infinity
Aslam Maxon
Hello,
This my first time posting to this news group. Anyways, we have a debate
here in our group over the definition of infinity. Here is a list of the
questions and thanks in advance for any help.
1. Is infinity a value or a concept ?
2. If it's a value, then can you give a mathematical expression that
will have infinity as a result ?
3. What is the value of any number such as 10 when divided by zero ?
Thanks
Maxon
Dann Corbit
Aslam Maxon <ma...@dlgesnw1.cr.usgs.gov> wrote in article
<3252B5...@dlgesnw1.cr.usgs.gov>...
> Hello,
>
> This my first time posting to this news group. Anyways, we have a debate
> here in our group over the definition of infinity. Here is a list of the
> questions and thanks in advance for any help.
>
> 1. Is infinity a value or a concept ?
others.
For instance,
There are an infinite number of integers.
There are an infinite number of real numbers.
There are more real numbers than integers, since you can't number them with
integers.
> 2. If it's a value, then can you give a mathematical expression that
> will have infinity as a result ?
> 3. What is the value of any number such as 10 when divided by zero ?
--
"I speak for myself and all of the lawyers of the world"
If I say something dumb, then they will have to sue themselves.
G. A. Edgar
> 1. Is infinity a value or a concept ?
What do you mean by "value"? In order to ask a precise question, you
need to know the meaning of the terms involved.
Infinity is not a real number, is that what you meant?
There are number systems that do have infinite members: were
you asking about other number systems?
> 2. If it's a value, then can you give a mathematical expression that
> will have infinity as a result ?
Such expressions are often seen in the subject of "calculus",
and more advanced mathematics. For example:
limit (as x approaches 0) 1/x^2 = infinity
> 3. What is the value of any number such as 10 when divided by zero ?
Well, 10/0 is not defined in the real number system. In certain other
number systems (for example, the Riemann sphere), 10/zero = infinity,
as you expected.
--
Gerald A. Edgar ed...@math.ohio-state.edu
la...@prancer.srv.ualberta.ca
G. A. Edgar (ed...@math.ohio-state.edu) wrote:
: > 1. Is infinity a value or a concept ?
The Riemann sphere is not a number system: In using this topological
compactification of the _set_ of complex numbers, you sacrifice the
algebraic properties: You will certainly agree that your equation
"10/zero = infinity" does _not_ imply "infinity*zero=10". Hence you are
not allowed to write such an equation.
You can define a (continous!) function f on the Riemann sphere by
{ 10/z z in C\{0}
f(z) = < 0 if z = infinity
{ infinity z = 0
and then state that infinity is the "value" of f(0): "f(0) = infinity".
--
Ulrich Lange Dept. of Chemical Engineering
University of Alberta
la...@gpu.srv.ualberta.ca Edmonton, Alberta, T6G 2G6, Canada
Jeffrey A. Young
In article <edgar-03109...@mathserv.mps.ohio-state.edu>,
G. A. Edgar <ed...@math.ohio-state.edu> wrote:
>
>> 1. Is infinity a value or a concept ?
>
>What do you mean by "value"? In order to ask a precise question, you
>need to know the meaning of the terms involved.
>
>Infinity is not a real number, is that what you meant?
Ouch! You accuse him of asking an imprecise question and then
give him an ambiguous answer. Please don't forget that 'real'
has an informal meaning too, which this questioner might easily
take as your meaning. So don't be surprised if he comes back
with "OK, so infinity is imaginary." :-)
>There are number systems that do have infinite members: were
>you asking about other number systems?
>
>> 2. If it's a value, then can you give a mathematical expression that
>> will have infinity as a result ?
>
>Such expressions are often seen in the subject of "calculus",
>and more advanced mathematics. For example:
>
> limit (as x approaches 0) 1/x^2 = infinity
This is an unfortunate notation that should be clarified as soon as
the concept of 'limit' is introduced. In no way does it mean that
'infinity' is a *result* of taking a limit. In fact what the notation
relates is the *lack* of any limit at a given value, that the value of
the function increases without bound as x moves toward zero. In
effect, it says "you cannot do that (take the limit) here (at zero)".
