history of mathematical & scientific revolutions

This is the syllabus for hon201. Find the links for course notes & submissions.

instructor

Dr. Josh Thompson | office - JAMR 2226 | email

classroom

hon 201 | 12:00 - 1:40 pm | TTh | Harden 309 | zoom

your daily attendance is required

texts

  • Dunham, William. Journey Through Genius: The Great Theorems of Mathematics.
  • Kuhn, Thomas S. The Structure of Scientific Revolutions.
  • Harari, Yuval Noah. Nexus: A Brief History of Information Networks from the Stone Age to AI.
  • Charniak, Eugene. AI & I: A Intellectual History of Artificial Intelligence.

office hours

  • Monday: 11am - 12pm
  • Tuesday: 2pm - 3pm
  • Wednesday: 11am - 12pm
  • Thursday: 2pm - 3pm

assessments

  • Weekly summaries - 20%
  • Two short response (in class) papers (weeks 4 and 10) — 20%
  • Midterm essay (week 8) — 20%
  • Participation and discussion leadership — 20%
  • Final paper & presentation (week 14) — 20%

grading scale

  • A: 94% - 100% | A- or B+: 93% - 88%
  • B: 83% – 87% | B- or C+: 83% - 78%
  • C: 73% – 77% | C- or D+: 73% - 68%
  • D: 63% – 67% | D-: 62% - 60%
  • F: Below 60% |

course description

This course investigates how human understanding transforms during major intellectual upheavals. We will use Thomas Kuhn’s philosophy of scientific paradigms to examine case studies in mathematics, science, information networks, and modern artificial intelligence. We explore how paradigms form, fracture, and redefine truth, knowledge, and power.

weekly schedule

unit i: paradigms, proof, and truth (weeks 1–4)

how communities construct frameworks of knowledge, establish axioms, and manage information networks.

week 1: what is a paradigm?

  • core themes: paradigms, normal science, scientific communities, and the myth of linear progress.
  • readings:
    • kuhn: ch. i–iii (introduction: a role for history, the route to normal science, the nature of normal science).
  • seminar focus: is science a steady accumulation of objective facts, or a series of community-governed worldviews?

week 2: axiomatic revolutions & the invention of proof

  • core themes: deductive certainty as humanity’s first formal truth network; the greek mathematical revolution.
  • readings:
    • dunham: ch. 1–2 (hippocrates’ quadrature of the lune, euclid’s proof of the pythagorean theorem).
    • harari: ch. 1–2 (the storytellers, inorganic networks and information vs. truth).
  • seminar focus: how does euclidean geometry establish a “paradigm” of proof? harari’s distinction between networks optimizing for order versus truth.

week 3: canons, bureaucracies, and information gatekeepers

  • core themes: institutionalizing truth; how early information networks preserve paradigms and suppress anomalies.
  • readings:
    • harari: ch. 3–4 (the canon: holy books and censorship).
    • kuhn: ch. iv–v (normal science as puzzle-solving, the priority of paradigms).
  • seminar focus: how sacred texts and scientific textbooks act as stabilizing instruments of “normal science.”

week 4: the archimedean leap & mathematical invention

  • core themes: extending paradigms through technical mastery; the boundary between calculation and physical reality.
  • readings:
    • dunham: ch. 4 (archimedes’ determination of circular area).
    • charniak: ch. 1 (early days & the dream of formal reasoning).
  • seminar focus: what role do exceptional tools (and exceptional figures) play in expanding a paradigm before it encounters conceptual limits?

unit ii: crisis, calculus, and the mechanization of knowledge (weeks 5–7)

the breakdown of old models, the calculus revolution, and the rise of mass-information networks.

week 5: anomaly, crisis, and the birth of calculus

  • core themes: incommensurability; infinitesimals as an “anomaly” that yielded immense practical power.
  • readings:
    • kuhn: ch. vi–viii (anomaly and the emergence of scientific discoveries, crisis and the emergence of scientific theories).
    • dunham: ch. 7 (the bernoullis and the harmonic series), ch. 8 (the epoch-making work of newton).
  • seminar focus: the leibniz-newton controversy: how competing paradigms fight for community acceptance when foundational rigor is initially incomplete.

