-
Appendix I Reference Answers
Unit 1 Mathematics
Part I
EST Reading
Reading 1
Section A Pre-reading Task
Warm-up Questions: Work in pairs and
discuss the following questions.
1. Who
is Bertrand Russell?
Bertrand Arthur
William Russell (b.1872
–
d.1970) was a British philosopher, logician,
essayist
and social critic best known
for his work in mathematical logic and analytic
philosophy. His most
influential
contributions
include
his
defense
of
logicism
(the
view
that
mathematics
is
in
some
important sense
reducible to logic), his refining of the predicate
calculus introduced by
Gottlob
Frege (which still forms the basis of
most contemporary logic), his defense of neutral
monism (the
view that the world
consists of just one type of substance that is
neither exclusively mental nor
exclusively
physical),
and
his
theories
of
definite
descriptions
and
logical
atomism.
Russell
is
generally
recognized
as
one
of
the
founders
of
modern
analytic
philosophy,
and
is
regularly
credited with being one of the most
important logicians of the twentieth century.
2. What is Russell’s
Paradox?
Russell discovered
the paradox that bears his name in 1901, while
working on his
Principles of
Mathematics
(1903). The
paradox arises in connection with the set of all
sets that are not members
of
themselves. Such a set, if it exists, will be a
member of itself if and only if it is not a member
of
itself.
The
paradox
is
significant
since,
using
classical
logic,
all
sentences
are
entailed
by
a
contradiction. Russell's discovery thus
prompted a large amount of work in logic, set
theory, and
the philosophy and
foundations of mathematics.
3. What effect did Russell’s Paradox
have on
Gottlob
Fregg’s
system?
At first Frege
observed that the consequences of Russell’s
paradox are not immediately clear.
For
example, ―Is it always permissible to
speak of the extension of a concept, of a class?
And if not,
how
do
we
recognize
the
exceptional
cases?
Can
we
always
infer
from
the
extension
of
one
concept’s coinciding with that of a
second, that every object wh
ich falls
under the first concept
also falls
under the second?
Because of
these kinds of worries, Frege eventually felt
forced to
abandon many of his views.
4. What is Russell’s response to the
paradox?
Russell's own
response to the paradox came with the development
of his theory of types in 1903.
It was
clear to Russell that some restrictions needed to
be placed upon the original comprehension
(or abstraction) axiom of naive set
theory, the axiom that formalizes the intuition
that any coherent
condition may be used
to determine a set (or class). Russell's basic
idea was that reference to sets
such as
the set of all sets that are not members of
themselves could be avoided by arranging all
sentences
into
a
hierarchy,
beginning
with
sentences
about
individuals
at
the
lowest
level,
sentences
about
sets
of
individuals
at
the
next
lowest
level,
sentences
about
sets
of
sets
of
individuals
at
the
next
lowest
level,
and
so
on
Using
a
vicious
circle
principle
similar
to
that
adopted
by the mathematician Henri Poincaré
,
and his own so-called
Russell
was
able
to
explain
why
the
unrestricted
comprehension
axiom
fails:
propositional
functions,
such as the function
would
involve
a
vicious
circle.
On
Russell's
view,
all
objects
for
which
a
given
condition
(or
predicate) holds must be at the same
level or of the same
5. Have you ever
heard of Zermelo-Fraenkel set theory.? Can you
give an account of it?
Contradictions
like
Russell’s
paradox
arose
from
what
was
later
called
the
unres
tricted
comprehension principle: the assumption
that, for any property
p
,
there is a set that contains all
and
only those sets that have
p
.
In Zermelo’s system, the comprehension principle
is eliminated in
favour of several much
more restrictive axioms:
a.
Axiom of extensionality. If two sets
have the same members, then they are identical.
b.
Axiom of
elementary sets. There exists a set with no
members: the null, or empty, set. For
any two objects a and b, there exists a
set (unit set) having as its only member a, as
well
as a set having as its only
members a and b.
c.
