Chapter 0
Reasoning
§ 0.1 REASONING.
Throughout the text a concerted effort will be made to encourage the
student to discern the difference in types of reasoning and the importance of
reasoning in mathematics. Broadly speaking, there are two types of reasoning
central to life, to science, and especially to mathematics. One reasons by
attempting to draw generalisations from patterns, from observed outcomes of
events, from an example, or a series of examples. The type of reasoning used in this manner is
called inductive reasoning. It is used to hypothesise, conjecture, and
form a basis for many theories that are fundamental building blocks to our
understanding of our lives, our world, and our universe. Regrettably, there is a basic flaw in this
form of reasoning. The existence of a
single instance of a contradictory result, a contradictory event, or such that
a pattern that seems to be really isn’t destroys the most parsimonious or elegant
theory.
For example, consider 1, 2, 4, 8, 16, . .
. We have seen such in our secondary or
primary school studies and some believe that the ellipsis (the three dots) causes there to be a pattern which
continues. There is no such cause
inherent in the commas, the ellipsis, nor in their combination. Consider 1.4142135623730950488016887242097 …
which is the decimal expansion for 
. We shall see (indeed prove in Math 255, Set Theory [the
subsequent course to this course]) that is an irrational
number so the decimal expansion is non-repeating hence does not have a
pattern! So the ellipsis only ‘suggests’
patterns where none such exist.
Mathematicians use inductive reasoning to
formulate ideas, conjectures, and hypotheses.
Such reasoning forms the foundation that one uses in order to
progress. However, it does not guarantee
that is right. In order to establish the
veracity of a claim one must prove it.
So, mathematics is built on a foundation, as we shall see, where there
are primary or atomic assumptions made and then the mathematician deduces a
result. The manner of reasoning so
outlined is called deductive reasoning. In deductive reasoning the emphasis is not
on events or particulars but on the general abstract thought process so that
from the general property whilst applying the laws of logic one can draw
necessary conclusions. It is not the
grey terminology that the foundationalist seeks but the concrete understanding
of a mathematical system or principle.
For example suppose one was told that for
the counting numbers that we shall generally call, ‘n,’ (1, 2, and so forth) we are allowed to assume
that we have the property that f(n) =
. From this assumption we can deduce that f(1) = 
. We can deduce that f(2) = 
. We can deduce that f(3) = 
; etcetera ad nausea.
So, we have formulated from the general to the specific that we have a
sequence (which many students have studied in secondary school, in the Analysis
sequence, but which will be rigorously defined in Mathematics 361, Real
Analysis) so that GIVEN f(n) = where n is a natural
number (defined in chapter 2) we have the sequence so that we have 1, 2, 4, 8, .
. . where the ellipsis indicates the
pattern continues because of the definition f(n) = where n is a natural
number rather than in spite of said definition or that because a pattern seems to be makes it be (which is utter
balderdash).
Many mathematics in the late twentieth
century began studying non-rigorous or non-closed systems such that they are
not deterministic as we have indicated previously. Such open or dynamic systems are ripe for study and are in every way,
shape, and form worthy of
attention. However, we shall approach
our study of mathematics from the deterministic or stable vantage point and
realise that once we understand the concrete, we can then study dynamic
(dynamical systems, differential equations, etc.) or stochastic (statistics,
measurement analysis, etc.).
Let us consider the following problem
which is quite amusing. It is a game as
are many games we play as youths where the object of the game is to solve the
puzzle and come up with the correct solution.
Example 0.1.1: The Circle Problem
Let C be a circle with radius one centred at
(0,0).
Set 1. Let A1 and A2 be
points on the circle. Call them
vertices. Connect all vertices with a
chord (in this case A1 and A2 are connected by a
chord). Consider the interior of the
circle. Let region be defined as an interior part of the circle such that a set
of chords separates it from other interior parts of the circle. The chord itself
is a ‘bound’ and as such is neither considered a part of nor a region. Count
the number of points (2), the number of chords, (1), and the number of regions
(2).
