defined by behavior. By moving out, the amoeba essentially

8 This book, page 17.

Fig. 1 The five fundamental steps of animal locomotion

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makes two statements. First, a profession of faith: “There will be food objects out there.” (The nonempty set in Zermelo’s Axiom of Choice.) Secondly, a declaration of hope: “I’ll get one.” (The non-specified element that Zermelo’s Axiom of Choice will let you pick.) The behavior defines the object as something reachable by locomotion and therefore something with time-space coordinates; this is existence. (The existence assertion of the axiom of choice.) Even if the time-space coordinates are yet unknown and not specified, this qualifies as numerical identity. This in pure form is Mammen’s choice category. It is also the first instance of intentionality, the aboutness that psychology is about in Franz Brentano’s classical definition.

Mammen’s sense category makes its entry in the figure when the active animal is guided—or, if passive, goaded—by an array of incoming sensory stimulation. Since the sensory display in itself would satisfy an—in principle endless—array of objects, rather than point to a particular object in the temporal-spatial matrix, it is called qualitative identity and distinguished from numerical identity. The theoreti-cal foundation of stimulus-response psychology lies with qualitative identity and the sense category. Basically, it is placing the cart before the horse.9

In the third stage, where the food object has come within reach, and—as vividly depicted by Herbert Spencer Jennings in Fig. 2—the amoeba starts grabbing for it, the choice and sense categories meet up, an intersection which is itself a choice category. (Axiom 11 in Mammen’s topology.)

Faith and hope done, now let’s turn to love at the other end of the sequence.

Love Is a Many-Splendored Thing

The two pillars of Mammen’s work are the human sense and the mathematical topology in which it can be formalized. The human sense, the title of the 1983 dissertation where it was first laid out, is a sense of numerical identity, particularity, and particular attachment unique for the human being. Love, as we saw, is now made the paradigmatic example of this “special structure in the world,” but it spreads its threads into the human world in a cascade of variations. Mammen mentions friendship and solidarity, parental love and bonds, family ties and obligations, inter-ests and affections, sentimental value and perezhivanie (lived emotional experi-ence), freedom and biography, personal belongings and private property, gifts and souvenirs, and history and cultural context; even science has its root here, why the human sense is also the “scientific sense.”10 In short, the whole portfolio without which “humans would not be humans.”11

How much of this is truly human, we may ask? Particular attachments are not completely unknown in the animal world. Prairie voles and barn owls pair for life;

9 Observe, however, that few psychological theories actually make that mistake; even theories of behaviorism usually get the horse placed right, Skinner’s operant behaviorism, for instance.10 This book, p. 42.11 This book, p.40.

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chimp mothers mourn and refuse to let go of their dead babies; killer whales form friendships for life; dogs recognize again their long-gone owners and can become perversely attached to particular toys; elephants remember people and past events like elephants, and so on. In other words, the animal world seems as full of love, attachment, and keeping track—and score—as the human world. Is Mammen defrauding animals of valid attributes to prove our human exclusivity? I think not, but the question is not irrelevant.

Nearly everyone on human record who has tried to home in on the specific human difference has been guilty of this self-serving stratagem, myself included. Arguably, Mammen was guilty too in his 1983 dissertation. While solely dealing with the human being and the human sense of particularity, and animals, when mentioned at all, merely served as a backdrop for our human exclusiveness, basically this clear and simple cut was made: humans have choice categorical access to the world, and animals have only sense categorical.12

12 Note that without this clear-cut oversimplification, Mammen would probably not have made his discovery in the first place, which makes it another example of how errors pave the way of scien-tific progress.

Fig. 2 Chasing amoeba. Herbert Spencer Jennings, Behavior of the Lower Organisms, New York, Columbia University Press, 1906

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If Mammen in his dissertation was sparing toward the animals, in the present book, however, this stinginess has been redressed in spades. Not only has the animal world been licensed to make choice categorical choices, the inanimate world has as well! With this radical largesse—The Full Monty, Mammen has called it after a 1997 British comedy drama13—two problems follow.

First, to avert panpsychism: how did the animal world emerge from the inanimate world? Secondly, to maintain human uniqueness: how did the human world emerge from the animal world? The already mentioned breach of symmetry provides the answer to the first question. About the second, Mammen says that “questions of the concrete and detailed emergence of ‘the human sense’ as a specific elaborated form of the duality already found in animals’ life are not central. What is central is that after the introduction of the specific human duality in relations to the world of objects, a new structure is found in these relations.”14 In his examination of this after-the-fact structural reality, Mammen in the dissertation followed Karl Marx’s astute analyses of the transformation of nature by concrete practical work and tool use and in the present book seems satisfied to stick with that: “A fundamental dif-ference between the duality in animals’ and humans’ relations to objects was that the sensory relation in humans to objects’ qualities or features was supported and framed by tools and particulars beyond the ‘naked’ senses shared with animals.”15

And why not? Not only Karl Marx, but up through the ages, Enlightenment icons like Benjamin Franklin have defined us as a tool-making animal, and up till this day, paleoanthropologists like Kenneth Oakley have named us Man the tool-maker. It is simply the time-tested standard story of human evolution; it is evidence based, too, and cannot be wrong. And neither can the tool-based and love-bonded duality- reality of the human being, Mammen describes so well.

Still I believe that the question of emergence is central and has elsewhere argued that tool use, amply present in the animal kingdom, itself, is not enough to tip the animal into the human being.16 A twist is needed, a Monsieur Jourdain twist, in fact. Whether right or wrong in this, pursuing such a twist will sharpen the understanding of Mammen’s discovery, and he has himself eased our path by coming up with the splendid concept of double-entry bookkeeping.

Double-Entry Bookkeeping

Now, if you would care to look again at Fig. 2, I trust you will agree that the amoeba chasing and trying to grab hold of the food object is engaging a thing with numerical identity and a trajectory in time and space. In other words, the amoeba’s action is all

13 ”The full MONTI. Preliminary answer to Niels Engelsted,” in Mammen, Engelsted, and et al., (2000), p. 327.14 This book, p. 43.15 This book, p. 55.16 Engelsted (2017).

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about grappling with particularity, and—by extension—that animals not just address sensory information, but address particularity as well. And, remember, when they address both, they speak particularity—Mammen’s Axiom 11. If they didn’t, they simply couldn’t be the grabbing creatures, they are. But if animals know par-ticularity intimately, they don’t know, they know. Like Monsieur Jourdain who spoke prose fluently, but did not know, the animals speak particularity fluently, but do not know. They remain enclosed in their animal state of innocence. Translated into Latin, knowing to know is consciousness, and consciousness has been considered the human prerogative long before Linnaeus named us Homo sapiens and filled our slot in his Systema Naturae with the admonition, Know Thyself! Even if human consciousness—and not least human language, its twin and walker—has invoked some rather idealistic and mystical non-explanations, rightfully criticized by Mammen in the book, this should tell us that we are on the right track.

