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5.1: Introduction - Mathematics


There is a fair bit of groundwork to cover before we get to the Incompleteness Theorem, and much of that groundwork is rather technical. We will then show that our language is rich enough to express several ideas that will be crucial in the construction of ( heta).

In Chapter 6 we will prove Gödel's Self-Reference Lemma and use that lemma to construct the sentence ( heta). We shall then state and prove the First Incompleteness Theorem, that there can be no decidable, consistent, complete set of axioms for (mathfrak{N}). We will finish the chapter with a discussion of Gödel's Second Incompleteness Theorem, which shows that no reasonably strong set of axioms can ever hope to prove its own consistency.


5.1: Math Calculations Introduction

  • Contributed by Ernstmeyer & Christman (Eds.)
  • Sourced from OpenRN
  • Accurately perform calculations using decimals, fractions, percentages, ratios, and/or proportions
  • Convert between the metric and household systems
  • Use military time
  • Use dimensional analysis
  • Accurately solve calculations related to conversions, dosages, liquid concentrations, reconstituted medications, weight-based medications, and intravenous infusions and evaluate final answer to ensure safe medication administration

The Institute of Medicine (IOM) has estimated that the average hospitalized patient experiences at least one medication error each day. Nurses are the last step in the medication administration process before the medication reaches the patient, so they bear the final responsibility to ensure the medication is safe. To safely prepare and administer medications, the nurse performs a variety of mathematical calculations, such as determining the number of tablets, calculating the amount of solution, and setting the rate of an intravenous infusion. [1]

Dosage calculation in clinical practice is more than just solving a math problem. Nurses must perform several tasks during drug calculations, such as reading drug labels for pertinent information, determining what information is needed to set up the math calculation, performing the math calculations, and then critically evaluating the answer to determine if it is within a safe dosage range for that specific patient. Finally, the nurse selects an appropriate measurement device to accurately measure the calculated dose or set the rate of administration. [2] This chapter will explain how to perform these tasks related to dosage calculations using authentic problems that a nurse commonly encounters in practice.


Exponents and Logarithms are related, let's find out how .

The exponent says how many times to use the number in a multiplication.

In this example: 2 3 = 2 × 2 × 2 = 8

(2 is used 3 times in a multiplication to get 8)

So a logarithm answers a question like this:

The logarithm tells us what the exponent is!

In that example the "base" is 2 and the "exponent" is 3:

So the logarithm answers the question:

What exponent do we need
(for one number to become another number)
?

The general case is:

So an exponent of 2 is needed to make 10 into 100, and:

So an exponent of 4 is needed to make 3 into 81, and:


5.1: Introduction - Mathematics

First day handout (pdf) (course contract & exam schedule)


Exams
Quiz 1 Quiz 2,3. Exam 1 Exam 2 Final
Quiz 1 Key April 17 March 4
ExA ExB Key
April 22
Ex2 Key
Practice for 2.3-4.2
FinalA Key A/B
Practice Exams
Term Quizzes Exam 1 Exam 2 Final
Spring `08 Quiz 1, Quiz 1 key,
Quiz 2, Quiz 2 key
1, 1 key
extra practice
2, 2 key
extra practice
Final, Final key
extra practice
(revised May 7, 2009)

Sample homework cover sheet or, something equivalent can be signed on the top of the first homework page

Homework Due Dates (READ the superscript notes 1,2,3. )

[1] Challenge problems are more difficult than those typically assigned on quizzes or exams. You should try these if you 1) want to get the most out of this course possible, 2) are an Applied Math major, or 3) intend to go to graduate school.

[2] Friday homework posting. Homework due on Fridays is gradually posted as we cover the material and is finalized by noon on the Monday before the homework is due. Homework is due 10am in class, 9:50am in my mailbox (E1 Rm210) no late homework is accepted.

[3] Watch all table rows! The required reading and suggested problems for the next class day may be in the 2nd or 3rd row from the top!


Math in Society: Mathematics for liberal arts majors

Publisher: Portland Community College Math Department

We dedicate this book to our students. May you have greater ease in paying for college and grow your proficiency and confidence in math.


Algebra I

It expands on material introduced in Pre-Algebra while introducing functions, systems of linear equations and trigonometry. The primary goal of Algebra I is for students to develop beyond understanding rudimentary mathematical operations to comprehend more abstract algebraic ideas.

Equations

  1. 1.1 Introduction to Equations
  2. 1.2 Solving Equations - Adding & Subtracting
  3. 1.3 Solving Equations - Multiplying & Dividing
  4. 1.4 Two Step Equations
  5. 1.5 Multistep & Distributive Part I
  6. 1.6 Multistep & Distributive Part II
  7. 1.7 Equations with Multiple Variables
  8. 1.8 Solving Percent Equations
  9. 1.9 More Applications

Linear Functions

  1. 2.1 Introduction to Linear Functions
  2. 2.2 Graphing Coordinate Point
  3. 2.3 Relations and Linear Functions Pt I
  4. 2.4 Relations and Linear Functions Pt II
  5. 2.5 Direct Variation
  6. 2.6 Slope & Rate of Change
  7. 2.7 Calculating Slope
  8. 2.8 Slope-Intercept Form
  9. 2.9 X and Y Intercepts & Standard Form
  10. 2.10 Calculate Slope using dydx and Intercepts
  11. 2.11 Equation of a Line
  12. 2.12 Special Lines
  13. 2.13 Cost vs Time Functions
  14. 2.14 Distance vs Time

