Math is Abstract?

This week we discuss "what do people mean when they say that math is abstract?"

We also look at the eye to gain some understanding about why we can or cannot see color when observing stars.

It is Friday! with a visit to Mono Lake, subject for a forthcoming blog. And the picture above of one giant sequoia redwood at the Mariposa Grove reflects a stop we made on the way to Yosemite. The other two pics below are from trails around Mammoth. Everyone is a great photographer with such scenery, just hold your camera steady.

If you want to come to Camp Mawks and need to settle any "ifs" or "buts" please do email or call. I like to hear what you are thinking.

And ... the t-shirts finally arrived, they are yours at our cost, and of course gratis when you arrive.

More Math:

What is on my mind at the moment are two viewpoints I have read about in the news about educating from grade school through undergraduate college. One view is that math, and techie terms and "concepts" should be started in kindergarden... in 1st grade, in 4th grade etc. Then there are those who say, no, this can lead to confusion since (and I totally disagree with the next statement) "not everyone can abstract."

My view is that math should be started up at every opportunity. It is never too early, and it is never too late. Some of us choose to go on in math, and all benefit from the mental exercises involved in learning to reason. In fact, we must exercise our brain and this involves reasoning.

Familiarity is one of the most important aspects of the process. Abstract? Just learn to do the computations. After a comfort zone is reached, we "abstract." What is that? Well you add more logical descriptions. You remember them. You recognize these descriptions as they are associated in other calculations. Effectively, you are doing math using a set of rules. Mathematics is a rule based system.

And everyone has a different view of the word "concept." I keep hearing that students must learn math concepts, or else. I often wonder whose concept will enable us. (Just do the calculations correctly, and let your understanding guide you.)

I note, as an example, that some flowers in nature have a set of five distinct petals in a geometry that is quite striking; amazingly enough, some rocks fracture with a similar geometry. In the case of the rock, the fractures might be along the underlying crystalline structure. And we know that the flowers instruction set, that is, its DNA or genes lead to its particular geometric petals. But how do the rocks and flowers seem to know the same rules?

So, what "concept" do we have for describing the why's of this similarity between the rock and the flower? There have been many answers over time. (As an example, we can look up the mathematician Rene Thom .)

Maybe, this formation of fives was the most efficient thing, energy wise to do. Maybe the chlorophyll light-needs in the plant required a certain angle to capture light ; and maybe the periodic table, at the same time, dictated some of the results. Mathematics tell us about the shortest distance between two points ( straight line! if on a flat table) and the least amount of light needed for a bud to flower. And nature, that we can observe, is enveloped in the language of mathematics.

We have been discussing least time and least energy principles from nature here; these processes have mathematical formulas to describe them that we learn in calculus. So here we are discussing "concepts ."

Most importantly: 1) do your computation correctly, step by step; that is what to get out of your instructor at all levels of math; 2) associate the verbal descriptions correctly, step by step.

We are good to go.

To begin, in Mammoth Lakes, we see more color in the night sky due to the darkness this far away from big city lights, and we are in high elevation so the atmosphere is thinner, there is less to hamper our sight.

The light sensitive part of the eye is the retina. The photoreceptors in the retina are called rods and cones. The retina contains about 6 to 7 million cones and about 120 million rods. The cones are concentrated near the center of the retina (near the optical axis of the eye). We use the cones to perceive color. Individual cones are sensitive to red, green, or blue light. Most of the cones, roughly 64%, are red sensitive. Next are the green (34%). Only about 2% are blue sensitive, but the blue cones are the most light sensitive.

Rods, on the other hand, are not sensitive to color. Rods are absent from the center of the retina (where the cones are) but dominant everywhere else in the retina. The rods are far more light sensitive than the cones. Thus, when it is dark, we are using our rods to see, and not the cones. That's why it is hard to see color in low light situations, and difficult to perceive the color of stars at night. The telescope gathers more light than our eyes alone. With the added light, the cones begin to respond again. Thus it is easier to perceive the color of stars, or nebulae, looking through a telescope. Its the light gathering power of the telescope, not the increased magnification that is responsible.