If, for a moment, we stay with language, its sentences can be parsed as subject- object relations, but also as subject-predicate compounds. In the latter, “a factual statement first identifies something it is ‘about’, and then adds what to say about it,” as Mammen explains.17 Which, of course, means that the subject-predicate com-pound is ideally suited to express the duality between an object’s choice categorical existence and its sense categorical appearance. The amoeba speaks in subject- predicate compounds when it chases the food object, compounds of existence and appearance, but as it cannot separate the two, it cannot know it speaks particularity. Animals “are simply not prepared for it,” Mammen writes, as they have not yet “established the human ‘double-entry bookkeeping’.”18 “It is evident,” he continues, “that if this duality of subject and predicate can’t be held separate and then again combined, factual statements would be impossible.”19 This operation—from whole to split to whole again, now knowing the old in a completely new way—is double- entry bookkeeping, and what Mammen here says of language could be equally said of human consciousness; it is simply the human sense that makes us the unique animal in the animal kingdom.

It is heavy stuff, agreed, so let’s summon up some foreign help and look at it from another angle.

The Hammerhead Falling Off

Mammen’s work could rightfully be called the topology of being, and this is precisely what Martin Heidegger has called his own thinking.20 Like Mammen, Heidegger was critical of the reign in philosophy and science of the universal, the global, and

17 This book, p. 46.18 Ibid.19 Ibid.20 ”Topologie des Seyns” in Heidegger, M.: Gesamtsausgabe 15, Frankfurt: Vittorio Klostermann, 1986, p. 344.

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the abstract to the neglect of the particular, the local, and the concrete. Like Mammen, Heidegger was concerned with the time and space of lived experience, and how it ties us into our familiar world with threads of local history and personal biography, with bonds of practical artifacts and sentimental keepsakes. Both also share an understanding of the truly human “conservatism” in “traditional values” and “in peoples’ relations of love, solidarity, friendship, faithfulness, owning, belonging, and reverence.”21 Finally, to catch this human reality, they have both introduced new elaborate codes, mathematical ciphers in Mammen’s case, verbal in Heidegger’s. It is to a string of the latter we now turn for supplementary elucidation.

With fingers in flight over the piano keyboard, music is flow, only when a note jarringly fails, it becomes score, and what was Inhanden-sein (ready-at-hand) becomes Vorhanden-sein (present-at-hand). The terms are Heidegger’s, not the example. Heidegger’s own signature example is the flow of hammering and the upset when the hammerhead breaks off. Basically, it is the flow of praxis turning into the stuttering scrutiny of theory, existence into rule and description, and a split in the subject-predicate compound Mammen was talking about.

The problem with most philosophy and science in Heidegger’s diagnosis has been that it has got stuck in the Vorhanden-sein mode, forgetting whence it came; or, as once said by Madame de Staël, that it “can only examine by division, [and] applies like a dissecting knife to dead nature, but … is a bad instrument to teach us to understand what is living.”22 But if Heidegger harbored a nostalgic yearning to regain the primordial flow of innocence—and I don’t know enough to say that he did—becoming a child again is obviously impossible. When praxis turns into theory, it cannot turn back again except as practice of theory, or theory of practice; for instance, as a book like Heidegger’s Being and Time about the primacy of the flow of praxis.

I believe this is Mammen’s point. Once the whole has been split into existence and appearance, humans are fated with a knowledge amoebas don’t have; further, humans can now use this double-entry bookkeeping to keep apart the sides in time and thereby learn the secrets of unfolding nature, as Mendel did with his peas in Mammen’s favorite example of how the human sense became also the scientific sense.

Heidegger’s subject is being. There are two kinds. Vorhanden-sein delivers the world as Seiendes, the things and objects of trade and science. Inhanden-sein delivers the world as Sein, the lived experience of living beings. Animals have lived experience too. In a famous essay, Thomas Nagel said it must be like something to be a bat, and of course it must. Shared by cats and gnats and our little friend, the amoeba, the Sein, or way of being, Heidegger talks about, is not a human prerogative. What is, is our

21 I hasten to add that Mammen—contrary to Heidegger—also sternly warns against the other side of this coin: “its perversions in hate, prejudices and xenophobia, chauvinism and expressions of supremacy, false generalizations, racism, discrimination, and exclusion.” This book, p. 54.22 Madame de Staël, (1814)/Le Van Baumer (1978), p. 475.

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ability to discover it. Meriting a special name, Heidegger called it Dasein and defined it as “that entity which in its Being has this very Being as an issue.”23

Though, admittedly, I’ll never make a living as cryptographer, it seems obvious to me that the reflective circularity in that little string of code refers to the human prerogative of knowing to know talked about above. Mammen’s human sense and Heidegger’s Dasein are in my view, if not twins, then at least family, working in parallel in the same field.24

Into Other Chapters and Different Translations

Heidegger’s preoccupation with tool use, and the hammerhead falling off as the Monsieur Jourdain moment that makes Dasein stand out, fits well with Mammen’s views on the emergence of the human sense. I believe, however, as said above, that a different Monsieur Jourdain moment had to arrive first to make it all work, and that the split in the subject-predicate compound had to be preceded by a split in the subject-object relation.

Ripping a different page from the chapters of Karl Marx, I have argued that, rather than work and working with tools, its immense importance untold, it was labor that was the decisive catalyst in the emergence of the human being. In labor, the subject is dispossessed of its object, and from Hegel on, this split—with subse-quent alienation—has been thought the root of human consciousness. As I have further suggested that this object transfer—in a fall from animal innocence through an Adam-and-Eve-like event—originated from prehuman mothers’ unselfish feed-ing of their young,25 Mammen and I have simply ended up in two different transla-tions of Paulus’ famous letter to the Corinthians. He in the modern version’s faith, hope, and love, but the greatest of these is love. Me in old King James’ faith, hope, and charity, but the greatest of these is charity. And while love is mostly sentiment, it can at least be said of charity—Paulus’ agape—that it captures the contours of concrete activity.

Such little things are important to those involved, of course, but whether work or labor was first, love or charity greatest, soon enough they were all there, contribut-ing to the human world, unique in precisely the way so richly described and astutely x-rayed in Mammen’s work.

23 Heidegger (1927/1962), p.68.24 Compare, for instance, the following quote with Mammen’s category of sense and its inability to capture choice categorical existence: “The senses do not enable us to cognize any entity in its Being; they merely serve to announce the ways in which ‘external’ Things within-the-world are useful or harmful for human creatures encumbered with bodies...they tell us nothing about entities in their Being.” Being and Time, 1927/1962.25 Engelsted (2017), pp. 89–102.

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The Final Word

Mathematics can say things with its sets and numbers that are beyond the words of ordinary language, in this Galileo was right. But words are not just mumbo jumbo and wandering in dark labyrinths; human language with its subject-object relations and subject-predicate compounds is the vehicle of human consciousness, and words can unfold worlds of life and beauty and depth quite different from the magic runes of mathematical formula. Had that not been the case, there could have been neither poetry, literature, nor psychology.

Now Mammen is leading the hardheads deep into this, our happy vale. Should we, the softheads, cry out in trepidation and alarm: Enemy at the gates! We should not. While resentment and regret would be the natural first reaction, upon further reflection, the invasion should be greeted as a welcome and long-awaited advance-ment. Not only can the hardheads now—as once the Mongols in China—at long last be introduced to these most important and tender matters, hitherto beyond their grasp, the delicate matters themselves can be sharply presented in new and enlight-ening exhibition cases. Lastly, and perhaps best of all, we psychologists can now have a hope, to finally gain the long-wanted respect and respectability that mathe-matics always bestows on a scientific field.

