tok mathematics
Introduction (258-261)
→ There can be one answer to a problem, or at at least one clearly defined set of answers to a problem
→ If there is more than one correct answer, it must be the properties of the numbers, not because of the different procedures generating different numbers
→ Mathematics is a language
→ Mathematics has a rule which stimulates the order in which operations must be done in order to correctly solve a problem, regardless of where the operations appear
→ The order of operations is an invention, not a discovery as it's a result of agreements among mathematicians
→ Convention - a procedure which is done because people have agreed to do it that way
→ Mnemonic - a word or phrase used as an aid to memory, it is not only limited to mathematics
→ Much of mathematics, if not all, is invented, rather than discovered, so it's more like the arts than natural science
Knowledge questions:
Should mathematics be defined as a language?
It depends how you look at it, it does have its own symbols, rules and grammar just like a language, allowing it to forward information quickly and easily just like a language. Some linguists do not consider it as a language though, since it's written rather than a spoken form of communication
Do any other areas of knowledge have a language or function as a language in the way that mathematics does?
Music has notation so that other artists understand the musical ideas, giving it a communicative function like mathematics
Chemistry uses chemical notations and formulas, giving it a function as a language to represent compounds and reactions, which is easily understood among chemists
How does the use of symbols in mathematics to convey meaning differ from the use of symbols in the arts to convey meaning? Are there similarities?
Symbols have a precise and agreed among mathematicians meanings, allowing for clear communication between one another. In the arts, symbols can be more interpretive and subjective, having different meanings based on context and perception of an individual. However, both mathematics and arts use symbols to communicate ideas and concepts
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Mathematics can be used to determine the identity of the creator of a work of art e.g. a computer science professor named Patrick Juola used a computer program and analysed a novel called „The Cuckoo’s Calling, by Robert Galbraith. The program determined that it was actually written by JK Rowling.
Mathematics can be also used to determine whether a painting is authentic e.g. a mathematician called Ingrid Daubechies developed a software program in order to analyse brushstrokes in paintings. The algorithm was able to identify the real painting of Van Gogh based on the brushstrokes.
Case study: Katherine Johnson
She worked on calculations which allowed the module to orbit the Earth and to re-enter safely. Her work, done by hand, was more accurate than the work done by computers.
Concept connection
The space race provides an example of how systematic power shapes the knowledge that is developed and which becomes the foundation for later knowledge development. The goal of the US was to put a man on the moon before the year 2000. It was achieved only because of the power of the US government financing the project. With this many people worked on solving problems that have never been solved before, in other words making brand new knowledge in many fields like science, mathematics and technology.
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Mathematics relies on having knowledge of many concepts describing how the world is organised.
Cognitive skills that must be mastered to make progress in mathematics are eg. recognising patterns and differences; using characteristics such as shape, size and height; organising data based on a logical structure, approximating
As you learn mathematics, you learn to apply these concepts in various scenarios, both theoretical and practical, which helps you understand how to utilise your mathematical skills in unordinary ways in real life.
Pure mathematics does not restrict us to things reflected in the real world -- negative numbers cannot be expressed in reality, but they have to exist in theoretical scenarios (e.g. in 4x+20=4, x MUST equal -4) this is an abstract, but necessary concept that allows for expanding our mathematical knowledge. Other examples are: parallel lines (they extend infinitely so we can only imagine them), infinity itself.
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Mathematical object that do not describe things that exist as physical object in real world:
imaginary numbers
parallel lines ( which extend to infinity)
the concept of infinity
INFINITE NUMBERS - there is an infinite amount of numbers, BUT there is NOT an infinite number of any object in the Universe ( not even atoms!!)
SCOPE OF MATHEMATICS:
Our ability to apply mathematics in an enormous range of situations to help provide solutions to an enormous amounts of situations
All of our cognitive abilities to recognize and organise relationships between physical objects which are strictly imagined
PERSPECTIVES:
STUDENT OF MATHEMATICS - following footsteps of mathematicians who developed the concepts and procedures which now make up the mathematical knowledge that someone who wishes to pursue mathematics as a career must have a foundation
PURE MATHEMATICS // APPLIED -> pure mathematician ( one who considers maths from the perspective of exploring the implications of existing mathematics) and an applied mathematician
Example:
Axioms as a starting place for making knowledge in mathematics:
Axioms are defining characteristics of mathematical systems
WHAT DOES IT MEAN? Once a system, like algebra, has been defined, there are certain characteristics of that system which define the system ( axioms, postulates). So the system is the game and axioms are like rules of a game.
