This post will ground a discussion of the existential implications of the differences between an education proper to humans - a conceptual education - and the pseudo-education foisted upon Americans by Pragmatist "educators," and increasingly spread by them to the rest of the world. When Ayn Rand discussed the issue in The Comprachicos, the Pragmatists were still in the initial stages of taking over. Today, many adult Americans don't even know that there is any workable alternative, to the Pragmatist dogma that education consists of (1) learning skills and facts by practice and rote, without conceptual understanding; and (2) of testing "educational achievement" by examinations during which the student has no time to think and no time to apply concepts - and the teacher has no choice but to "teach to the test" by imposing rote memorization on the student, and prohibiting conceptual thought lest the student think, and "waste time," on the eventual test.
A properly human, conceptual education has only the most limited contact with memorization. Most things that can be memorized, can be as fluently learned by repeated reasoning from first principles. I can multiply as fast as, and probably faster than, an adult who did memorize the multiplication tables in first grade. After reasoning out some specific result from first principles two or three times, the result will be remembered and retrieved faster, than if it had been memorized. And, having been understood, that result will be usable in solving problems beyond the grasp of the memorizer. (There are contexts where memorization is necessary - a surgeon cannot take the time to think about evolutionary anatomy while operating - but such contexts are rare.)
I had the enormous good fortune to have learned first grade arithmetic with the Łukasiewicz curriculum. Jan Łukasiewicz, an Aristotelian philosopher of mathematics best known for Aristotle’s Syllogistic from the Standpoint of Modern Formal Logic, had been, in 1919, independent Poland's first Minister of Education, and created its distinctly Aristotelian K-12 mathematics curriculum. This is what I remember of the sequence of ideas in first-grade Arithmetic:
1. Digital (finger) counting; cardinal and ordinal numbers; addition.
2. Doubling; odd and even numbers; halving of even numbers.
3. (Single digit) subtraction; zero and negative numbers; equations.
4. (Single digit) multiplication (beyond doubling) as repeated addition.
5. Exponentiation as repeated multiplication; powers of 10.
6. Number bases; place-order (Arabic) notation; the carry; multi-digit operations.
7. Multiplication shortcuts.
8. Modular (Clock) Arithmetic; telling time; clock arithmetic (time) operations.
Multiplication shortcuts took the place of memorized multiplication tables. All the shortcuts had a derivation from previously integrated concepts, so that I had a conceptual understanding of what I was doing:
x 2, AKA "doubling:" Add the starting number to itself.
x 10. Move the digits one place (base 10!) to the left and put a zero at the end (in the ones' place - the number of ones, after multiplication by the base, is zero.)
x 5. 5 is half of 10, so first multiply by 10, and then halve the result.
x 9. Multiply by 10 and subtract the original number from the result.
x 4. 4 is the second power of 2, so double twice.
x 3. 3 is (2 + 1,) so double and then add the original number again.
x 6. Multiply by 5 and add another instance of the original number.
x 8. Double thrice.
x 7. Multiply by 5 and add twice the original.
In a comparative test after the first grade, we (unlike a class of memorizers) would have understood what we were doing, although we would have been slower, than memorizers just recalling what they learned by rote. By the end of the second grade, we would have had enough accumulated practice to be equally fast - and incomparably superior in understanding what was going on, and in being able to figure out new calculation methods for new contexts.
The essential advantage of learning conceptually is that what one learns makes sense. And that makes multiplication, and all of Arithmetic, and all that comes after, natural and easy. There wasn't a single pupil in my first-grade class who could have thought, "I am not good at math."
Monday, January 17, 2011
Wednesday, January 05, 2011
Hard Atheism: Why a "God" is Impossible
Philosophers distinguish between "Soft Atheism" and "Hard Atheism."
"Soft Atheism" is the position that, as long as there is no evidence for the existence of a God, there is no more reason to believe in a God than to believe in gremlins or unicorns. In the absence of evidence, a reasonable human holds that gremlins, unicorns and gods are fictions that don't exist in reality.
"Hard Atheism" is the position that a belief in the existence of a God would contradict known facts of reality. Historically, "hard" Atheism was typically associated with the position that the properties attributed to the God or Gods of some specific religion were mutually contradictory, or stood in contradiction to known facts. This is not difficult to establish with respect to, for example, the attributes of omnipotence, omniscience and benevolence traditionally ascribed to God in Abrahamic monotheisms. But the attributes ascribed to God differ from religion to religion; a Hindu or a Deist need not ascribe to her God any of the traditional attributes from this list. Until the last half-century, hard Atheism was always associated with preclusion of some specific religious conception of God, rather than of God/Gods in general.
