Re: The modern mathematical concept of infinity is indefensible
FredJeffries
>
> Concepts are everywhere, but nowhere in nature.
> However they are in everbody's mind, aren't they?
>
Statements like this puzzle me.
Aren't people's minds part of nature?
Virgil
<ab951418-f69c-4aa0...@s9g2000prg.googlegroups.com>,
FredJeffries <FredJe...@gmail.com> wrote:
Certainly not in the same sense as their brains are.
Klaus Cammin
FredJeffries schrieb:
> Statements like this puzzle me.
Yeah, this was expressed oddly. ;-)
It was meant to mean 'outer reality' or just 'physical world without us'.
> Aren't people's minds part of nature?
Of course, but that wasn't my point, and it doesn't matter anyway.
Numbers are not part of the reality by the same token as trees are.
Viele Grüße
Klaus
Albrecht
Klaus Cammin schrieb:
"Tree" is a concept and alike no part of reality in your sense. Only
"this tree", "that tree" and "your christmas tree" are part of your
concept of reality.
Your concept of reality is poor.
Klaus Cammin
No, I just prefer to distinguish between that things and other things.
You're right, "tree" can be used as model for all trees. If there's a
specific tree, it satisfies our model of a tree (I'd call this
"dereferencing"). Another specific tree satisfies our model of a tree,
too, but this tree is another tree, in many aspects different.
Now do this with numbers. Is it consistent to asumme, that the specific
two in "two trees" is any different from the specific two in "two balls"?
No, hence there is no specific two, no thing that results from
dereferencing the concept of "two".
Viele Grüße
Klaus
Albrecht
Klaus Cammin schrieb:
> Albrecht schrieb:
> > Klaus Cammin schrieb:
> >> Hi Fred,
> >> FredJeffries schrieb:
> >>> Statements like this puzzle me.
> >> Yeah, this was expressed oddly. ;-)
> >> It was meant to mean 'outer reality' or just 'physical world without
> >> us'.
> >>> Aren't people's minds part of nature?
> >> Of course, but that wasn't my point, and it doesn't matter anyway.
> >> Numbers are not part of the reality by the same token as trees are.
> >
> > "Tree" is a concept and alike no part of reality in your sense. Only
> > "this tree", "that tree" and "your christmas tree" are part of your
> > concept of reality.
> >
> > Your concept of reality is poor.
>
> No, I just prefer to distinguish between that things and other things.
And how do you distinguish? By heart?
>
> You're right, "tree" can be used as model for all trees. If there's a
> specific tree, it satisfies our model of a tree (I'd call this
> "dereferencing"). Another specific tree satisfies our model of a tree,
> too, but this tree is another tree, in many aspects different.
>
> Now do this with numbers. Is it consistent to asumme, that the specific
> two in "two trees" is any different from the specific two in "two balls"?
> No, hence there is no specific two, no thing that results from
> dereferencing the concept of "two".
>
>
I don't claim that trees and numbers are same in any aspect. I do only
question about your aplomb to claim that natural numbers do not exist.
Do properties exist?
Virgil
<fbd3a2e7-8232-41ac...@35g2000pry.googlegroups.com>,
Albrecht <albs...@gmx.de> wrote:
However poor it might be, it is far richer than Albrecht's, which cannot
distinguish between the idea of a tree and the actuality of one.
Virgil
<1211f93f-ecc8-4410...@x6g2000pre.googlegroups.com>,
Albrecht <albs...@gmx.de> wrote:
> Klaus Cammin schrieb:
> > Albrecht schrieb:
> > > Klaus Cammin schrieb:
> > >> Hi Fred,
> > >> FredJeffries schrieb:
> > >>> Statements like this puzzle me.
> > >> Yeah, this was expressed oddly. ;-)
> > >> It was meant to mean 'outer reality' or just 'physical world without
> > >> us'.
> > >>> Aren't people's minds part of nature?
> > >> Of course, but that wasn't my point, and it doesn't matter anyway.
> > >> Numbers are not part of the reality by the same token as trees are.
> > >
> > > "Tree" is a concept and alike no part of reality in your sense. Only
> > > "this tree", "that tree" and "your christmas tree" are part of your
> > > concept of reality.
> > >
> > > Your concept of reality is poor.
