Logic and arithmetic can be considered in terms of operations that precede ready-made formal systems. The initial hypothesis is that their common foundation is the act of distinction, while different paths arise from the subsequent operations of choice and repetition.
The Stability of Mathematical Forms
Branches of mathematics are regarded as formal systems with axioms, rules of inference or transformation, and theorems. Yet their constructions are perceived as differing in stability: geometrical systems allow different variants, whereas the arithmetic of natural numbers seems almost inevitable.
The question of this stability leads to the conditions under which formal systems arise: what operations must already have occurred for logical constructions or counting to emerge?
The First Distinction
After a distinction is made, two sides and a boundary between them arise. At this level, there is not yet a language of propositions, truth and falsity, or numbers.
The minimal sequence is:
Undifferentiatedness → distinction → further operations.
In the proposed model, distinction precedes both logic and arithmetic.
Choice and Repetition
After distinction, two lines of development are possible:
- Choosing a side establishes the orientation “this / not this” and leads to logical constructions.
- Repeating what has been chosen leads to counting and arithmetic.
Choice emphasizes the active nature of the operation, which may be lost in the word “fixation.” The minimal operational set of this model consists of distinction, choice, and repetition.
Here, logic and arithmetic grow out of different operations following distinction; the question of which is primary is transferred to the common foundation of those operations.
Repetition of the Chosen and a New Distinction
Repetition of a choice stabilizes form: rules, habits, and stable structures. A new distinction creates new entities, new pairs of what has been distinguished, and deepens the structure.
Arithmetic arises through repetition of what has been chosen. A new distinction is not identical to that repetition.
The Basis of Counting
Counting is possible within a shared framework that has been chosen. Two apples and three pears do not form a homogeneous count of apples or of pears. To add them, a common basis must be chosen, for example, “fruit.”
Thus, counting presupposes a prior choice of what is to be regarded as one and the same with respect to the operation being performed.
Stabilization and Deepening
Two modes are distinguished:
- Stabilization and classification: distinction → choice → repetition. Form, rules, and counting arise.
- Deepening and investigation: distinction → new distinction → further detailing. New structures of distinctions arise.
In this model, forms and established “truths” are understood as the results of stabilized histories of choices.
Open Questions of the Model
- What makes the first distinction possible?
- What makes it possible to retain what has been chosen and repeat it?
- Is there a limit to the deepening of distinctions, or is detailing infinite in principle?
