TAMING THE INFINITE
"Mathematics did not spring into being fully formed. It grew from the cumulative efforts of many people, from many cultures, who spoke many languages. Mathematical Ideas that are still in use today go back to more tan 4000 years."
Chapter 1
Tokens, Tallies and Tablets.
All of the different fields of mathematics are based on one common thing: numbers. Numbers began as symbols and the current symbols we use as numbers came into use around 1500 years ago. The first example of these symbols, according to the book, are the tally marks that were found in a cave. These symbols varied, depending the cultures, from very simple to very complex (like the Babylonians and the Egyptians).
All of the different fields of mathematics are based on one common thing: numbers. Numbers began as symbols and the current symbols we use as numbers came into use around 1500 years ago. The first example of these symbols, according to the book, are the tally marks that were found in a cave. These symbols varied, depending the cultures, from very simple to very complex (like the Babylonians and the Egyptians).
Chapter 2
The Logic of Shape
Reading this chapter I came across many interesting things:
The first very interesting thing I came across was the second paragraph. It starts talking about how pictures leave more room for differences of interpretation than symbols. It mentions this because mathematicians use various types of visual reasoning, and some are pictures. So it hit me, I started thinking about an essay we read called The Rhetoric of the Image by Roland Barthes. I wonder if this essay has anything to do with this. I guess I will find that out later on.
The second very interesting thing was the Pythagoreans´ philosophy. I found it very interesting because the Pythagoreans understood that mathematics is about abstract concepts, and not about reality. They also believed that most of these abstract "things" were just a way to get to the ideal of an object. It is very true. For example, when we draw a circle we are actually trying to copy that abstract ideal in our heads that a circle is perfectly round. Of course our circle isn't perfect, but most of the time we try to perfect it. (Since I am not very good explaining things I did my best, trying to reach that ideal in my head. haha)
My third very interesting thing was kind of a big slap to my face. When I blogged about Euclid, I was complaining that people where mixing in three dimensions and two dimensions. I thought they hadn't been discovered yet. In page 33 it says that in Euclid's The Elements there a treatment to the geometry of two dimensions and of three dimensions (meaning space). I guess he did know about dimensions.
The fourth and last very interesting thing I found in the chapter was the initial statements. By initial statements I think it means the definitions. The book mentions that some of these initial statements can't themselves be proved (I mean, Euclid had to start somewhere). This makes me feel a whole lot better. I don't have to worry that much about understanding what a point is, since I guess even Euclid didn't have tha concept so clear.
I found many other interesting things in this chapter but these are the ones that really stood out.
Reading this chapter I came across many interesting things:
The first very interesting thing I came across was the second paragraph. It starts talking about how pictures leave more room for differences of interpretation than symbols. It mentions this because mathematicians use various types of visual reasoning, and some are pictures. So it hit me, I started thinking about an essay we read called The Rhetoric of the Image by Roland Barthes. I wonder if this essay has anything to do with this. I guess I will find that out later on.
The second very interesting thing was the Pythagoreans´ philosophy. I found it very interesting because the Pythagoreans understood that mathematics is about abstract concepts, and not about reality. They also believed that most of these abstract "things" were just a way to get to the ideal of an object. It is very true. For example, when we draw a circle we are actually trying to copy that abstract ideal in our heads that a circle is perfectly round. Of course our circle isn't perfect, but most of the time we try to perfect it. (Since I am not very good explaining things I did my best, trying to reach that ideal in my head. haha)
My third very interesting thing was kind of a big slap to my face. When I blogged about Euclid, I was complaining that people where mixing in three dimensions and two dimensions. I thought they hadn't been discovered yet. In page 33 it says that in Euclid's The Elements there a treatment to the geometry of two dimensions and of three dimensions (meaning space). I guess he did know about dimensions.
