On 14/08/2021 07:39, Roderick Stewart wrote:
> 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.

Is number a "property" of things? Depends on where one thing ends and
another begins. Hence, in recipes, it doesn't say "two beaten eggs" but
"two eggs, beaten".

--
Max Demian

In article <sf61dp$1rb2$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?

>>i.

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

It's not a joke. "i" is a name for the square root of -1, just as
"2" is a name for 1+1.

2 and i are both abstractions, but for some reason you seem happy
with one but not the other.

-- Richard

On Sat, 14 Aug 2021 11:50:45 +0100, Max Demian
<max_demian@bigfoot.com> wrote:

>On 14/08/2021 07:39, Roderick Stewart wrote:
>> 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.
>
>Is number a "property" of things? Depends on where one thing ends and
>another begins. Hence, in recipes, it doesn't say "two beaten eggs" but
>"two eggs, beaten".

Numbers have numerical properties. Magnitudes, factors, multiples,
relations to other numbers and so on. Geometrical shapes have
numerical properties, such as minimum or maximum values for distances
between shapes, packing densities and so on.

Relative magnitudes of numbers in theoretical space determine the
relative magnitudes of real objects in real space. If A is bigger than
B, then a person who possesses A objects will have more of them than
another person who only has B of them.

If a number is prime in theoretical space, then it will be impossible
to make equal shares out of that number of real objects in real space.

The ways in which shapes can fit together in theoretical space
determine how real physical shapes can fit together in real space.

The ways in which subatomic particles can fit together must be subject
to the same rules of geometry as everything else in real space, so it
may be that all of physical reality can only exist in particular
forms, determined by the properties of numbers.

Rod.

On Fri, 13 Aug 2021 21:14:23 +0100
Owen Rees <orees@hotmail.com> wrote:
>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.

I never said otherwise. I simply said they're a hack. Hacks usually work but
they're also ugly and often there's a better way of doing things.

>the relevant groups. Fractions are needed in order to have
>multiplicative inverses for the integral non-zero numbers in the

Fractions are needed to represent fractional parts of things. No need to get
carried away with some convoluted explanation.

On Fri, 13 Aug 2021 21:17:11 +0100
Owen Rees <orees@hotmail.com> wrote:
>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?

I can break a stick into 3 and physically show you a third. Can you show me i?
No, you can't, because it doesn't exist. Your argument is like saying "Write
down 3 without using the figure 3".

On Sat, 14 Aug 2021 12:46:28 +0000 (UTC)
richard@cogsci.ed.ac.uk (Richard Tobin) wrote:
>In article <sf61dp$1rb2$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?
>
>>>i.
>
>>You're about 3 posts too late for that joke.
>
>It's not a joke. "i" is a name for the square root of -1, just as
>"2" is a name for 1+1.
>
>2 and i are both abstractions, but for some reason you seem happy
>with one but not the other.

2 is not an abstraction in the sense I can show you two of something. Show me
'i' of something.

On 14/08/2021 16:11, mickspud@downthefarm.com wrote:
> On Sat, 14 Aug 2021 12:46:28 +0000 (UTC)
> richard@cogsci.ed.ac.uk (Richard Tobin) wrote:
>> In article <sf61dp$1rb2$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?
>>
>>>> i.
>>
>>> You're about 3 posts too late for that joke.
>>
>> It's not a joke. "i" is a name for the square root of -1, just as
>> "2" is a name for 1+1.
>>
>> 2 and i are both abstractions, but for some reason you seem happy
>> with one but not the other.
>
> 2 is not an abstraction in the sense I can show you two of something. Show me
> 'i' of something.

Can you show me minus three bananas?

In article <sf8mfe$a9s$1@gioia.aioe.org>, <mickspud@downthefarm.com> wrote:

>2 is not an abstraction in the sense I can show you two of something. Show me
>'i' of something.

Show me 2,473,845,582,972,592,398,117,937,832,969,322,411 of something.

-- Richard

On 14/08/2021 19:17, Norman Wells wrote:

> Can you show me minus three bananas?

I can show you minus twenty hanging baskets. There's a maniac round
here, keeps stealing hanging baskets. Lots of people have him on CCTV,
sneaking about in a striped jersey (yes, really). The net is closing in...

