Configuring a Concept. On Iteration and Infinity

The question asked in this paper concerns the relation between perception and the senses when the concept of infinity is formed, both in mathematics and in language. I suggest that I came across data that exemplifies the transition from the sensing of an Umwelt to a conceptual grasp.

During a playful reading experiment, five different groups of readers were presented with an excerpt of Le Petit Prince (in five different languages, at different occasions). They were asked to jot down what came to mind, and given 10 minutes for the task. They, unknowingly, produced high response numbers at textual segments with iterative structures (Bruche-Schulz 2014, 2013). An example is attached below.

Sensing that my ‘mind’ is directed to a something, does not produce awareness of my consciousness, but rather of the specific something, here signaling an iteration (shimmering, and trembling).

Cells, and higher organic units, are the building blocks of organs whose interplay results in the wholeness of an experiencing body, and the brain configures elements of experience ‘into resonant patters that form the basis of integral acts of life’ (Fuchs 2018: 169). As shown by the grammars of the world’s languages, human environments allow for the experience of a no-end. Infinity is experienced as an unending going-on in the realm of space (infinity in the sky), of human activity (unending movements, unending new possibilities), of emotive force (kindness, benevolence) and the like.

In spite of the diversity of the language typologies, the notion of ongoing processes is a semantic key notion shared by all, albeit in different intensities or clusters. All languages describe ‘infinity vs. finiteness’ as variations of iteration. Iteration may be unending (progressive, habitual), interrupted, completely stopped or negated (cf. Vanek 2012).

Since the beginning of the 20th century, Cantorian set theory made the concept of infinity into a central concept of mathematics – as indicator of variations of sets: actual infinities, transfinites, finites, and nested intersections thereof (Cantor 1885, in: 1966; see also Ferreiros 1995). (The ‘Absolute Infinity’ of the realm of God was still assumed to exist as underlying the particularities of a body, but remained outside the realm of mathematics.)

It will be suggested that Cantor‘s ‘Punktmannigfaltigkeiten’ (set(s) of points) correspond to the variations of the ‘ongoingness… and location in time’ as described in the grammars of the world‘s languages (Vanek 2012: 155). The question remains how the world of the senses is channeled into such thoughts. What can be said is this: The experience of a concept is a restatement of a felt sensing that confirms a something. ‘A feeling forced upon the mind … [is] strongly suggestive of thought’ (Peirce 1895, in 1998: 23).

reference
Bruche-Schulz, G., ‘Where semiosis begins when reading a text’, Sign System Studies, vol. 42, 2/3, 2014, pp. 330-358.
Bruche-Schulz, G., ‘On moving in conceptual space’, Breaking Down the Barriers, ed. Guangshun Cao et al., Academia Sinica, Taipei, Taiwan, 2013, pp. 49-77.
Cantor, Georg, ‘Über die verschiedenen Standpunkte in bezug auf das aktuell Unendliche‘ (1885), Abhandlungen mathematischen und philosophischen Inhalts, ed. E. Zermelo, Hildesheim: Georg Olms 1966, pp. 370-377.
Ferreirós, José, ‘“What Fermented in Me for Years”: Cantor‘s Discovery of Transfinite Numbers’, Historia Mathematica, vol. 22, pp. 33-42.
Fuchs, Thomas, Ecology of the Brain, Oxford: Oxford University Press, 2018.
Peirce, Charles S., ‘Of Reasoning in General’ (1895), The Essential Peirce, vol. 2, Bloomington: Indiana University Press, 1998, pp. 11-26.
Vanek, Norbert, ‘Language-specific perspectives in relation to time’, Space and Time in Languages and Cultures: Linguistic Diversity, eds. Filipovič, L. and Jaszczolt, K.M., Amsterdam: John Benjamins, 2012, pp. 135-156.

Gisela Bruche-Schulz

Independent Scholar, Berlin, Germany
gibrushu@gmail.com

Former Associate Professor of Linguistics, HKBU, Hong Kong. Main field of work: theory of grammar, discourse linguistics. Current research topics: concepts in mathematics and grammar (continuation, finiteness, infinity)

keywords
conceptual tools
and their bodily foundations