METAPHOR WITH GAME AND AXIOMS:
Rules of a game (so axioms) don't need any kind of argument or proof, they are the rules that have been agreed upon by the people in the game ( pure mathematicians)
What we know when we know the axioms of a system is the nature of the system itself
When we change the rules of a game (axioms), we get a completely new game (mathematical system). However, axioms are not chosen by mathematicians in the same way that rules are developed for games. The system is defined, and the axioms are the assumptions that automatically come along with that system, To change the axioms, we have to change the system first.
KNOWLEDGE QUESTION:
Does the knowledge we can develop about nature from mathematics differ in significant ways from the knowledge that we can gain about nature from the natural sciences or the arts?
Comparative Analysis:
Precision vs. Interpretation: Mathematics offers precise and logically rigorous knowledge. Natural sciences provide empirical, testable, and often dynamic understanding. The arts offer interpretive and expressive insights.
Universality vs. Context-Dependence: Mathematical knowledge is universal and abstract, applicable in various contexts without change. Scientific knowledge is empirical and context-dependent, evolving with new data. Artistic knowledge is subjective, varying across cultures and individuals.
Objective vs. Subjective: Mathematics and natural sciences strive for objectivity and replicability. The arts embrace subjectivity and personal interpretation, reflecting individual or collective experiences
The example of games serves as a metaphor, or an analogy, for how axioms work in mathematics. What other AOKs rely on metaphor or analogy to provide explanations?
1. Natural Sciences
Analogy: The "solar system" model of the atom.
Explanation: Electrons orbit the nucleus like planets orbit the sun, simplifying atomic structure.
2. Human Sciences
Metaphor: The "mind as a computer."
Explanation: The mind processes information like a computer, aiding in understanding cognition.
3. History
Metaphor: The "web of history."
Explanation: Events and people are interconnected like a web, illustrating historical interdependencies.
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Axioms (synonym: postulates) - defining characteristics of mathematical system:
axioms do not need any argument or proof, they are the agreed upon facts of the mathematical system,
the axioms of system are the nature of system itself,
to change an axiom we would have to change the system first,
we cannot define one set of axioms as ‘wrong’, just because it does not work in every system, an example:
The Triangle Sum Theorem states that the interior of a triangle adds up to 180°, however, when we move from the geometry of flat planes to the geometry of a sphere, the axioms change - the sum of angles spherical triangle varies with the size of the triangle and can be as large as 540°.
Axioms —-> Theorem
Axioms are the earliest premises in developing the arguments which then become theorems - established proofs. However, it’s important to remember that the theorem can also act as a premise in later proofs:
As you can see, connections between each level in the diagram above were established by using logic, therefore:
So long as the logic is valid, and if all the statements in the first column are true, then the statements in the second column are true, then the statement in the third column is true.
Euclidean geometry - the geometry of the flat plane (a flat surface that is infinitely large and with zero thickness)
Euclid’s Postulates form the entire basis for geometry:
a straight line segment can be drawn joining any two points.
Any straight line segment can be extended indefinitely in a straight line.
Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as centre.
All right angles are congruent
If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough. This postulate is equivalent to what is known as the parallel postulate.
To help us understand the role of axioms, the authors of the book recommend us to look at axioms like the rules of the game.
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DEEPER THINKING:
Axioms are assumptions which have been excused from any need or proof. They form the starting point for a long string of logical deductions which become new mathematics, which in turn forms the basis for the invention of still more mathematics
Mathematicians do not care- or need to know- whether the mathematical facts that they are working from are true or in any way related to reality (a claim made by Feynman)
By looking at Euclid’s axioms for geometry, we are accustomed to thinking that geometric figures are real-world objects. E.g. to build a house we need either square or rectangular walls.