Bill

On Sat, 14 Aug 2021 14:42:57 +0100, Roderick Stewart
<rjfs@escapetime.myzen.co.uk> wrote in
<62hfhgl9voufjhbb8fincll7c28rrbk8r6@4ax.com>:

>Numbers have numerical properties. Magnitudes, factors, multiples,
>relations to other numbers and so on. Geometrical shapes have
>numerical properties, such as minimum or maximum values for distances
>between shapes, packing densities and so on.
>
>Relative magnitudes of numbers in theoretical space determine the
>relative magnitudes of real objects in real space. If A is bigger than
>B, then a person who possesses A objects will have more of them than
>another person who only has B of them.

I think of that the other way round. People invented numbers and
relative magnitudes as a way to think about the real objects. The way
things work in the real world determines which formulae we use to reason
about the real world. There are plenty of other formulae but we tend not
to pay much attention to them unless we are mathematicians who consider
them to have some beauty (in the sense in which mathematicians use the
term).

>
>If a number is prime in theoretical space, then it will be impossible
>to make equal shares out of that number of real objects in real space.

That is a property of real space that we have chosen to model with
numbers so that we can deduce a property of the collection of real
objects by considering the numbers. The numbers do not create the
property, they are a model of it.

>
>The ways in which shapes can fit together in theoretical space
>determine how real physical shapes can fit together in real space.

We choose to describe shapes in theoretical space in order to be able to
manipulate the theoretical shapes in a way that predicts how the real
shapes will interact. The model does not control the reality, it is
chosen to be a model of it that is useful.
>
>The ways in which subatomic particles can fit together must be subject
>to the same rules of geometry as everything else in real space, so it
>may be that all of physical reality can only exist in particular
>forms, determined by the properties of numbers.

Once again, we are choosing models that give the best known predictions
of how physical things will interact.

Mathematicians have no trouble in creating models that do not reflect
the real world. For example. there is an old joke about the
mathematician's way of making toast - take it out of the toaster 10
seconds before it starts to smoke.

On Sat, 14 Aug 2021 15:08:52 -0000 (UTC), mickspud@downthefarm.com wrote
in <sf8ma4$7vn$1@gioia.aioe.org>:

>On Fri, 13 Aug 2021 21:14:23 +0100
>Owen Rees <orees@hotmail.com> wrote:

>>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.
>
>I never said otherwise. I simply said they're a hack. Hacks usually work but
>they're also ugly and often there's a better way of doing things.

In what way are complex numbers ugly? If you have a simpler way to
define the set of numbers needed to contain the zeros of all polynomials
with rational coefficients than the construction using i that has been
around for centuries then you should write it up - mathematicians love
simplicity and if you can produce something simpler than is currently
used then you could be a candidate for the Fields Medal.

Note also that from what I remember, engineers use complex numbers
because they make various formulae simpler so they are the better way to
handle that sort of work than the alternative formulae which are much
uglier and take a lot more work to process.

>
>>the relevant groups. Fractions are needed in order to have
>>multiplicative inverses for the integral non-zero numbers in the
>
>Fractions are needed to represent fractional parts of things. No need to get
>carried away with some convoluted explanation.
>

What do you mean by a fractional part of something? Do you expect to be
able to do arithmetic with fractions? If so, how does that work?

We may be taught various process in primary school for manipulating
fractions, but how do we know that those processes do what we are told
that they do?

On Sat, 14 Aug 2021 15:11:42 -0000 (UTC), mickspud@downthefarm.com wrote
in <sf8mfe$a9s$1@gioia.aioe.org>:

>On Sat, 14 Aug 2021 12:46:28 +0000 (UTC)
>richard@cogsci.ed.ac.uk (Richard Tobin) wrote:
>>In article <sf61dp$1rb2$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?
>>
>>>>i.
>>
>>>You're about 3 posts too late for that joke.
>>
>>It's not a joke. "i" is a name for the square root of -1, just as
>>"2" is a name for 1+1.
>>
>>2 and i are both abstractions, but for some reason you seem happy
>>with one but not the other.
>
>2 is not an abstraction in the sense I can show you two of something. Show me
>'i' of something.
>

2 is a much older abstraction but the idea that you can consider things
to be sufficiently like each other to be able to count them is still
creating a model then manipulating the model rather than the things
themselves.

