Federovici Silviu-Constantin

One possible way to understand how reasoning occurs in mathematics is to posit an ideal mind that mirrors our own thinking. Two contrasting paradigms can be identified in Turing’s view on computability and Brouwer’s intuitionism. While the Universal Machine captures mathematical development through mechanical processes, the Creating Subject theory situates this activity within constructive acts of mental experience. This paper investigates how idealizing cognition as subjective-experiential rather than deterministic-mechanical can help address certain limitations in Turing’s original conception. Drawing on van Atten (2008), I propose re-evaluating the tensions between these frameworks to enhance the classical perspective, outlining a hybrid model where mechanical computability is extended by the intuitionistic concept of mental experience. This synthesis demonstrates that experiential features such as temporal intuition, free decision, and reflexivity are essential for an accurate account of reasoning in mathematics.

Brouwer famously challenged the limits of mathematical knowledge by arguing that classical formalism obscures intuitive evidence. Hilbert, by contrast, considered that intuitive insights could safely be ignored as long as formal systems remained consistent and complete. Such a disagreement created a paradigmatic tension between intuitionism and formalism in how the foundations of mathematics should be regarded. This paper evaluates Hilbert’s eventual pragmatic dominance and explores, via a shared Kantian heritage, how intuitionistic insights might coexist with formal approaches. Focusing on axioms, the analysis reveals how neglecting certain epistemic values while admitting alternative forms of evidence shapes our understanding of mathematical limits.