Frege's view of numbers as objects
Tom Giles
2nd level concepts. Does his view point make sense? Should we really think
of numbers as objects rather than properties?
How exactly does frege's view relate to Peano axioms? Does it make sense to
apply such criteria to Frege's writings?
Neil W Rickert
>For Frege numbers were "self-subsistent objects" but were also related to
>2nd level concepts. Does his view point make sense? Should we really think
>of numbers as objects rather than properties?
As a mathematician, I certainly think of them as objects. Thinking of them
as properties would not fit ordinary mathematical discussion. I will allow
that they might be fictitious objects.
Owl
news:8gk4un$e...@ux.cs.niu.edu...
I suspect Tom had in mind something like treating a number as a property
of a set. For instance, you could have a 2-membered set, a 7-membered
set, and so on. Because there is a property of being 2-membered, doesn't
necessarily mean there must be an entity "2".
Peter H.M. Brooks
properties. If you consider that for any delta d>0, there are infinitely
many numbers between 2 and 2+d, what property would any of these numbers
correspond to?
--
Peter H.M. Brooks
"My choice of colours does not rest on any scientific theory; it is
based on observation, on feeling, on the experience of
my sensibility." Henri Matisse in Notes of a Painter, published in La
Grande Revue, 1908
http://www.psyche.demon.co.uk
Sent via Deja.com http://www.deja.com/
Before you buy.
Owen Holden
Neil W Rickert <ricke...@cs.niu.edu> wrote in message
news:8gk4un$e...@ux.cs.niu.edu...
> "Tom Giles" <thomas...@st-annes.ox.ac.uk> writes:
>
> >For Frege numbers were "self-subsistent objects" but were also related to
> >2nd level concepts. Does his view point make sense? Should we really
think
> >of numbers as objects rather than properties?
>
> As a mathematician, I certainly think of them as objects. Thinking of
them
> as properties would not fit ordinary mathematical discussion. I will
allow
> that they might be fictitious objects.
Why cant' we think of properties, predicates, as objects?
Carnap seems to say numbers are properties of properties.
Russell seems to say numbers are classes of classes, convenient fictions.
Quine seems to say numbers are sets of sets, abstract objects.
If our ontology begins with a description of what 'objects' are,
why can't we include; empirical (concrete), mathematical (abstract),
propositional (the true, the false), predicative (the contradictory ,
the universal, etc.)?
There seems to be an 'isomorphic' (if that is the right word) identiy
between; 0, null, false, contradiction, no-thing, etc.
even though these things are of different types.
Owen
Peter H.M. Brooks
"Owen Holden" <oho...@idirect.com> wrote:
>
>
> If our ontology begins with a description of what 'objects' are,
> why can't we include; empirical (concrete), mathematical (abstract),
> propositional (the true, the false), predicative (the contradictory ,
> the universal, etc.)?
>
'object'. What does a raven have in common with the number 2? Why not
just use your construction and say that you have 'objects', mathematical
entities, propositions etc.?
>
> There seems to be an 'isomorphic' (if that is the right word) identiy
> between; 0, null, false, contradiction, no-thing, etc.
> even though these things are of different types.
>
number of things, often the 'space' to show that there is neither a '0'
or a '1' in electronic transmission. 'false' is the state that a
proposition can be in, or the property of a proposition. A contradiction
is just that and has nothing to do with nothing - p & !p is not the same
as nothing at all, in fact since you can derive anything from this, you
could argue that it is everything. 'no-thing' is a reference to, for
example, a box that doesn't have an object in it.
So, I would say that they are all quite different things.
--
Peter H.M. Brooks
"Political language is designed to make lies sound truthful and murder
respectable,
and to give an appearance of solidity to pure wind."
George Orwell "Politics and the English Language"
Chris Menzel
> "Tom Giles" <thomas...@st-annes.ox.ac.uk> writes:
>
> >For Frege numbers were "self-subsistent objects" but were also related to
> >2nd level concepts. Does his view point make sense? Should we really think
> >of numbers as objects rather than properties?
>
> As a mathematician, I certainly think of them as objects. Thinking of them
> as properties would not fit ordinary mathematical discussion. I will allow
> that they might be fictitious objects.
Who sez properties can't be thought of as objects?
Chris Menzel
Neil W Rickert
>Neil W Rickert <ricke...@cs.niu.edu> wrote in message
>news:8gk4un$e...@ux.cs.niu.edu...
>> >For Frege numbers were "self-subsistent objects" but were also related
>to
>> >2nd level concepts. Does his view point make sense? Should we really
>think
>> >of numbers as objects rather than properties?
>> As a mathematician, I certainly think of them as objects. Thinking of
>them
>> as properties would not fit ordinary mathematical discussion. I will
>allow
>> that they might be fictitious objects.
>I suspect Tom had in mind something like treating a number as a property
>of a set. For instance, you could have a 2-membered set, a 7-membered
>set, and so on. Because there is a property of being 2-membered, doesn't
>necessarily mean there must be an entity "2".
However, that does not work very well. To a mathematician, a set of
numbers is one of the primary examples of a set.
Neil W Rickert
>On 25 May 2000 16:11:51 -0500, Neil W Rickert <ricke...@cs.niu.edu> said:
>> "Tom Giles" <thomas...@st-annes.ox.ac.uk> writes:
>> >For Frege numbers were "self-subsistent objects" but were also related to
>> >2nd level concepts. Does his view point make sense? Should we really think
>> >of numbers as objects rather than properties?
>> As a mathematician, I certainly think of them as objects. Thinking of them
>> as properties would not fit ordinary mathematical discussion. I will allow
>> that they might be fictitious objects.
>Who sez properties can't be thought of as objects?
If numbers are properties, what are they properties of?
A measurement is a property of whatever is measured. A count is
a property of whatever is counted. And yes, I agree that we can
treat these properties as objects in their own right.
But a number is neither a measurement nor a count. They are objects
we have created by abstraction from measurements and counts. They
are ideal objects that we can study, so as to learn useful relations
that would apply to measurements and counts.
