The Zero Theorem Quotes

Just solving certain theorems makes waves in the Platonic over-space. Pump lots of power through a grid tuned carefully in accordance with the right parameters—which fall naturally out of the geometry curve I mentioned, which in turn falls easily out of the Turing theorem—and you can actually amplify these waves, until they rip honking great holes in spacetime and let congruent segments of otherwise-separate universes merge. You really don’t want to be standing at ground zero when that happens.

The theorem states that in zero-sum games in which the players’ interests are strictly opposed (one’s gain is the other’s loss), one player should attempt to minimize his opponent’s maximum payoff while his opponent attempts to maximize his own minimum payoff. When they do so, the surprising conclusion is that the minimum of the maximum (minimax) payoffs equals the maximum of the minimum (maximin) payoffs. The general proof of the minimax theorem is quite complicated, but the result is useful and worth remembering. If all you want to know is the gain of one player or the loss of the other when both play their best mixes, you need only compute the best mix for one of them and determine its result.

It will be noticed that the fundamental theorem proved above bears some remarkable resemblances to the second law of thermodynamics. Both are properties of populations, or aggregates, true irrespective of the nature of the units which compose them; both are statistical laws; each requires the constant increase of a measurable quantity, in the one case the entropy of a physical system and in the other the fitness, measured by m, of a biological population. As in the physical world we can conceive the theoretical systems in which dissipative forces are wholly absent, and in which the entropy consequently remains constant, so we can conceive, though we need not expect to find, biological populations in which the genetic variance is absolutely zero, and in which fitness does not increase. Professor Eddington has recently remarked that 'The law that entropy always increases—the second law of thermodynamics—holds, I think, the supreme position among the laws of nature'. It is not a little instructive that so similar a law should hold the supreme position among the biological sciences. While it is possible that both may ultimately be absorbed by some more general principle, for the present we should note that the laws as they stand present profound differences—-(1) The systems considered in thermodynamics are permanent; species on the contrary are liable to extinction, although biological improvement must be expected to occur up to the end of their existence. (2) Fitness, although measured by a uniform method, is qualitatively different for every different organism, whereas entropy, like temperature, is taken to have the same meaning for all physical systems. (3) Fitness may be increased or decreased by changes in the environment, without reacting quantitatively upon that environment. (4) Entropy changes are exceptional in the physical world in being irreversible, while irreversible evolutionary changes form no exception among biological phenomena. Finally, (5) entropy changes lead to a progressive disorganization of the physical world, at least from the human standpoint of the utilization of energy, while evolutionary changes are generally recognized as producing progressively higher organization in the organic world.

The Fundamental Theorem of Algebra If is a polynomial of degree , then has at least one zero in the complex number domain. In other words, there is at least one complex number such that .

In other words, the subjective perception of a copy of you in a typical parallel universe is a seemingly random sequence of wins and losses, behaving as if generated through a random process with probabilities of 50% for each outcome. This experiment can be made more rigorous if you take notes on a piece of pater, writing "1" every time you win and "0" every time you lose, and place a decimal point in front of it all. For example, if you lose, lose, win, lose, win,win,win, lose,lose and win, you'd write ".0010111001." But this is just what real numbers between zero and one look like when written out in binary, the way computers usually write them on the hard drive! If you imagine repeating the Quantum Cards experiment infinitely many times, your piece of paper would have infinitely many digits written on it, so you can match each parallel universe with a number between zero and one. Now what Borel's theorem proves is that almost all of these numbers have 50% of their decimals equal to 0 and 50% equal to 1, so this means that almost all of the parallel universes have you winning 50% of the time and losing 50% of the time. It's not just that the percentages come out right. The number ".010101010101..." has 50% of its digits equal to 0 but clearly isn't random, since it has a simple pattern. Borel's theorem can be generalized to show that almost all numbers have random-looking digits with no patterns whatsoever. This means that in almost all Level III parallel universes, your sequence of wins and losses will also be totally random, without any pattern, so that all that can be predicted is that you'll win 50% of the time.

The Banach-Tarski Theorem is an astonishing result. We have decomposed a ball into finitely many pieces, moved around the pieces without changing their size or shape, and then reassembled them into two balls of the same size as the original. I think the theorem teaches us something important about the notion of volume. As noted earlier, it is an immediate consequence of the theorem that some of the Banach-Tarski pieces must lack definite volumes and, therefore, that not every subset of the unit ball can have a well-defined volume. A little more precisely, the theorem teaches us that there is no way of assigning volumes to the Banach-Tarski pieces while preserving three-dimensional versions of the principles we called Uniformity and (finite) Additivity in chapter 7. (Proof: Suppose that each of the (finitely many) Banach-Tarski pieces has a definite finite volume. Since the pieces are disjoint, and since their union is the original ball, Additivity entails that the sum of the volumes of the pieces must equal the volume of the original ball. But Uniformity ensures that the volume of each piece is unchanged as we move it around. Since the reassembled pieces are disjoint, and since their union is two balls, Additivity entails that the sum of their volumes must be twice the volume of the original ball. But since the volume of the original ball is finite and greater than zero, it is impossible for the sum of the pieces to equal both the volume of the original ball and twice the volume of the original ball.) If I were to assign the Banach-Tarski Theorem a paradoxicality grade of the kind we used in chapter 3, I would assign it an 8. The theorem teaches us that although the notion of volume is well-behaved when we focus on ordinary objects, there are limits to how far it can be extended when we consider certain extraordinary objects - objects that can only be shown to exist by assuming the Axiom of Choice.