The notation is a convenience that is too easily misinterpreted.
(Likewise with 'limit (as x approaches infinity)'.)
Jeff
G. A. Edgar
In article <5313jr$13...@pulp.ucs.ualberta.ca>, Ulrich Lange
<la...@gpu.srv.ualberta.ca> wrote:
>
> The Riemann sphere is not a number system: In using this topological
> compactification of the _set_ of complex numbers, you sacrifice the
> algebraic properties: You will certainly agree that your equation
> "10/zero = infinity" does _not_ imply "infinity*zero=10". Hence you are
> not allowed to write such an equation.
>
I agree the Riemann sphere is not a field, but I did not say it
was. I said it is a number system. Mathematicians do, indeed,
write 10/0=infinity in that system, and it makes perfectly
good sense. Perhaps *YOU* are not allowed to write such an equation
as 10/0=infinity, but *I* am... You are correct when you say
"10/zero = infinity" does _not_ imply "infinity*zero=10".
> You can define a (continous!) function f on the Riemann sphere by
>
> { 10/z z in C\{0}
> f(z) = < 0 if z = infinity
> { infinity z = 0
>
> and then state that infinity is the "value" of f(0): "f(0) = infinity".
Correct.
U Lange
: <la...@gpu.srv.ualberta.ca> wrote:
:
: >
: > The Riemann sphere is not a number system: In using this topological
: > compactification of the _set_ of complex numbers, you sacrifice the
: > algebraic properties: You will certainly agree that your equation
: > "10/zero = infinity" does _not_ imply "infinity*zero=10". Hence you are
: > not allowed to write such an equation.
: >
:
: I agree the Riemann sphere is not a field, but I did not say it
: was. I said it is a number system.
What`s your definition of a "number system", then? Your equation is not
allowed in a ring as well. Is it allowed that "number systems" contain
elements for which the algebraic operations are not defined? Then
everything is a "number system" since I can simply add any elements I
like to an algebraic structure.
Is there a distinction between your term "number system" and "set" (resp.
"topological structure")?
: Mathematicians do, indeed,
: write 10/0=infinity in that system, and it makes perfectly
: good sense.
Of course, if you define it as a _notation_ for f(0)=infinity (which you
didn`t). If you interpret it as just a notation, in what sense is
"infinity" a "value", then? (_You_ were the one who wanted the original
poster to be more "precise"!)
: Perhaps *YOU* are not allowed to write such an equation
: as 10/0=infinity, but *I* am...
What exactly do you mean here? I don`t think that the truth of some part
of mathematics should depend on the person who is doing it.
Vincent Johns
(posted & emailed)
Dann Corbit <a-c...@microsoft.com> wrote:
>
> Aslam Maxon <ma...@dlgesnw1.cr.usgs.gov> wrote in article
> <3252B5...@dlgesnw1.cr.usgs.gov>...
> > Hello,
> >
> > This my first time posting to this news group. Anyways, we have a debate
> > here in our group over the definition of infinity. Here is a list of the
> > questions and thanks in advance for any help.
> >
> > ...
> > 3. What is the value of any number such as 10 when divided by zero ?
> Undefined.
Although "undefined" is a stock answer to this question, IMHO it seems
a bit wimpy, giving the impression that one has no idea. Dividing 10 by
zero does not give any result that is a real number, but dividing 10
by a sequence of smaller and smaller numbers (approaching zero) will
give a sequence of larger and larger results (said to "approach
infinity",
although what they are really doing is getting larger without bound).
In the sense of a limit, it seems to me reasonable to say that 10/0 is
infinite (as a shorthand way of describing this limit). A physical
model
of this might be estimating the total number of very small things in
a 10-liter container.