Thinking more deeply, we can realise Euclid’s geometry is the geometry of the flat plane. Remembering that the definition of a flat plane is a flat surface that is infinitely large and with zero thickness. We can clearly state that such a thing does not exist. Nothing in the universe has zero thickness, it cannot exist as a physical object
Other geometric figures pose the same problem. Consider the rectangle: a 4-sided polygon where all interior angles are 90’. We have to realise that a rectangle is a part of plane geometry- it is not infinite but it does have zero thickness. It is considered to be absolutely perfect: no flaws, perfect four exactly 90’ angles- not even a tiniest smidgen more or less. In the real world we are not physically able to create a perfect rectangle, the angles would be just slightly more or less than 90’. We have tools that help us to get closer, and with computers we can get pretty close, but if we print it on paper, flaws will become visible
That level of precision is not necessary for us to be able to use the principles of plane geometry in the real world. Our wall is not going to fall down because of tiny errors in measurement, so long as, when we create our real world rectangle, the corners are as close to 90’ as possible
APPLIED MATHEMATICS:
Pure mathematics proceeds within its own boundaries as an intellectual pursuit without needing any direct connection to the real world
Much mathematics does apply to the real world, and it gives us another perspective on how and why we make knowledge in mathematics: applied mathematics
There are people working as mathematicians whose objective is the development of the mathematics needed in order to solve problems in the real world.
An example of this kind of applied mathematician is one working on developing ‘unbreakable’ codes and on how to break ‘unbreakable’ codes. There are quite a few such mathematicians in the world and most of them are working for governments for military purposes
There are specific titles for mathematicians who do these jobs. E.g. cryptographers (the people who create encryption systems) or cryptanalysts (the people breaking encryption systems)
CONCEPT CONNECTION:
The work that Alan Turing and the other cryptanalysts did (The Enigma machine- a machine used during World War II in order to assign impossible to decode by a human- being and of various (17 000) combinations coded letters) during the Second World War was incredibly valuable to the world. It is credited with saving thousands of lives and with shortening the war by several years. Also the value to the world of computer technology is incalculable
In the twenty- first century. pure mathematics, as well as, applied one are of great importance and valued deeply
By the number of nobel prizes given to different mathematicians, such as Andrew WIles we can understand that their outstanding work is valued, no matter if it has great utility in the world
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Different perspectives on mathematics
→ different perspectives on mathematics result in different kinds of mathematical knowledge: different mathematical systems represent differing mathematical perspectives
→ branches of mathematics build one on other in the same way that new mathematics builds up on existing mathematics
- each branch can be seen as a perspective, because it looks on mathematical questions in different ways, using different methods and generating different kinds of knowledge.
each new branch of mathematics takes a different perspective from the one before it; it allows us to map how something changes over tiny increments of time
Invention or discovery
→ Richard Feynman: thought of mathematics as the ‘language of nature’- if we want to understand the nature, we must learn its language - PLATONISM
Platonism: mathematics exists independently of humans, in the same way that animals and oceans and atoms and molecules exist - mathematics is like atoms and molecules, because we cannot see them
The belief in named after Plato, because he believed in two objects: the material and the immaterial (something that is not made up of matter; it has no physical mass; Plato meant that numbers and other mathematical ‘objects’ exist in the same kind of plane as the human soul) - if we draw a square, than the drawing is a material object, but the actual square is an immaterial object
→ In Platonism all the development of mathematics is the discovery of mathematical principles - view held by many mathematics, but a contentious one
→ opposite view held by mathematics: it is a human invention which has been developed as a means of describing or modelling reality - a useful tool, which we use for solving problems and which we expand by pursuing the logical development of ideas
Perspective that they cannot be both right , we have no way to establish which belief is right – our understanding of what we know is completely different depending on which perspective is correct:
Platonism: part of what we know when we know mathematics is the immaterial world, a reality outside ourselves
anti-Platonist view: our knowledge of mathematics does not reveal anything to us about a reality outside ourselves that would continue to exist if all humanity suddenly blinked out of existence
Three different perspectives in mathematics:
Of the people who study maths, defined by the purpose
Of the content of mathematics, defined by the focus of different branches
Of the different perspectives of the relationship of mathematics to reality
Considering them helps us understand the complexity of mathematics and the diversity of the ways in which our mathematical knowledge develops
Methods and tools
Methods of pure mathematics
Cumulative nature of mathematics: increasing by one addition after another → mathematical knowledge is cumulative as it builds up, so that all the earlier knowledge continues to exist alongside the newer knowledge: new maths is created out of older maths
‘If-then’ structure of mathematics: if this is true, then this must also be true
Infinite nature of mathematics: infinite number of numbers, and a geometric plane, for example, extends infinitely in all directions → we can never demonstrate a mathematical claim by trying out all the possibilities
Deduction had to be used by the Pythagoras
→ Proof: rigorous proof (making new knowledge in mathematics out of series of logical deductions)
Mathematicians begin with a conjecture (an idea that one or more mathematicians think is true, bo which has not yet been through the process of proof.
