The history of the universe, if the physicists are to be believed, traces the development of complexity. The big bang supposedly exploded some kind of incredibly energetic super atom. From there, various subatomic particles found expression, then some of the lighter elements, then heavier elements were cooked in stars and all of this was eventually followed by the development of complex organic molecules and living organisms, such as ourselves. Thus, we see a movement from a singularity to multiplicity, from simplicity to complexity.
When solving an equation, we see movement in the reverse, or largely in the reverse. It may start off being of moderate length. It may lengthen and become terribly long, several lines or even pages, consisting of numerous pieces. Then starts the process of simplification, in which we move numbers around, divide here and there, knock off this and that. And we simplify and simplify, knocking off piece after piece until we arrive at THE ANSWER, THE SINGULARITY.
Thus, scientific and mathematical thinking, or a least a large chunk of it, can be seen as a specious attempt to reverse the flow of cosmic history, at least in the imagination. When this thinking moves beyond the realm of the imagination, and is used to make smokestacks and manufacture automobiles, is it any surprise that this is resulting in the destruction of our species?
Thursday, January 31, 2013
Sunday, January 27, 2013
A supplemental note on the Andorian basis for mathematics
Much like the study of chemical reactions, which has been discussed in an earlier post, mathematical reasoning, or at least a great deal of mathematical reasoning, involves an elaborate "and/or" dance. It is often necessary to begin with what appear to be interminably long equations. Variables are shifted around, moved from one side of the equation to the other, actualizing the "and" on one side of the equation, the "or" on the side of the equation that is losing a variable. It is lengthened and shortened until, lo and behold, an answer has been arrived at, which is generally one number. A lengthy equation with numerous constituent parts, is eventually peeled away until all has been compressed into one number. Thus, while the "and" and "or" is involved in mathematical problem solving, we can say that the "and" eventually triumphs when a solution is arrived at. Or is it a triumph of the "or", with a solution being reached by a peeling away of layers. Most likely a fine balance, as is required for good health. Problem solving, like health, requires a balance of the "and" and the "or".
Saturday, January 26, 2013
A brief transition from mathematics to physics
And we can say that because the laws of physics are expressed in equations, the "and" is present in all the laws of physics. And unlike mathematical equations, equations expressing the laws of physics are not entirely self referential. Rather, they connect to the world. And the "and" is present in their connection to the world.
Thus, the "and" is present in Newton's law, Force = Mass times Acceleration, Distance = 1/2 acceleration times time squared, etc. Thus, the "and" connects Force to Mass, Force to Acceleration, and Mass to Acceleration (for Force divided by Mass = Acceleration. Each member of each equation is in some way connected.
But the "or" is also present in each law, for the quantity described by each member is not the same, and the "or" separates differences in quantity (see last post). Thus, Force does not equal Acceleration, it equals Mass times Acceleration.
Thus, the "and" and the "or" are present in each of the laws of physics.
Everything is related. Everything is different.
Thus, the "and" is present in Newton's law, Force = Mass times Acceleration, Distance = 1/2 acceleration times time squared, etc. Thus, the "and" connects Force to Mass, Force to Acceleration, and Mass to Acceleration (for Force divided by Mass = Acceleration. Each member of each equation is in some way connected.
But the "or" is also present in each law, for the quantity described by each member is not the same, and the "or" separates differences in quantity (see last post). Thus, Force does not equal Acceleration, it equals Mass times Acceleration.
Thus, the "and" and the "or" are present in each of the laws of physics.
Everything is related. Everything is different.
Some more (simple) philosophy of mathematics
We have already said that the "and" is what makes possible the operations of addition and multiplication, while the "or" underlies subtraction and division.
Similarly, the "and" underlies all quantity, or all numbers greater than one. The number two would not be possible without the "and" conjoining separate units. While the "and" makes quantity possible, the "or" makes possible differences in quantity.
We can take that all of this a step further, and say that mathematics consists largely of equations. And equations, in their simplest form, consist of two sides, a left side and a right side separated by an "=" sign. Statements of equality essentially join together the two sides, saying their the same. Thus, we can say that the "and" makes possible all mathematical equations.
