MrSpud_k9@j5ei2on7o.org wrote:
> No they're not. Either the square root of -1 doesn't exist in which case
> complex numbers are simply a hack to get over the mathematical equivalent of
> an energy barrier, or sqr(-1) does exist and the whole basis of our
> mathematics is wrong. You can't have it both ways.

You have created a false opposition. The square root of -1 does
indeed exist in the sense that you can mathematically define
properties for it, and for a long time now it has played a
significant part in the ongoing extension of mathematics.

There are far weirder and more abstract objects in mathematics
than something as mundane and well understood as the centuries
old and now entirely uncontroversial object "sqrt(-1)".

#Paul

MrSpud_Fbyeq@ov00mymhsmu.eu wrote:
> I'm not talking about mathematics that does a good enough job of predicting
> how various laws of physics pan out over time, I'm talking about the fundamental
> underpinnings of reality itself. If mathematics is incomplete as Godel showed
> it was and it needs various hacks such as complex numbers to work plus
> oddities such as pi being transcendental when it describes a real physical
> ratio, reality itself cannot be based on mathematics as we know it.

Complex numbers are not hacks (and neither are quaternions,
or octonions, for that matter). And pi is called transcendental
because that is what mathematicians call numbers like that,
so as to distinguish from other more mundane numbers like
"rational" or whatever; It's not an oddity, just a label.

#Paul

On Thu, 12 Aug 2021 14:25:09 +0100
news20k.noreply@threeformcow.myzen.co.uk (#Paul) wrote:
>MrSpud_k9@j5ei2on7o.org wrote:
>> No they're not. Either the square root of -1 doesn't exist in which case
>> complex numbers are simply a hack to get over the mathematical equivalent of
>
>> an energy barrier, or sqr(-1) does exist and the whole basis of our
>> mathematics is wrong. You can't have it both ways.
>
>You have created a false opposition. The square root of -1 does
>indeed exist in the sense that you can mathematically define
>properties for it, and for a long time now it has played a
>significant part in the ongoing extension of mathematics.
>
>There are far weirder and more abstract objects in mathematics
>than something as mundane and well understood as the centuries
>old and now entirely uncontroversial object "sqrt(-1)".

Oh ok. Whats its value then? And don't just say 'i'.

On Thu, 12 Aug 2021 14:16:23 +0100
news20k.noreply@threeformcow.myzen.co.uk (#Paul) wrote:
>MrSpud_Fbyeq@ov00mymhsmu.eu wrote:
>> I'm not talking about mathematics that does a good enough job of predicting
>> how various laws of physics pan out over time, I'm talking about the
>fundamental
>> underpinnings of reality itself. If mathematics is incomplete as Godel showed
>
>> it was and it needs various hacks such as complex numbers to work plus
>> oddities such as pi being transcendental when it describes a real physical
>> ratio, reality itself cannot be based on mathematics as we know it.
>
>Complex numbers are not hacks (and neither are quaternions,

Of course they're hacks. If a "number" has a value that cannot be written
down and can only ever be used as an algebreic value that has to be cancelled
out of an equation at the end to get a real value then that is virtually the
definition of a hack. Perhaps the human mind simply isn't smart enough to think
of a way to do the maths without using such a hack, who knows, but hack it is.

On 12/08/2021 15:54, mickspud@downthefarm.com wrote:
> On Thu, 12 Aug 2021 14:25:09 +0100
> news20k.noreply@threeformcow.myzen.co.uk (#Paul) wrote:
>> MrSpud_k9@j5ei2on7o.org wrote:
>>> No they're not. Either the square root of -1 doesn't exist in which case
>>> complex numbers are simply a hack to get over the mathematical equivalent of
>>
>>> an energy barrier, or sqr(-1) does exist and the whole basis of our
>>> mathematics is wrong. You can't have it both ways.
>>
>> You have created a false opposition. The square root of -1 does
>> indeed exist in the sense that you can mathematically define
>> properties for it, and for a long time now it has played a
>> significant part in the ongoing extension of mathematics.
>>
>> There are far weirder and more abstract objects in mathematics
>> than something as mundane and well understood as the centuries
>> old and now entirely uncontroversial object "sqrt(-1)".
>
> Oh ok. Whats its value then? And don't just say 'i'.

