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Re: [HM] Dedekind's objection to the Newtonian concept of number.
Posted:
Jun 26, 2006 2:13 PM
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Prof. Lueneburg wrote:
<<Dedekind is also saying in the footnote that you are quoting that
you need the reals to construct the complex numbers. >>
Does he? I am not sure I follow you. You refer here, I suppose, to the
second sentence in the footnote. My understanding of the first sentence
is that, since there are no "complex magnitudes" it would not be
possible to define a complex number as the ratio of a "complex
magnitude" to a unit. And if that was his meaning, I say he is wrong.
The second sentence is less clear to me. What does he mean by
"Verhaeltniss"? Ratio? Is he saying that ratios of magnitudes cannot
be understood without understanding irrational _numbers_? And what
does the expression "umgekehrt" refer to? What is "on the contrary"?
Let me suggest the purpose of the footnote. With his method, D. is able
to define only real numbers. Complex numbers must be built up from
pairs of reals. This is no embarrassment for him, as he has built up
the reals from rationals and the rationals from wholes. It is something
of an embarrassment for the Newtonian approach if it cannot encompass
complex numbers, but must also build them from pairs of reals.
But it seems to me that the Newtonian approach can, by reasonable
extension of the notion of magnitude, include the complex numbers,
which, rather than being built up from, can be decomposed into, reals,
rationals, wholes, as well as angles and directions.
Complex numbers appear first with Cardano as algebraic forms, a + bi.
Cardano says he doesn't know what they are but "let's use them anyway"
and calls them _quantitas sophistica_. So from the outset a complex
number was _two_ numbers, however these numbers might be understood.
But consider, if you will, an alternative history. Hamilton in the
paper I mentioned before works explicitly with proportions of
directions, e.g. north is to east as east is to south. Hamilton's
recitation is a bit dreamy, but if you follow what he says you can
hardly help but see these ratios as complex numbers. Now the Irish are
not the only dreamers (just the best), so suppose some renaissance
Italian dreamer had tried to solve some problem with proportions of
directions and the notion of a ratio of directions had been established
when Cardano stumbled on "complex numbers". They would not be called
"complex" of course. The correlation of _quantitas sophistica_ to these
ratios of directions could then have been established by the time Newton
came along and he would naturally have used them as the basis of his
definition. Now, accepting all that pseudo-history, would Dedekind
have written his paper? For certain he would, because the foundations
would still be quite as shaky as they actually were, but he would not
have appended this footnote. What difference would that have made?
None, that I can see, because he "demands" that numbers be defined
arithmetically.
<< In order to multiply two directed segments >>
I do not _multiply_ two directed lines. I do not need to and am not
sure what meaning will attach to such a term. We speak of "multiplying"
magnitudes by irrational numbers, but we do so only with an extreme
extension of the meaning of the word "multiply", i.e. "to make multiple".
<< you stretch and you rotate. Stretching means using the positive reals
and rotating means using the reals modulo 2pi:>> Well once one has
defined complex numbers as ratios of directed magnitudes and applied
them to lines (i.e. what we loosely call multiplying), then, of course
one gets stretching and rotating. But why do you insist that a complex
number should be _two_ things. Look at the advantage of regarding it as
_one_ thing, i.e. a ratio. In my original message I mentioned these
things, i.e. stretching and rotating, only in hopes of clarifying to the
reader what properties of complex numbers are involved, but "stretching
and rotating" have nothing to do with the _definition_.
<<Dedekind knew what he was talking about.>>
The man said a great many things, and although a careful and deep
thinker, surely he was wrong sometime.
<<I ask you again: Where do you get the directed segments from? >>
And I answer again: "from the same place Newton gets the line he uses
to represent real numbers." In any case, I consider this "foundational"
question a separate issue.
<< I need a definition I can work with mathematically. I need
foundation, I'm afraid.>>
What? You ask me in an e-mail to give what a great mathematician was
reluctant to do at all? And before supper I will remove my own appendix.
But I gave you a definition already in my first message:
"So from this view a complex number is the ratio of a directed line
segment to a directed line segment not in the same line which is taken
as a unit."
I add, for completeness, that the plain must be oriented,
clockwise/counterclockwise. I have already admitted this is a
considerable extension of the Newtonian notion, but this can hardly be a
valid criticism, considering the enormously complex machinery Dedekind
is about to introduce.
<<You asked of the status of such a demand in mathematics. >> Heinz! A
joke.
And now I leave you because I must scrub for surgery and then dinner.
Bob
Robert Eldon Taylor
philologos at mindspring dot com
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