Bibliography

Engelsted, N. (1989). What is the psyche and how did it get into the world. In N. Engelsted, L. Hem & J. Mammen (Eds.), Essays in general psychology. Seven Danish contributions (pp.  13–48). Århus, Denmark: Aarhus University Press. http://engelsted.net/almenbiblio/biblioengelsted/whatispsyche.pdf

Engelsted, N. (2017). Catching up with Aristotle. A Journey in Quest of General Psychology. New  York, NY: Springer. Doi https://doi.org/10.1007/978–3–319-51,088-0.

Engelsted, N. (1984). The leap from animal to human [in Danish], Copenhagen, Denmark: Dansk Psykologisk Forlag. http://engelsted.net/springet.htm

Galileo Galilei. (1623). The Assayer. http://web.stanford.edu/~jsabol/certainty/readings/Galileo-Assayer.pdf

Heidegger, M. (1927/1962). Being and time. London, UK: SCM Press.Heidegger, M. (1986). Gesamtsausgabe 15. Frankfurt, Germany: Vittorio

Klostermann.Le Van Baumer, F. (1978). Main Currents of Western Thought. Hanover, Germany:

Yale University Press.Mammen, J. (1983). The human sense. An essay on the domain of psychology. [In

Danish]. Copenhagen, Denmark:Dansk psykologisk Forlag. http://engelsted.net/almenbiblio/bibliomammen/DMSheletext.pdf

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Mammen, J., Bertelsen, P. Engelsted, N., Flor, J. R., Hansen, T., Hoffmann-Jørgensen, J., ... & Schultz, E. (2000). The topology of psyche. The mathematical foundation for the theory of Sense- and Choice-categories. Letters to the society for theoretical psychology. [In Danish], Psykologisk Skriftserie, Psykologisk Institut, Aarhus Universitet, Vol. 25, No. 1. http://engelsted.net/almenbiblio/bib-liomammen/PsykensTopologi.pdf

Penrose, R. (2004). The Road to Reality. New York: Alfred A. Knopf.

Commentary 1: Faith, Hope, and Love

109© The Author(s) 2017 J. Mammen, A New Logical Foundation for Psychology, SpringerBriefs in Psychology, https://doi.org/10.1007/978-3-319-67783-5

This note relates to only one of Jens Mammen’s many themes: the construction of a “new mathematics, tailored for psychology,” of which the book shows us only “a small, but important, part” (p. 58). In a nutshell, Mammen proposes to consider an ambient set of “objects” and to endow it with two sets of subsets: the subsets representing (or simply called) “sense categories” and those representing (called) “choice categories.” The two sets of subsets are disjoint, except for the empty set that belongs to both. The nonempty sense categories are infinite, and together with the empty set, they form a certain topology. The choice categories may be finite, and a choice category always contains a one-element choice category. The set of choice categories is required to be closed under finite intersections and unions only. The only relationship between the two types of categories is that the intersection of a choice category with a sense category is a choice category.

This is a surprisingly subtle axiomatic construction, with many possible interpre-tations and directions of specialization (pp. 87–88). A critical evaluation of this con-struction is beyond the aims of this note. Rather, as a psychophysicist, I am interested in possible operational meanings of this and other similarly abstract constructions. How, by what empirical procedures, does one acquire knowledge of a sense category or a choice category? The book does not provide much detail in this respect. We read, however, that “[i]n 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” (Footnote 2 on p. 63). I will assume that I can translate (or generalize) the term “decision” into “response,” meaning any form or aspect of observable behavior (including physiological reactions). This will place the discussion in a familiar conceptual framework: whatever the theoretical picture one wants to construct in psychology, one has to ultimately relate it to recordable behavior under recordable conditions. That this time-honored position does no vio-

Commentary 2: On Random Variability of Responses – A Note on Jens Mammen’s Book

Ehtibar N. DzhafarovPurdue University, West Lafayette, IN, USAehtibar@purdue.edu

110

lence to Mammen’s theory is further evidenced by his saying that the representation of categories by subsets of a set of objects “could also be seen as an ‘ecological’ generalization of classic experimental psychophysics” (ibid).

The admission that the content (equivalently, meaning) of a category is reflected in, if not determined by, recordable responses given by a person or organism to certain sets of objects has an obvious consequence: if the responses change, the contents (meanings) of the categories generally change too. This leads me to this commentary’s departure point: most responses do change, from one instance of presentation to another or from one respondent to another. If one presents many times the same pair of physically very close (or even identical) color patches to a person and asks whether they are the same or different in color, the response will sometimes be yes and sometimes no. If many people are asked whether they trust a certain political party, the response will differ from one person to another. These two types of variability (famously labeled by Thurstone, 1927, as Case I and Case II, respectively) are so ubiquitous that they can be called fundamental. A philosopher may disagree that the latter term can be used in psychology in the same sense in which it is used in relation to quantum mechanics, but this makes little difference: in most psychological situations, the best one can hope to do is to evaluate and, with the help of a theory, predict probabilities of occurrences of various responses, rather than responses themselves.

Why is this stochastic variability worth mentioning? Am I not talking about trivi-alities of an experiment, “statistical noise” or “errors of measurement”? Is it not some-thing to be carefully isolated from an “ideal” picture, like the one proposed by Mammen, and in no way affecting its essential features? I do not think so. The most conspicuous difference between modern psychophysics and psychophysics predating the 1950s is that in the former the probabilistic aspects of responses are treated as an essential and irreducible part of the experimental paradigms one deals with, such as detection, discrimination, or identification. The same can be said about many other areas of psychology, especially decision making, whether low level or high level. Random variables associated with typical responses used in experiments are simple and well understood, especially categorical ones (those with a finite number of pos-sible values), and among those, especially dichotomous ones, such as yes/no choices. As a result, not only are not they a nuisance factor obscuring an ideal theoretical pic-ture, their probability distributions provide critical information helping one to con-struct such pictures, being in many cases the only such information available.

Thus, if a person is presented elements of a Mammen’s set of objects Ù pairwise, and for each ordered pair (x, y) says whether y is greater than x in a designated respect (such as brightness or beauty), then the probability of the positive response γ transforms the set of objects into a structured space (Ù, γ). (Of course, if we speak of a realistic experiment, each pair should be presented many times, and probabili-ties should be statistically estimated from frequencies and theoretically interpo-lated/extrapolated to all possible pairs—all well-known difficult issues I am going to gloss over as they are not specific to the present discussion.) Due to the complete transparency of the function γ, one can investigate the properties of this space with-out first understanding the physical properties of the objects and their perceptual

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representations. Thus, one can first relabel the objects to get rid of all “true dupli-cates” (using Mammen’s term but in relation to γ only): this means that if

g gx y x y1 2, ,( ) = ( )

for all y, then x1 and x2 are “lumped” and considered the same object. For y-objects the “lumping” is similar. Following the tradition, one defines a match (point of sub-jective equality) for x as the object y for which γ(x, y) = 1/2, with a match for y defined symmetrically. One can then check if the space is “regular” (Dzhafarov & Dzhafarov, 2010), which in relation to (Ù, γ) means that (after the “lumping”) x is matched by one and only one y and vice versa. It is easy to see then that in a regular space x matches y if and only if y matches x, and this allows one to further relabel either the x-objects or the y-objects to bring the space to a canonical form in which γ (x, y) equals 1

2 if and only if x = y.The quantities

D x y x y, ,( ) = ( ) -| |g 1

2

are called psychometric increments (from x to y). By observing (or postulating) vari-ous properties of D, one can keep imposing on the space of objects progressively more informative (more restrictive) structures. Thus, it is clear that D (x, x) = 0 and that D (x, y) > 0 if x and y are distinct. We say that sequences x1, x2,… and x x1 2¢ ¢ ¼, , converge to each other if D x xn n, ¢( ) converges to zero (as n → ∞). One can

further assume (provided no contradicting empirical evidence is found) that for any sequences {xn} and xn

¢{ } converging to each other and any sequences {yn} and yn¢{ }

converging to each other:

D x y D x yn n n n, ,( ) - ( ) ®¢ ¢ 0.