If you can cope with the concept of having minus three bananas, you
should be able to cope with i.

mickspud@downthefarm.com wrote:
>>There are far weirder and more abstract objects in mathematics
>>than something as mundane and well understood as the centuries
>>old and now entirely uncontroversial object "sqrt(-1)".
>
> Oh ok. Whats its value then? And don't just say 'i'.

Ok, ok. In engineering, it is not uncommon to use "j"
instead, presumably to avoid confusion with electrical
current, which is often already being used to denote "i".
Does that help? But really, any symbol would do, as long as
the reader understands that it stands for the mathematical
object used to represent "sqrt(-1)".

But if you dont want to consider "i" a number, then fine,
don't. There might be a distinction you want to make
between a bare "i" and the imaginary number "1i", for
example.

But whatever *you* chose to call it has no bearing on its
mathematical definition, utility, or meaning. And
mathematicians, engineers, physicists, &etc will continue to
call objects like 5+3i (or even just 1i) "complex numbers".
Your dislike of either the naming or the concept does not
mean that its definitions or uses are flawed.

Everything in mathematics is a careful construction designed
to follow specific rules and operations. Sometimes the names
chosen might seem weird, misleading, or obscure. But it's
the rules and operations that mathematical objects -
including numbers - follow that really matter ... not the
naming. Sometimes the same thing can even be defined in
different ways. Would you prefer it if I called it - IIRC - a
pseudo-scalar? :-)

#Paul

On Thu, 12 Aug 2021 23:25:02 +0100
news20k.noreply@threeformcow.myzen.co.uk (#Paul) wrote:
>mickspud@downthefarm.com wrote:
>>>There are far weirder and more abstract objects in mathematics
>>>than something as mundane and well understood as the centuries
>>>old and now entirely uncontroversial object "sqrt(-1)".
>>
>> Oh ok. Whats its value then? And don't just say 'i'.
>
>Ok, ok. In engineering, it is not uncommon to use "j"
>instead, presumably to avoid confusion with electrical
>current, which is often already being used to denote "i".
>Does that help? But really, any symbol would do, as long as
>the reader understands that it stands for the mathematical
>object used to represent "sqrt(-1)".

Whats its value?

>But if you dont want to consider "i" a number, then fine,
>don't. There might be a distinction you want to make
>between a bare "i" and the imaginary number "1i", for
>example.
>
>But whatever *you* chose to call it has no bearing on its
>mathematical definition, utility, or meaning. And

I never said otherwise.

>mathematicians, engineers, physicists, &etc will continue to
>call objects like 5+3i (or even just 1i) "complex numbers".
>Your dislike of either the naming or the concept does not
>mean that its definitions or uses are flawed.

I have no opinion either way , if it works it works. I'm simply saying its
a hack, probably because the human mind isn't smart enough to do the same
with real numbers.

>Everything in mathematics is a careful construction designed
>to follow specific rules and operations. Sometimes the names
>chosen might seem weird, misleading, or obscure. But it's
>the rules and operations that mathematical objects -
>including numbers - follow that really matter ... not the
>naming. Sometimes the same thing can even be defined in
>different ways. Would you prefer it if I called it - IIRC - a
>pseudo-scalar? :-)

Whats its value? sqrt(-1) isn't a value, its an operation.

On 13/08/2021 10:29, mickspud@downthefarm.com wrote:
> On Thu, 12 Aug 2021 23:25:02 +0100
> news20k.noreply@threeformcow.myzen.co.uk (#Paul) wrote:
>> mickspud@downthefarm.com wrote:
>>>> There are far weirder and more abstract objects in mathematics
>>>> than something as mundane and well understood as the centuries
>>>> old and now entirely uncontroversial object "sqrt(-1)".
>>>
>>> Oh ok. Whats its value then? And don't just say 'i'.
>>
>> Ok, ok. In engineering, it is not uncommon to use "j"
>> instead, presumably to avoid confusion with electrical
>> current, which is often already being used to denote "i".
>> Does that help? But really, any symbol would do, as long as
>> the reader understands that it stands for the mathematical
>> object used to represent "sqrt(-1)".
>
> Whats its value?

