I come from a background in philosophy and have taken a few courses on logic, mostly with an eye for applications in philosophy. I am quite fine in the realm of mathematical logic, but lack in terms of more basic areas of math such as elementary calculus, probability, and statistics (including others).
I have no problem doing calculations, solving equations, computing probabilities etc. But, as part of my never-ending philosophical inkling, as well as background in formal logic, I am finding it hard to pin down the exact pattern of reasoning that takes place when one is 'solving an equation', which causes me some conceptual problems, or at least irritation.
I know the definition of solving an equation is "to find all the values of the variables satisfying the equation", but I was wondering whether there is a way to spell out the pattern of reasoning involved in solving an equation more formally – and also exactly how to spell out the "unknown" component of a variable, because I feel that this expression is ambiguous between that of a free variable (without any assignment of values), and that of some variable to which a value has been assigned (either implicitly or explicitly) but of which we are (currently) ignorant of its particular value.
For example, if asked to solve a basic equation such as the following, "$4x = 16$", I am thinking of the pattern of reasoning along the lines of (very informal sketch):
(1) We start by assuming that there is an $x$ such that $4x = 16$ (we existentially quantify the variables in the open sentence '$4x = 16$').
(2) Then we perform existential instantiation and use the symbol '$x$' to denote whatever satisfies the equation '$4x = 16$' (I would take this to be what mathematicians usually mean by an 'unknown', i.e., a "dummy name" / a variable having a value assigned to it but in which we are ignorant of its particular value – or am I completely wrong?)
(3) By performing a few arithmetical calculations (which are valid due to the laws of arithmetic – we perform a certain number of universal instantiation of these laws, which works because we are treating '$x$' as denoting a particular quantity now, and not just as a free variable without a value) we find that $x = 4$ (we have found the value of our unknown).
(4) Thus, if there is a value for '$x$' such that '$4x = 16$' is true, then that value is equal to $4$. I.e, under the assumption that some value, '$x$', satisfies the equation '$4x = 16$', we find that it must be the case that $x = 4$.
(5) Thus, the solution set to the equation '$4x = 16$' is $\{4\}$.
I think what might confuse me when I see mathematical writing is the potential shift in treating '$x$' as a free variable to which no value has been assigned, and then suddenly treating it as denoting an 'unknown' quantity which (supposedly) makes some equation true in the above deductions.
I would like to hear whether I'm completely off, whether or not this is a weird way to understand the process of solving an equation, or whether you find my whole idea pointless. If there are other ways to think about it, I would like to hear those as well!
Side note: I am also a bit confused when an equality ('$=$') is defined as a "statement claiming that two expressions are equal" (which I have come across many, many times while trying to better my mathematical abilities recently) – this certainly does not seem true if the equality contains free variables, in which it does not express a statement (before we have assigned values to the variables, that is)? Am I completely wrong here? Wouldn't a definition such as 'an equality is the result of writing the equals sign ('$=$') between two expressions', or something of the like, be more appropriate?
Thanks in advance!
Best Answer
To solve an equation is to provide a characterization of its solutions.
When someone asks you to "solve" the equation $4x=16$ (over the set of real numbers), they are asking for a characterization of "the set of real numbers $x$ such that $4x=16$." A "characterization" of a set of just a description of that set, and it is generally understood that a characterization should be simple or useful.
Of course, we can easily give a trivial characterization of the set described above—for example, we can simply say: "The set of real numbers $x$ such that $4x=16$ is the set of real numbers $x$ such that $4x=16$." However, this characterization is not simple or useful. A much more desirable characterization is the following: "The set of real numbers $x$ such that $4x=16$ consists precisely of the number $4$", or, symbolically: $$\{x \in \mathbb{R} : 4x = 16 \} = \{4 \}.$$
Characterizing consists of proving necessity and sufficiency.
Mathematicians characterize all sorts of sets, not just the solution sets of equations. In general, if you wans to characterize the set $S$ as the set of things satisfying the property $P$, you must prove two things:
In the example of characterizing the solution set of $4x = 16$ as the set containing $4$ and nothing but $4$, this boils down to proving the following two things:
The 5 steps that you outline in your question cover the "necessity" step, but they do not complete the "sufficient" step. To finish the proof of this characterization, you need to do one final step: proving that being equal to $4$ is sufficient for satisfying $4x = 16$. This step is pretty easy—you simply evaluate $4 \cdot 4$ and notice that it is equal to $16$.
I will also add that it does not matter which order you perform the proofs of necessity and sufficiency—you are free to check that $4$ solves the equation before you prove that $4$ is the only solution.
Additionally, I point out that these two steps will sometimes appear in various disguises. For example, you might prove the "uniqueness" and the "existence" of some mathematical object. Or, if you are trying to prove that set $X$ is equal to set $Y$, you might go through the two steps of proving that $X \subseteq Y$ and that $X \supseteq Y$. These examples are manifestations of the fundamental notions of necessity and sufficiency.
Proving necessity and sufficiency simultaneously
As Rob Arthan suggested in his comment, a very common way for solving simple equations like $4x = 16$ involves proving necessity and sufficiency at the same time, instead of doing them one at a time. If you are careful about how you carry out the arithmetic operations in step (3) of your proof (in particular, if you make sure to check that they are all reversible), you can prove necessity and sufficiency at the same time. However, if you are proving characterizations of other kinds of sets, you may be forced to split up the two steps.