Nature, Characteristics and Definition of Mathematics

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Mathematics????

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Some Math MythsMath = terrifyingMath = static (Greeks, Newton,…)Math other sciencesMath = solitaryMath = impractical major as career preparation 3Angel Rathnabai

Math has an Image ProblemMathematicians — smart, but “from another world”Actually, math lurks behind the curtain of popular subjects: simulation, forecasting, data mining, networks, polling, design, optimization, synthetic environmentsMathematicians have no monopoly on the practice of math — nearly everyone in science and technology uses it

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Intellectual Foci of the SciencesField The Study of …

Astronomy Origin and evolution of stars, galaxies, and the universe

Biology Genomics, organisms, ecologies

Chemistry Molecular structure of matterEarth Sciences Geology, Oceanography,

Atmospheric Sciences, GeophysicsMaterials Science Micro- and macrostructure of

materialsPhysics Space-time, energy, atomic and

nuclear structure of matterMathematical Sciences Patterns, structures, abstract models

of reality

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Views about Mathematics

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Student’s View of mathematics (Schoenfeld, 1992)

Mathematics problems have one and only one right answer.Mathematics is facts and rules with one way to get the right answer. You find the rule and get the answer. Usually, the rule to use is the one your teacher just taught you.You don’t need to understand why the rules work.If you don’t solve a problem in five minutes, then you’ll never solve it. Give up.Only geniuses discover or create mathematics, so if you forget something, you’ll never be able to figure it out on your own.Mathematics problems have little to do with the real world. In the real world, do what make sense. In mathematics, follow the rules.Mathematics is arithmetic 7Angel Rathnabai

Parent’s View of mathematics (Schoenfeld, 1992)

Mathematics is about numbers and arithmetic, unbending accuracy and infallible rules.The students should know the basics.Mathematics is an innate ability. Mathematics is difficult, and so, students should not be expected to do too much.

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Teacher’s View of mathematicsRichard Skemp (1976) : there are two effectively different subjects being taught under the same name “mathematics”.

Instrumental MathematicsIt consists of a limited number of rules without reasons

Relational MathematicsIt is knowing both what to do and why. It involves building up conceptual structures or schemas from which a learner can produce an unlimited number of rules to fit an unlimited sets of situation. 9Angel Rathnabai

Comprehensive View of Mathematics

Mathematics is not arithmetic.

Mathematics is problem posing and problem solving.

Meaningful problems take a long time to pose as well as to solve. They stimulate curiosity about mathematics, not just about the answer to a problem. They engage a variety of students’ ideas and skills. They lead students to thinking about how the world work from a mathematical point of view and to think about how mathematics itself works. They open up discussion to a variety of contributions from multiple participants.

Cont…

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Mathematics is the activity of finding and studying patterns and relationships.Mathematical activity includes perceiving, describing, discriminating, classifying, and explaining patterns everywhere in number, data, and space, and even in patterns themselves.

Mathematics is a language.Mathematics is also used to communicate about patterns.

Mathematics is doing mathematics.The process of ‘doing’ mathematics is far more than just calculation or deduction; it involves observation of patterns, testing of conjectures, and estimation of results.

Cont…

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Mathematics is a path to independent thinking.Mathematics is an area in which even young children can pose and solve a problem and have confidence that the solution is correct not because the teacher says it is, but because its inner logic is so clear.

Mathematics is a way of thinking and a tool for thinking.

Mathematics is a changing body of knowledge, an ever-expanding collection of related ideas.

Mathematics is useful for everyone.

Cont…

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Meaning of Mathematics

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Mathematics – Meaning‘máthēma’ - "that which is learnt", "what one gets to know" máthēma is derived from ‘manthano’, while the modern Greek equivalent is ‘mathaino’ - "to learn“adjective is ‘mathēmatikós’ meaning "related to learning" or "studious“In Latin, and in English until around 1700, the term mathematics more commonly meant "astrology" (or sometimes "astronomy") rather than "mathematics"; the meaning gradually changed to its present one from about 1500 to 1800. In English, the noun mathematics takes singular verb forms. It is often shortened to maths or, in English-speaking North America, math.

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Definition of Mathematics

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Definition – Mathematics‘“The abstract science which investigates deductively the conclusions implicit in the elementary conceptions of spatial and numerical relations, and which includes as its main divisions geometry, arithmetic, and algebra”

- Oxford English Dictionary, 1933

“The study of the measurement, properties, and relationships of quantities and sets, using numbers and symbols”

- American Heritage Dictionary, 2000

“The science of structure, order, and relation that has evolved from elemental practices of counting, measuring, and describing the shapes of objects”

- Encyclopedia Britannica 16Angel Rathnabai

Definition – Mathematics“The science of quantity”

- Aristotle“The science of indirect measurement” (1851).