> >
> > No, I just prefer to distinguish between that things and other things.
>
> And how do you distinguish? By heart?
By the five senses between physical objects, by the intellect, for
mental ones, and both for mixed sets.
>
> I don't claim that trees and numbers are same in any aspect. I do only
> question about your aplomb to claim that natural numbers do not exist.
Numbers do not have an existence detectable by any of the five senses.
But the number of objects of a particular type may be.
>
> Do properties exist?
To the mind, yes. As physical objects themselves, no.
And separate physical objects can be separated by their properties.
Mitch Harris
> Albrecht <albst...@gmx.de> wrote:
>
> > I don't claim that trees and numbers are same in any aspect. I do only
> > question about your aplomb to claim that natural numbers do not exist.
>
> Numbers do not have an existence detectable by any of the five senses.
Actually, yes, they d, at least small ones. Look up 'subitizing', read
'The Number Sense' by Stephen Dehaene. and there was a paper in the
eighties in Science that (supposedly) determined that cats can
recognize the number 4 immediately.
All this means though is that (oversimplifying) there are biological
mechanisms for quickly recognizing small numbers. (just like there are
biological mechanisms for recognizing 'red' or drum patterns.)
Mitch
Virgil
<8420dfa2-d5eb-4d5d...@v5g2000prm.googlegroups.com>,
Mitch Harris <mah...@gmail.com> wrote:
> On Jan 29, 3:58 pm, Virgil <Vir...@gmale.com> wrote:
> > Albrecht <albst...@gmx.de> wrote:
> >
> > > I don't claim that trees and numbers are same in any aspect. I do only
> > > question about your aplomb to claim that natural numbers do not exist.
> >
> > Numbers do not have an existence detectable by any of the five senses.
>
> Actually, yes, they d, at least small ones. Look up 'subitizing', read
> 'The Number Sense' by Stephen Dehaene. and there was a paper in the
> eighties in Science that (supposedly) determined that cats can
> recognize the number 4 immediately.
Is it the "number" 4 or the property of four-ness that cats recognize?
To me they are distinct.
>
> All this means though is that (oversimplifying) there are biological
> mechanisms for quickly recognizing small numbers.
I grant mechanisms for fast recognition of and distinguishing between
differing small frequencies, but deny that distinguishing threeness from
fourness is the same as distinguishing the number 3 from the number 4 as
abstract ideas.
Mitch Harris
> Mitch Harris <maha...@gmail.com> wrote:
> > On Jan 29, 3:58 pm, Virgil <Vir...@gmale.com> wrote:
>
> > > Numbers do not have an existence detectable by any of the five senses.
>
> > Actually, yes, they d, at least small ones. Look up 'subitizing', read
> > 'The Number Sense' by Stephen Dehaene. and there was a paper in the
> > eighties in Science that (supposedly) determined that cats can
> > recognize the number 4 immediately.
>
> Is it the "number" 4 or the property of four-ness that cats recognize?
I couldn't say (poor memory). Something with the word 'four' was
involved. And cats.
> To me they are distinct.
Can you elaborate in what way you find them distinct?
> > All this means though is that (oversimplifying) there are biological
> > mechanisms for quickly recognizing small numbers.
>
> I grant mechanisms for fast recognition of and distinguishing between
> differing small frequencies, but deny that distinguishing threeness from
> fourness is the same as distinguishing the number 3 from the number 4 as
> abstract ideas.
OK, but can you be more specific what you find that distinction to be?
Mitch
Virgil
<9ec0ca98-616c-410d...@u13g2000yqg.googlegroups.com>,
Mitch Harris <mah...@gmail.com> wrote:
> On Jan 29, 9:25 pm, Virgil <Vir...@gmale.com> wrote:
> > Mitch Harris <maha...@gmail.com> wrote:
> > > On Jan 29, 3:58 pm, Virgil <Vir...@gmale.com> wrote:
> >
> > > > Numbers do not have an existence detectable by any of the five senses.
> >
> > > Actually, yes, they d, at least small ones. Look up 'subitizing', read
> > > 'The Number Sense' by Stephen Dehaene. and there was a paper in the
> > > eighties in Science that (supposedly) determined that cats can
> > > recognize the number 4 immediately.
> >
> > Is it the "number" 4 or the property of four-ness that cats recognize?