Axiom
of
separation.
For
any
well-formed
property
p
and
any
set
S,
there
is
a
set,
S
1
,
containing all and only the members of
S that have this property. That is, already
existing
sets can be partitioned or
separated into parts by well-formed properties.
d.
Power-set
axiom.
If S is a set, then there exists
a set, S
1
, that contains all
and only the
subsets of S.
e.
Union axiom.
If S is a set (of sets), then there is a set
containing all and only the members
of
the sets contained in S.
f.
Axiom of choice. If S is a nonempty set
containing sets no two of which have common
members, then there exists a set that
contains exactly one member from each member of S.
g.
Axiom
of
infinity.
There
exists
at
least
one
set
that
contains
an
infinite
number
of
members.
With
the
exception
of
(b),
all
these
axioms
allow
new
sets
to
be
constructed
from
already-
constructed sets by
carefully constrained operations; the method
embodies what has come to be
known as
the ―iterative‖ conception of a set.
Section C Post-reading Task
Reading Comprehension
1.
Directions: Work on your own and fill in the
blanks with the main idea.
Part 1
(Para. 1): Brief introduction to Russell’s
paradox
Part 2 (Paras.
2-
5): The effect of Russell’s paradox
on Gottlob Frege’s system
.
Para. 2: Russell’s paradox dealt a
heavy blow to Frege’s attempts to develop a
foundation
for all of mathematics using
symbolic logic.
Para. 3: An
illustration of Russell’s paradox in terms of
sets
Para. 4: Contradiction
found in the set.
Para. 5: Frege
noticed the devastating effect of
Russell’s paradox on his system and inability
to solve it.
Part 3 (Paras.
6-
8): Solutions offered by
mathematicians to Russel’s paradox
Para. 6: Russell’s own response to the
paradox with his
Para. 7:
Zermelo's solution to Russell's paradox
Para. 8: What became of the effort to
develop a logical foundation for all of
mathematics?
Part 4 (Para. 9):
Correspondence between Russell and Frege on the
paradox
2. Directions: Work
in pairs and discuss the following questions.
1)
W
hat is the basic idea of
Russell’s paradox?
Russell's
paradox is the most famous of the logical or set-
theoretical paradoxes. The paradox
arises
within
naive
set
theory
by
considering
the
set
of
all
sets
that
are
not
members
of
themselves. Such a set
appears to be a member of itself if and only if it
is not a member of
itself, hence the
paradox.
2
）
How to
explain Russell’s paradox in terms of
sets?
Some sets, such as the
set of all teacups, are not members of themselves.
Other sets, such
as the set of all non-
teacups, are members of themselves. Call the set
of all sets that are not
members of
themselves S. If S is a member of itself, then by
definition it must not be a
member of
itself. Similarly, if S is not a member of itself,
then by definition it must be a
member
of itself.
3
）
Can
you explain the contradiction found in the sets
related to Russell’s
paradox
The contradiction arises in
the logic of sets or classes. Some classes (or
sets) seem to be
members of themselves,
while some do not. The class of
all
classes is itself a class, and so
it seems to be in itself. The null or
empty class, however, must
not
be a member of itself.
However, suppose that we can form a
class of
all
classes (or
sets) that, like the null class,
are
not
included in themselves.
The paradox arises from asking the question of
whether
this
class is in itself. It is if and only
if it is not.
4
）
Is Russell’s
own response to the paradox workable?
Russell's response to the paradox is
contained in his so-called theory of types. His
basic
idea
is
that
we
can
avoid
reference
to
S
(the
set
of
all
sets
that
are
not
members
of
themselves)
by
arranging
all
sentences
into
a
hierarchy.
This
hierarchy
will
consist
of
sentences (at the lowest level) about
individuals, sentences (at the next lowest level)
about
sets of individuals, sentences
(at the next lowest level) about sets of sets of
individuals, etc.