This so-called intrinsic uniform convergence property allows one to impose on Ù a structure called uniformity, which in turn induces a certain form of topology.

In a series of publications (see Dzhafarov, 2011, for an overview), it has been shown how, with the addition of one other property whose description I omit as it is more technical, the notion of the dissimilarity function D can be used to impose on Ù a metric structure and develop a full-fledged geometry. The operational part of the development in most of these publications is different from the one presented here: the dissimilarity function there is based on mathematically more complex but operationally more versatile “same-different” judgments rather than “greater-less” judgments (Dzhafarov & Colonius, 1999, 2007). One can also define D through other procedures, such as adjustment of y until it seems matching x (Dzhafarov & Perry, 2014). All of these procedures deal with randomly varying responses. We need not go further into details to appreciate the fact that this random variability, far from being a nuisance factor, is part and parcel of the construction of the “ideal” pictures of the kind presented in Mammen’s book.

Commentary 2: On Random Variability of Responses – A Note on Jens Mammen’s Book

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My other illustration of the necessity to treat random variability as essential part of one’s theoretical construction relates to the issue of (probabilistic) contextuality. Mammen touches on this issue very briefly but prominently on pp. 23–24, when discussing quantum entanglement as having dealt a “deathblow” to the idea of the universe being driven by local mechanical interactions. This particular paradigm of contextuality is schematically shown below.

Two particles (say, electrons), created and “entangled” within a certain state called “singular,” move away from each other, and at some moment of time, Anna and Curt measure their spins, Anna in the left electron and Curt in the right one. Spin is a property of an elementary particle (here, electron) tied to a choice of an axis (direction) in space: thus, for the left particle, one can speak of its spin in direc-tion 1 or in direction 3, and Anna always chooses one of these two axes. Similarly, Curt always chooses one of the two axes labeled 2 and 4 for the right particle. The labeling of the axes in the two particles is chosen so that the possible pairs of the axes simultaneously chosen by Anna and Curt form a cycle.

1c1 2

c2

4

c4

3c3

.

Each such a pair of the axes is called a context. So in the paradigm considered, we have four axes combined into four contexts, c1, c2, c3, and c4.

In electrons, the outcomes of measuring a spin along any axis is binary, and one can denote these outcomes arbitrarily, say, “yes” and “no.” Let us denote by Rq

c the yes/no random variable representing the result of measuring spin along axis q in context c. We have therefore eight random variables, as shown in the matrix below:

EPR B ( )é

ë

êêêêêê

ù

û

úúúúú

c c c c

R R

R R

R R

R R

1 2 3 4

11

14

21

22

32

33

43

44

1

2

3

4 úú

.

Commentary 2: On Random Variability of Responses – A Note on Jens Mammen’s Book

113

The abbreviation EPR stands for Einstein, Podolsky, and Rosen (1935), and B stand either for Bohm, who adapted the EPR problem to spins (Bohm & Aharonov, 1957), or Bell (1964, 1966), who famously investigated it. The ques-tion that interested Bell and two generations of quantum physicists after him, when formulated in the language of the representation above, is this (Dzhafarov & Kujala, 2014, 2017b): can one present all the random variables in the system as functions of one and the same (“hidden”) random variable, so that any two random variables measuring spin for the same axis in two different context are the same? In other words, can one find a random variable R such that every Rq

c in the system can be presented as some function fq (R) (without the superscript c)? As it turns out, the axes 1, 2, 3, and 4 can be chosen so that the answer to this question is negative. For instance, in accordance with the laws of quantum mechanics, coplanar axes 1, 2, 3, and 4 forming the respective angles 0, π/4, π/2, and −π/4 with a horizontal line yield

contextc

R Yes R No

R Yes

R No

1

21

21

11

11

1 2

2 2 8 2 2 8 1 2

= ( )= =

= +( ) -( )=

,

/ / /

22 2 8 2 2 8 1 2

1 2 1 2

44

-( ) +( )

é

ë

êêêêêêêêê

ù

û

úúúúúúúúú

=

/ / /

/ /

contextc ,11

2 2 8 2 2 8 1 2

2 2 8 2 2

44

44

14

14

( )= =

= -( ) +( )= +( ) -(

R Yes R No

R Yes

R No

/ / /

/ ))

é

ë

êêêêêêêêê

ù

û

úúúúúúúúú

= ( )=

/ /

/ /

8 1 2

1 2 1 2

2 32

22

contextc

R Yes R,

222

32

32

2 2 8 2 2 8 1 2

2 2 8 2 2 8 1 2

1 2 1

=

= +( ) -( )= -( ) +( )

No

R Yes

R No

/ / /

/ / /

/ / 22

3 43

43

43

33

é

ë

êêêêêêêêê

ù

û

úúúúúúúúú

= ( )= =

=

contextc

R Yes R No

R Ye

,

ss

R No

2 2 8 2 2 8 1 2

2 2 8 2 2 8 1 2

1 2 1 233

+( ) -( )= -( ) +( )

é

ë

êêêêêêêê

/ / /

/ / /

/ /êê

ù

û

úúúúúúúúú

Here, the first matrix corresponds to context c1, in which Anna chooses axis 1 and Curt chooses axis 2. The cell for R Yes1

1 = and R No21 = shows the joint prob-

ability of these two outcomes, and the cell for R Yes11 = on the margin shows the

“marginal” probability of this outcome, irrespective of the other measurement. Other cells in this and other contexts are interpreted analogously. It can be shown that even though every pair Rq

c and Rqc¢ measuring spin along the same axis in

different contexts are identically distributed (i.e., in the parlance of quantum mechanics, the system is “non-signaling”), there is no way of presenting all eight random variables as functions of a single background variable R that would make Rq

c and Rqc¢ indistinguishable (i.e., always equal to each other). At least for some

axes q, the variables Rqc and Rq

c¢ must be presented as different functions f Rqc ( )

and f Rqc¢ ( ) . One can say that the two random variables have different identity due

to different contexts, and this difference cannot be ignored even though it does not

Commentary 2: On Random Variability of Responses – A Note on Jens Mammen’s Book

114

translate into different distributions. We say in cases like this that the system of random variables is contextual, in the traditional quantum-mechanical understanding of the term.

