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January 20

Rotating a point by a distance

What is the term used to mean "rotate a point around a circle by a specific distance"? For example, I have a circle with a radius of 50 inches. I want to rotate a point on the edge of the circle clockwise 2 inches. Everything I find in textbooks and online refers to rotating by a number of degrees or radians, not by a distance. I have found references to measuring the distance rotated after rotating by a specific number of degrees, which is not exactly what I am looking for. 97.82.165.112 (talk) 15:44, 20 January 2023 (UTC)

So, the reason why radians exist is that radians is a distance, or almost so. For a unit circle of radius 1, the length of an arc of X radians is... X. Radius * angle in radians = length of arc. So if you have a 2 inch arc, on a circle of radius 50, that's an angle of 2/50 = 0.04 radians, or about 2.29 degrees. --Jayron32 16:57, 20 January 2023 (UTC)

If the issue is not one of formulas but of terminology, you might say that the point is shifted along an arc of the given circle having a specific length. (The term "distance" may be interpreted in the context of planar geometry as meaning Euclidean distance.)  --Lambiam 17:16, 20 January 2023 (UTC)

Thanks. Searching for rotation by arc length brings up references to converting arc length to radians (as Jayron suggested) and then rotating by the angle. 97.82.165.112 (talk) 18:00, 20 January 2023 (UTC)

January 21

Relevance of colours in magic T-hexagon diagram

Are the colours relevant?

I wish to vectorise this diagram. Would anyone know if the colours are mathematically relevant (or purely aesthetic)? Thanks, cmɢʟee⎆τたうaʟκかっぱ 18:33, 21 January 2023 (UTC)

They were helpful in explaining restrictions on the search space that made its discovery possible, as described on this archived page: "The Magic T-Hexagon of Side 2". This specific one was the first found. I do not see a relevance extending beyond its use as an aid in explaining the search process on that archived page.  --Lambiam 21:35, 21 January 2023 (UTC)

Thanks a lot, Lambiam. cmɢʟee⎆τたうaʟκかっぱ 16:50, 22 January 2023 (UTC)

January 23

Unlucky sum of three primes

Is there a number that can be written as the sum of three odd primes in exactly 13 different ways?  --Lambiam 10:46, 23 January 2023 (UTC)

If you don't find an answer here, it sounds like your question may make a good video topic for either a Numberphile, Mathologer, or Matt Parker video. Maybe you could write to them and see if they put it in the queue. --Jayron32 13:11, 23 January 2023 (UTC)

Rule of inference vs. logical consequence

I think I understand the distinction between material conditional and logical consequence: A logically implies B iff A → B is a tautology, right? But I must be confused, because this seems the same as the relationship between material conditional and rule of inference. (The article on rule of inference even seems to equate rules of inference with entailment in this section, and our article on logical consequence seems to take entailment as a synonym for logical consequence.) Yet Hilbert systems for propositional logic make a clear distinction between an axiom, which seems in this context to be a statement that some propositional formula is a tautology, and an inference rule. What am I missing? -Amcbride (talk) 16:03, 23 January 2023 (UTC) (EDIT: Just realized that middle section of rule of inference says it's using the sequent notation, which I'm used to seeing as entailment, specifically to emphasize the distinction between axioms and rules of inference... but in my confusion, this notation just seems to blur the distinction further.)

Inference rules and tautologies are very closely related. Let's consider modus ponens: $P,P\to Q\vdash Q$. There's a corresponding tautology: $(P\wedge (P\to Q))\to Q$. In general, any rule of inference gives rise to a tautology by writing that the conjunction of the hypotheses imply the conclusion -- this is called soundness of the rules of inference. Conversely, for tautologies of the form $A\to B$, you can show $A\vdash B$ by a sequence of rules of inference -- this is completeness of the rules of inference.

An important thing to understand is that tautologies are statements in the logical language, but rules of inference are not. Rules of inference are methods of combining statements to generate new statements.

Axioms are statements, but are generally not tautologies. They are additional statements which we are asserting to be true.--2600:4040:7B33:6E00:5D8:EC09:6C4:6CA0 (talk) 17:14, 23 January 2023 (UTC)

Thanks. I think part of my confusion is that, although a tautology itself is a statement in an object language, the statement that a given tautology is indeed a tautology is a metalanguage statement. Right? It's that latter animal, the assertion that a given statement is a tautology, that I'm having trouble distinguishing from a rule of inference. (And you say axioms are not generally tautologies, but for propositional logic, they are, right?) -Amcbride (talk) 17:53, 23 January 2023 (UTC)

(edit conflict) I may not be the best person to answer this, since I, like you, have been confused by the way this is defined and discussed in the literature. Here is my try:

Given a logic (a formal system), one should distinguish between the theory $\Theta$ of the logic, formed by the sentences that can be proved as theorems using its rules of inference, and its "necessity" $\Delta$, being the set of sentences that are true in all models of the logic. Normally, one should only allow ground terms here or variables that are bound by a quantifier; otherwise the notion of a sentence being true is unclear. If the logic is sound, $\Theta \subseteq \Delta ,$ that is, all provable sentences are true in all models. If the logic is complete, the converse inclusion $\Delta \subseteq \Theta$ holds. The easiest way (IMO) to think of the material conditional (aka "material implication") is as just a formula in the formal language of the logic, assuming it has an implication connective such as $\to$, in which case it is a sentence $\sigma$ of the form $P\to Q.$ Depending on the logic and on the antecedent and consequent of $\sigma$, it may or may not be an element of $\Theta ,$ and it may or may not be an element of $\Delta .$ All four cases are possible. But if it is a member of $\Delta ,$ it gets awarded a special status, variously known as "strict implication", or "entailment", or "logical consequence". The latter term is confusing; this notion of "consequence" is in terms of the models of the logic, and is not related to its deductive system with its rules of inference. It is a semantic notion; using double turnstile notation, we can express it as $\vDash P\to Q.$  --Lambiam 17:18, 23 January 2023 (UTC)

Thank you. This gives me a lot to read. I think part of my problem may be that I don't know enough about model theory or proof theory to disentangle ideas that belong with one or the other. -Amcbride (talk) 17:59, 23 January 2023 (UTC)