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Mathematics: Misconceptions

For ClubEd (4/25/13)

Brian Katz

on 21 August 2017

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Transcript of Mathematics: Misconceptions

For many people, math is the only field in which knowledge is absolutely certain and all questions have answers.
Mathematical Truth is subtle and debatable, like all knowledge constructed by humans.
Some truths are out of reach.
Math and Science are asking different questions.
Is math the
"science of patterns"?
The Riemann Hypothesis
Many people think of math as one of the sciences...
Math is not about numbers.
Math is not about shapes.
Math is not about procedures.
Numbers and Shapes
The ability to count is a fundamental piece of consciousness. This skill transforms into mathematics when we explore the nature of numbers.
Many procedures are useful and important, but practicing procedures transforms into mathematics only when we start asking how they work and what they mean.
Geometric objects are part of our lives, so it is useful to know their names and properties, but the study of shape becomes mathematics when we use it to ask about our fundamental assumptions.
A scientist is convinced by consistent data.
Humans have found that the first 10,000,000,000,000 values of the Riemann Zeta Function are simple zeros, and yet this puts us no closer to having proved it than knowing one value.
A Scientist is trying to describe what IS about the world and objects within it. This knowledge is ALWAYS realized as a framework for connecting structures and causes.
In the language of William Perry, people believe that math allows them to hold onto a Dualist dream.
Do we create or discover math?
Kurt Goedel proved that any formal system (that allows for counting) contains unprovable truths - statements that are true but could NEVER be proved.
Non-mathematical Question: What are the decimal representations of the numbers {1/2,1/3,1/4}?
Mathematical Question: Some fractions of the form 1/n have finite decimal representations and other have infinitely long ones. How can we tell by looking at n which one it will be?
Non-mathematical Question: What is the product of these two numbers?
Mathematical Question: Why is the product of two negative numbers a positive number?
Non-mathematical Question: If the side length of a square is 2ft, what is its perimeter?
Mathematical Question: How many sides does a Mobius strip have? What will happen if I cut it in half?
... but this connection is strained.
A mathematician is not convinced by any number of examples.
Science is Inductive.
Mathematics is Deductive.
(PS: If every human on earth checked one such value every hour, it would still take us months so verify the data up to now!)
A mathematician is trying to explore the consequences of a structure, which may tell us about the possible worlds and objects within it.
Are mathematical objects "real"?
What is the firm foundation for mathematical truth?
How can the human mind come to know something other than through experience?
Goedel's Theorem implies that we can NEVER know everything about truth from inside a system!
(PS: This doesn't even mention how we just assume our axioms to be true!)
So, what is mathematics?
Some of my students decided that "mathematician" didn't communicate what I do. They suggested that I instead call myself a "word-ninja".
I might define mathematics as the study of abstract structure, pattern, and relation.
A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.

-GH Hardy
What is mathematics? Is today's mathematics just a matter of huge calculations, with the mathematician as a kind of zookeeper, making sure the precious computers are fed and watered? Mostly, it's different. Mathematics is not what you expect it to be, you turn your back for a moment and it's changed. It's certainly not just a fixed body of knowledge, its growth is not confined to inventing new numbers, and its hidden tendrils pervade every aspect of modern life.

-Ian Stewart
Mathematics expresses values that reflect the cosmos, including orderliness, balance, harmony, logic, and abstract beauty.

-Deepak Chopra
Euclid's first common notion is this: Things which are equal to the same things are equal to each other. That's a rule of mathematical reasoning and its true because it works - has done and always will do. In his book Euclid says this is self evident. You see there it is even in that 2000 year old book of mechanical law it is the self evident truth that things which are equal to the same things are equal to each other.

-"Abraham Lincoln"
In the course of my law reading I constantly came upon the word “demonstrate”. I thought at first that I understood its meaning, but soon became satisfied that I did not. I said to myself, What do I do when I demonstrate more than when I reason or prove? How does demonstration differ from any other proof?

I consulted Webster’s Dictionary. They told of ‘certain proof,’ ‘proof beyond the possibility of doubt’; but I could form no idea of what sort of proof that was. I thought a great many things were proved beyond the possibility of doubt, without recourse to any such extraordinary process of reasoning as I understood demonstration to be. I consulted all the dictionaries and books of reference I could find, but with no better results. You might as well have defined blue to a blind man.

At last I said,- Lincoln, you never can make a lawyer if you do not understand what demonstrate means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what demonstrate means, and went back to my law studies.

-Abraham Lincoln
Definition: Mathematics, from the Greek 'mathematikos' meaning "inclined to learn"
In what ways are these beliefs contributing to the exclusion of some groups of students from mathematics?
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