The initial exercise was to find the solution(s) for an equation and and not to determine the truth of falsity of any statement.
In the most usual formulation of arithmetic, infinity is not a valid number, and so 5\times\infty-5\times\infty=5 is not a mathematical statement under that formulation. There are other formulations of arithmetic that do include infinite numbers, but in some of those, infinity isn’t one specific number, but a category of them, and you’d need to specify which infinity you mean, and then that statement would be meaningful but false. In yet other formulations, there is one and only one number called “infinity”, but in those formulations, the expression 5\times\infty-5\times\infty is indeterminate (that is, like \frac{0}{0}, it can equal anything).
This has been slightly overtaken by events, but I though it still worth posting.
It is worth noting that the fallacy of the excluded middle is not at variance with Aristotelian logic. It is just a particular misuse of that logic. If you assert
A \iff \neg B
and there is a circumstance where both might actually be true (or false) together you have made an error in logic. Fallacy of the excluded middle is all about a rhetorical argument that sneaks in the assumption without actually stating it. Often blandly asserting the options, and without drawing breath, steam-rolling on with whatever agenda is being argued. (Insert your favourite political polemic. “You are either with us or against us” is a good start.)
Probably going way too far down the rabbit hole now.
An answer to the broader question is simply that the structures we use are useful. This applies to Aristotelian logic, and the various mathematical systems we develop. There are an infinitude of useless systems we could posit. We basically ignore them.
Logic that allows us to do something useful and progress our knowledge and reasoning ability needs to be self consistent. How self consistent is an interesting side question, but the last couple of millennia of progress have been based on a pretty simple set of rules that demand a clear idea of consistency.
Same with mathematics. We can come up with all sorts of ways of defining systems. Most of the systems we might define are boring. As in they allow no development of ideas of structure and tend to just quietly collapse into nothingness.
The exciting ones tend to follow a pattern whereby properties that exist follow a set of rules for self consistency and application that builds interesting and sometimes powerful structures. Some map into real life in very useful ways. What we generally consider mathematics in the wider sense embodies this big time.
Thus, in order for one system to be actually useful we find that the whole thing sits atop a basis of self consistency. How we build our modern mathematical systems has evolved over the centuries. There has always been enthusiasm for at least some structure, usually down to basic axioms and logical structures. (Famously we know that there are unexpected and interesting limitations. But it doesn’t undermine the core basics.
Trying to build a logic that manages some sort of inconsistency is an interesting side gig for some logicians. But such work has little to no relevance for day to day mathematics.)
The conventional mechanics for building day to day mathematics is not interesting to most people, but it is there non-the-less. And mathematicians care about it a lot. The era of “New Math” saw bewildered young students introduced to sets and the concept of cardinality. Sadly, something to be soon forgotten, unless they go on to study tertiary level mathematics, and want to bore their friends. The whole New Math exercise was a bit weird, but a well intentioned attempt to justify numbers as something other than a rote learnt sequence of words.
For the OP, the why of the original question and its result is as before. If we unpick the steps a little more, we can see the interplay between the overarching logical structure we use, Aristolien logic, and the specific mathematical structure the question is posed in. Millennia of experience tells us that not adopting the LEM, and using well defined logic just ends up in a morass of useless nonsensical rhetoric. If you want powerful tools to attack difficult problems, you base them on solid foundations.
The OP’s question was explicitly posed in the real numbers. So, not integers, not complex numbers, and importantly not some modulo integer space, (within which as noted a few times above we can come up with a meaningful interpretation.)
We thus explicitly bring in the rules for arithmetic and algebra that come with the reals, as well as the axioms that define the reals, these add to our common core machinery of logical discourse.
The question poses an expression that is well formed in this space.
The question then poses a well understood question that is also mapped to the space in a well formed and understood manner. The question posed being essentially one of: apply valid algebraic manipulations to the first expression in such a manner that yields a value for x where original expression evaluates to zero.
There is an implied assumption that the initial expression is consistent with the space we are posing the question over, and thus the manipulations are valid.
So we apply some apparently legal operations to the first expression and rather than find the desired result, we slam into an expression that contradicts one of the axioms of the real numbers. That 0 = 5. This violates the assumption that the question was well posed in the first place.
Thus the answer is not that the expression has no solutions. The answer is that the expression is invalid in the defined space it was posed in. To reuse a famous line, it isn’t even wrong. The expression has no anything.
There is the interesting question as to whether applying the algebraic manipulations were valid in the first place. But they either were, or they were not. Either way, the expression still isn’t valid.