If you want to say that numbers are fictitious objects, or convenient
fictions, I would have no problem with that. But to say that numbers
are properties is to confuse numbers with counts or measurements.
Neil W Rickert
>Neil W Rickert <ricke...@cs.niu.edu> wrote in message
>news:8gk4un$e...@ux.cs.niu.edu...
>> >For Frege numbers were "self-subsistent objects" but were also related to
>> >2nd level concepts. Does his view point make sense? Should we really
>think
>> >of numbers as objects rather than properties?
>> As a mathematician, I certainly think of them as objects. Thinking of
>them
>> as properties would not fit ordinary mathematical discussion. I will
>allow
>> that they might be fictitious objects.
>Why cant' we think of properties, predicates, as objects?
I presume you can.
But if you want to say that numbers are properties, you have to first
have other things for them to be properties of. What are those other
things?
>Carnap seems to say numbers are properties of properties.
>Russell seems to say numbers are classes of classes, convenient fictions.
>Quine seems to say numbers are sets of sets, abstract objects.
I will take "convenient fictions" as the best of the above.
>If our ontology begins with a description of what 'objects' are,
But it doesn't.
>why can't we include; empirical (concrete), mathematical (abstract),
>propositional (the true, the false), predicative (the contradictory ,
>the universal, etc.)?
>There seems to be an 'isomorphic' (if that is the right word) identiy
>between; 0, null, false, contradiction, no-thing, etc.
>even though these things are of different types.
I would prefer to say that we often find it convenient to identify
these. But at other times we prefer to treat them as distinct.
Owen Holden
> cme...@philebus.tamu.edu (Chris Menzel) writes:
>
> >On 25 May 2000 16:11:51 -0500, Neil W Rickert <ricke...@cs.niu.edu>
said:
>
> >> >For Frege numbers were "self-subsistent objects" but were also related
to
> >> >2nd level concepts. Does his view point make sense? Should we really
think
> >> >of numbers as objects rather than properties?
>
> >> As a mathematician, I certainly think of them as objects. Thinking of
them
> >> as properties would not fit ordinary mathematical discussion. I will
allow
> >> that they might be fictitious objects.
>
>
> If numbers are properties, what are they properties of?
Things, of course .
> A measurement is a property of whatever is measured. A count is
> a property of whatever is counted. And yes, I agree that we can
> treat these properties as objects in their own right.
>
> But a number is neither a measurement nor a count. They are objects
> we have created by abstraction from measurements and counts.
What theory of numbers does this?
> They
> are ideal objects that we can study, so as to learn useful relations
> that would apply to measurements and counts.
>
> If you want to say that numbers are fictitious objects, or convenient
> fictions, I would have no problem with that. But to say that numbers
> are properties is to confuse numbers with counts or measurements.
Do you think you have made yourself clear ??
What is this talk of counts and measurements?
Carnap clearly states that numbers are properties of properties.
Do you think Carnap is confused here?
Owen
Neil W Rickert
>Neil W Rickert <ricke...@cs.niu.edu> wrote in message
>news:8gm42l$i...@ux.cs.niu.edu...
>> cme...@philebus.tamu.edu (Chris Menzel) writes:
>> >Who sez properties can't be thought of as objects?
>> If numbers are properties, what are they properties of?
>Things, of course .
I can think of things which have numerical properties. I cannot
think of anything that has a number as a property.
>Carnap clearly states that numbers are properties of properties.
>Do you think Carnap is confused here?
Yes. But he has a lot of company in that.
Neil W Rickert
>Neil W Rickert wrote:
>> I can think of things which have numerical properties. I cannot
>> think of anything that has a number as a property.
>>
>Neil...
>Aristotle would turn over in his grave. The sought after property
>category he would call 'quantity'. Things have quantity.
But a number is not a quantity.
LindaGee
Neil W Rickert wrote in message <8gnaj3$2...@ux.cs.niu.edu>...
That's right a number is a reference or referral to a quantity. Still, I
would say that in the case of a referent that is being used as an
abstracting device and therefore is stripped to the minimum of property
attritubes, which would be just the one (its quantity), that it and its only
property are identical with each other. So the referent for a number is
identical to it's one and only property attributed to it. The identifiyer
or number always maintains the property attributed to that referenced
numbers only property (quantity). That is why both mathematics and music
are such wonderful devices compared to all of the gray areas that exist
within other forms of communication. At least in my opinion. Still, I am a
poet...and try to treat my words and wording with as much precision as can
possibly be obtained. Probably because I have such an appreciation about
this aspect of the maths....of itself....
Linda
Sci.Phil.Meta.
Owleye
Neil W Rickert wrote:
> I can think of things which have numerical properties. I cannot
> think of anything that has a number as a property.
>
Neil...
Aristotle would turn over in his grave. The sought after property
category he would call 'quantity'. Things have quantity.
I think it might be better to say, for example, that many animals have
the property of "two of something." They are bi-polar in many
attributes. Perhaps you don't agree with this?
owleye
caveat...@my-deja.com
Neil W Rickert <ricke...@cs.niu.edu> wrote:
> Owleye <owl...@earthlink.net> writes:
> >Neil W Rickert wrote:
>
> >> I can think of things which have numerical properties. I cannot
> >> think of anything that has a number as a property.
> >>
>
> >Neil...
>
> >Aristotle would turn over in his grave. The sought after property
> >category he would call 'quantity'. Things have quantity.
>
I agree with Mr. Rickert. There was a similar controversy among the
Medieval scholastics. It was suggested that a number was an *ens
rationis*, or a "being of reason," i.e. a convenient fiction. It was
claimed that numbers existed as abstractions in the mind of man but had
no "substantial" existence, meaning there was no substrate in
which something like Four-ness could inhere, since its existence and
definition would then not be *per se*, but rather dependent upon the co-
existence of three other entities (that all happened to share or
participate in some kind of quality or predication). It was posited
that numbers were modal states that were conceived by means of the
percipient's rational faculty.
