What would physics be like without relativity? What would mathematics be like without group theory? Philosophy without Objectivism? Music without Bach's influence?

All of these were historical near-misses. You don't even know about the ones that were hit.

Additivity is the simple and intuitive principle that it is better to know than to remain ignorant. We define knowledge as the sum of accumulated ideas over time minus those we have forgotten. That is,

Knowledge = Σ_time Ideas(time) - Loss(time)

First, let me explain what I mean when I call these monumental ideas “near misses”: they were ideas that were nearly lost to society through various forms of censorship. Relativity, as is well known, was popularized by Einstein (less well known figures, including Poincaré and Lorentz, also contributed to the idea). However, the very notion of relativity was considered absurd and discarded for six years before it was vindicated. Group theory in its modern form was developed by the young mathematicians Abel and Galois. Abel's idea, like Abel himself (who lived in poverty his whole life), was continuously overlooked, and his big break (in the form of his first academic position) came two days after his death of tuberculosis at the age of 26. Galois fared no better: his ideas, revolutionary as they were, were constantly underestimated and scorned by mathematicians as lacking rigor, and he himself was expelled from school, incarcerated, and ultimately died in a duel at the age of 20. The Fountainhead, Ayn Rand's famous book on her philosophy (which would later become known as Objectivism), was rejected by 12 publishers before it ever saw the light of day. Finally, Bach himself was simply overlooked for nearly a century after his death, known more as an organist than composer, until Felix Mendelssohn discovered and popularized his music.

They are ideas that society nearly lost through various forms of censorship, deliberate or incidental. Had they been truly lost, like a vast multitude of ideas that neither I nor anyone else will ever be able to name, the world would be a much worse place for it. This is a consequence of additivity.

When an idea is in our knowledge base, such as that of a tree having branches, we “know” it. When we truly forget this idea, it leaves our knowledge base; the only way to add it back is to re-learn it, although we tend not to truly forget things as to place them outside of what we can consciously recall. Nevertheless, the act of forgetting is fairly independent of the act of learning, and thus does not factor into our discussion.

Without taking forgetfulness into account, it should be clear that knowledge increases monotonically with incoming ideas; no such thing as a “negative idea” can exist. Thus, ideas are quite literally additive. The logical consequence of this is to gain exposure to as many ideas as possible in order to maximize the size of your knowledge base. Moreover, by the Universality principles previously discussed, the wider the knowledge base, the further the range of ideas one can synthesize. This has a compounding effect.

Thus, while we will argue that ideas have an inestimable objective worth in the next two chapters, additivity dictates that this worth is never negative. There's literally no such thing as a “bad idea” - except, perhaps, the suppression of other ideas.