Copernicus: Realization that we (Earth) are not the center of the universe (what we see on a clear night, and beyond)
Newton: His laws of motion
Perfecting the steam engine: Thermodynamics and statistical mechanics
Electricity and Magnetism: Maxwell's equations and Einstein's special relativity (which reconciled Maxwells equations with Newton's laws of motion)
Quantum mechanics: Rooting out the continuum (measurement is always grainy)
The information age: Measuring information and building computers
(Color seen by birds is magic. Birds see much that we must use scientific observation followed by sophisticated math before we realize just how magical is the knowledge of these "inferior" beings.)
Meriam Webster Dictionary
How do we visualize?
How do we calculate?
How might we extend our abilities?
Jean Piaget studied human mental developments that we acquire as we mature. The most advanced thinking skills he found ("formal operations"; they lead to human science and math) are insights that let us better understand multiple influences and things that require multiple dimensions to see graphically.
|Watch the desert sky from your sleeping bag for several nights in a row. Note how the stars move across the sky during the night. Note they don't rise and set quite the same times as the year progresses. Note how the moon moves, rises and sets. Then: assume the orbits and spins described by Copernicus and answer the question: Are the spin of the Earth, the orbit of the moon about the Earth and the orbit of the Earth about the Sun all in the same direction, or is one different from the other two? How many distinct possibilities exists? Which one is it?|
(laws of motion)
|Measurements (and calculations
Ratios and proportions...
Measurements that vary continuously
Rates of change
...the above leads to Newton's "fluxions" (calculus).Measurements with several components
Sorting the relevant from the irrelevant
|What is the likely result
of the fist on the jaw?
What is the direction, up or down, of the acceleration of a freely bouncing ball at the bottommost point of its bounce, that is, at the instant its velocity changes from down to up?
|Recognizing multiple variables
Eliminating subtle contradictions
Recognizing statistical relationships
...What's the difference between heat and temperature?
must avoid the "gambler's fallacies.
|Statistical relationships: "If a majority of people understood statistics, there would be no gambling casinos and no lotteries." Consider: A hustler presents you with three boxes into one of which he places a valuable prize, but you don't know which one. You get to choose one box and keep the prize if it's there. You choose, but he doesn't tell you if you won. Instead, he says to you "I'm going to open one of the boxes and show you what's inside." He does (and you see that there's nothing inside), and then says, "Now, would you like to stay with your first choice, or would you like to switch." Should you stay or switch?|
(dynamics of charged particles)
|Relating multiple variables
...It helps a lot to develop Boolean intuition.Multiple dimensional space visualization
...Maxwell's equations are in 3D, relativity in 4D.?????????
|If we have developed good Boolean intuition; the answer to this question is obvious: In a set of cards each card has a number on one side and a letter on the other. Four cards are lying on a table. They show an "I", an "N", a "6", and a "3". Someone suggests the hypothesis: If a card has a vowel on one side then it has an odd number on the other side. The problem is to determine which cards must be turned over to test the hypothesis. No card is to be turned over unless necessary to test the hypothesis.|
|Recognize mathematical abstractions.
...Keith Devlin's fourth level of abstraction
(patterns of patterns of...)
|Quantum mechanics describes a universe for which human evolution did not provide us perceptions. When do we interpret the abstractions of quantum mechanics too much in terms of our familiar perceptions?|
|Recognize validity of information.
Recognize sufficiency of information.
Use measures of information.
...band pass, disk capacity, CPU speed,,,
|Information is something (words, mathematical equations, images, metaphors, pattern descriptions, etc) that represents something in ways a human brain (or an extension of a human brain, such as a computer) can recognize. Information is necessary to select from alternatives so that actions we take result in what we want to happen (more probable, anyway). It can err by being wrong or by being insufficient. Why are maps so often full of errors, errors that can get us lost?|
|Each of these had to await sharpening
of human intellectual skills beyond those necessary for survival in prehistoric
worlds and cultures.
