All the different actually infinite cut-sets exist in the Mindscape, and all the real numbers are already there, whether or not they can be finitely named or constructed. Just as chemistry was unified and simplified when it was realized that every chemical compound is made of atoms, mathematics was dramatically unified when it was realized that every object of mathematics can be taken to be the same kind of thing.
There are now other ways than set theory to unify mathematics, but before set theory there was no such unifying concept. Since the advent of set theory, once can correctly say that all mathematicians are exploring the same mental universe. The transfinite ordinal numbers can be thought of as arising through counting.
We also need a first ordinal to start with, called 0. Strictly speaking, the second principle for generating ordinals give us 0, since zero is the first ordinal after the empty sequence.
Infinity and the Mind: The Science and Philosophy of the Infinite
In any case, once we have zero, the first principle can be repeatedly applied to get the ordinal numbers 0, 1, 2, …. And to use the tools of symbolic logic to investigate an empirically existing phenomenon is not to commit a category mistake, any more than it is a category mistake to look at living cells through the inanimate lenses of a microscope. We have a primitive concept of infinity. This concept is inspired, I suspect, by the same deep substrate of mind that conditions religious thought. Set theory could even be viewed as a form of exact theology.
By means of the set-theoretic analysis of Absolute Infinity, we attain knowledge of many lower infinities—the transfinite ordinals and cardinals. As a Parmenidean monist, Zeno viewed space as an undivided whole that cannot really be broken down into parts.
We can find scattered locations in space, but space is always more than the sum of these from an Absolutely Continuous tract of space, but there will always be a residue of leftover space, of continuous little pieces, infinitesimal intervals over which the actual motion takes place. This view of space has been held by several philosophers since Zeno, notably C.
In calculus one discusses different sorts of calculation processes C , , whose outcome depends on what numbers are fed in. This is called the limit process. That is, if one believes that the whole world is a dream, an illusion, an image in some Mind… if one believes this, then it is hard to account for the identity between the numbers that different people extract from the world. If I go into the woods and count the branches on a particular dead oak that looks like a dinosaur, and if you do the same tomorrow, then our numbers will certainly agree.
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You can leave your home for fourteen years, and when you go back, your house is still the seventh on the right. The only way to account for the numerical identities among the worlds you and I dream up must be that in some sense you and I are really the same person.
In itself, the string of symbols G M , M , M is not really a name. Of course, theses definitions involve more words, but the assumption is that ultimately any description should be reducible to extremely simple statements about, let us say, writing down strokes. It is not hard to imagine how one would complete the name G M , M , M by adding a description of M, of multiplication, of the Ackerman generalized exponential, and so on. We denote this expanded name by [ G M , M , M ]. The idea is that [ G M , M , M ] might be something like the last few pages, along with a few more pages of further amplification.
Given this full description, anyone would be able to figure out what number is being named, to the point of being able to come up with a list of G M , M , M strokes, if there were no limitation on time. Names like [the googolth prime number] or [ G M , M , M ] are what we might call constructive names. Trying to find the number named by each of these names involves searching through all the numbers until the right sort of number is found. But in each case, the search could be fruitless.
To avoid paradox, one has to accept the fact that the names u 0 and W are really not names. The symbols n-a-m-e-a-b-l-e point to the concept, but they do not really reach it. Just as the Absolute lies beyond any possible description, the notion of nameable in a lifetime lies beyond any ration human description. Is it a final solution to the Berry paradox? Not really. He feels that the library includes everything, and one of those marvellous Borgesian lists ensues:.
In a sense a library like this is useless. Randomly selecting a book from the Library of Babel is equivalent to sitting down and randomly typing pages.
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Looking at the titles of the books would be of no help, for a book called The Continuum Problem might turn out to be about, say, astral travel. We will consider a universe that continues expanding forever after an initial singular state. Our own universe may very well be like this.
Given an infinite future, with no future collapses to rub everything out, might look for an irreducible infinity in the form of a random sequence. But there is the problem that you will not be around forever, so you will not be able to produce an infinite sequence of digits. To avoid this, you might build a coin-flipping machine. To keep the machine running, you supply it with a couple of repair robots, who are also capable of repairing and even building copies of themselves.
If we have an everlasting universe with an infinite amount of matter in it, then there is no theoretical reason why such an immortal coin-flipper could not be set up.
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If the coin-flipper is unbiased, it is at least logically possible that nothing but heads will come up from now on. But we would expect it to be more likely to produce a sequence of zeros and ones answering to no finite description.
A universe in which at least one such random coin-flipper existed would not have any finite description. If every possible universe exists, then there is no need to account for the special peculiarities of this universe e. If every possible universe exists, then there is no need to explain any peculiarity. Why is there an ant on my screen? No reason—there is another universe exactly the same except with no ant.
This situation is a bit like the Total Library. In a sense, the Total Library contains no information at all! Even if some perfectly accurate description U of the universe exist; it seems likely that if we represented U by the clumsy expedient of putting numbers in books, then this representation of U would not fit in the universe. Of course, the most efficient representation of U is the universe itself, so at least one representation of U exists. But could we ever hope to have a desk-top or pocket-sized model of the universe? Only if matter.
For if there is some smallest size particle, then any object in the universe will have less particles than the universe, and thus cannot serve as a scale model.
Infinity and the Mind: The Science and Philosophy of the Infinite by Rudy Rucker
But if there is no smallest particle size, then any portion of matter contains the same infinitely many particles, so it would be possible for some small region of matter to look exactly like the entire universe. And this is not so easy to know. In point of fact, there can be no finite complete description of truth. Truth is undefinable. As we will see, this will be our way out of the Liar paradox.
The thinkers of the Industrial Revolution liked to regard the universe as a vast preprogrammed machine. It was optimistically predicted that soon scientists would know all the rules, all the programs. Of course, anyone can say that science does not have all the answers. In order to grasp the implications of such a programmatic approach to human knowledge, I would like to run a little thought-experiment here, an alternate-world fantasy. There is no complete set MT of axioms for mathematical truth.
Any system of knowledge about the world is, and must remain, fundamentally incomplete, eternally subject to revision. The analogy is quite close.
The Pythagoreans learned that no ratio of natural numbers could fully describe the relation between the diagonal and the side of a square. That is, he has shown that the set of all true statements about mathematics is finitely unnameable, and thus essentially random and infinite. Although this theorem can be stated and proved in a rigorously mathematical way, what it seems to say is that rational thought can never penetrate to the final, ultimate truth.
But if mathematicians cannot ever fully understand something as simple as number theory, then it is certainly too much to expect that science will ever expose any ultimate secret of the universe. Scientists are thus left in a position somewhat like K. Endlessly we hurry up and down corridors, meeting people, knocking on doors, conducting our investigations. But the ultimate success will never be ours.