Embedded Files
Knowledge Questions:
Why does mathematics enjoy a privileged status in many education systems?
What is meant by the term “proof” in mathematics, and how is this different similar to or different from what is meant by this term in other areas of knowledge?
Think, Pair, Share: Should Maths be a compulsory part of the IBDP? Justify your answer.
Mathematical proof: "A rigorous mathematical argument which unequivocally demonstrates the truth of a
given proposition. A mathematical statement that has been proven is called a theorem."
Mathmatical conjecture is "A proposition which is consistent with known data, but has neither been verified nor shown to be false."
For ToK purposes, we can ask what degree and type of proof is required for a mathematical statement to be considered knowledge. Does computerised proof count? Want about proof by exhaustion?
Read the article, opposite, about computersed proof.
Then have a go....
The ____ colour theorem – 'proof' by exhaustion
Draw an imaginary map with as many countries as you like
Colour in the map with one colour per country
Every country must be a different colour to every coutry that it borders
Try to use as few colours as you possible can
How few colours do you need to use so that no adjoining countries are the same colour?
You never need more than ____ colours (the ? colour theorem) – but this has never been conclusively proven. Rather, computers have "proven" it by exhaustion (effectively, inductive reasoning). Mathematicians disagree about whether this counts as proof, and therefore knowledge.
What do you think? How does this relate to the previous knowledge question, about the reality of numbers?
Intuition - A basic way of knowing. Link here
How do we know basic mathematical truths? How do you know that 2+2=4? How could you convince someone who disagreed?
"Basic math is another thing that can’t be proven. Its truths are known by intuition. Someone once took me to task on this, suggesting he could scientifically prove two plus two equals four. He took two apples and put them together with two more apples to give a total of four. That was his “scientific” proof.
The math wasn’t proven in this case, though; it was simply exemplified with different tokens. A token is some physical representation—a sound, a mark of ink on a piece of paper, an object—that represents the unseen type, in this case, a number. Let me illustrate.
I could write “two plus two equals four,” or “2 + 2 = 4,” or substitute apples as my tokens instead of words or numerals. In each case, the math is demonstrated—restated with different tokens—not actually proven.
Math is obvious because of our intuition. As long as one knows what the symbols in the equation 2 + 2 = 4 represent—the numerals and the mathematical signs—a moment’s reflection shows that the truth of the equation is self-evident. Indeed, if you disagreed, I would be at a complete loss to prove it to you. Either you see it, or you don’t."
Read article on the nature of Mathematical proof and the nature of objectivity and subjectivity
Page updated
Report abuse