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"Ignorance is a state of mind, stored in neurons, not the environment. The red ball does not know that we are ignorant of it. A probability is a way of quantifying a state of mind. Our ignorance then obeys useful mathematical properties—Bayesian probability theory—allowing us to systematically reduce our ignorance through observation. How would you go about reducing ignorance if there were no way to measure ignorance? What, indeed, is the advantage of not quantifying our ignorance, once we understand that quantifying ignorance reflects a choice about how to think effectively, and not a physical property of red and white balls?"
I want to propose a short note for this priceless observation. Maybe I'm overreacting, and it's not as significant as I see it. Apologies if that is so.
Your conjecture presupposes a unidirectional linear, absolute, and static structure of knowledge - a minimal perspective or not generally applicable. It seems as if you have forgotten about the phenomenon, "a fresh pair of eyes." Literally meaning employing someone who has not gone the same road as you did (a person "more" ignorant of the problem at hand than you) to help you get out of your informational dead-end.
You have fallen into the same trap as the philosophers who believed that there is a formula to ultimate and absolute knowledge and ultimate state of mind, who prophesized that if only we find and follow this formula, everyone will attain eternal happiness. I'm personally skeptical about measuring the quality, quantity, and practical applicability of knowledge or ignorance. Let alone the questions about what really matters and their mutual interaction. But, unfortunately, your mode of thinking will most likely lead to the same premises and methods found in totalitarian regimes, and ultimately to inability to adapt and to intellectual stagnation. If I were to choose one argument against measuring knowledge, it would be that this will preclude the invention of the "ultimate" knowledge elixir and, as a result, will retain random factors in knowledge-seeking.
But to indulge your theory a little further, let me mention a few other predictions. For example, the fabric of knowledge is probably not linear or unidirectional, and it probably has local limits (dead-ends). And probably our perception of truths depends on time and our condition. And also, moving in one direction may increase ignorance in the opposite, so to speak. Of this, we have countless accounts.
When I think about epistemology, I sometimes remember Little Gidding. I think it has a very peculiar relationship with your discovery:
We shall not cease from exploration
And the end of all our exploring
Will be to arrive where we started
And know the place for the first time.