Now, there are different schools of thought concerning what mathematical equations really say. It has been said that they don't say anything about the world. Rather, they are entirely self referential, with the statement on one side simply being another way of expressing the statement on the other. And if the two sides of an equation are not really different but are really the same thing, then we are arguably not tying together two different things, and the "and" is not operative, for in order for the "and" to be operative, there must be at least two things.
While there may be some validity to this point of view, we cannot say the "and" is not present at all, as the "and" underlies all quantity. Moreover, while the quantity on each side of an equation maybe the same, we can't say that all ways of naming the same quantity have the same meaning. The quantity (4+2) may be the same as (5+1) but it is hard to argue that we mean the same thing when we say (4+2) as when we say (5+1). At the very least, it would seem, an equation would would tie together two putatively different quantities by showing that they are in fact the same. And once again, in this tying together of putatively different meanings, the "and" is present.
It can also be argued that an equation ties together two different sets, one on each side of the equation. This is a relatively dynamic picture of what happens in an equation. In the equation 4 plus two equals six, according to this school of thought, you have a set of six units on the right side of the equation, and two sets on the left side, one consisting of four units and the other of two. These two sets are joined together, and when you count the total number of units on the left side (6), you see that the quantity is the same as what is on the right side. The "and" is prominently featured in this conception, and as noted above, it is dynamic. Things are happening. Things are being joined together or, in the case of subtraction, wrenched apart.
While we usually think of mathematics consisting of statements of equality, it can also consist of statements of inequality, such as four plus two does not equal 7, or 4+2<7. The "or" would appear to be present in such statements, distinguishing between the different quantities.
We can attempt to compare the strength of the "and" and the "or" in the mathematical realm by asking whether there are more possible statements of equality or inequality. It would appear at first blush that there are more possible statements of inequality. Four plus two can only equal 6. It can't equal 7, 8, 9 etc. Thus, it may seem that for each statement of equality, the number of statements of inequality is infinite. However, there are an infinite number of numbers that add up to 6 (3 plus 3, 2 1/2 + 3 1/2, 2 1/3 + 3 2/3 etc.) And if we wished to compare the number of statements of inequality with the number of statements of equality by mapping each statement of inequality to a statement of equality, since the number of both is infinite, it is always possible to find a statement of equality to map against a statement of inequality.
Thus, we can say that the "and" and the "or" are equally strong in the mathematical realm. And there we have another statement of equality!!
Similarly, we can say that the "and" is always present in a statement of inequality. For even when we say that two numbers are different, we are linking them together when we compare them.
Similarly, the "and" underlies all quantity, or all numbers greater than one. The number two would not be possible without the "and" conjoining separate units. While the "and" makes quantity possible, the "or" makes possible differences in quantity.
We can take that all of this a step further, and say that mathematics consists largely of equations. And equations, in their simplest form, consist of two sides, a left side and a right side separated by an "=" sign. Statements of equality essentially join together the two sides, saying their the same. Thus, we can say that the "and" makes possible all mathematical equations.
Now, there are different schools of thought concerning what mathematical equations really say. It has been said that they don't say anything about the world. Rather, they are entirely self referential, with the statement on one side simply being another way of expressing the statement on the other. And if the two sides of an equation are not really different but are really the same thing, then we are arguably not tying together two different things, and the "and" is not operative, for in order for the "and" to be operative, there must be at least two things.
While there may be some validity to this point of view, we cannot say the "and" is not present at all, as the "and" underlies all quantity. Moreover, while the quantity on each side of an equation maybe the same, we can't say that all ways of naming the same quantity have the same meaning. The quantity (4+2) may be the same as (5+1) but it is hard to argue that we mean the same thing when we say (4+2) as when we say (5+1). At the very least, it would seem, an equation would would tie together two putatively different quantities by showing that they are in fact the same. And once again, in this tying together of putatively different meanings, the "and" is present.
