Introduction to knowledge

General knowledge system
	All our human knowledge comes from experience perceived by our senses. Seeing with our eyes (visual), hearing with our ears (auditory), feeling with our hands, body and internally (kinesthetically), smell with our nose (olfactory) and taste with our mouth and tongue (gustatory). In the past this gathered data and information was classified and determined. Eventually this must have resulted into a language-system by which communication was possible. Preserved knowledge from ancient civilizations is mostly visual in the form of wall-paintings or carvings and later recorded as codified symbols now interpreted as written texts. Archeologists also expose the physical trails and objects of past civilizations and interpret these findings which gives us information about systems they have used. Most of our currently used knowledge and culture was originated in the old Greek and Roman society. Philosophy is said to be the mother of all (liberal) arts or sciences. Artes liberales is the Latin for liberal arts. The canon of the seven liberal arts dates from the fifth century AD and has its roots in Roman times. The first three sciences, grammatica, rhetorica and dialectica or logics are called the trivium which is the path that leads to wisdom. The remaining four were arithmetica, geometria, astronomia and musica, usually referred to as the quadrivium, the fourfold way. Until around 1200 AD these sciences were the foundation of any study at a university. The library classification systems (i.e. UDC) and educational systems today still have many similarities and comparisons to these 'original' arts. Although our educational focus has shifted to more pragmatic, and specialized forms related to service professions taught in classical form, there are still craftsmen who learn their trade by tradition from a master. In general I distinguish three kinds of learning-contexts: Collective, classical group-presentation, from within an educational system Individual, (mostly one to one) adaptive learning of crafts or skills, from professional or educational background. Individual, autodidact learning from personal reading, practice and/or experience, inspired by personal motives. In the ideal form, al learning is initiated by individual demand and personal motives (in contrast to our conventional linear educational system). Maybe not everyone could handle the needed responsibility and self-activation in today's high-tech information-society. We need to be tremendously curious, perceptive, and on top of that aware of manipulation-techniques of (mass-)media and be able to select and abstract from enormous amounts of (un)wanted audio-visual input. We can distinguish three kinds of literacy: Language-literacy (spoken and written); Media-literacy (what kinds of manipulation techniques are used) Computer-literacy (the ability to work with and use computers) The most basic didactical distinction in general learning is Benjamin Bloom's three domains: Cognitive learning (thoughts), such as teaching someone to add fractions. Affective learning (feelings, values), such as teaching someone to not want to smoke. Physical or motor learning (actions), such as teaching someone to touch type. As an underlying principle I would like to add Intentional Learning, or intrinsic manifested curiosity, the really wanting to know, and asking for more as a (auto)didactical principle. By dedicated use of this principle, we create a positive empowered knowledge system in the mind and body. Cognitive learning is the primary focus as everything has to start with a thought. The major levels of cognitive learning can be classified as memorizing, understanding, and applying. Most content can be learned at any of these three levels of learning. Memorization; This is rote learning. It entails learners encoding facts or information in the form of an association between a stimulus and a response, such as a name, date, event, place or symbol. The behavior that indicates that this kind of learning has occurred is stating (or "regurgitating"), usually verbatim. Understanding; This is meaningful learning. It entails learners relating a new idea to relevant prior knowledge. The behaviors that indicate that this kind of learning has occurred include comparing and contrasting, making analogies, making inferences, elaborating, and analyzing (as to parts and/or kinds), among others. Application; This is learning to generalize to new situations, or transfer learning. It entails learners identifying critical commonalities across situations, such as predicting the effects of price increases. The behavior that indicates that this kind of learning has occurred is successfully applying a generality (the critical commonalities) to a diversity of previously unencountered situations. It is useful to identify three types of content that can be learned on the application level: Concepts Procedures Principles A concept is a group or class of particulars which have something in common. A procedure is an ordered sequence of steps for accomplishing some goal. A principle is a relationship between two or more changes. It can be a causal, correlational, or natural-order relationship. A good rule for identifying these kinds of content: Concepts are concerned with grouping things into categories. What? Procedures are concerned with how to do something. How? Principles are concerned with predictions and explanations. Why? It is also helpful to keep in mind that these three types of content can be learned at any of the three levels of learning. (Source: Methods of instruction by Charles M. Reigeluth) Education and learning has been one of my primary themes in the past years. With the growing use of Information & Communication Technology (ICT) and new media in education numerous possibilities arise: once the learning-module is produced and digitalized the distribution-costs and multiplication-cost could be theoretically brought down to zero by the use of Internet. Although this is a slightly different subject, thinking about these place- and time-independent opportunities with Internet, and not only related to education!, might just be one of the things to clear away some of our past burdens of the industrial society. Knowing how we can learn, and using a supportive system of thought strengthening in our environment will bring us closer to fulfilling our dreams. Aspects of manifesting your personal dream is cooping with obstacles and solving 'problems' between the current situation and the (visualized) wanted situation. It is my intention to provide methods (including reflections and techniques) for these necessary steps to take. The first step of this ambitious plan is my personal research. By publishing and sharing this research and findings in raw form on the Internet I start the process of presenting these ideas in a more tangible form. I hope you (reader) enjoy this and I'm looking forward to see your feedback.
Some Quotes:
	There do exist enquiring minds, which long for the truth of the heart, seek it, strive to solve the problems set by life, try to penetrate to the essence of things and phenomena and to penetrate into themselves. If a man reasons and thinks soundly, no matter which path he follows in solving these problems, he must inevitably arrive back at himself, and begin with the solution of the problem of what he is himself and what his place is in the world around him. For without this knowledge, he will have no focal point in his search. Socrates’ words, “Know thyself” remain for all those who seek true knowledge and being. [Gurdjieff] "Life isn't about finding yourself. Life is about creating yourself." - George Bernard Shaw "The marble not yet carved can hold the form of every thought the greatest artist has." - Michelangelo "A mediocre idea that generates enthusiasm will go further than a great idea that inspires no one." - Mary Kay Ash "If you can dream it, you can do it." - Walt Disney "The place to improve the world is first in one's own heart and head and hands." - Robert M. Pirsig [Toine Fennis, MetaMagazine.com 2001]