Newton: He needed to be able to meaningly measure quantities that vary constantly--in a "state of flux." So he invented his "theory of fluxions." Today, we call it "calculus." Can't skip it completely: we need the elementary concepts.
Heat Engines: We must recognize interactions of multiple variables, sort through them, sift the relevant from the irrelevant, relate them mathematically, and not fall into those traps that we call " gambler's fallacies."
E & M: Vectors...and other multi-component measures. This is a really big step based on a rather simple (but difficult) concept.
QM: The realization that our simple math isn't quite a perfect match to the real world..
computers: Bringing it all together: Human interaction with human
environment...and it's statistical in all its roots.
Confusing all with some
Selecting corroborations while rejecting disconfirmations
Fixating on irrelevancies
Forging ahead oblivious to fatal logical contradictions and errors
Seeing scalars when looking at multicomponent measures
Seeing gambler's fallacies in place of statistical realities
Seeing an additive relationship when it's really multiplicative
Not seeing orders of magnitude: confusing millions, billions, trillions...
Seeing ratio and proportion as additive relationships
(Most of these derive directly from Piaget's "formal operations.")
In Europe, a woman was near death from a very bad disease, a special kind of cancer. There was one drug that the doctors thought might save her. It was a form of radium that a druggist in the same town had recently discovered. The drug was expensive to make, but the druggist was charging ten times what the drug cost him to make. He paid $200 for the radium and charged $2000 for a small dose of the drug. The sick woman's husband Heinz went to everyone he knew to borrow the money, but he could only get together about $1000 which was half of what it cost. He told the druggist that his wife was dying, and asked him to sell it cheaper or let him pay later. But the druggist said, "No, I discovered the drug and I'm going to make money from it." Heinz got desparate and broke into the man's sotre to steal the drug for his wife. Should the husband have done that? Was it right or wrong?
First, think about how you would answer.
Then think about how someone else would probably answer, but pick someone who's way of looking at such things is really important. For example in the 2004 Presidential elections, how do you think John Kerry would answer, and how do you think George W Bush would answer?
Then look at the six categories of moral development* that this question was designed to evaluate.
I first saw this in the Summer 1983 issue of Daedalus, in an article by Lawrence Kohlberg & Carol Gilligan entitled "The Adolescent as Philosopher." Adolescence is the time we develop our abilities to understand the kinds of abstractions that lead to science and mathematics. (Note that what at the end of the 20th century was called "physics" was called "philosophy" at the end of the 19th century.) So studying how our mental abilities improve during adolescence tells us a lot about how we can or cannot use science and math, about how our abilities to philosophize improve.
Here lie the best competencies that the human mind is capable of achieving. Do we see this philosophy as being "ivory towered and quite out of touch..." or it is a personal intellectual power source which we can actually use to our advantage.
"We can, and we've got to, do better than this."
Theodore Seuss Geisel
On the linked page, Kohlberg's higest levels of morality are described
as being "defined in terms of conformity to shared standards, rights, or
duties apart from supporting authority." Stage 5 refers to "rules
which seem to have a rational basis," and stage 6 leads to "self-condemnation
and guilt" if internal values are not adhered to.
I suggest that Stages 5 & 6 stem simply from a broader and more complete recognition of social relationships and implications of mutual reciprocity. They consider a much broader range of reality than do the lower stages. "Shared standards" are really rather rare at Kohlberg Level III: too few share standards that arise from recognizing rights of a broad spectrum of different people--making "conformity" here a misleading term. "Seem to have a rational basis," suggests to me that some imperatives haven't been recognized, imperatives that make the viewpoint more rational. Rationality is not an either/or process. And "guilt" is a common misperception of the feelings of people who understand the implications of mutual reciprocity when they see lack of understanding steering society toward disaster--that's not guilt; it's frustration.
"Ethical-Logical Comprehensiveness" describes the more complete moralities.