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On 18 Sep 2006 02:25:21 -0700, "Roger Sindreu"
<Roger.Sindreu@xxxxxxxxx> wrote:
>
>David C. Ullrich ha escrit:
>> On 16 Sep 2006 09:17:13 -0700, "Roger Sindreu"
>> <Roger.Sindreu@xxxxxxxxx> wrote:
>>
>> >
>> >David C. Ullrich ha escrit:
>> >> On 14 Sep 2006 15:20:52 -0700, "Roger Sindreu"
>> >> <Roger.Sindreu@xxxxxxxxx> wrote:
>> >>
>> >> >> definition (a) is not really a definition at all,
>> >> >> because it doesn't say what the square root of -1 _is_,
>> >> >> nor does it give any reason for thinking there is such aI
>> >axioms are consistent.
>> >
>> >*(See Godel's theorems)
>>
>> Well duh.
>>
>> Actually saying that there's no way to show PA is consistent
>> is a considerable oversimplification of the story. But never
>> mind that.
>
>I said "can't" with an *, because I also think this is a
>oversimplification of the story.
>
>> How does one show that the existence of sqrt(-1)
>> is consistent, _if_ the axioms that are generally accepted
>> as a basis for mathematics are assumed to be consistent.
>> Which of course is what is meant when one says "consistent
>> with the other axioms". It _is_ very easy to show that
>> the existence of sqrt(-1) is consistent with the other
>> axioms for a field - one does this by exhibiting a model.
>> The simplest way to do that is to use the ordered pairs
>> of reals as a model.
>
>I don't have much knowledge about that, actually.
You don't have much knowledge of the idea that one shows
a theory consistent by showing it has a model? I guess
there's nothing wrong with that, but probably you should
avoid trying to make points by citing Godel's theorem
until after you've learned some basic mathematical logic.
>> >> >> NOTE: In any given context the concept has only _one_
>> >> >> definition! You seem to be suggesting that (a) and (b)
>> >> >> are both definitions - they can't both be the definitions,
>> >> >> at least not in one given context. The "official" definition
>> >> >> these days is (b).
>> >> >
>> >> >I think both definitions are accepted at the same time, since it is
>> >> >usual to see, i^2 = -1, and manipulate complex numbers with cos and sin.
>> >>
>> >> Of course both (a) and (b) are true facts! But they cannot both
>> >> be the _definition_ of what a complex number is; a given concept
>> >> has at most one definition.
>> >>
>> >> (That's in a given context; this doesn't preclude one person
>> >> from taking (a) as the definition and another person from
>> >> taking (b) as the definition. But you can't take them both
>> >> as "the" definition simultaneously.)
>> >>
>> >
>> >Just because they are two sentences, you can't say they are two
>> >definitions. Even if they are: Big definition= definition_a &
>> >definition_b. And that's a single definition.
>>
>> First, you're contradicting yourself - _you_ said that
>> noth definitions were accepted at the same time.
>>
>> Second, this is _not_ done. It simply doesn't happen
>> that people give both a and b as part of one definition
>> of the complex numbers.
>
>I didn't say that both definitions were accepted at the same time,
>actually I started asking if they were.
And then in your _latest_ post you said that they _were_ both
parts of one definition.
> But it is simply that your
>explanation that a complex number is sometimes a), and sometimes b)
>doesn't sound mathematically rigorous to me.
Huh?????????
In your original post you didn't ask about mathematical rigor,
you asked a question about the _history_.
>I would like that all true concepts were either axioms, either
>consequences, nothing else. Not sometimes axioms, and sometimes not
>axioms.
Well, that's not the way it is. In an actual mathemtatical treatment
of complex numbers they are defined as in (b), or by something
equivalent. In a less mathematically inclined treatment people
don't worry about what sqrt(-1) is, they just say it's
"imaginary" and use it without worrying about justifying its
existence.
>Note, that my intention was not to start an argumentation, but to ask
>people who just knew the answer. As some explanations, didn't convince
>me, I just pointed out the parts that were weird to me.
>
>It is now easier for me to find a clear answer somewhere else.
Lucky you. The clear answer you find, if correct, will be what I said.
>Anyhow, cheers for your explanations.
>
>Roger
************************
David C. Ullrich
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