It can also be argued that an equation ties together two different sets, one on each side of the equation. This is a relatively dynamic picture of what happens in an equation. In the equation 4 plus two equals six, according to this school of thought, you have a set of six units on the right side of the equation, and two sets on the left side, one consisting of four units and the other of two. These two sets are joined together, and when you count the total number of units on the left side (6), you see that the quantity is the same as what is on the right side. The "and" is prominently featured in this conception, and as noted above, it is dynamic. Things are happening. Things are being joined together or, in the case of subtraction, wrenched apart.
While we usually think of mathematics consisting of statements of equality, it can also consist of statements of inequality, such as four plus two does not equal 7, or 4+2<7. The "or" would appear to be present in such statements, distinguishing between the different quantities.
We can attempt to compare the strength of the "and" and the "or" in the mathematical realm by asking whether there are more possible statements of equality or inequality. It would appear at first blush that there are more possible statements of inequality. Four plus two can only equal 6. It can't equal 7, 8, 9 etc. Thus, it may seem that for each statement of equality, the number of statements of inequality is infinite. However, there are an infinite number of numbers that add up to 6 (3 plus 3, 2 1/2 + 3 1/2, 2 1/3 + 3 2/3 etc.) And if we wished to compare the number of statements of inequality with the number of statements of equality by mapping each statement of inequality to a statement of equality, since the number of both is infinite, it is always possible to find a statement of equality to map against a statement of inequality.
Thus, we can say that the "and" and the "or" are equally strong in the mathematical realm. And there we have another statement of equality!!
Similarly, we can say that the "and" is always present in a statement of inequality. For even when we say that two numbers are different, we are linking them together when we compare them.
Sunday, January 20, 2013
Knowledge
Knowledge, of course, is what is known. To know something is to understand it. To understand it is to perceive something about it or within it so that it has meaning.
We, or more accurately I, can take some examples. I know some Spanish. My vocabulary is sufficient to put together sentences, and I can often express what I want to say when I am in a Spanish speaking country. However, my knowledge of Spanish is not such that I can understand two Spanish speakers engaging in a conversation. If I didn't understand any Spanish, what I would hear would be a plethora of usually indistinguishable syllables. My level of comprehension is sufficient for me to pick up a word here and a phrase there. The remainder is this plethora that my cats probably hear. Thus, there are different levels of knowing and different levels of understanding. When we know a language well, what we hear is not a jumble of syllables but utterances that have meaning. The entire character of the experience is changed. And we feel more at ease as we understand what is being said. Since I don't understand most of what is said, I feel a certain lack of comfort in Spanish speaking countries, which would explain why I would like to learn more Spanish. Through this example we see how knowledge involves a relationship with something, a connectedness with something, that allows us to see, to derive meaning. Knowing a language does not increase our familiarity with rocks and animals or the laws of physics. It does increase our familiarity with certain sounds, and if sounds are things, we can say that knowledge increases our familiarity with part of the world (i.e. things in the world.) It goes without saying that the meanings we have decided to ascribe to words is entirely arbitrary. There is no logical reason that the word "the" should mean "the" rather than "of" or "dog".
The act of learning involves the most intense unity between the learner and the learned, a revealing that peels aware the barrier between the two. (It should go without saying that the "and" is operative here. Knowledge is what follows the act of learning. The unity may be less intense, but there is a comfort, a dwelling, a sense of contentment that results from knowledge. What was previously drivel is now understood.
We crave the act of revealing, the moving away of barriers, and the comfort, the dwelling that comes with knowledge. It is partly a kind of insecurity that motivates the desire to learn and understand. I am in a Spanish speaking country and don't understand anything that is being said around me. I have some interest in physics because I believe I will feel more comfortable with the world if I know something about how things work. (Of course, there are various other reasons why we learn, the most obvious being that we are forced to by our parents, guardians and teachers. Education succeeds when we are done with its compulsory aspect and still wish to learn.)
And this, in part, is what motivates the desire to learn. But the desire to learn is also motivated by a desire to control, to gain power and autonomy over what is learned. To solve a problem, whether it be a difficult equation or to learn a language is to conquer it, to gain mastery over it, to subsume it. Once again, the "and" is involved in this merging. And this conquest, while resulting in an immediate sensation of exhilaration, is also what causes the sense of comfort alluded to earlier.
