Euclidean geometry can be easily visualized; this is the argument adduced for the unique position of Euclidean geometry in mathematics. It has been argued that mathematics is not only a science of implications but that it has to establish preference for one particular axiomatic system. Whereas physics bases this choice on observation and experimentation, i.e., on applicability to reality, mathematics bases it on visualization, the analogue to perception in a theoretical science. Accordingly, mathematicians may work with the non-Euclidean geometries, but in contrast to Euclidean geometry, which is said to be "intuitively understood," these systems consist of nothing but "logical relations" or "artificial manifolds". They belong to the field of analytic geometry, the study of manifolds and equations between variables, but not to geometry in the real sense which has a visual significance.
Hans ReichenbachRelated topics
analogue applicability argument base choice contrast experimentation field geometry mathematics nothing perception position real science sense significance study unique visual work relations visualizationRelated quotes
Arthropods and vertebrates share some broad features of general organization - elongated, bilaterally symmetrical bodies, with sensory organs up front, excretory structures in the back, and some form of segmentation along the major axis. But the geometry of major internal organs could hardly be more different... Arthropods concentrate their nervous system on their ventral (belly) side as two major cords running along the bottom surface of the animal. The mouth also opens on the ventral side, with the esophagus passing between the two nerve cords, and the stomach and remainder of the digestive tube running along the body above the nerve cords. In vertebrates, and with maximal contrast, the central nervous system runs along the dorsal (top) surface as a single tube culminating in a bulbous brain at the front end. The entire digestive system then runs along the body axis below the nerve cord.
Stephen Jay Gould
First of all I want to remind you of the essential features of models. In my opinion they are: (i) drawing up a list of the variables to be considered; (ii) drawing up a list of the equations or relations the variables have to obey and (iii) testing the validity of the equations, which implies the estimation of their coefficients, if any. As a consequence of especially (iii) we may have to revise (i) and (ii) so as to arrive at a satisfactory degree of realism of the theory embodied in the model. Then, the model may be used for various purposes, that is, for the solution of various problems. The advantages of models are, on one hand, that they force us to present a "complete" theory by which I mean a theory taking into account all relevant phenomena and relations and, on the other hand, the confrontation with observation, that is, reality. Of course these remarks are far from new.
Jan Tinbergen
It was the White Race who produced men like Columbus who crossed the unknown Atlantic; men like Magellan who first circumnavigated the globe; men like Michaelangelo, Leonardo da Vinci, Rembrandt, Velazquez, Bernini, Rubens, Raphael and thousands of other geniuses who created beautiful and exquisite productions in the fields of sculpture and painting; geniuses like Beethoven, Bach, Wagner and Verdi who created beautiful music; men like James Watt who invented the steam engine; men like Daimler who invented and built the reciprocating internal combustion engine; production geniuses like Henry Ford, inventors like Thomas Edison; such a prodigal genius as Nikola Tesla in the field of physics and electricity; literary geniuses like Shakespeare, Goethe and thousands of others, untold geniuses in the fields of mathematics, in the fields of chemistry and physics.
Ben Klassen
... a rather different class of applications of the idea of best allocation of scarce resources... usually referred to as the theory of optimal economic growth. In most studies of this kind made in the countries with market economies there is not an identifiable client to whom the findings are submitted as policy recommendations. Nor is there an obvious choice of objective function, such as cost minimization or profit maximization in the studies addressed to individual enterprises. The field has more of a speculative character. The models studied usually contain only a few highly aggregated variables. One considers alternative objective functions that incorporate or emphasize various strands of ethical, political, or social thought. These objectives are then tried out to see what future paths of the economy they imply under equally simplified assumptions of technology or resource availability.
Tjalling Koopmans
In the field of non-Euclidean geometry, Riemann... began by calling attention to a distinction that seems obvious once it is pointed out: the distinction between an unbounded straight line and an infinite line. The distinction between unboundedness and infiniteness is readily illustrated. A circle is an unbounded figure in that it never comes to an end, and yet it is of finite length. On the other hand, the usual Euclidean concept of a straight line is also unbounded in that it never reaches an end but is of infinite length.
...he proposed to replace the infiniteness of the Euclidean straight line by the condition that it is merely unbounded. He also proposed to adopt a new parallel axiom... In brief, there are no parallel lines. This ... had been tried... in conjunction with the infiniteness of the straight line and had led to contradictions. However... Riemann found that he could construct another consistent non-Euclidean geometry.
Morris Kline