The fourth and last very interesting thing I found in the chapter was the initial statements. By initial statements I think it means the definitions. The book mentions that some of these initial statements can't themselves be proved (I mean, Euclid had to start somewhere). This makes me feel a whole lot better. I don't have to worry that much about understanding what a point is, since I guess even Euclid didn't have tha concept so clear.
I found many other interesting things in this chapter but these are the ones that really stood out.
Chapter 3
Notations and Numbers
As I mentioned before most of the ancient civilizations had their own way of writing numerals. In this chapter we go through the Romans, Greeks, Indian, Arabic Italian and the Mayas.
What I found interesting about this chapter is that most of these numerals are related either because of political matters or because of trading between each other. They each adopt their numerals to other countries, and as a result we have numerals that are much like the ones we have today.
This chapter also introduces us to the negative numbers which at the beginning didn’t actually mean that a value was less than zero but that it was a subtraction (according to the Chinese it represented debt).
The one thing that really got me mad about this chapter is a mistake often made (even by the History Channel): Mayans did not live in South America, only in CENTRAL America. South America did have indigenous tribes, but they were not Maya, one example is the Incas.
As I mentioned before most of the ancient civilizations had their own way of writing numerals. In this chapter we go through the Romans, Greeks, Indian, Arabic Italian and the Mayas.
What I found interesting about this chapter is that most of these numerals are related either because of political matters or because of trading between each other. They each adopt their numerals to other countries, and as a result we have numerals that are much like the ones we have today.
This chapter also introduces us to the negative numbers which at the beginning didn’t actually mean that a value was less than zero but that it was a subtraction (according to the Chinese it represented debt).
The one thing that really got me mad about this chapter is a mistake often made (even by the History Channel): Mayans did not live in South America, only in CENTRAL America. South America did have indigenous tribes, but they were not Maya, one example is the Incas.
Chapter 4
Lure Of The Unknown
This chapter was mostly about algebra, what it means and how it originated. According to the book algebra is numbers represented by letters.
The Babylonians had a great effect on algebra. They invented the linear equations (ax + b = 0) and the quadratic equations (ax2 + bx + c = 0). Babylonian schools, much like our schools are believed to have taught only procedure (the what we do, but not the how).
Al-Jabr. This was something that had been on my mind for quite a while. Since the first week at MPC I had wondered what Al-Jabr was, and I finally found out. Al-Jabr means algebra in Arabic, it is the adding equal amounts to both sides of the equation.
One of the biggest breakthroughs was made by Renaissance Italy, which was to solve 3 types of quadratic equations.
The + and – symbols also arrived through commerce, and from then on many other symbols were invented like: x, =, <. >. ( ), √ . It is amazing to think that so many things were invented by commerce. Commerce is such great part of our lives and we don't even know the influence it has had on many things.
The parts that I found the most interesting was the story of Girolamo Cardano published a book with the 3 types of quadratic equations. And the reason why Robert Recorde used = to represent equality; he said that he could not think of no two things more alike than two parallel lines.
This chapter was mostly about algebra, what it means and how it originated. According to the book algebra is numbers represented by letters.
The Babylonians had a great effect on algebra. They invented the linear equations (ax + b = 0) and the quadratic equations (ax2 + bx + c = 0). Babylonian schools, much like our schools are believed to have taught only procedure (the what we do, but not the how).
Al-Jabr. This was something that had been on my mind for quite a while. Since the first week at MPC I had wondered what Al-Jabr was, and I finally found out. Al-Jabr means algebra in Arabic, it is the adding equal amounts to both sides of the equation.
One of the biggest breakthroughs was made by Renaissance Italy, which was to solve 3 types of quadratic equations.
The + and – symbols also arrived through commerce, and from then on many other symbols were invented like: x, =, <. >. ( ), √ . It is amazing to think that so many things were invented by commerce. Commerce is such great part of our lives and we don't even know the influence it has had on many things.
The parts that I found the most interesting was the story of Girolamo Cardano published a book with the 3 types of quadratic equations. And the reason why Robert Recorde used = to represent equality; he said that he could not think of no two things more alike than two parallel lines.
