General > General Technical Chat
"Veritasium" (YT) - "The Big Misconception About Electricity" ?
HuronKing:
--- Quote from: penfold on April 11, 2022, 01:49:21 pm ---But, I do think that natural suspicion is a very reasonable thing to have, because, by intrinsically imbueing significance to the imaginary numbers, it conflicts with both the modern mathematical definitions (no numbers are natural) and the 17th century (some numbers are natural) views. But I can live with a third "engineering maths" definition of "anything goes".
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Please show me where modern mathematical definitions say "no numbers are natural." I don't understand what this means. ???
Mathematicians of the 17th century barely understood calculus. Descartes, to his credit, laid the foundation with analytic geometry but he didn't know how to take a derivative. He was close, but he still wouldn't be able to pass a 1st semester calculus course with the extent of his knowledge. In a very big sense, you and I are WAY smarter than Descartes. You and I can solve differential equations and integrals Descartes could never even imagine. 8)
Nature is more complex than just what we can count on our fingers and toes and it often violates our intuition. Yet, for some reason, nature seems to obey logical mathematical rules.
As another example, what is 0 times infinity? 0 divided by 0? Infinity divided by 0?
https://en.wikipedia.org/wiki/L%27H%C3%B4pital%27s_rule
Answers to these questions are impossibilities for (pre-calculus) 17th century mathematics - yet they do have answers... under the right conditions and their answers directly correlates to physical meaning:
https://lhospitalsrule.weebly.com/real-world-applications.html
There is a huge amount of things in mathematics that ought to philosophically bother you, not just the complex numbers, if you must restrict yourself to only what you can count on your fingers and toes. And maybe the idea of solving an equation which asks "what's infinity divided by 0" does bother you too. ;D
Neither mathematics nor engineering compels us to restrict ourselves to what is perceptible to our intuition. Our intuition is often wrong. And we have lasting evidence of what relying on faulty intuition gets us (bad terminology for mathematical definitions for starters ;) ). Follow the logic, bravely, and see where it takes you (hopefully not back to 1657).
--- Quote ---TBH, its the same gripe I have with Poynting, it is a mathematical theorem of Maxwell's equations that we cannot contemplate avoiding, but it doesn't necesarily agree wiyh everything else when considered as a physical process... but the reason I let it slide is that it is a million times easier to explain than what might actually be going on... we'd first have to descover that, but I sruggle slightly in philosophically accepting it at DC along-side deBroglie (suggesting a wave with zero frequency carries zero momentum). Its just an internal pondering no rejection of the theories.
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Because we have to operate within the paradigms of the theory. You have to be careful when trying to extend classical electromagnetism (Heaviside didn't know about photons) to quantum physics.
TimFox:
I was taught that the "natural numbers" are the positive integers (not including zero). An older name is "counting numbers".
Some mathematicians include 0 in the set of natural numbers, but others prefer "whole numbers" or "non-negative integers" for the set including 0.
Historically, the number zero is a later concept than the other integers.
By the way, the "rational numbers" are not ones that are less icky or more sane than "irrational numbers", they are numbers that can be expressed as a "ratio" of two integers.
When communicating with other technical people, it is salutary to use standard names for well-understood concepts.
SiliconWizard:
--- Quote from: TimFox on April 11, 2022, 04:58:24 pm ---I was taught that the "natural numbers" are the positive integers (not including zero). An older name is "counting numbers".
Some mathematicians include 0 in the set of natural numbers, but others prefer "whole numbers" or "non-negative integers" for the set including 0.
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I have myself learned that the set N of natural numbers included zero, and the notation to exclude zero from it was writing N*. (Same for Z and Z*...)
(Btw, do you consider zero to be part of Z?)
But it seems to differ depending on where you have learned. From a lot of material I read over the years, I'm under the impression that N excluding zero is a common notation in the USA, while using N and N* seems more common in Europe. Just personal observation here.
As to irrational numbers, they are in essence *roots* of some equations, that can't be expressed as rational numbers. So we can only express them implicitly via some equation.
sqrt(2) as usually defined is the principal root in R of the equation x^2 = 2. Likewise, i is the principal root in C of the equation x^2 = -1. In both cases, they are defined are roots of some equation.
Whether you consider them something "real" or just a "handy trick" is more philosophical than technical.
TimFox:
Yes. The word "irrational" just means that the root in question cannot be expressed as the ratio of two integers. (Pi is not 22/7.)
penfold:
--- Quote from: HuronKing on April 11, 2022, 04:47:37 pm ---
--- Quote from: penfold on April 11, 2022, 01:49:21 pm ---But, I do think that natural suspicion is a very reasonable thing to have, because, by intrinsically imbueing significance to the imaginary numbers, it conflicts with both the modern mathematical definitions (no numbers are natural) and the 17th century (some numbers are natural) views. But I can live with a third "engineering maths" definition of "anything goes".
--- End quote ---
Please show me where modern mathematical definitions say "no numbers are natural." I don't understand what this means. ???
[...]
--- End quote ---
My apologies, I wasn't paying attention, I didn't mean natural mathematically, I meant natural, philosophically, in the sense of being of or directly or closely related to natural things, say, quantities of countable things or something perceivable to humans: a number of sticks in a pot would represent the same quantity to two people regardless of the language or abstraction thereof. Rational numbers in the same way, as they can be formed (often, for most quantities) from fractions of units. Irrational numbers... that's a whole other discussion.
I can't recall immediately a good reference and I'm away from home for the week so it'll be a little while before I can dig for the right citation. Any generic set theory and mathematical logic textbook should give an idea how the more axomatic and less physical significance of "numbers" overtakes in a more modern sense. Think about how you might word it if you were to describe the equation or process using words, i.e. are you directly multiplying a length or are you multiplying numbers that represent the number of unit lengths, then what is the result and how would you then represent that physically, is there a measurement process used in between and how do you get from the written number to the physical quantity... its unfortunately one of those things that takes a lot of reading of lots of different books and single explanatory references aren't very common.
But, your reference to l'Hopital sums it up so nicely, as you lead into it with "As another example, what is 0 times infinity? 0 divided by 0? Infinity divided by 0?", suggests you havn't quite understood the question yourself, l'Hoptital it would give the value to a function that contains terms that individually tend to those values... not of the pure numbers themselves necesarily.
The complexities of nature are kinda irrelevent to the maths, the maths describes only our observations and patterns amongst them, it all exists within the artificial construct of logic that is related to human reasoning, nature just does its own thing.
--- Quote from: HuronKing on April 11, 2022, 04:47:37 pm ---[...]
Because we have to operate within the paradigms of the theory. You have to be careful when trying to extend classical electromagnetism (Heaviside didn't know about photons) to quantum physics.
--- End quote ---
Yeah... exactly... they both agree mathematically, but rely on very different implications towards physical processes, so it becomes a question of observeable quantities - so at the same time as apprechiating the limits of the theories one must also be careful of what the maths implies about physical processes - so when we are so quick to say that Poynting explains something, (rhetorical question) are we simultaneously saying that it is the genuine underlying physical process? I suspect you still havn't worked out that my gripe is not with maths itself, but with how people are so quick to ignore the fact it is only describing links between the observations etc, and whilst can (and has) predict(ed) other physical phenomena, the purely mathematical proof does not itself proove something physically.
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