theoretically you can introduce new words to be more precise, for example eskimos have many words for snow, from (1) that is because they use polysynthesis: a base word is attached to many different suffixes which change its meaning. So, while in the English language we might have a sentence describing snow, fusional languages such as the Eskimo-Aleut family will have long, complex words.
So, you may solve the ambiguity problem by introducing a huge numbers of words, that could help a LLM, but humans need a relatively small vocabulary tailored to everyday use. So the tradeoff is reducing ambiguity and not increasing a lot the vocabulary.
I think that when law is introduced the consequences are not clear so the small print is used to introduce modifications to the rules, so the problem is about adapting a rule to the everyday use of it.
I think you need a scalar product to define volume. But fuzzy thinking an analogy: Think friendship, define three concepts related to friendship and that they are orthogonal, now establish some numeric scale to measure each concept, then you have just defined a volume to measure the degree of friendship. Unfortunately there is no canonical way to establish those orthogonal features, but they could be obtained applying a linear model to big data sets of measures of friendship.
> Like, you can inscribe a circle inside a square and have it touch all sides without extending outside the square, and you can similarly “inscribe” a sphere inside a cube such that the sphere touches all sides of the cube without extending outside the cube. Does this hold true in higher dimensions?
Of course, points with only one nonzero coordinates xi in $\{1,-1}$ are in both the sphere and the sides of the cube, and those are the minimal distance points from the sides of the cube to the center of the circle (since they are orthogonal to the hyperplane xi=+1, or xi=-1 that contains it). That also shows that no side of the cube extends outside of the circle. Some more math: A well known result in math is that the minimum distance from the origin O to a linear subspace of R^n is attained at points P such that OP is orthogonal to the direction of the subspace. In this case, since the linear space is a hyperplane there is just one orthogonal direction to that hyperplane, and there is just one point in the intersection of the hyperplane and the line defined by the point O and the orthogonal direction to the hyperplane, and that intersection is just the points we alluded before.
What language don't require type annotations to achieve good performance?, more specifically tell me any programming language that can beat sbcl at speed without using type annotations.
So, you may solve the ambiguity problem by introducing a huge numbers of words, that could help a LLM, but humans need a relatively small vocabulary tailored to everyday use. So the tradeoff is reducing ambiguity and not increasing a lot the vocabulary.
(1) https://readable.com/blog/do-inuits-really-have-50-words-for...