(We are not dealing here with the notion of knowledge as "enlightenment" or "wisdom". Those who see it in those terms are generally unable to explain what it is and resort to the rationalization that it somehow transcends definition. That's fine. We won't talk about it.
Still, you may say, "Is knowledge of a language really knowledge?" My answer is simple. "Try saying that to a linguist." To say that one type of knowledge is less worthy than another is, quite frankly, elitist. Language is the filter through which we view the world, and viewing it through different filters undoubtedly broadens our aspect of reality. Over the last 120 or so years, others have written about this subject at great length, and I will not burden the record further. )
We, or more accurately I, can take some examples. I know some Spanish. My vocabulary is sufficient to put together sentences, and I can often express what I want to say when I am in a Spanish speaking country. However, my knowledge of Spanish is not such that I can understand two Spanish speakers engaging in a conversation. If I didn't understand any Spanish, what I would hear would be a plethora of usually indistinguishable syllables. My level of comprehension is sufficient for me to pick up a word here and a phrase there. The remainder is this plethora that my cats probably hear. Thus, there are different levels of knowing and different levels of understanding. When we know a language well, what we hear is not a jumble of syllables but utterances that have meaning. The entire character of the experience is changed. And we feel more at ease as we understand what is being said. Since I don't understand most of what is said, I feel a certain lack of comfort in Spanish speaking countries, which would explain why I would like to learn more Spanish. Through this example we see how knowledge involves a relationship with something, a connectedness with something, that allows us to see, to derive meaning. Knowing a language does not increase our familiarity with rocks and animals or the laws of physics. It does increase our familiarity with certain sounds, and if sounds are things, we can say that knowledge increases our familiarity with part of the world (i.e. things in the world.) It goes without saying that the meanings we have decided to ascribe to words is entirely arbitrary. There is no logical reason that the word "the" should mean "the" rather than "of" or "dog".
The act of learning involves the most intense unity between the learner and the learned, a revealing that peels aware the barrier between the two. (It should go without saying that the "and" is operative here. Knowledge is what follows the act of learning. The unity may be less intense, but there is a comfort, a dwelling, a sense of contentment that results from knowledge. What was previously drivel is now understood.
We crave the act of revealing, the moving away of barriers, and the comfort, the dwelling that comes with knowledge. It is partly a kind of insecurity that motivates the desire to learn and understand. I am in a Spanish speaking country and don't understand anything that is being said around me. I have some interest in physics because I believe I will feel more comfortable with the world if I know something about how things work. (Of course, there are various other reasons why we learn, the most obvious being that we are forced to by our parents, guardians and teachers. Education succeeds when we are done with its compulsory aspect and still wish to learn.)
And this, in part, is what motivates the desire to learn. But the desire to learn is also motivated by a desire to control, to gain power and autonomy over what is learned. To solve a problem, whether it be a difficult equation or to learn a language is to conquer it, to gain mastery over it, to subsume it. Once again, the "and" is involved in this merging. And this conquest, while resulting in an immediate sensation of exhilaration, is also what causes the sense of comfort alluded to earlier.
(We are not dealing here with the notion of knowledge as "enlightenment" or "wisdom". Those who see it in those terms are generally unable to explain what it is and resort to the rationalization that it somehow transcends definition. That's fine. We won't talk about it.
Still, you may say, "Is knowledge of a language really knowledge?" My answer is simple. "Try saying that to a linguist." To say that one type of knowledge is less worthy than another is, quite frankly, elitist. Language is the filter through which we view the world, and viewing it through different filters undoubtedly broadens our aspect of reality. Over the last 120 or so years, others have written about this subject at great length, and I will not burden the record further. )
Einstein
Of course, Einstein demonstrated a kind of connectedness, a primordiality of the "and". Space and time are no longer completely independent characteristics of the universe. They are integrally connected. When one object accelerates, time for that object runs more slowly than it does for a stationary object. (That object or person would not experience time as running more slowly, but from the standpoint of the stationary observer, it does.) And acceleration causes the shape of space, literally, to change. The object accelerating itself becomes smaller in the eyes of the stationary observer, and "circles" in this world would no longer have the same circumference that they have in a stationary world. Thus, as distance covered over time increases, time itself moves more slowly in the eyes of the stationary overserver.
