## Mathematics Reasoning Text

Mathematics deals with the definition, properties and manipulation of “objects” such as numbers, lines and shapes. Are these mathematical objects real? The ancient Greeks argued that they are an ideal reality independent of time and space. Their properties exist independently of us and are just waiting to be discovered. However, others like Albert Einstein have argued that we invent these objects (as concepts). They are the products of our thinking and have only the properties we ascribe to them. Some even argue that there are no mathematical objects at all; all we have is an elaborate game with a set of rules and formulas expressed in symbols. Regardless of the view of mathematics, all would agree that mathematics is essential to our understanding and manipulation of the world around us.

How the field of mathematics has developed and continues to develop is related to the different views of the nature of mathematics and can be illustrated (in part) by examining various common forms of mathematical reasoning. In other words, what thinking processes do mathematicians and people who use mathematics, like scientists, employ? The forms of mathematical reasoning discussed here are somewhat simplified and include the following:

• reasoning by deduction
• reasoning by induction
• reasoning by analogy/models

An understanding of mathematical reasoning will help to answer the questions: how is mathematical knowledge created? how is it tested? how is it used?

All Mathematical Reasoning documents, including text and exercises, are available for download in the Download Centre.

“How can it be that mathematics, being after all a product of human thought independent of experience, is so admirably adapted to the objects of reality?”Albert Einstein

## Reasoning by Deduction

Reasoning by deduction is the process of starting with a more general or accepted statement and inferring a less general or more specific conclusion. This reasoning is the opposite of induction—starting with specific instances and inferring a general statement or conclusion.

Reasoning by deduction is generally considered a more powerful argument because it is possible to construct a valid argument so that the conclusion is necessarily true. [Validity is a property of arguments and truth is a property of propositions or statements (e.g., premises or conclusions)]. In other words, a well-developed system of logic exists that ensures the conclusion must be true, if the premises—starting statements—are true. The search for absolute truths has long been a tradition in mathematics and uses an example of reasoning by deduction known as deductive logic.

This emphasis on absolute truth and logic has been challenged by a variety of scholars—mathematicians and philosophers. Mathematical knowledge, like scientific knowledge, is socially constructed and is fallible. Some argue that the emphasis should be less about truth (which we may never know) and more about creating ideas that can be tested. The creating and testing of hypotheses uses reasoning by deduction not only to create hypothesis but also to test them.

## Deductive Logic - a Description

Mathematics has traditionally been viewed as a study of logical systems—a set of axioms and rules for creating concepts or theorems from the axioms—in which truth is transmitted from premises to conclusions. For any particular system, such as geometry, certain fundamental but unproven assumptions (called axioms or postulates) are accepted as absolutely true and rules of logic are used to deduce new concepts or theorems. These new concepts add to the accepted knowledge and can be used to further expand the store of mathematical knowledge in a particular field.

 axioms (unproven assumptions) and/or concepts/theorems (previously proven from axioms) → new concepts/theorems (new proof)

For example, more than two thousand years ago, Euclid used known properties (axioms and proven postulates) of lines, angles and shapes to prove the Pythagorean theorem—in a right-angle triangle, the square of the hypotenuse equals the sum of the squares of the other two sides.

This kind of thinking from accepted truths to specific new concepts is one example of reasoning by deduction, called deductive logic. We can translate this deductive logic into a general form, with a simple example, as follows:

 General form Example If p then q* If a number if divisible by 2, then the number is an even number. p 6 is divisible by 2 therefore q therefore 6 is an even number

* Where p represents the hypothesis and q represents the conclusion.

Notice that the conclusion or product of this reasoning must be no less certain than the starting hypothesis; that is, if the hypothesis is true then the conclusion must be true. In mathematical proofs, the hypotheses are the axioms (assumed to be true) or previously proven concepts or theorems. This means that mathematics can include absolute proofs and claim to discover new truths based on previous or accepted truths.

The method described above is sometimes referred to as a direct proof. However, mathematicians very often use deductive reasoning in a proof by contradiction (also known as “reductio ad absurdum”—Latin, meaning reduction to the absurd). In this method, we assume the opposite of what we are trying to prove and then derive a contradiction or error. This then shows that the assumption is false and, therefore, the original hypothesis is true.

A well-known example of proof by contradiction is the proof that the square root of two is irrational—cannot be expressed as a fraction a/b where a and b are non-zero integers with no common factor; that is, as a simple fraction. The proof by contradiction is outlined below.

(1) If the square root of 2 is rational (assuming the opposite of the premise), then (2) This conclusion can be shown to be incorrect (3) Therefore, the square root of 2 is irrational.

In general symbols, proof by contradiction follows the outline:

 General form Notes If –p then q* assume the opposite of p, denoted by –p –q the conclusion q is shown to be false, denoted by –q therefore p the assumed opposite hypothesis, –p, is false because the conclusion is shown to be false; therefore p is true

* Where p represents the hypothesis and q represents the conclusion.

