5.5.2 Full Model Response Notes

IB May 2026 (M26) TOK Title #5 Model Response

To what extent do you agree with the claim that “all things are numbers” (Pythagoras)? Discuss with reference to the arts and the human sciences.

1. The essay below is written as a teaching draft to illustrate the structure, tone, and depth of analysis expected in a high-scoring Theory of Knowledge essay.
2. It includes call-outs after each paragraph that explain why particular choices were made and how they align with the IB assessment criteria.
3. In a formal submission, you would need to provide proper references and citations (using MLA, APA, or the referencing style your school/IB requires).

Introduction

When Pythagoras claimed that “all things are numbers,” he was pointing to a world where patterns, ratios, and proportions underpinned reality. In TOK, the challenge is to ask whether everything in knowledge can truly be reduced to numbers. At a literal level, the claim is too strong: not all things in the arts or human sciences can be quantified without loss. Yet at a structural level, numbers often provide the scaffolding that holds knowledge together. I argue that Pythagoras’s claim is true to a considerable extent: numbers are indispensable tools, but they do not capture the whole of meaning, interpretation, or lived experience.

Arts I: Where Numbers Generate Form

Music is the most obvious case where numbers create knowledge. Frequency ratios explain consonance and dissonance, beats per minute define rhythm, and digital recordings reduce sound to binary code. Even in visual art, proportion, symmetry, and perspective, from the golden ratio in Da Vinci’s Vitruvian Man to geometric precision in Islamic mosaics, are grounded in numbers. In these cases, numbers are not just measurements after the fact but the building blocks of the art itself. To this extent, the claim that “all things are numbers” feels persuasive.

Arts II: Where Numbers Cannot Capture Meaning

But numbers fall silent when it comes to interpretation. A Chopin nocturne can be scored in notes and tempos, yet the reason it makes someone cry is not in the numbers. Abstract art goes further, sometimes created to resist reduction altogether. A Jackson Pollock canvas may be analysed in terms of paint viscosity or fractal geometry, but its meaning lies in subjective response. Numbers can chart structure, but they cannot exhaust significance. In the arts, numbers are powerful lenses, but they never give the whole view.

Human Sciences I: Numbers As Authority

In the human sciences, numbers often stand as the gold standard of knowledge. Economists model markets with supply and demand curves; sociologists track inequality with the Gini coefficient; psychologists run experiments that turn behaviour into data points. Governments even use “happiness indexes,” asking citizens to rate life satisfaction from 0–10. These figures make comparison possible and policy actionable. Numbers, here, transform individual experiences into generalizable knowledge. In this sense, Pythagoras seems vindicated: numbers are what make messy social phenomena tractable.

Human Sciences II: Numbers As Reduction

Despite this, quantification has limits. Imagine rating your friendship as “8/10.” The number is convenient, but it erases the complexity of trust, history, and shared experience. Human sciences face the same issue: surveys and scales capture slices of reality but flatten nuance. GDP per capita, for example, suggests Ireland is one of the richest countries in the world, yet average wages are far lower, the number hides inequality. Numbers can simplify to the point of distortion. They are useful, but not sufficient for genuine understanding.

Cross-AOK comparison

The comparison reveals a pattern. In the arts, numbers describe structure but cannot explain meaning. In the human sciences, numbers give authority but risk reduction. Across both, numbers are necessary but not sufficient. They provide clarity, comparability, and structure, yet they leave out interpretation, intention, and context. Pythagoras was right to see numbers everywhere, but wrong to call them everything.

Conclusion

I agree to a considerable but not total extent with Pythagoras. Numbers are foundational: they generate music, shape visual form, allow social science to generalise, and give knowledge legitimacy. But “all things” is too sweeping. Numbers do not capture why art moves us, why people act irrationally, or why meaning shifts with culture. The deeper lesson is that knowledge always involves a balance: numbers as skeleton, meaning as flesh. Without numbers, knowledge collapses into vagueness; without meaning, numbers are lifeless. Pythagoras was right that numbers matter everywhere, but the essence of “all things” cannot be reduced to them alone.