If one tries to use "infinity" too similarly to a real number, one can
get into trouble; for example, expressions like zero times infinity
or infinity minus infinity are indeterminate unless one can get some
information about the processes leading to them. An example of this
might
be trying to calculate the speed of a car based on its location at
almost
identical times; you need more information. On the other hand, some
arithmetic involving "infinity" is relatively safe; "infinity" plus
"infinity" equals "infinity", and one divided by "infinity" equals
"zero".
(Proving this involves using the limits of unbounded sequences.)
Please be aware that many people disagree with my terminology (watch
this
space for other views), but not everyone does, and it has the advantage
over "undefined" of giving an intuitive impression of a large number or
quantity.
-- Vincent Johns
Andras Malatinszky
>: Perhaps *YOU* are not allowed to write such an equation
>: as 10/0=infinity, but *I* am...
>
that this is the right time to stop the debate because you're not likely to
get any more intelligent responses from this point on.
Tony
Aslam Maxon <ma...@dlgesnw1.cr.usgs.gov> wrote in article
<3252B5...@dlgesnw1.cr.usgs.gov>...
> This my first time posting to this news group. Anyways, we have a debate
> here in our group over the definition of infinity. Here is a list of the
> questions and thanks in advance for any help.
>
> 1. Is infinity a value or a concept ?
Infinity is not a value, it's just a convention to represent this idea (or
concept).
numt...@tiac.net
fai...@usa.pipeline.com(Andras Malatinszky) wrote:
You have quoted the wrong person. PROFESSOR Edgar could not
have possible said what you attributed to him.....
You can lead a horse's ass to knowledge, but you can't make him think.
Terry Moore
In article <325549...@pop1.backbone.ou.edu>, Vincent Johns
<vjo...@pop1.backbone.ou.edu> wrote:
Dividing 10 by
> zero does not give any result that is a real number, but dividing 10
> by a sequence of smaller and smaller numbers (approaching zero) will
> give a sequence of larger and larger results (said to "approach
> infinity",
> although what they are really doing is getting larger without bound).
> In the sense of a limit, it seems to me reasonable to say that 10/0 is
> infinite (as a shorthand way of describing this limit).
What of the divisors contain both negative and positive reals?
If you add jsut one point at infinity, this doesn't matter. But if
there are +oo and -oo, it does.
On the other hand, some
> arithmetic involving "infinity" is relatively safe; "infinity" plus
> "infinity" equals "infinity",
Not if oo = -oo (as is often accepted in complex analysis).
> Please be aware that many people disagree with my terminology (watch
> this
> space for other views),
I have just proved this statement correct.
Terry Moore, Statistics Department, Massey University, New Zealand.
Imagine a person with a gift of ridicule [He might say] First that a
negative quantity has no logarithm; secondly that a negative quantity has
no square root; thirdly that the first non-existent is to the second as the
circumference of a circle is to the diameter. Augustus de Morgan
Terry Moore
In article <533fpr$e...@pulp.ucs.ualberta.ca>, la...@gpu1.srv.ualberta.ca (U
> : I agree the Riemann sphere is not a field, but I did not say it
> : was. I said it is a number system.
>
> What`s your definition of a "number system", then? Your equation is not
> allowed in a ring as well. Is it allowed that "number systems" contain
> elements for which the algebraic operations are not defined? Then
> everything is a "number system" since I can simply add any elements I
> like to an algebraic structure.
Well, in a field division by zero is not defined. (Isn't that
where we started?). So what is wrong with having other
elements with undefined operations?
> What exactly do you mean here? I don`t think that the truth of some part
> of mathematics should depend on the person who is doing it.
It is often convenient to change the rules. The only requirements
anre that the rules be clearly explained, and the rules are change
frivolously.
la...@prancer.srv.ualberta.ca
Terry Moore (T.M...@massey.ac.nz) wrote:
: In article <533fpr$e...@pulp.ucs.ualberta.ca>, la...@gpu1.srv.ualberta.ca (U
: > : I agree the Riemann sphere is not a field, but I did not say it
: > : was. I said it is a number system.
