I'm putting this in the Free Thoughts section because it is, after all, rather trivial. I'm just curious about something. It has been apparent to me for a long time that many people don't really understand some of the very basic concepts which support our understanding of the universe. For that reason, I'm going to ask a few very basic questions here and see what happens. 1. What is a dimension? 2. What is a unit of measurement? 3. What is a model (in terms of science and particularly physics)? 4. What is a hypothesis? 5. What is a theory? 6. What is a law? Don't just post a link or paste in something someone else said that you believe covers it. In your own words, answer those questions.
Good idea. As a mathematician, I will answer questions 1 and 5 from a mathematical point of view. 1. The dimension (of a vector space) is the number of vectors in it's basis Please Register or Log in to view the hidden image! 5. In a logical language \( \mathbb{L}_0 \) given the following axioms and rule: A1 : \( (\alpha \rightarrow (\beta \rightarrow \alpha)) \) A2 : \( (((\alpha \rightarrow (\beta \rightarrow \gamma)) \rightarrow ((\alpha \rightarrow \beta) \rightarrow(\alpha \rightarrow \gamma))) \) A3 : \( ((\neg \beta \rightarrow \neg \alpha) \rightarrow (\alpha \rightarrow \beta)) \) \( \alpha , \beta , \gamma \in Form(\mathbb{L}_0) \) Rule: Modus Ponens: From \( \alpha \) and \( (\alpha \rightarrow \beta) \) infer \( \beta \) . \( \forall \alpha , \beta \in Form(\mathbb{L}_0) \), for any \( \Gamma \subset Form(\mathbb{L}_0) \), we say \( \alpha \) is provable from \( \Gamma \) if there is a finite sequence \( \alpha _1 , \alpha _2 , \ldots , \alpha _n \in Form(\mathbb{L}_0) \) such that for each \( 1 \leq i \leq n \) either 1: \( \alpha _i \) is an axiom 2: \( \alpha _i \in \Gamma \) , or 3: \( \exists j<k<i \) such that \( \alpha _j = (\alpha _k \rightarrow \alpha _i) \) OR \( \alpha _k = (\alpha _j \rightarrow \alpha _i) \) AND \( \alpha _n = \alpha \) We then write \( \Gamma \vdash \alpha \) FINALLY... we define a Theorem to be any \( \alpha \in Form( \mathbb{L}_0) \) such that, \( \Gamma \vdash \alpha \) where \( \Gamma = \emptyset \) Sorry if I overformalised, but hey, I am a mathematician.
A unit of measurement depends on the thing being measured. Units like Amperes or Coulombs are arbitrary units "normalized" to a region of space and time where they act over arbitrary units of space and time. In the case of earth's surface the units of distance are a fixed portion of a geodesic distance - which has recently been modified to fit other physical units more exactly. Units of time are completely arbitrary, though. A model is a "naive algorithm" with which to develop a better algorithm. An algorithm is a hypothesis which is proved by confirming that it halts, so a theory is a program that is known to halt, or yield an answer that fits the model. So it all comes down to having units for the "computer model" to compute with, and so calculate a reasonable answer... For instance the hypothesis that the distance from the equator to the North Pole remains constant can be verified by measuring it at different times, using a fixed length of arbitrary units, called metres.
