Online research in philosophy

“Anyone who considers arithmetical methods of producing random digits is, of course, in a state of sin.”1 John von Neumann’s famous dictum points an accusing finger at all who set their ordered minds to engender disorder. Much as in times past thieves, pimps, and actors carried on their profession with an uneasy conscience, so in this day scientists who devise random number generators suffer pangs of guilt. George Marsaglia, perhaps the preeminent worker in the field, quips when he asks his colleagues, “Who among us has not sinned?” Marsaglia’s work at the Supercomputer Computations Research Institute at Florida State University is well-known. Inasmuch as Marsaglia’s design and testing of random number generators depends on computation, and inasmuch as computation is fundamentally arithmetical, Marsaglia is by von Neumann’s own account a sinner. Working as he does on a supercomputer, Marsaglia is in fact a gross sinner. This he freely admits. Writing of the best random number generators he is aware of, Marsaglia states, “they are the result of arithmetic methods and those using them must, as all sinners must, face Redemption [sic] Day. But perhaps with better understanding we can postpone it.”.

We review briefly the attempts to define random sequences $(\S0)$ . These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence ( $\S\S1-3$ and 5); the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests $(\S4)$.

Number Relationship between Numbers identified Similar to Pi, E, and O. Could be called a new group such that the first number when raised to the number of units or places before the final digit equals the value of the final number.

For the given logical calculus we investigate the proportion of the number of true formulas of a certain length n to the number of all formulas of such length. We are especially interested in asymptotic behavior of this fraction when n tends to infinity. If the limit exists it is represented by a real number between 0 and 1 which we may call the density of truth for the investigated logic. In this paper we apply this approach to the intuitionistic logic of one variable with implication and negation. The result is obtained by reducing the problem to the same one of Dummett's intermediate linear logic of one variable (see [2]). Actually, this paper shows the exact density of intuitionistic logic and demonstrates that it covers a substantial part (more than 93%) of classical prepositional calculus. Despite using strictly mathematical means to solve all discussed problems, this paper in fact, may have a philosophical impact on understanding how much the phenomenon of truth is sporadic or frequent in random mathematics sentences.

Is the number an absolute intelligible reality? The author investigates the number and its nature in Plotinus. works trying to solve the following question: what number is considered intelligible - the number in general or the number in particular? Three answers are given over this study. Thus, if the number is generally defined as intelligible (as Plotinus sometimes does), than the number in general is an intelligible reality (a general intelligible number, therefore, exists). On the other hand, if we make a distinction between numbers (the plural) and number (the singular), it seems that, for Plotinus, only the particular number could be considered clearly intelligible, while the number as a generic reality is not so. Actually, the final solution comes out from the agreement between these two divergent theses. This agreement is based on the idea of the total number: a number that is in the same time particular and general, a number which is the object of the final part of the present study.

We show how to determine the k-th bit of Chaitin’s algorithmically random real number Ω by solving k instances of the halting problem. From this we then reduce the problem of determining the k-th bit of Ω to determining whether a certain Diophantine equation with two parameters, k and N , has solutions for an odd or an even number of values of N . We also demonstrate two further examples of Ω in number theory: an exponential Diophantine equation with a parameter k which has an odd number of solutions iff the k-th bit of Ω is 1, and a polynomial of positive integer variables and a parameter k that takes on an odd number of positive values iff the k-th bit of Ω is 1.

This thesis is presented in the hope that it will resonate with mathematicians and others who are interested in analysis concepts and pure number theory.

Almost everyone has an intuitive notion of what a random number is. For example, consider these two series of binary digits: 01010101010101010101 01101100110111100010 The first is obviously constructed according to a simple rule; it consists of the number 01 repeated ten times. If one were asked to speculate on how the series might continue, one could predict with considerable confidence that the next two digits would be 0 and 1. Inspection of the second series of digits yields no such comprehensive pattern. There is no obvious rule governing the formation of the number, and there is no rational way to guess the succeeding digits. The arrangement seems haphazard; in other words, the sequence appears to be a random assortment of 0's and 1's.

many symbols. We define o, as the probability that an arbitrary machine be circular and we prove that o, is a random number that goes beyond..