Over the centuries, people have found a need to do more than just count
things so have added new abstractions to make it easier to reason about
them.

I am part of the way through reading "The Laws of Thought" by George
Boole. It is very heavy going as might be expected for a book written by
a Victorian clergyman, much more so than one mighr expect considering
the relative simplicity of Boolean algebra. It is fascinating to see the
early presentation of ideas that have since been reworked and presented
in a much simpler and easy to use form.

On Sat, 14 Aug 2021 15:10:48 -0000 (UTC), mickspud@downthefarm.com wrote
in <sf8mdo$9ka$1@gioia.aioe.org>:

>On Fri, 13 Aug 2021 21:17:11 +0100
>Owen Rees <orees@hotmail.com> wrote:
>>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?
>
>I can break a stick into 3 and physically show you a third. Can you show me i?
>No, you can't, because it doesn't exist. Your argument is like saying "Write
>down 3 without using the figure 3".

You may be able to break a stick into three smaller sticks but when you
show me one it will just be a stick smaller that the one you used to
have (and breaking it into three identical parts will be impossible for
a real stick).

I vaguely remember learning to divide whole numbers in primary school
before learning about fractions or at least how to do arithmetic with
them. If it did not divide exactly, there was a remainder.

You do not need the concept of "a third" in order to divide something
into three parts.

On 14/08/2021 21:43, Owen Rees wrote:

<snipped, but I agree with your argument>

> For example. there is an old joke about the
> mathematician's way of making toast - take it out of the toaster 10
> seconds before it starts to smoke.
>
:-)

That's better than the one I remember:
Q. How does a mathematician deal with constipation?
A. He works it out with a pencil.

Jim

On Sat, 14 Aug 2021 21:43:59 +0100, Owen Rees <orees@hotmail.com>
wrote:

>>Relative magnitudes of numbers in theoretical space determine the
>>relative magnitudes of real objects in real space. If A is bigger than
>>B, then a person who possesses A objects will have more of them than
>>another person who only has B of them.
>
>I think of that the other way round. People invented numbers and
>relative magnitudes as a way to think about the real objects. The way
>things work in the real world determines which formulae we use to reason
>about the real world.

No, you've got it the wrong way round. It's an easy trap to fall into,
but you're assuming that the order in which we think of things or
become aware of them is the same as the order in which they occur,
which is not necessarily so. Mathematics doesn't work the same way as
history. Apples don't fall out of trees as a consequence of Newton's
laws; it's the other way round - apples were falling out of trees long
before Newton was born, and the mathematical detail of how they fall
was just waiting for him to discover it.

Our view of the universe isn't the only one. Causality flows its own
way. The impossibility of equal shares from a prime number (to pick
just one of an infinite number of examples) existed as a true fact
long before the evolution of anything with the brains to realise it.
Mathematical truth is universally true, and is not made true by our
invention of symbols. Nobody invented or created any of the
mathematical constants; they were there before us, and we discovered
what they already were.

Rod.

In article <sf8mfe$a9s$1@gioia.aioe.org>, <mickspud@downthefarm.com>
wrote:
> >2 and i are both abstractions, but for some reason you seem happy with
> >one but not the other.

> 2 is not an abstraction in the sense I can show you two of something.

You can show '2' of 'something' you've defined in a way that allows that to
be the case.

When you say "Here are two polar bears" you are defning a class "polar
bears" that requires some aspects of each instance but ignores others to
suit your convenience.

Are 8 cells in the womb of a polar bear 'another one' polar bear? Depends
on what definition you find useful for a purpose.

So, yes, '2' is - in maths - an abstraction.