Other "beings of reason" included certain Euclidean definitions
like: "a point is that which has no parts. A line is breadthless
length." Such things could not exist in the physical world, but could
be discussed as ideas. And then there were "impossible" notions like
square circles, which were inherently contradictory.
As for the previous suggestion that "null", "0", and "false", are, for
all intents and purposes *isomorphic*, I would just like to say that as
a Java programmer I personally found it humourous.
Cheers!
caveat_lector
Owleye
Neil W Rickert wrote:
> Owleye <owl...@earthlink.net> writes:
> >Neil W Rickert wrote:
>
> >> I can think of things which have numerical properties. I cannot
> >> think of anything that has a number as a property.
> >>
>
> >Neil...
>
> >Aristotle would turn over in his grave. The sought after property
> >category he would call 'quantity'. Things have quantity.
>
> But a number is not a quantity.
Neil...
So what is a quantity such that numbers aren't among them?
owleye
Neil W Rickert
>> >Neil...
>Neil...
A quantity is a measurement. We use numbers to represent
quantities. But the numbers themselves are not quantities.
The prime number theorem is not about quantities.
Seth Russell
> A quantity is a measurement. We use numbers to represent
> quantities. But the numbers themselves are not quantities.
So Archimedes Plutonium was correct after all, natural
numbers come first and integers should be derived
from them, not the other way around like mathematicians
usually do.
Seth Russell
http://robustai.net/ai/word_of_emouth.htm
Click on the button ... see if you can catch me live!
http://robustai.net/JournalOfMyLife/users/SethRussell.html
Http://RobustAi.net/Ai/Conjecture.htm
Owleye
Neil W Rickert wrote:
> A quantity is a measurement. We use numbers to represent
> quantities. But the numbers themselves are not quantities.
>
> The prime number theorem is not about quantities.
I think it is wrong to say that a quantity is a measurement. It is
better to say that measurements are quantitative (when they are).
In any case, what do you mean by 'represent' such that numbers (can be
used to) represent quantities? I suspect you mean more than it being a
symbol. If I count the _number_ of toes on your right foot, thereby
measuring its quantity, am I representing that quantity as the number
five or am I actually saying you have five toes on your right foot?
Mathematics as you wish to think about it is about abstract objects, but
this does not exclude quantity from consideration. Indeed, we can say
that a set of numbers has a quantity.
owleye
Neil W Rickert
>Neil W Rickert wrote:
>> A quantity is a measurement. We use numbers to represent
>> quantities. But the numbers themselves are not quantities.
>So Archimedes Plutonium was correct after all, natural
>numbers come first and integers should be derived
>from them, not the other way around like mathematicians
>usually do.
I don't know how you get that from my statement. Natural numbers
are numbers, not quantities. They are just as much inventions as
are negative numbers, complex numbers, or uncountable ordinals.
Once mathematicians are into inventing, it is their choice as to the
appropriate order of construction.
Neil W Rickert
>Neil W Rickert wrote:
>> A quantity is a measurement. We use numbers to represent
>> quantities. But the numbers themselves are not quantities.
>> The prime number theorem is not about quantities.
>I think it is wrong to say that a quantity is a measurement. It is
>better to say that measurements are quantitative (when they are).
>In any case, what do you mean by 'represent' such that numbers (can be
>used to) represent quantities?
We use mathematics to model real world problems. In that modelling we use
numbers to represent quantities (among other things).
> I suspect you mean more than it being a
>symbol.
I suppose that, technically, the numeral is the symbol. In solving
real world problems, the numeral might represent a quantity. In
abstract mathematics, the numerals are used to represent numbers.
> If I count the _number_ of toes on your right foot, thereby
>measuring its quantity, am I representing that quantity as the number
>five or am I actually saying you have five toes on your right foot?
I hope you are reporting it as 5 toes. We should not forget the unit
when we deal with quantities.
>Mathematics as you wish to think about it is about abstract objects, but
>this does not exclude quantity from consideration. Indeed, we can say
>that a set of numbers has a quantity.
Agreed.
Owleye
Neil W Rickert wrote:
> We use mathematics to model real world problems. In that modelling we use
> numbers to represent quantities (among other things).
Fair enough. But we can inquire further what such a representation is.
Would you say our model of the world (the constituents of which can be said
to represent it) _corresponds_ in some way to the world. Is it like a
mapping? Am I mapping the abstract domain of numbers onto or into the world
in some meaningful way? Or is it merely an "idealization" of the world, or,
it is how we would like to to be, or it is how it works best for us, or it is
some other?
> I suppose that, technically, the numeral is the symbol. In solving
> real world problems, the numeral might represent a quantity. In
> abstract mathematics, the numerals are used to represent numbers.
Yes. I had in mind that your view was that numbers are abstract objects, and
that they are not what we use to stand for them. Thus, in the last sentence
it would probably have been better had you substituted 'refer to' or 'stand
for' in place of 'represent' just to get clear the sense of representation
you have in mind.
> > If I count the _number_ of toes on your right foot, thereby
> >measuring its quantity, am I representing that quantity as the number
> >five or am I actually saying you have five toes on your right foot?
>
> I hope you are reporting it as 5 toes. We should not forget the unit
> when we deal with quantities.
I'm not sure I follow you here. Perhaps you are cautioning me about the
problem of "addition" or something like it, where adding one heap of sand to
another does not make two heaps of sand. If so, I certainly agree that we
have to be careful with our application of quantity (number) to things so as
to make sure we are talking about something "essential" to it. It was for
this reason I chose "fiveness" as the number of toes on a foot. This
"fiveness" has the further basis that presupposes an essential unity. The
model should, in my mind at least, have some rationale -- some basis or
reason for our holding it to be a good one other than that it just happens to
work.
Good discussion, I believe. Good as well that posts are brief.
owleye
Tom Giles
The (mistaken) view that Frege views numbers as properties comes from some
lack of clarity in the Grundlagen.