>
> I couldn't say (poor memory). Something with the word 'four' was
> involved. And cats.
>
>
> > To me they are distinct.
>
> Can you elaborate in what way you find them distinct?
The sort of fourness that cats, as well as people, recognize
instinctively does not add or multiply well, as neither they nor we
appear instinctively to recognize either eightness nor sixteenness.
Mitch Harris
> In article
> <9ec0ca98-616c-410d-836b-debb80220...@u13g2000yqg.googlegroups.com>,
> Mitch Harris <maha...@gmail.com> wrote:
>
>
>
> > On Jan 29, 9:25 pm, Virgil <Vir...@gmale.com> wrote:
> > > Mitch Harris <maha...@gmail.com> wrote:
> > > > On Jan 29, 3:58 pm, Virgil <Vir...@gmale.com> wrote:
>
> > > > > Numbers do not have an existence detectable by any of the five senses.
>
> > > > Actually, yes, they d, at least small ones. Look up 'subitizing', read
> > > > 'The Number Sense' by Stephen Dehaene. and there was a paper in the
> > > > eighties in Science that (supposedly) determined that cats can
> > > > recognize the number 4 immediately.
>
> > > Is it the "number" 4 or the property of four-ness that cats recognize?
>
> > I couldn't say (poor memory). Something with the word 'four' was
> > involved. And cats.
>
> > > To me they are distinct.
>
> > Can you elaborate in what way you find them distinct?
>
> The sort of fourness that cats, as well as people, recognize
> instinctively does not add or multiply well, as neither they nor we
> appear instinctively to recognize either eightness nor sixteenness.
I don't get it. How does that provide a distinction between the
'number' 4 and the 'property' four-ness?
So the cats (or necessarily people either) aren't great at
manipulating their concept of (something to do with) 4. I just think
the understanding of the property or number is just not as deep.
What's the distinction?
Mitch
Virgil
<9aded5bb-f4d7-4794...@k36g2000pri.googlegroups.com>,
Mitch Harris <mah...@gmail.com> wrote:
The "property of fourness" as recognized by cats does not include being
a perfect square or a composite number or a power of two, etc., or other
properties of 4 that are dependent on it being a member of a number
system.
Mitch Harris
OK. I see your distinction. I had trouble distinguishing because there
was nothing in the locutions 'number 4' or 'property of fourness' that
gave any hint either way.
Your explanation of 'property of fourness' applies just as well to
most people, and I'm guessing is really a restriction to the simple
unadorned idea of the ability count exactly 4 things (which -is-
dependent on 4 being the member of a number system).
As interesting as this distinction is, I don't think that anything
that we've been trying to convince Albrecht and Han of has anything in
it beyond just plain old counting (is that 'property of fourness' or
'number 4'?).
Mitch
Virgil
<fcd6b3ac-22a1-42f9...@f20g2000yqg.googlegroups.com>,
Mitch Harris <mah...@gmail.com> wrote:
Convincing Han of anything he does not already believe is difficult.
Convincing Albrecht of anything he does not already believe is
impossible.
FredJeffries
> Hi Fred,
>
> FredJeffries schrieb:
>
> > Statements like this puzzle me.
>
> Yeah, this was expressed oddly. ;-)
> It was meant to mean 'outer reality' or just 'physical world without us'.
But, for better or worse, we ARE part of the physical world. Until we
meet an alien intelligence, there is no understanding of the 'physical
world' that doesn't involve us because we are the ones doing the
understanding. We can't understand the 'physical world' without
understanding the 'us' that is understanding the physical world.
Human minds are part of/created by/... human beings who are part of
the physical world. Infinite sets and the 47 other varieties of
mathematical infinity are creations of/discoveries of human minds.
MY crank thesis: you ain't gonna' be able to understand human minds if
you can't understand how they create/discover things. Therefore an
understanding of Cantorian Set Theory in particular is indispensable
in the study of Psychology.
Klaus Cammin
FredJeffries schrieb:
> [...] we ARE part of the physical world. [...]
I guess we don't have a disagreement here. Apparently I'm quite incapable
or even unwilling to avoid the pitfalls when using a wider or more narrow
notion of 'nature' or 'us'. If someone objects to that, I can nothing but
agree. 'us' may include 'any creature with a brain capable of understanding
numbers', but the phrases sound weird with that term.