A "hard Atheism" independent of any specific religious conception of God can only be based on a contradiction between reality and those attributes that are shared by every conceivable God of any conceivable religion. There are two such attributes: consciousness and non-measurability. A measurable entity cannot be "transcendent," and an entity incapable of awareness cannot be "spiritual" in any religious sense of those terms. Thus, a demonstration that the same entity cannot be simultaneously non-measurable and aware would be a foundation of a Hard Atheism independent of any specific religion's list of the attributes of its God. It would be the foundation of a simultaneously "hard" and universal Atheism.
The steps in demonstrating the contradiction are as follows:
(1) Mass and energy are measurable. To be non-measurable, a God cannot be physical in the sense of being composed of, or containing, matter or energy. A God is non-physical.
(2) To be aware is to be aware of the identity of something. To be aware, an aware God must have information about the identity of whatever it is ware of. A God must have information.
(3) The quantity of information in an entity, is given by Shannon's Law applied to the range of possible states of its physical attributes. For that quantity to be non-zero, the entity must have physical attributes with measurable properties. But only physical entities composed of matter and energy can have measurable properties and states. Therefore only physical entities composed of matter and energy can contain or carry information. (For more detail on this, an explanation of the need in every data network protocol stack, for a physical layer to carry the information, can be found in any standard textbook on data networks.)
(4) Therefore only a physical entity can be aware.
(5) Therefore a God, being non-physical, cannot be aware. QED.
Thus Shannon's Law precludes the possibility of there being a God who is simultaneously non-measurable and aware. If one understands Shannon's Law, and the evidence on which it stands, one must be a Hard Atheist - or incoherent.
"Soft Atheism" is the position that, as long as there is no evidence for the existence of a God, there is no more reason to believe in a God than to believe in gremlins or unicorns. In the absence of evidence, a reasonable human holds that gremlins, unicorns and gods are fictions that don't exist in reality.
"Hard Atheism" is the position that a belief in the existence of a God would contradict known facts of reality. Historically, "hard" Atheism was typically associated with the position that the properties attributed to the God or Gods of some specific religion were mutually contradictory, or stood in contradiction to known facts. This is not difficult to establish with respect to, for example, the attributes of omnipotence, omniscience and benevolence traditionally ascribed to God in Abrahamic monotheisms. But the attributes ascribed to God differ from religion to religion; a Hindu or a Deist need not ascribe to her God any of the traditional attributes from this list. Until the last half-century, hard Atheism was always associated with preclusion of some specific religious conception of God, rather than of God/Gods in general.
A "hard Atheism" independent of any specific religious conception of God can only be based on a contradiction between reality and those attributes that are shared by every conceivable God of any conceivable religion. There are two such attributes: consciousness and non-measurability. A measurable entity cannot be "transcendent," and an entity incapable of awareness cannot be "spiritual" in any religious sense of those terms. Thus, a demonstration that the same entity cannot be simultaneously non-measurable and aware would be a foundation of a Hard Atheism independent of any specific religion's list of the attributes of its God. It would be the foundation of a simultaneously "hard" and universal Atheism.
The steps in demonstrating the contradiction are as follows:
(1) Mass and energy are measurable. To be non-measurable, a God cannot be physical in the sense of being composed of, or containing, matter or energy. A God is non-physical.
(2) To be aware is to be aware of the identity of something. To be aware, an aware God must have information about the identity of whatever it is ware of. A God must have information.
(3) The quantity of information in an entity, is given by Shannon's Law applied to the range of possible states of its physical attributes. For that quantity to be non-zero, the entity must have physical attributes with measurable properties. But only physical entities composed of matter and energy can have measurable properties and states. Therefore only physical entities composed of matter and energy can contain or carry information. (For more detail on this, an explanation of the need in every data network protocol stack, for a physical layer to carry the information, can be found in any standard textbook on data networks.)
(4) Therefore only a physical entity can be aware.
(5) Therefore a God, being non-physical, cannot be aware. QED.
Thus Shannon's Law precludes the possibility of there being a God who is simultaneously non-measurable and aware. If one understands Shannon's Law, and the evidence on which it stands, one must be a Hard Atheist - or incoherent.
Subscribe to:
Posts (Atom)