Contextuality is a deep concept that cannot be confined to nonlocality and entan-glement only. For example, the KCBS (Klyachko, Can, Binicioglu, & Shumovsky, 2008) system with five axes pairwise used in five contexts has essentially the same mathematical structure:

KCBS( )é

ë

êêêêê

c c c c c

R R

R R

R R

R R

R R

1 2 3 4 5

11

15

21

22

32

33

43

44

54

55

1

2

3

4

5

êêêê

ù

û

úúúúúúúú

,

and lends itself to essentially the same contextuality analysis, even though all the measurements are performed on a single particle. The same applies to the SZ-LG (Suppes & Zanotti, 1981; Leggett & Garg, 1985) system with a single particle mea-sured at three fixed moments of time grouped pairwise into three contexts:

SZ LG ( )é

ë

êêêêê

ù

û

úúúúú

c c c

q R R

q R R

q R R

1 2 3

1 11

13

2 21

22

3 32

33

.

Returning to the EPR-B paradigm, it is clear that one can create a system of the same formal structure outside quantum mechanics. The axes 1, 2, 3, and 4 can be replaced, e.g., with yes/no questions asked of a person or many people. Mammen has written a delightful essay (Mammen, 2016) in which Anna and Curt instead of measuring spins answer questions asked of them simultaneously in two different Danish cities (which is the reason I use these names instead of the traditional “Alice” and “Bob”). The scenario Mammen considers is as follows:

Commentary 2: On Random Variability of Responses – A Note on Jens Mammen’s Book

115

contextc

R Yes R NoR YesR No

1

21

21

11

11

1 2

1 2 0 1 20 1 2 1 2

1 2

= ( )= =

==

,

/ // /

/ 11 2

414

44

44

14

/

é

ë

êêêêêêêê

ù

û

úúúúúúúú

= ( )= =

=

contextc

R Yes R NoR Ye

,

ssR No

contextc

0 1 2 1 21 2 0 1 21 2 1 2

14

2

/ // // /

=

é

ë

êêêêêêêê

ù

û

úúúúúúúú

= 33 2

1 2 0 1 20 1 2 1 2

1 2 1 2

22

22

32

32

,( )= =

==

é

ë

êêêê

R Yes R NoR YesR No

/ // /

/ /

êêêêê

ù

û

úúúúúúúú

= ( )= =

=

contextc

R Yes R NoR YesR

3

43

43

33

3 4

1 2 0 1 2

,

/ /

333 0 1 2 1 2

1 2 1 2=

é

ë

êêêêêêêê

ù

û

úúúúúúúú

No / // /

In the quantum-mechanical literature, this system is referred to as a PR box (after Popescu & Rohrlich, 1994). It is a system with maximal algebraically possible con-textuality, and it cannot be realized by any quantum-mechanical system. Of course, nor can this or any other contextual system be realized by the humorous situation described in Mammen’s essay. However, there seems to be no a priori reasons why a contextual system cannot be realized if the paired questions are asked of the same person. The point that is important in the present discussion is that any chance of finding a contextual system in biological or social behavior is contingent on the system being stochastic: a deterministic system is always noncontextual.

One safe generalization about biological and social behavior is that a response to object x is always directly (causally) influenced by any other object in spatial- temporal proximity of x. If a person answers question 1, the distribution of her responses will be different depending on whether together with 1 she is asked question 2 or question 4. This has been demonstrated in numerous attempts to recre-ate the formal structure of EPR-B, SZ-LG, and other similar systems in human behavior (see Dzhafarov, Zhang, & Kujala, 2015, for an overview). This means that unlike in the traditional quantum-mechanical definition, we cannot stipulate “non- signaling” as a necessary condition for contextuality analysis, unless we want the result of this analysis to be trivially predetermined. Fortunately, there are compel-ling reasons for and natural ways of generalizing the definition of (non)contextual systems to include those with “signaling” (Dzhafarov & Kujala, 2015, 2017a, 2017b; Kujala, Dzhafarov, Larsson, 2015). In regard to the EPR-B system, the gen-eralized definition would be: The system is noncontextual if each of the eight random variables Rq

c in it can be presented as functions of one and the same “hidden” vari-able R, so that f Rq

c ( ) and f Rqc¢ ( ) coincide with the maximal possible probability

(this maximal probability being 1 if and only if Rqc and Rq

c¢ are identically distributed). The statement that is relevant to this note, however, remains valid: a deterministic

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system cannot be contextual, whether “signaling” or not. Therefore any procedure, such as averaging, that would eliminate variability as a nuisance factor would also eliminate contextuality.

References

Bell, J. (1964). On the Einstein-Podolsky-Rosen paradox. Physics, 1, 195–200.Bell, J. (1966). On the problem of hidden variables in quantum mechanics. Review

of Modern Physics, 38, 447–453.Bohm, D., & Aharonov, Y. (1957). Discussion of experimental proof for the paradox

of Einstein, Rosen and Podolski. Physical Review, 108, 1070–1076.Dzhafarov, E.  N. (2011). Mathematical foundations of Universal Fechnerian

Scaling. In B.  Berglund, G.  B. Rossi, J.  Townsend, & L.  Pendrill (Eds.), Measurements with persons (pp. 185–210). New York: Psychology Press.

Dzhafarov, E. N., & Colonius, H. (1999). Fechnerian metrics in unidimensional and multidimensional stimulus spaces. Psychonomic Bulletin and Review, 6, 239–268.

Dzhafarov, E. N., & Colonius, H. (2007). Dissimilarity cumulation theory and subjective metrics. Journal of Mathematical Psychology, 51, 290–304.

Dzhafarov, E.  N., & Dzhafarov, D.  D. (2010). Sorites without vagueness II: Comparative sorites. Theoria, 76, 25–53.

Dzhafarov, E. N., & Kujala, J. V. (2014). Contextuality is about identity of random variables. Physica Scripta, T163, 014009. (available as arXiv:1405.2116.)

Dzhafarov, E. N., & Kujala, J. V. (2015). Conversations on contextuality. In E. N. Dzhafarov, S. Jordan, R. Zhang, & V. H. Cervantes (Eds.), Contextuality from quantum physics to psychology (pp. 1–22). New Jersey: World Scientific. (available as arXiv:1508.00862.)

Dzhafarov, E. N., & Kujala, J. V. (2017a). Contextuality-by-default 2.0: Systems with binary random variables. In J. A. de Barros, B. Coecke, & E. Pothos (Eds.), Lecture Notes in Computer Science (Vol. 10106, pp.  16–32.) (available as arXiv:1604.04799.)

Dzhafarov, E.N., & Kujala, J.V. (2017b). Probabilistic foundations of contextuality. To be published in Fortschritte der Physik – Progress of Physics 65, 1600040 (1–11). (available as arXiv:1604.08412.)

Dzhafarov, E. N., & Perry, L. (2014). Perceptual matching and sorites: Experimental study of an ancient Greek paradox. Attention, Perception, and Psychophysics, 76, 2441–2464.

Dzhafarov, E.  N., Zhang, R., & Kujala, J.  V. (2015). Is there contextuality in behavioral and social systems? Philosophical Transactions of the Royal Society A, 374, 20150099. (available as arXiv:1504.07422.)

Einstein, A., Podolsky, B., & Rosen, N. (1935). Can quantum-mechanical descrip-tion of physical reality be considered complete? Physical Review, 47, 777–780.

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Klyachko, A. A., Can, M. A., Binicioglu, S., & Shumovsky, A. S. (2008). A simple test for hidden variables in spin-1 system. Physical Review Letters, 101, 020403.

Kujala, J. V., Dzhafarov, E. N., & Larsson, J.-Å. (2015). Necessary and sufficient conditions for extended noncontextuality in a broad class of quantum mechanical systems. Physical Review Letters, 115, 150401. (available as arXiv:1407.2886).

