Section 2.2 Deductive Reasoning
Motivating Questions.
In this section, we will explore the following questions.
What is deductive reasoning? How does it differ from inductive reasoning?
How is deductive reasoning employed in mathematics?
What are some strengths and weaknesses of deductive reasoning?
Formal mathematical reasoning is deductive (defined momentarily), and begins with axioms, which are statements that should be self-evident and taken to be true. Note that while axioms are not always explicitly stated, they can be when necessary.
Investigation 2.2.1.
The most famous set of axioms are Euclid’s postulates for geometry, defined in
The Elements, which not only shaped thousands of years of geometry, but solidified the deductive approach to doing and explaining mathematics that we will explore in this unit. At the beginning of Book I of
The Elements, Euclid identified five postulates and five axioms.
Euclid’s postulates are:
A straight line segment can be drawn joining any two points.
Any straight line segment can be extended indefinitely in a straight line.
Any circle can be described given a center and (radial) distance.
All right angles are equal to one another.
If a straight line intersecting two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the side on which the angles are less than two right angles.
Euclid’s axioms (or common notions) are:
Things which are equal to the same thing are also equal to one another.
If equals are added to equals, the wholes are equal.
If equals are subtracted from equals, the remainders are equal.
Things which coincide with one another are equal to one another.
The whole is greater than the part.
The desired qualities of a system of axioms are:
consistency: we cannot deduce contradictory propositions, such as "God exists" and "God does not exist" from the same set of axioms
simplicity: we have as few axioms as possible, and they are no more complicated than they need to be
completeness: the system can answer every question we can think to ask
In your groups, discuss Euclid’s postulates and common notions, perhaps in view of the desired qualities of an axiomatic system. What strikes you as being interesting or noteworthy? Make a list of at least 2-3 observations. Then consider: on what axioms or assumptions do you make decisions (e.g., about how to spend your time, resources, etc)?
The process of
deductive reasoning in mathematics begins from a set of generally agreed-upon axioms of
set theory and uses logic to make inevitable conclusions from those axioms. These conclusions are generally called
theorems. They are usually given as conditional statements of the form “If
\(P\text{,}\) then
\(Q\text{,}\)” where
\(P\) and
\(Q\) are sensible statements. Moreover, since most deductive statements are presented in conditional form, their scope is generally limited. That is, the statement “if it is Monday, then we have math class” is only making a claim about what happens on Mondays; it says nothing whatsoever about any other day of the week. We will explore the consequences of this more in
Chapter 3.
The author Lewis Carroll loved logic puzzles (he was actually a mathematics professor!), and wrote many of them. Here is one, axiomatized for easy reference.
Axiom 2.2.2. (Carroll).
Consider the axioms:
If a kitten loves fish, then it is teachable.
Every kitten without a tail will play with a gorilla.
All kittens with whiskers love fish.
If a kitten has green eyes, then it is not teachable.
All kittens have whiskers or do not have tails.
Once you have a deductive argument that (generally) begins from your premises and reasons, step-by-step, to your conclusion, you can write out the argument in a short essay known as a proof. For our purposes, a proof is just a convicing argument. It should be written at a level appropriate to the reader and clearly lay out the steps necessary for a reader who accepts your hypotheses to believe the conclusion. As an example, consider the following.
Example 2.2.3.
All kittens with whiskers are teachable.
Proof. Suppose we have a kitten with whiskers. Let’s call him Arthur. By
Axiom 2.2.2 #3, Arthur loves fish. Since Arthur loves fish,
Axiom 2.2.2 #1 implies that Arthur is teachable.
Since Arthur was an arbitrary kitten, we conclude that all kittens with whiskers are teachable.
There are several observations which are worth a moment of our time in the proof in
Example 2.2.3.
We first note that the proof is written using standard conventions of academic writing, including complete sentences, proper punctuation and capitalization, etc. This is important! In order to convince someone that your argument is valid, they need to be able to read it.
While the statement to be proved is not written as “if \(P\text{,}\) then \(Q\)”, it can be stated that way: “If \(A\) is a kitten with whiskers, then \(A\) is teachable”. Thus, our proof begins by considering an arbitrary kitten with whiskers, who we name Arthur. We observe, however, that there is nothing special about Arthur that figures into our proof in a meaningful way, so the argument will apply just as well to any kitten with whiskers we may find.
In each step we take throughout the proof, we refer to the specific axiom from
Axiom 2.2.2 that allow us to take that step. It is valuable to be able to do this, but generally we do not specifically refer to the axioms by number. This is to improve the readability of the proof.
Finally, note that our proof concludes with a conclusion: all kittens with whiskers are teachable. This is good practice and sends an unmistakable signal to the reader that you are done.
Theorem 2.2.4.
If a kitten has green eyes, then it does not love fish.
Theorem 2.2.5.
If a kitten has a tail, then it does not have green eyes.
Theorem 2.2.6.
Every kitten with green eyes will play with a gorilla.
Activity 2.2.7.
Compare and contrast the structures of the proofs of the preceding theorems. Can you clarify the general reasoning patterns you used to prove them?
Discussion 2.2.8.
We have now explored both inductive and deductive reasoning. How are they similar? How are they different? How might you decide which type of reasoning to employ in a given situation? What are their strengths and weaknesses?
Conclusion.
In this section, we explored deductive reasoning, which begins from accepted axioms and premises and then reasons, step by logical step, toward a conclusion. This is the primary form of reasoning used in mathematics. We saw that while conclusions reached via deductive reasoning are generally tighter and more certain, there are still some drawbacks.
The main drawback of deductive reasoning involves scope. We must begin with axioms, so the axioms must be well-chosen and sensible. However, if one disagrees with the choice of a set of axioms, then one must be willing to set aside any results deduced from them (or, at least, deduced from the particular axioms with which one disagrees).
A second drawback having to do with scope concerns the premises of a conditional statement. In particular, if the premises of a statement are not satisfied, the statement makes no assertion whatsoever (though, as we will see in
Chapter 3, there is still a
consistent way to assign truth values to statements whose premises are not satisfied).
Exercises Exercises
1.
Invent one or two additional theorems that can be deduced from
Axiom 2.2.2. Prove them.
2.
While deductive proofs are crucial for the advancement of mathematical knowledge, they can often be complex and hard to understand, even for experts. An extreme example of this arose in the 1970s via the proof of the
four color theorem. Read about this theorem and the controversy surrounding its proof, and write a one-paragraph summary. What is the current state of the theorem?