Negative numbers and zero are equally "imaginary". And fractions.

--
Max Demian

In article <sf5e1b$1632$1@gioia.aioe.org>, <mickspud@downthefarm.com> wrote:

>Whats its value? sqrt(-1) isn't a value, its an operation.

You're just making up terms. You could call square root an operation,
but sqrt(-1) certainly isn't.

-- Richard

In article <R8SdnU0N0caePmv9nZ2dnUU78LmdnZ2d@brightview.co.uk>,
Max Demian <max_demian@bigfoot.com> wrote:

>> I think first you have to define what you mean by "exist". I suspect we
>> might be straying into Bertrand Russell territory here.

>He wrote a book trying to prove that 1+1=2 and failed...

Russell and Whitehead's Principia famously *does* prove that 1+1=2,

See 110.643 on page 86 of Vol 2:

https://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath&cc=umhistmath&idno=aat3201.0002.001&frm=frameset&view=image&seq=126

and note the comment after the proof.

-- Richard

"Richard Tobin" <richard@cogsci.ed.ac.uk> wrote in message
news:sf5jjr$ea3$1@macpro.inf.ed.ac.uk...
> In article <R8SdnU0N0caePmv9nZ2dnUU78LmdnZ2d@brightview.co.uk>,
> Max Demian <max_demian@bigfoot.com> wrote:
>
>>> I think first you have to define what you mean by "exist". I suspect we
>>> might be straying into Bertrand Russell territory here.
>
>>He wrote a book trying to prove that 1+1=2 and failed...
>
> Russell and Whitehead's Principia famously *does* prove that 1+1=2,
>
> See 110.643 on page 86 of Vol 2:
>
> https://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath&cc=umhistmath&idno=aat3201.0002.001&frm=frameset&view=image&seq=126
>
> and note the comment after the proof.

I did maths at A level and as part of Elec Eng at university. But that looks
like fluent Martian. There's lots of Greek letters and other symbols (*)
which I've never seen before. I presume that "*110.64" etc are line numbers
that are referred to later on.

(*) backwards E, Greek xi, |- symbol, backwards C, back-quote `

On 13/08/2021 12:04, Richard Tobin wrote:
> In article <R8SdnU0N0caePmv9nZ2dnUU78LmdnZ2d@brightview.co.uk>,
> Max Demian <max_demian@bigfoot.com> wrote:
>
>>> I think first you have to define what you mean by "exist". I suspect we
>>> might be straying into Bertrand Russell territory here.
>
>> He wrote a book trying to prove that 1+1=2 and failed...
>
> Russell and Whitehead's Principia famously *does* prove that 1+1=2,
>
> See 110.643 on page 86 of Vol 2:
>
> https://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath&cc=umhistmath&idno=aat3201.0002.001&frm=frameset&view=image&seq=126
>
> and note the comment after the proof.

I got lost on page 770.

--
Max Demian

On 13/08/2021 11:28, Max Demian wrote:
> On 13/08/2021 10:29, mickspud@downthefarm.com wrote:
>> On Thu, 12 Aug 2021 23:25:02 +0100
>> news20k.noreply@threeformcow.myzen.co.uk (#Paul) wrote:
>>> mickspud@downthefarm.com wrote:
>>>>> There are far weirder and more abstract objects in mathematics
>>>>> than something as mundane and well understood as the centuries
>>>>> old and now entirely uncontroversial object "sqrt(-1)".
>>>>
>>>> Oh ok. Whats its value then? And don't just say 'i'.
>>>
>>> Ok, ok. In engineering, it is not uncommon to use "j"
>>> instead, presumably to avoid confusion with electrical
>>> current, which is often already being used to denote "i".
>>> Does that help? But really, any symbol would do, as long as
>>> the reader understands that it stands for the mathematical
>>> object used to represent "sqrt(-1)".
>>
>> Whats its value?
>
> Negative numbers and zero are equally "imaginary". And fractions.

