We have all seen on the Internet questions like, "If $1=5$, $2=6$,$3=7$, and $4=8$, then $5=?$". I don't really understand what those questions are supposed to mean. I suppose one way to analyze them would be to state that the question formally means: "Find the solution set of the open sentence $((1=5 \land 2=6 \land 3=7 \land 4=8) \rightarrow 5=x)$" But, this would be a bad formalization, because, since the antecedent is false, any number $x$ would be a solution to the open sentence. So, what is the best way to analyze and formalize those questions where you are asked what a certain number equals given some other numerical equalities? Or is there no such thing, and those questions are not actually mathematical, and don't have a definite answer which you can unambiguously justify?
A formal analysis of those questions where you are asked what a certain number equals given other equations.
3$\begingroup$ I have never seen Internet questions like that. $\endgroup$– JonathanZ supports MonicaCOct 16, 2022 at 2:09
$\begingroup$ They are not well defined. Roughly speaking, one can ask for the "simplest" function $f$ such that $f(1)=5,f(2)=6,f(3)=7,f(4)=8$ , interpret the equality above to "mean" this function assignment, and then say that $f(x)=x+4$ is "the simplest" such function , so $f(5)=9$. However, unless "simplest" is not defined, you're in trouble. Let's say you were in an exam, however, and you know only so many functions (or only so many families of functions). Then, you may assign to each function a "complexity", and the question of "simplest function" becomes well-defined. Unfortunately, that still... $\endgroup$– Sarvesh Ravichandran IyerOct 20, 2022 at 3:36
$\begingroup$ ... results in some degree of arbitrariness, but in a local conversation, if people are willing to agree upon a "complexity" function, such a question is well-defined. Typically, however, these are just "thrown out" to the public, and of course there is no universal agreement on simplicity. Therefore, while it is possible to make this rigorous, people would not do it for a video, 100%! (Note : I don't know the set-theoretic consequences of such assignments, but I'll assume that they can be safely executed). $\endgroup$– Sarvesh Ravichandran IyerOct 20, 2022 at 3:38
I think you are asking two separate questions, both of which can't really be answered.
The first is to formalize the statement "$1 = 5$". I think what the writer is trying to say here is that the first number in a sequence is $5$, then that the second is $6$, and so on.
The second question asks for a general method to solve a problem
where you are asked what a certain number equals given some other numerical equalities
There is no such method. Try searching this site for "predict next". Here's one hit: Predict next number from a series