#Worldview matters
Worldview matters
If this math professor is like many that I studied under, he probably doesn't know anything about metallurgy or agriculture or a host of other applied fields in any practical detail. He probably wouldn't know about crop rotation or how to domesticate corn or how to carve wood with stone tools or how to start a fire or how to knap flint.
But, what a math professor would have, is a scientific and modern worldview.
He would think in terms of a spherical earth in a heliocentric solar system containing other planets in a big galaxy full of stars in a bigger universe of finite age. He would see mountains and valleys and oceans in the context of plate tectonics. He would know the basic concept of what a volcano was and would be aware of the possibilities of meteors and comets. He would see rocks as assemblages of certain kinds of minerals.
He might not be good at predicting weather, but would understand that it flows from chaotic deterministic systems, that air is a mixture of gases, and that water evaporates, is carried in clouds and produces rain when the temperature and pressure are right.
He would have a germ theory of disease, a crude Mendelian concept of genetic heredity, and understand the concepts of atoms and molecules. He would have a sense of how predator-prey models work and the basic mechanisms behind evolution.
He would naturally think in terms of currency, the time value of money, and supply and demand. He would recognize core concepts like the non-zero sum nature of economic interaction. He would be alert to the possibility of domestication of plants and animals.
His natural philosophy would be more Aristotle than Plato. He would understand the scientific method and be able to advocate for it.
He would know instinctively what kinds of social organization and etiquette and diplomacy lead to productive interaction and which do not. He would know how to teach people to operate productively in much larger groups than they had ever imagined.
He would have a good knack for distinguishing what kind of investigations and new ideas were likely to be fruitful and which were not.
For example, wheels were used in toys by the Sumerians and Aztecs for centuries before anyone thought to make an industrial scale version with economic advantage, but our professor would see those opportunities to generalize existing knowledge that might seem like dead ends to others.
Also, the math professor would be literate, would insist on training children to be literate from a young age when it is easiest to learn (in the real world people didn't start to learn to read and write for the first time until middle school or high school age and these skills were restricted to a narrow class of scholars until the late Bronze Age or early Iron Age, the Minoans and earliest Chinese dynasties, for example, had no literate elementary school age children). This technology could provide huge advantages.
These worldview differences wouldn't have direct application, but would set his band of people on a straight and narrow path towards unprecedented progress relative to those of people with a muddled and inaccurate understanding of the big picture of how the world worked.
