George Packard began his little lecture:

The history of humankind and of modern civilization can be described by the history of mathematical discoveries. Each new mathematical understanding has almost immediately led to significant technological advances and “progress”.

First, let’s consider the history of numbers. It’s a long story. From the simple counting numbers – from markings inscribed on pieces of bone dated by archeologists at 15,000 to 20,000 years old – to fractions, to negative numbers, squares, square roots, exponential numbers, irrational numbers, imaginary numbers, and yes, the transcendental numbers (pi, phi, and the rest) – these “discoveries” all plot out a fine map of man’s history. Each “discovery” enabled a leap forward in the advance of technology and science.

The “discovery” of pi, the constant ratio between the diameter and the circumference of a circle, was apparently”discovered” or “revealed” to the Egyptian builders of the Great Pyramid. At about the same time the “discovery”of the golden proportion phi. This was almost 5,000 years ago. Knowlege of pi and phi enabled the Egyptians to build the Great Pyramid of Cheops, the Pyramids that still stand at Giza, on the 30th parallel, in Egypt.

The number zero, “discovered” in India around 500 B.C. and expressed in the Hindi-Arabic numbers around 500 AD, enabled all computations, as well as the so-called algebra, to blossum. The Olmecs in Central America were already using symbols for zero in their mathematics about 500 B.C. The Mesopotamians used a zero in their sexagesimal number system around 2,000 BC; they also used the Pythagorean theorem.

Other mathematical landmarks would be:

Euclid’s Elements—500 B.C. Euclid’s system of logic and proof was the foundation of science.

Archimede’s mathematics : the lever and the principles of mechanics around 250 BC

Cartesian geometry and the mathematics and science of Isaac Newton circa 1640

Binary Numbers, developed by Leibniz in 1679 – which later enabled computers to enter our world.

Non-Euclidian geometry

Fractals

George Packard would stop at this point, and look out over his class, the assemblage of teen-aged heads and hair-do’s that sat before him, several of them leaning on their propped elbows in listening poses..

As he was about to resume his little lecture a hand rose up from the back of the room. Young Pola was raising her hand.

“Yes Pola”

“Mr. Packard, I mean like why did you put all the parenthesisis marks each time on the word discovery? Huh?”

“Hey Pola that’s a good question. Does anybody out there know why?

Packard scanned the bodies in the room and noticed they were were now paying attention. They were stirring in their seats, sort of roused from their slumber.

“Well, I did that because each of these “discoveries” was present and operative in the world before man “discovered” them. The plants were already growing in phi proportions, the planets revolving in great ellipses, the stars retreating and the universe expanding according to these mathematical underpinnings long before man came along.”

“That’s true, Mr. Packard!” yelled out someone from the back.

George Packard loved to present these interesting things to the kids each year. Maybe some of them might be encouraged and brightened. Here and there anyway.

So thought George Packard as he closed his notebook and packed it in after another day at L.A. High.

What a fantastic post. George Packard was obviously a great teacher. I love math history.

Thank you so much.

There are 10 types of people in the world; those that understand binary and those that don’t.

(Yes, I know it’s an oldie, but I couldn’t resist).

That’s a good one!

Mr. Packard, like why is Planck’s constant so important? Do I need to know about it to get on a bus? (Notice how, with uncertainty, I return to the bus fetish…)

Dear Prospero,

An explanation of Planck’s constant, together with $1.50, will gain you admission on the bus of your choice. I’ve puzzled over Planck’s constant myself and can’t get a good handle on it. I’ve thought about getting hold of Max Planck himself, but he’s six feet under and probably wasn’t much fun to grab hold of in his better days anyway.

Imagine the “discoveries” of the future… Mr. Packard’s wet dreams… the mathematical formulae that describe how consciousness builds solid form from fields of quantum probability… and other things so far out we don’t even have words or concepts for them yet.

Yes. Our consciousness could be the missing link in understanding what is going on at the quatum level. I do not grasp quantum theory – and apparently no one does. Neither do I know what consciousness is – even though I think I possess it.

Perhaps the mathematics can instruct us.