Science, Decoded · The Mathematical Rosetta Stone for the AI Age

Σ is an addition loop.
Π is a multiplication loop.

You don’t need to solve every equation. You need to understand what it’s telling you. Physics, biology and AI papers often use the same mathematical machinery to describe very different things. This is a curriculum for learning to recognise it, and to tell when a resemblance is real.

\sum_{n=0}^{4} 3n

is the same instruction as

let sum = 0;
for (let n = 0; n <= 4; n++) {
  sum += 3 * n;
}
// sum === 30

\prod_{n=1}^{4} 2n

is the same instruction as

let product = 1;
for (let n = 1; n <= 4; n++) {
  product *= 2 * n;
}
// product === 384

The subscript sets where the counter starts and the superscript where it stops. The expression after the symbol is the loop body. Once you see that, $-\sum_i p_i \log p_i$ in an information-theory paper and $\sum_i w_i x_i$ in a neural-network paper stop looking like a wall. Each is a loop with a different body.

Start anywhere

Begin with the Σ loop

Module 1 opens with the realisation this curriculum is built around. Or jump straight to a symbol you keep seeing.

Start Module 1 Browse symbols

Reference

The Rosetta Stone

30 symbols you meet in research papers. Each has a plain-English reading, a code analogy, a visual, and what it means in each field. Search by symbol or by description.

Flagship exercise

Three papers, one question

Do neural networks converge on shared structure as they scale? Map the papers as Cartographer, attack the synthesis as Adversary, and keep what survives as Editor.

Three levels of connection

Every cross-disciplinary claim on this site is sorted into one of three levels. Most interesting-sounding connections live at Level 1. The hard part of interdisciplinary research is saying honestly which level you have reached.

Analogy

Two phenomena resemble one another. That can suggest where to look, but resemblance is not evidence of a common mechanism.

Shared mathematical structure

Both can be described by related equations or formal models. This is substantive, but the same equation can arise from different causes.

Shared mechanism or tested prediction

A proposed link names a mechanism, or makes specific predictions that survive testing. These are two different achievements, and this curriculum keeps them apart.

Evidence labels

Every substantive claim carries one of six labels. Learn to apply them to everything you read, including what AI models tell you.

EstablishedEmpirical regularityMathematical modelTheoretical proposalAnalogySpeculation

The eight modules

About eight weeks at two or three focused hours each. The order is a suggestion, not a deadline: prerequisites are advisory, nothing is locked, and there are no grades.

01Mathematics~2.5 h

Learning to Read Equations

Variables, functions, exponents, logarithms, Σ and Π as loops, and a full reading of the neural scaling-law equation L(N) = L∞ + A·N^−α.

02Physics~2.5 h

The Physics of Systems

Forces, energy, conservation, entropy, waves, fields, units and symmetry, read through F = ma, E = mc² and λ = c/f, with their assumptions stated.

03Biology~2.5 h

Biology as an Information System

DNA to protein, regulatory networks, natural selection, homeostasis, neurons and metabolic scaling, with a hard line between evolution and gradient descent.

04Mathematics~3 h

Probability, Entropy, and Information

Probability, Bayes, distributions, expectation, Shannon entropy, cross-entropy and likelihood: the mathematics behind every LM loss curve, with its look-alikes in physics and genomics.

05Complexity~2.5 h

Complexity and Systems Thinking

Feedback loops, networks, power laws and phase transitions, and why two systems that share a power law need not share a mechanism.

06AI / ML~3 h

The Mathematics Inside Modern AI

Vectors, dot products, gradients, loss and attention: the handful of operations that most ML equations are built from, plus where the textbook versions stop matching production systems.

07Complexity~3 h

The Physics–Biology–AI Connection Lab

Real papers as case studies in grading connections across physics, biology and AI, including the Kaplan thread and why the AdS/CFT ↔ Constitutional AI analogy stays an analogy.

08Epistemics~2.5 h

The Scientific Skeptic’s Toolkit

The working toolkit for evaluating claims: evidence vs. hypothesis, confounding, overfitting, extrapolation, preprints, benchmark contamination, and how LLMs fabricate plausible syntheses.

Tools & labs

Rosetta Stone30 symbols, decoded

Equation InspectorTap any symbol in an equation

Three Papers, One QuestionCartographer · Adversary · Editor

Paper Reading LabGuided reads of real papers

Connection AnalyzerAnalogy, structure or mechanism?

Skeptic's ToolkitFind where the argument breaks

Prompt LabCopy-ready research prompts

Media LibraryVerified videos, courses, books

Notebook & PathNotes, bookmarks, schedule

How this was built

The design borrows from integrated-science programs that teach shared quantitative structure across disciplines (Princeton’s Integrated Science Curriculum, Northwestern’s Integrated Science Program) and from the Santa Fe Institute’s Complexity Explorer. Free resources are linked in the Media Library, and every video and URL there was checked on 30 October 2026.

Everything you write, including notes, reflections, bookmarks and worksheets, is saved only in this browser. No account, no server and no AI model sits behind this site. Where an AI could help, the site gives you a prompt to take to Claude or ChatGPT and says plainly that nothing was generated automatically.

Σ is an addition loop.Π is a multiplication loop.