"essence"

The term 'essence' from philosophy does not refer to a verbal construct; it refers to that which make a thing what it is - a single hypothesized and putative characteristic that supposedly can be abstracted to the object level and subsequently labeled with a word. You must understand the background for this point of view - that of Cartesian or Platonic dualism - that there exists a world of ideals, of which physical instances are but imperfect copies.

From a general semantics point of view, in which "what is going on" (that which the parabola represents), cannot be directly accessed or known, we are limited to object level and verbal level abstractions. Any statement about the "essence" of something is the verbal expression of an abstraction, and consequently, one made by an abstractor.

Therefore, Anything we say is an essence is removed from the event level by at least two levels of abstraction, and indicates a connection made by the abstractor in question between his or her symbolic environment and his or her object world. Although there is a great proportion of commonality in the symbolic environments of two native speakers of the same language, there are always individual differences sufficient that we can infer that EVERY mapping from any person's object level experience to his or her symbolic environment and the union of all such symbolic environments is unique.

What it boils down to is this: Any expression of an abstraction that we label as an essence gets "projected" onto the what is going on, and this projection may or may not correspond to any "actual" (independent of any observers) putative "structure" in the what is going on. Naive realists believe such structures exist, but this is a metaphysical question, and general semantics does not deal in metaphysical questions. General semantics is strictly "modern open applied epistemology"; as such it deals only with epistemological questions - how we know. We can make models and we can corroborate them, but we cannot confirm them. We can certainly disconfirm them.

So, if you begin with the postulate that something has an essence, general semantics would ask "how can you know that?", and that must be implemented by extensionally oriented empirical testing. Once you have made an abstraction this is no longer absolute - it's relative to the observer. Once you have expressed that abstraction in a formulation, you have made another abstraction - relative to the abstractor's experience with the general symbolic environment (his or her idiosyncratic symbolic environment).

Simply put, the term 'essence' traditionally refers to putative unique characteristics of ideals in "Plato's heaven" on a one-to-one basis. Each "ideal" has a unique essence or essential property. Saying what that is, however, is another topic.

As a practical matter, under the rubric of general semantics, in common speech, a formulation asserted to be an expression of the essence of some named thing indicates some characteristic or structure that a particular abstractor wants to focus on together with his or her judgement that this structure is more relevant (under that person's experience) than other structures that can be abstracted. Communicating that structure typically involves the use of metaphor in ordinary language. It can involve the use of precise definition in mathematical language.

Since the structures of mathematics may be considered to be elements of "Plato's heaven", we may choose to consider that such precise definitions describe the essence of such an object - provided no other definitions can be shown to be equivalent, as this would violate the uniqueness constraint.

For example, "continuity of a function" can be described or defined in many different ways, so none of these ways would be a formulation of "the" essence of continuity. If you are a young mathematician, and you can think up a number of new ways to define continuity, AND you show that they are equivalent to the other ways, we can infer that you are abstracting from a non-verbal appreciation of structure, and that this "object" (silent level) experience is a construct from experience. If we take the view that the essence of continuity exists in Plato's heaven, and that, through your experience, you have been able to respond to that world, we could say that your object is an abstraction from the essence in Plato's heaven of what we label "continuity".

Philosophers of mathematics are divided on the question; some say that we "discover" such structures, and the above reading would be consistent with them. Others, on the other hand, say that we "create" such structures in the act of defining them, and we discover relations among our creations using verbal relations. In such cases, the "object" the young mathematician experiences, and from which he or she creates new definitions may be a composition of his or her experiences, but it cannot be said to be the direct abstraction of characteristics from some single putative event level structure (hypothesized to exist independently of any observer).

From my perspective general semantics would say that an "essence" is simply a single characteristic abstracted to a "highest" level. If you object to "highest", recall that abstraction principally involves leaving out characteristics (the unconnected strings). If you have abstracted a finite number of characteristics, and you continue to successively abstract fewer and fewer, you eventually get to a single characteristic, and then you can go no further, unless you return to some lower level first. Since there are only a finite number of words, every verbal abstraction sequence not involving substituting (dropping back) eventually comes to a highest level.