Chapter 1 Review
One of the single most important questions in philosophy.
Why do things change?
or
What is... "the problem of Change"?
Who was Heraclitus and Parmenides, and what did they believe about change?
5th century BC philosophers who disagreed on the meaning of "change."
Heraclitus = "Everything Changes!"
Parmenides = "Nothing Changes!"
What is a "mathematical" or "genuine" line? How does this relate to Plato's Forms?
Mathematical lines = Lines that are perfect - having length but no breadth (thickness).
Plato's Forms are ideas of the perfect line.
How is the similarity between mathematical objects and the physical world a "double-edged sword"?
We can use mathematical (theoretical) ideas to represent physical problems, but figuring out how to use theoretical ideas to represent real life problems is not always easy!!
What are two ways that time is represented spatially?
_____ "in time."
Represent time in history with a timeline and can represent time as distance between two moments.
Mathematically with a line representing seconds or minutes.
(See examples on page 30 in book).
Another word for change.
Motion
Who is Zeno?
Zeno of Elea = Student and defender of Parmenides.
Proposed a thought experiment to show that motion simply cannot occur.... leading to his famous paradox.
Why is it strange to say that mathematical lines area made out of points?
Lines are made up of points, but points have no size.
In what way can we "see" numbers?
We can see numbers when we place them on a number line.
What are the two most fundamental features of the physical world?
Who is considered the first Greek philosopher, scientist, and mathematician?
Thales
What is Zeno's Paradox?
In order to travel any distance, you would first have to travel half of that distance, and then half of that distance... and so on.
How many points make up a line?
What two realms does the number line unite?
The realm of numbers and the realm of points.
(Read quote on page 25. ".... wedded bliss")
What are the two most fundamental features of the mathematical world?
Points and Numbers
What was Thales' answer to the problem of change?
Everything is made of water
Was Aristotle able to solve Zeno's Paradox?
Nope!
Why is no point immediately next to any other point on a line?
We can always zoom in closer between two points and find another point between any two points.
Why is no number immediately next to any other number on the number line?
There is always another number between two numbers no matter how close in value the two numbers are.
What is the connection between the physical world and the mathematical world?
Pythagoras was a student of Thales. He was Greek philosopher and mathematician. Famous for the Pythagorean Theorem and musical interval discovery.
Motto: "All is number"
What is Aristotle's little motto about motion and nature?
Ignorance of motion is ignorance of nature.
What branch of mathematics can be called the "Science of Space"?
Geometry
Numbers and lines share the strangeness of what two important (and related) concepts?
continuity and infinity
(Discuss what each word means)
Draw the triangle diagram that shows the relation between numbers, shapes, and space. Explain what this diagram represents.
See Figure 2.5 on page 33 in book.
Similarity between physical space and numbers allows us to quantify physical things... because of this we can discover important truths about the world by calculating numbers.