I should also note, firstly, that i am *only* talking about the Natural
numbers, which is what Frege discusses in the Grundlagen. For Frege natural
numbers are numbers which answer the question "how many?"; thus we include
0, since this is a respectable answer to "how many bespectacled green
monsters, with huge long tentacles, are there in my room at this precise
moment?". Now Aristotle & the scholastics talked of quantity, so i am
restricting my discussion to what they called "discrete quantity", not
"continuous quantity". Real numbers answer questions of "how much?" and
this is continuous quantity. This may go some way to explaining why Neil
Ricket said "a quantity is a measurement" and was not wrong. [by the above
explanation a measurement is the answer to the two question "how many?" and
"how much?"]; but why he was wrong in the sense that he means quantity as an
answer to "how much?"
Secondly, Frege argues in the Grundlagen that the problem with Mill's view
of numbers as the number of ways an agglomeration of objects (like the
object which is all the chairs in the room) is mistaken, since we cannot
associate only one number with an agglomeration. Consider a pack of cards,
he says, this is one deck, but also 4 suits and 52 separate cards. Thus he
suggests that a statement of number is "makes an assertion about a
concept"(Gl. p59).
He goes on to say things like "If i say 'the king's carriage is drawn by
four horses', then i assign the number four to the concept 'horse that draws
the king's carriage' (Gl p59). So the third point he makes is that numbers
are *not* properties. We could view them as properties of the extension of
a concept, i.e. the number of objects which fall under the concept, but this
is not what Frege means (Gl. p68). Rather, he means the property of a
concept in question, is that of having a number associated with it. So
Frege, as a mathematician, like Neil, views them as abstract objects (rather
than 'fictious' ones, which might imply their properties could change in the
future, etc..)
On this account Frege does not seem to be making any contradictory
statements like asserting that something is both a concept and an object.
My fourth point is that Frege distinguishes sharply between concepts (or
predicates, which are the manifestation of concepts in language) and
objects, which is an important theory. He says things are objects if they
are complete, whole, or saturated and concepts if they are not. The point
is an object is never true or false, but a statement of concept is. Then
different objects satisfy different concepts, making true and false
statements. For example, take the concept of transparicy and form the
predicate "... is transparent". Then a window would fall under the concept,
by satisfying the predicate, but an apple wouldn't, because the statement
"an apple is transparent" is false. [He later developed this theory to
explain the meaning of concepts. Their meaning comes from our (total)
understanding of the conditions to be met by an object for it to satisfy the
predicate]. This theory also explains the "identity between 0, null,
false, contradiction, no-thing,..". 0 and Nothing are quantifiers which
assert that no object can make some sentence true (Consider "Nothing is
larger than itself"). The concept relating to a false statement, i.e. a
contradiction, has associated with its extension the number 0; if we were to
construct a set of objects which satisfied a contradiction it would be the
null set. Etc., etc.
One of Frege's arguments that numbers are objects not concepts is that
advanced by Neil Ricket that "thinking of them as properties would not fit
ordinary mathematical discussion." But i am interested in arguments that
suggest that numbers are properties and how plausible they are. Is it
correct that Carnap thought of them as 2nd level concepts, against Frege?
What exactly did Russell mean by a class? how different is it from a set?
If numbers are sets of sets, for Quine, are they really abstract objects,
and is this theory also subject to Russell's paradox? What are the ZF axioms
& what do they say about numbers? What about the Peano axioms? Is it
possible to have two competing formularisations of numbers satisfying either
of those two systems? Is this credible? What did Banacerraf argue?
Also, i wonder whether Frege would agree with Neil Ricket's claim that "We
use numbers to represent quantities. But the numbers themselves are not
quantities." He would not, of course, this if we take 'numbers' to mean the
symbols representing quantities. But what of the more interesting
intepretation, where we take numbers to be abstract objects? Are numbers
then actually specific "quantites"? Perhaps rather we should interpret them
as quantifiers, as Frege does. He says that when we say "¬ExFx" we mean that
the extension of F has the number 0 associated with it.
Ideas?
Neil W Rickert
>Neil W Rickert wrote:
>> We use mathematics to model real world problems. In that modelling we use
>> numbers to represent quantities (among other things).
>Fair enough. But we can inquire further what such a representation is.
>Would you say our model of the world (the constituents of which can be said
>to represent it) _corresponds_ in some way to the world. Is it like a
>mapping? Am I mapping the abstract domain of numbers onto or into the world
>in some meaningful way? Or is it merely an "idealization" of the world, or,
>it is how we would like to to be, or it is how it works best for us, or it is
>some other?
Yes, our model of the world corresponds in some way to the world.
But that is a rather weak statement. We can say that it is _like_ a
mapping, but that is weaker than saying it is a mapping. Now, we are
not mapping the numbers onto or into the world. If anything, it is
the other way around. We are 'mapping' aspects of the world into the
domain of numbers. However, it is not really a mapping, at least not
in the mathematical sense. We certainly idealize when we do this.
When we deal with quantities (counts, measurements, etc), we apply
certain procedures (computation, for example). In some sense, the
numbers are an abstract system we have invented so as to allow us to
test and study these procedures in an idealized setting.
>> I suppose that, technically, the numeral is the symbol. In solving
>> real world problems, the numeral might represent a quantity. In
>> abstract mathematics, the numerals are used to represent numbers.
>Yes. I had in mind that your view was that numbers are abstract objects, and
>that they are not what we use to stand for them. Thus, in the last sentence
>it would probably have been better had you substituted 'refer to' or 'stand
>for' in place of 'represent' just to get clear the sense of representation
>you have in mind.
I have no problem with that.
>> > If I count the _number_ of toes on your right foot, thereby
>> >measuring its quantity, am I representing that quantity as the number
>> >five or am I actually saying you have five toes on your right foot?
>> I hope you are reporting it as 5 toes. We should not forget the unit
>> when we deal with quantities.
>I'm not sure I follow you here.
Just that when we are dealing with quantities, the unit being used is
part of the specification of the quantities. A numerical
specification, with out corresponding units (whether explicit or
contextually implied) is not a quantity.
