Seminar za edukacijsku matematiku

U okviru Seminara za edukacijsku matematiku, u ponedjeljak 25. svibnja 2026. s početkom u 17:15 u prostoriji 101

profesor Bharath Sriraman (University of Montana)

održat će predavanje pod naslovom 

Near Misses in Mathematics and Its Learning Frameworks.

Predavanje će se održati i online, putem Zooma. Poveznica za praćenje:
https://zoom.us/j/97213729879?pwd=wFInSb3X1Twt6bFK4ad43e56wddzai.1

Meeting ID: 972 1372 9879
Passcode: 376030

ABSTRACT

The identity 1 + 2 + 3 + ・ ・ ・ = − 1/12 is true under zeta regularization, false under any totally regular summation method, and indispensable to quantum field theory and string theory.

How does a mathematical community come to understand an object this strange? This talk argues that mathematical progress is driven by near-misses: objects that fill every slot of an available conceptual frame except the one slot that turns out to be structurally necessary.

Drawing on Minsky’s notion of frames and uniframes (Society of Mind, 1986) and Lonergan’s theory of insight (Insight, 1957), we trace two parallel near-miss chains. The first runs through the irrationals— √2 (quadratic, periodic continued fraction) → e (patterned but not

periodic) → π (no known pattern) → Liouville numbers (μ = ∞)—with Roth’s theorem (1955) emerging as the uniframe the chain was pointing toward. The second runs through divergent series, from Grandi (1703) and the factorial series, through Fourier and Heaviside’s

correct-but-banned calculations, to the rehabilitations of Cesaro, Abel, and Borel, and finally to Hardy’s axiomatization (1949). Three independent derivations of − 1/12 —heuristic, zetaregularized, and via smoothed cutoffs—illustrate Lonergan’s distinction between insight and judgment, and locate the identity precisely as a fact about analytic continuation rather than about addition. The pedagogical conclusion: near-misses are not curiosities to be mentioned after the main theorem. They are the cognitive provocation that makes the main theorem necessary.

Keywords: divergent series; continued fractions; irrationality measure; Roth’s theorem; zeta

regularization; Minsky frames; Lonergan; mathematical pedagogy.