Jim

--
Please use the address on the audiomisc page if you wish to email me.
Electronics https://www.st-andrews.ac.uk/~www_pa/Scots_Guide/intro/electron.htm
biog http://jcgl.orpheusweb.co.uk/history/ups_and_downs.html
Audio Misc http://www.audiomisc.co.uk/index.html

On Sun, 15 Aug 2021 08:36:42 +0100, Roderick Stewart
<rjfs@escapetime.myzen.co.uk> wrote in
<eefhhgduub3lvt1aj5lapckd6d0mudbv50@4ax.com>:

>On Sat, 14 Aug 2021 21:43:59 +0100, Owen Rees <orees@hotmail.com>
>wrote:
>
>>>Relative magnitudes of numbers in theoretical space determine the
>>>relative magnitudes of real objects in real space. If A is bigger than
>>>B, then a person who possesses A objects will have more of them than
>>>another person who only has B of them.
>>
>>I think of that the other way round. People invented numbers and
>>relative magnitudes as a way to think about the real objects. The way
>>things work in the real world determines which formulae we use to reason
>>about the real world.
>
>No, you've got it the wrong way round. It's an easy trap to fall into,
>but you're assuming that the order in which we think of things or
>become aware of them is the same as the order in which they occur,
>which is not necessarily so. Mathematics doesn't work the same way as
>history. Apples don't fall out of trees as a consequence of Newton's
>laws; it's the other way round - apples were falling out of trees long
>before Newton was born, and the mathematical detail of how they fall
>was just waiting for him to discover it.

I think that some of that is what I was trying to say but did not
express as clearly as I could have. I think that we are agreeing that
the world works the way it does and would continue to do so if
mathematics did not exist. The existence of any sort of formulae that
let us predict how physical things will behave does not in itself
exercise any control over the real world. People choose the formulae
that are the best available fit to the world, we cannot cause the world
to conform to a formula we might find more convenient.

>
>Our view of the universe isn't the only one. Causality flows its own
>way. The impossibility of equal shares from a prime number (to pick
>just one of an infinite number of examples) existed as a true fact
>long before the evolution of anything with the brains to realise it.
>Mathematical truth is universally true, and is not made true by our
>invention of symbols. Nobody invented or created any of the
>mathematical constants; they were there before us, and we discovered
>what they already were.

There are interesting philosophical points about invention or discovery
there.

On Sat, 14 Aug 2021 19:17:20 +0100
Norman Wells <hex@unseen.ac.am> wrote:
>On 14/08/2021 16:11, mickspud@downthefarm.com wrote:
>> On Sat, 14 Aug 2021 12:46:28 +0000 (UTC)
>> richard@cogsci.ed.ac.uk (Richard Tobin) wrote:
>>> In article <sf61dp$1rb2$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?
>>>
>>>>> i.
>>>
>>>> You're about 3 posts too late for that joke.
>>>
>>> It's not a joke. "i" is a name for the square root of -1, just as
>>> "2" is a name for 1+1.
>>>
>>> 2 and i are both abstractions, but for some reason you seem happy
>>> with one but not the other.
>>
>> 2 is not an abstraction in the sense I can show you two of something. Show me
>
>> 'i' of something.
>
>Can you show me minus three bananas?

I can show you minus 3 feet - dig a hole.

On Sat, 14 Aug 2021 19:36:23 +0000 (UTC)
richard@cogsci.ed.ac.uk (Richard Tobin) wrote:
>In article <sf8mfe$a9s$1@gioia.aioe.org>, <mickspud@downthefarm.com> wrote:
>
>>2 is not an abstraction in the sense I can show you two of something. Show me
>>'i' of something.
>
>Show me 2,473,845,582,972,592,398,117,937,832,969,322,411 of something.

Go to a beach and start counting the grains of sand.

On Sat, 14 Aug 2021 22:03:38 +0100
Owen Rees <orees@hotmail.com> wrote:
>On Sat, 14 Aug 2021 15:08:52 -0000 (UTC), mickspud@downthefarm.com wrote
>in <sf8ma4$7vn$1@gioia.aioe.org>:
>
>>On Fri, 13 Aug 2021 21:14:23 +0100
>>Owen Rees <orees@hotmail.com> wrote:
>
>>>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.
>>
>>I never said otherwise. I simply said they're a hack. Hacks usually work but
>>they're also ugly and often there's a better way of doing things.
>
>In what way are complex numbers ugly? If you have a simpler way to

Using a value that cannot exist can hardly be said to be elegant.