My basic view is: numbers (and mathematical objects) are made by brains to
let the creature cope with the world. For me it's a little too far driven
if you place that process out of a brain into the exterior world (exterior
to the creature in question!) and let them have an independent existence.
But apparently Albrecht wants to have that. For him e.g. natural numbers
are something like natural laws, as prior discussions revealed, and his
criticism to the contemporary notion of infinity relies on that, i.e. that
in his view transfinite numbers are something "unnatural" and violates
basic rules. Of course your objection applies here, too.
And there's no clue that anything of it is true. Even if there were such
laws, as proven a math independent of them is possible. Our contemporary
understanding of natural numbers does in no way harm any natural
requirement.
> Human minds are part of/created by/... human beings who are part of
> the physical world. Infinite sets and the 47 other varieties of
47? ;-)
> mathematical infinity are creations of/discoveries of human minds.
Yeah, normally mathematicians avoid the question, whether mathematical
objects "are out there". It's not very important in their realm. Your use
of "creations/disscoveries" interchangably point to that.
> MY crank thesis: you ain't gonna' be able to understand human minds if
> you can't understand how they create/discover things. Therefore an
> understanding of Cantorian Set Theory in particular is indispensable
> in the study of Psychology.
I like the idea of psychologicans understanding Cantor. ;-)
Better than reasoning about big boops bumping over the hill ...
Viele Grüße
Klaus
Aatu Koskensilta
> MY crank thesis: you ain't gonna' be able to understand human minds if
> you can't understand how they create/discover things. Therefore an
> understanding of Cantorian Set Theory in particular is indispensable
> in the study of Psychology.
Is the study of Captain Planet and the Planeteers equally
indispensable in the study of Psychology?
--
Aatu Koskensilta (aatu.kos...@uta.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
FredJeffries
> FredJeffries <FredJeffr...@gmail.com> writes:
> > MY crank thesis: you ain't gonna' be able to understand human minds if
> > you can't understand how they create/discover things. Therefore an
> > understanding of Cantorian Set Theory in particular is indispensable
> > in the study of Psychology.
>
> Is the study of Captain Planet and the Planeteers equally
> indispensable in the study of Psychology?
>
The study of how the mind creates Captain Planet and suspends
disbelief enough to enjoy Captain Planet certainly is. I can imagine
that some study of Captain Planet Himself is necessary for the latter.
Like I said, it's a crank theory, the best anti-anti-Cantorian one I
can come up with at the moment.
Aatu Koskensilta
> My basic view is: numbers (and mathematical objects) are made by
> brains to let the creature cope with the world. For me it's a little
> too far driven if you place that process out of a brain into the
> exterior world (exterior to the creature in question!) and let them
> have an independent existence.
Paintings, computers, books (as physical objects), stone axes, houses,
hamburgers, and so on are made by us in a perfectly clear sense. Talk
of abstract objects of mathematics, jurisprudence, art, computer
science, as having been made by us on the other hand is necessarily
metaphorical. There is nothing wrong in this metaphor, but if
stretched it breaks down. For example, we can't sensibly ask of a real
number when it was made or by whom, in contrast to a painting, a
hamburger or a house; we can't ask where the aleph-tenth level of the
constructible universe resides (and the purported answer "In the
brains of set-theorists." is either complete non-sense or must again
be interpreted metaphorically); and so on. The metaphor of various
abstract objects as something we make is very suggestive and appealing
for all sorts of reasons, but of not much help in a context such as
that of the present (sub)thread. What we need is instead a
non-metaphorical explanation and spelling our of the idea(s)
underlying this metaphor.
Aatu Koskensilta
> The study of how the mind creates Captain Planet and suspends
> disbelief enough to enjoy Captain Planet certainly is. I can imagine
> that some study of Captain Planet Himself is necessary for the
> latter.
Following this sober-headed line of thought we are then led to
conclude that some study of pretty much everything and anything is
indispensable in the study of psychology.
FredJeffries
> Hi Fred,
>
> FredJeffries schrieb:
>
> > [...] we ARE part of the physical world. [...]
>
> I guess we don't have a disagreement here. Apparently I'm quite incapable
> or even unwilling to avoid the pitfalls when using a wider or more narrow
> notion of 'nature' or 'us'. If someone objects to that, I can nothing but
> agree. 'us' may include 'any creature with a brain capable of understanding
> numbers', but the phrases sound weird with that term.