Inequalities & Absolute Value Functions

  1. 3.1 Introduction to Inequalities & Absolute Value Functions
  2. 3.2 Writing Solution Sets
  3. 3.3 Graphing 1D
  4. 3.4 Solving Multistep
  5. 3.5 Compound Inequalities
  6. 3.6 Solving Compound Inequalities
  7. 3.7 Inequalities in 2D
  8. 3.8 Absolute Value
  9. 3.9 Graphing Absolute Value Functions

Exponential Functions

  1. 4.1 Introduction to Exponents
  2. 4.2 The Product Property
  3. 4.3 The Quotient Property
  4. 4.4 Zero and Negative Exponents
  5. 4.5 Fractional Exponents
  6. 4.6 Power of a Power Property
  7. 4.7 Power of a Product Property
  8. 4.8 Power of a Fraction
  9. 4.9 Order of Operations with Exponents – Numeric
  10. 4.10 Simplifying Algebraic Expressions with Exponents
  11. 4.11 Scientific Notation Part I
  12. 4.12 Scientific Notation Part II
  13. 4.13 Scientific Notation Part III
  14. 4.14 Exponential Decay
  15. 4.15 Exponential Growth

Polynomials

  1. 5.1 Introduction to Inequalities
  2. 5.2 Adding Polynomials
  3. 5.3 Subtracting Polynomials
  4. 5.4 Adding & Subtracting Polynomials With More Than One Variable
  5. 5.5 Multiplying Polynomials
  6. 5.6 Common Factoring
  7. 5.7 Factoring Polynomials Using The GCF
  8. 5.8 Factoring Perfect Square Trinomials
  9. 5.9 Factoring Difference Of Squares
  10. 5.10 Factoring Trinomials
  11. 5.11 Solving Equations Using Factoring
  12. 5.12 Graphing The Cubic Function

Quadratic Functions

  1. 6.1 Introduction To Quadratic Functions
  2. 6.2 Graphing Quadratic Functions
  3. 6.3 Graphing Quadratic Function - Vertex Form
  4. 6.4 Solving Equations with Square Roots
  5. 6.5 Solving by Completing the Square
  6. 6.6 Converting Quadratic Functions Into Vertex Form Using Completing the Square
  7. 6.7 The Quadratic Formula
  8. 6.8 Graphing Quadratic Inequalities
  9. 6.9 Applications Of Quadratics

Rational Functions

  1. 7.1 Introduction to Rational Functions
  2. 7.2 Simplifying Rational Expressions
  3. 7.3 Adding & Subtracting Rational Expressions
  4. 7.4 Multiplying Rational Expressions
  5. 7.5 Dividing Rational Expressions
  6. 7.6 Graph of a Rational Function

Radical Functions

  1. 8.1 Introduction to Radical Functions
  2. 8.2 Prime Factors
  3. 8.3 Square Roots
  4. 8.4 Simplifying Radicals – Numerical
  5. 8.5 Simplifying Radicals – Algebraic
  6. 8.6 Adding & Subtracting Radicals
  7. 8.7 Multiplying Radicals
  8. 8.8 Dividing Radicals
  9. 8.9 Solving Radical Equations
  10. 8.10 Graphing Radical Functions

Transformations

  1. 9.1 Introduction to Transformations
  2. 9.2 Domain of Parent Functions
  3. 9.3 Range of Parent Functions
  4. 9.4 Translations
  5. 9.5 Reflections
  6. 9.6 Vertical Stretches & Compressions
  7. 9.7 Horizontal Stretches & Compressions
  8. 9.8 Summary on Transformations
  9. 9.9 Multiple Transformations

Systems of Equations & Inequalities

  1. 10.1 Introduction to Systems of Linear Equations & Inequalities
  2. 10.2 Graphing Systems of Linear Equations
  3. 10.3 Graphing Inequalities
  4. 10.4 Graphing Systems of Inequalities
  5. 10.5 Evaluating Expressions using Substitution
  6. 10.6 Solving a System of Equations by Substitution
  7. 10.7 Solving Systems of Equations by Elimination
  8. 10.8 Identifying Types of Systems using Equations
  9. 10.9 Writing an Equation
  10. 10.10 Applications of Systems of Equations & Inequations

Trigonometry

  1. 11.1 Introduction to Trigonometry
  2. 11.2 Sine Ratio
  3. 11.3 Cosine Ration
  4. 11.4 Tangent Ratio

Probability

  1. 12.1 Introduction to Probability
  2. 12.2 Simple Probability
  3. 12.3 Fundamental Counting Principle
  4. 12.4 Independent Events
  5. 12.5 Dependent Events
  6. 12.6 Compound Probability
  7. 12.7 Experimental & Theoretical Probability
  8. 12.8 Set Theory & Venn Diagrams
  9. 12.9 Set Theory - Intersection and Union

A Practical Example: Tap and Tank

Let us use a tap to fill a tank.

The input (before integration) is the flow rate from the tap.

We can integrate that flow (add up all the little bits of water) to give us the volume of water in the tank.

Imagine a Constant Flow Rate of 1:

With a flow rate of 1, the tank volume increases by x. That is Integration!