>define the set of numbers needed to contain the zeros of all polynomials
>with rational coefficients than the construction using i that has been
>around for centuries then you should write it up - mathematicians love
>simplicity and if you can produce something simpler than is currently
>used then you could be a candidate for the Fields Medal.

As I've already said twice - though apparently being able to read properly is
optional on this group - the human brain probably isn't up to that task.

>>Fractions are needed to represent fractional parts of things. No need to get
>>carried away with some convoluted explanation.
>>
>
>What do you mean by a fractional part of something? Do you expect to be
>able to do arithmetic with fractions? If so, how does that work?

Break a stick in 2 - you have 2 halves. Is that too complex for you?
Pun intended.

On Sat, 14 Aug 2021 22:21:06 +0100
Owen Rees <orees@hotmail.com> wrote:
>On Sat, 14 Aug 2021 15:11:42 -0000 (UTC), mickspud@downthefarm.com wrote
>in <sf8mfe$a9s$1@gioia.aioe.org>:
>>2 is not an abstraction in the sense I can show you two of something. Show me
>>'i' of something.
>>
>
>2 is a much older abstraction but the idea that you can consider things
>to be sufficiently like each other to be able to count them is still
>creating a model then manipulating the model rather than the things
>themselves.

Having 2 of something is not an abstraction. The only abstract part about it
is what name you give it and what written figure you use to represent it with.
Even some animals can understand simple integers.

On Sat, 14 Aug 2021 22:38:16 +0100
Owen Rees <orees@hotmail.com> wrote:
>On Sat, 14 Aug 2021 15:10:48 -0000 (UTC), mickspud@downthefarm.com wrote
>in <sf8mdo$9ka$1@gioia.aioe.org>:
>
>>On Fri, 13 Aug 2021 21:17:11 +0100
>>Owen Rees <orees@hotmail.com> wrote:
>>>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?
>>
>>I can break a stick into 3 and physically show you a third. Can you show me i?
>
>>No, you can't, because it doesn't exist. Your argument is like saying "Write
>>down 3 without using the figure 3".
>
>You may be able to break a stick into three smaller sticks but when you
>show me one it will just be a stick smaller that the one you used to
>have (and breaking it into three identical parts will be impossible for
>a real stick).

And? Its still 1/3rd of the original stick.

On Sat, 14 Aug 2021 17:51:25 +0100
Jim Lesurf <noise@audiomisc.co.uk> wrote:
>In article <sf8mfe$a9s$1@gioia.aioe.org>, <mickspud@downthefarm.com>
>wrote:
>> >2 and i are both abstractions, but for some reason you seem happy with
>> >one but not the other.
>
>> 2 is not an abstraction in the sense I can show you two of something.
>
>You can show '2' of 'something' you've defined in a way that allows that to
>be the case.
>
>When you say "Here are two polar bears" you are defning a class "polar
>bears" that requires some aspects of each instance but ignores others to
>suit your convenience.

Bollocks. You can have a polar bear and a tin of beans, that would still be 2
objects. You're conflating the definition of discrete objects with counting
because you're trying to be clever.

In article <sfd62r$1d77$1@gioia.aioe.org>, <mickspud@downthefarm.com> wrote:

>>>2 is not an abstraction in the sense I can show you two of something. Show me
>>>'i' of something.

>>Show me 2,473,845,582,972,592,398,117,937,832,969,322,411 of something.

>Go to a beach and start counting the grains of sand.

There aren't that many.

There are estimated to be only 10^80 baryons in the universe. Do you
believe in numbers bigger than that?

-- Richard

In article <sfd69d$1g7m$1@gioia.aioe.org>, <mickspud@downthefarm.com> wrote:

>Having 2 of something is not an abstraction.

Of course it is. That you see something in common between 2 cows and
2 pigs requires abstracting away from the animals themselves. It's
just an abstraction you're familiar with.

>The only abstract part about it
>is what name you give it and what written figure you use to represent it with.
>Even some animals can understand simple integers.

The definition of "abstraction" is not "something that animals can't do".

-- Richard