Inductive Reasoning

11 Key excerpts on "Inductive Reasoning"

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  Journey into Discrete Mathematics

  • Owen D. Byer, Deirdre L. Smeltzer, Kenneth L. Wantz(Authors)
  • 2018(Publication Date)
  • American Mathematical Society

    (Publisher)

  Before looking at the Principle of Mathematical Induction in detail, though, we will examine the role of induction more generally in science and mathematics. 6.1 Inductive and Deductive Thinking Within the physical sciences (biology, chemistry, physics, astronomy), inductive rea-soning is used on an everyday basis to derive a general conclusion after examining many specific cases. If after many observations, it becomes clear that objects that go up in the air (a ball, a skateboarder, a frisbee) return to the earth, one might conclude that “what goes up must come down”—i.e., the law of gravity holds. Similarly, noting 133 134 Chapter 6 Induction that plant growth rates are directly related to the amount of water the plant absorbs leads (inductively) to the conclusion that water is critical to plant growth. Inductive Reasoning is used outside of science also. After listening many times to older speakers of English, a young child concludes that the way to make a verb past tense is to add “d” or “ed”: played, walked, pushed, and so on. Whenever a general principle is reached after observing multiple specific instances, induction has been used. Deductive reasoning, on the other hand, is the application of general theory to a specific situation. That is, the use of deduction involves determining logical conse-quences of an established fact. The Pythagorean Theorem can be proven in complete generality: A triangle is a right triangle if and only if the square of the longest side is equal to the sum of the squares of the two shorter sides. Because this general statement can be proven, any specific example of a triangle in the plane must follow this rule; if the shorter sides of a right triangle have lengths 3 and 4, the longest side must have length 5. Many other examples of deductive reasoning were presented in Chapter 4. In deductive reasoning, if all of the premises are true and valid rules of inference are applied, then the conclusion must also be true.

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  Modern Mathematics

  Made Simple

  • Patrick Murphy(Author)
  • 2014(Publication Date)
  • Made Simple

    (Publisher)

  ELEMENTARY ΜΑΤΉΕΜΑΉΟΑί LOGIC In general we use two difierent styles of reasoning in mathematics—namely, Inductive Reasoning and deductive reasoning. If we reach a conclusion based on our own observations or experiences then we are using Inductive Reasoning. For example, having noticed that each time we add two odd numbers the result is an even number we conclude, inductive-ly, that this may always be true, but of course we have not really proved this result for all possible pairs of odd numbers. Deductive reasoning starts with assumptions and then reaches a conclusion based on those assumptions. In short, we generally discover a mathematical result by Inductive Reasoning but we prove it by deductive reasoning. Inductive Reasoning Inductive Reasoning is the creative part of mathematical activity and it is usually the first step towards finding out what might be worth proving. We have already referred to the example of the addition of two odd numbers always resulting in an even number, so let us pursue this in a little more detail. We experiment by listing a few results: 1 + 3 = 4 3 4-5 = 8 5 + 7 = 12 7 4- 9 = 16 9 + 11 = 20 and so on. Now, thinking inductively about the above list of results we might believe that the result of adding two odd numbers was always a multiple of 4. Are we entitled to suggest that this result is always true? Not on the basis of these five observations but then how many observations must be made before conclud-ing inductively that this result must be true: 100, 1000?—it will have to be a matter of experience which eventually decides. Interestingly, the thousandth example in the above pattern of examples would be 1999 + 2001 = 4000 so are we now able to say that two odd numbers have a sum which is a multiple of 4? The answer is *no' because we have not varied the choice of numbers sufficiently.

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  Identifying, Assessing and Supporting Learners with Dyscalculia

  • Kathleen Kelly(Author)
  • 2020(Publication Date)
  • Corwin UK

    (Publisher)

  Models can also be used to demonstrate how a formula such as ‘the area of a triangle = base times perpendicular height divided by 2’ is derived, and practice can be given in proving this (e.g. by exploring squares, rectangles and parallelograms and dividing them into two triangles). Teacher modelling of procedures and solutions, asking pupils probing questions (e.g. how do you know?), providing opportunities to explain thinking and justify answers, and setting open-ended problems to be solved can all help to develop mathematical thinking and the language used in deductive reasoning. Activities to develop deductive reasoning are included in the structured lessons for one-to-one or small-group teaching (see items 6, 7, 8 and 9 on the lesson plan proforma in Chapter 14). Inductive Reasoning Inductive Reasoning is based on conclusions drawn from observations rather than derived from facts. It involves reasoning from specific examples to generate an overall rule, although it does not provide a proof as the rule found may not apply in all cases. It is often associated with finding patterns and relationships among numbers or figures. It can be used to discover properties (e.g. of shapes) and to disprove conjectures by producing counter-arguments. Inductive and deductive reasoning are considered to be linked, and through developing proficiency in Inductive Reasoning students can start to use deductive reasoning (Murawska and Zollman, 2015). It is important, therefore, to provide sufficient opportunities for pupils to examine examples and look for patterns in order to produce a generalisation. At primary age (5–11) the focus is normally on Inductive Reasoning in order to develop the skills needed for deductive reasoning at secondary age (11+)