With a flow rate of 1 liter per second, the volume increases by 1 liter every second, so would increase by 10 liters after 10 seconds, 60 liters after 60 seconds, etc.

The flow rate stays at 1, and the volume increases by x

And it works the other way too:

If the tank volume increases by x, then the flow rate must be 1.

This shows that integrals and derivatives are opposites!

Now For An Increasing Flow Rate

Imagine the flow starts at 0 and gradually increases (maybe a motor is slowly opening the tap):

As the flow rate increases, the tank fills up faster and faster:

  • Integration: With a flow rate of 2x, the tank volume increases by x 2
  • Derivative: If the tank volume increases by x 2 , then the flow rate must be 2x

We can write it down this way:

The integral of the flow rate 2x tells us the volume of water:

The derivative of the volume x 2 +C gives us back the flow rate:

And hey, we even get a nice explanation of that "C" value . maybe the tank already has water in it!

  • The flow still increases the volume by the same amount
  • And the increase in volume can give us back the flow rate.

Which teaches us to always remember "+C".


How to Do it

Step 1 is usually easy, we just have to prove it is true for n=1

Step 2 is best done this way:

  • Assume it is true for n=k
  • Prove it is true for n=k+1 (we can use the n=k case as a fact.)

It is like saying "IF we can make a domino fall, WILL the next one fall?"

Step 2 can often be tricky, we may need to use imaginative tricks to make it work!

Example: is 3 n &minus1 a multiple of 2?

Is that true? Let us find out.

1. Show it is true for n=1

Yes 2 is a multiple of 2. That was easy.

2. Assume it is true for n=k

(Hang on! How do we know that? We don't!
It is an assumption . that we treat
as a fact for the rest of this example)

Now, prove that 3 k+1 &minus1 is a multiple of 2

3 k+1 is also 3×3 k

And then split into and

And each of these are multiples of 2

  • 2×3 k is a multiple of 2 (we are multiplying by 2)
  • 3 k &minus1 is true (we said that in the assumption above)

Did you see how we used the 3 k &minus1 case as being true, even though we had not proved it? That is OK, because we are relying on the Domino Effect .

. we are asking if any domino falls will the next one fall?

So we take it as a fact (temporarily) that the "n=k" domino falls (i.e. 3 k &minus1 is true), and see if that means the "n=k+1" domino will also fall.


Ex 5.1 Class 9 Maths Question 1.
Which of the following statements are true and which are false? Give reasons for your answers.
(i) Only one line can pass through a single point.
(ii) There are an infinite number of lines which pass through two distinct points.
(iii) A terminated line can be produced indefinitely on both the sides.
(iv) If two circles are equal, then their radii are equal.
(v) In figure, if AB – PQ and PQ = XY, then AB = XY.

Solution:
(i) False
Reason : If we mark a point O on the surface of a paper. Using pencil and scale, we can draw infinite number of straight lines passing
through O.

(ii) False
Reason : In the following figure, there are many straight lines passing through P. There are many lines, passing through Q. But there is one and only one line which is passing through P as well as Q.

(iii) True
Reason: The postulate 2 says that “A terminated line can be produced indefinitely.”

(iv) True
Reason : Superimposing the region of one circle on the other, we find them coinciding. So, their centres and boundaries coincide.
Thus, their radii will coincide or equal.

(v) True
Reason : According to Euclid’s axiom, things which are equal to the same thing are equal to one another.

Ex 5.1 Class 9 Maths Question 2.
Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they and how might you define them?
(i) Parallel lines
(ii) Perpendicular lines
(iii) Line segment
(iv) Radius of a circle
(v) Square
Solution:
Yes, we need to have an idea about the terms like point, line, ray, angle, plane, circle and quadrilateral, etc. before defining the required terms.
Definitions of the required terms are given below:

(i) Parallel Lines:
Two lines l and m in a plane are said to be parallel, if they have no common point and we write them as l ॥ m.

(ii) Perpendicular Lines:
Two lines p and q lying in the same plane are said to be perpendicular if they form a right angle and we write them as p ⊥ q.

(iii) Line Segment:
A line segment is a part of line and having a definite length. It has two end-points. In the figure, a line segment is shown having end points A and B. It is written as (overline < AB >) or (overline < BA >).

(iv) Radius of a circle :
The distance from the centre to a point on the circle is called the radius of the circle. In the figure, P is centre and Q is a point on the circle, then PQ is the radius.

(v) Square :
A quadrilateral in which all the four angles are right angles and all the four sides are equal is called a square. Given figure, PQRS is a square.

Ex 5.1 Class 9 Maths Question 3.
Consider two ‘postulates’ given below
(i) Given any two distinct points A and B, there exists a third point C which is in between A and B.
(ii) There exist atleast three points that are not on the same line.
Do these postulates contain any undefined terms? Are these postulates consistent? Do they follow from Euclid’s postulates? Explain.
Solution:
Yes, these postulates contain undefined terms such as ‘Point and Line’. Also, these postulates are consistent because they deal with two different situations as
(i) says that given two points A and B, there is a point C lying on the line in between them. Whereas
(ii) says that, given points A and B, you can take point C not lying on the line through A and B.
No, these postulates do not follow from Euclid’s postulates, however they follow from the axiom, “Given two distinct points, there is a unique line that passes through them.”