Correction: *all* numbers are imaginary.

--
Max Demian

In article <aLudnWeiE4P-94v8nZ2dnUU78QOdnZ2d@brightview.co.uk>,
Max Demian <max_demian@bigfoot.com> wrote:

>> Negative numbers and zero are equally "imaginary". And fractions.

>Correction: *all* numbers are imaginary.

All numbers are an abstraction.

The square root of -1 has an interesting property. It's always true
that if x^2 = y, then (-x)^2 also = y [*]. So if i^2 = -1, (-i) = -1
too. But unlike, say, 2 and -2, there's no way to distinguish between
-i and i. If you take any equation involving i, and replace it
everywhere with (-i), the equation is still true.

[*] because -x is -1 * x so (-x) * (-x) is (-1 * x) * (-1 * x) which
is 1 * x^2 (because multiplication is distributive and commutative).

-- Richard

NY <me@privacy.invalid> wrote:
> "Richard Tobin" <richard@cogsci.ed.ac.uk> wrote in message
> news:sf5jjr$ea3$1@macpro.inf.ed.ac.uk...
>> In article <R8SdnU0N0caePmv9nZ2dnUU78LmdnZ2d@brightview.co.uk>,
>> Max Demian <max_demian@bigfoot.com> wrote:
>>
>>>> I think first you have to define what you mean by "exist". I suspect we
>>>> might be straying into Bertrand Russell territory here.
>>
>>> He wrote a book trying to prove that 1+1=2 and failed...
>>
>> Russell and Whitehead's Principia famously *does* prove that 1+1=2,
>>
>> See 110.643 on page 86 of Vol 2:
>>
>> https://quod.lib.umich.edu/cgi/t/text/pageviewer-idx?c=umhistmath&cc=umhistmath&idno=aat3201.0002.001&frm=frameset&view=image&seq=126
>>
>> and note the comment after the proof.
>
> I did maths at A level and as part of Elec Eng at university. But that looks
> like fluent Martian. There's lots of Greek letters and other symbols (*)
> which I've never seen before. I presume that "*110.64" etc are line numbers
> that are referred to later on.
>
> (*) backwards E, Greek xi, |- symbol, backwards C, back-quote `
>
>

Wikipedia’s list of logic symbols covers most of that. IIRC I had met
existential quantification doing A level maths but not turnstile until at
least undergraduate level.

The numbers refer to previously proved results rather than lines as such.

On Fri, 13 Aug 2021 11:28:38 +0100
Max Demian <max_demian@bigfoot.com> wrote:
>On 13/08/2021 10:29, mickspud@downthefarm.com wrote:
>> On Thu, 12 Aug 2021 23:25:02 +0100
>> news20k.noreply@threeformcow.myzen.co.uk (#Paul) wrote:
>>> mickspud@downthefarm.com wrote:
>>>>> There are far weirder and more abstract objects in mathematics
>>>>> than something as mundane and well understood as the centuries
>>>>> old and now entirely uncontroversial object "sqrt(-1)".
>>>>
>>>> Oh ok. Whats its value then? And don't just say 'i'.
>>>
>>> Ok, ok. In engineering, it is not uncommon to use "j"
>>> instead, presumably to avoid confusion with electrical
>>> current, which is often already being used to denote "i".
>>> Does that help? But really, any symbol would do, as long as
>>> the reader understands that it stands for the mathematical
>>> object used to represent "sqrt(-1)".
>>
>> Whats its value?
>
>Negative numbers and zero are equally "imaginary". And fractions.

No they're not. Negative numbers are simply a vector in the opposite
direction and if breaking something in 2 doesn't give me 2 halves what does
it give?

On Fri, 13 Aug 2021 10:46:39 +0000 (UTC)
richard@cogsci.ed.ac.uk (Richard Tobin) wrote:
>In article <sf5e1b$1632$1@gioia.aioe.org>, <mickspud@downthefarm.com> wrote:
>
>>Whats its value? sqrt(-1) isn't a value, its an operation.
>
>You're just making up terms. You could call square root an operation,
>but sqrt(-1) certainly isn't.

Whats its value?

