@astrolite1: Whenever I say 1+1=2, do I imply 2+2=4? Not immediately. The first statement establishes the conditions under which further implications become possible. A question to raise; why hold onto these rules? This will be a crucial question for the ending paragraphs. This is a problem of existence, in mind or utterance, we must raise a thought about cognition as actualization (Brouwer’s intuitionism), or the unfolding of certain mathematical rules (Wittgenstein). We may say something like 1+1=2 only implies 2+2=4 now on two conditions (1) the acceptance of the rule and (2) the cognition of the acceptance of the rule applied to a new formation. This needs to be pressed but we’ll press on logicians who may say that the implication is already independent of cognition. (mathematical platonism) Only then must we ask what it is if something is implied from 1+1=2 following the rule governed system and you say it exists independently from cognition then what is it? A concept: semantic existence, can attempt to resolve this problem: it is the mode of existence belonging to rule governed possibilities that are not actual cognitive events or independently existing objects. They exist as structured possibilities of actualization within a semantic existence, to now completely reframe our “We may..” passage we can say that the formal implication of 2+2=4 from a rule governed arithmetic doesn’t, by itself, explain the ontological status of that implication or the cognitive event by which it’s actualized. We aren't saying semantic existence is itself (circular) but that a proposition such as 2+2=4, relative to the prior institution of arithmetic, possesses semantic existence before it possesses cognitive existence. It’s neither merely thought or independently real; it’s available for actualization through the continued operation of the rule governed practice (Peano's axioms) As for formal logic itself it successfully characterizes valid inferential relations but remains neutral regarding the ontological status of those relations. Semantic existence is introduced to account for this missing ontological register by explaining how implications can be available for cognition without reducing them either to mental events or to timeless abstract objects. The collection of semantic existences creates the semantic field To return to our question on why hold onto these rules. Arithmetic has this field due to mathematics' own ready-made commitments (that come from practices, human stipulations within these practices) which are not inherent qualities of mathematics but give mathematics its “royal science.”^1 The Peano framework itself creates a type of mathematics no different than what we may call parodies of mathematics, this puts us in conversation with nomadic science, further showcasing what mathematics could become as a field (the word choice here is deliberate) To avoid "the possible" the "field" of mathematics functions similar to a virtual multiplicity (Deleuze), our first two paragraphs work within that but do not take this structure for granted as royal science aspires to do. Peano arithmetic is one actualization of mathematical potentiality, not the exhaustion of mathematics itself, parodies become philosophically productive regardless if they take themselves seriously (as what is serious is what is royal) So 1+1=2 doesn’t "imply" 2+2=4 by platonist necessity, nor by immediate cognitive presence. The initial stipulation institutes a semantic field (virtual space) which royal science codifies, and which a mind can subsequently traverse and actualize 1. "Royal, or State, science only tolerates and appropriates stone cutting by means of templates under conditions that restore the primacy of the fixed model of form, mathematical figures… Royal science only tolerates and appropriates perspective if it is static, subjected to a central black hole divesting it of its heuristic and ambulatory capacities." ATP p.365 #philosophy #mathematics #deleuzeandguatarri

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Monday 27 July 2026 14:00:40 GMT