A proof by contradiction is considered to be as rigourous (logically valid) as a direct proof.

Some Pitfalls of Deductive Logic

• If one or more of the starting premises is not true, then the conclusion cannot be true even if a valid argument is used.
• If there is an error in logic (an invalid argument), then no reliable conclusion can be made regardless of whether the premises are true.

## Creating and Testing Hypotheses (text)

Although mathematics has traditionally been defined in terms of deductive logic based on rigorous proofs, many modern philosophers and mathematics educators argue that this is not the only way mathematical knowledge is created. Furthermore, it is not always necessary or even possible to start with first principles (axioms). Rigorous proofs have their place in mathematics; however, some alternate views of mathematics propose several key points:

• initial assumptions or hypotheses may not be known to be true
• hypotheses or conjectures need to be tested by trying to disprove them with counterexamples—examples that shows the hypothesis to be false
• more/less plausible claims can, in some cases, replace the true/false claims required in deductive logic.

Based on these points, problem solving in mathematics is not restricted to rigid rules (as in deductive logic). Hypotheses can be made from informed guesses based on trial-and-error or from a stated mathematical claim. Hypotheses can also be obtained by making deductions (i.e., “If….then….” statements) based on the implications of known mathematical facts.

Regardless of the origin of the hypothesis, predictions are made and tested by trying to find counterexamples that would falsify the hypothesis. This plan is based on the philosophy of Karl Popper who argued that you can never prove a hypothesis to be true because it is not possible to test every example; however, one counterexample can prove the hypothesis to be false.

Counterexamples serve either to completely reject the hypothesis, or to refine it by restricting or revising definitions and conditions. If no counterexamples can be found, the hypothesis becomes plausible (but certainly not proven no matter how many supporting instances are found). A plausible hypothesis may also be broken down into sub-concepts that are subjected to additional testing and possible refinements.

The process of making and testing hypotheses can be “messy”; that is, jumping between testing, evaluating and revising. Some would argue that mathematics does not develop in a linear, logical fashion.

Example:

For any two integers, x and y, how does the sum of the squares compare with the square of the sum?

Hypothesis: The square of a number increases very rapidly as the size of the number increases, therefore the square of the sum is greater than the sum of the squares; i.e., (x + y)2 > x2 + y2.

Testing:

 x y (x + y)2 x2 + y2 Evaluation 1 2 9 5 verified 0 2 4 4 false; counterexample -2 -3 25 13 verified -2 3 1 13 false; counterexample

Revised Hypothesis:

If both x and y are non-zero positive or negative integers, then the square of the sum is greater than the sum of the squares;

i.e., (x + y)2 > x2 + y2.

According to the limited testing, it appears that the revised hypothesis is plausible. Other questions may arise from this hypothesis; for example, does it apply to real numbers? to imaginary numbers? to matrices?

This is not the only deductive reasoning possible. Using concepts from algebra, reasoning by deduction may also be possible to prove the revised hypothesis.

Some Pitfalls of Creating and Testing Hypotheses

• inadequate testing: insufficient effort to find counterexamples
• incomplete/inadequate conditions for revised hypotheses

## Reasoning by Deduction - Exercise

1. Reasoning by deduction uses a/an __i__ argument that ensures the conclusion must be __ii__ if the premises are __iii__.
2. The choice that best completes the above statement is
 i ii iii A. general probable true B. valid true true C. accepted valid reasonable D. specific testable valid
1. Complete the conclusion for the following deductive argument.
2. If an integer is an even number, then its square is also even.
3. 4 is an even number.
4. Therefore, ….
5. One of Euclid’s axioms, considered to be absolutely true, is very commonly used and can be expressed in symbols as
6. If A = B and B = C, then ….
7. A simple and logical completion of this statement is ...
8. Deductive logic can also be presented in the form of premises leading to a logical conclusion. For example,
9. Premise 1: All men are mortal.
10. Premise 2: Aristotle is a man.
11. Conclusion: Aristotle is mortal.
12. Complete the conclusion from the following premises.
13. Premise 1: All plane figures enclosed by straight lines are polygons.
14. Premise 2: A square is plane figure with four straight sides in a closed path.
15. If a valid argument is used and the conclusion is shown to be false, then the premise or starting hypothesis must be
16. A. inconclusive
17. B. probable
18. C. false
19. D. true
20. Euclid, a Greek mathematician, is credited with one of the first proofs by contradiction of the premise, “There are infinitely many prime numbers.”
21. What is the opposite premise; that is, the first step in the proof by contradiction?
22. Consider the following claim: if a2 – b2 = 1, then a and b cannot be positive integers. Complete a proof by contradiction of this claim.
23. Counter-examples do not constitute a proof but they are useful in both falsifying and revising hypotheses. Consider the hypothesis: “All numbers that are not positive are negative.” Identify a counterexample and revise the hypothesis if necessary.
24. Hypothesis:
25. 26. Test this hypothesis and revise if necessary.

