24 June 2015
Of course, the logic of mathematics makes possible to reach a result (e.g. a mathematical proof), but even if you accept the axioms as a given and unchangeable base, the possibility doesn’t means the necessity. Even a few elements can result “hyper astronomically” huge number of combinations (to borrow Quine’s term). Thus searching for an answer (e.g. examining whether a theorem is true) can be interpretable as (a random) walk in the phase space of mathematics. Perhaps we are convinced that there is a certain mountain pike in this virtual landscape, but we do not know the path to it; or we even don’t know whether the hill exists at all. Obviously, this image is more or less misleading, since there aren’t existing routes originally, and we have to build them–and in some cases this activity constructs the target itself. Furthermore, not only the phase space of a certain mathematics is enormously huge, but the phase space of mathematics based on different sets of axioms and considerations are similarly large as well.
It is an interesting question that to what extent overlaps different mathematics (=mathematics based different set of axioms and rules) each other. And since the “phase space” describes mathematics as an n-dimensional landscape where every point is defined by certain parameters, we can try to define them using another method. Ad analogiam: Remember Descartes’s idea to mathematize geometry.
Arithmetic is compressible: if you know the adding rules, then you can calculate the result of 2+2 directly, omitting the steps of 1+1=2; 2+1=3; 3+1=4. Simply speaking, there aren’t intermediate steps.
Opposite to it, in case of a mathematical theorem you cannot omit any part of the proofing jumping from the starting point directly to the end, thus it is history dependent and the first step is essential to reach the second one, etc. This uncompressible method is strongly resembles for the cellular automation’s operation mode.
Cellular automata (CA) is based on simple rules that determine the status of a certain cell (=a certain point of the landscape or cellular space, if you prefer) taking into consideration certain cells’ intermediate states. It is an uncompressible process: You cannot tell the result without executing the process itself.
It seems to be plausible that we can build a CA to generate point by point the path to any certain proof: After all, we can adapt the rules which guides the work of cellular automata to the objective to be achieved. What is more, probably we could construct a “universal mathematical CA” (UCA) to execute the whole mathematics. Or, we could build other UCAs to examine their ways of work. Perhaps it would result totally different mathematics.
17 June 2015
The big data approach can be applied to mathematics at least in two ways.
1. First, traditional mathematics is small data “science”: it manages only a small amount of data and tries to find more or less direct connections between certain features using a kind of deductive logics (which replaces causality in mathematics). E. g. we know the Euler theorem d^2=R(R – 2r) which describes the distance (d) between the circumcentre (R=circumradius) and incentre (r=inradius) in a triangle in geometry. Obviously, it is a proofed theorem, so we understand the cause of the correlation between these data. But why don’t try to adapt the big data approach and why we don’t try to analyze all the possible geometrical data to find new, although unproven, connections? Similarly, we could examine the distribution of prime numbers taking into consideration not only their places on the number line, but all the accessible data about numbers from HCNs (highly composite numbers) to triangle numbers to any other features to discover connections even we aren’t able to prove them.
2. There is another way to apply big data approach to a new level. Reverse mathematics is a program to examine which sets of axioms are necessary to build the foundations of mathematics. I.e. how should we choose a small amount of starting points to get a certain solution? It is, in accordance with its name, a reverse approach to the traditional mathematical way of thought which moves from a small set of axioms to theorems and which is a small data approach. But we can apply the big data “philosophy” more or less imitating Google’s solution examining different combinations of an enormously huge amount of possible axioms to create different data landscapes. Perhaps it would lead to a new kind of metamathematics.
10 June 2015
Teleological systems have two interpretations in theology: We can argue either that the Universe itself is or only some of its parts are teleological. (ibid, p. 59) Examples of biological teleological systems (e. g. eyes) are used to verify the existence of a Creator. But even accepting the teleological nature of the eyes (or human body, or planetary systems, etc.), we wouldn’t get answered whether He created the whole Universe as a machine to achieve a purpose and its parts serves His will or only our Universe’s parts are created to fulfil a task. In other words: even the verification the created nature of the eyes wouldn’t verify that we live in a created Universe.
