Cuts Without Knives: 8 Mathematical and Logical Cuts (Without Platonism)

Mathematics and logic are often imagined as the domain of pre-existing truths. Numbers, sets, propositions — these are treated as eternal, awaiting discovery, independent of the mind. Cuts in this realm are thought to be passive observations: we uncover distinctions that already exist.

This is a mistake.

As in perception and language, distinctions in mathematics and logic are actualised through perspectival cuts. Numbers do not exist as discrete entities awaiting enumeration; propositions do not exist as fixed truths awaiting proof. They emerge from relational structures brought into intelligibility through the act of cutting, each formal operation a gesture that co-actualises pattern and observer.

Consider a simple logical distinction: “A implies B.” There is no prior world of propositions separate from the act of distinguishing this relation. The implication is realised in the context of the formal cut, a perspectival act that draws relations into intelligibility. Similarly, a mathematical structure, whether a set, a function, or a category, is not “out there” to be discovered. It is co-actualised by the formal distinctions applied within a field of potentialities.

This has several profound consequences:

• Mathematical objects are traces of actualisation, not Platonic forms. They exist relationally, not independently.

• Formal multiplicity does not require prior parts; structures emerge through the act of cutting relations.

• Proof and demonstration are not discoveries of eternal truth, but perspectival operations that actualise distinctions, revealing what patterns are intelligible in the chosen relational frame.

In this light, mathematics and logic become generalised domains of cuts, demonstrating that perspectival actualisation is not limited to perception, language, or social systems. The same principles — distinction without difference, multiplicity without parts, field co-emergent with the cut — govern the formal as well as the experiential.

In the next post, we will begin Phase IV: the consequences of abandoning the knife. When Cutting Becomes Violence will explore where the knife metaphor is dangerous, and how recognising the relational, non-destructive nature of cuts transforms our understanding of ethics, politics, and responsibility.

For now, hold this principle firmly: numbers, propositions, structures — they do not exist prior to the cut. Each formal act of distinction is a co-actualisation of pattern and possibility. The knife is absent; the cut illuminates.