1. B

2. 4² or 16 is an even number

3. A = C

4. Conclusion: A square is a polygon.

5. C

6. There are a finite number of prime numbers.

7. (1) If a2 – b2 = 1, then a and b are positive integers

(2) a2 - b2 = (a - b)(a + b) = 1

either a - b = 1 and a + b = 1

or a - b = -1 and a + b = -1

solving these equations gives either a = 1 and b = 0 or a = -1 and b = 0

Both solutions contradict the premise in (1).

(3) Therefore, the original claim is true.

8. Zero is neither positive nor negative. All nonzero numbers that are not positive are negative.

9.

 x Evaluation 2 1.4 verified 16 4 verified 1 1 false; counterexample 0.5 0.07 false; counterexample

Revised hypothesis: ## Reasoning by Induction

Reasoning by induction is the process of starting with a number of specific instances and creating a general statement or hypothesis. This reasoning is the opposite of deduction—starting with a more general or accepted statement and inferring a less general or more specific instance

Consider the following simple example. Suppose you measure the angles of several different triangles, calculate the sum of the angles, and get 180°. This might lead you to conclude that the sum of the angles for all triangles is 180°. How certain is this hypothesis? The answer to this question depends on the number of different triangles observed. If only a few triangles were measured, then the hypothesis is less probable or certain compared with a large number of triangles measured. In all cases, it is best to word the created concept (hypothesis) using some degree of probability; e.g., “possibly 180°”, “very likely 180°”.

Reasoning by induction is not logically valid because it is usually not possible to observe every known case. Therefore, any claim can never be absolutely true as in deductive logic. Nevertheless, reasoning by induction does play an important role in mathematics because it is often part of the discovery process that produces hypotheses. These hypotheses can then be put to the test using deductive reasoning.

Some Pitfalls of Inductive Reasoning

• inadequate generalization: too few cases or too narrow a range of possibilities have been considered
• selective treatment of evidence: consciously or unconsciously choosing only examples that support the hypothesis that is created

## Inductive Reasoning Exercise

1. Reasoning by induction
2. A. uses patterns to create logically valid proofs
3. B. claims to use accepted truths to create new truths
4. C. develops a general conclusion based on general concepts
5. D. develops a general conclusion based on specific observations
6. Inductive reasoning does not create
7. A. tentative hypotheses
8. B. absolute truths
9. C. probable answers
10. D. supporting evidence
11. Choose the best general conclusion that follows from this specific observation:
12. Every time I touch snow, it feels cold.
13. A. All snow is cold.
14. B. Some snow is cold.
15. C. My hands may be cold.
16. D. My hands are made of snow.
17. In the thirteenth century, Italian mathematician Fibonacci presented a series of numbers that is found in many aspects of nature; for example, in the arrangement of the seeds on a sunflower head. These seeds are observed in two sets of spiral rows, where one curves to the left and the other curves to the right. Complete the following pattern of seeds observed in a sunflower seed head: 1, 2, 3, 5, 8, , ,
18. A. 8, 5, 3
19. B. 13, 34, 89
20. C. 13, 21, 34
21. D. 40, 320, 12 800
22. (a) Using the number 5 as your starting number, complete the following steps.
23. Choose a number.
24. Multiply it by 3.
26. Divide by 3.
27. Subtract 1.
28. What is the new number that you obtained?
29. (b) Choose another number (pick any number that you wish) and repeat the steps from part (a). What is your starting number and the new number that you obtained?
30. (c) Create a generalization that describes how your new number compares with your starting number.
31. (d) Make a hypothesis and then prove it algebraically by repeating the steps in part (a). (Hint: Remember to simplify your algebraic process.)
32. Use the following table of evidence to answer the questions below.
 Length (m) Triangles 1 2 3 Side a 3 9 8 Side b 4 12 15 Side c 5 15 17
1. (a) Describe a relation between the lengths of the sides of these triangles.
2. (b) Write a generalization, in words and symbols, describing how the squares of the length of each of the sides are related.
3. (c) What is the name of this relationship?

