Having just discussed the objective nature of an idea, it may seem as if we are arguing that ideas are easily stratified into “good” and “bad”, and indeed, we do argue that some ideas are inherently “better” than others, in that they possess a wider and more powerful range of applications. However, to judge the true worth of an idea by our definition, one would require knowing the worth of every idea in the first idea's image - that is, every idea that could conceivably be generated from the starting point of our original idea. However, doing this requires knowing the worth of every idea in those ideas' images as well, and so forth. If the network of ideas existing in the universe is strongly connected, as it would seem to be, this literally would require knowing every possible unit of knowledge in the universe. Even if this were not the case, there are an infinite number of ideas in the universe (Proof: suppose there were not: all ideas must then derive a finite number of ideas, for if the cardinality of one idea's image is infinite, the cardinality of all ideas combined must be infinite as well. Consider the idea of a polynomial equation, say x² + 4x + 2. We may derive new ideas from this idea by adding a constant. However, there are infinitely many constants which may be used in this manner, each of them resulting in a new idea. Thus, the image of a polynomial equation is infinite, contradicting our assumption, and thus the image of a universe containing polynomial equations cannot be finite). True objective knowledge of the universe thus requires infinite knowledge. Perhaps such knowledge may be considered the province of whatever omniscient deity (or lack thereof) you believe in, but it certainly is not ours.

Even if we did have infinite knowledge, we would still be limited by the inherent inability to deduce the answers to certain problems. For example, it is impossible to tell whether a computer program will terminate on all possible inputs (the Halting Problem), and there are entire axiomatic systems containing undecidable propositions (as shown by Gödel's Theorem of Incompleteness).

Thus, we must accept that our knowledge of the universe and our comprehension of ideas is not perfect, and, as the cardinality of our knowledge base is concerned, is in fact quite infinitesimal. It is, however, better than nothing, and thus we must estimate as best we are able if we wish to know anything at all.

There are a number of practical applications of this principle. First, it suggests that we should always provide for contingencies in our plans, as we do not possess complete knowledge about the scenarios we are venturing into. Among other things, this is yet another reason why it is prudent to acquire breadth as well as depth of knowledge. Second, it means that we may be very often wrong about things. However, the task of rigorously purging all inaccuracies from our knowledge base is impossible, and we should stop berating ourselves for failure to do so. The words “I don't know” are not preferable, but neither are they shameful, just as the thought “one day I shall no longer exist” is disquieting but not inherently evil.

Skepticism seems like the appropriate response to such inherently intractable ignorance, when it is, in fact, the exact opposite, for when we are unsure of an idea, the worst course of action is to discard it out of hand. In fact, nothing kills creativity more easily than snap judgments. If we do not truly understand every facet of the universe, we must always consider the possibility that we are wrong. Because ideas are additive and have a natural tendency to spread if not suppressed, the better course of action is to avoid interfering with their expression, granting them the maximal extent of their natural capacity for exposure.

Thus,

Think: Do not allow others to judge ideas for you and do not judge ideas for others in turn.

Keep an open mind.

Welcome ideas into the abode of your mind and you will find they make eager guests.

Finally, we must acknowledge that our perception tints the ideas we perceive, and that, unlike the idea itself, there may be ambiguous or conflicting representations of the idea within reality. Consider a photograph of a tree. If we have knowledge of the scene within the photograph, we are likely to identify the photograph with the original tree and to treat the tree in the photograph as if it is the actual tree. However, suppose we changed the exposure on the camera and photographed the tree a second time. The photograph looks different, and yet we would still identify the tree in the photograph with the physical concept and the abstract idea of the tree in reality. We thus have two differing representations of the tree, but we are able to identify them using the same concept. This is another factor confounding our ability to truly perceive the objective nature of the world.