It is then possible
to refer to all objects for which a given
condition (or predicate) holds
only if
they are all at the same level or of the same
the
test,
this
system
served
as
vehicle
for
the
first
formalizations
of
the
foundations
of
mathematics;
it
is
still
used
in
some
philosophical
investigations
and
in
branches
of
computer science.
5
）
Do
you know Zermelo-Fraenkel set theory?
In mathematics,
Zermelo
–
Fraenkel set theory
with the axiom of choice, named after
mathematicians Ernst Zermelo and
Abraham Fraenkel and commonly abbreviated ZFC, is
one of several axiomatic systems that
were proposed in the early twentieth century to
formulate a theory of sets without the
paradoxes of naive set theory like Russell's
paradox.
Specifically, ZFC does not
allow unrestricted comprehension. Today ZFC is the
standard
form of axiomatic set theory
and as such is the most common foundation of
mathematics.
3.
Directions:
Read
the
following
passage
carefully
and
fill
in
the
blanks
with
the
words
you’ve learned in the
text.
Russell's
own
response
to
the
paradox
came
with
the
development
of
his
theory
of
types
in
1903.
It
was
clear
to
Russell
that
some
restrictions
needed
to
be
placed
upon
the
original
comprehension
(or
abstraction)
axiom
of
naive
set
theory,
the
axiom
that
formalizes
the
intuition that any coherent condition
may be used to determine a set (or class).
Russell's basic
idea was that reference
to sets such as the
set of all sets
that are not members of themselves
could be avoided by arranging all
sentences into a hierarchy, beginning with
sentences about
individuals
at
the
lowest
level,
sentences
about
sets
of
individuals
at
the
next
lowest
level,
sentences about sets
of sets of individuals at the next lowest level,
and so on. Using a vicious
circle
principle similar to that adopted by the
mathematician Henri Poincaré
, and his
own so-
called
class
theory
of
classes,
Russell
was
able
to
explain
why
the
unrestricted
comprehension axiom fails:
propositional functions, such as the function
be
applied
to
themselves
since
self-application
would
involve
a
vicious
circle.
On
Russell's
view, all objects
for which a given condition (or predicate) holds
must be at the same level or of
the
same
Vocabulary and Structure
1. Directions: Give the correct form of
the word according to the indication in the
brackets.
Then complete the sentences
using the right form for each word. Use each word
once.
1)
The math
may not have been new, but Duchin enjoyed the
process of discovery, and she got
to
work collaboratively with half a dozen other math
whizzes.
2)
Packages can be sealed and can contain
personal correspondence if it relates to the
contents
of the package.
3)
New research
indicates that the brain region may prefer
symbolic notation to other numeric
representations.
4)
To
do
this,
an
ideal
model
based
on
the
equality
paradigm
was
constructed
and
then
compared with a neutral
model reflecting the further education system as
it existed before
the Act took effect.
5)
Is this not in
flagrant contradiction to Einstein's rule that
signals do not travel faster than the
velocity of light?
6)
Sequential
organization
has
the
major
advantage
that
the
records
are
stored
in
a
logical
order,
presumably
that
sequence
to
which
the
records
are
normally
required
for
printing
and
for soft copy reports.
7)
The mathematical description of a zero-
sum two-person game is not difficult to construct,
and
determining
the
optimal
strategies
and
the
value
of
the
game
is
computationally
straightforward.
8)
The
proof
we
now
know
required
the
development
of
an
entire
field
of
mathematics
that
was unknown in Fermat's time.
9)
Williams
adds
that
many
courses
in
geometry,
―the
one
high
school
class
that
demands
formal
reasoning,‖ have already been ―gutted‖
and are no longer proof
-based.
10)
The
concept
of
total
aircraft
ownership
will
become
increasingly
important
should
the
traditional trade structure be unable
to cover the expanse of technologies economically.
2. Directions: Complete the
sentences with the words given in the brackets.
Change the form
if necessary.