Ex 5.1 Class 9 Maths Question 4.
If a point C lies between two points A and B such that AC = BC, then prove that AC = (frac < 1 >< 2 >) AB, explain by drawing the figure.
Solution:
We have,

AC = BC [Given]
∴ AC + AC = BC + AC
[If equals added to equals then wholes are equal]
or 2AC = AB [∵ AC + BC = AB]
or AC = (frac < 1 > < 2 >AB)

Ex 5.1 Class 9 Maths Question 5.
In question 4, point C is called a mid-point of line segment AB. Prove that every line segment has one and only one mid-point.
Solution:
Let the given line AB is having two mid points ‘C’ and ‘D’.

AC = (frac < 1 > < 2 >AB) ……(i)
and AD = (frac < 1 > < 2 >AB) ……(ii)
Subtracting (i) from (ii), we have
AD – AC = (frac < 1 > < 2 >AB-frac < 1 > < 2 >AB)
or AD – AC = 0 or CD = 0
∴ C and D coincide.
Thus, every line segment has one and only one mid-point.

Ex 5.1 Class 9 Maths Question 6.
In figure, if AC = BD, then prove that AB = CD.

Solution:
Given: AC = BD
⇒ AB + BC = BC + CD
Subtracting BC from both sides, we get
AB + BC – BC = BC + CD – BC
[When equals are subtracted from equals, remainders are equal]
⇒ AB = CD

Ex 5.1 Class 9 Maths Question 7.
Why is axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that, the question is not about the fifth postulate.)
Solution:
As statement is true in all the situations. Hence, it is considered a ‘universal truth.’

NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid’s Geometry (युक्लिड के ज्यामिति का परिचय) (Hindi Medium) Ex 5.1





NCERT Solutions for Class 9 Maths Chapter 5 Introduction to Euclid Geometry Ex 5.2

Ex 5.2 Class 9 Maths Question 1.
How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?
Solution:
We can write Euclid’s fifth postulate as ‘Two distinct intersecting lines cannot be parallel to the same line.’

Ex 5.2 Class 9 Maths Question 2.
Does Euclid’s fifth postulate imply the existence of parallel lines ? Explain.
Solution:
Yes. If a straight line l falls on two lines m and n such that sum of the interior angles on one side of l is two right angles, then by Euclid’s fifth postulate, lines m and n will not meet on this side of l. Also, we know that the sum of the interior angles on the other side of the line l will be two right angles too. Thus, they will not meet on the other side also.

∴ The lines m and n never meet, i.e, They are parallel.


Contents

The history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, which is shared by many animals, [14] was probably that of numbers: the realization that a collection of two apples and a collection of two oranges (for example) have something in common, namely the quantity of their members.

As evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have also recognized how to count abstract quantities, like time—days, seasons, or years. [15] [16]

Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic, algebra and geometry for taxation and other financial calculations, for building and construction, and for astronomy. [17] The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. [18] Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry. [19] It is in Babylonian mathematics that elementary arithmetic (addition, subtraction, multiplication and division) first appear in the archaeological record. The Babylonians also possessed a place-value system and used a sexagesimal numeral system [19] which is still in use today for measuring angles and time. [20]

Beginning in the 6th century BC with the Pythagoreans, with Greek mathematics the Ancient Greeks began a systematic study of mathematics as a subject in its own right. [21] Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom, theorem, and proof. His book, Elements, is widely considered the most successful and influential textbook of all time. [22] The greatest mathematician of antiquity is often held to be Archimedes (c. 287–212 BC) of Syracuse. [23] He developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus. [24] Other notable achievements of Greek mathematics are conic sections (Apollonius of Perga, 3rd century BC), [25] trigonometry (Hipparchus of Nicaea, 2nd century BC), [26] and the beginnings of algebra (Diophantus, 3rd century AD). [27]

The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. [28] Other notable developments of Indian mathematics include the modern definition and approximation of sine and cosine, [28] and an early form of infinite series.

During the Golden Age of Islam, especially during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics. The most notable achievement of Islamic mathematics was the development of algebra. Other achievements of the Islamic period include advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. [29] [30] Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī.

During the early modern period, mathematics began to develop at an accelerating pace in Western Europe. The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. [31] Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries. [32] Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, [33] who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show in part that any consistent axiomatic system—if powerful enough to describe arithmetic—will contain true propositions that cannot be proved. [34]

Mathematics has since been greatly extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today. According to Mikhail B. Sevryuk, in the January 2006 issue of the Bulletin of the American Mathematical Society, "The number of papers and books included in the Mathematical Reviews database since 1940 (the first year of operation of MR) is now more than 1.9 million, and more than 75 thousand items are added to the database each year. The overwhelming majority of works in this ocean contain new mathematical theorems and their proofs." [35]

Etymology

The word mathematics comes from Ancient Greek máthēma ( μάθημα ), meaning "that which is learnt," [36] "what one gets to know," hence also "study" and "science". The word for "mathematics" came to have the narrower and more technical meaning "mathematical study" even in Classical times. [37] Its adjective is mathēmatikós ( μαθηματικός ), meaning "related to learning" or "studious," which likewise further came to mean "mathematical." In particular, mathēmatikḗ tékhnē ( μαθηματικὴ τέχνη Latin: ars mathematica) meant "the mathematical art."

