Abstract Frege argues that number is so unlike the things we accept as properties of external objects it cannot be such a property. In particular, (1) arbitrary in way qualities are not, and (2) not predicated its subjects are. Most Aristotle scholars suppose either has refuted Aristotle's theory or avoids Frege's objections by making numbers objects. This led some to conclude accounts arithmet...

In the debate on Frege Point, ‘Spinoza thesis’ is often mentioned. But Leibniz kept out. Yet, this topic, Spinoza and shared a fairly similar goal. They sought to root assertive force in conceptual activity of subject. Leibniz, unlike Spinoza, wanted also build coherent theory propositions. Propositions are for him always provided with force. what affirmed by propositions logic only possibility...

According to what used be the standard view (Poincaré, Wang, etc.), although Frege endorses, and Kant denies, claim that arithmetic is reducible logic, there not a substantive disagreement between them because their conceptions of logic are too different. In his “Frege, Kant, Logic in Logicism,” John MacFarlane aims establish do share enough conception for this substantive, judicable dispute. m...

[This is the penultimate version of an essay whose final version appears in Philosophia Mathematica Vol 15 #3 (2007) pp 321-346]

We give a general reduction of lengths-of-proofs lower bounds for constant depth Frege systems in DeMorgan language augmented by a connective counting modulo a prime p (the so called AC[p] Frege systems) to computational complexity lower bounds for search tasks involving search trees branching upon values of maps on the vector space of low degree polynomials over Fp. In 1988 Ajtai [2] proved th...

We study initial cuts of models of weak two-sorted Bounded Arithmetics with respect to the strength of their theories and show that these theories are stronger than the original one. More explicitly we will see that polylogarithmic cuts of models of V are models of VNC by formalizing a proof of Nepomnjascij’s Theorem in such cuts. This is a strengthening of a result by Paris and Wilkie [17] and...

The Hajj os Calculus is a simple, nondeterministic procedure which generates the class of non-3-colorable graphs. Mansseld and Welsch MW] posed the question of whether there exist graphs which require exponential-sized Hajj os constructions. Unless NP 6 = coNP, there must exist graphs which require exponential-sized constructions, but to date, little progress has been made on this question, des...