Mathematics is the study of pattern - abstract pattern that places concepts in a systematized relationship to one another, expressed in a symbolic system that we can manipulate using reason alone, with no necessary reference to the world.

Mathematics and the Real World

- Galileo Galilei - “The universe was written in the mathematical language!”

- G.H. Hardy - “Math is all around us and we must discover and/or observe it!”

Pure and Applied Mathematics

Fermat's Last Theorem

Kurt Godel

**AOK**

**Mathematics**

Sheauwn, Joise, Lanse, Eldon

Math as a Foundation

Mathematics asks the central questions of all areas of knowledge: How do we know?, What methods can we use to investigate?, What justifications can we offer for our knowledge claims?

Mathematics is often spoken as a building, constructed on firm foundations. IF the foundations are solid and unshakeable, the construction that is built on top rests secure.

And so, as we enter mathematics, we are going into an area of clarity, abstraction, and according to many brilliant mathematicians, “ultimate beauty”.

Euclid's Axioms:

TOK?!!!!!: Did we discover Mathematics or did we invent it?

- Regardless of whether we consider math to be a part of the universe or to be our invention, it is still undeniable that mathematical equations can describe the physical universe.

- Pi - describes circles/spheres

- Euler’s constant - describes radioactive decay, spread of epidemics and more.

Common Notions:

Postulates:

-Things which are equal to the same thing are also equal to one another.

-If equals be added to equals. the wholes are equal.

-If equals be subtracted from equals, the remainders are equal.

-Things which coincide with one another are equal to one another.

-The whole is greater than the part.

-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 center.

-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.

Pure mathematics refers to the math that is studied with no consideration of necessity or application.

Researchers of pure math (this includes abstract fields like algebra or geometry) only develop pure math for the sake of the principles of mathematics and they are not concerned with the direct applications of their labour.

Applied mathematics refers to the math studied for the sake of application.

Researchers of applied math (fields that include information theory, scientific computing) focus on developing math tools to enhance research in other areas.

Methods: Shared knowledge

- Math is a shared knowledge.

- Individual mathematicians contribute their work to a communal pool where other mathematicians review it and contributes their review to the pool. They build on each other’s works.

Pure Mathematics

Applied Mathematics

Mathematics developed from the study of numbers by ancient Egyptians, Babylonians, and Chinese mathematicians, then through the ancient Greeks, and then through further developments in Arabia and China, and onwards to Europe and now, math has developed into an internationally shared exchange.

- Verifying the truth of that axiom would require someone to accompany the line forever, to ensure that it never intersects the first line. They tried to prove the fifth postulate as if it were a theorem but it failed.

- Lobachevsky replaced Euclid’s fifth postulate with the idea that through a point P next to a given line, at least two lines exist that are parallel to the line.

Fifth Postulate

Therefore, Mathematics...

It is more considered in terms of coherence check: the Euclidean system or the Riemann system are both internally coherent, both with bodies of statements free from contradiction.

is no longer considered in terms of correspondence check for the accuracy of its statements in reference to the world.

No mathematical result enters the realm of mathematics until it becomes public knowledge: it must undergo peer review just as the sciences does it.

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However, this collective knowledge in maths acts differently from the sciences: In the sciences, they examine each other’s works to eliminate error only but in the maths, once a knowledge has been proven, it is proven forever.

And today, math knowledge has increased greatly. At the beginning of the 20th century, there were about 12 subject domains related to math and as it ended, there were about 60 to 70.

And what kind of people are these mathematicians?

Ways Of Knowing

180°

Sense Perception

120°

“The point of mathematics is that in it, we have always got rid of the particular instance, and even any particular sorts of entities… The certainty of mathematics depends on its complete abstract generality.”

90°

Math conclusions do not rest on sense perception

We know that working with right triangles can satisfy the equation, a2 + b2 = c2

When Fermat postulated was that no trios of integers can satisfy equations such as a3 + b3 = c3

or a4 + b4 = c4

and so forth, for powers greater than squares.

Many mathematicians tried and failed to find a proof.