Similarly, one of the two main schools of thought in Pythagoreanism was known as the mathēmatikoi (μαθηματικοί)—which at the time meant "learners" rather than "mathematicians" in the modern sense. [38]

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. This has resulted in several mistranslations. For example, Saint Augustine's warning that Christians should beware of mathematici, meaning astrologers, is sometimes mistranslated as a condemnation of mathematicians. [39]

The apparent plural form in English, like the French plural form les mathématiques (and the less commonly used singular derivative la mathématique), goes back to the Latin neuter plural mathematica (Cicero), based on the Greek plural ta mathēmatiká ( τὰ μαθηματικά ), used by Aristotle (384–322 BC), and meaning roughly "all things mathematical", although it is plausible that English borrowed only the adjective mathematic(al) and formed the noun mathematics anew, after the pattern of physics and metaphysics, which were inherited from Greek. [40] In English, the noun mathematics takes a singular verb. It is often shortened to maths or, in North America, math. [41]

Mathematics has no generally accepted definition. [6] [7] Aristotle defined mathematics as "the science of quantity" and this definition prevailed until the 18th century. However, Aristotle also noted a focus on quantity alone may not distinguish mathematics from sciences like physics in his view, abstraction and studying quantity as a property "separable in thought" from real instances set mathematics apart. [42]

In the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions. [43]

A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable. [6] There is not even consensus on whether mathematics is an art or a science. [7] Some just say, "Mathematics is what mathematicians do." [6]

Three leading types

Three leading types of definition of mathematics today are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought. [44] All have severe flaws, none has widespread acceptance, and no reconciliation seems possible. [44]

Logicist definitions

An early definition of mathematics in terms of logic was that of Benjamin Peirce (1870): "the science that draws necessary conclusions." [45] In the Principia Mathematica, Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proved entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's (1903) "All Mathematics is Symbolic Logic." [46]

Intuitionist definitions

Intuitionist definitions, developing from the philosophy of mathematician L. E. J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other." [44] A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proved to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct. Intuitionists also reject the law of excluded middle (i.e., P ∨ ¬ P ). While this stance does force them to reject one common version of proof by contradiction as a viable proof method, namely the inference of P from ¬ P → ⊥ , they are still able to infer ¬ P from P → ⊥ . For them, ¬ ( ¬ P ) is a strictly weaker statement than P . [47]

Formalist definitions

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems". [48] A formal system is a set of symbols, or tokens, and some rules on how the tokens are to be combined into formulas. In formal systems, the word axiom has a special meaning different from the ordinary meaning of "a self-evident truth", and is used to refer to a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

Mathematics as science

The German mathematician Carl Friedrich Gauss referred to mathematics as "the Queen of the Sciences". [49] More recently, Marcus du Sautoy has called mathematics "the Queen of Science . the main driving force behind scientific discovery". [50] The philosopher Karl Popper observed that "most mathematical theories are, like those of physics and biology, hypothetico-deductive: pure mathematics therefore turns out to be much closer to the natural sciences whose hypotheses are conjectures, than it seemed even recently." [51] Popper also noted that "I shall certainly admit a system as empirical or scientific only if it is capable of being tested by experience." [52]

Several authors consider that mathematics is not a science because it does not rely on empirical evidence. [53] [54] [55] [56]

Mathematics shares much in common with many fields in the physical sciences, notably the exploration of the logical consequences of assumptions. Intuition and experimentation also play a role in the formulation of conjectures in both mathematics and the (other) sciences. Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics.

The opinions of mathematicians on this matter are varied. Many mathematicians [57] feel that to call their area a science is to downplay the importance of its aesthetic side, and its history in the traditional seven liberal arts others feel that to ignore its connection to the sciences is to turn a blind eye to the fact that the interface between mathematics and its applications in science and engineering has driven much development in mathematics. [58] One way this difference of viewpoint plays out is in the philosophical debate as to whether mathematics is created (as in art) or discovered (as in science). In practice, mathematicians are typically grouped with scientists at the gross level but separated at finer levels. This is one of many issues considered in the philosophy of mathematics. [59]

Mathematics arises from many different kinds of problems. At first these were found in commerce, land measurement, architecture and later astronomy today, all sciences suggest problems studied by mathematicians, and many problems arise within mathematics itself. For example, the physicist Richard Feynman invented the path integral formulation of quantum mechanics using a combination of mathematical reasoning and physical insight, and today's string theory, a still-developing scientific theory which attempts to unify the four fundamental forces of nature, continues to inspire new mathematics. [60]

Some mathematics is relevant only in the area that inspired it, and is applied to solve further problems in that area. But often mathematics inspired by one area proves useful in many areas, and joins the general stock of mathematical concepts. A distinction is often made between pure mathematics and applied mathematics. However pure mathematics topics often turn out to have applications, e.g. number theory in cryptography.

This remarkable fact, that even the "purest" mathematics often turns out to have practical applications, is what the physicist Eugene Wigner has named "the unreasonable effectiveness of mathematics". [13] The philosopher of mathematics Mark Steiner has written extensively on this matter and acknowledges that the applicability of mathematics constitutes “a challenge to naturalism.” [61] For the philosopher of mathematics Mary Leng, the fact that the physical world acts in accordance with the dictates of non-causal mathematical entities existing beyond the universe is "a happy coincidence". [62] On the other hand, for some anti-realists, connections, which are acquired among mathematical things, just mirror the connections acquiring among objects in the universe, so that there is no "happy coincidence". [62]

As in most areas of study, the explosion of knowledge in the scientific age has led to specialization: there are now hundreds of specialized areas in mathematics and the latest Mathematics Subject Classification runs to 46 pages. [63] Several areas of applied mathematics have merged with related traditions outside of mathematics and become disciplines in their own right, including statistics, operations research, and computer science.

