# Such a box *cannot* exist To begin with a bit of a story. At the end of the 19<sup>th</sup> century the mathematician Frege was developing a theory which could be used as a basis of all of mathematics. This theory was based on the idea of a set. A set is a simple object that either contains or does not contain every object. In Frege's construction of set theory any proposition about objects such as *"It is green"* or *"It is an even number"* would have a set that contained exactly all the things that satisfied it. So you could have the *set of all green things* or the *set of all even numbers*. This is a very straight forward system. However there is an issue. The philosopher Bertrand Russel put forth a proposition > It is a set that does not contain itself. Since this is indeed a proposition, in Frege's theory there must be a set that contains exactly all sets that do not contain themselves. The problem then is > Does that set contain itself? If the set does contain itself then it fails the proposition so it can't be in the set. But if it doesn't then it passes the proposition and must be in the set. It's a paradox, there can be no correct answer to this question (technically *both* answers are correct, but that's another thing). This spelled doom for Frege's theory. As it turns out a theory of sets must either allow paradoxes like above or there must be valid propositions which do not have sets. Later set theories take the latter route. --- Now returning to the question at hand. Let's say we have a box which produces the correct answer to every question that has a correct answer. We can produce a question modeled after Russel's proposition: > Does the box produce a negative answer to this question? This question has a correct answer. Either the box does or doesn't produce a negative answer. However the box cannot produce the correct answer. If it answers "No", then it is wrong because "No" is a negative answer. If it produces "Yes", then it is wrong because "Yes" is not a negative answer. The only way for the box to be correct is for it to provide no answer at all. Any box must reject some sorts of answerable questions, if it wishes to always produce correct answers. If it is fine with producing wrong answers then it can accept every question, no problem, but such a box is probably not very useful. More generally *any* box that produces answers to our questions, must either sometimes produce the wrong answer, or sometimes produce no answer at all. And this leaves space for philosophers. Philosophers can of course answer the question quite easily. If the box is consistent, meaning it always produces correct answers, then the answer to this question is "No, it can't." easily provided by philosophers. There are other questions too that the box can't answer but we can. That doesn't mean that humans are particularly special or anything. There are questions philosophy can never answer as well and the box might be able to cover those, but the fact that there are still questions the box can't resolve for us. --- # The box in real life I think very interesting to this question is the fact that at one point in history there almost was "the box" ... sort of. In the early 20<sup>th</sup> century there was some legitimate concern that mathematicians may find a universal algorithm which could solve any mathematical question. If found then it could be executed by a person with a little training, or later a computer, to solve any mathematical problem. Mathematicians would be obsolete! This would be "the box" for mathematics. However using the same structure above (albeit complicated enough I won't get into the details) it was shown that any algorithm must either sometimes produce the wrong answer or fail to produce an answer at all. (This is called the [Entscheidungsproblem](https://en.wikipedia.org/wiki/Entscheidungsproblem).) And later still this same structure was used to show even deeper truths. Gödel demonstrated, using the same sort of self reference as Russel, that any system of math that can express basic arithmetic either allows you to prove some false statements to be true or is unable to prove some true statements ([Gödel's Incompleteness Theorems](https://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems)). This result is considered by many to be one of the crowning achievements of modern mathematics. There are of course partial solutions. Boxes that can solve many mathematical problems, but mathematicians are still around and busier than ever. So I think the future for philosophers bodes well. --- # Isn't this some sort of contradiction? So one criticism I have received in the comments is that the question, while it may appear to be easily answerable contains a subtle contradiction, and actually can't be answered. And well this is exactly true. The question as provided is contradictory. And so this is going to get a little technical, and I apologize, but feel free to ask for clarifications. Let's start by showing why this question is contradictory. > Does the box produce a negative answer to this question? Now here "the box" is a box that produces the correct answer to every question with a correct answer, so we can rephrase this as: > Does a box that produces the correct answer to every question with a correct answer produce a negative answer to this question? That's a mouthful so we can simplify this: > Is the correct answer to this question a negative answer? And this is clearly contradictory. There is no correct answer. We might as well be asking *"What is the last digit of pi?"* or other nonsense. So what is the point then. The point is that this serves as a "proof by contradiction". We are assuming as part of our framing that such a box does exist. If the box does exist then the question, does have an answer since we can observe the box and get the answer. However as we have just shown this question is unanswerable. This is the contradiction and so our framing must be incorrect. Such a box is impossible. And this is the exact same in the case of the other problems I presented. Russel's proposition is a nonsense proposition when "set" is taken to mean "Frege set", (and the other ones two but I haven't discussed them in detail). The point is to take a premise which may seem fine and to tease out a contradiction.