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by jongjong
1 day ago
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I think part of the confusion is that the base isn't always clear. A lot of times, when first exposed to logs, we are shown only base 10 or base e so it's just written as the word log or ln... This omission obscures the meaning and versatility of the function. Log and ln are essentially presented as separate constructs. Which is retarded. Personally I only remember being briefly exposed to log and ln in school. I don't recall doing any complex algebra with them. So when I started doing algebra with them at university, I had to kind of re-learn from scratch. I asked Claude about the weird way I was taught logs and it said this, which I found interesting: There's a broader pattern here that shows up all over maths: notation and procedures get taught as the primary object, with the meaning treated as optional enrichment that "advanced" students might appreciate later. For a lot of people it's backwards — the meaning is the cheap, load-bearing thing, and the procedures are what should be derived from it. Logs are maybe the most egregant case because the gap between "incomprehensible button" and "obvious once stated" is a single plain-English sentence. |
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Mathematics is an interesting intellectual sport but it should not be allowed to stand in the way of obtaining sensible information about physical processes.
The purpose of computing [i.e. using mathematics] is insight, not numbers.
Does anyone believe that the difference between the Lebesgue and Riemann integrals can have physical significance, and that whether say, an airplane would or would not fly could depend on this difference? If such were claimed, I should not care to fly in that plane.
What this implies is that Mathematics is merely a formalism i.e. set of symbols/operations/rules manipulated using logic. Think of it as just a language which is succinct, precise and standardized. Only when it explains something in the physical world does it have meaning else it is merely a game.
Quotes from Albert Einstein;
Imagination is more important than knowledge. Knowledge is limited. Imagination encircles the world.
Logic will get you from A to Z; imagination will get you everywhere.
Any fool can know. The point is to understand.
The mere formulation of a problem is far more essential than its solution, which may be merely a matter of mathematical or experimental skills. To raise new questions, new possibilities, to regard old problems from a new angle requires creative imagination and marks real advances in science.
If I had an hour to solve a problem, I'd spend 55 minutes thinking about the problem and five minutes thinking about solutions.
What this implies is that it is important to exercise imagination to understand different perspectives of a problem eg. historical/existing/new ones, and reformulate it properly before focusing on mere formalisms.
In the 1960s the "New Math" movement was instituted to teach "Mathematics the Modern Way" which though criticized, took root and has permeated mathematics education to this day to the detriment of the students and the field - https://en.wikipedia.org/wiki/New_Math Read the "Reception" and "Legacy" sections in particular.
So to understand mathematics, look to history when a particular concept first came into existence to explain/solve a particular problem and then look at the formalism standardized/employed today to generalize it. Now we can understand the motivation behind the current formalism better and its applications to the real world. This is called the "Genetic Method" of teaching mathematics - https://en.wikipedia.org/wiki/Genetic_method
For example; logarithms were invented by Napier long before exponentiation was formally defined i.e. he had no concept of a "base"! The resources i had linked to above detail his construction. So what does it mean when today we define logarithm as inverse of exponentiation?
It means logarithm is merely a table lookup from one set of numbers to another where one is a geometric series and the other a arithmetic series and its sole motivation was to map multiplication/division/etc. of large numbers onto addition/subtraction of smaller numbers. In our current mathematics we have just formalized it using the language of functions i.e. exponentiation/logarithms.