>
> My basic view is: numbers (and mathematical objects) are made by brains to
> let the creature cope with the world. For me it's a little too far driven
> if you place that process out of a brain into the exterior world (exterior
> to the creature in question!) and let them have an independent existence.
I do not wish to place the process outside of the brain. But the brain
itself and its processes are part of that exterior world and cannot
exist apart from that exterior world.
As an anti-anti-Cantorian crank I object to the (dualistic?) idea that
concepts and ideas, being (mere) products of our minds, do not have an
existence in the real world.
FredJeffries
> FredJeffries <FredJeffr...@gmail.com> writes:
> > The study of how the mind creates Captain Planet and suspends
> > disbelief enough to enjoy Captain Planet certainly is. I can imagine
> > that some study of Captain Planet Himself is necessary for the
> > latter.
>
> Following this sober-headed line of thought we are then led to
> conclude that some study of pretty much everything and anything is
> indispensable in the study of psychology.
>
Having never studied psychology I have the perfect credentials to
assert what is indispensable, as opposed to our anti-Cantorian crank
friends who seem to have studied at least some mathematics.
herbzet
FredJeffries wrote:
> Aatu Koskensilta wrote:
> > FredJeffries writes:
> > > MY crank thesis: you ain't gonna' be able to understand human minds if
> > > you can't understand how they create/discover things. Therefore an
> > > understanding of Cantorian Set Theory in particular is indispensable
> > > in the study of Psychology.
> >
> > Is the study of Captain Planet and the Planeteers equally
> > indispensable in the study of Psychology?
> >
>
> The study of how the mind creates Captain Planet and suspends
> disbelief enough to enjoy Captain Planet certainly is. I can imagine
> that some study of Captain Planet Himself is necessary for the latter.
>
> Like I said, it's a crank theory, the best anti-anti-Cantorian one I
> can come up with at the moment.
Good work!
It's always exciting to be around at the inception of a new field of study.
Anti-crank-crankery -- consider the possibilites!
--
hz
David R Tribble
>> The study of how the mind creates Captain Planet and suspends
>> disbelief enough to enjoy Captain Planet certainly is. I can imagine
>> that some study of Captain Planet Himself is necessary for the
>> latter.
>
Aatu Koskensilta wrote:
>> Following this sober-headed line of thought we are then led to
>> conclude that some study of pretty much everything and anything is
>> indispensable in the study of psychology.
>
More correctly, what is indespenable is the study of how human
minds create and manipulate abstractions (metaphors) for pretty
much everything and anything.
Once we discover how a person conceives of a cat, for example,
we pretty much know how he also conceives of a dog, a horse,
and perhaps all animals in general.
David Formosa (aka ? the Platypus)
> In article
><ab951418-f69c-4aa0...@s9g2000prg.googlegroups.com>,
> FredJeffries <FredJe...@gmail.com> wrote:
>> Aren't people's minds part of nature?
>
> Certainly not in the same sense as their brains are.
Descartes thought so. Though other thinkers have had diffrent ideas.
FredJeffries
>
> Paintings, computers, books (as physical objects), stone axes, houses,
> hamburgers, and so on are made by us in a perfectly clear sense.
Thank you for giving me an idea for my Second anti-anti-Cantorian
cranks' proof of the reality of the actual infinite:
The artistic technique of Perspective (see for instance
http://www.getty.edu/art/exhibitions/geometry/images/linear_perspective.jpg
http://www.mcs.csuhayward.edu/~malek/Illusions/RedBlue/Vieux/Raphael/3draphael3.jpg
http://www.homeschoolarts.com/perspective.htm )
inspired the development of the mathematics of Projective Geometry. We
all know that projective geometry uses points and lines at infinity.
It has been conceded above that paintings are physical objects.
Therefore the points and lines at infinity involved in the creation of
these paintings have a real existence and are indispensable to the
creation of the great works of art.
Please note: Myself having absolutely no artistic abilities I have the
perfect qualifications to pronounce on what is indispensable to the
creation of works of art.