Figure 0.1.1
Hypothesize as to the general number of points,
chords, and regions. How confident are
you with your predictions? Do you think
your hypotheses are true or false? Why?
Set 2. Let A1, A2, and A3 be points on
the circle. Call them vertices. Connect all vertices with a chord (in this
case A1 and A2 are connected by a chord, A1
and A3 are connected by a chord, and A3 and A2
are connected by a chord). Consider the
interior of the circle. Let region be defined as an interior part of the circle
such that a set of chords separates it
from other interior parts of the circle. Count the number of points (3), the
number of chords, (3), and the number of regions (4).
Figure 0.1.2
Hypothesize as to the general number of points,
chords, and regions. How confident are
you with your predictions? Do you think
your hypotheses are true or false? Why?
Set 3. Let A1, A2, A3, and A4
be points on the circle. Call them
vertices. Connect all vertices with a
chord (in this case all vertices are connected by a chord. Consider the interior of the circle. Let
region be defined as an interior part of the circle such that a set of chords
separates it from other interior parts of the circle. Count the number of
points (4), the number of chords, (6), and the number of regions (8).
Figure 0.1.3
Hypothesize as to the general number of points,
chords, and regions. How confident are
you with your predictions? Do you think
your hypotheses are true or false? Why?
Exercise (see exercise set): Do this for set 5, 6, 7, and 8.
Hypothesize as to the general number of points, chords, and regions. How confident are you with your predictions? Do you think your hypotheses are true or
false? Why?
Have you determined any flaw in your reasoning?
If so, what flaw; if not, why?
Let us consider the following problem
which is also quite illuminating. It is
a problem that one could easily find in any high school algebra book. The
object of the exercise is of course to solve the puzzle and construct the
correct solution.
Example 0.1.2: The Polynomial Problem
Consider f(n)
= n2 + n + 41 " n Î N (usually in
high school it would be stated as “consider f(n) = n2 + n + 41 for each n
being 1, 2, 3, 4, and so forth [never ending]”).
Set 1. Consider f(1). It is 12 +
1 + 41 = 43. What kind of natural number is it? Hypothesize as to the general
rule for f(n). How confident are you with your
prediction? Do you think your hypothesis
is true or false? Why?
Set 2. Consider f(2). It is 22 +
2 + 41 = 4 + 2 + 41 = 47. What kind of natural number is it? Hypothesize as to
the general rule for f(n). How confident are you with your
prediction? Do you think your hypothesis
is true or false? Why?
Set 3. Consider f(3). It is 32 +
3 + 41 = 9 + 3 + 41 = 53. What kind of natural number is it? Hypothesize as to
the general rule for f(n). How confident are you with your
prediction? Do you think your hypothesis
is true or false? Why?
Set 4. Consider f(4). It is 42 +
4 + 41 = 16 + 4 + 41 = 61. What kind of natural number is it? Hypothesize as to
the general rule for f(n). How confident are you with your
prediction? Do you think your hypothesis
is true or false? Why?
Set 5. Consider f(5). It is 52 +
5 + 41 = 25 + 5 + 41 = 71. What kind of natural number is it? Hypothesize as to
the general rule for f(n). How confident are you with your
prediction? Do you think your hypothesis
is true or false? Why?
Exercise (see exercise set): Do this for set 6, 7, 8, 9, 10,
and 11. Hypothesize as to the general number of points, chords, and
regions. How confident are you with your
predictions? Do you think your hypotheses
are true or false? Why?
Have you determined any flaw in your reasoning?
If so, what flaw; if not, why?
Each
of the examples are rather facile once on understands the basic problem.
Nonetheless, note that some of the patterns one might opine exists either
exists or does not. There is no
guarantee as to the veracity of the claim without some manner of proof (which
is what we will be studying later).
