# „We must... maintain that mathematical geometry is not a science of space insofar as we understand by space a visual structure that can be filled with objects - it is a pure theory of manifolds.“

1891 - 1953

### „Geometry can in no way be viewed... as a branch of mathematics; instead, geometry relates to something already given in nature, namely, space. I... realized that there must be a branch of mathematics which yields in a purely abstract way laws similar to geometry.“

—  Hermann Grassmann German polymath, linguist and mathematician 1809 - 1877
Forward, as quoted by Mario Livio, Is God a Mathematician? (2009)

### „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 Reichenbach American philosopher 1891 - 1953

### „Whereas the conception of space and time as a four-dimensional manifold has been very fruitful for mathematical physicists, its effect in the field of epistemology has been only to confuse the issue. Calling time the fourth dimension gives it an air of mystery. One might think that time can now be conceived as a kind of space and try in vain to add visually a fourth dimension to the three dimensions of space. It is essential to guard against such a misunderstanding of mathematical concepts. If we add time to space as a fourth dimension it does not lose any of its peculiar character as time.... Musical tones can be ordered according to volume and pitch and are thus brought into a two dimensional manifold. Similarly colors can be determined by the three basic colors red, green and blue... Such an ordering does not change either tones or colors; it is merely a mathematical expression of something that we have known and visualized for a long time. Our schematization of time as a fourth dimension therefore does not imply any changes in the conception of time.... the space of visualization is only one of many possible forms that add content to the conceptual frame. We would therefore not call the representation of the tone manifold by a plane the visual representation of the two dimensional tone manifold.“

—  Hans Reichenbach American philosopher 1891 - 1953

### „(…) the bottom line is that if you believe in an external reality independent of humans, then you must also believe that our physical reality is a mathematical structure. Nothing else has a baggage-free description. In other words, we all live in a gigantic mathematical object—one that’s more elaborate than a dodecahedron, and probably also more complex than objects with intimidating names such as Calabi-Yau manifolds, tensor bundles and Hilbert spaces, which appear in today’s most advanced physics theories. Everything in our world is purely mathematical—including you.“

—  Max Tegmark Swedish-American cosmologist 1967

### „In a mathematical sense, space is manifoldness, or combination of numbers. Physical space is known as the 3-dimension system. There is the 4-dimension system, there is the 10-dimension system.“

—  Charles Proteus Steinmetz Mathematician and electrical engineer 1865 - 1923

### „The philosophy of the foundations of probability must be divorced from mathematics and statistics, exactly as the discussion of our intuitive space concept is now divorced from geometry.“

—  William Feller Croatian-American mathematician 1906 - 1970
Introduction, The Nature of Probability Theory, p. 3.

### „The confidence placed in physical theory owes much to its possessing the same kind of excellence from which pure geometry and pure mathematics in general derive their interest, and for the sake of which they are cultivated.... We cannot truly account for our acceptance of such theories without endorsing our acknowledgement of a beauty that exhilarates and a profundity that entrances us.“

—  Michael Polanyi Hungarian-British polymath 1891 - 1976
p. 15

### „1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves.2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra.3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry.Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics.“

—  Nicholas Murray Butler American philosopher, diplomat, and educator 1862 - 1947
Editor's Introduction, The Teaching of Elementary Mathematics https://books.google.com/books?id=NKoAAAAAMAAJ (1906) by David Eugene Smith

### „1. The human mind is so constructed that it must see every perception in a time-relation—in an order—and every perception of an object in a space-relation—as outside or beside our perceiving selves.2. These necessary time-relations are reducible to Number, and they are studied in the theory of number, arithmetic and algebra.3. These necessary space-relations are reducible to Position and Form, and they are studied in geometry.Mathematics, therefore, studies an aspect of all knowing, and reveals to us the universe as it presents itself, in one form, to mind. To apprehend this and to be conversant with the higher developments of mathematical reasoning, are to have at hand the means of vitalizing all teaching of elementary mathematics.“