week 6: the printing press and the distributed revolution

  • core themes: how media technology reshapes scientific communities and institutional authority.
  • readings:
    • harari: ch. 5–6 (the printing press: the scientific revolution vs. the witch hunts).
    • kuhn: ch. ix–x (the nature and necessity of scientific revolutions, revolutions as changes of world view).
  • seminar focus: the dual nature of open information networks: how the press facilitated both modern peer-reviewed science and widespread mass delusions.

week 7: the transfinite shock — when math broke its own paradigm

  • core themes: foundational crises; non-constructive proofs; cantor’s revolution of multiple infinities.
  • readings:
    • dunham: ch. 11–12 (cantor and the transfinite realm, the uncountability of the continuum).
    • kuhn: ch. xi–xii (the invisibility of revolutions, the resolution of revolutions).
  • seminar focus: resistance to cantor’s transfinite arithmetic & incommensurability.

unit iii: the symbolic ai era & the limits of discrete logic (weeks 8–10)

the rise of classical ai (gofai), symbolic representations, and the “brittleness” barrier.

week 8: classical ai as normal science

  • core themes: symbolic ai, expert systems, discrete logic tokens, and parsing natural language.
  • readings:
    • charniak: ch. 2–4 (rule-based systems, natural language, and classical search).
    • kuhn: ch. xiii (progress through revolutions).
  • seminar focus: the classical ai worldview: conceptualizing human cognition as the deterministic manipulation of formal symbols.

week 9: the anomaly of common sense & the ai winters

  • core themes: the frame problem, brittle knowledge bases, combinatorial explosion, and the crisis of classical ai.
  • readings:
    • charniak: ch. 5–6 (the wall: why hand-crafted rules failed).
    • harari: ch. 7 (totalitarianism and centralized information loops).
  • seminar focus: top-down feature engineering vs. with noisy, continuous real-world data.

week 10: midterm colloquium — mapping revolutions across disciplines

  • seminar format: collaborative student workshops and presentations analyzing historical inflection points:
    • case 1: euclidean geometry \(\rightarrow\) non-euclidean geometry & transfinite sets.
    • case 2: classical determinism \(\rightarrow\) statistical mechanics & quantum probability.
    • case 3: symbolic logic / expert systems \(\rightarrow\) statistical learning & neural representations.

unit iv: the deep learning revolution & the algorithmic nexus (weeks 11–14)

vector representations, emergent intelligence, self-correcting networks, and post-human epistemologies.

week 11: the paradigm shift: from discrete logic to continuous vectors

  • core themes: neural networks, backpropagation, continuous vector spaces, high-dimensional geometry, and embeddings.
  • readings:
    • charniak: ch. 7–9 (the statistical turn, word embeddings, and the return of the perceptron).
  • seminar focus: charniak’s central thesis: why continuous mathematics (vectors, gradients, linear algebra) succeeded where discrete logical manipulation failed.

week 12: the incommensurability of the “black box”

  • core themes: empirical success vs. theoretical explainability; are neural networks a new scientific paradigm or an engineering shortcut?
  • readings:
    • charniak: ch. 10–12 (transformers, large language models, and the open frontier).
    • kuhn: postscript - 1969 (exemplars, tacit knowledge, and disciplinary matrices).
  • seminar focus: if an ai model reliably predicts physical phenomena or generates mathematical proofs without explicit human-readable axioms, does it still fit kuhn’s definition of scientific understanding?

week 13: nexus in the silicon age — inorganic intelligence and social power

  • core themes: autonomous algorithms, feedback loops, the breakdown of self-correcting mechanisms, and truth vs. engagement.
  • readings:
    • harari: ch. 8–10 (the algorithmic bureaucracy, inscrutable agents, and the threat to democracy).
  • seminar focus: how autonomous ai agents differ from previous information tools (e.g., printing press, radio); the risks of closed-loop algorithmic governance.

week 14: final synthesis & the future of inquiry

  • core themes: the next scientific revolution: will future paradigms be authored by machines?
  • readings:
    • harari: ch. 11 (conclusion: restoring self-correction).
  • seminar focus: concluding roundtable and capstone project presentations.

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