to convert conjecture into a theorem, the mathematicians must subject it to the process of rigorous (=all the possibilities that the proof could be wrong, have been excluded proof, and proof is absolutely certain)
Knowledge questions:
Is the knowledge generated in different disciplines of the natural and human sciences related in the same way that the knowledge in different disciplines of mathematics is related?
not related in the same way due to:
In the sciences, integration involves empirical application and context-dependent connections, whereas mathematics integrates through intrinsic, logical coherence.
Sciences validate through experiments and observations, while mathematics relies on proofs and logical deduction.
Types of Knowledge: Scientific knowledge is empirical and revisable, while mathematical knowledge is abstract and certain.
Discovery Methods: Sciences use experimentation and observation, while mathematics relies on logical reasoning and abstraction.
Consider the other AOKs: Is the question of their invention or discovery problematic in the same way that it is problematic in mathematics?
What is meant by the term ‘proof’ in mathematics, and how is this similar to, or different, from what is meant by this term in other areas of knowledge?
A proof in mathematics often consists of a logical set of steps that validates the truth of a general statement beyond any doubt.
Mathematics: Rigorous logical deduction from axioms.
Other Areas: Validation may involve empirical evidence, interpretive analysis, or consensus. - difficult to achieve absolute certainty
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CASE STUDY: The Pythagorean Theorem
For the Pythagoras Theorem to become absolutely certain, Pythagoras had to demonstrate that his conjecture was true for every single possible right triangle. Once he could do that, the conjecture was proven and became the Pythagoras Theorem. In this case, the proof can be used as a tool in conveying the logic which ensures the absolute certainty of the proof.
If Pythagoras conjecture was correct, then the area of squares a and b would equal exactly the area of square c. However, simply adding them up would demonstrate that claim but only for this specific triangle.
The question is: How can we show that the area of the orange and green squares combined is equal to the area of the yellow square?
What we can do is create a new figure by moving the square with side a over to the other side of the triangle. A new triangle has formed in the empty space and it is exactly the same as the original triangle.
DEEPER THINKING:
Reason and imagination
To develop his theorem, Pyhtagoras had to use a fair amount of creative thinking as well as deductive reasoning.
Similarly, thinking back to the cats, mice and fish problem you had to use cognitive tools such as
memory → knowledge of language (what the cats, mice and fish stand for, recognize the mathematical symbols)
reasoning (if 3 cats equal 30, then 1 cat must equal 10)
imagination (imagining cats, mice and fish as symbols of numbers)
CONNECTION TO THE CORE THEME:
Imagination and intuition
Imagination: the ability to put pieces of the known together to come up with something new
Intuition: the ability of the mind to work on a problem on the unconscious level, using all of the knowledge previously gathered; trained intuition is much more than just a guess!
Mathematics is an act of imagination about the nature of reality, and imagination is not merely a method of making knowledge in mathematics; mathematics is a system which helps us to imagine the universe.
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Case study:
Three- body problem involves taking the initial positions and velocities (or momenta) of three point masses and calculating their subsequent trajectories using Newton's laws of motion and Newton's law of universal gravitation. The problem for NASA was that celestial bodies in space exert gravitational forces on each other. The solution to the two-body problem had been fairly easily solved, so NASA knew, for example, how the Earth and the Moon were each affected by the gravitational force from the other. Sending a spaceship to the Moon would necessarily introduce a third body into the system, and NASA needed to know how the gravity from the Earth and the Moon would affect the ship’s trajectory. Richard Arenstorf was working at the Army Ballistics Missile Agency and he solved three-body problem for Apollo 11 case. Arenstorf knew precisely what the problem was when he set out to solve it. He was able to solve theproblem for the exact case at hand: the case in which the three bodies were the Earth, the Moon and the Apollo spacecraft. The problem has not been solved for all cases.
