" In mathematics, the numbers have no meaning assigned to them." Unfortunately for most Americans ... and most of mankind ... that is all too true. After grade 2, "numbers" are curriculum-divorced from their conceptual meanings, and the *school* versions of "mathematics" reduce to senseless manipulations of numeric formulas.
Many adults who have been reared through that kind of "education" .... in that kind of "mathematics" ... fail to perceive that all numbers are *abstracts* that respectively speak of classes of exemplary instances.
" 6 + 4i is just a number, just like 153 is a number, just like 1x10^2 + 5*10^1 + 3*10^0 is a number." "6 + 4i" means the (6,4) long-lat location in the coordinate plane or on the globe ... or the full-planar vector that carries (0,0) to (6,4). "153" means means a specific point on the (Peano) line of Arabic numerals. The "power analysis" of that numeral is a 3-dimensional vector whose base-8 value is 107.
Numbers are devoid of meanings only for persons who were so-educated (or blindly or mis-educated) ... to deal with numeric formalities, without actually knowing what those formaities are talking about. We are scholastically taught that "applying" mathematics to the real world is an aspect of "mathematics." It is even more true (though untaught) that *abstracting* mathematics from the real world is a crucial aspect of mathematics. In fact, that is what enables us to "apply" mathematics ... and to grasp the common-sensibility of its formalities.
Cordially, Clyde - -------------------------------------------------- From: "Robert Hansen" <email@example.com> Sent: Friday, November 30, 2012 7:38 PM To: <firstname.lastname@example.org> Subject: Re: Some important demonstrations on negative numbers
> > On Nov 30, 2012, at 6:51 PM, Joe Niederberger <email@example.com> > wrote: > >>> Joe, numbers are not quantities. Quantities are physical, they include a >>> number, a unit, a direction, if needed, and for goodness sake, a >>> freaking context. >> >> My goodness. I was speaking of the good old world, as Dickens might say, >> not your personal universe. >> >> Here is Isaac Newton: >> "Algebraic quantities are of two sorts, affirmative and negative; an >> affirmative quantity is greater than nothing, and is known by this sign >> +; a negative quantity is less than nothing, and is known by this >> sign -." >> >> He follows up with a money example. >> >> Here is Euler: >> "The calculation of imaginary quantities is of the greatest importance." >> >> Lest you think taking number as "quantity" is merely archaic usage, check >> these: >> >> * http://oxforddictionaries.com/definition/english/mathematics >> * http://dictionary.reference.com/browse/mathematics?s=t&ld=1122 >> * http://en.wikipedia.org/wiki/Mathematics >> * http://en.wikipedia.org/wiki/Quantity#Quantity_in_mathematics > > If you like, it is archaic usage. More importantly, it is dead wrong. > > >> >> I understand the distinction you are pointing to, though; nice as it is, >> it doesn't seem particularly germane in this context, that of >> understanding negative numbers and their rules. > > It isn't just nice, it is the truth, and it is quite germane to > understanding negative numbers. And it is dead on in the discussion of > "common sense". > > >> >> Saying that a negative number "is a mathematical concept" (well, by >> golly, its abstract!) does nothing to explain what it is. How are they >> different from the whole numbers a child already knows about? What good >> are they? Why are the rules (esp. the infamous one) such as they are? >> >> What's your lesson look like? What are the key points for a child? > > It looks like doing a lot of math with negative numbers. There are no key > points. The child exits with (4 - 10) being every bit the same as (10 - > 4), mathematically speaking. What you are referring to is not negative > numbers, but the application of negative numbers to real world situations. > I already stated that application of math is healthy, but without the > math, application of the math is moot. > > Let me put it this way, and please think about it. Physics is the > application of math and all application of math is physics. That is broad, > I know, but hear me out. When you apply mathematics, whether it be to > physics, or adding apples or something just practical like accounting, you > are modeling. There is a context involved. This involves the assignment of > meaning to the numbers, whether those assignments are explicit or just > taken for granted. In mathematics, the numbers have no meaning assigned to > them. 6 + 4i is just a number, just like 153 is a number, just like 1x10^2 > + 5*10^1 + 3*10^0 is a number. Thus, anytime you apply meaning to the > numbers you are applying mathematics. I am going to call that physics, for > the hell of it, and because it implies that there is more going on then > just assigning meaning to the numbers. If it is accounting, then there are > certain laws that must be obeyed. If it is just apples then there is at > least the law of conservation of apples. An! > d if it is physics, well, you know what I am thinking. > > So you see, there is mathematics, and there is the application of > mathematics. > > >> >> R.H. says: >>> Mathematics deals only with the number part of all that. >> >> And which part is that? Does your concept of number include separable >> components as well? What makes a real number real, but an imaginary >> number a figment of the imagination? > > They are both figments of the imagination. We use the terms "real" and > "imaginary" only to distinguish them. > > >> >> And now, just for fun, some people who want to get real about math: >> http://web.maths.unsw.edu.au/~jim/structmath.html >> http://web.maths.unsw.edu.au/~jim/manifesto.html > > Ha. > >> >> Cheers, >> Joe N