120 deductive/inductive reasoning

and the definition of a

REAL NUMBER

was subject to

much debate.

In 1872 J

ULIUS

D

EDEKIND

had the very simple and

elegant idea to simply define the irrationals to be the

gaps in the rational number line. He noted that each

“gap,” like the square root of 2 for example, divides the

line of rationals into two pieces—a left piece and a right

piece. One can focus one’s attention on just the left

piece (for those points that are not in it constitute the

right piece) and this left piece Lsatisfies the following

three properties:

1. It is not empty, nor is it the whole set of points.

2. If ais a number in L, and b<a, then balso belongs

to L.

3. If ais a number in L, it is possible to find another

number calso in Lbut slightly larger than a.

Dedekind simply defined a real number to be any subset

Lof the rational numbers satisfying these three proper-

ties. Such a set is today known as a Dedekind cut.

Every rational number rdefines a cut. One can

check that the set r* = {a ∈ Q : a < r} satisfies the three

properties. Thus the set of Dedekind cuts “contains”

all the rational numbers as sets of this type. It also con-

tains other types of numbers. For example, the square

root of 2 is given by the set:

L= {a ∈ Q : ais negative, or ais positive and a2<2}

One can check that any union of cuts, in the context

of

SET THEORY

, is again a cut. With this surprisingly

simple definition of a real number, Dedekind was

able to prove all the properties of the real-number

system required for establishing the soundness of cal-

culus. In particular, he was able to show that any col-

lection of real numbers with an upper

BOUND

necessarily possesses a least upper bound. (This least

upper bound is the union of all the cuts listed in the

collection.)

deductive/inductive reasoning In the scientific meth-

od, there are two general processes for establishing

results. The first, called inductive reasoning, arrives at

general conclusions by observing specific examples,

identifying trends, and generalizing. “The sun has

always risen in the past, therefore it will rise tomor-

row,” for example, illustrates this mode of reasoning.

The inductive process relies on discerning patterns

but does not attempt to prove that the patterns

observed apply to all cases. (Maybe the sun will not

rise tomorrow.) For this reason, a conclusion drawn by

the inductive process is called a conjecture or an edu-

cated guess. If there is just one case for which the con-

clusion does not hold, then the conjecture is false. Such

a case is called a

COUNTEREXAMPLE

.

To illustrate, in the mid-1700s L

EONHARD

E

ULER

observed that the product of two consecutive integers

plus 41 seems always to yield a

PRIME

number. For exam-

ple, 2 ×3 + 41 = 47 is prime, as is 23 × 24 + 41 = 593

and 37 ×38 + 41 = 1447. By inductive reasoning, we

would conclude that n×(n+ 1) + 41 is always prime.

However, this is a false conclusion. The case n=40 pro-

vides a counterexample: 40 ×41 + 41 = 41 ×41 = 1681

is not prime. (Curiously n×(n+ 1) + 41 is prime for all

values n between –40 and 39.)

Many intelligence tests ask participants to identify

“the next number in the sequence.” These questions

rely on inductive reasoning, but are not mathematically

sound. For example, given the challenge:

What number comes next in the sequence:

24 6?

any answer is actually acceptable (although the test

designers clearly expect the answer “8”). One can

check that the

POLYNOMIAL

for example, has values 2, 4, and 6 when n equals 1, 2,

and 3, respectively, and value awhen nequals 4. Setting

ato be an arbitrary value of your choice gives justifica-

tion to any answer to this problem. (This particular

polynomial was devised using L

AGRANGE

’

S FORMULA

.)

On the other hand, deductive reasoning works to

prove a specific conclusion from one or more general

statements using logical reasoning (as given by

FORMAL

LOGIC

) and valid

ARGUMENT

s. For example, given the

statements, “All cows eat grass” and “Daisy is a cow,”