What perhaps muddies the water is that, at first sight, the expression looks a bit like a simple well understood expression that might commonly have a solution, or be valid without a solution. Hence some earlier comments about odd and even order (polynomial) functions. If it were say 5x - 4x = 5, it would trivially be an odd order polynomial, and guaranteed a solution. Or say 4x^2 = -4 which is even order, thus is not guaranteed a solution, and here doesn’t. (Or you might be trying to solve a set of linear equations. Then the resulting logic is much the same.)
Where this whole construction of logic is really important is in proving things. The external structure of logic around our chosen mathematical system provides the machinery by which we build the mathematics up. As noted above, almost all logicians and mathematicians are comfortable that a series of steps that leads to a contraction is a valid mechanism to prove the starting point is false, and use this as a tactic for proving the converse true. It is one of the most commonly used tools in the toolbox.
People have been trying to figure out what is infinity is for thousands of years. I don’t think we will come to a definite conclusion this morning. As Chronos points out, there are many different categories of infinities. Nonetheless, I wholeheartedly agree that you can’t have a good argument with someone unless you agree on some definitions first.
It’s worth noting that it’s impossible to prove that any nontrivial mathematical system is actually consistent. We think that standard mathematics is consistent, we really hope it is, we’ve used it for a very long time without ever finding any inconsistencies… but we can never be sure that it’s actually consistent.
Aspects of New Math were overambitious and didn’t work well, but most of it is now just the standard math that’s taught to everyone. When you hear people aged 60 or younger complaining about “New Math” and “Why can’t we just do it the way I learned in school?”, the “way they learned it in school” is New Math.
No, the equation in the OP is perfectly valid and always false, and that validly means that it has no solutions.
I thought arithmetic was proven consistent- with the problem being that consistency also implied that there are well-formed arithmetic statements that can’t be proved or disproved
Consistency is one of the necessary conditions for there to exist well-formed statements that can’t be proven nor disproven. That’s easy to see, because an inconsistent system can prove literally anything. But arithmetic has never been proven consistent, and in fact there’s a corollary to Gödel’s Theorem that it can’t be proven consistent (and hence, neither can any other system that’s complicated enough to support arithmetic, such as Euclidean geometry).
What can sometimes be proven is that two systems are equally consistent. For instance, spherical geometry, hyperbolic geometry, and Euclidean geometry are all equally consistent, and ZF set theory (without the Axiom of Choice) is equally consistent to ZFC set theory (with the Axiom of Choice). That is to say, if one of those systems is inconsistent, so is the other. In practice, since we usually start out trusting one of the systems, this is in effect an argument to trust the other, but it doesn’t actually solve the problem of trusting the first system.
There are also some mathematical systems that are proven to be both consistent and complete, but, as a result of Gödel’s Theorem, we know that that can only be true of systems too simple to encompass arithmetic. In other words, they fall into the very large category of boring systems that nobody much cares about.
Well, yeah — but if the statement we’re trying to determine the truth or falsity of just so happens to involve the solution(s) to an equation, it can be possible to do one by doing the other.
You cannot separate the two. Stating n is a solution is equivalent to saying 5n - 5n is a true statement.
It’s equivalent to saying that it’s a true statement for that particular value of the variable.
I mistyped. Should have read.
… 5n - 5n = 5 is a true statement.
Yes, I thought it was implied that n was the value of the variable that solves the equation.
Is proof by contradiction the only way to determine that 5n - 5n = 5 has no solution(s)?
Here’s a proof that at least appears not to use proof by contradiction: If x is any real number, then 5x - 5x = 0 \neq 5. Hence, there is no real number x for which 5x-5x = 5, that is, there are no real solutions of that equation.
Hidden in the words here is the passage from “\forall x, \neg P(x)” to “\neg\exists x, P(x),” that is, if for all x the statement P(x) is false, then there is no x for which P(x) is true. However, I have a sneaking suspicion that this is not constructively valid, that it requires the LEM. I’m not a constructivist, and constructive logic is subtle.
What does it even mean when someone says 0 = 5 has no solutions? Does 2 + 2 = 4 have a solution?
But that’s what you wrote isn’t it? Do you disagree that 5x-5x is equal to 0 (of course for the given limits you want to place for x)? Or are we going to have to define what equals means? Or what minus means? How far back do we need to go for you?
I never wrote that (“0 = 5 has no solutions”) until that last post when I questioned it. Correct me if I’m wrong. Previous to that Thudlow_Boink and perhaps others had said it. I wrote that 5x - 5x = 5 has no solutions. How can you have a solution to an equation that has no variables?
Will you state that 5x-5x is 0?
Yes, of course.
And the solution for the equation 5x - 5x = 0 is the set of all real numbers.