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Fri, 24 Nov 2023
Math SE report 2023-09: Sense and reference, Wason tasks, what is a sequence?
Proving there is only one proof?OP asks:
This was actually from back in July, when there was a fairly substantive answer. But it left out what I thought was a simpler, non-substantive answer: For a given theorem !!T!! it's actually quite simple to prove that there is (or isn't) only one proof of !!T!!: just generate all possible proofs in order by length until you find the shortest proofs of !!T!!, and then stop before you generate anything longer than those. There are difficult and subtle issues in provability theory, but this isn't one of them. I say “non-substantive” because it doesn't address any of the possibly interesting questions of why a theorem would have only one proof, or multiple proofs, or what those proofs would look like, or anything like that. It just answers the question given: is it possible to prove that there is only one shortest proof. So depending on what OP was looking for, it might be very unsatisfying. Or it might be hugely enlightening, to discover that this seemingly complicated question actually has a simple answer, just because proofs can be systematically enumerated. This comes in handy in more interesting contexts. Gödel showed that arithmetic contains a theorem whose shortest proof is at least one million steps long! He did it by constructing an arithmetic formula !!G!! which can be interpreted as saying:
If !!G!! is false, it can be proved (in less than one million steps) and our system is inconsistent. So assuming that our axioms are consistent, then !!G!! is true and either:
Which is it? It can't be (1) because there is a proof of !!G!!: simply generate every single proof of one million steps or fewer, and check at the last line of each one to make sure that it is not !!G!!. So it must be (2). What counts as a sequence, and how would we know that it isn't deceiving?This is a philosophical question: What is a sequence, really? And:
And several other related questions that are actually rather subtle: Is a sequence defined by its elements, or by some external rule? If the former how can you know when a sequence is linear, when you can only hope to examine a finite prefix? I this is a great question because I think a sequence, properly construed, is both a rule and its elements. The definition says that a sequence of elements of !!S!! is simply a function !!f:\Bbb N\to S!!. This definition is a sort of spherical cow: it's a nice, simple model that captures many of the mathematical essentials of the thing being modeled. It works well for many purposes, but you get into trouble if you forget that it's just a model. It captures the denotation, but not the sense. I wouldn't yak so much about this if it wasn't so often forgotten. But the sense is the interesting part. If you forget about it, you lose the ability to ask questions like
If all you have is the denotation, there's only one way to answer this question:
and there is nothing further to say about it. The question is pointless and the answer is useless. Sometimes the meaning is hidden a little deeper. Not this time. If we push down into the denotation, hoping for meaning, we find nothing but more emptiness:
We could keep going down this road, but it goes nowhere and having gotten to the end we would have seen nothing worth seeing. But we do ask and answer this kind of question all the time. For example:
Are sequences !!S_1!! and !!S_2!! the same sequence? Yes, yes, of course they are, don't focus on the answer. Focus on the question! What is this question actually asking? The real essence of the question is not about the denotation, about just the elements. Rather: we're given descriptions of two possible computations, and the question is asking if these two computations will arrive at the same results in each case. That's the real question. Well, I started this blog article back in October and it's still not ready because I got stuck writing about this question. I think the answer I gave on SE is pretty good, OP asked what is essentially a philosophical question and the backbone of my answer is on the level of philosophy rather than mathematics. [ Addendum: On review, I am pleasantly surprised that this section of the blog post turned out both coherent and relevant. I really expected it to be neither. A Thanksgiving miracle! ] Can inequalities be added the way that equations can be added?OP says:
I have a theory that if someone is having trouble with the intuitive meaning of some mathematical property, it's a good idea to turn it into a question about fair allocation of resources, or who has more of some commodity, because human brains are good at monkey tasks like seeing who got cheated when the bananas were shared out. About ten years ago someone asked for an intuitive explanation of why you could add !!\frac a2!! to both sides of !!\frac a2 < \frac b2!! to get !!\frac a2+\frac a2 < \frac a2 + \frac b2!!. I said:
I tried something similar this time around:
Then the baker also trades your bag !!b!! for a bigger bag !!d!!.
Someday I'll write up a whole blog article about this idea, that puzzles in arithmetic sometimes become intuitively obvious when you turn them into questions about money or commodities, and that puzzles in logic sometimes become intuitively obvious when you turn them into questions about contract and rule compliance. I don't remember why I decided to replace the djinn with a baker this time around. The cookies stayed the same though. I like cookies. Here's another cookie example, this time to explain why !!1\div 0.5 = 2!!. What is the difference between "for all" and "there exists" in set builder notation?This is the same sort of thing again. OP was was asking about $$B = \{n \in \mathbb{N} : \forall x \in \mathbb{N} \text{ and } n=2^x\}$$ but attempting to understand this is trying to swallow two pills at once. One pill is the logic part (what role is the !!\forall!! playing) and the other pill is the arithmetic part having to do with powers of !!2!!. If you're trying to understand the logic part and you don't have an instantaneous understanding of powers of !!2!!, it can be helpful to simplify matters by replacing the arithmetic with something you understand intuitively. In place of the relation !!a = 2^b!! I like to use the relation “!!a!! is the mother of !!b!!”, which everyone already knows. Are infinities included in the closure of the real set !!\overline{\mathbb{R}}!!This is a good question by the Chip Buchholtz criterion: The answer is much longer than the question was. OP wants to know if the closure of !!\Bbb R!! is just !!\Bbb R!! or if it's some larger set like !![-\infty, \infty]!!. They are running up against the idea that topological closure is not an absolute notion; it only makes sense in the context of an enclosing space. I tried to draw an analogy between the closure and the complement of a set: Does the complement of the real numbers include the number !!i!!? Well, it depends on the context. OP preferred someone else's answer, and I did too, saying:
I try to make things very explicit, but the downside of that is that it makes my answers longer, and shorter is generally better than longer. Sometimes it works, and sometimes it doesn't. Vacuous falsehood - does it exist, and are there examples?I really liked this question because I learned something from it. It brought me up short: “Huh,” I said. “I never thought about that.” Three people downvoted the question, I have no idea why. I didn't know what a vacuous falsity would be either but I decided that since the negation of a vacuous truth would be false it was probably the first thing to look at. I pulled out my stock example of vacuous truth, which is:
This is true, because all rubies are red, but vacuously so because I don't own any rubies. Since this is a vacuous truth, negating it ought to give us a vacuous falsity, if there is such a thing:
This is indeed false. And not in the way one would expect! A more typical false claim of this type would be:
This is also false, in rather a different way. It's false, but not vacuously so, because to disprove it you have to get my belts out of the closet and examine them. Now though I'm not sure I gave the right explanation in my answer. I said:
But is this the right analogy? I could have gone the other way:
Ah well, this article has been drying out on the shelf for a month now, I'm making an editorial decision to publish it without thinking about it any more. [Other articles in category /math/se] permanent link |