1. D
2. B
3. A
4. C
5. (a) 6
6. (b) For example, if the starting number chosen was 7, then the new number will be 8.
7. (c) The new number solution is one more than the original number.
8. (d) Hypothesis: If x = the original number, then the new number will be one more than the original number after completing the following series of steps.
9. Chose a number. x
10. Multiply it by 3. 3x
11. Add 6 . 3x + 6
12. Divide by 3. 3x + 6
13. 3
14. Subtract 1. 3x + 6 - 1
15. 3
16. Simplying: 3 (x + 2) -1 = (x + 2) – 1 = x + 1
17. 3
18. (a) If you square side a and add it to the square of side b, then take the square root of this sum, you get the same number as given for side c.
19. (b) The square of side c is equal to the sum of squares of sides a and b.
20. c2 = a2 + b2
21. (c)Pythagorean Theorem, c2 = a2 + b2

## Reasoning by Analogy/Models

The way we think, the way we express ourselves and the way we reach conclusions are very often based on analogies. An analogy is a comparison between two things or processes; one of which is usually familiar and understood and the other requires explanation or understanding. In logic, an argument by analogy is a form of an inductive argument in which the conclusion has some degree of probability but can never be certain (true). Formal logic, in particular reasoning by deduction, is a central part of mathematics, but it is not the only reasoning. Analogies are often the source of many new mathematical conjectures (hypotheses).

For example, an exploration of solid geometry can be initiated by an analogy to plane geometry. Specifically, we can hypothesize that some characteristics of a tetrahedron, such as finding its centre of gravity, may be analogous to the procedure for finding the centre of gravity of a triangle—the point of intersection of the three medians (a straight line from a vertex to the midpoint of the opposite side). Of course there is no guarantee that an analogy will be useful or even that one exists, but looking for and using analogies can provide many useful suggestions and lead to new knowledge.

Fundamentally, an analogy is about making connections between a source (familiar thing) and a target (unfamiliar thing). Implicit or explicit similarities between the source and the target lead to an attempted transfer of understanding of properties or processes. A model, whether it is physical or mental, is a kind of analogy often used in mathematics and the sciences. Mathematical models are widely used to represent some area of interest whether that be stock markets, traffic patterns, weather, climate or any of a myriad of other human or natural phenomena.

No analogy or model is ever perfect. We try to obtain as many direct correspondences between properties of the source and the analogy/model as possible but we need to be aware of the limitations; for example, what the analogy/model does not include.

Some Pitfalls of Using Anologies or Models

• overgeneralization: inappropriate transfer of meaning from the source to the target
• weak analogies/models: the degree of similarity is low compared with the degree of difference; like “comparing apples and oranges”
• over reliance on intuition: just because something seems intuitively reasonable does not make it so.

“I had a scheme, which I still use today when someone is explaining something that I’m trying to understand: I keep making up examples. For instance, the mathematicians would come in with a terrific theorem, and they’re all excited. As they are telling me the conditions of the theorem, I construct something which fits all the conditions. You know, you have a set (one ball) – disjoint (two balls). Then the balls turn colors, grow hairs, or whatever, in my head as they put more conditions on. Finally, they state the theorem, which is some dumb thing about the ball which isn’t true for my hairy green ball, so I say, ‘False!’ ”

— Richard Feynman, Nobel Prize winner in physics and one the world’s greatest theoretical physicists

## Tools for Mathematics Reasoning

Deductive and inductive reasoning have a long history in mathematics; however, this does not mean that these two basic types of reasoning are used rigourously and explicitly every time one does mathematics. A formal proof by deductive logic is a cornerstone of the foundation of mathematics. However, even for mathematicians, it is not possible or desirable to start from first principles (axioms) every time a problem is tackled. In most cases, previous proofs are accepted as facts to form a body of knowledge. This body of knowledge is used to create new knowledge, test knowledge and use it in applications such as problem solving. All of these activities—create, test and use—involve deductive and inductive reasoning, whether these activities are done explicitly or implicitly.

Although there are many fields of mathematical knowledge, the concepts and procedures from basic arithmetic, algebra, geometry and statistics are the tools often used in mathematical reasoning.

• arithmetic: deals with numbers, their properties and operations
• algebra: deals with symbols and equations—for showing relations, generalizing or modeling
• geometry: deals with size, shape and relative position of figures
• statistics: deals with the collection and analysis of large quantities of data or evidence

Some Pitfalls of Mathematical Tools

• arithmetic: inadequate consideration of constraints or conditions
• algebra: confusion between symbols used as unknowns and symbols for variables
• geometry: inaccurate use of concepts or lack of internal consistency in applying concepts to a figure
• statistics: insufficient size and/or relevancy of data or evidence.

## Mathematical Content and Processes

There are ways that the philosophies of mathematics are infused in two guiding documents important to Alberta curricula: The Western and Northern Canadian Protocol for Mathematics (WNCP, 2006; 2007) and the National Council of Teachers of Mathematics (NCTM) Standards (2000). Each of these documents includes an explicit focus on mathematical reasoning, which is considered one of several mathematical processes. A summary of this infusion is provided in Mathematical Content Processes.

## Assessing Mathematical Reasoning

The assessment rubric for mathematical reasoning (Appendix B) addresses problem solving, considers students' ability to break a problem into manageable sub-goals and their willingness to test, evaluate and try various strategies.