You can find out about the mystery notes here.
Chaos Theory continued:
Chaos theory is a field of study in mathematics, with applications in several disciplines including meteorology, physics, engineering, economics, biology, and philosophy. Chaos theory studies the behavior of dynamical systems that are highly sensitive to initial conditions—a paradigm popularly referred to as the butterfly effect. Small differences in initial conditions (such as those due to rounding errors in numerical computation) yield widely diverging outcomes for such dynamical systems, rendering long-term prediction impossible in general. This happens even though these systems are deterministic, meaning that their future behavior is fully determined by their initial conditions, with no random elements involved. In other words, the deterministic nature of these systems does not make them predictable. This behavior is known as deterministic chaos, or simply chaos. This was summarized by Edward Lorenz as follows:
Chaos: When the present determines the future, but the approximate present does not approximately determine the future.
Chaotic behavior can be observed in many natural systems, such as weather. Explanation of such behavior may be sought through analysis of a chaotic mathematical model, or through analytical techniques such as recurrence plots and Poincaré maps.
Okay, I could be wrong about the Ivey Business School. It could be anyone taking a course involving any one of the disciplines mentioned above.
Another interesting article about using symbols as a means for counting and record keeping can be found here. It is titled “Tokens of plenty: how an ancient counting system evolved into writing and the concept of abstract numbers.”
A quote from the article:
One, two, three, four…. We learn to count at such an early age that we tend to take the notion of abstract numbers for granted. We know the word “two” and the symbol “2” express a quantity that can be attached to apples, oranges or any other object. We readily forget the mental leap required to go from counting specific things such as apples to the abstract concept of number as an expression of quantity.
Just such a leap may have occurred roughly 5,000 years ago among people living in ancient Mesopotamia, a fertile region watered by the Tigris and Euphrates rivers in the Middle East. Ten thousand years ago, counting was a concrete affair. Residents of small agricultural settlements kept track of their goods by maintaining stores of baked clay tokens–one token for each item, different shapes for different types of items. A marble-sized clay sphere would stand for a bushel of grain, a cylinder for an animal, an egg-shaped token for a jar of oil.
Hmmm…baked clay? The new note includes a brick.
Maybe this isn’t about any of the above but a way to show how people can take the same thing and turn it into whatever they choose. Just how many theories we can conjure up and mangle the original data or marketing test? LOL!
One of the more interesting things I happened to find was a book written by Ivar’s Peterson called “The Jungles of Randomness: A Mathematical Safari”
I wish he had been my math teacher. I can’t stop reading it! Here is a quote from his book:
Mathematics encompasses the joy of solving puzzles, the exhilaration of subduing stubborn problems, the thrill of discerning patterns and making sense of apparent nonsense, and the immense satisfaction of nailing down an eternal truth. It is above all a human enterprise, one that is sometimes pursued simply for its own sake with nary a practical application in mind and sometimes inspired by a worldly concern but invariably pushed into untrodden territory. Mathematical research continually introduces new ideas and uncovers intriguing connections between old, well-established notions. Chance observations and informed guesses develop into entirely new fields of inquiry. Almost miraculously, links to the rest of the world inevitably follow.
With its system of theorems, proofs, and logical necessity, mathematics offers a kind of certainty. The tricky part lies in establishing meaningful connections between the abstract mathematical world that we create in our minds and the everyday world in which we live. When we find such links, mathematics can deliver accurate descriptions, yield workable solutions to real-world problems, and generate precise predictions. By making connections, we breathe life into the abstractions and symbols of the mathematicians’ games.
Intriguingly, the mathematics of randomness, chaos, and order also furnishes what may be a vital escape from absolute certainty—an opportunity to exercise free will in a deterministic universe. Indeed, in the interplay of order and disorder that makes life interesting, we appear perpetually poised in a state of enticingly precarious perplexity. The universe is neither so crazy that we can’t understand it at all nor so predictable that there’s nothing left for us to discover.
Isn’t that beautiful writing? Math just not be so scary after all. 🙂
Because of that, I am still sticking to my numbers, chaos and pattern theory.