—  David Eugene Smith American mathematician 1860 - 1944
David Eugene Smith, "Editor's Introduction," in: The Teaching of Elementary Mathematics https://books.google.com/books?id=NKoAAAAAMAAJ (1906)

### „Geometry is that of mathematical science which is devoted to consideration of form and size, and may be said to be the best and surest guide to study of all sciences in which ideas of dimension or space are involved. Almost all the knowledge required by navigators, architects, surveyors, engineers, and opticians, in their respective occupations, is deduced from geometry and branches of mathematics. All works of art constructed according to the rules which geometry involves; and we find the same laws observed in the works of nature. The study of mathematics, generally, is also of great importance in cultivating habits of exact reasoning; and in this respect it forms a useful auxiliary to logic.“

—  Robert Chambers (publisher, born 1802) Scottish publisher and writer 1802 - 1871
Robert Chambers, Chambers's Information for the People (1875) Vol. 2 https://books.google.com/books?id=vNpTAAAAYAAJ

### „The one intelligible theory of the universe is that of objective idealism, that matter is effete mind, inveterate habits becoming physical laws. But before this can be accepted it must show itself capable of explaining the tridimensionality of space, the laws of motion, and the general characteristics of the universe, with mathematical clearness and precision; for no less should be demanded of every Philosophy.“

—  Charles Sanders Peirce American philosopher, logician, mathematician, and scientist 1839 - 1914

### „My presentation of a general theory of living systems will employ two sorts of spaces in which they may exist, physical or geographical space and conceptual or abstract space...The characteristics and constraints of physical space affect the action of all concrete systems, living and nonliving... Physical space is a common space because it is the only space in which all concrete systems, living and nonliving, exist (though some may exist in other spaces simultaneously). Physical space is shared by all scientific observers, and all scientific data must be collected in it. This is equally true for natural science and behavioral science.“

—  James Grier Miller biologist 1916 - 2002
p. 9-10; As cited in: Kenneth D. Bailey (1994) Sociology and the New Systems Theory: Toward a Theoretical Synthesis. p. 262

### „Quantum theory also tells us that the world is not simply objective; somehow it's something more subtle than that. In some sense it is veiled from us, but it has a structure that we can understand.“

—  John Polkinghorne physicist and priest 1930
Divine Action: An Interview with John Polkinghorne http://www.aril.org/polkinghorne.htm by Lyndon F. Harris in Cross Currents, Spring 1998, Vol. 48 Issue 1.

### „We can not gaze into space and understand that space has no end. There is no brick wall at the end of space, because if there was, what would be behind the brick wall? It just goes on and on……never ending. The thought of infinity hurts my brain – I can not understand it. The Bible teaches us that God is eternal, without beginning and without end. We being finite, can not understand a world without time. Time is just a dimension that God has created and subjected man to Eternity and infinity are too much for our tiny brains to comprehend.“

—  Ray Comfort New Zealand-born Christian minister and evangelist 1949

### „But how can we avoid the use of human language? The... symbol. Only by using a symbolic language not yet usurped by those vague ideas of space, time, continuity which have their origin in intuition and tend to obscure pure reason—only thus may we hope to build mathematics on the solid foundation of logic.“

—  Tobias Dantzig American mathematician 1884 - 1956
p, 125

### „The concepts which now prove to be fundamental to our understanding of nature—a space which is finite; a space which is empty, so that one point [of our 'material' world] differs from another solely in the properties of space itself; four-dimensional, seven- and more dimensional spaces; a space which for ever expands; a sequence of events which follows the laws of probability instead of the law of causation—or alternatively, a sequence of events which can only be fully and consistently described by going outside of space and time—all these concepts seem to my mind to be structures of pure thought, incapable of realisation in any sense which would properly be described as material.“

—  James Jeans British mathematician and astronomer 1877 - 1946
p. 122, 1937 ed.