Chapter 5
Eternal Triangles
Trigonometry = measuring triangles, but it is actually useful for all mathematics.
Origins of trigonometry:
The basic problems were the calculation and properties of a triangle (sides and angles). Trigonometry relies on a number of special functions (cosine, sine and tangent started by George Joachim Rhaeticus). It seems to have originated in astronomy using angles to measure distances.
The most important book in trigonometry was Mathematical Syntaxis of Ptolemy. Early trigonometric was used by Hindu astronomers and mathematics.
Some of the most important functions in mathematics:
Trigonometry = measuring triangles, but it is actually useful for all mathematics.
Origins of trigonometry:
The basic problems were the calculation and properties of a triangle (sides and angles). Trigonometry relies on a number of special functions (cosine, sine and tangent started by George Joachim Rhaeticus). It seems to have originated in astronomy using angles to measure distances.
The most important book in trigonometry was Mathematical Syntaxis of Ptolemy. Early trigonometric was used by Hindu astronomers and mathematics.
Some of the most important functions in mathematics:
- The logarithm (Log x, logxy =logx + logy)
- Nasperian logarithms
- Base 10 logarithms
- Number e
Chapter 6
Curves and Coordinates
Fermat: Discovered the connection between algebra and geometry. He was the first person to describe coordinates.
Descartes: He gave us the modern notion of coordinates, and also the second and third dimension.
One of the important applications of coordinates in mathematics are that they help us represent functions graphically.
I had never thought of rotating a picture on the computer as using a coordinate plane.
The greatest breakthroughs often hinge upon making some unexpected connection between previously distinct topics.
Fermat: Discovered the connection between algebra and geometry. He was the first person to describe coordinates.
Descartes: He gave us the modern notion of coordinates, and also the second and third dimension.
One of the important applications of coordinates in mathematics are that they help us represent functions graphically.
I had never thought of rotating a picture on the computer as using a coordinate plane.
The greatest breakthroughs often hinge upon making some unexpected connection between previously distinct topics.
Chapter 7Patterns In Numbers
The first number theory was made by Euclid. Every number is built by primes. Diophantus: 3-4-5 triangle 32 + 42 = 52 5-12-13 triangle I have always been amazed by these triangles. Fermat: 2 is the only even prime Gauss:
|
Chapter 8
The System of the World
Calculus was invented by Isaac Newton. After its discovery, mathematical patterns governed almost everything in the physical world.
Calculus is the mathematics of instantaneous rates of change – how rapidly is some particular quantity changing at this very instant?
Calculus is divided into:
Calculus was invented by Isaac Newton. After its discovery, mathematical patterns governed almost everything in the physical world.
Calculus is the mathematics of instantaneous rates of change – how rapidly is some particular quantity changing at this very instant?
Calculus is divided into:
Differential Calculus:
|
Integral calculus:
|
Physical astronomical background:
Mathematical background:
- Planets: From Ptolemy, to Copernicus, to Brahe, to Kepler, to Galileo.
- Kepler´s laws of planetary motions that are still in use today:
- Planets move around the sun in elliptical orbits.
- Planets sweep out equal areas in equal times.
- The square of the period of revolution of any planet is proportional to the cube of its average distance from the sun
Mathematical background:
- Leibniz: introduces integrals ∫.
- Newton: law of gravity. Main law of motion: acceleration multiplied by its mass is equal to the force that acts on the body.
Chapter 9
Patterns in Nature
There are two types of differential equations:
D`Alamberts PDE: wove equation, superposition of symmetrically placed waves.
These wavelengths are what guide the overtones or harmonics used in instruments (half a chord is an octave higher)
Applications of PDE:
Contributions of the ODE`s in mechanics:
There are two types of differential equations:
- Ordinary (ODE): refers to an unknown function Y of a single variable X.
- Partial (PDE): refers to an unknown function Y or two or more variables, where X and Y are coordinates in the plane and T is time.
