Does the Algebraic Manipulation of an Equation Generate an Ontologically Different Equation?

Question

I have recently been relearning algebra, so that I can have a better foundation for studying set theory and other more advanced topics. My true interest is in philosophy of mathematics. I have had many questions about mathematics over the years that have never been answered. In particular, here's one that has been bothering me recently:

Does the algebraic manipulation of an equation generate a new equation, or is it merely the same equation with a different visible form? For algebra to "work," it would seem to have to be the same equation, just with a different form. Yet there are examples of valid algebraic manipulation which do indeed change the actual equation completely. If I multiply both sides of a linear equation by x, then I generate a quadratic equation. Also, in solving a rational equation, we eliminate the denominators from the equation by multiplying by a common denominator. Are we still dealing with the same rational equation after doing this procedure? If I were to take a snapshot of the equation after eliminating the denominator, and present this picture to somebody, and ask them if this were a rational equation, they would say of course not.

I guess here I am employing a distinction between the form a thing takes, and what that actual thing is, ontologically. Two equations can have different forms, but can be identical ontologically. Liquid water and ice are both H2O, for example, but have different visible forms.

Thanks, Kyle

Answer 1

This is just a particular instance of the more general distinction between names and referents: e.g. "Kurt Godel" and "The person who first proved the the consistency of CH" are different names with the same referent. So the short answer to your question is:

As strings of symbols, "$x+1=2$" and "$x=1$" are clearly different. However, they have the same referent. And the relation between name and referent here is no more - or less - complicated than it is in natural language.

EDIT: My interpretation of equations as names as opposed to statements is not unobjectionable - see the comments below - but I do think it's ultimately correct here (a sentence is really just a name for its meaning, after all :P).

That said, it's worth saying a bit about what makes this stand out from general natural language issues (and why this question may be appropriate here as opposed to only philosophy.stackexchange):

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There are a couple of important points to make here, one common to mathematics and natural language and the other fairly specific to mathematics. I'll start with the commonality.

• Just like with natural language, it's often quite hard to pin down what sort of thing the referent of a mathematical expression actually is.

Take "$x^4=12$." The most obvious way to interpret this string of symbols as referring to some object is via solution sets: "$x^4=12$" is a name for the set of all numbers whose fourth power is twelve. However, there's a serious issue here: what's the context? $x^4=12$ defines one solution set over $\mathbb{R}$ and another one over $\mathbb{C}$ (and yet another one over $\mathbb{Q}$, and so on). Moreover, it's often mathematically useful to think of "$x^4=12$" as deliberately ambiguous (we sometimes want to consider the "same" polynomial in different fields).

Now this isn't unique to mathematics. For example, "the president" is similarly domain-vague (president of what?). But where we see some arguably unique behavior is when we look at the interaction between this kind of vagueness and the second important point:

• Unlike with natural language in general, we have explicit rules for manipulating mathematical names which are given axiomatically as opposed to inferred through experience.

All of language is of course subject to a general "usage/meaning feedback loop," but in mathematics this axiomatic structure leads to a level of clarity (initiallly, at least) which as far as I can tell is unique. Somehow, in mathematics - when we really "look under the hood" - we manage to coherently precisely manipulate vagueries. This is really quite neat!

Answer 2

Suppose you have a quadratic equation with a negative power

$$n^2-4+\frac{6}{n}=0$$ and you want to solve it with the quadratic equation $$n=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$

as it turns out, this is really a cubic equation with a different face $$n^3-4n+6=0$$

which can be solved.

This is just an example but I think it makes the point that manipulation does not change the basic equation, just the view.