: >
: > What`s your definition of a "number system", then? Your equation is not
: > allowed in a ring as well. Is it allowed that "number systems" contain
: > elements for which the algebraic operations are not defined? Then
: > everything is a "number system" since I can simply add any elements I
: > like to an algebraic structure.
: Well, in a field division by zero is not defined. (Isn't that
: where we started?). So what is wrong with having other
: elements with undefined operations?
Division is not an operation of its own. In a field, both addition and
multiplication are well defined for zero. Zero does not belong to the
multiplicative group, it has no multiplicative inverse, that`s all.
Besides, even if you insist on considering division an operation of its
own, it makes IMHO a difference if a particular algebraic operation is
undefined for an element, or if _all_ algebraic operations are undefined.
Therefore, I would consider zero still a "number", but I wouldn`t consider
the point "infinity" on the Riemann sphere a "number" (That`s where I
started).
Jan Rosenzweig
Terry Moore wrote:
>
> In article <539ogr$s...@pulp.ucs.ualberta.ca>, la...@prancer.srv.ualberta.ca
> () wrote:
> >
> > Terry Moore (T.M...@massey.ac.nz) wrote:
>
> > : Well, in a field division by zero is not defined. (Isn't that
> > : where we started?). So what is wrong with having other
> > : elements with undefined operations?
> >
> > Division is not an operation of its own. In a field, both addition and
> > multiplication are well defined for zero. Zero does not belong to the
> > multiplicative group, it has no multiplicative inverse, that`s all.
> >
> > Besides, even if you insist on considering division an operation of its
> > own, it makes IMHO a difference if a particular algebraic operation is
> > undefined for an element, or if _all_ algebraic operations are undefined.
> >
> > Therefore, I would consider zero still a "number", but I wouldn`t consider
> > the point "infinity" on the Riemann sphere a "number" (That`s where I
> > started).
>
> convenient to avoid infinity (algebraically simpler) in some
> cases. It is convenient to allow infinity and put up with more
> complicated algebra in other cases. For example, in complex
> analysis, a function that is analytic everywhere, including at
> infinity, is a constant. For this to work we have to be able
> to treat infinity on a par with the other numbers. (Well, almost.
> We use the reciprocal function to tame the point at infinity.)
>
Did any of you ever read A. Robinson's 'Non-standard Analysis'? It
deals with certain extensions of real numbers (arythmetic and topology
of infinitesimally small 'numbers') and gives some serious applications.
Not that it has anything to do with what you were talking about, but
it shows that one can actually discuss something like that SERIOUSLY
and actually apply it.
By the way, the word 'number' doesn't actually mean anything - words
like 'real number', 'complex number', 'ordinal number', 'cardinal
number' etc have a meaning. Let me ask you both: is \omega (the first
infinite ordinal) a number? How about \aleph_0 ? Or i?
--
Jan Rosenzweig
e-mail: ro...@math.mcgill.ca
office: home:
Department of Mathematics and Statistics 539 Rue Prince Arthur O.
Burnside Hall, room 1132, mbox F-10 Montreal
805 Rue Sherbrooke O. Quebec H2X 1T6
Montreal, Quebec H3A 2K6
Terry Moore
In article <539ogr$s...@pulp.ucs.ualberta.ca>, la...@prancer.srv.ualberta.ca
() wrote:
>
> Terry Moore (T.M...@massey.ac.nz) wrote:
> : Well, in a field division by zero is not defined. (Isn't that
> : where we started?). So what is wrong with having other
> : elements with undefined operations?
>
> Division is not an operation of its own. In a field, both addition and
> multiplication are well defined for zero. Zero does not belong to the
> multiplicative group, it has no multiplicative inverse, that`s all.
>
> Besides, even if you insist on considering division an operation of its
> own, it makes IMHO a difference if a particular algebraic operation is
> undefined for an element, or if _all_ algebraic operations are undefined.
>
> Therefore, I would consider zero still a "number", but I wouldn`t consider
> the point "infinity" on the Riemann sphere a "number" (That`s where I
> started).
