The Raskin Center

The Effectiveness of Mathematics

Archive Notice: This page is part of the Jef Raskin historical archive, preserved for its academic and historical significance.

An unpublished essay by Jef Raskin

In 1960, the physicist Eugene Wigner published an essay titled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.” The question Wigner posed was simple and unanswerable: why does abstract mathematics — developed by mathematicians for purely internal reasons, with no reference to the physical world — turn out to describe the physical world so precisely?

Maxwell’s equations for electromagnetism were developed as mathematical structures. They turned out to predict the existence of radio waves before radio waves were discovered. The non-Euclidean geometry developed in the nineteenth century as a purely abstract exercise turned out to be exactly the geometry of spacetime in general relativity. Group theory, developed by mathematicians pursuing questions in abstract algebra, turned out to be the precise language for describing quantum mechanical symmetries.

This is strange. Wigner called it unreasonable. Raskin found it worth examining at length.

The Question

Mathematics is, on one account, a human invention — a system of symbols, rules, and deductions that exists in minds and on paper. Physical reality is, on the same account, something entirely independent of human thought. Why should a human invention map onto something independent of human thought with such precision?

Several possible answers present themselves, none entirely satisfying.

Mathematics is discovered, not invented. On the Platonist view, mathematical structures exist independently of human minds, and mathematicians discover them the way scientists discover physical facts. If this is correct, then mathematical structures and physical structures are both real, and their alignment is not coincidental — it reflects a deep feature of reality. But this answer pushes the question back rather than answering it. Why does physical reality have the structure of the mathematical universe?

Physical reality is mathematical. On a more radical view, associated with physicist Max Tegmark’s mathematical universe hypothesis, physical reality is not just described by mathematics — it is a mathematical structure. The universe we inhabit is, on this view, a particular mathematical object. The effectiveness of mathematics is not mysterious because mathematics and physics are, at some level, the same thing. This is appealing and nearly impossible to test.

We select for effective mathematics. A more deflationary answer: of the vast number of mathematical structures that have been developed, we notice and remember the ones that turned out to be physically applicable. The failures — mathematical systems developed in hopes of physical application that led nowhere — are forgotten. What looks like an unreasonably high hit rate is actually a publication bias operating across centuries.

Mathematics reflects cognitive structure, not physical structure. Perhaps mathematical effectiveness tells us about how human minds organize reality rather than about reality itself. Human brains have evolved to find certain patterns salient, to categorize in certain ways, to abstract in certain directions. Mathematics is the formal extension of these cognitive tendencies. Physical reality, as humans perceive and measure it, is already shaped by the same cognitive tendencies. The alignment is real, but it is an alignment between cognitive tools and cognitively-shaped physical models, not between pure thought and raw reality.

Raskin’s Position

Raskin was trained as a mathematician and philosopher before he was a computer scientist, and he approached this question with both rigor and skepticism toward all simple answers. His essay does not arrive at a clean resolution — Wigner’s question does not permit one — but it does develop a particular emphasis.

Raskin was interested in the implications of the question for how we think about the relationship between formal systems and practice. The unreasonable effectiveness of mathematics in physics is a specific case of a broader question: when and why do formal, abstract systems turn out to be useful tools for understanding and acting in the world?

This question had direct relevance to his work in interface design. Raskin had long argued that interface quality should be measured rather than estimated — that formal tools like GOMS analysis and Fitts’s law could provide quantitative predictions of interface performance more reliable than designer intuition. This was a small-scale version of the claim that mathematical structures could model human behavior.

If Wigner’s question has a satisfying answer — if there is a reason why formal systems are effective — that answer has implications for interface design. If the effectiveness is mysterious but real, it still licenses the use of formal tools in design while leaving their success unexplained.

The Relevance to Rigorous Design

Raskin’s engagement with the Wigner question reflected a consistent theme in his thinking: the preference for formal, quantifiable accounts over informal, intuitive ones. He was skeptical of design decisions made on the basis of what felt right. He wanted measurements.

This is not the dominant mode of design practice. Interface design is typically a qualitative discipline — designers make decisions based on experience, aesthetic judgment, user observation, and heuristic principles. Raskin agreed that these inputs were valuable. He thought they were insufficient.

The formal tools available in his time — GOMS, Fitts’s law, keystroke-level models — were crude compared to the precision of mathematical physics. But they were better than nothing. They made specific, testable predictions. They could be compared to measurement. And when they were applied, they sometimes produced results that surprised experienced designers — showing that interfaces that felt fast were slower than ones that felt awkward, that menus that felt organized took longer to navigate than ones that felt cluttered.

This was, in Raskin’s view, the value of formal thinking applied to practical problems: it can contradict intuition. And when formal analysis contradicts intuition, at least the question becomes specific. You can run an experiment. You can settle the dispute.

This page is part of the Jef Raskin Archive, preserving the unpublished writings of the creator of the Macintosh project at Apple.