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According to the viewpoint which I suggest to call computational rationalism, the first of these questions should be answered in the affirmative, while the second remains open, being the subject of a vivid controversy. To be ranked among computational rationalists, it is not necessary to have a solution of the latter question, but it is indispensable to be aware of the problem, acknowledge its import, and tend to finding an answer.
The vision of mathematical order of the world is what the modern computational rationalism shares with its predecessors — ancient, medieval, and later (not computational yet). The very phrase “ordered in number” is taken from the popular biblical verse (Wisdom of Salomon Book, 11:21): omnia in mensura et numero at pondere disposuisti. This means: “Thou hast ordered all things in measure and number and weight”. (This sentence is redundant in meanings, according to a rule of biblical stylistics; saying “in number” would suffice alone, when measure and weight are just applications of numbers.) There is a splendid host, indeed, of rationalist thinkers fascinated with this idea, from its first authors, Pythagoras and Plato (who influenced the Book of Salomon) through Augustine, Aquinas, Da Vinci, Copernicus, Ficino, Cusanus, Galileo, Kepler, Leibniz, Newton, up to Georg Cantor who lifted the notion of number up to infinitely many infinite dimensions.
In modern rationalism of computational brand there appear two completely novel points, one concerning practical applications, the other one connected with a deep theoretical problem, as hinted with the second question in the title.
Its practical implications arise from combining the idea “omnia in numero” with titanic powers of digital computers. If all things are defined numerically, then all things, i.e. systems, processes, relations, etc. in the universe can be simulated with computers. Even the universe itself in its entirety, provided a sufficiently strong mathematics and gigantic resources of energy. Thus the collective human reason, which amounts to global civilization, owing to mathematical skills and wonderful technology, can match the creative divine-like potential of Plato’s dēmiourgós (as praised in the dialogue “Timaios”).
While some modern computational rationalists believe in such a perspective, as possible at least in principle, other ones remain sceptical about that. This difference of opinions is deeply rooted in some worldview considerations.
The belief in the possibility of overall simulation stems from what the physicist Ed Fredkin has called digital philosophy . There is quite a number of eminent physicist and computer scientists to endorse this point, often connected with the contention that the physical universe may be a giant digital machine to compute its own evolution (among the first to claim so was Konnrad Zuse, the German constructor who pioneered efficient digital computers about 1940, before British and American constructions). Then “in numero” means “in natural numbers”, that is, in the mode of computing fully available for digital machines, not engaging into the inexplorable realm of continuum of real numbers.
Now it is in order to recall that in the uncountably infinite set of reals there is the set of uncomputable numbers whose existence has been proved by Alan Turing in 1936 with the help of Cantorian diagonal argument. This is a mathematical fact. Now the question arises about the nature of physical reality: are there in it any functions having uncomputable numbers as values? If the answer would be in the negative, then digital computers should suffice to computatioanlly map with simulations, or modelling, the landscape of physical universe.
Were there in the universe functions with uncomputable values, then some regions of it would be inaccessible for digital computing. Then — here a new factor would step in: a real help might be expected from analog computing, somehow neglected at the present stage of computer science, but not devoid of the chance of a successful come back. Some problems have been discovered which can be tackled with analog computers alone, but in the moment there remains open the question of how much are these issues significant for the progress of science.
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