In the first place, Coase specified three crucial conditions for his conclusion to hold. These were: a legal framework establishing liability for actions, presumably supported by governmental authority; perfect information; and zero transaction costs (including organization costs and the costs of making side-payments). It is absolutely clear that none of these conditions is met in world politics. World government does not exist, making property rights and rules of legal liability fragile; information is extremely costly and often held unequally by different actors; transaction costs, including costs of organization and side-payments, are often very high. Thus an inversion of the Coase theorem would seem more appropriate to our subject. In the absence of the conditions that Coase specified, coordination will often be thwarted by dilemmas of collective action.

Finally, the third law of thermodynamics, initially formulated in 1906 as Walther Nernst’s (1864–1941) heat theorem, states that all processes come to a stop (and entropy shows no change) only when the temperature nears absolute zero (–273°C).

The fact of zero He added nonstop: Did you know that zero was not used throughout human history! Until 781 A.D, when it was first embodied and used in arithmetic equations by the Arab scholar Al-Khwarizmi, the founder of algebra. Algorithms took their name from him, and they are algorithmic arithmetic equations that you have to follow as they are and you will inevitably get the result, the inevitable result. And before that, across tens and perhaps hundreds of thousands of years, humans refused to deal with zero. While the first reference to it was in the Sumerian civilization, where inscriptions were found three thousand years ago in Iraq, in which the Sumerians indicated the existence of something before the one, they refused to deal with it, define it and give it any value or effect, they refused to consider it a number. All these civilizations, some of which we are still unable to decipher many of their codes, such as the Pharaonic civilization that refused to deal with zero! We see them as smart enough to build the pyramids with their miraculous geometry and to calculate the orbits of stars and planets with extreme accuracy, but they are very stupid for not defining zero in a way that they can deal with, and use it in arithmetic operations, how strange this really is! But in fact, they did not ignore it, but gave it its true value, and refused to build their civilizations on an unknown and unknown illusion, and on a wrong arithmetical frame of reference. Throughout their history, humans have looked at zero as the unknown, they refused to define it and include it in their calculations and equations, not because it has no effect, but because its true effect is unknown, and remaining unknown is better than giving it a false effect. Like the wrong frame of reference, if you rely on it, you will inevitably get a wrong result, and you will fall into the inevitability of error, and if you ignore it, your chance of getting it right remains. Throughout their history, humans have preferred to ignore zero, not knowing its true impact, while we simply decided to deal with it, and even rely on it. Today we build all our ideas, our civilization, our software, mathematics, physics, everything, on the basis that 1 + 0 equals one, because we need to find the effect of zero so that our equations succeed, and our lives succeed with, but what if 1 + 0 equals infinity?! Why did we ignore the zero in summation, and did not ignore it in multiplication?! 1×0 equals zero, why not one? What is the reason? He answered himself: There is no inevitable reason, we are not forced. Humans have lived throughout their ages without zero, and it did not mean anything to them. Even when we were unable to devise any result that fits our theorems for the quotient of one by zero, then we admitted and said unknown, and ignored it, but we ignored the logic that a thousand pieces of evidence may not prove me right, and one proof that proves me wrong. Not doing our math tables in the case of division, blowing them up completely, and with that, we decided to go ahead and built everything on that foundation. We have separated the arithmetic tables in detail at our will, to fit our calculations, and somehow separate the whole universe around us to fit these tables, despite their obvious flaws. And if we decide that the result of one multiplied by zero is one instead of zero, and we reconstruct the whole world on this basis, what will happen? He answered himself: Nothing, we will also succeed, the world, our software, our thoughts, our dealings, and everything around us will be reset according to the new arithmetic tables. After a few hundred years, humans will no longer be able to understand that one multiplied by zero equals zero, but that it must be one because everything is built on this basis.

wide range of behavioral relations, and is now an umbrella term for the science of logical decision making in humans, animals, and computers. Modern game theory began with the idea regarding the existence of mixed-strategy equilibria in two-person zero-sum games and its proof by John von Neumann. Von Neumann's original proof used the Brouwer fixed-point theorem on continuous mappings into compact convex sets, which became a standard method in game theory and mathematical economics. His paper was