On Sep 1, 2012, at 2:27 AM, Andrzej Kozlowski wrote:
> > On 31 Aug 2012, at 09:57, John Doty <firstname.lastname@example.org> wrote: > >> On Thursday, August 30, 2012 2:09:19 AM UTC-6, Alexei Boulbitch wrote: >> >>> Let me just point out that the origin of this interesting and passionate discussion was the >>> question of what should be the content and tools of the mathematical education for students in >>> non-mathematical specialities at present, observing that since long computers have become the >>> reality of our world. >> >> The nature of mathematics is very relevant here. If mathematics is the product of a magical sense that detects objects in a supernatural world, then it is impossible to scientifically approach the problem of teaching mathematics. But if mathematics is a product of the natural processes of cognition that humans use to find their way in the world, teaching mathematics is similar to other kinds of teaching. Then, the methods and insights of cognitive science are likely to be of use in answering your question. >> > > The question of how human beings learn mathematics and whether the natural (not "super-natural") world is governed by laws which can be described by means of mathematics are two quite different (although related) questions. Mathematical "pPlatonism" is only concerned with the latter question, it asserts that mathematical theorems (in some sense) hold in the real world and not just in human minds. How human beings learn and discover them is a rather different matter. > > To idea that human beings acquire mathematics as part of a "natural processes of cognition that humans use to find their way in the world" and at the same time mathematics "has nothing to do" with this world, sounds to me like a very feeble one, but since the issue is metaphysical, probably harmless. However the idea that by using "insights of cognitive science" you can turn people who have trouble adding fractions into Riemann's and Poincare's sounds to me about as "useful" as Marx's idea that once you "abolish" private property all human beings will become geniuses and every kitchen maid will be able to run the State. And, if you are not claiming that everyone can be a Newton, Gauss, Riemann, or Poincare etc. (with proper, "cognitive science" guided education) then why not? Is it possible that mathematical ability may be largely inherited? Why, what evolutionary purpose would that serve? > > > Andrzej Kozlowski >
I do not know how long this answer would be, so my advise to the serious reader is to stop reading this right now and rather read the New York Times.
Now, that I have just the attention of nut cases like myself, I can dive into it :-)
If I go back about 4.5 billion years and look the Solar System, I see just a mixture of different kind of materials, mixed together by electromagnetism, gravitation, weak interaction and the strong interaction. Earth is about 2000K at the surface, there is no sign of life as we know it and consequently (??) no mathematical axioms, theorems of any kind. Just reality. Is this Solar System with its hot Earth unique at that time ? Hardly. There are millions of it, mass-produced by cosmic forces just in the Milky way Galaxy. Even the Milky Way Galaxy is not unique at that time, millions or even billions of similar Galaxies could have been seen by an intelligent eye, if such eye would be in existence 4.5 billion years ago. Then what is the real ingredient, that if we just give 4.5 billion years to this Solar system, to make about 18 rotation around the nucleus of this Milky way Galaxy, in a Galaxy where seemingly nothing happens, except some fat stars blow themselves up from time to time, that life appears and accidentally/ultimately human beings with mathematical ideas in their head came to existence. Can something be created that is not there to start with ? Here is short story from S. Lem:
<nyissz> One day Trurl the constructor put together a machine that could create anything starting with n. When it was ready, he tried it out, ordering it to make needles, then nankeens and negligees, which it did, then nail the lot to narghiles filled with nepenthe and numerous other narcotics. The machine carried out his instructions to the letter. Still not completely sure of its ability, he had it produce, one after the other, nimbuses, noodles, nuclei, neutrons, naphtha, noses, nymphs, naiads, and natrium. 'This last it could not do, and Trurl, considerably irritated, demanded an explanation.
"Never heard of it," said the machine.
"What? But it's only sodium. You know, the metal, the element..."
"Sodium starts with an s, and I work only in n."
"But in Latin it's natrium."