Googol and Googolplex

My kids (from 3 and 5 years old) use the large number word Googolplex frequently in their conversations. I mistakenly explained a Googolplex as a 1 with 100 zero's (which I now know is only a Googol). To set the record and their education straight I thought it would be nice to actually look-up the origin and explanation of this intriguing word which baffles anyone trying to name a large number (just try to name a larger number from the top of your mind...). [Toine]

Googol and Googolplex

Some time before 1940, mathematician Edward Kasner asked his (at the time) nine-year-old nephew to name the number 10¹⁰⁰, and the nephew invented the name googol. Kasner then named 10^googol with the name googolplex.

googol is a class-2 number, and is reasonably easy to comprehend. You can even write it out:

10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000

The number of particles (protons, electrons, photons, etc.) in the universe has been estimated as 10⁸⁰, so while a googol is an awful lot bigger, it's still roughly the same idea.

Googolplex on the other hand, is 10^googol, or:

10^{10000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000}

and that's a number far too big to relate to anything physical — or is it?

This description comes from Don Page, and is highly paraphrased:

In quantum physics one talks about distinct and indistinct states, overlapping wave functions and many other things, and one of the important principles that comes out of it the concept of how much time it takes for a system with many possible states to spontaneously assume a certain, given state. For simplicity it is imagined that the system is inside an impermeable, rigid box (which thus prevents anything from going in or out and thus isolates the system). If you're outside the box you have no idea what's inside, and any state can spontaneously occur (even a higher-energy state). For example, if the system is a proton and a neutron, the possible states include: A hydrogen atom, a hydrogen atom with the electron in a higher "shell", a separate proton and electron, a neutron, or many other things. The amount of time you might have to wait for one of these states to occur depends on the number of possible states, which depends on the number of particles inside.

Cosmologists and physicists often use these concepts to discuss what might happen inside black holes. The inside of a black hole is isolated from us (nothing can get out) and so one can talk about all the possible states that might exist inside the black hole and how long it might take for a certain state to arise. In theory, if left inside a rigid nonpermeable box for long enough (so that nothing new can go into the black hole) the contents of the box will at some point in the future cease to be a black hole and instead be real matter.

The formula for the number of possible states is a really big exponential function with Planck constants in it and other such things. I don't have the exact formula but we do know that if the black hole had a mass equal to the Milky Way Galaxy, plus the Andromeda galaxy, and Magellanic clouds, and a few other similar globular clusters in our region, then the number of states, and the amount of time you'd have to wait, is somewhere around a googolplex. Its such a big number that there's no way to distinguish what units should be used. It could be a googolplex seconds or a googolplex millenia. Whatever the units, thats the amount of time you'd have to wait for the contents of an impermeable, rigid box containing a black hole with the mass of the local galactic neighborhood to spontaneously become identical to the local galactic neighborhood as we know it.

The original (and much better) description is below.

Reference:

Edward Kasner and James Newman, Mathematics and the Imagination, Penguin, 1940.
[Source: http://www.mrob.com/pub/math/ln-notes1.html#googol ]

How to Get A Googolplex
So I finally convinced you that you cannot imagine something close to Googolplex? This was my belief, too, until I was taught a lesson by Don Page, don@page.phys.ualberta.ca, a gravitational physicist. But beware, it's deep into physics.

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Stephen Hawking, Nature 248 (1974), pp. 30-31, and Communications of Mathematical Physics 43 (1975), p. 199ff, found that the dimensionless entropy (i.e., the entropy after dividing by Boltzmann's constant k) of a black hole is 1/4 its area in Planck units, or 4*pi times its mass squared in Planck units for a Schwarzschild (i.e., uncharged and nonrotating) black hole.

Planck units are those one gets when one multiplies the appropriate powers of Planck's reduced constant h-bar=h/(2*pi), the speed of light c, and Newton's gravitational constant G to get a quantity of whatever dimension one wants.