- Auguste Comte'sThree leading types of definition of mathematics are called:

O Logicist, “The science that draws necessary conclusions" (1870).

- Benjamin Peirce's

O Intuitionist, "All Mathematics is Symbolic Logic"(1903).

- Russell'sO Formalist,

"Mathematics is the mental activity which consists in carrying out constructs one after the other“

- L.E.J. Brouwer17Angel Rathnabai

Definition – Mathematics“The subject in which we never know what we are talking about, nor wheather what we are saying is true”.

- Bertrand Russell (1901)“Mathematics is the indispensible instrument of all physical researches”.

- Kant “Mathematics is the queen of sciences and arithmetic is queen of all mathematics”.

- Gauss “Mathematics is the gateway and key to all sciences”.

- Bacon“Mathematics is the language of physical sciences and certainly no more marvelous language was created by the mind of man”

- Lindsay“Mathematics is a way to settle in the mind a habit of reasoning”.

- Locke“Mathematics is engaged, in fact, in the profound study of art and the expression of beauty”.

- J.B.Shaw 18Angel Rathnabai

Nature of Mathematics

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Nature of Mathemati

cs

A science of

discovery

An intellectual game

The art of drawing

conclusions

A tool subject

A system of logical processes

An intuitive method

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Characteristics of Mathematics

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Characteristics of

Mathematics

Logical sequence

Structure

Precision and accuracy

AbstractnessMathematical

Language and

Symbolism

Applicability

Generalisation and

classification

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Development of

Mathematics

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An Ancient Subject

Pythagorus of Samos569-475 B.C. (?)

musician, geometer, first “pure” mathematician

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A Modern SubjectKaren Uhlenbeck, 1942 - University of TexasPartial differential equations and mathematical physics

National Academy of SciencesNational Medal of Science, 2000

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Major Subfields of Math Sciences

Subfield The Study of …

Foundations Axiomatic underpinnings of mathematics

Algebra and Combinatorics Structures, relations between discrete objects

Topology and Geometry Spatial structures, patterns, shapes

Number Theory Properties of numbers and polynomials

Analysis Theory of functions, extensions and generalizations of the calculus

Probability Randomness and stochastic phenomena

Statistics Collection, analysis, and application of data

Applied Mathematics Modeling, analyzing and optimizing systems

Computational Mathematics Computer-based, experimental mathematics

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Math as the Hub of Science

Mathematical Sciences

Physical Sciences

Chemistry

Biological

Sciences

Engineering and

Technology

Agriculture

Economics

Psychology

FinanceCommunication

Geosciences

Computers

Arts MusicArcheolo

gy

Logic

Philosophy

Linguistics

Management

Geography

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Math in Society and Technology

Problem/Application Contribution from Mathematics

MRI and CAT Imaging Integral transforms, geometry

Internet: search engines, compression Graph theory, linear algebra, wavelets

Financial options valuation Black-Scholes model and Monte Carlo simulation

Global reconnaissance Signal processing, image processing, data mining

Confidentiality and integrity Number theory, cryptology/combinatorics

Modeling of atmosphere and oceans Wavelets, statistics, numerical analysis

Analysis of the human genome Data mining, pattern recognition, discrete algorithms

Rational drug design Data mining, statistics, optimization

Digital entertainment, animation Signal processing, geometric and graphic algorithms

Aerodynamic design Differential equations, optimization

Earthquake analysis and prediction Statistics, dynamical systems/turbulence

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Few Applications

of Mathematics

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Applied Math - What’s playing in classrooms and

labs near you?Functional Genomics Finance CryptographySimulation of physical systems (e.g., airplanes, tokomaks, hurricanes)Simulation of discrete systems (e.g., traffic flow, networks, battlefields)Immersive visualization

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Functional GenomicsThe Protein Folding Problem A hidden Markov model for sequence analysis

d1 d2 d3 d4

I0 I2 I3 I4I1

m0 m1 m2 m3 m4 m5

Start End

m= match state (output), I = insert state (output), d= delete state (no output)

sandwich proteinimmunoglobulinPDB: 7FAB

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Image Processing: Inpaintings Scratch

Removal Disocclusion

Graffiti Removal

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Synthetic Images

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What’s Math got to do with it

Picture courtesy of Doug Roble, Digital Domains.

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Simulation Example: Aerodynamics

Airflows over wing models can be computed and visualized

1999 Gordon Bell Prize

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Simulation Example: Bioinformatics

Proteins fold in a way that minimizes configuration energy

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“Experimental Mathematics”

"There will be opened a gateway and a road to a large and excellent science into which minds more piercing than mine shall penetrate to recesses still deeper." Galileo (1564-1642) on “experimental mathematics”

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