Brouwer–Hilbert on the Limits of Mathematical Knowledge
DOI:
https://doi.org/10.24193/subbphil.2025.sp.iss.02Keywords:
philosophy of mathematics, Brouwer-Hilbert controversy, epistemic limits, Kantian heritageAbstract
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.
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