Bon voyage: Long-lasting learning

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The world is complex! But our nine reasoning tools help us master and enjoy the complexity. Spanning fields of knowledge, the tools connect disparate facts and ideas and promote long-lasting learning.

An analogy for the value of connected knowledge is an infinite two-dimensional lattice of dots: a percolation lattice [21]. Every dot marks a piece of knowledge—a fact or an idea. Now add bonds between neighboring pieces of knowledge, with a probability p_bond for each bond. The following figures show examples of finite lattices starting at p_bond = 0.4. Marked in bold is the largest cluster—the largest connected set of dots. As p_bond increases, this cluster unifies an ever-larger fraction of the lattice of knowledge.

An infinite lattice might hold many infinite clusters, and the measure analogous to the size of the largest cluster is the fraction of dots belonging to an infinite cluster. This fraction \(f_{\infty}\) is, like the number of infinite clusters, zero until p_bond reaches the critical probability 0.5. Then it rises above zero and, as p_bond rises, eventually reaches 1.

For long-lasting learning, the pieces of knowledge should support each other through their connections. For when we remember a fact or use an idea, we activate connected facts and ideas and solidify them in our minds.

Our knowledge lives best in infinite, self-supporting clusters. But if we learn facts and ideas in isolation, we make dots without bonds. Then p_bond falls and, with it, the membership in infinite clusters. If p_bond falls too much, the infinite clusters simply vanish.

So, for long-lasting learning and understanding, make bonds; connect each new fact and idea to what you already know. This way of thinking will help you learn in one year what took me two or twenty. Use your reasoning tools to weave a richly connected, durable tapestry of knowledge. Bon voyage as you learn and discover new ideas and their fascinating connections!

Only connect! That was the whole of her sermon… Live in fragments no longer.

—E. M. Forster [16]