In article <sf5usp$mmb$1@gioia.aioe.org>, <mickspud@downthefarm.com> wrote:
>On Fri, 13 Aug 2021 10:46:39 +0000 (UTC)

>>>Whats its value? sqrt(-1) isn't a value, its an operation.

>>You're just making up terms. You could call square root an operation,
>>but sqrt(-1) certainly isn't.

>Whats its value?

i.

-- Richard

On Fri, 13 Aug 2021 14:50:52 +0000 (UTC)
richard@cogsci.ed.ac.uk (Richard Tobin) wrote:
>In article <sf5usp$mmb$1@gioia.aioe.org>, <mickspud@downthefarm.com> wrote:
>>On Fri, 13 Aug 2021 10:46:39 +0000 (UTC)
>
>>>>Whats its value? sqrt(-1) isn't a value, its an operation.
>
>>>You're just making up terms. You could call square root an operation,
>>>but sqrt(-1) certainly isn't.
>
>>Whats its value?
>
>i.

You're about 3 posts too late for that joke.

On 13/08/2021 15:16, mickspud@downthefarm.com wrote:
> On Fri, 13 Aug 2021 11:28:38 +0100
> Max Demian <max_demian@bigfoot.com> wrote:
>> On 13/08/2021 10:29, mickspud@downthefarm.com wrote:
>>> On Thu, 12 Aug 2021 23:25:02 +0100
>>> news20k.noreply@threeformcow.myzen.co.uk (#Paul) wrote:
>>>> mickspud@downthefarm.com wrote:
>>>>>> There are far weirder and more abstract objects in mathematics
>>>>>> than something as mundane and well understood as the centuries
>>>>>> old and now entirely uncontroversial object "sqrt(-1)".
>>>>>
>>>>> Oh ok. Whats its value then? And don't just say 'i'.
>>>>
>>>> Ok, ok. In engineering, it is not uncommon to use "j"
>>>> instead, presumably to avoid confusion with electrical
>>>> current, which is often already being used to denote "i".
>>>> Does that help? But really, any symbol would do, as long as
>>>> the reader understands that it stands for the mathematical
>>>> object used to represent "sqrt(-1)".
>>>
>>> Whats its value?
>>
>> Negative numbers and zero are equally "imaginary". And fractions.
>
> No they're not. Negative numbers are simply a vector in the opposite
> direction and if breaking something in 2 doesn't give me 2 halves what does
> it give?

A negative means you are owed something. A fraction is one number
divided by another (apart from *irrational* numbers; let's not go there).

--
Max Demian

On Fri, 13 Aug 2021 13:11:35 +0000 (UTC), richard@cogsci.ed.ac.uk
(Richard Tobin) wrote in <sf5r27$hob$1@macpro.inf.ed.ac.uk>:

>[*] because -x is -1 * x so (-x) * (-x) is (-1 * x) * (-1 * x) which
>is 1 * x^2 (because multiplication is distributive and commutative).

Multiplication over real numbers and over complex numbers[1] is
commutative and associative. The distributive property applies to two
operators - multiplication distributes over addition for those sets:
x*(y+z) = x*y + x*z.

Note that multiplication of matrices is associative but not commutative
so you cannot just say 'multiplication' without defining the set over
which it is operating.

[1] Not an exhaustive list! Those who want more on this topic should
look up groups, especially abelian groups, and then progress on to rings
and fields in theur study of abstract algebra.

On Fri, 13 Aug 2021 16:48:45 +0100, Max Demian <max_demian@bigfoot.com>
wrote in <4M-dnbq_UYL-CYv8nZ2dnUU78VWdnZ2d@brightview.co.uk>:

