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If I think about a variable's value as the result of a measurement or counting, it makes sense to me to think that such a measurement wouldn't magically square itself. That measurement can only be squared if it's put into an appropriate context such as an equation.

That is the train of thought that lead me to wonder about the question.

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    "variable" in that sense usually means some magnitude that is function of... time, etc. In the Wiki's entry linked above, "measurement or counting" is used to express intuitively the difference between continuous and discrete magnitudes. What does it mean that "a measurement wouldn't magically square itself"? Commented Aug 23, 2023 at 13:19
  • Having said that, an equation is a linguistic expression suitable to describe facts and relations between objects. Commented Aug 23, 2023 at 16:24
  • Thanks for the comment. I figure you're saying "continuous" and "discrete" is more formal and maybe more accurate than "measured" or "counted". I also recognize that I was indeed referring to variables as magnitudes and not as matrices, functions… Finally, I was thinking about the area of a square; the base requires the height for the squaring to happen. The mere measurement of the base cannot yield the area without putting it in the context of a multiplication or squaring. Of course, this is all intuitive. I don't know if this makes sense formally.
    – Gabriel
    Commented Aug 24, 2023 at 4:53

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At its core, an emergent property is a phenomenon that arises from simpler interactions but is not predictable or deducible from the sum of its parts alone. In the context of equations, one could argue that the relationship described by the equation is the emergent property, arising from the interaction of its variables.

Take, for instance, the equation for kinetic energy:

KE = ½mv2

The kinetic energy of an object isn't something that is inherent in its mass or its velocity alone. Instead, it emerges when you consider both the mass and the velocity squared in the context of this equation. The equation provides a framework for understanding how these variables interact to produce a new concept: kinetic energy.

However, it's also worth noting that equations are human constructs. They are tools we've developed to describe and predict the behavior of the universe around us. In this sense, the "emergence" is not so much a property of the universe, but rather a product of our understanding and description of it.

While it's not traditional to describe equations as emergent properties, your perspective offers a fresh lens through which we can appreciate the intricate dance of variables and the relationships they form. It's a reminder that mathematics, at its heart, is a language we've crafted to describe the complex symphony of interactions in the universe.

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The notion of an emergent property is physical, so an equation, not being physical, cannot be an emergent property, and I presume what you are really asking is whether there is some aspect of phrases or abstract objects analogous to emergent properties, and whether equations are analogous to emergent properties in this way.

If you are viewing equations as phrases, then I suppose that there is something akin to emergence in language since phrases often have meanings that cannot be inferred from their parts. For example you see someone with a black eye and ask "what did the other guy look like?" There is nothing in the words that suggests a fist fight, but that is implied by informal convention and the context. However, equations are not emergent in this sense. In fact formal languages are specifically designed to prevent this from happening--the meaning of an expression in a formal language is exactly what it is defined to be by formal conventions.

If you are viewing an equation as an abstract object--an abstract proposition, for example--then I can't think of any analog to emergence that would apply. Assuming that the variables are even components of the abstract object (which is itself controversial), they are related by necessary relationships.

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