European Rationalism * "Our Pub" Library
by Witold Marciszewski |

Let Einstein and Gödel, pictured at this photo (talking during a walk in Princeton), symbolize two great streams of thought that have met with each other to form our modern view on the dynamic and ordered complexity of the world, including civilization as the world's most convoluted realm. As for Einstein, his general theory of relativity (1917), supported with Hubble's astronomical observations, brought about an astonishing reverse of our worldview on the universe. The firm centuries-old belief in the static universe (initially shared by Einstein himself) has dramatically given way to the picture of the universe evolving towards a giant dynamic complexity. While Einstein and Hubble discovered the dynamics of universe, Gödel revealed a boundless dynamics of the mathematical mind. Insofar as mathematics forms a core of scientific thinking - according to the Mathesis Universalis claim - other sciences get endowed with such a limitless potential too. This enouraging fact is too often disregarded by prophets of epistemological pessimism who falsely perceive Gödel's theorem (on undecidability of arithmetic, 1931) as proclaiming helplessnes of human mind in the face of unsolvable questions. In fact, this theorem means only so much that in some hard cases it is a computer that gets helpless as being, by definition, devoid of creativity. On the other hand, human beings enjoy creative insights to solve hard problems on their own. Gödel's result is to the effect that at any stage of mathematical inquiry there are problems which cannot be solved in mechanical manner (the only way available for computer) with those devices (axioms and rules) which exist at the given stage of the theory in question. However, new axioms or rules can be find out to solve some problems so far undecidable. And when new undecidable cases arise, again the inventive spirit of humans can succeed in devising new means to go ahead, and so on. Gödel hinted at some efficient strategies for such inquiries, which can be successively adopted without limitations.
These two revelations, one regarding the universe, the other concerning mind, take us closer to materialization of the old dream about Mathesis Universalis. However, to figure out how that happens, we should have a look at two other mental breakthroughs, due to the founding fathers of mathematical logic and the theory of infinite sets. Logic helps to perceive the nature of mechanical reasoning (owed to logical formalism); set theory reveals power and boldness of the human mind to address infinity.
On the left we see Erhard Weigel, who in the period 1652-1699 was a mathematics professor at Jena University, on the right Gottlob Frege who was a mathematics professor at the same University between 1879 and 1918. Such a striking concidence of their professional careers would not deserve so much attention, did not they share a great intellectual design, to wit Mathesis Universalis. In Weigel's time such a design was just a vision of what ought to be done, while Frege's time was ripe enough to materialize such a vision. This idea inspired a lot of great thinkers. Among them was Rene Descartes who made it popular in the 17th century, and with his analytical geometry made a significant step towards a partial realization of the programme. However, in a further progress he was blocked by his dislike of formal logic, while just the formal-logical approach has proved the way from conception to completion. It was Weigel who more efficiently tried to combine
Aristotelian logic with Eucklidean mathematics as a step towards Universal
Mathematics -- below refwerred to as He transmitted Weigl's idea through centuries up to Frege
who explicitly declares his intention to continue Leibniz's project of MU as
involving both the ideal langage (lingua charactersitica) and formal logical
calculus (calculus ratiocinator). For Frege's intention has been cerried
out, first, in the work with much speaking title
The title which Frege gave to his study in three phrases renders three main intentions of the Leibnizian MU.
All these features of Frege's work are essential for each system of contemporary symbolic logic, and this is a way in which symbolic logic may be deemed as MU revisited. With examining how is MU related to Cantor's settheoretical ideas, we shall make a next step towards the informational worldview -- as a modern embodiment of MU vision.
Georg Cantor's (1845-1918) role in creating the modern MU
version is tremendous. While Frege devises a conceptual notation and an
efficient calculus for reasoning (Leibniz's
Leibniz's considered a number of concepts for the role of
conceptual primitives (called also semantic universals). However, he failed
to notice that it is the notion of Two sets are said to be equinumerous if there is
one-to-one correspondence between their elements (as, say, between fingers
of left and right hand). A composition of these two ideas results in
defining
The notion of equinumerosity plays a decisive role in making us aware of singularities of infinite sets. A set is infinite then and only then when it has a proper part equinumerous with the whole set; say the set of prime numbers being equinumerous with the set of all natural numbers in which it is included, i.e., forms a subset. However, not every infinite set having an infinite subset is equinumerous with it. Cantor was the first who noticed and proved the astonishing fact that the infinite set of natural numbers, being a subset of infinite set of reals, does not equal the latter: there are more real numbers than natural numbers. * Now, to render the relation between MU as a legacy handed over to us by earlier ages, and its modern realization in our time, let us observe the following. Some items of the original project which for the lack of
relevant devices could not be carried out by its originators, have been
completed with modern logic. We have got the conceptual language of symbolic
logic, and very efficient calculus for deduction, as well as the method of
successiwe introducing more complex notions on the basis of simpler
primitives. The Cantorian notion of set as the fundamental primitive idea,
symbolic languages of Frege, but also those of Peano, Russell, Hilbert,
£ukasiewicz etc., exemplify the accomplishment of However, the idea of universal mathematically shaped knowledge must have deeply changed in the face of dramatic changes in our picture of the world. The universe as seen by modern people is no longer a closed static reality. Instead, it appears to us as an open and dynamic system tending to ever greater complexity. This understading started with the discovery of the universe's evolution as reflected in Einstein's general relativity, hence its significance for the modern MU. While exploring ever new realms of the universe, that display a growing complexity, we need more and more advanced mathematical models, as exemplified with the history of relativity and quanta. This continously moves the frontiers of modern MU ahead, and shows that it is not likely to reach whenever a final stage. On the other hand, Gödel teaches us optimistically that the dynamics of human mind is able to keep up with the dynamically growing complexity of research problems. This is a corrolary to Gödel's statement that at any phase of the development of arithmetic there are problems not solvable in it in a mechanical way, but the mind's creative intuition can devise fresh axioms and rules to create new means of mechanical problem-solving. However, when the word "intuition" appears, one may wonder if we remain true to the very MU idea, since MU should have provide us with means of solving any problem whatever in a computational ("calculesmu"), that is, mechanical manner. |