> Perhaps you are cautioning me about the
>problem of "addition" or something like it, where adding one heap of sand to
>another does not make two heaps of sand.
That's a different kind of problem, one that applies mainly to
measurements (floating point quantities) rather than to count
(integral quantities). Our number system is idealized such as to
prevent these sorites problems that arise with quantities.
>Good discussion, I believe. Good as well that posts are brief.
Yes, I agree.
Neil W Rickert
>To clarify what i meant, Frege does view numbers as objects not properties.
>The (mistaken) view that Frege views numbers as properties comes from some
>lack of clarity in the Grundlagen.
>I should also note, firstly, that i am *only* talking about the Natural
>numbers, which is what Frege discusses in the Grundlagen. For Frege natural
>numbers are numbers which answer the question "how many?"; thus we include
>0, since this is a respectable answer to "how many bespectacled green
>monsters, with huge long tentacles, are there in my room at this precise
>moment?". Now Aristotle & the scholastics talked of quantity, so i am
>restricting my discussion to what they called "discrete quantity", not
>"continuous quantity". Real numbers answer questions of "how much?" and
>this is continuous quantity. This may go some way to explaining why Neil
>Ricket said "a quantity is a measurement" and was not wrong. [by the above
>explanation a measurement is the answer to the two question "how many?" and
>"how much?"]; but why he was wrong in the sense that he means quantity as an
>answer to "how much?"
I was being deliberately ambiguous. A count is a form of
measurement. I assumed that the natural numbers were the main
concern here. But since that had not been explicit, I tried to word
things in ways that could also apply to real numbers.
>Secondly, Frege argues in the Grundlagen that the problem with Mill's view
>of numbers as the number of ways an agglomeration of objects (like the
>object which is all the chairs in the room) is mistaken, since we cannot
>associate only one number with an agglomeration. Consider a pack of cards,
>he says, this is one deck, but also 4 suits and 52 separate cards. Thus he
>suggests that a statement of number is "makes an assertion about a
>concept"(Gl. p59).
>He goes on to say things like "If i say 'the king's carriage is drawn by
>four horses', then i assign the number four to the concept 'horse that draws
>the king's carriage' (Gl p59). So the third point he makes is that numbers
>are *not* properties. We could view them as properties of the extension of
>a concept, i.e. the number of objects which fall under the concept, but this
>is not what Frege means (Gl. p68). Rather, he means the property of a
>concept in question, is that of having a number associated with it. So
>Frege, as a mathematician, like Neil, views them as abstract objects (rather
>than 'fictious' ones, which might imply their properties could change in the
>future, etc..)
However, some people would say that Plato's world of perfect forms
does not exist. In some sense, abstract objects are fictitious.
The way I look at it, the distinction does not matter. As
mathematicians, we talk about properties of numbers. But if you
examine what our theorems are about, you will see that we are mainly
concerned with relations between numbers, and with procedures that
can be applied to numbers. These relations and procedures are really
idealizations of relations and procedures that can exist between
quantities. So the numbers are really an idealized system to allow
us to study relations and procedures that may be useful in every day
life. For this purpose, it does not matter whether the numbers
themselves are fictions.
>One of Frege's arguments that numbers are objects not concepts is that
>advanced by Neil Ricket that "thinking of them as properties would not fit
>ordinary mathematical discussion." But i am interested in arguments that
>suggest that numbers are properties and how plausible they are. Is it
>correct that Carnap thought of them as 2nd level concepts, against Frege?
>What exactly did Russell mean by a class? how different is it from a set?
>If numbers are sets of sets, for Quine, are they really abstract objects,
>and is this theory also subject to Russell's paradox? What are the ZF axioms
>& what do they say about numbers? What about the Peano axioms? Is it
>possible to have two competing formularisations of numbers satisfying either
>of those two systems? Is this credible? What did Banacerraf argue?
I think there is a background for all of this. Traditional
epistemology defines knowledge as justified true belief. But beliefs
have to be about something. So if numbers are fictions, and if
mathematics is concerned with statements about numbers, then
mathematics has no knowledge content.
I prefer to think of knowledge as effective procedures. If the
numbers are there primarly to discover, test, and evaluate effective
procedures that have application to quantities, then mathematics is
rich in knowledge content regardless of whether numbers are
fictions.
In any case, the discussion has been interesting. If somebody can
make a case for a different view of numbers, that too will be
of interest.
Owleye
Tom Giles wrote:
To clarify what i meant, Frege does view numbers as objects not properties.
The (mistaken) view that Frege views numbers as properties comes from some
lack of clarity in the Grundlagen.
Since I was a rather latecomer to this discussion and only did so on the prima facie rash statement of Nick's and that my understanding of Frege is rather limited, I'm not sure I'll be able to add much to the discussion. Nonetheless, nothing ventured....
I should also note, firstly, that i am *only* talking about the Natural
numbers, which is what Frege discusses in the Grundlagen. For Frege natural
numbers are numbers which answer the question "how many?"; thus we include
0, since this is a respectable answer to "how many bespectacled green
monsters, with huge long tentacles, are there in my room at this precise
moment?". Now Aristotle & the scholastics talked of quantity, so i am
restricting my discussion to what they called "discrete quantity", not
"continuous quantity". Real numbers answer questions of "how much?" and
this is continuous quantity. This may go some way to explaining why Neil
Ricket said "a quantity is a measurement" and was not wrong. [by the above
explanation a measurement is the answer to the two question "how many?" and
"how much?"]; but why he was wrong in the sense that he means quantity as an
answer to "how much?"
Sounds good. Perhaps I would have preferred "a quantity is the result of a measurement."
Secondly, Frege argues in the Grundlagen that the problem with Mill's view
of numbers as the number of ways an agglomeration of objects (like the
object which is all the chairs in the room) is mistaken, since we cannot
associate only one number with an agglomeration. Consider a pack of cards,
he says, this is one deck, but also 4 suits and 52 separate cards. Thus he
suggests that a statement of number is "makes an assertion about a
concept"(Gl. p59).