E.g., the Planck unit of area, or simply the Planck area, is (h-bar*G)/(c^3)=2.612x10^{-70} square meters, the Planck length is the square root of this, or 1.616x10^{-35} meters, the Planck time is the Planck length divided by the speed of light, or Sqrt[h-bar*G/c^5]=5.391x10^{-44} seconds, and the Planck mass is Sqrt[h-bar*c/G]=2.177x10^{-8} kilograms.) Thus one would need a Schwarzschild black hole of mass Sqrt[10^{100}/(4*pi)]=2.821x10^{49} Planck masses, or 6.140x10^{41} kg, to have an entropy of one Googol. Since the sun has a mass of about 1.989x10^{30} kg or 9.137x10^{37} Planck masses, one needs about 3.087x10^{11} solar masses to give a black hole with an entropy of one Googol.

If the entropy of a black hole represents the (natural) logarithm of the number of states of similar macroscopic configurations (as it does for other thermal systems, and as I shall assume is also true for black holes, though there is some controversy about this), then a black hole with an entropy of one Googol would have e^{googol}=10^{googol/(ln10)} states. To get a Googolplex of states, the entropy needs to be larger than a Googol by a factor of ln10, and so the mass of the black hole needs to be larger by the square root of ln10, or about 4.685x10^{11} solar masses to give a Googolplex of macroscopically similar states. This is a bit more than three times the 1.4x10^{11} solar masses that C. W. Allen, Astrophysical Quantities, Third Edition (The Athlone Press, University of London, 1976), p. 282, attributes to our Galaxy, but Allen, p. 287, attributes 10^{11.5} solar masses to the Andromeda Galaxy (M31), so if one adds that, and perhaps one or more smaller galaxies in our Local Group, such the Large Magellanic Cloud, the Small Magellanic Cloud, M32, etc. (each of which are a few percent of the mass of our Galaxy), one should get to a mass such that the corresponding black hole would have a Googolplex of states with similar macroscopic configurations. Certainly adding the Maffei Galaxy, with a mass of 10^{11.3} solar masses according to Allen, would put one over what is needed for this, unless Allen's numbers are significantly overestimated. (I would guess that dark matter would make the actual masses rather larger than the estimates Allen quotes, but I don't know the latest figures on the masses of components of our Local Group.)

Thus if one takes some (probably moderately large) fraction of the mass of our Local Group of galaxies, puts it into a black hole, and asks how many states there are with a similar macroscopic appearance, one would get a Googolplex.

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Obviously, Mr. Page is into big numbers, too. "The number of states in a black hole with a mass roughly equivalent to the Andromeda Galaxy". And you've got a Googolplex. Couldn't be simpler, could it?

However, after a user's complaint that it is a little too technical even for an advanced physics student,

Huh? I took physics for two years, and calculus for two years, but your
explanation that there is indeed enough matter in the
universe to comprise a Googolplex floored me.

Mr. Page decided to add another paragraph.

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This result, plus the assumption that a black hole is a nearly ideal mixer of macroscopic information, would imply that if one put this black hole into a hypothetical rigid nonpermeable box, a few million light years in size, and looked at the contents once a year, it would look like our present Local Group for the first time again after roughly a Googolplex of years. In other words, a Googolplex is a quantitative measure of an extension (to a much larger system) of the temporal periods expressed by such colloquial phrases as "until the cows come home," which would give recurrence times that would presumably be long in human terms but much shorter than a Googolplex if these smaller systems (e.g., of cows and their home) would actually last long enough.

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You might be amused to note that in Information Loss in Black Holes and/or Conscious Beings? to be published in Heat Kernel Techniques and Quantum Gravity, edited by S. A. Fulling (Discourses in Mathematics and Its Applications, No. 4, Texas A&M University Department of Mathematics, College Station, Texas, 1995) (University of Alberta report Alberta-Thy-36-94, Nov. 25, 1994), hep-th/9411193, I estimated a quantum Poincare recurrence time for the quantum state of an extremely hypothetical rigid nonpermeable box containing a black hole with the mass of what may be the entire universe in one of Andrei Linde's stachastic inflationary models and got 10^(10^{10^[10^(10^1.1)]}) Planck times, millenia, or whatever. As I wrote in the following line, "So far as I know, these are the longest finite times that have so far been explicitly calculated by any physicist."

Don Page
[Source: http://www.fpx.de/fp/Fun/Googolplex/GetAGoogol.html]

Dont mistake Googolplex with Googleplex which is searchengine Google headquarters in Mountain View, California, (USA). I assume they liked the alliteration and reference to this large number when naming the complex. [Toine Fennis]

MetaMagazine articles (collect & share)

This MetaMagazine Blog is my effort of collecting and sharing articles and resources I find useful or interesting. These articles are used as background for MetaBook.
My personal (related) websites:

• MetaMagazine.com (Personal Research Magazine)
  
• MetaMagazine.nl (Dutch MetaMagazine & META Method)
  
• MetaBook.nl / MetaBoek.nl (Publishing)
  
• MetaBooks.info (Ebooks, Software and webmaster resources)
  
• MetaHelpdesk.com (Articles categorized under subjects)

These are just some of the websites I am using. [Toine Fennis]