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  Handbook of Mathematical Induction

  Theory and Applications

  • David S. Gunderson(Author)
  • 2014(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Part I Theory This page intentionally left blank This page intentionally left blank Chapter 1 What is mathematical induction? Induction makes you feel guilty for getting something out of nothing, and it is artificial, but it is one of the greatest ideas of civilization . —Herbert S. Wilf, MAA address, Baltimore, 10 Jan. 1998. 1.1 Introduction In the sciences and in philosophy, essentially two types of inference are used, deduc-tive and inductive. Deductive inference is usually based on the strict rules of logic and in most settings, deductive logic is irrefutable. Inductive Reasoning is the act of guessing a pattern or rule or predicting future behavior based on past experience. For example, for the average person, the sun has risen every day of that person’s life; it might seem safe to then conclude that the sun will rise again tomorrow. However, one can not prove beyond a shadow of a doubt that the sun will rise tomorrow. There may be a certain set of circumstances that prevent the sun rising tomorrow. Guessing a larger pattern based upon smaller patterns in observations is called empirical induction . (See Chapter 6 for more on empirical induction.) Proving that the larger pattern always holds is another matter. For example, after a number of experiments with force, one might conclude that Newton’s second “law” of motion f = ma holds; nobody actually proved that f = ma always holds, and in fact, this “law” has recently been shown to be flawed (see nearly any modern text in physics, e.g. , [56, p.76]). Another type of induction is more reliable: Mathematical induction is a form of reasoning that proves , without a doubt, some particular rule or pattern, usually infinite. The process of mathematical induction uses two steps. The first step is the “base step”: some simple cases are established. The second step is called 1

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  Reasoning, Communication And Connections In Mathematics: Yearbook 2012, Association Of Mathematics Educators

  Yearbook 2012, Association of Mathematics Educators

  • Berinderjeet Kaur, Tin Lam Toh(Authors)
  • 2012(Publication Date)
  • World Scientific

    (Publisher)

  But mathematics is a deductive science and cannot accept Inductive Reasoning as a valid form of proof. Moreover in everyday reasoning, if there are reasonable-enough justifications from past experiences, a conjecture is often accepted as truth. This is contrary to providing a mathematical proof where a conjecture is not considered to have the same rigour of truth as a proven theorem. Ernest (1984) suggested that the prevalence of the inductive proof scheme could be due to the confusion over the name induction. Ernest also suggested that it is important for teachers to carefully highlight the distinction between heuristic Inductive Reasoning and the proof by mathematical induction as a rigorous deductive proof. He further suggested that to distinguish the two, the heuristic Inductive Reasoning be called by some other name like method of generalisation other than induction. This may not be a problem in Singapore, because students are not told that the heuristic Inductive Reasoning method is also known as induction. So when Junior College students mention the word “induction” it is commonly taken by the teachers and students to refer to mathematical induction. It is more likely that the exhibition of the inductive proof scheme by students here is due to their misconception that the generalisation from patterns constitutes a valid justification. This, as mentioned, may be a carry-over Students’ Reasoning Errors in Writing Proof 231 effect from everyday reasoning, or could be due to their prior experience with questions that require students to deduce an answer from patterns, which they have encountered in secondary schools. JT is one student who had accepted the inductive proof scheme as a valid proof because of his prior experiences in secondary school. I showed JT Figure 2 and asked, I: Is this justifiable? Does this prove? JT: Hmm… It is logical.

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  The Notion of mathematical proof: Key rules and considerations

  • Olga Moreira(Author)
  • 2023(Publication Date)
  • Arcler Press

    (Publisher)