For those who are mathematically inclined, there is often a definite aesthetic aspect to much of mathematics. Many mathematicians talk about the elegance of mathematics, its intrinsic aesthetics and inner beauty. Simplicity and generality are valued. There is beauty in a simple and elegant proof, such as Euclid's proof that there are infinitely many prime numbers, and in an elegant numerical method that speeds calculation, such as the fast Fourier transform. G. H. Hardy in A Mathematician's Apology expressed the belief that these aesthetic considerations are, in themselves, sufficient to justify the study of pure mathematics. He identified criteria such as significance, unexpectedness, inevitability, and economy as factors that contribute to a mathematical aesthetic. [64] Mathematical research often seeks critical features of a mathematical object. A theorem expressed as a characterization of the object by these features is the prize. Examples of particularly succinct and revelatory mathematical arguments have been published in Proofs from THE BOOK.

The popularity of recreational mathematics is another sign of the pleasure many find in solving mathematical questions. And at the other social extreme, philosophers continue to find problems in philosophy of mathematics, such as the nature of mathematical proof. [65]

Most of the mathematical notation in use today was not invented until the 16th century. [66] Before that, mathematics was written out in words, limiting mathematical discovery. [67] Euler (1707–1783) was responsible for many of the notations in use today. Modern notation makes mathematics much easier for the professional, but beginners often find it daunting. According to Barbara Oakley, this can be attributed to the fact that mathematical ideas are both more abstract and more encrypted than those of natural language. [68] Unlike natural language, where people can often equate a word (such as cow) with the physical object it corresponds to, mathematical symbols are abstract, lacking any physical analog. [69] Mathematical symbols are also more highly encrypted than regular words, meaning a single symbol can encode a number of different operations or ideas. [70]

Mathematical language can be difficult to understand for beginners because even common terms, such as or and only, have a more precise meaning than they have in everyday speech, and other terms such as open and field refer to specific mathematical ideas, not covered by their laymen's meanings. Mathematical language also includes many technical terms such as homeomorphism and integrable that have no meaning outside of mathematics. Additionally, shorthand phrases such as iff for "if and only if" belong to mathematical jargon. There is a reason for special notation and technical vocabulary: mathematics requires more precision than everyday speech. Mathematicians refer to this precision of language and logic as "rigor".

Mathematical proof is fundamentally a matter of rigor. Mathematicians want their theorems to follow from axioms by means of systematic reasoning. This is to avoid mistaken "theorems", based on fallible intuitions, of which many instances have occurred in the history of the subject. [b] The level of rigor expected in mathematics has varied over time: the Greeks expected detailed arguments, but at the time of Isaac Newton the methods employed were less rigorous. Problems inherent in the definitions used by Newton would lead to a resurgence of careful analysis and formal proof in the 19th century. Misunderstanding the rigor is a cause for some of the common misconceptions of mathematics. Today, mathematicians continue to argue among themselves about computer-assisted proofs. Since large computations are hard to verify, such proofs may be erroneous if the used computer program is erroneous. [c] [71] On the other hand, proof assistants allow verifying all details that cannot be given in a hand-written proof, and provide certainty of the correctness of long proofs such as that of the Feit–Thompson theorem. [d]

Axioms in traditional thought were "self-evident truths", but that conception is problematic. [72] At a formal level, an axiom is just a string of symbols, which has an intrinsic meaning only in the context of all derivable formulas of an axiomatic system. It was the goal of Hilbert's program to put all of mathematics on a firm axiomatic basis, but according to Gödel's incompleteness theorem every (sufficiently powerful) axiomatic system has undecidable formulas and so a final axiomatization of mathematics is impossible. Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory. [73]

Mathematics can, broadly speaking, be subdivided into the study of quantity, structure, space, and change (i.e. arithmetic, algebra, geometry, and analysis). In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty. While some areas might seem unrelated, the Langlands program has found connections between areas previously thought unconnected, such as Galois groups, Riemann surfaces and number theory.

Discrete mathematics conventionally groups together the fields of mathematics which study mathematical structures that are fundamentally discrete rather than continuous.

Foundations and philosophy

In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed. Mathematical logic includes the mathematical study of logic and the applications of formal logic to other areas of mathematics set theory is the branch of mathematics that studies sets or collections of objects. The phrase "crisis of foundations" describes the search for a rigorous foundation for mathematics that took place from approximately 1900 to 1930. [74] Some disagreement about the foundations of mathematics continues to the present day. The crisis of foundations was stimulated by a number of controversies at the time, including the controversy over Cantor's set theory and the Brouwer–Hilbert controversy.

Mathematical logic is concerned with setting mathematics within a rigorous axiomatic framework, and studying the implications of such a framework. As such, it is home to Gödel's incompleteness theorems which (informally) imply that any effective formal system that contains basic arithmetic, if sound (meaning that all theorems that can be proved are true), is necessarily incomplete (meaning that there are true theorems which cannot be proved in that system). Whatever finite collection of number-theoretical axioms is taken as a foundation, Gödel showed how to construct a formal statement that is a true number-theoretical fact, but which does not follow from those axioms. Therefore, no formal system is a complete axiomatization of full number theory. Modern logic is divided into recursion theory, model theory, and proof theory, and is closely linked to theoretical computer science, [75] as well as to category theory. In the context of recursion theory, the impossibility of a full axiomatization of number theory can also be formally demonstrated as a consequence of the MRDP theorem.