This is a good enough reason for tossing both Kant's and Mill's intuition-based mathematics.
He goes on to say things like "If i say 'the king's carriage is drawn by
four horses', then i assign the number four to the concept 'horse that draws
the king's carriage' (Gl p59). So the third point he makes is that numbers
are *not* properties. We could view them as properties of the extension of
a concept, i.e. the number of objects which fall under the concept, but this
is not what Frege means (Gl. p68). Rather, he means the property of a
concept in question, is that of having a number associated with it. So
Frege, as a mathematician, like Neil, views them as abstract objects (rather
than 'fictious' ones, which might imply their properties could change in the
future, etc..)
What does Frege mean by 'assign'? If four is not already in the concept
'horse that draws the king's carriage' is there some other kind of thinking
that is non-conceptual that permits me to assign four to that concept?
Since he is in the conceptual realm it strikes me that concepts by the
possibility of assignment are synthetically determined. I question
this only because we ordinarily restrict this sort of thinking to concrete
particulars out of which we've abstracted a concept and you are expressingly
forbidding it to drag numbers along for the ride, so to speak. (This
is a partially premature response, however. See next.)
Tom continues...
On this account Frege does not seem to be making any contradictory statements like asserting that something is both a concept and an object.
My fourth point is that Frege distinguishes sharply between concepts (or
predicates, which are the manifestation of concepts in language) and
objects, which is an important theory. He says things are objects if they
are complete, whole, or saturated and concepts if they are not. The point
is an object is never true or false, but a statement of concept is. Then
different objects satisfy different concepts, making true and false
statements. For example, take the concept of transparicy and form the
predicate "... is transparent". Then a window would fall under the concept,
by satisfying the predicate, but an apple wouldn't, because the statement
"an apple is transparent" is false. [He later developed this theory to
explain the meaning of concepts. Their meaning comes from our (total)
understanding of the conditions to be met by an object for it to satisfy the
predicate]. This theory also explains the "identity between 0, null,
false, contradiction, no-thing,..". 0 and Nothing are quantifiers which
assert that no object can make some sentence true (Consider "Nothing is
larger than itself"). The concept relating to a false statement, i.e. a
contradiction, has associated with its extension the number 0; if we were to
construct a set of objects which satisfied a contradiction it would be the
null set. Etc., etc.
I can appreciate this "truth conditions" aspect of concepts. If I combine this with your prior points, the concept of a (full) deck of cards doesn't really include its possessing 52 of them. This is merely what we assign to a full deck of cards once we've established its truth conditions. Perhaps one may add that we "discover" that it has 52 cards. This makes sense. If I were creating an object model of a deck of cards I probably wouldn't need to have the number 52 in it.
On the remainder of your interesting post, I'll need to spend more time on it to further the discussion along.
owleye
Owleye
Neil W Rickert wrote:
> Yes, our model of the world corresponds in some way to the world.
> But that is a rather weak statement. We can say that it is _like_ a
> mapping, but that is weaker than saying it is a mapping. Now, we are
> not mapping the numbers onto or into the world. If anything, it is
> the other way around. We are 'mapping' aspects of the world into the
> domain of numbers. However, it is not really a mapping, at least not
> in the mathematical sense. We certainly idealize when we do this.
A person like Penrose wants to make more of it than just an analogy. He thinks,
for example, that the mathematical formulation of quantum states that has within
it the "quantity" sqrt(-1) _is_ the sought after reality. In the days when we
spoke about wave-particle dualism (and we may still do for all I know) I always
had difficulty thinking of the world as comprised of waves. Instead this was
merely the (mathematical) form that matter takes under certain conditions. In
the case of light, this makes me want to say that photons are merely "leaks" in
the fabric of the world that make some impression on us. Thus, mathematics is
helpful to us in knowing something about the world, but not reality itself.
> When we deal with quantities (counts, measurements, etc), we apply
> certain procedures (computation, for example). In some sense, the
> numbers are an abstract system we have invented so as to allow us to
> test and study these procedures in an idealized setting.
True enough. Mathematicians and perhaps others do not really think that reality
goes through a computational process to achieve its next state. Indeed, there is
nothing in mathematics that requires it to take time to add numbers together, for
example, despite the way some intuitionists think.
> Just that when we are dealing with quantities, the unit being used is
> part of the specification of the quantities. A numerical
> specification, with out corresponding units (whether explicit or
> contextually implied) is not a quantity.
Agreed.
> That's a different kind of problem, one that applies mainly to
> measurements (floating point quantities) rather than to count
> (integral quantities). Our number system is idealized such as to
> prevent these sorites problems that arise with quantities.
Perhaps, but I take your point that numbers (as quantities) don't exist
absolutely.
owleye
David Libert
"Tom Giles" (thomas...@st-annes.ox.ac.uk) writes:
> To clarify what i meant, Frege does view numbers as objects not properties.
> The (mistaken) view that Frege views numbers as properties comes from some
> lack of clarity in the Grundlagen.
> I should also note, firstly, that i am *only* talking about the Natural
> numbers, which is what Frege discusses in the Grundlagen. For Frege natural
> numbers are numbers which answer the question "how many?"; thus we include
> 0, since this is a respectable answer to "how many bespectacled green
> monsters, with huge long tentacles, are there in my room at this precise
> moment?". Now Aristotle & the scholastics talked of quantity, so i am
> restricting my discussion to what they called "discrete quantity", not
> "continuous quantity". Real numbers answer questions of "how much?" and
> this is continuous quantity. This may go some way to explaining why Neil
> Ricket said "a quantity is a measurement" and was not wrong. [by the above
> explanation a measurement is the answer to the two question "how many?" and
> "how much?"]; but why he was wrong in the sense that he means quantity as an
> answer to "how much?"
This is a good distinction to make. Frege's basic discussion of
number is natural numbers (how many). Frege's intended system had its
own version of set theory, so ignoring inconsistency for a moment, the
construction from natural numbers up to higher number systems including
reals could follow the usual set theoretic modelling. Frege's main
focus will be natural numbers.
