Logicism

Logical and mathematical truths are timeless — they hold in every possible world and at every moment, because their content is exhausted by the logical relations that constitute them. The logicist treats time as a domain to be quantified over (by formalised tense-logic or by the temporal indices that physics supplies) rather than as anything internal to the truths of arithmetic. Mathematical existence is non-temporal: the prime numbers were neither created in 1900 nor will they pass away. The infinite extent attributed to time here reflects the unbounded reach of the formal systems within which mathematical reasoning is conducted, not a substantive cosmological claim.

The logicist treats space in much the way it treats time: as a domain to be described by formal theories (geometry, topology, analysis) whose own truths are ultimately logical. Russell and Whitehead's reduction of mathematics to logic was supposed to ground spatial reasoning along with arithmetic, by reconstructing geometric objects as logical constructions out of relations. The logicist is largely silent on the metaphysics of physical space, treating that question as belonging to physics rather than to the foundations of mathematics. Where logicism speaks of space at all, it speaks of formally specifiable structures, not of an extended natural medium.

Logicism treats matter as emergent and largely peripheral to its proper subject matter. The logical objects on which the programme insists — numbers as extensions of concepts, classes as logical constructions — are not material in any substantive sense; they are abstract entities given by logical definition. The empirical world of material bodies is the proper domain of the natural sciences, on which the logicist takes no distinctive stand. What matters for the foundations of mathematics is the formal structure that logic exhibits, not the substance of any particular physical realisation.

The logicist treats the mathematical knower as a rational mind whose grasp of logical structure is the warrant for arithmetical truth. The observer is not the embodied, situated subject of phenomenology but an idealised reasoner who follows definitions and inference rules wherever they lead. Frege's anti-psychologism is decisive here: the laws of logic are not contingent features of human cognition but norms binding any rational thinker, and the mathematician's task is to make explicit what those norms require. Knowledge is gained by proof from logical premises, and its authority does not depend on intuition, sensation, or convention.

Energy belongs to the physical sciences and is not, on the logicist view, a foundational notion in its own right. To the extent that logicism addresses energy at all, it does so by treating the mathematics in which physical energy is expressed — real analysis, differential equations, group theory — as itself reducible to logic. The Hilbert-style 'energy' of a formal system, its proof-theoretic strength, is closer to the logicist's natural concern than the energy of physics. Energy is therefore treated as a derived, emergent notion that the underlying logical-mathematical framework helps to make precise.

Logical and mathematical truths are necessary, substantival, and discrete. They are not made up by human convention but discovered; the logician's task is to make explicit what was already implicit in the structure of thought.