- In 1993, a mathematician named Andrew Wiles have finally proved Fermat's Last Theorem by presenting his 150-page paper at a conference as a "traditional proof"

- Proofs have to rely on intuitive arguments which can be easily translated by trained mathematicians into rigorous deductive chains. Proofs are usually presented this way because too much formality would obscure its main points, much like watching a movie frame by frame would distract the viewer from following its storyline.

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Mathematics

as an Area of Knowledge

Mathematics is a very broad area of knowledge. It shows something of humanity, the fascination, the challenge, the creativity, the aspiration, the disappointments and sense of triumph. It also exhibits characteristics of level of care and development, peer review and the difficulties of new and complex works.

The final characteristic that Math exhibit is that it develops through time. Challenges, theories and products can last over centuries.

Mathematics has flaws and limitations as well. If mathematical truth depends on coherence of all the statements within the mathematical system, what are the implications for mathematics if it is found to contain contradictions? Bertrand Russell and Alfred North Whitehead started to attempt constructing the real number system using mathematical sets. They discovered in 1901 that there's a contradiction regarding those sets which are, or are not, members of themselves.

#MATHswag

Russell's paradox had implication for all mathematics is an intellectual game played by its own internal rules, and expected to be complete and free of contradiction, then what claim to knowledge can it have if there is an inconsistency within it?

Way of Knowing: Language

- mathematical justification does not rely on language

- however mathematical growth does

- everyday language is how mathematical concepts and knowledge is shared

- as math has grown, it has taken on the characteristics of a language

“We cannot speak of single or isolated numbers. The essence of number is always relative, not absolute. A single number is only a single place in a general systematic order. It has no being of its own, no self-contained reality. Its meaning is defined by the position it occupies in the whole numerical system. We conceive it as a new and powerful symbolism which, for all scientific purposes, is infinitely superior to the symbolism of speech. for what we find here are no longer detached words but terms that proceed according to one and the same fundamental plan and that, therefore, show us a clear and definite structural law.”

- Emst Cassirer

Characteristics

of

mathematical "language"

ACTIVITY

precise and explicit

compact

transformable without loss of meaning

abstract and conceptual

it can lead to new conclusions

Interview with Miles Davenport

Miles believes that, like physics, biology can be given simple through mathematics and that, to do this, all the different subsidiary rules and classifications first need to be cut away. He says that mathematics tends to unify our observations by using a formal language to describe basic rules of behavior.

“We will be wrong at first, but only by making testable, quantitative predictions can we advance in the field. The discipline that mathematical analysis imposes on the field is the need to state how different factors will interact, and what outcomes we predict. Because the predictions are quantitative, they can then be rigorously tested, and the models refined. As a result, we learn more about the system in question. At the end of the day, Newtonian physics is a simplification and an abstraction, but those simple rules took us a long way in understanding and predicting the world around us.”

Ways of knowing:

Intuition & Imagination

- mathematics practitioners use intuition and imagination akin to those of other fields of knowledge

Intuition: gives fast and rough grasp of pattern. But cognitive scientists claim it to be faulty. It may happen that creative thinking follows counter-intuitive routes. In math, it may be more useful to look at imagination.

Imagination: "the creative capacity to reassemble familiar components into new ones or project beyond them into fresh conceptualization." It exists in math, as it does in every other area of knowledge, even though it is different than imagination in literature LOL! Mathematicians speak as if their mathematical objects and concepts are real, "Lucy has 10 apples in one hand and 67 oranges in the other", the subject matter is alread in a world of imagination.

“Do mathematicians imagine differently than poets, painters, philosophers, or novelists? Are the cognitive operations necessary for solving problems, writing equations, and engaging in abstract mathematical thought the same as those used in other sorts of creative endeavour? Are the semiotic differences among words, numbers, and diagrams as distinct as they seem? How have accounts of the imagination (philosophical, psychological, physiological, neurological, literary, aesthetic) positioned it in relation to the kind of knowledge that mathematics is thought to provide?”

Mathematicians confirm that they use propositional imaging: they imagine that a statement is true, as scientists do with a hypothesis.

But in mathematics, not only is the generation of the statement imagined, but the testing and solving also require imagination since the statement is not played out in the real world.