We are free to make definitions that are useful. It is certainly
convenient to avoid infinity (algebraically simpler) in some
cases. It is convenient to allow infinity and put up with more
complicated algebra in other cases. For example, in complex
analysis, a function that is analytic everywhere, including at
infinity, is a constant. For this to work we have to be able
to treat infinity on a par with the other numbers. (Well, almost.
We use the reciprocal function to tame the point at infinity.)
Terry Moore, Statistics Department, Massey University, New Zealand.
Douglas J. Zare
Jeffrey A. Young <jyo...@ultra0.rdrc.rpi.edu> wrote:
>In article <edgar-03109...@mathserv.mps.ohio-state.edu>,
>G. A. Edgar <ed...@math.ohio-state.edu> wrote:
>>Such expressions are often seen in the subject of "calculus",
>>and more advanced mathematics. For example:
>>
>> limit (as x approaches 0) 1/x^2 = infinity
>
>This is an unfortunate notation that should be clarified as soon as
>the concept of 'limit' is introduced. In no way does it mean that
>'infinity' is a *result* of taking a limit.
Sure it does.
>In fact what the notation
>relates is the *lack* of any limit at a given value, that the value of
>the function increases without bound as x moves toward zero. In
>effect, it says "you cannot do that (take the limit) here (at zero)".
>The notation is a convenience that is too easily misinterpreted.
>(Likewise with 'limit (as x approaches infinity)'.)
That is one possible interpretation, but I would be surprised if any
mathematician used it without mentioning it specifically as a nonstandard
convention. There are natural compactifications of R, R^2, C, etc. such as
the one-point compactification to S^1, the two-point compactification to
I, the real projective plane, H^2 union the circle at infinity/the unit
disk, and the Riemann sphere. In general, compactification is too strong;
we just need extra points which are limits of sequences of points which
did not converge before.
Infinity in this context is usually a point added to the domain. The
limit as x goes to infinity usually can be taken to mean that any function
which agrees with the given one, is defined at the specified "infinite"
point oo, and is continuous at oo, must necessarily have the stated value
at oo.
Similarly, one can consider compactifications of the range of the
function. One again looks for a unique value taken by continuous
extensions.
Because of the nature of this discussion, I have been more precise than is
usually done. The above agrees with the conventions given to students
taking calculus who are not assumed to be able to deal with point-set
topology.
Examples:
1) The limit as x goes to oo of sin x does not exist:
In either the one-point compactification or the two-point
compactification, there is no extension of sin x. Hence, we usually say
that the limit does not exist.
2) The limit as x goes to 0 of 1/x is oo:
Here the domain is R\{0}, and we are taking a partial compactification to
all of R. The domain is simultaneously compactified by the one-point
compactification. There is a unique extension which is continuous, and
that has value oo at 0.
3) The limit as x goes to 0+ of 1/x is +oo:
Here the domain is R\{0}, and the partial compactification is to add a
point called 0+ a basis of whose neighborhoods looks like
{{0+}U(0,r)|r>0}. The range is compactified to the two-point
compactification of R, i.e., we add a point called +oo and a point called
-oo. The only possible value of an extension which is continuous at 0+ is
+oo.
4) Radial limits of xy/(x^2+y^2):
The domain is R^2\{(0,0)}. The domain now is extended to include a point
in each direction, e.g. 0_a, whose neighborhoods include 0_a and a line
segment extending from (0,0) in direction a. The range does not need to be
changed. Then there is a unique continuous extension to this domain.
Sometimes it is not clear what the appropriate extension of the domain
should be, and different extensions can lead to quite different results.
Although the above often can be viewed as merely a language, it helps to
face the many related difficulties as a single problem.
Douglas Zare
http://www.cco.caltech.edu/~zare
U Lange
Jan Rosenzweig (ro...@math.mcgill.ca) wrote:
: Terry Moore wrote:
: > la...@prancer.srv.ualberta.ca wrote:
: > > Therefore, I would consider zero still a "number", but I wouldn`t consider
: > > the point "infinity" on the Riemann sphere a "number" (That`s where I
: > > started).