Leggett, A., & Garg, A. (1985). Quantum mechanics versus macroscopic realism: Is the flux there when nobody looks? Physical Review Letters, 54, 857.

Mammen, J. (2016). A conversation about entanglement – A science fictive dia-logue. http://vbn.aau.dk/files/260767691/A_Conversation_about_Entanglement

Popescu, S., & Rohrlich, D. (1994). Quantum nonlocality as an axiom. Foundations of Physics, 24, 379–385.

Suppes, P., & Zanotti, M. (1981). When are probabilistic explanations possible? Synthese, 48, 191–199.

Thurstone, L. L. (1927). Psychophysical analysis. American Journal of Psychology, 38, 368–389.

Commentary 2: On Random Variability of Responses – A Note on Jens Mammen’s Book

119© The Author(s) 2017 J. Mammen, A New Logical Foundation for Psychology, SpringerBriefs in Psychology, https://doi.org/10.1007/978-3-319-67783-5

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 catego-

ries so that one object is in the first and the other object in the other one.

Ax. 5 (perfectness): No sense category contains just one object.Ax. 6: No nonempty choice category is a sense category.Ax. 7: There exists a nonempty choice category.Ax. 8: Any nonempty 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 mathemati-cal “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 cate-gories. 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

Appendix: Proof of Independence of Axioms for Sense and Choice Categories

120

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}.26

All except Ax. 3: Ù = R; S = all open intervals in R and Ø; C = Ø and {1}.27

All except Ax. 4: Ù = R; S = all intersections of Q and O; C = Ø and {1}.28

All except Ax. 5: Ù = R; S = O and all unions of O and {2}; C = Ø and {1}.29

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.30

All except Ax. 9: Ù = R; S = O; C = Ø and all unions of a finite nonempty 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 < ∞.31

All except Ax. 10: Ù = R; S = O; C = Ø, {1} and {2}.32

All except Ax. 11: Ù = R; S = O; C = {1}.33

26 The intersection of two disjunct open intervals is Ø which is not in S.27 The union of two disjunct open intervals is not an open interval and hence not in S.28 Irrational numbers are not member of a subset in S.29 {2} is the union of Ø in S and {2} and hence itself in S.30 No member in C is finite.31 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. 11.32 {1,2}, the union of {1} and {2}, is not a choice category.33 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).

Appendix: Proof of Independence of Axioms for Sense and Choice Categories

121© The Author(s) 2017 J. Mammen, A New Logical Foundation for Psychology, SpringerBriefs in Psychology, https://doi.org/10.1007/978-3-319-67783-5

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Einstein, A., Podolsky, P., & Rosen, N. (1935). Can quantum-mechanical descriptions of physical reality be considered complete? Physical Review, 47, 777–780.

Engelsted, N. (1987). Englen og dyret [The Angel and the animal]. Copenhagen, Denmark: Tidens Forlag. http://engelsted.net/engelogdyr.htm

Engelsted, N. (1989). Personlighedens almene grundlag I & II [The general foundation of per-sonality, Vol. I & II]. Aarhus, Denmark: Aarhus Universitetsforlag. http://engelsted.net/person-lighedensalmenegrundlag.htm

Engelsted, N. (2017). Catching Up With Aristotle. A Journey in Quest of General Psychology. New York, NY: Springer. https://doi.org/10.1007/978-3-319-51088-0

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Gibson, J. J. (1966). The senses considered as perceptual systems. Boston, MA: Houghton Mifflin.Gibson, J. J. (1979). The ecological approach to visual perception. Boston, MA: Houghton Mifflin.Hewitt, E. (1943). A problem of set-theoretic topology. Duke Mathematical Journal, 10, 309–333.Mammen, J., Bertelsen, P., Engelsted, N., Flor, J. R., Hansen, T., Hoffmann-Jørgensen, J.,... &

Schultz, E. (2000) Psykens topologi. Det matematiske grundlag for teorien om sanse- og udvalgskategorier. Breve til Selskabet for Teoretisk Psykologi [The topology of psyche. The mathematical foundation for the theory of sense and choice categories. Letters to The Society

34 All works by Mammen can be downloaded freely from http://engelsted.net/mammenbibliografi.htm (until 2011) or from the links listed.

122

for Theoretical Psychology]. Psykologisk Skriftserie, Psykologisk Institut, Aarhus Universitet, (Vol. 25, pp. 353–360). http://vbn.aau.dk/files/252809502/MaximalPerfectTopologies.pdf

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difference? Cognitive Development, 15, 169–184.Krøjgaard, P. (2007). Comparing infants’ use of featural and spatiotemporal information in an object

individuation task using a new event-monitoring design. Developmental Science, 10(6), 892–909.Krøjgaard, P. (2009). The human ability to single out and track specific objects through space and

time: Origin and application. In H. Høgh-Olesen, J. Tønnesvang, & P. Bertelsen (Eds.), Human characteristics. Evolutionary perspectives on human mind and kind (pp. 89–116). Cambridge: Cambridge Scholars Publishers.

Krøjgaard, P. (2016). Keeping track of individuals: Insights from developmental psychology. Integrative Psychological & Behavioral Science, 50(2), 264–276. https://doi.org/10.1007/s12124-015-9340-4

Krøjgaard, P. (2017). On the theorem of correspondence. Integrative Psychological & Behavioral Science, 51(1), 29–38. https://doi.org/10.1007/s12124-016-9363-5

Kuratowski, K. (1961). Introduction to set theory and topology. Oxford/Warszawa: Pergamon/PWN.

Leontiev, A. N. (1978). Activity, consciousness, and personality. Englewood Cliffs, NJ: Prentice- Hall. https://www.marxists.org/archive/leontev/index.htm

Leontiev, A.  N. (1981). Problems of the development of the mind. Moscow, Russia: Progress Publishers. https://www.marxists.org/archive/leontev/index.htm

Leontiev, A. N. (1982). Psychologie des Abbilds [Psychology of the perceptual image]. Forum Kritische Psychologie, 9, 5–9 (translated from a Russian manuscript, 1975).

Lewin, K. (1935). The conflict between Aristotelian and Galilean modes of thought in contempo-rary psychology. Journal of General Psychology, 5, 141–177. Reprinted in: K. Lewin: A dynamic theory of personality. New York, NY: McGraw-Hill, pp. 1–42. (German original, 1931).

Mammen, J. (1967). Om identitet og karakterisering [On identity and characterization]. In M. Brun (Ed.), Fænomenologien til debat [Phenomenology at issue] (pp. 16–19). Copenhagen: Denmark’s Educational University/Copenhagen University (1975).

Mammen, J.  (1986). Erkendelsen som objektrelation [Cognition as object relation]. Psyke & Logos, 7(1), 178–202 (Summary, p. 207).

Mammen, J. (1989). The relationship between subject and object from the perspective of activity theory. In N. Engelsted, L. Hem, & J. Mammen (Eds.), Essays in general psychology. Seven Danish contributions (pp. 71–94). Aarhus, Denmark: Aarhus University Press.

Mammen, J. (1993). The elements of psychology. I. In N. Engelsted, M. Hedegaard, B. Karpatschof, & A. Mortensen (Eds.), The societal subject (pp. 29–44). Aarhus, Denmark: Aarhus University Press.

