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6: Analogy - Mathematics

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6: Analogy - Mathematics

Mathematics and Analogical Reasoning

The goal of this conference is to investigate the role of mathematics as a heuristic device for analogical reasoning in science and philosophy.

Empirical science relies heavily on mathematics. Mathematical models enable physicists to simulate dynamical ‘dumb hole’ analogues to gravitational black holes (Curiel 2019 Dardashti, Thébault, and Winsberg 2017 Gryb, Palacios, and Thébault 2019), chemists to study the behaviour of molecules, medical researchers to examine the spreading rate of diseases, and biologists to understand changes in animal populations. There is no doubt that mathematics is an indispensable scientific tool.

However, this is not the only way in which mathematics contributes to scientific progress. Sometimes particular mathematical structures serve scientists as heuristic devices in their own right by giving indications of structural similarities between otherwise unrelated physical systems (Steiner 1989, 1998). For example, scientists and engineers use physical analogies, such as scale models of bridges, along with other kinds of models, such as mathematical models, computer models, and model organisms. Such models are used for a variety of purposes but crucial to their utility is that they resemble or are in some sense analogous to the intended target system (Bartha 2010, 2016). Of particular interest here are mathematical analogies between two apparently different systems. For example, the logistic equation in ecology models the behaviour of a population growing until it approaches carrying capacity. This same equation crops up in many other places as well. In economics it is the diffusion of innovations equation and in chemistry it describes autocatalytic reactions. These three different physical systems apparently share a common mathematical core and this common core serves as the basis for fruitful analogies, such as the prediction of behaviour of chemical reactions because of known features of population growth (Colyvan 2002 Colyvan and Ginzburg 2010). Thus, a shared mathematical structure can offer valuable insights into the physical structures of otherwise unconnected systems .

In philosophical discourse, it is often not singular mathematical formulas, but the entire structure of mathematics that is used as an argumentative point of reference . Especially in the last ten years there has been a surge of publications using mathematics as a heuristic device (Brown 2010) in order to better understand long-standing metaphysical and epistemological problems in different philosophical fields. Metaethicists in particular have started to exploit local structural parallels between mathematics and morality in order to corroborate (Baker 2016 Clarke-Doane 2012, 2014 Enoch 2011 Roberts 2016) or undermine (Berry forthcoming, 2018 Leng 2016) realist views of morality, but mathematics has also been argued to share relevant features with the domains of logic (Leitgeb 2010 Schechter 2010, 2013), modality (Clarke-Doane 2019 Jonas 2017), and even theism (Jonas 2018 Wielenberg 2016). The analogies featuring in those arguments share a common form: a local analogy between mathematics and another meta-empirical domain is identified, from which a global conclusion is drawn about one or both of the domains. Thus, shared features between mathematics and other meta-empirical domains can offer new insights into the metaphysical structures of otherwise unrelated domains.

Questions we aim to address at the conference include (but are not limited to):

How can a positive mathematical analogy generate support for a particular theoretical view about otherwise disconnected physical systems?

Can we be sure that epistemic lessons from one domain carry over to another domain, given that there are always known points of disanalogy? If so, how?

Does the fact that shared mathematical structures can generate new scientific insights have a bearing on (enhanced) indispensability arguments for mathematical realism?

How can a mathematical analogy generate understanding of one system given our understanding of the model system?

What is an adequate methodology for analogical reasoning about meta-empirical domains (like mathematics or ethics)?

Are the mathematical background assumptions of recent arguments featuring mathematical analogies plausible (specifically in light of recent pluralist developments in set theory)?


Contents

Volume I: Induction and analogy in mathematics Edit

Polya begins Volume I with a discussion on induction, not mathematical induction, but as a way of guessing new results. He shows how the chance observations of a few results of the form 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 3 + 7, etc., may prompt a sharp mind to formulate the conjecture that every even number greater than 4 can be represented as the sum of two odd prime numbers. This is the well known Goldbach's conjecture. The first problem in the first chapter is to guess the rule according to which the successive terms of the following sequence are chosen: 11, 31, 41, 61, 71, 101, 131, . . . In the next chapter the techniques of generalization, specialization and analogy are presented as possible strategies for plausible reasoning. In the remaining chapters, these ideas are illustrated by discussing the discovery of several results in various fields of mathematics like number theory, geometry, etc. and also in physical sciences.