: >
: > We are free to make definitions that are useful. It is certainly
: > convenient to avoid infinity (algebraically simpler) in some
: > cases. It is convenient to allow infinity and put up with more
: > complicated algebra in other cases.
:
: Did any of you ever read A. Robinson's 'Non-standard Analysis'? It
: deals with certain extensions of real numbers (arythmetic and topology
: of infinitesimally small 'numbers') and gives some serious applications.
: Not that it has anything to do with what you were talking about, but
: it shows that one can actually discuss something like that SERIOUSLY
: and actually apply it.
I didn`t read Robinson, but I from the little I understand of non-standard
analysis, I certainly agree that it makes sense and may have serious
applications.
I also agree with Terry's statement. Of course, function theory on
the Riemann sphere is in many respects much simpler than avoiding the
inclusion of the point at infinity. Thus the definition of the point
infinity and the _conventions_ which extend the meaning of the algebraic
operations from C to the Riemann sphere are certainly useful.
But Imho the names by which we call these useful objects should reflect
the characteristic properties they share with less abstract entities.
Thus, I consider the term "point infinity on the Riemann sphere" far more
appropriate than the term "the number infinity in the Riemann number
system".
: By the way, the word 'number' doesn't actually mean anything - words
: like 'real number', 'complex number', 'ordinal number', 'cardinal
: number' etc have a meaning. Let me ask you both: is \omega (the first
: infinite ordinal) a number? How about \aleph_0 ? Or i?
We have an intuitive notion of what a "number" is and what we can do
with it (Please don`t misunderstand this: I am definitely _not_ an
intuitionist). In this sense "number" does mean something, although there
is of course no exact definition of the term. You give examples of exact
definitions for particular number systems, which extend our notion of what
we can do with "numbers" (Adding and multiplying (i), counting (\aleph_0)
etc.) in some way. I would consider all of your examples numbers.
But what about the point "infinity" on the Riemann sphere, does it really
have any "number-like" properties?
Bill Dubuque
This thread originated in a query as to whether infinity or 1/0
could be admitted as a "value", and soon drifted into discussion
of the Riemann sphere and other topological manifestations of
infinity via compactification. Below I point out a couple of
marvelous references on these topics; further I would like to
bring to your attention a much wider perspective on such topics,
namely that of existential closure as studied in model theory.
There is a beautiful exposition of points at infinity, projective
closure, compactifications, modifications, etc. in [FM], Chapter 7,
Points at Infinity, by H. Behnke and H. Grauert. This is volume III
in the excellent "Fundamentals of Mathematics" series, MIT Press,
which deserves to be on the bookshelf of every mathematician.
A much deeper appreciation of the methodology behind these constructions
can be had by studying them from a model-theoretic perspective, in
particular from the standpoint of existential closure and model
completion. Kenneth Manders has written a series of thought
provoking papers from this perspective -- see the Math Reviews below.
I'll close with an excerpt from the introduction to Manders 1989:
"The systematic adjunction of roots, or solutions to other simple
conditions, as in formation of the complex numbers by adjoining
imaginaries, or in adjunction of points "at infinity" in traditional
geometry, may be analysed as _existential closure_ and _model
completion_. 'Existential closure' refers to a class of processes
which attempt to round off a domain and simplify its theory by
adjoining elements -- more properly, it refers to the formal
relationship that obtains in such a process. 'Model completion' is
one of the terms employed when this process is successful. The
formation of the complex numbers, and the move from affine to
projective geometry, are successes of this kind. Thus, the theory of
existential closure gives a theoretical basis of Hilbert's "method
of ideal elements." I now sketch the theory of existential closure,
to bring out when, how, and in what sense existential closure gives
conceptual simplification."
-Bill Dubuque
[FM] Fundamentals of mathematics. Vol. III. Analysis.
Edited by H. Behnke, F. Bachmann, K. Fladt and W. Suss.
Translated from the second German edition by S. H. Gould.