Mammen, J.  (1996). Den menneskelige sans. Et essay om psykologiens genstandsområde [The human sense. An essay on the object of psychology]. Copenhagen, Denmark: Dansk Psykologisk Forlag (1st ed. 1983, 2nd ed. 1989).

Mammen, J. (1997). Om frihed og frihedsgrader [About freedom and degrees of freedom]. Forum for Antropologisk Psykologi/Journal of Anthropological Psychology, 2, 94–100.

Mammen, J. (2000). Om ånd og natur i psykologien eller om Descartes’ problem [On spirit and nature in psychology or on Descartes’ problem]. In H. Høgh-Olesen (Ed.), Ånd og natur. Ti almenpsykologiske essays ad modum Katzenelson (pp. 175–200). Dansk Psykologisk Forlag: Copenhagen, Denmark.

Mammen, J.  (2002). Mapping the subject: The renewal of scientific psychology. Journal of Anthropological Psychology, 11, 77–89.

Mammen, J. (2008a). What is a concept? Journal of Anthropological Psychology, 19, 25–27.Mammen, J. (2008b). Om betingelserne for objektiv erkendelse af kvantitative forhold. En praktisk-

realistisk teori [About the conditions for objective quantitative knowledge. A practical-realistic theory]. I: A. B. Christiansen, T. Hetmar, J. Ivanouw & H. Kirkeby (red.). Metodologiske indb-lik og udsyn. Festskrift til Benny Karpatschof. Copenhagen, Denmark: Forlag for Psykologisk Forskningsmetode, pp. 97–115.

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References

125© The Author(s) 2017 J. Mammen, A New Logical Foundation for Psychology, SpringerBriefs in Psychology, https://doi.org/10.1007/978-3-319-67783-5

AAbduction, 66Academic community, 14Affections, 51, 52Aristotelian, 47Aristotle, 5, 6, 18, 24, 32, 35, 38Artificial stimuli, 15Aspect, A., 23Asymmetric decidability, 58, 62Asymmetry, 2, 6, 10, 14, 30, 31, 37, 38, 44,

49, 58, 60, 62, 63, 73, 79Awareness, 90Axiom of choice, 3, 58–60, 69, 85, 86Axiom of comprehension, 61Axiomatic method, 66Axiomatic systems

choice categories (see Choice categories)concept of a subspace in Ù, 80duality, 55–58finite choice category, 80induced topology, 80local sense categories, 80locally open subset, 80mathematics, 58–65mini-universe, 81mutual relations, 66–67natural science, 57objects, 56, 65, 66principle of correspondence, 81sense categories (see Sense categories)structure of sense and choice categories,

66–67

BBiological sciences, 57Bjerg, A., xiBohr, N., 17, 21, 23, 24Boltzmann, L., 25Boundary, 60, 72

CCantor, G., 75Cardinality, 61, 67, 86Causal atomism, 23CERN, 17Choice categories, 55–88

asymmetry, 79axiom TS C, 78concept, 78definition, 63extensional relations, 78finite nonempty, 80first-order equivalent language, 78idealized and realistic models, 86–88infinite, 77interaction/cooperation, 79, 82, 83intersection and union, 78, 79nonempty choice category, 78qualities and sensory criteria, 77structure, 66–67time, 83–84

Choice function, 59Christian theology, 6, 9, 27Closed sets, 60, 72, 73, 79

Index

126

Coexistence, 39, 40, 52, 53, 93Communities, 53Compact, 75Compact-open topology, 83Complementarity, 11, 87Complete, 3, 18, 20, 24–26, 84–87Completeness, 85, 86Connected, 57, 65, 75, 83, 90Conservatism, 54Cooperation and mutual completion, 63Copenhagen interpretation, 21Copernican revolution, 17Cornish, 27Correspondence, 58, 62, 81, 83, 84, 86Countable basis, 85, 87Crossley, J.N., 61Cultural psychology, 27Cultural-historical school, 16Culture, 53, 54

DDecidability, 57, 58

mathematics, 60Decidable category, 85Decision stress

analytical tool, 89awareness, 90choice categories, 90experimental neurosis, 90extensional method, 89framed by choice categories, 89indirect description, 89philosophy and psychology, 90qualia, 90random generator, 89

Dahl, M., xiDecidable categories, 84, 85, 87Dedekind, R., 75Deductive power, 66Descartes, R., 6–8, 12, 14, 16, 22, 43, 44Differential calculus, 21, 23Digital computer, 87Discrete universe, 80Discriminate sensory, 74Dualism, 11Duality

subject-object relations, 55rules, 73

EEinstein, A., 23Emergence, 16, 43

Emotions, 51, 52Engelsted, N., 7, 9, 14, 18, 32, 34,

38, 40Epiphenomena, 20Epiphenomenalism, 18Epistemology, 21, 58Euclid, 66Existence, 9, 10, 14–17, 24, 29, 32, 35, 39, 40,

43, 51, 53, 56, 57, 59–62, 70, 77, 78, 81, 86, 87, 91

Existentialism, 8, 24, 32Experimental neurosis, 90Experimental situations, 81Extensional approach, 63Extensional method

and sense and choice categoriesasymmetric decidability, 62asymmetric structures, 64, 65conceptual tool, 62cooperation and mutual

completion, 63decidability, 62extensional, 63extensional approach, 63general topology, 61intensional, 63isomorphism, 62numerical identity, 63ontological and epistemological

basis, 64psychology, 64qualitative identity, 63selections or identifications, 63sensing and sensory decisions, 62theory of sets, 62topology, 62

Exterior, 72

FFascism, 27, 54Fechner, G.T., 13Fechner’s law, 13Fetishism, 90Fifth quasi-dimension, 51Finite choice category, 80Finite induction, 68First-order equivalent, 67First-order equivalent language, 78First-order language, 67, 86Food source, 32Fox, R.H., 83Framing. See Sensory perceptionsFreudo-Marxism, 9

Index

127

GGagliano, M., 34Galilean, 47Galilei, G., 5Gegenstand, 32Geisteswissenschaft, 13Geisteswissenschaften, 10–12General topology, 61Geometrical topology, 75Gibson, J.J., 2Gödel, K., 60, 61, 66, 87Goethe, J.W., 17Grangier, P., 24

HHausdorff, F., 59, 70, 71, 84, 88Hausdorff space, 85

open sets, 73–77Hereditary, 82Hermeneutics, 9, 10Hewitt, E., 85, 87Hoffmann-Jørgensen, J., 85–87Hoffmann’s conjecture, 86Holism, 16, 17, 24Human privilege, 9Human situation

characteristics of life, 37co-existence, 39, 40cognitive and practical capacities, 39description, 37distal relations, 44dual relations, 39emergence, 43entropic asymmetry, 37functionalism and pragmatism, 40general conditions of life, 38gifts, 40, 42historical depth, 39individual manifestations, 40intentional beings, 38interface and interspace, 38lasting/permanent relations, 39mammals and birds, 42particularity and sentimental

value, 42phylogenesis, 43precursors, human sense, 43preliminary conclusions, 43, 44privileges, 39resources, 40symmetry break, 37time and space, 39tools, 42

Humanismatoms of causation, 7dualism, 11eternal emptiness, 17freedom of speech, 10fundamental principles, 8Geisteswissenschaften,