Theoretical computer science includes computability theory, computational complexity theory, and information theory. Computability theory examines the limitations of various theoretical models of the computer, including the most well-known model—the Turing machine. Complexity theory is the study of tractability by computer some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with the rapid advancement of computer hardware. A famous problem is the " P = NP? " problem, one of the Millennium Prize Problems. [76] Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence deals with concepts such as compression and entropy.

Pure mathematics

Number systems and number theory

The study of quantity starts with numbers, first the familiar natural numbers N > and integers Z > ("whole numbers") and arithmetical operations on them, which are characterized in arithmetic. The deeper properties of integers are studied in number theory, from which come such popular results as Fermat's Last Theorem. The twin prime conjecture and Goldbach's conjecture are two unsolved problems in number theory.

As the number system is further developed, the integers are recognized as a subset of the rational numbers Q > ("fractions"). These, in turn, are contained within the real numbers, R > which are used to represent limits of sequences of rational numbers and continuous quantities. Real numbers are generalized to the complex numbers C > . According to the fundamental theorem of algebra, all polynomial equations in one unknown with complex coefficients have a solution in the complex numbers, regardless of degree of the polynomial. N , Z , Q , R , mathbb , mathbb , mathbb > and C > are the first steps of a hierarchy of numbers that goes on to include quaternions and octonions. Consideration of the natural numbers also leads to the transfinite numbers, which formalize the concept of "infinity". Another area of study is the size of sets, which is described with the cardinal numbers. These include the aleph numbers, which allow meaningful comparison of the size of infinitely large sets.

Structure

Many mathematical objects, such as sets of numbers and functions, exhibit internal structure as a consequence of operations or relations that are defined on the set. Mathematics then studies properties of those sets that can be expressed in terms of that structure for instance number theory studies properties of the set of integers that can be expressed in terms of arithmetic operations. Moreover, it frequently happens that different such structured sets (or structures) exhibit similar properties, which makes it possible, by a further step of abstraction, to state axioms for a class of structures, and then study at once the whole class of structures satisfying these axioms. Thus one can study groups, rings, fields and other abstract systems together such studies (for structures defined by algebraic operations) constitute the domain of abstract algebra.

By its great generality, abstract algebra can often be applied to seemingly unrelated problems for instance a number of ancient problems concerning compass and straightedge constructions were finally solved using Galois theory, which involves field theory and group theory. Another example of an algebraic theory is linear algebra, which is the general study of vector spaces, whose elements called vectors have both quantity and direction, and can be used to model (relations between) points in space. This is one example of the phenomenon that the originally unrelated areas of geometry and algebra have very strong interactions in modern mathematics. Combinatorics studies ways of enumerating the number of objects that fit a given structure.

( 1 , 2 , 3 ) ( 1 , 3 , 2 ) ( 2 , 1 , 3 ) ( 2 , 3 , 1 ) ( 3 , 1 , 2 ) ( 3 , 2 , 1 ) (1,2,3)&(1,3,2)(2,1,3)&(2,3,1)(3,1,2)&(3,2,1)end>>
Combinatorics Number theory Group theory Graph theory Order theory Algebra

Space

The study of space originates with geometry—in particular, Euclidean geometry, which combines space and numbers, and encompasses the well-known Pythagorean theorem. Trigonometry is the branch of mathematics that deals with relationships between the sides and the angles of triangles and with the trigonometric functions. The modern study of space generalizes these ideas to include higher-dimensional geometry, non-Euclidean geometries (which play a central role in general relativity) and topology. Quantity and space both play a role in analytic geometry, differential geometry, and algebraic geometry. Convex and discrete geometry were developed to solve problems in number theory and functional analysis but now are pursued with an eye on applications in optimization and computer science. Within differential geometry are the concepts of fiber bundles and calculus on manifolds, in particular, vector and tensor calculus. Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space. Lie groups are used to study space, structure, and change. Topology in all its many ramifications may have been the greatest growth area in 20th-century mathematics it includes point-set topology, set-theoretic topology, algebraic topology and differential topology. In particular, instances of modern-day topology are metrizability theory, axiomatic set theory, homotopy theory, and Morse theory. Topology also includes the now solved Poincaré conjecture, and the still unsolved areas of the Hodge conjecture. Other results in geometry and topology, including the four color theorem and Kepler conjecture, have been proven only with the help of computers.

Change

Understanding and describing change is a common theme in the natural sciences, and calculus was developed as a tool to investigate it. Functions arise here as a central concept describing a changing quantity. The rigorous study of real numbers and functions of a real variable is known as real analysis, with complex analysis the equivalent field for the complex numbers. Functional analysis focuses attention on (typically infinite-dimensional) spaces of functions. One of many applications of functional analysis is quantum mechanics. Many problems lead naturally to relationships between a quantity and its rate of change, and these are studied as differential equations. Many phenomena in nature can be described by dynamical systems chaos theory makes precise the ways in which many of these systems exhibit unpredictable yet still deterministic behavior.