"Look, old boy," said the machine, "if I could do everything starting with n in every possible language, I'd be a Machine That Could Do Everything in the Whole Alphabet, since any item you care to mention undoubtedly starts with n in one foreign language or another. It's not that easy. I can't go beyond what you programmed. So no sodium." <nyassz>
Then we have to take it to heart, that whatever we see around us today, or any time in the future, had to be there in the Solar System 4.5 billion years ago, but not necessarily in the same shape and form we see it today. Even mathematical ideas should have been there embodied in some manner. A process, called evolution built them up to the present form in the last 4.5 billion years. Yeah, but what is evolution ? Well there is many good description of it, most originate from biology, although astrophysics and astronomy also have used the word evolution to describe the "life cycle" of astronomical objects and.or processes with no connection to earthly biological life. Even the Solar System as a whole has been going through some 'evolutionary process' in the last 4.5 billion years. The question is if today's objects and ideas are already embodied at any time in the past, then how it is done, what shape and form the embodiment takes place, what are the laws governing this embodiment, etc.,,
On the big scale we can see galaxy clusters placed on filaments, showing a not apparent distribution in space, governed by gravity and electromagnetism. On scale of the individual elemental particles it is all the strong and weak nuclear forces with some play of electromagnetism. Neither scale shows anything that would seem an obvious component for this embodiment. Then we might have to do a salto mortale and make all those that look essential inessential, and whatever it is inessential to make essential. /I told you I will be a not case !! / Let's assume for a moment that all the material Reality that we see around us is inessential, that it is not essential, it is a byproduct of a process and try the grasp that something that is essential and present at every level of this inessential material Reality.
I call it the Information Ocean. Where does it exist ? Above the Planck scale, but below the elemental particle level. /Above the elemental particle level are the nuclear, atomic, chemical, nano, biological, cell, multi cell, etc.../ It is the Information Ocean that creates matter. But why ? Because it stuck between the Plank scale and the elementary particles' scale. It is the Information Ocean that creates such powerful attributes as spin, charge and mass. According to Landau, a proton in the nucleus has a speed of c/4, where c is the speed of light. Even in the nucleus of the Hydrogen atom, that consist only of one proton, this lonly proton achieves this c/4 speed. Every atom in every one of us contains minimum one proton and those protons are moving with c/4 in their own confinement. The speed of light was known and measured before the proton was known and measured. But which one is the more important for the embodiment, proton or photon ? Looks like both are equally important and that is shown with the c/4 speed of the proton in the nucleus. When the Information Ocean creates a proton with spin, charge and mass and the forces attached to it to govern it in space and time at the elementary particle level, some kind of balance is broken. To compensate for this brokenness another entity has to be created, but in this entity the mass, charge and spin might not be separated, but rather tied together closer to the Plank scale and show only one of the component from the three /or more/ and we might perceive it is "dark matter" or "dark energy".
The Information Ocean is where all the information can be found. Past present future. No exception. It is the Information Ocean that builds up the material world, the material Reality to help himself out from his prison stuck between the Plank scale and the elementary particle scale, like baron M=FCnchhausen, who pulled himself out from the mud by grabbing his own hair. Material Reality is for Information Ocean as the hair is for baron M=FCnchhausen, to pull up. There are some consequences. One of them is, that all what Information Ocean builds it builds from information first. It is not by chance that the meaning of a point is getting very fuzzy as we approach the Planck scale from the macro worlds. Already at the elementary particle level it is very troublesome to speak about the location of an electron, or making bets where the electron can be found just in a moment. In the case of a photon we cannot even attach a local coordinate system to it, and in every other coordinate systems the photon speed is c, independently how these coordinate systems are moving to each other.
So when the Bible says that God separated Earth from haven, the real meaning of it is that material Reality everywhere in the Universe was separated by Information Ocean from another reality that is between Information Ocean and the Plank scale.
As Information Ocean created the elementary particles from information, it retained a connection to them. We can look every elementary particle as a chunk of structured information, and this structure gives birth to the different attributes of the elementary particles. Every elementary particle has connection to every other particle via those connections the Information Ocean retained for itself. Quantum mechanical exhibition of it is the entanglement.
If I look the World as a place where Information Ocean tries to express itself by creating and building a scaffolding for itself, then I can see clearly that this material Reality skeleton is there exactly for that purpose for Information Ocean to come to the macro world from its present level. Then it is obvious that mathematical ideas are there at 2000K Earth 4.5 billion years ago. Every structure from the electron to a galaxy cluster is the exoskeleton of Information Ocean and Information Ocean is where all the information can be found, among the many the mathematical ideas too.