  Within the work in the classroom environment, usually, students mostly experience some memorized algorithmic reasoning [32] or with some formal proofs, a sequence of formulas in the formal language [33] instead of the actual creative mathematical reasoning within a complex mathematical inquiry that enables understanding and construction of a new knowledge. The Notion of Mathematical Proof: Key Rules and Considerations 44 The ability to make some generalizations and to provide appropriate reasoning are both considered as high-level mathematical skills. Pedemonte further [34] studied certain relations between the level of the generalization and reasoning skills. However, generalization as such and concurrently as a process appears to be required for construction of mathematical induction, although, on the other hand, some relation was pointed out as a structural distance between them. Therefore, just the open-ended problems are considered suitable ways to develop required reasoning skills by students. Thus, the abductive reasoning serves for passing from the inductive to deductive reasoning [26]. It is not clear how the classifications or characteristics of generalization and justification arising from the studies that have focused on generalizing a number patterns apply to other mathematics topics or tasks. Nor do these frameworks make clear other reasoning processes that enable generalization and justification to occur [35]. Combinatorics has a special place in the field of mathematics and mathematics education. It is considered as a source of frequent and hard difficulties of students at various levels [15]. The fact is that just the combinatorial thinking presents the process of creating various possible combinations of ideas and cognitive operations [36]. The use of combinatorial thinking in combinatorial situations means developing a special ability to create abstract models and to find the structure of a set of outcomes [37].

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  Introduction to Mathematics with Maple

  • P Adams, K Smith;R V??born??;;(Authors)
  • 2004(Publication Date)
  • WSPC

    (Publisher)

  Chapter 5 Mat hernat ical Induction In this chapter we study proof by induction and prove some im- portant inequalities, particularly the arithmetic-geometric mean inequality. In order to employ induction for defining new objects we prove the so called recursion theorems. Basic properties of pow- ers with rational exponents are also established in this chapter. 5.1 Inductive Reasoning The process of deriving general conclusions from particular facts is called induction. It is often used in the natural sciences. For example, an ornithol- ogist watches birds of a certain species and then draws conclusions about the behaviour of all members of that species. General laws of motion were discovered from the motion of planets in the solar system. The following example shows that we encounter Inductive Reasoning also in mathematics. Example 5.1 numbers: Let us consider the numbers n5 - n for the first few natural n n5-n 1 0 2 30 3 240 4 1020 5 3120 6 7770 7 16800 It seems likely that for every n E N the number n5 - n is a multiple of 10. 129 130 Introduction to Mathematics with Maple The reasoning in the above example does not give us the feeling of cast- iron certainty which mathematical arguments usually have. It may not be true for n = 8, though you can easily check that it is. Even if you have used a computer to check the first billion natural numbers, that does not prove that it is true for all natural numbers. Indeed, basing arguments on a finite number of examples is an uncertain procedure, and it can lead to serious mistakes, as we shall shortly see in Example 5.2. In everyday life, and in the natural sciences, our conclusions are subject to further observations and experiments (devised to check the conclusions). In mathematics this additional check is missing, and there is yet another important difference. In Example 5.1 we observed a few particular cases, but we made conclusions about the validity of the formula for infinitely many cases.

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  Mathematical Reasoning

  Patterns, Problems, Conjectures, and Proofs

  • Raymond Nickerson(Author)
  • 2011(Publication Date)
  • Psychology Press

    (Publisher)

  6

  CHAPTER

  Informal Reasoning in Mathematics

  A mathematician’s work is mostly a tangle of guesswork, analogy, wishful thinking and frustration, and proof, far from being the core of discovery, is more often than not a way of making sure that our minds are not playing tricks. (Rota, 1981)

  The characterization of mathematics as a deductive discipline is accurate but incomplete. It represents the finished and polished consequences of the work of mathematicians, but it does not adequately represent the doing of mathematics. It describes theorem proofs but not theorem proving. Moreover, the history of mathematics is not the emotionless chronology of inventions of evermore esoteric formalisms that some people imagine it to be. It has its full share of color, mystery, and intrigue.

  That the process of mathematical discovery is not revealed in the finished proofs that mathematicians publish was pointed out by Evariste Galois, the brilliant French mathematician who, after inventing group theory, died in a duel at the age of 21. It has been convincingly documented by Polya (1954a, 1954b) and Lakatos (1976). In addition to deducing the implications of axioms, mathematicians also invent new axiomatic systems, and this cannot be done by deductive reasoning alone. As Polish-American mathematician Stanislav Ulam (1976) puts it, “In mathematics itself, all is not a question of rigor, but rather, at the start, of reasoned intuition and imagination, and, also, repeated guessing” (p. 154). Rucker (1982) makes essentially the same point: “In the initial stages of research, mathematicians do not seem to function like theorem-proving machines. Instead, they use some sort of mathematical intuition to ‘see’ the universe of mathematics and determine by a sort of empirical process what is true. This alone is not enough, of course. Once one has discovered a mathematical truth, one tries to find a proof for it” (p. 208).