10–12hermeneutics, 9, 10language, 9, 10mathematics, 17mechanicism, 5–7Naturwissenschaften, 10–12philosophy, 6, 8, 9religion, 9thinking and communication, 10two-front war, 8

Humanistic liberation, 7

IIdeal and realistic models

sense and choice categories, 86–88

Ideal dimensions, 92Idiosyncrasy, 90Ilyenkov, E. V., 51, 92Incompleteness, 17, 20–22, 61, 87Induced topology, 80Infinity, 32, 57, 69, 75, 76, 82Interaction/cooperation

sense and choice categories, 82Interior, 14, 71, 72Isomorphism, 62Iterative/recursive process, 60

KKant, I., 8, 13, 24, 35, 46, 53, 70Kierkegaard, S., 8, 17Krøjgaard, P., 48, 81Kuratowski, K., 17, 61Kvorning, J., xi

LLeibniz, G.W., 5, 46, 70, 74, 77Leontiev, A.N., 2, 10, 50, 51Lewin, K., 47Life

asymmetric relation, 30contradiction, 29decisions, 30living organisms, 31

Index

128

Life (cont.)plants and animals

asymmetric metabolic relation, 31duality of relations, 33–35food source, 32fundamental difference, situation, 32individual death, 31intentionality, 32interface and interspace, 31numerical and qualitative identity, 35organism’s growth, 31photosynthesis, 31reproduction, 32surplus energy, 33

second law of thermodynamics, 29Linguistic turn, 9, 27Local sense categories, 80Locally open subset, 80Logic

and mathematics, 84–86Logical schism, 6Löwenheim-Skolem-Tarski theorem, 67Lower and higher mental functions, 14, 15

MMadsen, I., xiMainstream mathematical topology, 75Mandelbrot set, 60, 62Marx, K., 51Material atomism, 24Material identity, 42Mathematical formalism, 20Mathematical mapping, 77Mathematics

asymmetric decidability, 58axiom of choice, 58–60axiom of comprehension, 61classical to mechanistic processes, 58concept of infinity, 57correspondence/backward compatibility, 58criteria and rules, 57decidability, 57, 58, 60definitions, 57epistemology, 58existence, 60–61extensional method, 61–65and logic, 84–86mapping, 60–61necessary concepts, 57sense and choice categories, 61–65topology, 60

Maximal perfect topology, 85, 87Maxwell, J.C., 30

Maxwell’s demon, 30Mechanicism

borderline case, 18, 25causal atomism, 23causal laws, 23characteristics of life, 25conceptual frames, 18consciousness, 20, 21differential calculus, 23human life and psyche, 19material atomism, 24mathematical formalism, 20matter-form duality, 24principles, 18quantum mechanics, 18, 21,

23, 24thermodynamics, 18, 24

Mechanistic conceptual frame, 57Mendel, G., 48, 91Merleau-Ponty, M., 77Method of double stimulation, 15Metric space, 87Mini-universe, 81Mironenko, I., 2, 40, 46, 52, 81, 87Modern mathematics, 22Moore, G.H., 59, 85, 86Motivation, 51, 52Müller, J.-W., 27

NNatural science

psychology’s compatibility, 57subject-object relations, 57–58

Naturwissenschaft, 13Naturwissenschaften, 10–12Neumann, A., 52Neurology, 12Newton, I., 5, 20Nietzche, F., 8Nominalism, 18Non-algorithmic, 87, 89No nonempty sense category, 76Non-algorithmic decisions, 89Numerical identity, 24, 35, 40, 42, 46, 63, 65,

70, 84, 91

OObjects

definition, 56, 65featural/property space, 65features/presence, 56geographical space, 65

Index

129

numerical identity, 65selection, 56sensory criterion, 56

Ontological mathematics, 58Open interval, 59Open sets, 60, 63, 78–80, 87

Hausdorff space, 73–77in topological space, 71–73

Otto, S.L., 9Ownership, 39, 42, 50

PPanpsychism, 26, 27Parametric space, 87Particulars, 13, 21, 23, 35, 39–42, 44–53, 55,

56, 63, 65, 67, 70, 91, 92Pavlovian physiology, 16Peirce’s concept, 66Penrose, R., 87Perezhivanie, 16Perfect, 74–77, 85, 87Personal sense, 51Philosophy, 6, 8, 9Podolsky, P., 23Political system, 16Populism, 27Practical limitations, 77Practical sensory interface, 74Prigogine, I., 29, 38Principle of correspondence, 81Probabilistic ecology, 47Psychiatry, 12Psychology

Alsace/Elsass, 2axiomatic system, 3ecological tradition, 2knowledge, 1mathematics, 3mechanicism, 3modus vivendi, 2scientific landscape, 2scientific tradition, 2social sciences, 3

Psychophysics, 1, 14, 63, 89

QQualia, 90Qualitative identity, 46, 47, 50, 55, 63,

70, 84Quantum mechanics, 6, 7, 18, 19, 21, 23,

24, 81Quantum physics, 21, 44, 77

RRandom generator, 89Rasch, G., 49Rosen, N., 23Rubber sheet topology, 75Russell, B., 61Russian Activity Theory, 2

SScientific knowledge

critical evaluation, 45double-entry bookkeeping, 45, 46, 48factual statements, 46humans and animals, 45inductive capacity, 47inductive generalizations, 47numerical and qualitative identity, 46particularity, primary data, 47physics and psychology, 47single particular object, 47subject-predicate structure, 46time and space, 46

Sense and meaning, 50, 51Sense categories

axiomatics/axiomatic method, 70axioms, 67capacities, 69definition, 63disjunct, 70extensional properties, 67finite induction, 68first-order equivalent, 67first-order language, 67idealized and realistic models, 86–88identity of the indiscernibles, 70interaction/cooperation, 82–83intersections, 69open sets, 71–77sets of objects, 67structure, 66–67time, 83–84universe of objects, 67

Sensory criterion, 56Sensory perceptions

collection of objects, 48features, 49interpretation, 48mechanical/electronic devises, 49minerals, 49objects’ identity, 48specific objectivity, 49

Sentiments, 51, 52Social constructionism, 16

Index

130

Social interaction, 14Standard topology, 77Stengers, I., 29, 38Strawson, P.F., 35, 46, 70Subject-object relations

duality in animal and human, 55with natural science, 57–58

Subspace, 80, 81Superindividual psyche, 27

TTheory of interaction, 13Theory of sets, 62Thermodynamics, 6, 18, 24, 25, 29, 30Time

sense and choice categories, 83–84Time reversible, 6Topological space, 60, 62, 75, 78, 79

closed sets, 71–73open sets, 71

Topologymathematics, 60

Traditional humanities, 1Traditional values, 54Two cultures

allied against mechanicism, 91coordinate system, 92duality, 91

hermeneutics and linguistic discourses, 91

profit, 92sense categories, 91solid theory and methods, 92theoretical definition, 92

Two-dimensional coordinate system, 60

UUniversals, 5–7, 10, 18, 23, 24, 26, 31, 35, 46,

55, 59, 91

VValsiner, J., 14, 16, 27, 53van Douwen, E.K., 85, 87Volksgeist, 27Vygotsky, L.S., 14–16, 47

WWittgenstein, L., 9, 27

ZZeno’s paradox, 20Zermelo, E., 59Zorn’s Lemma, 85

Index