Example 0.1.3: The Hard Polynomial Problem
Consider f(n)
= 991n2 + 1 " n Î N
Set 1. Consider f(1). It is 992. What kind
of natural number is it? Hypothesize as to the general rule for f(n).
How confident are you with your prediction? Do you think your hypothesis is true or
false? Why?
Set 2. Consider f(2). It is 3,965. What kind
of natural number is it? Hypothesize as to the general rule for f(n).
How confident are you with your prediction? Do you think your hypothesis is true or
false? Why?
Set 3. Consider f(3). It is 8,920. What kind
of natural number is it? Hypothesize as to the general rule for f(n).
How confident are you with your prediction? Do you think your hypothesis is true or
false? Why?
Set 4. Consider f(4). It is 15,857. What
kind of natural number is it? Hypothesize as to the general rule for f(n).
How confident are you with your prediction? Do you think your hypothesis is true or
false? Why?
Set 5. Consider f(5). It is 24,776. What
kind of natural number is it? Hypothesize as to the general rule for f(n).
How confident are you with your prediction? Do you think your hypothesis is true or
false? Why?
Exercise (see exercise set): Do this for set 6, 7, 8, 9, 10,
and 11. Hypothesize as to the general number of points, chords, and
regions. How confident are you with your
predictions? Do you think your
hypotheses are true or false? Why?
§ 0.1 EXERCISES.
In this exercise set (and this set only) the
solutions are provided on the following page.
1. Do
the circle problem for set 5. Hypothesize as to the general number of points,
chords, and regions. How confident are
you with your predictions? Do you think
your hypotheses are true or false? Why?
2. Do
the circle problem for set 6, 7, and 8. Hypothesize as to the general number of
points, chords, and regions. How
confident are you with your predictions?
Do you think your hypotheses are true or false? Why?
3. Do
the polynomial problem for set 6, 7, 8, 9, 10, and 11. Hypothesize as to the
general number of points, chords, and regions.
How confident are you with your predictions? Do you think your hypotheses are true or
false? Why?
4. Do
the polynomial problem for set 15, 20, 25, 30, 35, 40, and 45. Hypothesize as
to the general number of points, chords, and regions. How confident are you with your
predictions? Do you think your
hypotheses are true or false? Why?
5. Do
the hard polynomial problem for set 6, 7, 8, 9, 10, and 11. Hypothesize as to
the general number of points, chords, and regions. How confident are you with your
predictions? Do you think your
hypotheses are true or false? Why?
6. Do
the polynomial problem for set 100, 200, 1,000, 10,000, 1,000,000, and
1,000,000,000. Hypothesize as to the general number of points, chords, and
regions. How confident are you with your
predictions? Do you think your
hypotheses are true or false? Why?
Hint 1.
The Circle Problem
Consider n = 6
Hint 2.
The Polynomial Problem
Consider f(40)
Solution 1[1]: For n = 6 the conjecture about regions fails for 25 = 32,
but one will get 30 or 31 depending on the placement of the vertices (31 if the
vertices are not equidistant around the circle, 30 if they are).
Solution 2[2]: For n = 40 the conjecture f(n)
is prime fails since f(40) = 412.
Solution 3[3]: For n = 12,055,735,790,331,359,447,442,538,767 the conjecture f(n) is not a perfect square fails since
f(12,055,735,790,331,359,447,442,538,767)
= 1028.
The
point of the exercise id to realise that inductive patterns are not generally
true even if there are many examples with which to point to as evidence of a
pattern. It is the case that many people
get tired with checking each and every case (with the natural numbers on cannot
check each and every case) so one skips and jumps about checking a certain
number of cases (a heuristic – a rule of thumbs - by which one satisfies
oneself with the pattern) and then
claims ‘truth.’ For truth to exist for a
human there must be a proof to accompany the claim. Proof is a central part of mathematics and
will occupy our attention for the remainder of the course.