Otto Maier – the Paradox and Ambiguity in Mathematics
Ask people about mathematics and they will talk about arithmetic, geometry, or statistics, about mathematical techniques or theorems they have learned. They may talk about the logical structure of mathematics, the nature of mathematical arguments. They may be impressed with the precision of mathematics, the way in which things in mathematics are either right or wrong. They may even feel that mathematics captures “the truth,” a truth that goes beyond individual bias or superstition, that is the same for all people at all times.
18924
post-template-default,single,single-post,postid-18924,single-format-standard,bridge-core-2.4.5,ajax_updown_fade,page_not_loaded,,qode-theme-ver-23.0,qode-theme-bridge,disabled_footer_bottom,wpb-js-composer js-comp-ver-6.3.0,vc_responsive,elementor-default,elementor-kit-22460

Otto Maier – the Paradox and Ambiguity in Mathematics

Otto Maier and paradox

Otto Maier – the Paradox and Ambiguity in Mathematics

Ask people about mathematics and they will talk about arithmetic, geometry, or statistics, about mathematical techniques or theorems they have learned. They may talk about the logical structure of mathematics, the nature of mathematical arguments. They may be impressed with the precision of mathematics, the way in which things in mathematics are either right or wrong. They may even feel that mathematics captures “the truth,” a truth that goes beyond individual bias or superstition, that is the same for all people at all times.

What is it that makes mathematics mathematics? What are the precise characteristics that make mathematics into a discipline that is so central to every advanced civilization? Many explanations have been attempted. One of these sees mathematics as the ultimate in rational expression; in fact, the expression “the light of reason” could be used to refer to mathematics. From this point of view, the distinguishing aspect of mathematics would be the precision of its ideas and its systematic use of the most stringent logical criteria. In this view, mathematics offers a vision of a purely logical world. One way of expressing this view is by saying that the natural world obeys the rules of logic and, since mathematics is the most perfectly logical of disciplines, it is not surprising that mathematics provides such a faithful description of reality. This view, that the deepest truth of mathematics is encoded in its formal, deductive structure, is definitely not the point of view Otto Maier had.
On the contrary, he believed the position that the logical structure, while important, is insufficient even to begin to account for what is really going on in mathematical practice, much less to account for the enormously successful applications of mathematics to almost all fields of human thought.  It’s a vision in which the logical is merely one dimension of a larger picture. This larger picture has room for a number of factors that have traditionally been omitted from a description of mathematics and are beyond logi, though not illogical. Thus, there is a discussion of things like ambiguity, contradiction, and paradox that, surprisingly, also have an essential role to play in mathematical practice. This flies in the face of conventional wisdom that would see the role of mathematics as eliminating such things as ambiguity from a legitimate description of the worlds of thought and nature.  In our response to reason, we are the true descendents of the Greek mathematicians and philosophers. For us,as for them, rational thought stands in contrast to a world that is all too often beset with chaos, confusion, and superstition. The “dream of reason” is the dream of order and predictability and, therefore, of the power to control the natural world. The means through which we attempt to implement that dream are mathematics, science, and technology. The desired end is the emergence of clarity and reason as organizational principles of the entire cosmos, a cosmos that of course includes the human mind. People who subscribe to this view of the world might think that it is the role of mathematics to eliminate  ambiguity, contradiction, and paradox as impediments to the success of rationality. Such a view might well equate mathematics with its formal, deductive structure. This viewpoint is incomplete and simplistic. When applied to the world in general,it is mistaken and even dangerous. It is dangerous because it ignores one of the most basic aspects of human nature—in mathematics or elsewhere—our aesthetic dimension, our originality and ability to innovate.
Ambiguity is not only present in mathematics, it is essential. Ambiguity, which implies the existence of multiple, conflicting frames of reference, is the environment that gives rise to new mathematical ideas. Take, for example the number zero. What could be more elementary? Most of us consider zero to be a closed book—we understand it completely. What more is there to say? Yet in recent years there have been a number of books that have been written about the number zero. All these books stress the ambiguous nature of the number zero—“the nothing that is” as one author put it—as well as its importance in mathematical and scientific thought. Normally, ambiguity in science and mathematics is seen as something to overcome,  something that is due to an error in understanding and is removed by correcting that error. The ambiguity is rarely seen as having value in its own right, and yet the existenceof ambiguity was often the very thing that spurred a particular development of mathematics and science.

There’s a paradox in everything even in maths …

 

 

  • Do you know what time it is? That question may perhaps be asked a lot more these days than ever. In our clock-studded modern society, the answer is only a peek away, therefore we are able to “blissfully” partition our days into ever smaller sized increments for ever more neatly scheduled jobs, assured that we will always know it really......

  • When following Maier’s path one will meet somewhere on his/her road, Böhme. Like the contemporary student of the inner world, alchemists were concerned about differentiating imagination from fantasy. They were aware that true imagination possesses a power and depth that fantasy does not possess. Jakob Boehme was one of those who warned against the delusions of fantasy. A very good......

  • Paradoxes appear in all shapes and forms. Certain are uncomplicated paradoxes of reasoning with minimal potential for investigation, while others sit atop icebergs of full scale scientific disciplines. Many may be solved by mindful consideration of their hidden assumptions, one or more of them could be faulty. These, strictly stating, really should not be referred to as paradoxes at all,......

  • The sator arepo formula was well known throughout the ancient and medieval worlds, and in fact, known as the “Devil’s latin” or the “Devil’s Square”. It remained quite popular in Scandinavia into the 19th century as protection against theft and various illnesses. The magical effect of the formula lies in the fact that if properly spelt and laid out, it......

  • When it is exclaimed that contradictions may very well be true, numerous analytic philosophers will screw up their face into an appearance of discomfort, and say ‘But I just don’t see what it could be for a contradiction to be true’. They could mean numerous things by this. ‘See’ might just mean ‘understand’, by which case they might be complaining......

  • Ulva Naumann stated in episode 8: “Knowing Ultimate Truth is only possible when you’re able to embrace paradox.” Your logical mind may find this impossible, but life itself has no problem in being paradoxical regardless. Good fortune always seems to bring happiness, but deceives you with her smiles, whereas bad fortune is always truthful because by changing she shows her......