One of the buzz words in the sciences that study complicated things is “Emergence”. As you build up a complex situation step by step, it appears that thresholds of complexity can be reached that herald the appearance of new structures and new types of behavior which were not present in the building blocks of that complexity. The world wide web or the stock market or human consciousness seem to be phenomena of this sort. They exhibit collective behavior which is more than the sum of their parts. If you reduce them to their elementary components, then the essence of the complex behavior disappears. Such phenomena are common in physics too. A collective property of a liquid, like viscosity, which describes its resistance to flowing, emerges when a large number of molecules combine. It is real but you won’t find a little bit of viscosity on each atom of hydrogen and oxygen in your cup of tea.
Emergence is itself a complex, and occasionally controversial subject. Philosophers and scientists attempt to define and distinguish between different types of emergence, while a few even dispute whether it really exists. One of the problems is that the most interesting scientific examples, like consciousness or life, are not understood and so there is an unfortunate extra layer of uncertainty attached to the cases used as exemplars. Here mathematics can help. It gives rise to many interesting emergent structures that are well defined and suggest ways in which to create whole families of new examples.
Take finite collections of positive numbers like 1,2,3,6,7,9. Then no matter how large they are, they will not possess the properties that “emerge” when a collection of numbers becomes infinite. As Georg Cantor first showed clearly in the 19th century, infinite collections of numbers possess properties not shared by any finite subset of them, no matter how large they are. Infinity is not just a big number. Add one to it and it stays the same, subtract infinity from it and its stays the same. The whole is not only bigger than its parts, it also possesses qualitatively different “emergent” features from any of its parts.
Many other examples can be found in topology, where the overall structure of an object can be strikingly different from its local structure. The most familiar is the Mobius strip. We make one by taking a thin rectangular strip of paper and gluing the ends together after adding a single twist to the paper. It is possible to make up that strip of paper by sticking together small rectangles of paper in a patchwork. The creation of the Mobius strip then looks like a type of emergent structure. All rectangles that were put together to make the strip have two faces. But when the ends are twisted and stuck together the Mobius strip that results has only one face. Again the whole has a property not shared by its parts.