Applying this distinction between the created (and teleological) parts and the created system as a whole, we get the following variants:
1. Both the World and the human race (and every part of the World) are created. Both the whole system and its parts have purposes (=ad analogiam a car). It can be regarded as the traditional theological point of view.
2. Opposite to it, we can state that neither the Universe nor its parts are teleological. It is the traditional atheist approach.
3. The World is created by a mighty entity, but the development of this World’s parts is regulated by laws independent of the Creator: For example, He constructed the Universe including the laws of evolution, and then the evolution resulted us. This it can be interpreted as a machine where the whole system (supposedly) has a purpose, but it is not true for its parts (and it cannot be true, since the Creator cannot be able to affect the results of the evolutionary process. Obviously, there are theologians who support this approach). It is another question whether the Creator would be able to find another solution for intelligence creation instead of the use of evolution. A parallel can be drawn between this model and the internet. According to Hubert L. Dreyfus, Ford’s automobile was a tool to support human mobility, and although it had some unintended effects (e.g. the liberalization of sex), it was a teleological machine. But because of its protean nature, the internet doesn’t have a “purpose”: It is a framework of opportunities. (On the Internet, p. 1-2)
4. The (or some) parts of the Universe are teleological systems, but the Universe itself isn’t. This model presupposes a Creator who is not outside his Universe, but a part of it. It seems to be the more exciting variant, since it introduces an inferior (Demiurge-like) Creator. To give an example, Polish SF-writer and thinker Stanislaw Lem played with the idea of a Cosmos in his book entitled Fantastyka I Futurologia (1970) where the actual state of our Universe observed by us was influenced by cosmoengineering activities of an intelligent race lived billion years ago. According to this story, their aim was to influence the density of intelligence in the Universe, thus our World satisfies the criteria of a system with teleological parts but without an overall teleological system.
And what is even more exciting: originally it was held that a machine (either a mechanical construction or a world) can be a teleological system which contains teleological parts (=car-like machine), but we have another metaphor now: An internet-like system where the teleological considerations aren’t valid on the level of its parts. The ultimate question is given: What other kind of machines and worlds can be imaginable?
02 June 2015
Obviously, demiurgology ask not exactly the same questions as religions. To give an example, Christian theology distinguishes three kinds of arguments for the existence of God: The cosmological, the design and the ontological arguments (Rowe, ibid. p. 19). The first one concludes from the existence of the universe the existence of a Creator; the design argument is based on the presupposition that the order of the world was created by Him. Both of them are applicable to the field of demiurgology–but the case of the ontological argument is different, since it states that we can conclude the existence of a creator by a deductive logic. But the natural sciences’ integral part is induction and the feedback to reality by experiments.
To give another example, the design argument presupposes the validity of the cosmological argument. After all, it is held by theology that the world was created by God, who is responsible for its order and structure, too. In other words: the existence of a created world is a precondition of the existence of order and structure. In the case of demiurgology, it is imaginable that our world is constructed by a superior cosmologist. But it doesn’t make inevitable his/her ability to create the laws/structure of the constructed world. Using a metaphor to demonstrate that sometimes the creation and the design can be independent: A sculptor makes the sculpture, but not the marble.
Or, we can mention as another example again, that according to Kant, one who isn’t perfect is unable to recognize whether our world is perfect (John Hedley Brooke: Science and Religion, p. 281), but a perfect world is only a subset of the possible worlds which can be created by someone in demiurgology.
Similarly, it isn’t a necessity from our point of view, neither a world creator’s all-powerfulness nor his/her omniscience: he/she is perhaps incompetent. And what is even more important: While the ontological argument isn’t interpretable in demiurgology, it perhaps offers opportunities to study some problems that are uninterpretable from a theologian’s approach.