1. Researchers in algorithmic information theory have recently studied concepts of randomness based on sequences of outcomes of processes. In this context randomness refers to the absence of pattern in such a sequence. They have also analyzed what it means to have a pattern and have used a concept from computer science, the Turing machine, to give this a precise meaning. Using these notions, they have been able to show that almost all numbers are random in the measure theoretic sense that in the numeric interval from 0 to 1, the measure of the subset of the random numbers is one and the measure of the subset of the nonrandom numbers is zero. In this sense, patterns are unusual; randomness is the norm.
  2. Physicists have recently recorded extensive lists of the outcomes of "beam-splitting" experiments – the quantum theoretic collapse of an electron's wave function into one or the other of two states. They have recorded these as numerical sequences – a 1 for the outcome "spin up" and a 0 for the outcome "spin down." Thus, for example, a list of eight recorded states could read "spin up, spin up, spin down, spin up, spin up, spin down, spin down, spin down." These would be numerically coded as 11011000. Statisticians have developed batteries of tests of randomness that can be applied to sequences of binary digits; such tests are critical in the development of secure codes. When these tests were applied to the lists of bits generated by the beam splitting experiments, the lists performed better than sequences of bits generated by all computer-based pseudo-random generators that were tested. This strongly suggests that randomness is an inherent property of particles at the quantum level.
  3. There are abundant examples of randomness in the natural world that could be interpreted as purposeful. For example: (i) The human body has roughly 1013 cells. Delivery of nutrients to them and removal of waste products is accomplished by random molecular motion using the process of diffusion. (ii) Diffusion also maintains stable air pressure in a closed setting like a room. (iii) When plants and animals reproduce, their offspring are not genetic copies of their parents – for instance, the DNA of each human sperm or egg can assume one of over 8 million patterns. Random selection of which patterns actually are instantiated as offspring enables plant and animal species to adapt to dynamically changing ecosystems; random mutations allow for a form of creativity in which new genetic patterns allow species to explore the possibilities available to them.
  4. Even though randomness can be understood as serving purposeful functions as in 3) above, some dynamical systems (e.g., some Julia Sets and Limit Sets) have the property that they transform random inputs into stable outcomes. There are many such examples, for instance, genetic algorithms, neural nets, game theory with random inputs, and quantum randomness. As David Bartholomew puts it, "God can have it both ways" – randomness and order.
  5. The following theorems and understandings in mathematics suggest that transformation of randomness into order is a fundamental property of aspects of the natural world:

    i. The Law of Large Numbers
    ii. The Central Limit Theorem
    iii. Poisson Law of Small Numbers

    That is, randomness and orderliness are not mutually exclusive; in subtle ways, randomness at one level gives rise to order at a higher level. As formulated by the 17th century physicist Pierre Louis Maupertuis, The Principle of the Least Action, a fundamental principle of physics says, "Nature is thrifty in all its actions." The principle has often been used to argue that nature can be thought of being teleological, although this claim has been controversial. An analogous principle may apply to randomness, that nature both uses randomness and transforms it into order. That is, there may be fundamental natural processes involving randomness that can be interpreted as pointing to purposefulness in nature.