>On 13/08/2021 15:16, mickspud@downthefarm.com wrote:
>> On Fri, 13 Aug 2021 11:28:38 +0100
>> Max Demian <max_demian@bigfoot.com> wrote:
>>> On 13/08/2021 10:29, mickspud@downthefarm.com wrote:
>>>> On Thu, 12 Aug 2021 23:25:02 +0100
>>>> news20k.noreply@threeformcow.myzen.co.uk (#Paul) wrote:
>>>>> mickspud@downthefarm.com wrote:
>>>>>>> There are far weirder and more abstract objects in mathematics
>>>>>>> than something as mundane and well understood as the centuries
>>>>>>> old and now entirely uncontroversial object "sqrt(-1)".
>>>>>>
>>>>>> Oh ok. Whats its value then? And don't just say 'i'.
>>>>>
>>>>> Ok, ok. In engineering, it is not uncommon to use "j"
>>>>> instead, presumably to avoid confusion with electrical
>>>>> current, which is often already being used to denote "i".
>>>>> Does that help? But really, any symbol would do, as long as
>>>>> the reader understands that it stands for the mathematical
>>>>> object used to represent "sqrt(-1)".
>>>>
>>>> Whats its value?

It is the simplest thing to add to the real numbers in order to
construct the algebraic numbers. Having created complex numbers they
turn out to be useful to scientists and engineers in modelling things
they are interested in.

That is its value in the sense of utility. Why does it need to have a
numeric value other than just being what it is?

>>>
>>> Negative numbers and zero are equally "imaginary". And fractions.
>>
>> No they're not. Negative numbers are simply a vector in the opposite
>> direction and if breaking something in 2 doesn't give me 2 halves what does
>> it give?

Negative numbers (what kind of number?) are not vectors in the sense in
which that term is used by mathematicians (or physicists and engineers).
Vectors are another invented concept that turns out to be useful in
modelling various physical things but which are studied by
mathematicians as part of abstract algebra. Vector spaces have some well
defined properties that make them useful to people who are not just
interested in the mathematical beauty of the structure.

>
>A negative means you are owed something. A fraction is one number
>divided by another (apart from *irrational* numbers; let's not go there).

Negative numbers can be used to model the idea of being owed something
but that is not an essential part of their nature.

Negative numbers are the additive inverses of the positive numbers in
the relevant groups. Fractions are needed in order to have
multiplicative inverses for the integral non-zero numbers in the
relevant groups but once you have introduced some rational numbers you
really have to include them all to have any sort of coherent structure.

On Thu, 12 Aug 2021 14:56:56 -0000 (UTC), mickspud@downthefarm.com wrote
in <sf3cro$1gbb$1@gioia.aioe.org>:

>Of course they're hacks. If a "number" has a value that cannot be written
>down and can only ever be used as an algebreic value that has to be cancelled
>out of an equation at the end to get a real value then that is virtually the
>definition of a hack. Perhaps the human mind simply isn't smart enough to think
>of a way to do the maths without using such a hack, who knows, but hack it is.

Can you write down the value of one third without using a division
operator or some other notation introduced for the specific purpose of
writing down such mumbers?

On Fri, 13 Aug 2021 13:49:33 +0100, Max Demian
<max_demian@bigfoot.com> wrote:

>> Negative numbers and zero are equally "imaginary". And fractions.
>
>Correction: *all* numbers are imaginary.

But real things depend on their properties.

Rod.

On Fri, 13 Aug 2021 11:28:38 +0100, Max Demian <max_demian@bigfoot.com> wrote:

>On 13/08/2021 10:29, mickspud@downthefarm.com wrote:
>> On Thu, 12 Aug 2021 23:25:02 +0100
>> news20k.noreply@threeformcow.myzen.co.uk (#Paul) wrote:
>>> mickspud@downthefarm.com wrote:
>>>>> There are far weirder and more abstract objects in mathematics
>>>>> than something as mundane and well understood as the centuries
>>>>> old and now entirely uncontroversial object "sqrt(-1)".
>>>>
>>>> Oh ok. Whats its value then? And don't just say 'i'.
>>>
>>> Ok, ok. In engineering, it is not uncommon to use "j"
>>> instead, presumably to avoid confusion with electrical
>>> current, which is often already being used to denote "i".
>>> Does that help? But really, any symbol would do, as long as
>>> the reader understands that it stands for the mathematical
>>> object used to represent "sqrt(-1)".
>>
>> Whats its value?
>
>Negative numbers and zero are equally "imaginary". And fractions.

and infinity
--

Martin in Zuid Holland