> Secondly, Frege argues in the Grundlagen that the problem with Mill's view
> of numbers as the number of ways an agglomeration of objects (like the
> object which is all the chairs in the room) is mistaken, since we cannot
> associate only one number with an agglomeration. Consider a pack of cards,
> he says, this is one deck, but also 4 suits and 52 separate cards. Thus he
> suggests that a statement of number is "makes an assertion about a
> concept"(Gl. p59).
>
> He goes on to say things like "If i say 'the king's carriage is drawn by
> four horses', then i assign the number four to the concept 'horse that draws
> the king's carriage' (Gl p59). So the third point he makes is that numbers
> are *not* properties. We could view them as properties of the extension of
> a concept, i.e. the number of objects which fall under the concept, but this
> is not what Frege means (Gl. p68). Rather, he means the property of a
> concept in question, is that of having a number associated with it. So
> Frege, as a mathematician, like Neil, views them as abstract objects (rather
> than 'fictious' ones, which might imply their properties could change in the
> future, etc..)
Frege in _Grundgesetze der Arithmetic_ (The Basic Laws of
Arithmetic) proposed that numbers are extensions, that is objects of a
particular sort.
> On this account Frege does not seem to be making any contradictory
> statements like asserting that something is both a concept and an object.
Yes, numbers are objects, not concepts (for Frege).
> My fourth point is that Frege distinguishes sharply between concepts (or
> predicates, which are the manifestation of concepts in language) and
> objects, which is an important theory. He says things are objects if they
> are complete, whole, or saturated and concepts if they are not.
Yes.
> The point
> is an object is never true or false, but a statement of concept is.
Though for Frege, the truth values the True and the False, are
objects.
> Then
> different objects satisfy different concepts, making true and false
> statements. For example, take the concept of transparicy and form the
> predicate "... is transparent". Then a window would fall under the concept,
> by satisfying the predicate, but an apple wouldn't, because the statement
> "an apple is transparent" is false.
Concepts for Frege are particular examples of first level functions,
that is they apply to objects and produce objects, specifically in the
case of concepts truth values.
> [He later developed this theory to
> explain the meaning of concepts. Their meaning comes from our (total)
> understanding of the conditions to be met by an object for it to satisfy the
> predicate]. This theory also explains the "identity between 0, null,
> false, contradiction, no-thing,..". 0 and Nothing are quantifiers which
> assert that no object can make some sentence true (Consider "Nothing is
> larger than itself"). The concept relating to a false statement, i.e. a
> contradiction, has associated with its extension the number 0; if we were to
> construct a set of objects which satisfied a contradiction it would be the
> null set. Etc., etc.
>
> One of Frege's arguments that numbers are objects not concepts is that
> advanced by Neil Ricket that "thinking of them as properties would not fit
> ordinary mathematical discussion."
Yes, in mathematics we want to form constructions over numbers and
manipulate numbers, treating them as "objects", understood intuitively.
Frege's theory allows this by taking numbers as Frege's technical notion
of objects.
> But i am interested in arguments that
> suggest that numbers are properties and how plausible they are. Is it
> correct that Carnap thought of them as 2nd level concepts, against Frege?
But Frege's version of numbers still allows a coded version of this
second level cincepts too. In Frege first level concepts have
extensions, which are objects. Numbers, also objects, can apply to
these extensions to yield truth values.
The advantage of Frege's approach is to allow further constructions
over numbers. How does Carnap handle this?
> What exactly did Russell mean by a class? how different is it from a set?
In Principia Mathematica there are propositions, which have truth
values. Distinct propositions may have the same truth value. Russell's
higher types have propositional functions, which map elements from a
lower type to propositions. Since propositions have truth values these
are remiscent of sets over lower type. On the other hand, since
different propositions can have the same truth value, these higher types
of Russell are intensional, unlike sets which are extensional.
PM provides an alternative to this. At each type level classes will
be discussed, an extensional version of the propositional function.
These are an example of a general doctrine of Russell. The most obvious
way to introduce new terms to a language is by explicit definition of
the new terms. Russell also maintains there is a less direct method, by
contextual definitions. We do not define what the new terms are in
isolation, instead we define how each of the contexts that use those
terms are defined. Then whenever our language wants to make any use of
the term we have a definition for the context. So the extended language
can be taken as abbreviations of the respective contexts.
This is how Russell and Whitehead introduced classes. For each
proposotional function they "have" a corresponding class, taking each
element of the lower type to the corresponding truth value. These
classes however are extensional. This is done by introducing a symbol
for the class of a propositional function. The contexts defined for
classes are membership and = of classes. Membership is defined by the
corresponding propositional function application. = between classes is
defined by coextensiveness.
So quantification over classes is accomplished by quantification over
the corresponding propositional functions. All = and memberships are
interpreted by the contextual definitions. So the full language of
classes can be interpreted in the language of propostional functions.
But these classes don't "really exist". The basic system of types is
the propositional functions. This is similar to ZF really only having
sets, but limited language about classes over sets can be interpreted
into ZF. The class version over PM is more powerful though, as PM can
code quantification over classes.
> If numbers are sets of sets, for Quine, are they really abstract objects,
> and is this theory also subject to Russell's paradox?
Quine has his own version of set theory, New Foundations (NF). NF has
a universal set. Numbers are defined as the set of all those sets
having that number of elements. NF proves such sets exist: every set
has a number representing its "size". On the other hand, there is no
Russell's paradox in NF. There is no Russell set in NF, NF's
restrictions on set formation don't allow the Russell set.
> What are the ZF axioms
> & what do they say about numbers?
The ZF axioms allow the building of sets from lower rank sets, tracing
back at the base to the empty set. A set always has rank strictly
higher than the rank of its members (ranks in general being ordinals).
In particular there is no universal set in ZF. Also there are no sets
corresponding to the NF definition of number.
