well done David K. As someone who studied a bit of Math, I know you are right on here.
Zangelbert,
Your q’s about David K’s definition of complex numbers are all answered in any good math book on the subject. They are also answered in his last paragraph, if you understood it. But if you knew enough to know what the words he was using mean, you would also know the answer to your own q’s. There are subtleties, which is why you have to read an actual book.
why the rah-rah cheering? i made a plain argument from definitions. you are welcome to show exactly what is wrong with it.
by the way, full disclosure: if we’re going to play the authority game, not that it counts for a whit but just to allay your fears i graduated summa cum laude in pure mathematics and did quite of bit of undergrad research in topology so i do know what David K. means about injective homomorphisms and isomorphic fields. yet i am speaking plainly about definitions because that is most likely be understood by more people and it is simpler anyway.
again, if a real number is defined as (x,0) and a complex number is an ordered pair of real numbers, then we have a FUNDAMENTAL ABSURDITY. no amount of fancy math lingo will make that go away.
in fact there are several issues at play here, and one is that Mathis is making this argument about complex numbers within an argument about why non-Euclidean geometry is useful for fudging in PHYSICS. so “if we are only interested in the structure of these numbers fields” is neither here nor there, as Mathis is actually making this argument because he is interested in more than just the mathematical abstraction. he is interested in how mathematical formalism is misused in physics.
Revised History of Ancient Kingdoms - A Complete Chronology by Newton is a magnet for critics because Newton uses the Bible as a outline…So of course, people that do not believe in the Bible are going to criticique this book very harshly.
With all due respect, those two paragraphs contradict each other. No one with the credentials you claim could have written that second paragraph, sophistry and straw man that it is.
The way it goes it that one defines real numbers, or assumes real numbers as given. Then one examines the set S of all ordered pairs of real numbers and 4 ways of operating on those ordered pairs which are called, for a reason, “adding” “subtracting” “multiplying”, and “dividing” those ordered pairs. Obviously, these are NOT the same operations as adding etc real numbers. One then defines “real numbers within the set S” as being those ordered pairs whose second element is zero. These are NOT the same as the real numbers we began with. But they are called “real numbers” loosely, and for a reason.
Certainly your summa cum laude original research will reveal to you why those ordered pairs are called by the confusing name of real numbers.
“looks like you MUST equivocate to make this work!”
Yes. I explained that in my last post.
Dave, look at what i wrote: i am using David K’s definitions and coming up with an absurdity. neither you nor him have addressed that, instead embracing "loose"ness, “confusing” terms, and equivocation as if these were somehow virtues. then you appealing to authority again by saying “i would have learned that” and “it’s in the textbook”. i am not arguing about what’s in the textbook, i am arguing about the definitions David K specifically gave.
the real question here is “why equivocate in a rigorous field”? why not call those ordered pairs of real numbers something different than “real numbers”? what kind of person would CHOOSE TO SYSTEMATICALLY EQUIVOCATE besides a habitual fudger? think about it: equivocation is the biggest red flag that can be raised in intellectual discourse, yet here in mathematics we see it EMBRACED. this ought to be cause for alarm.
Equivocation simplifies things, which is the point. Someone just needs to know what is equivocated with what and when.
simplifies the task of fudging, yes. that is Mathis’s point about non-Euclidean geometry. read him to see his case made, as it is not mine.
Anyone who studies math beyond calculus knows the answer to this q. The “original reals” are isomorphic to the “reals in the set S”, so that every result about one is true of the other. In any mathematical discussion, it is quite clear from the context which set is intended. This is embraced as a result of a feature of the human mind, that it prefers simplicity to constant lugging around of detail that is not germane to the discussion at hand, [even though the detail is neccesary for the logical underpinnings]. This has nothing ot do with statism.
If you say you disagree with David K’s definition of real and complex numbers, but agree with mine, then Im glad we agree on something. In any case, Mathis is dead in the water in his attack on complex numbers.
not quite: the square root of -1 is undefined in the reals, but defined in S.
“not quite: the square root of -1 is undefined in the reals, but defined in S.”
The mathematicians here are laughing at you, son, more and more as you continue.
In both sets the statement “there is no real number whose square is minus one”, is true. The square root of minus one exists in S, but not in the reals in S.
you just equivocated again: multiplication has a different meaning in the original reals vs. in the reals in S, as i think you will agree. so we cannot say that “square” or :“square root” mean the same thing in both.
A recursive definition works just fine; (((((x,0),0),0),0),0) is reduced to (x,0) the same way that 333/999 is simplified to 1/3. It only seems absurd because you failed to consider the actual recursion.
it’s equivocations all the way down! hallelujah!
you wave your wand and it is so! but seriously, how do you figure an infinite set “reduces” to a two-set?
The first number of the set is x, the second defined as 0. No matter how far down you go, you end up with x starting the set and every other one being zero. It matters as much as the infinite zeros to the left and right of a number.
Nobody said they mean the same thing in both. We said that “square root” has a meaning in both [though of course different, since it is talking about different operations on different sets], and that a sentence containing the phrase square root is true about the reals if and only if it is true about the reals in S.
Zangelbert, I reluctantly end this dialogue now, because I fear you are so deficient in elementary abstract math that your time will be better spent reviewing and/or learning for the first time about these fascinating topics.
My last post to you, sir.