From Leibniz to Mandelbrot
Benoit Mandelbrot, in his monumental The Fractal Geometry of Nature, wrote:
“...To sample Leibniz’ scientific works is a sobering experience. Next to calculus, and to other thoughts that have been carried out to completion, the number and variety of premonitory thrusts is overwhelming. We saw examples in “packing,”... My Leibniz mania is further reinforced by finding that for one moment its hero attached importance to geometric scaling. In “Euclidis Prota”..., which is an attempt to tighten Euclid’s axioms, he states,...: “I have diverse definitions for the straight line. The straight line is a curve, any part of which is similar to the whole, and it alone has this property, not only among curves but among sets.” This claim can be proved today” (Mandelbrot 1977: 419).
received inspiration from Leibniz while he, as an IBM fellow, was trying to
complete his ingenious fractal geometry.
Among the genius ideas of Leibniz, it was that of self
similarity, together with the principle of continuity: “natura non facit
saltus”, that inspired Mandelbrot most.
As Mandelbrot admits here, Leibniz, following his metaphysical line,
actually heralded the beginning of topology.
As for the above mentioned “packing”, Leibniz told to his friend de
Bosses to imagine a circle, then to inscribe within it three congruent circles
with maximum radius; the latter smaller circles could be filled with three
even smaller circles by the same procedure.
This process can be continued infinitely, thus giving a good image
of self similarity.
Likewise, Leibniz’s improvement of Euclid’s axiom contains the same
concept. The statement that “the
straight line is a curve, any part
of which is similar to the whole...” was really an idea which preceded the
birth of topology well over two centuries.
All these episodes tell us that with how keen interest Leibniz saw
the wonder of the nature’s infinity.
And what astonishes us more was that he who knew the nature’s infinity
and its self similarity better than anyone, was at the same time the man who
frankly held that we had to be humble enough to admit, as we will see later,
that our reason naturally fell always short of this nature’s infinity, and
that the confidence that the nature was rational in the sense it had a
priori law was something always for us to believe in.
Mathematica and Physica
Anyone familiar with the work of Mandelbrot would agree that his major aim is to make mathematics only one more step closer to the nature itself. To Mandelbrot, self-similarity is an important clue the nature reveals to mathematics. Mandelbrot’s book contains very interesting record of an experiment once given by an English statistician, L. F. Richardson, on measuring various coast lines’ lengths. It would seem that their lengths differ according to the measure one scales them with; finer the measure nearer to the true value of length. It is true as long as one suffices with rough approximation, as this true value is never actually reachable. But in this experiment, Richardson found an impressive case of “error” in scaling the nature.
Two countries sharing a common border line, like Spain and Portugal, claim different lengths to their “common border”. Is this an accidental error removable if one uses finer and finer measure in scaling? Or to put it theoretically, can one get as accurate length one desires as one uses an infinitely minute measure? Obviously not; for thus doing, one would end up with infinitely long coast or border length. The problem lies in one’s measure used in scaling. Using a straight line as a measure is not suitable in scaling a natural configuration like coast line or land surface. On the contrary, this method of using a straight line for a measure works well when one scales an “artificial” object like the length of the embankment of the river Thames or the acreage of a stadium; thus revealing the sharp opposition between the nature itself and man’s factitious artificiality.
This was the starting point for Mandelbrot to articulate fractal geometry which tried to generalize non-integer as well as integer dimensions. A straight line with dimension one is not altogether appropriate in measuring the configurations created by the nature itself. Mandelbrot’s attempt itself makes us realize the imperfection of our mathematical knowledge, which is often supposed to be the most perfect and exemplar humans can ever acquire. The above example shows that we can have a good reason to believe that Mandelbrot is in the opinion that mathematics is not at once an almighty tool to grasp the nature, much less equivalent to the nature itself.
Equally in physics, it was a German physicist, Herbert Breger, who found another important implication in Leibnizian philosophy. Above all, he also stressed on the importance of Leibnizian metaphysical concept of “infinity” as well as of “possibility” to natural scientists and their theory building:
der Tat konstatiert Leibniz, daß die Physik in ihrer Gesamtheit niemals eine
vollkommene Wissenschaft sei werde.
Damit ist aber nur gemeint, daß sich nicht alle Erfahrungen von der
Natur in wissenschaftlichen Gesetzen fassen lassen.
Die Gesetzlichkeit der Phänomene ist nach Leibniz das Unterpfang dafür,
daß die Phänomene kein bloßer Traum sind.
Die Lösung des Dilemmas von Individualität und Gesetzlichkeit der Natur
wird durch zwei Begriffe erreicht, die bei Leibniz verschiedentlich eine Schlüsselrolle
spielen: Unendlichkeit und Möglichkeit”(Weizsäcker et. al.: 1989: 81).
In other words, the nature is in its every aspect a unique whole by Leibniz, and natural sciences will never be able to cease their effort to bridge the gap between observed facts and their theorized laws. As long as the observed facts exist, they can serve as a sure ground (das Unterpfang) to believe that such phenomena are not mere illusions. But this does not mean that humans can obtain from them the natural scientific laws all at once. Breger argues that “error factors(die Störfaktoren)” and their “contaminating effects(die Dreckeffekte)” intrinsic in every observation or experiment should be taken as the essential separating line which marks the realm of the “possible”, to which mathematics and physics alike belong, and the uniqueness, that is “perfection”, of the nature itself.
Isn’t this a critically important statement also to those in the social and human sciences? It is generally believed that these latter can be sciences so long as they comply with the exemplar of natural sciences, which in their turn have been believed the most exact and once for all universal law giver.