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What does a variable refer to in a formula? If it is a free variable, it has no reference, yet it exists as an element of the formula.

In an unassigned formula, what is the semantic meaning of a variable occurrence? Or simply, do we need to perform 'assignment' to give the formula a semantic meaning? The 'formula' is simply a string of symbols that does not take a semantic meaning until after assignment?

Under a given interpretation, do we simply take a formula and treat it as a 'template' for a meaningful assertion?

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    Does this answer your question? What are free variables and what does it mean for a statement to contain one?
    – user14511
    Dec 6, 2022 at 16:19
  • Formulae express concepts (and not conversely, what you seem to suggest: you don't give meaning to formulae, you grant formulae of meaning). The meaning you ask for is the same as a language statement without a predicate, and with placeholders: the price of apples and the weight of oranges would be Ax+By. If the predicate is are the same, then the formula would be: Ax-By=0. 2x-3/n has no meaning except a pure mathematical one.
    – RodolfoAP
    Dec 6, 2022 at 18:09

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A formula is a machine: you put something raw in at one end, crank the handle, and get something useful out the other end. A free variable is a hopper where you put raw numbers so they can be processed.

This is a linguistic confusion, mainly. I mean, when you drink a cup of tea, you don't actually drink the cup, right? You drink the tea, and the cup is just there so the tea goes where it's supposed to. A variable without a reference is like a teacup in a cupboard; it just holds a space until the next time you need to use it.

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Its the variables that are unassigned or free, not the formula.

Formula come later. You begin with variables. If all your variables are assigned then your formula cannot be free, if you insist on labeling your formula unassigned or free. The thing is, the terms don't apply to formulae.

What do it means for a variable to be free? It means you haven't decided which natural data it represents. Yes, its as ridiculous as it sounds. Why will you proceed making a formula if you are not even clear about the terms of formula? and How? You shouldn't and you can't.

Do it happen that people make formula with unassigned variables? Yes, a lot. Examples:

  1. Dark Energy. The data for how much force of gravity exist in universe do not match with Theory of Relativity. The data suggests more gravity. What scientists then did is put a free-floating variable, labeled it Dark Energy and still have an equation.

The right way to approach is to not propose any equation till you have your act together. What is that Dark Energy? Is it ever observed? Can it be measured independently? The answer is its an imaginary thing nobody ever observed. It means anything the scientist want it to mean, depending on situation.

  1. Delta as in Delta-X. The Basis of Derivates in Mathematics.

Nobody ever decided what delta is. Its said to be infinitesimally small. Well, if its infinitely small then its zero, isn't it? Its said to be smaller than any number - including fractions - but greater than zero. What?

Its not that I can arbitrary take a value for its length. Say, I decide to measure a thing at intervals of 1000th of its length. No, as per theory of derivates as soon as I decide on any number - 1000th in this case - I am wrong. I cannot put any number on it.

I can think of many more from the top of my head, such as Virtual Particles, Energy Borrowed From Field (once a particle is in an energy pit) etc. But it would become a physics answer then.

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