So ZFC (ZF + Axiom of Choice (AC)) defines cardinal numbers as
special examples of sets of that many elements. For example, the number
0 is the empty set. The number 1 is a one element set of a specific
form. The number 5 is a 5 element of the corresponding form. ZFC
proves that every set is bijective with exactly one set of the special
form, so this form is used to define numbers by taking for each size a
representative set of that size.
ZF (no AC) proves that the above definition works for all finite
sets. ZFC proves it works for all sets, including infinite ones. In
fact this last implication reverses over ZF, ZF proves the above
definition provides a number for every set <-> AC. So this definition
breaks down without AC (at least for infinite cardinals, for finite its
ok).
There is another way to define cardinal numbers in ZF. This
definition associates to each set a cardinal number measuring its size
(though a different system of numbers than the ZFC version above). This
is a good definition in that two sets get assigned the same number iff
they are bijective with each other. On the other hand, these numbers
don't have their own cardinality: ie now the set representing 5
does not have 5 members. This method is using Scott's trick.
It would be possible to amalgamate the two methods, for example using
the ZFC style over finite, and the ZF over infinite, doing this all in
ZF. So you could arrange that 5 has 5 elements. But you can only do
this so far over ZF alone. At infinite levels you have to include cases
where the numbers are not representative sets of size their own number.
I seem to recall reading in something by Jech that it has been proven
that if ZF is consistent then there is a model of ZF in which there is
no definable method of assigning to every set X a set N(X) s.t.
for all X,Y N(X) = N(Y) <-> X and Y are bijective with each other
and such that for all X X is bijective with N(X).
To revisit Frege, Carnap and PM more explicitly, each of these is
similar in its own way to the NF definition of number. Frege's 1 is the
extension which maps those objects which are themselves extensions
mapping exactly one element to the True, to the True, and mapping all
other objects to the False. Similarly other numbers, 2 maping
extensions with two objects going to the True to the True etc.
Carnap's 1 would be the property of sets applying to all those sets
having one member (well, assuming what you wrote above and that Carnap
is doing it in the obvious way along those lines, I had not heard
Carnap's version before).
PM's numbers are classes, having as members propositional functions
from the next level down which map exactly the corresponding number of
yet lower elements to true propositions. The awkward thing in PM is
each type level has its own copy of a number. Ie there are infinitely
many 1 's, one on each type level. Each type level has its own copy of
the number system, and you can't use one to count things from another
type level, else you would violate PM's type restrictions.
> What about the Peano axioms?
PA does not say what numbers are, it just says how the act under S
(successor), + and *. (Also in Peano's original second order version
under sets of numbers). This is how PA can interpret into these other
theories, since PA didn't say what numbers are, the answers the other
theories provide can't conflict with PA. As long as those theories
interpret the functions over numbers correctly its ok. Those other than
Frege do this. Frege, you have to decide how to discuss interpreting
into an inconsistent theory. Before Frege knew his theory was
inconsistent, he could already prove his theory proved the
interpretation of all the PA axioms.
> Is it
> possible to have two competing formularisations of numbers satisfying either
> of those two systems? Is this credible?
I would say yes. Each theory may have its merits.
> What did Banacerraf argue?
I don't know anything about what he has written so you sure got me
there.
> Also, i wonder whether Frege would agree with Neil Ricket's claim that "We
> use numbers to represent quantities. But the numbers themselves are not
> quantities." He would not, of course, this if we take 'numbers' to mean the
> symbols representing quantities. But what of the more interesting
> intepretation, where we take numbers to be abstract objects? Are numbers
> then actually specific "quantites"? Perhaps rather we should interpret them
> as quantifiers, as Frege does. He says that when we say "¬ExFx" we mean that
> the extension of F has the number 0 associated with it.
I don't understand this distinction between numbers and quantities. I
think Frege considered numbers to be really existing abstract objects.
--
David Libert (ah...@freenet.carleton.ca)
1. I used to be conceited but now I am perfect.
2. "So self-quoting doesn't seem so bad." -- David Libert
3. "So don't be a morron." -- Marek Drobnik bd308 rhetorical salvo IRC sig
Andrew Boucher
Tom Giles wrote:
> He [Frege] says things are objects if they
> are complete, whole, or saturated and concepts if they are not.
I'd be honest to say I 'm not sure what Frege means, because I always found
him confusing on this point. Sure, object/concept is supposed to be a
dichotomy, but when it comes down to it, I don't find it works. According to
Frege, something is an object if it is indicated by a term with the single
definite article--but then *everything* would seem to be an object, because
everything can be indicated by a term with a single definite article.
Now Frege in "On Concept and Object" says:
<<The three words 'the concept "horse" ' do designate an object, but on
that very account they do not designate a concept, as I am using the word.
This is full accord what the criterion I gave--that the singular definite
article always indicates an object, whereas the indefinite article accompanies
a concept-word.>>
Let me explain where I think Frege is slipping up by using
subject/predicate, which tends to be a bit more comprehensible. Subjects and
predicates are distinct. (Here there is indeed a dichotomy.) Anything which
can be referred to by a subject is an object. But that doesn't mean that
objects/predicates or objects/concepts (where concepts are in some way
associated to predicates) are distinct; indeed, predicates (and concepts) are a
subset of objects, because predicates can be referred to by a subject.
The only possible way in which you can save object/concept is by saying
that object = the sense of a singular term, and concept = the sense of a
predicate. That is, there is no singular term whose sense is the same as the
sense of "is bald". And I am perfectly willing to grant this to Frege. But it
would be very disappointing, because then he's not really talking about objects
as we usually do. I, for instance, am not a sense, so I would not be an
object...
But then again, I can hardly claim to be an expert on Frege!!
Ruslan Makarov
Neil W Rickert <ricke...@cs.niu.edu> сообщил в новостях
следующее:8gm3f5$i...@ux.cs.niu.edu...
> However, that does not work very well. To a mathematician, a set of
> numbers is one of the primary examples of a set.
A math. set is an abstract notion and treats its content regardless of the
nature of things
set elements are simply objects