Applied mathematics

Applied mathematics concerns itself with mathematical methods that are typically used in science, engineering, business, and industry. Thus, "applied mathematics" is a mathematical science with specialized knowledge. The term applied mathematics also describes the professional specialty in which mathematicians work on practical problems as a profession focused on practical problems, applied mathematics focuses on the "formulation, study, and use of mathematical models" in science, engineering, and other areas of mathematical practice.

In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics, where mathematics is developed primarily for its own sake. Thus, the activity of applied mathematics is vitally connected with research in pure mathematics.

Statistics and other decision sciences

Applied mathematics has significant overlap with the discipline of statistics, whose theory is formulated mathematically, especially with probability theory. Statisticians (working as part of a research project) "create data that makes sense" with random sampling and with randomized experiments [77] the design of a statistical sample or experiment specifies the analysis of the data (before the data becomes available). When reconsidering data from experiments and samples or when analyzing data from observational studies, statisticians "make sense of the data" using the art of modelling and the theory of inference—with model selection and estimation the estimated models and consequential predictions should be tested on new data. [e]

Statistical theory studies decision problems such as minimizing the risk (expected loss) of a statistical action, such as using a procedure in, for example, parameter estimation, hypothesis testing, and selecting the best. In these traditional areas of mathematical statistics, a statistical-decision problem is formulated by minimizing an objective function, like expected loss or cost, under specific constraints: For example, designing a survey often involves minimizing the cost of estimating a population mean with a given level of confidence. [78] Because of its use of optimization, the mathematical theory of statistics shares concerns with other decision sciences, such as operations research, control theory, and mathematical economics. [79]

Computational mathematics

Computational mathematics proposes and studies methods for solving mathematical problems that are typically too large for human numerical capacity. Numerical analysis studies methods for problems in analysis using functional analysis and approximation theory numerical analysis includes the study of approximation and discretisation broadly with special concern for rounding errors. Numerical analysis and, more broadly, scientific computing also study non-analytic topics of mathematical science, especially algorithmic matrix and graph theory. Other areas of computational mathematics include computer algebra and symbolic computation.

Arguably the most prestigious award in mathematics is the Fields Medal, [80] [81] established in 1936 and awarded every four years (except around World War II) to as many as four individuals. The Fields Medal is often considered a mathematical equivalent to the Nobel Prize.

The Wolf Prize in Mathematics, instituted in 1978, recognizes lifetime achievement, and another major international award, the Abel Prize, was instituted in 2003. The Chern Medal was introduced in 2010 to recognize lifetime achievement. These accolades are awarded in recognition of a particular body of work, which may be innovational, or provide a solution to an outstanding problem in an established field.

A famous list of 23 open problems, called "Hilbert's problems", was compiled in 1900 by German mathematician David Hilbert. This list achieved great celebrity among mathematicians, and at least nine of the problems have now been solved. A new list of seven important problems, titled the "Millennium Prize Problems", was published in 2000. Only one of them, the Riemann hypothesis, duplicates one of Hilbert's problems. A solution to any of these problems carries a 1 million dollar reward. Currently, only one of these problems, the Poincaré conjecture, has been solved.


5.1: Introduction - Mathematics

1. Symmetric Linear Systems 1.1 Introduction 1.2 Gaussian Elimination 1.3 Positive Definite Matrices 1.4 Minimum Principles 1.5 Eigenvalues and Dynamical Systems 1.6 A Review of Matrix Theory 2. Equilibrium Equations 2.1 A Framework for the Applications 2.2 Constraints and Lagrange Multipliers 2.3 Electrical Networks 2.4 Structures in Equilibrium 2.5 Least Squares Estimation and the Kalman Filter 3. Equilibrium in the Continuous Case 3.1 One-dimensional Problems 3.2 Differential Equations of Equilibrium 3.3 Laplace's Equation and Potential Flow 3.4 Vector Calculus in Three Dimensions 3.5 Equilibrium of Fluids and Solids 3.6 Calculus of Variations 4. Analytical Methods 4.1 Fourier Series and Orthogonal Expansions 4.2 Discrete Fourier Series and Convolution 4.3 Fourier Integrals 4.4 Complex Variables and Conformal Mapping 4.5 Complex Integration 5. Numerical Methods 5.1 Linear and Nonlinear Equations 5.2 Orthogonalization and Eigenvalue Problems 5.3 Semi-direct and Iterative Methods 5.4 The Finite Element Method 5.5 The Fast Fourier Transform 6. Initial-Value Problems 6.1 Ordinary Differential Equations 6.2 Stability and the Phase Plane and Chaos 6.3 The Laplace Transform and the z-Transform 6.4 The Heat Equation vs. the Wave Equation 6.5 Difference Methods for Initial-Value Problems 6.6 Nonlinear Conservation Laws 7. Network Flows and Combinatorics 7.1 Spanning Trees and Shortest Paths 7.2 The Marriage Problem 7.3 Matching Algorithms 7.4 Maximal Flow in a Network 8. Optimization 8.1 Introduction to Linear Programming 8.2 The Simplex Method and Karmarkar's Method 8.3 Duality in Linear Programming 8.4 Saddle Points (Minimax) and Game Theory 8.5 Nonlinear Optimization Software for Scientific Computing References and Acknowledgements Solutions to Selected Exercises Index


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