1)
The
key
to
unraveling
such
apparent
paradoxes
is
to
characterize
the
initial
set
of
possibilities
(
meaning
before
you
receive
any
extra
information)
and
then
to
eliminate possibilities based on that
extra information.
2)
Indeed, this
separation of meaning is reflected by the
definition of
with a distinct sense
reserved for its use when pertaining to that of
solutions.
3)
The
resulting radical pollution control program
outlined by Nixon, calling for a 90 per cent
reduction in vehicle emissions by 1980,
not only led to him being credited (albeit
briefly)
as policy initiator of an
environmental clean-up but also provided him with
the chance to
deal a blow to one of his
most important opponents in the 1972 elections,
Edmund Muskie.
4)
While most of us are used to
representing physical objects in the terms of one,
two, or three
dimensions
(or
four,
if
one
considers
time
)
,
Mandelbrot
came
up
with
a
way
of
representing
another
―dimension‖
of
an
object
—
that
is,
its
degree
of
roughness
and
irregularity.
5)
In this work
he was led to topology, a still new kind of
mathematics related to geometry,
and to
the study of shapes (compact manifolds) of all
dimensions.
6)
If there is no allowable string which
spans the whole graph, then we can search in the
same
way as
described above, but wherever the
required path does not exist in the tree, check if
that position in the tree is flagged
for end-of-word.
7)
During
the
past
century,
steps
forward
in
physics
have
often
come
in
the
form
of
newly
found particles; in
engineering, more complex devices;
in
astronomy, farther planets and
stars;
in biology, rarer genes; and in chemistry, more
useful materials and medications.
8)
A second
reason for measurements is the more theoretical,
put by Love as
of numerical relations
between the quantities that can be measured to
serve as
a basis for
the inductive determination of the form
of the intrinsic energy function.
9)
Thus the
optimum conditions for coastal terrace development
would seem to be areas with
small
tidal
ranges.
Finally,
tidal
range
is
an
important
factor
in
the
generation
of
tidal
currents
which may locally become of geomorphological
importance.
10)
The
original
double
entrance
doors
to
the
booking
hall
had
been
replaced
by
an
utterly
incongruous picture window as had
adjacent booking hall and waiting room windows.
3. Directions:
Reorder the disordered parts of a sentence to make
a complete sentence.
1)
A simple way to describe topology is as
a 'rubber sheet geometry' - topologists study
those
properties of shapes that remain
the same when the shapes are stretched or
compressed.
2)
Since
the
mid-1990s
scientists
have
floated
the
idea
that
representations
of
numeric
quantities,
whether
expressed
as
digits
or
as
written
words,
are
codified
by
the
parietal
cortex, a higher-processing region in
the brain.
3)
As
activity
was
monitored,
located
just
above
the
forehead,
researchers
noted
changes
under the assumption that the brain
reduces activity as it becomes accustomed to a
stimulus
and then reactivates when a
novel stimulus is presented.
4)
That has not
stopped physicists from devising new algorithms
for the devices, which can
calculate a
lot faster than ordinary
computers
—
in fact,
exponentially faster, in quite a literal
sense.
5)
Such a device would be made of
metamaterial, a thicket of metal rings or other
shapes that
bends light in funny ways.
4. Directions: Change the
following sentences into nominalized ones.
1)
The passage of night could be marked by
the appearance of 18 of these stars.
2)
The full proof
of Fermat's Last Theorem is contained in these two
papers.
3)
The
concept of fixed-length hours, however, did not
originate until the Hellenistic period.
4)
There
is
a
probability
that
my
first
sock
is
red
because
only
one
of
the
remaining
three
socks is red.
5)
The
importance
of
accurate
data
in
quantitative
modeling
is
central
to
using
Bayes's
theorem to calculate the probability of
the existence of God.
Discourse
Understanding
-
-
-
-
-
-
-
-
-
上一篇：英语谚语成语
下一篇：Lesson 1 Of Studies--Bacon