Thus, space and time are connected, and so is empty space and matter. Einstein showed that the the mass of physical objects (gravity) has the same affect on space and time as acceleration. In the presence of massive objects time slows down (at least from the standpoint of an observer who is not near a massive object.) The shape of space changes to the extent that a straight line, from the view point of someone on earth, would not be straight to an observer viewing it in outer space. Light itself, and the speed of light is one of the essential parameters, the boundary of this world.
Similarly, he showed that matter and energy are largely the same. Matter is potential energy and can be converted to energy (e=mc squared). And light, the electromagnetic force, is connected to it all, setting the parameters through which space, acceleration and time must be viewed. According to physicist Brian Greene, we always move through space-time at the speed of light. When we accelerate through space, some of that motion is, in effect, diverted from the time access, slowing down time. Thus, space, time, matter, gravity, and energy are all connected.
Connectedness, connectedness, connectedness. The "and" underlies more than was first thought. While he failed in his efforts to show a grand unifying theory of the universe, might it be because of the strength of the "or"? Things can only be so connected?
Thus, space and time are connected, and so is empty space and matter. Einstein showed that the the mass of physical objects (gravity) has the same affect on space and time as acceleration. In the presence of massive objects time slows down (at least from the standpoint of an observer who is not near a massive object.) The shape of space changes to the extent that a straight line, from the view point of someone on earth, would not be straight to an observer viewing it in outer space. Light itself, and the speed of light is one of the essential parameters, the boundary of this world.
Similarly, he showed that matter and energy are largely the same. Matter is potential energy and can be converted to energy (e=mc squared). And light, the electromagnetic force, is connected to it all, setting the parameters through which space, acceleration and time must be viewed. According to physicist Brian Greene, we always move through space-time at the speed of light. When we accelerate through space, some of that motion is, in effect, diverted from the time access, slowing down time. Thus, space, time, matter, gravity, and energy are all connected.
Connectedness, connectedness, connectedness. The "and" underlies more than was first thought. While he failed in his efforts to show a grand unifying theory of the universe, might it be because of the strength of the "or"? Things can only be so connected?
Another look at multiplication - nothing that original
On second thought, multiplication does not have to involve creating something from nothing. It can involve dividing something into parts and then adding them up. Take 5 x 3. We can either start with 5 units, and add two more sets of 5 units to this set, creating, as it were, these additional two sets out of nothing. But we can also start off with a set of 5 units, slice each member into three parts, and then add these smaller units together. This, of course, involves the "or" (division), which is overcome by the "and", addition.
This probably comes closer to describing what occurred during the birth of the universe, which started out as one tiny superatom, broke apart, parts gathered to form new elements, scattered, broke apart again, combined in massive suns to form new heavier elements, and eventually, more explosions, more scattering, dust coalescing to form planets, complex molecules being formed, eventually indescribably complex organic molecules etc. The and/or dance that I have described at length in my earlier posts.
So yes...multiplication was undoubtedly key. But this involved division and addition. And as the universe continues to expand and things drift apart, members are subtracted from each set.
Thus, the and/or, is, in its own beautiful way, the unifying principle that underlies all creation.
This probably comes closer to describing what occurred during the birth of the universe, which started out as one tiny superatom, broke apart, parts gathered to form new elements, scattered, broke apart again, combined in massive suns to form new heavier elements, and eventually, more explosions, more scattering, dust coalescing to form planets, complex molecules being formed, eventually indescribably complex organic molecules etc. The and/or dance that I have described at length in my earlier posts.
So yes...multiplication was undoubtedly key. But this involved division and addition. And as the universe continues to expand and things drift apart, members are subtracted from each set.
Thus, the and/or, is, in its own beautiful way, the unifying principle that underlies all creation.
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