1. Discovered or Invented?
Mathematics
Discovered or Invented?

2. The Numbers of Life
How do you thing Fibonacci came up with this sequence of numbers? Does it have any real application or is it just “neato”?
What did you think about the idea that this sequence is a natural occurrence rather than something that was created by man?
Did you have an opportunity to examine the websites?

3. Fibonacci Numbers The Golden Rectangle
Fibonacci Numbers

4. A Little More Nature
The Golden Rectangle
Nautilus Shell

5. A Curious Incident
Did you notice how much math Christopher saw in the world? Take about 5 minutes to find as many as you can.
What are some examples of things that he noted?
Why do most of us overlook things like this?

6. Fractals
Let’s take a little bit to watch a video explaining fractals and their “discovery” as well as their application.
Are there any other occurrences of fractals that you can think of in the world around you?
I happen to have another SHORT video exploring fractals in another fashion that I had to to you because of school blockage issues.
What does all of this tell you?

7. Chaos Theory
Chaos Theory is the idea that seemingly chaotic can actually be formalized into mathematical models and studied/predicted.
The “catch” is that small influences in the beginning state of the system can cause incredibly different, seemingly random, outcomes. The same magnitude of influences later in the process of the system have seemingly insignificant impact on the final result.
One mathematician actually named this the “butterfly effect,” claiming that the flap of a butterfly’s wings on one side of the world could theoretically impact the weather on the other side of the world.

8. Chaos Theory
Chaotic behavior is noted in natural systems, such as weather (think about predicting the weather, especially hurricanes). It is also used in Quantum Mechanics.
Fractals and Chaos Theory are very closely related in that looking at patterns and developing complex mathematical models to replicate those patterns is involved in both.
In what sense might chaos (non-linear dynamical systems) theory suggest a limit to the applicability of mathematics to the real world?

9. Is Mathematics the Language of the Universe?
What did you think about this reading?
What does it tell you about mathematics?
After considering it and all of the other stuff that you have learned, take a minute to come up with an answer to the following quesiton:
We can use mathematics successfully to model real-world processes. Is this because we create mathematics to mirror the world or because the world is intrinsically mathematical?

10. A Few More Questions
Some major advances in physics, for example, discoveries of elementary particles, have come about through arguments involving the beauty, elegance or symmetry of the underlying mathematics. What does this tell us about the relationship between the natural sciences, mathematics and the natural world?
Is mathematics better defined by its method or by its subject matter?
In the light of the questions above, is mathematics invented or discovered?

11. Homework
Read
“What Does “Existence” Mean in Math?”