Unless Lucy does really have 10 apples and 67 oranges in her hands, in which case woohoo big hands.

CANTOR

- invented set theory at the end of the nineteenth century. He considered infinity, placing him outside of real-world though, then he considered multiple infinities. WOAH WHAT A BADASS!!!!!!!! To prove his ideas of multiple infinities, he used his special power, reductio ad absurdum, and contradicted himself. Obviously this worked and he proved his ideas.

- Clearly, Cantor was using creativity and explored paths other had not. Also clearly, he operated in a world separate from ours where we still act like little babies and count using our fingers. What a hero.

Way of Knowing:

Reason

- in all other areas of knowing, reason is applied one way or another, while in math, reason is free from sense perception and functions independently

AYYYYYEEEEE MEMORIAL DAY MONDAY

- deductive reasoning takes old knowledge, and creates new knowledge.

But for reasoning to work, there needs to be old knowledge.

Premises?

Validity: applies to the reasoning process, if the reasoning has been done according to the rules, it is valid and free of contradictions.

Truth: applies to the content of the statements as checked in different truth checks.

true original premise + valid reasoning

= true conclusion

SUPER DEEP QUESTION

How can we establish those first true statements, so that all of what we reason from them is also true?

“ A mathematician is a person who can find analogies between theorems; a better mathematician is one who can see analogies between proofs and the best mathematicians can notice analogies between theories. One can imagine that the ultimate mathematician is one who can see analogies between analogies.

Stefan Banach”

“ You see, the way that mathematics progresses is you trivialize everything! The way it progresses is that you take a result that originally required an immense effort, and you reduce it to a trivial corollary of a more general theory!” ~Kurt Godel

Mathematics in Social Context

Mathematics is created by individual mathematicians and reviewed by peers in a methodology of shared knowledge.

Math has an intimate relationship with nature

But...

Mathematics

Universal or Cultural?

“Mathematics transcends all cultures and binds us,” ~Janna Levin professor of astronomy and physics

“Abstract knowledge may seem to have nothing to do with any of us and yet has to do with all of us” ~Janna Levin

Cultural Dominance of “western” mathematics

‘Western” Mathematics (Not really western in its merged traditions) is mainly well know through colonization.

Western Math is now a days taught all over the world

Social Attitudes toward Mathematics

Math often regarded as intellectually challenging,

comes in for its share of controversy.

National level professors emphasizes on what math should teach

An American Professor says that Math has the status of a difficult subject.

Activity for Everyone

Whoever gets this question in less than 10 seconds will get a special Dollarama chocolate.

The question is: What is the sum of the consecutive numbers from 1-31?

The timer will start now...

Let's check in the calculator to see if I'm right...

#yay math

such universe

much math

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In 1931 when he was 25 years old, Kurt Godel published “Godel’s Incompleteness Theorem” - States the dream of having mathematics reach a state of completeness is impossible to achieve

Kurt Godel was an Austrian, and later American, logician, mathematician, and philosopher

- Mathematics is a knowledge shared universally. It is known all around the globe.

Alan Bishop, who has specialized in Mathematics ltural values, has identified six forms of recurring mathematical ideas spanning vastly different cultures.

- Counting: Thousands of counting systems exist

- Locating: Navigating and placing things in relation to one another

- Measuring: Units of measurement

- Designing: Geometrical shapes

- Playing: Many games involve features such as puzzles, paradoxes, rules and gambling

- Explaining: People find a reason why for things happen. It can possibly be explaine through math (Patterns, numbers, geometrical shapes, etc)

Western Mathematics spread through trade and taught through various education systems.

The Values of Mathematics were also taught

These values are rationalism and objectivism

"a way of perceiving the world as if it were composed of discrete objects, about to be remove and abstracted...from their context"

Knowledge Questions

Knowledge Questions

To what extend does Mathematics describe the real world?

Is Math the most certain area of knowledge

Does Math need a language to be understood

Knowledge issues that stem from this AOK

If in mathematics, we build on other people’s work and we don’t try to find error in them then how do we know that the basis of our works is true?

How has this AOK shaped our knowledge perspective

We learn that sometimes, there can only be a right or wrong like in math