Evolution = The way how Information Ocean expressing itself via the material Reality exoskeleton scaffolding.
There are peaks and volleys in evolution. It is not a strait line, because the structures created has to find each other, has to work together and has to play with each other. That takes time. So, evolutions flattens out and speeds up from time to time.
Now if I am looking biological beings, than there is the brain that has a unique purpose. Everyone will tell you that the brain is the seat for human creativity and ingenuity, etc=85. I see it differently. The brain is that exoskeleton piece that is "evolutionary" suited for direct communication to and from the Information Ocean. See my definition above for evolution. If I am looking just human beings, it is obvious that everyone has a totally different brain structure, even if for first glance - from a distance of a few meters - every exposed human brain looks alike. These differences are obvious for everyone - except politicians, that is why we have the democratic one person one vote -, and it manifests itself in different capabilities. From my point of view these manifestations are not really important. Only one difference is important and it is the capability of the brain to have a direct connection to Information Ocean. When do we experience such direct connection ? In three cases:
- In dreams. In my dreams in 1988 I could speak fluently in English, although I hardly could put a full sentence together in real life. - In the creative moment. My favorite example for it is how the idea of the book of "One Hundred Year Solitude" came to the mind of Gabriel Grac=EDa M=E1rguez. He was driving in Mexico on the transAmerican highway with his wife, and suddenly as the road turned, he saw an incredible green valley. As he looked down into the valley, the whole story of the book came into his mind. He said later it was so clear in his brain that he could write down the first 3 chapters without even thinking about it. He went back, closed himself into a room and as they gave him food he gave out the pages of manuscript to print. It took him 3 months to write down the book, that took less than a millisecond to appear in his mind. This show how incredible high is the speed between the brain and the Information Ocean, because in his brain it was not like a book, but rather as a film or movie. - in the "flow" as described by Mih=E1ly Cs=EDkszentmih=E1lyi.
Consciousness dampens this direct communication channel. Consciousness makes this direct channel noisy and effectively shuts it down. I would even say, that a conscious brain is a brain that lost its high speed connection to Information Ocean. It is basically on an autopilot to do the daily chores without to much creativity or thought. A conscious brain is an algorithmic brain, preprogrammed like Trurl's machine to do all things starts with "n".
To open a channel to Information Ocean, all those individual drops of exocapsulated informations where the exoskeletons are the protons, electron, etc=85 has to entangle in the brain. It is entangled in the = brain, because of the structure of the brain and Penrose with Hameroff = wrote extensively about it. I agree with the mechanism, but have an = opposite view on its purpose. Penrose and Hameroff thinks that the = entangled water is in the microtubules where computation takes place = and consciousness arise. My thinking is, that consciousness is there = with the same process, but as a fragmented process all the time. The = entanglement in the brain becomes a communication channel to Information = Ocean when consciousness steps aside, with that it allows the = entanglement to be enhanced and that enhancement opens the channel. = When the conscious mind comes back - the person wakes up, the creative = moment passed, the flow is broken -, the entanglement gets fragmented = and the channel to Information Ocean closes.
Can this channel activity to Information Ocean enhanced and learned ? = Well with more dreaming sure. An example of it is Benzol with his = snake. To get to the stage of flow is also a help, but this channel is = a rather "slow" speed, compared to the speed of the other two, but still = much higher than what a conscious mind is able to produce. I still = remember my physics professor, Miklos Dede, who were able to get to this = flow by will. Whenever a hard question was posted to him, he looked = somewhere to the upper right, his bright eyes fixed there on something, = then came back to reality and told the best answer in a very fluent way, = like if he would read it from a book, without hesitation. Of course on = the long run genetical engineering is the tool to make this channel = enhancement possible in the everyday life.
Mathematica can be a helper application to clear the road that leads to = less conscious thinking, that is helping the brain to get to the = creative moment stage. But just as with almost everything else practice = makes the Master, and to experience flow, mastery is a requirement.