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  Critical Thinking and Informal Logic

  • Timothy Crews Anderson(Author)
  • 2021(Publication Date)
  • Humanities E-Books

    (Publisher)

  Chapter 3: Inductive Reasoning 3.1 Introduction to Inductive Reasoning As was mentioned in the introduction, in a very real sense this book does not introduce much that the average person does not already do. Nowhere is that more true than with Inductive Reasoning. Recall that an inductive argument is an argument in which it is claimed that the premises provide reasons that sup-port the probable truth of the conclusion . It turns out that this sort of reasoning comprises the vast majority of human thinking and provides the support for most beliefs. Indeed, any belief that comes as a result of experience is supported by Inductive Reasoning and thus is a matter of probability. An explication of the reasons for this would require a more in-depth voyage into philosophy than is possible here, but for those who are interested, follow these links to a discussion of the problem of induction and the philosophy of David Hume . For current purposes, it should suffice to say that an inductive argument expresses an inference in which the conclusion goes beyond what is implicit in the premises. Contrast this with a valid deductive argument in which the conclu-sion can be inferred merely by unpacking what is already stated in the premises. For this reason, while a valid deductive argument gives absolute certainty that its conclusion is true (assuming that the premises are true), an inductive argu-ment can at best only suggest the likelihood that its conclusion is true. 3.2 Evaluation of Inductive Arguments In inductive arguments, it is claimed only that the conclusion is probably true, so evaluating them is not as simple as it is for a deductive argument. The success of an inductive inference must be measured across a spectrum and not in the either-valid-or-invalid system that works for deduction. Another way to think of this is that while successful and unsuccessful deductive argument forms are different

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  Decision Making

  A Behavioral Economic Approach

  • Michal Skorepa(Author)
  • 2017(Publication Date)
  • Red Globe Press

    (Publisher)

  PART II Models and Findings CHAPTER 7 Inductive Reasoning I – Generalization and Sampling ■ Can similarity lead us to violate fundamental laws of probabilistic reasoning? ■ Do experimenters make fools of people by abusing the rules of human communication? ■ Or do we follow probabilities when real money is involved? ■ Is there an unintentional systematic pattern in correct answers to multiple-choice questions? Compared with the first six chapters, this chapter takes a radically different perspective – from now on we will look at specific types of decision-making tasks and at the ways that people cope with them. We start with inductive rea-soning. Among the most fundamental types of Inductive Reasoning tasks are generalization and sampling, which form the subject matter of this chapter. In decision-making practice, these tasks tend to occur in various disguises rather than in the pure and simple forms discussed here. This chapter can thus be viewed as a warm-up for Chapter 8, which discusses Inductive Reasoning phenomena more directly linked to everyday life. Selected classes of inductive tasks Inductive Reasoning occurs in many classes of tasks (Sternberg, 2001). Some of the more important classes of tasks within induction are classification, iden-tification of causes of a phenomenon, generalization, sampling, forecasting, estimation of the probability of an event, correlation assessment, and hypothe-sis testing. When making distinctions among various classes of inductive tasks, however, it should be kept in mind that there are many tasks that fit into more than one class or that are on the borderline between two or more classes. Moreover, even tasks in a single class may differ from each other on many dimensions (how specific the input information is, how specific the response from the individual should be, speed and quality of feedback on the correctness of the judgments). 193

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  Understanding Mathematical Proof

  • John Taylor, Rowan Garnier(Authors)
  • 2016(Publication Date)
  • Chapman and Hall/CRC

    (Publisher)

  Despite its name, any proof by mathematical induction uses deductive rea-soning, which is the basis of all mathematical proofs; it does not rely on the kind of Inductive Reasoning described in section 1.3. 303 304 To motivate the idea behind the method, suppose we are given the following instruction. • If you write down an integer k , then you must also write down the integer k + 1 . What do we need to do to satisfy the instruction? The simple answer is: nothing at all! The statement is a conditional of the form W ( k ) ⇒ W ( k + 1) where W ( k ) stands for ‘you write down k ’. A conditional statement is true whenever its antecedent is false, so the statement above is true when you write nothing at all. Now let’s add another instruction. Consider the following two instructions. 1. Write down the number 1 . 2. If you write down an integer k , then you must also write down the integer k + 1 . What do we now need to do to satisfy these instructions? Firstly, to obey the first instruction, we need to write down ‘1’. Then the second instruction ‘kicks in’: because we have now written down ‘1’ we then need to write down ‘2’. But once we have written down ‘2’, to follow the second instruction, we also need to write down ‘3’. Then we have written ‘3’ so, to follow the second instruction, we need to write ‘4’, then ‘5’, then ‘6’, and so on. In other words, to follow both instructions, we would need to write down all the positive integers. Of course, since Z + is infinite, we cannot actually do this. So, although we cannot actually carry out both instructions, they carry within them a ‘process’ for ‘generating’ all positive integers. This idea is formalised within the natural numbers by the Axiom of Induction, which we give below. The natural numbers can be defined by five relatively simple axioms, now called the Peano Axioms, after the 19th-century Italian mathematician Giuseppe Peano, who presented them in a book in 1889.

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