D`Alamberts PDE: wove equation, superposition of symmetrically placed waves.
These wavelengths are what guide the overtones or harmonics used in instruments (half a chord is an octave higher)
Applications of PDE:
- Music (drums and chord)
- Electromagnetism
- Potential theory (gravitational attraction)
- Heat flow (heat equation)
- Fluid dynamics.
Contributions of the ODE`s in mechanics:
- Pairing off coordinates
- Energy.
Chapter 10
Impossible Quantities
Number systems:
Square root of -1 = imaginary number
The first to run into square roots of negative numbers where the renaissance algebraists with the solution of cubic equations. One can find solutions with imaginary numbers, these could occur when solutions did exits.
Number systems:
- Natural numbers: 1, 2, 3…
- Integers: … -1, 0, 1 …
- Rational numbers: p/q
- Real numbers: decimals, √, ∏, e…
Square root of -1 = imaginary number
The first to run into square roots of negative numbers where the renaissance algebraists with the solution of cubic equations. One can find solutions with imaginary numbers, these could occur when solutions did exits.
Chapter 11
Firm Foundations
Formulas are used to model physical reality.
Mathematicians started doubting the definition and use of a function. So in the 19th century they started separating the concepts.
Even though they sorted some thing out, mathematicians still assumed most of the function properties.
“Limits (according to Newton) are about what certain quantities approach as some other number approaches infinity to zero.”
Formulas are used to model physical reality.
Mathematicians started doubting the definition and use of a function. So in the 19th century they started separating the concepts.
- Meaning of terms.
- Representation of a function.
- Properties of a function.
- Representations that guarantee properties.
Even though they sorted some thing out, mathematicians still assumed most of the function properties.
“Limits (according to Newton) are about what certain quantities approach as some other number approaches infinity to zero.”
Chapter 12
Impossible Triangles
Euclid was the only known geometry, until projective geometry appeared, but this geometry had to do more with art.
I don't get the Desargues Theorem.
Similar triangles: having the same angles but different size.
Non Euclidean geometry: natural geometry of a curved surface.
Euclid was the only known geometry, until projective geometry appeared, but this geometry had to do more with art.
I don't get the Desargues Theorem.
Similar triangles: having the same angles but different size.
Non Euclidean geometry: natural geometry of a curved surface.
Chapter 13
The Rise of Symmetry
Group theory: frame work for studying symmetry.
Started with the Babylonians.
It all started by trying to prove the algebraic solutions for equations of the 5th degree.
Abel later proved that it was impossible.
Then Galois realized that the solutions for algebraic equations had to do with symmetry.
Group theory: frame work for studying symmetry.
Started with the Babylonians.
It all started by trying to prove the algebraic solutions for equations of the 5th degree.
Abel later proved that it was impossible.
Then Galois realized that the solutions for algebraic equations had to do with symmetry.
Chapter 14
Algebra Comes of Age
Algebra started to change due to Lie groups and number theory.
Euclidean group: transformations that don’t affect distances and angles
Kinds of geometry:
LIE group: most significant symmetries (satisfies algebraic identities and topological manifolds, transformations in many variables).
Other types of algebraic systems:
Algebra started to change due to Lie groups and number theory.
Euclidean group: transformations that don’t affect distances and angles
- Translation
- Rotation
- Reflection
- Glide reflection
Kinds of geometry:
- Elliptic geometry
- Hyperbolic geometry
- Projective geometry
LIE group: most significant symmetries (satisfies algebraic identities and topological manifolds, transformations in many variables).
Other types of algebraic systems:
- Ring
- Field
- Algebra
Chapter 15
Rubber Sheet Geometry
Geometry = measurement
In the 19th century rubber sheet geometry was created: a flexible geometry.
In other words, topology.
The most important aspect of this geometry is continuity.
What shape is this thing?
Geometry = measurement
In the 19th century rubber sheet geometry was created: a flexible geometry.