Reprint of the 1974 edition. MIT Press,
Cambridge, Mass.-London, 1983. xiii+541 pp. ISBN: 0-262-52095-8 00A05
------------------------------------------------------------------------------
91b:03009 03A05 00A30 03C99
Manders, Kenneth (1-PITT)
Domain extension and the philosophy of mathematics. (English)
J. Philos. 86 (1989), no. 10, 553--562.
------------------------------------------------------------------------------
The author has embarked, starting with a previous article, on an important and
novel philosophical program in which model-theoretical methods are employed in
analyzing the role of new concepts in mathematics and the very nature of
mathematical understanding. As the author has previously emphasized,
"theoretical understanding (is) the primary intellectual goal of mathematical
activity; the reliability-theoretical program in logic, and much of the
associated philosophical work, is based on a severely defective understanding
of mathematical knowledge" [cf. the author, in Logic colloquium '85 (Orsay,
1985), 193--211, North-Holland, Amsterdam, 1987; MR 89a:03013].
In the article under review, using the logic of existential closure, the
author analyzes how domain extension can enhance understandability of a prior
subject matter. The analysis explains why complex numbers, geometric points at
infinity, and similar mathematical objects are useful, and thereby opens up
perspectives for philosophical views of mathematics which account for the
significance of specific mathematical structures.
The article is self-contained and can be read without any prior knowledge of
model theory. The author carefully leads the reader through the underlying
ideas for constructing existentially closed structures, apparently with a
forcing construction in mind. A fuller technical account is promised for a
forthcoming companion paper. The article terminates with a discussion of the
epistemological consequences of model-completion-like domain extensions, with
particular emphasis on the resulting increase in mathematical understanding
through conceptual unification.
Reviewed by Yehuda Rav
------------------------------------------------------------------------------
89a:03013 03A05 00A25
Manders, Kenneth L. (1-PITT-Q)
Logic and conceptual relationships in mathematics. (English) Logic
colloquium '85 (Orsay, 1985), 193--211,
Stud. Logic Found. Math., 122,
North-Holland, Amsterdam-New York, 1987.
------------------------------------------------------------------------------
In this very interesting paper the author argues that traditional epistemology
of mathematics has overemphasised the notion of reliability. He urges that
more attention should be paid to the role of "making things clear by using new
concepts", for example the use of Galois theory to reveal properties of
equations. The bulk of the paper develops, with reference to a wide range of
examples, the notions of "conceptual setting, supposed to capture the idea of
a mathematical area or topic; presentation of a conceptual setting, supposed
to capture in formal terms inferential structure or, generally, intellectual
organisation of a subject area; reconceptualisation relationships, supposed to
capture the idea of connections between areas and their effects on
presentations, and accessibility properties of a conceptual setting, supposed
to capture reasons why certain matters are clear and accessible to inquiry in
that setting".
{For the entire collection see MR 88b:03003}.
Reviewed by Jamie Tappenden
Cited in: 91b:03009
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86h:03070 03C65 51D99 51G05
Manders, Kenneth L. (1-PITT)
Interpretations and the model theory of the classical geometries. (English)
Models and sets (Aachen, 1983), 297--330,
Lecture Notes in Math., 1103, Springer, Berlin-New York, 1984.
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The notions of model completeness, model completion and prime model are
discussed for Euclidean and other classical geometries. The main result is
that if the theory of a class $F$ of infinite skew fields is model complete
then the theory of projective planes over the skew fields from the class $F$
is the model companion of the theory of affine planes over the same class of
skew fields. It is essential that both geometrical theories are formulated in
terms of collinearity. There are generalisations of this theorem to higher
dimensions and other geometries. A modification for ordered spaces is also
given.
{For the entire collection see MR 85k:03002a}.
Reviewed by L. W. Szczerba
Bill Dubuque
This thread originated in a query as to whether infinity or 1/0
could be admitted as a "value", and soon drifted into discussion
of the Riemann sphere and other topological manifestations of
infinity via compactification. Below I point out a couple of
bring to your attention a much more general view of such topics,