Volume II: Patterns of Plausible Inference Edit

This volume attempts to formulate certain patterns of plausible reasoning. The relation of these patterns with the calculus of probability are also investigated. Their relation to mathematical invention and instruction are also discussed. The following are some of the patterns of plausible inference discussed by Polya.


6: Analogy - Mathematics

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by Kristi Youmans Using Critical Thinking Skills in the Math Classroom There’s NOTHING more unique than these math books! The analogies can be used to teach thinking skills to all students, as anchor activities, or as enrichment pages can be laminated and placed in learning centers, assigned as homework, or used as “Warm Ups” before math class begins. Students learn to think critically by completing visual and verbal math analogies, and they must find very close relationships for each analogy. Each one challenges their mathematical thinking just like verbal/language analogies challenge reading students. Students choose the very best answer possible, and they must also clearly state the relationship for the analogy. They may not use the exact language that the teacher uses, but they must know the relationship and be able to say it in their own words. The math analogies correlate to national standards identified by the National Council of Teachers of Mathematics in 6th-8th grade: Number and Operations, Geometry, Measurement, Algebra, Data Analysis and Probability, Communication, Connections, Problem Solving, Reasoning and Proof, and Representations


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Universitas Islam Riau
Indonesia

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Analogical reasoning ability of mathematics education students at six state islamic universities (UIN) in Indonesia

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References

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Analogy Questions and Answers

Q1. Clock : Time : : Thermometer : ?

B. Temperature

D. Radiation

Q2. Gun : Bullet : : Chimney : ?

Q3. Candle : Wax : : Paper : ?

Q4. Paw : Cat : : Hoof : ?

A. Elephant

Q5. Flower : Bud : : Plant : ?

Q6. Genuine : Authentic : : Mirage : ?


Analogy - Solved Examples

In the following question, choose the pair/group of words that show the same relationship as given at the top of every pair/group.

In winter we feel cold similarly in summer we feel hot.

In the following question, choose the pair/group of words that show the same relationship as given at the top of every pair/group.

Teacher’s job is teaching similarly driver’s job is driving.

In the following question, choose the pair/group of words that show the same relationship as given at the top of every pair/group.

In swimming pool we swim hence option A is the answer.

In the following question, choose the pair/group of words that show the same relationship as given at the top of every pair/group.

Down is opposite of up. Hence, option C is correct.

In the following question, choose the pair/group of words that show the same relationship as given at the top of every pair/group.

Writer’s job is to write story similarly poet’s job is to make a poetry.

In the following question, choose the pair/group of words that show the same relationship as given at the top of every pair/group.

Pakistan is a neighbouring country of India similarly Canada is related to the USA.

In the following question, choose the pair/group of words that show the same relationship as given at the top of every pair/group.

Option B is the correct answer.

In the following question, choose the pair/group of words that show the same relationship as given at the top of every pair/group.

Q 8 &minus Bible : Bhagwad Geeta

Option C is correct because brinjal and lady finger are of same category i.e. vegetables.

In the following question, choose the pair/group of words that show the same relationship as given at the top of every pair/group.

As football is a sport similarly odia is a language.

In the following question, choose the pair/group of words that show the same relationship as given at the top of every pair/group.


6: Analogy - Mathematics

Analogies compare or contrast words that have a relationship. Learning the relationship between sets of words and how to create analogies improves our use of language. Learning about analogies also:

· develops understanding of the nature of various kinds of relationships

· helps us identify and analyze relationships

· develops and refines our understanding of the specific vocabulary and concepts that are used in analogies

· develops critical thinking abilities

Dry is similar in meaning to arid, just as lost is similar in meaning to mislaid.