In other words, topology.
The most important aspect of this geometry is continuity.
- 3 dimensions
- 2 dimensions
What shape is this thing?
Chapter 16
The Fourth Dimension
4 dimension: Time is its interpretation.
Space: is a collection of objects together with a notion of the distance between them.
People before didn’t believe in 4th dimensional spaces. By trying to find a 3-dimensional algebra, William Rowan Hamilton stumbled upon the 4-dimensional algebra, a system called Quaternions. After Hamilton´s Quaternions, other mathematicians started considering other dimensions: Grassman´s hyper complex numbers and Gibb´s vectors.
All of these led to the same description of a vector (x, y, z).
Later geometers tried to explain this geometrically (differential geometry).
The important part of multidimensional space is to use imagination.
4 dimension: Time is its interpretation.
Space: is a collection of objects together with a notion of the distance between them.
People before didn’t believe in 4th dimensional spaces. By trying to find a 3-dimensional algebra, William Rowan Hamilton stumbled upon the 4-dimensional algebra, a system called Quaternions. After Hamilton´s Quaternions, other mathematicians started considering other dimensions: Grassman´s hyper complex numbers and Gibb´s vectors.
All of these led to the same description of a vector (x, y, z).
Later geometers tried to explain this geometrically (differential geometry).
The important part of multidimensional space is to use imagination.
Chapter 17
The Shape of Logic
Peano´s axioms for whole numbers:
What are numbers?
Numbers are just a conceptual concept. A 2 isnt really a two. They are just symbols for thing they represent.
A set is a collection of objects. In mathematics usually a collection of numbers in brackets.
Mathematics was thought to be true and logically consistent. No one could even think of denying it, until Gödel.
Gödel proved that mathematics could not be proved or disproved. He also proved that mathematics is incomplete. That no system of axioms rich enough to formalize mathematics can be logically complete. Some problems don’t have solutions.
Peano´s axioms for whole numbers:
- Exists a number 0
- Every number has a successor (n + 1)
- If P(n) is a property of numbers such that P(0) is true, and whenever P(n) is true than P(s(n)) is true, then P(n) is true for every n.
What are numbers?
Numbers are just a conceptual concept. A 2 isnt really a two. They are just symbols for thing they represent.
A set is a collection of objects. In mathematics usually a collection of numbers in brackets.
Mathematics was thought to be true and logically consistent. No one could even think of denying it, until Gödel.
Gödel proved that mathematics could not be proved or disproved. He also proved that mathematics is incomplete. That no system of axioms rich enough to formalize mathematics can be logically complete. Some problems don’t have solutions.
Chapter 18
How Likely is That?
Probability studies the chances associated with random events.
Here is where the famous branch STATISTICS comes out, which analyses real world data.
All probabilities lie between 1 and 0. 1 being certain and 0 being impossible.
Probability studies the chances associated with random events.
Here is where the famous branch STATISTICS comes out, which analyses real world data.
All probabilities lie between 1 and 0. 1 being certain and 0 being impossible.
Chapter 19
Number Crunching
Ever since the beginnings of mathematics we have tried to look for devices to help us calculate.
Ever since the beginnings of mathematics we have tried to look for devices to help us calculate.
Mathematics helped design computers and now computers help solve mathematics, and even open new fields for it. (It is a never ending loop).
Chapter 20
Chaos and Complexity
Newton´s differential equations are deterministic:
Once the initial state of the system is known, its future is determined uniquely for all time.
Theoretical monsters:
These helped mathematics break away from the traditional shapes. If we see, nature is full of complex and irregular structures (clouds, trees, rivers, etcetera).
Newton´s differential equations are deterministic:
Once the initial state of the system is known, its future is determined uniquely for all time.
Theoretical monsters:
These helped mathematics break away from the traditional shapes. If we see, nature is full of complex and irregular structures (clouds, trees, rivers, etcetera).
My opinion:
Haz clic aquĆ para modificar.