Explanation in applied mathematics:
Proof - the mechanism for explaining how mathematicians are absolutely certain
Solving a specific case ( like in case study no rigorous proof but the scientist needed to demonstrate that he is absolutely certain) - it is based on deductive reasoning and peer review.
Problems in applied mathematics:
- Problems are known before mathematicians exist to solve them. Example above 👆👆👆
- Pure mathematicians creating mathematics that can be used to solve real-life problems ( however, it wasn't intentional). For example: probability theory was developed by a mathematician who was a compulsive gambler - Girolamo Cardano. Cardano thought he had calculated the odds of being able to throw a double six with two dice at least once in 24 throws, but it didn’t work. He called in Blaise Pascal and Pierre de Fermat in 1654, and together they laid the foundations for modern probability theory. One very practical, but unanticipated, application of this mathematics is for use in the insurance industry. Insurers used to think that they should sell few policies because each one was of high risk, but probability theory demonstrates that the opposite is true: it’s better to sell many policies because, as shown by the law of large numbers, the bigger the number, the better the prediction.
Tool in mathematics:
- In the past: pencil and paper, ruler, mathematical compass and protractor and cognitive ability of human's mind.
- Now: electronic calculator, computer which exceed the ability to process mathematical concepts beyond human's mind ability. But the basic methods have been unchanged for many centuries: logical deduction and rigorous proof remain the primary means by which mathematical knowledge is extended.
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There are cases in which pure mathematicians create mathematics which can be used later in order to solve real-world problems. Examples of these cases include:
-Irish mathematician came up with the mathematics for something called quaternions. Quaternions are representations of what happens if the complex number system is extended into the fourth dimension – a dimension which we, as three-dimensional beings, cannot experience. After 150 years, they turned out to have an application in robotics and gaming;
-Mathematics about probability theory began to be developed in the seventeenth century by a mathematician who was also a compulsive gambler. He thought he had calculated the odds of being able to throw a double six with two dice at least once in 24 throws, but it didn’t work. He consulted it with a friend and together they laid the foundations for modern probability theory. One very practical, but unanticipated, application of this mathematics is for use in the insurance industry.
Mathematics does not rely on the kind of tools that are used in laboratory science, and for many centuries, the tools that were used were quite simple: a pencil and paper, a ruler and the cognitive abilities of the human mind have been the tools which created much of the world’s maths. Although in recent decades, the electronic calculator and the computer have been added to the repertoire, logical deduction and rigorous proof remain the primary means by which mathematical knowledge is extended.
Ethical concerns with regard to developing mathematical knowledge:
Ethical concerns as they relate to mathematics are different from the ethical concerns for all other areas of knowledge, because only in mathematics is absolute certainty attainable as the standard for knowledge. In order to employ the methods of mathematics ethically, mathematicians must take whatever actions necessary to attempt to attain it.
AMBIGUITY
There are problems for which it can be difficult to determine what to do. PEDMAS says, for instance, that we do division and multiplication before addition and subtraction, but there is not actually a hard rule that we always do division before multiplication, and, in fact, many mathematicians prefer that multiplication comes first.
Ambiguity is a problem in mathematics because in pure mathematics, the standard is absolute certainty. We can’t have absolute certainty if any of the premises in the proof are ambiguous, or if the final statement of the claim is ambiguous.
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Ethical concerns with regard to applying mathematical knowledge:
In the late twenty century a problem arose, which challenges the assumption that absolute certainty is possible. Computers can now generate proofs that that are so complicated that can not be checked by a human minds to the level of absolute certainty
The fact that modern technology has made possible mathematical work that cannot be done by a human or humans has created a problem for existing values and ethical practice.
Mathematicians are struggling with the idea that absolute certainty may have to be dropped as the standard which would make mathematics much more like the natural science where absolute certainty has never been possible since it is based on inductive reasoning.
Ethics play role in applied mathematics: as new technologies arise new ethical questions and new problems also arise
Mathematicians consider mathematics to be elegant and beautiful.
Richard Feynman says: ,,mathematics is being the only way we can truly appreciate nature, even for a moment”
Mathematics has the ability to express so exactly the world of nature and the world of imagination makes it elegant and takes us back to the question whether mathematics is invented or discovered