Kind is the opposite of cruel, just as happy is the opposite of sad.

CHAPTER : BOOK :: fender : automobile

A chapter is part of a book, just as a fender is part of an automobile

Mirrors are characteristically smooth, just as sandpaper is characteristically rough.

POLKA : DANCE :: frog : amphibian

BIRD : CARDINAL :: house : igloo

A polka may be classified as a dance, just as a frog may be classified as an amphibian.

A cardinal may be classified as a bird, just as a igloo may be classified as an house.

A gift can cause joy, just as rain can cause a flood.

The function of a knife is to cut, just as the function of a shovel is to dig.

A fish can be found in the sea, just as a moose can be found in a forest.

CHUCKLE : LAUGH :: whimper : cry

Chuckle and laugh have similar meanings, but they differ in degree in the same way that whimper and cry have similar meanings but differ in degree

Performer and Related Object

CASHIER : CASH :: plumber : pipe

A cashier works with cash, just as a plumber works with pipe.

Performer and Related Action

AUTHOR : WRITE :: chef : cook

You expect an author to write, just as you expect a chef to cook

Action and Related Object

You boil an egg, just as you throw a ball. (In these items the object always receives the action.)

Finding the relationship between analogies:

1. Identify the relationship between the capitalized pair of words.

2. Look for that relationship in the pairs of words in the answer choices. Eliminate those that do not have that relationship.

3. Choose the pair of words whose relationship and word order match those of the capitalized pair.


MAT Practice &mdash Analogies Involving Mathematics

At least a few of the 120 analogies (questions) on your MAT will test your knowledge of mathematics and your mathematical skills. MAT mathematics content is limited to basic arithmetic, number theory, descriptive statistics, algebra and geometry.

On this page are five MAT-style practice analogies involving mathematics. Further down the page you'll find an analysis of each question.

Directions: In each of the following questions, you will find three terms in upper-case letters and, in parentheses, four answer options lettered (a) , (b) , (c) and (d) . Select the answer option that best completes the analogy with the three other terms.

( a. XXIII, b. XVI, c. LVIII, d. XLII) : XCII :: XVI : LXIV

QUADRILATERAL : OCTAGON :: ( a. septagon, b. square, c. rhombus, d. triangle) : HEXAGON

2 2 : SQUARE ROOT OF 64 :: CUBE ROOT OF 64 : ( a. 2 2 , b. 2 3 , c. 3 2 , d. 3 3 )

( a. 4, b. 8, c. 16, d. 32) : GALLON :: 2 : PINT

1 &ndash1 : ( a. &ndash40, b. 50, c. 100, d. 1,000) :: 5 &ndash1 : 20

The correct answer is (a) . XXIII (23) is one fourth of XCII (92), while XVI (16) is one fourth of LXIV (64). (Analogy type: mathematical)

The correct answer is (d) . A quadrilateral (four-sided geometric shape) contains half the number of sides as an octagon (eight-sided shape). Similarly, a triangle (three-sides shape) contains half the number of sides as a hexagon (six-sided shape). (Analogy type: mathematical)

The correct answer is (b) . 2 2 = 4, which is one half the square root of 64 (8). The cube root of 64 is 4, which is one half of 2 3 (8). (Analogy type: mathematical)

The correct answer is (c) . There are 2 cups to a pint, 2 pints to a quart, and 4 quarts to a gallon. Therefore, there are 16 cups to a gallon. (Analogy type: mathematical)

The correct answer is (c) . 1 &ndash1 = 1/1, or 1, which is 1/100 of 100. 5 &ndash1 = 1/5, which is 1/100 of 20. (Analogy type: mathematical)

stewart/images/icon_new_window.png" /> | GMAC | LSAC | Pearson — Mathematics Teaching in the Middle School

"Together, the contributed articles form a rich and provocative collection, challenging some widely held ideas. They deserve careful reading by psychologists, mathematics educators, and philosophers of mathematics. Lyn English has done an outstanding job organizing varied viewpoints into a volume that is readable and well structured."
— Contemporary Psychology