Albrecht
FredJeffries schrieb:
> On Feb 2, 10:18 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
> >
> > Paintings, computers, books (as physical objects), stone axes, houses,
> > hamburgers, and so on are made by us in a perfectly clear sense.
>
> Thank you for giving me an idea for my Second anti-anti-Cantorian
> cranks' proof of the reality of the actual infinite:
>
> The artistic technique of Perspective (see for instance
>
> http://www.getty.edu/art/exhibitions/geometry/images/linear_perspective.jpg
>
> http://www.mcs.csuhayward.edu/~malek/Illusions/RedBlue/Vieux/Raphael/3draphael3.jpg
>
> http://www.homeschoolarts.com/perspective.htm )
>
> inspired the development of the mathematics of Projective Geometry. We
> all know that projective geometry uses points and lines at infinity.
> It has been conceded above that paintings are physical objects.
> Therefore the points and lines at infinity involved in the creation of
> these paintings have a real existence and are indispensable to the
> creation of the great works of art.
>
This kind of concept of infinity doesn't harm anything. There is no
problem with that. Nobody (in any case I) argues against that use of
infinity.
Alike the use of infinity in analysis, the limit doesn't make any
problem.
Try to understand the different concepts of potential and actual
infinity.
Best regards
Albrecht S. Storz
Virgil
<619034f4-a691-4828...@m40g2000yqh.googlegroups.com>,
Albrecht <albs...@gmx.de> wrote:
We do a good deal better than Storz does.
At least we manage to avoid the self-contradiction of potentially
infinite sets.
Mitch Harris
> FredJeffries schrieb:
> > On Feb 2, 10:18 am, Aatu Koskensilta <aatu.koskensi...@uta.fi> wrote:
>
> > We all know that projective geometry uses points and lines at infinity.
> > It has been conceded above that paintings are physical objects.
> > Therefore the points and lines at infinity involved in the creation of
> > these paintings have a real existence and are indispensable to the
> > creation of the great works of art.
>
> This kind of concept of infinity doesn't harm anything. There is no
> problem with that. Nobody (in any case I) argues against that use of
> infinity.
Well, I won't argue against it, but I will say that I find aleph_0
quite a bit less 'magical' than the infinity in analysis.
You're OK with 'lim x->oo 1/x' and instantaneous velocity but not the
number of numbers?
> Alike the use of infinity in analysis, the limit doesn't make any
> problem.
>
> Try to understand the different concepts of potential and actual
> infinity.
So the infinity of calculus is which one of these, potetntial or
actual? Or is it another way entirely?
Mitch
Michael Stemper
>Try to understand the different concepts of potential and actual
>infinity.
Please provide mathematical definitions of the terms:
- "potential infinity"
- "actual infinity"
--
Michael F. Stemper
#include <Standard_Disclaimer>
Indians scattered on dawn's highway bleeding;
Ghosts crowd the young child's fragile eggshell mind.
MoeBlee
wrote:
> In article <619034f4-a691-4828-9e9a-16427e395...@m40g2000yqh.googlegroups.com>, Albrecht <albst...@gmx.de> writes:
>
> >Try to understand the different concepts of potential and actual
> >infinity.
>
> Please provide mathematical definitions of the terms:
> - "potential infinity"
> - "actual infinity"
Albrecht has no idea what a mathematical definition is. You might as
well ask a two-year old for an explanation of international monetary
exchange rates.
MoeBlee
Albrecht
Michael Stemper schrieb:
> In article <619034f4-a691-4828...@m40g2000yqh.googlegroups.com>, Albrecht <albs...@gmx.de> writes:
>
> >Try to understand the different concepts of potential and actual
> >infinity.
>
> Please provide mathematical definitions of the terms:
> - "potential infinity"
> - "actual infinity"
>
> --
You are free to inform yourself in internet and books about that
issue. I'm not the one who invents this concepts nor I have to define
them.
David Formosa (aka ? the Platypus)
> Michael Stemper schrieb:
[...]
>> Please provide mathematical definitions of the terms:
>> - "potential infinity"
>> - "actual infinity"
> You are free to inform yourself in internet and books about that
> issue. I'm not the one who invents this concepts nor I have to define
> them.
Your the one who is using it, so how do we know the defintion that you using
corrsponds to the one that a random person on the internet has posted?
Virgil
Actually, one can be reasonably sure the it isn't the same as any
standard definition.