The question of continuity and its links to infinitesimal analysis as advanced by A. Cauchy in particular was thus an early concern for Dedekind. Dedekind lamented that,. Even the most rigorous expositions of the differential calculus do not base their proofs upon continuity but, with more or less consciousness of the fact, they either appeal to geometric notions or those suggested by geometry, or depend upon theorems which are never established in a purely arithmetic manner.
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An analogy could be made between a straight line with an arbitrarily chosen origin o and the rational number series. In such a model, every rational number is represented by one point on this line. Naturally, mathematicians have been left wondering what continuity actually refers to. Rather, a precise account of continuity requires a definition of the following sort:.
If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one [point] which produces this division of all points into two classes, this severing of the straight line into two portions. Whenever we have a cut A 1 , A 2 , in which A 1 is the lower limit of the number and A 2 is the upper limit neither of which is produced by a rational number then mathematicians must create a new irrational number, which should be regarded as completely defined by the cut A 1 , A 2.
In effect, every irrational number becomes an interval on a line in which the upper and lower limits of the interval are the same. These examples demonstrate the fundamental connection between the principle of continuity and infinitesimal analysis, Dedekind claimed; they also indicate the manner in which continuity can be arithmetically presented and thus severed from any geometrical roots. I may say that I am glad if every one finds the above principle so obvious and so in harmony with his own idea of a line; for I am utterly unable to adduce any proof of its correctness, nor has any one the power.
It is not empirical or contingent upon the observation of motion in space.
It is an assertion—it is an exercise in logic. Because it is a created concept, Dedekind concluded:. If space has at all a real existence it is not necessary for it to be continuous; many of its properties would remain the same even were it discontinuous. As Cooke explains, Descartes was among the first to interpret the product of two lengths as a line as opposed to an area. His analytic geometry was not, however, without its problems. Guarantee that a curve containing points on both sides of a line must intersect the line.
But, as the Pythagoreans had shown, the numerical version of this theorem was false: the point of intersection might very well not correspond to any number. It was incorrect to call the intersection an irrational number, since there was no articulated theory of irrational magnitudes that allowed them to be added or multiplied like numbers. Any such definition first of all begged the question of the existence of the object defined; and if existence is granted by appeal to geometry, the rules for treating lengths as numbers still needed to be formulated and proved correct.
Such was the situation that confronted Dedekind. They adopt such constructed rules as axiomatically true statements about the systems within which they work.
Basics of Linear Spaces
In arguing for such an account of mathematics, Dedekind was engaged in the process of separating mathematical constructs from both scientific theorizing and physical existence. It exists as a constructed mathematical tool rendered axiomatically true by fiat, and used in mathematical puzzle-solving, though not necessarily in scientific theorizing. It is empirically justifiable; it makes sense to believe that space is a continuous fabric of some sort of material ether. It makes sense to believe that space exists everywhere—that there are no gaps of non-existence from this end of the room to the other.
It also makes sense to believe that there are no gaps in the continuity of time; there are likely no moments of non-existence that follow moments of existence. In so far as we trust our sensory organs, we are justified, therefore, in believing in the continuity of space and time. Hence, we are justified in basing our mathematical description of physical phenomena upon the assumption that continuity exists as a physical fact. Calculus, he argued, involves the art of describing the rate of change in motion of a point or an object.
Note that, for Clifford, mathematics is fundamentally empirical, as well as conventional—the former being due to the fact that we can know nothing beyond the provisions of our sensory organs, and the latter being due to the fact that all of our knowledge, no matter how well justified, is ultimately limited by the finite perceptual capacities of our sensory organs.
All knowledge is therefore empirically derived and open to future revision.
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Empirical experience observation as the basis of all mathematical knowledge;. Throughout the s, Clifford felt the need to discuss and explain these beliefs in great detail. His popular lectures and his Common Sense of the Exact Sciences published posthumously in were aimed at non-specialist audiences for whom the notion of continuity would have been new, or at least unfamiliar, terrain. In part this was because Clifford was working within a society that favoured and lauded popularizations of science and mathematics.
Scientists and popular lecturers such as T. Huxley, John Tyndall, and even P. Tait who popularized links between science and religion had made it the duty of experts to explain their expertise in palatable and popular forms, even when writing for expert audiences. Clifford was hypersensitive to this fact given his employment at UCL, an institution that had been established with the aim of extending the scope of higher education to include students who had emerged from families not traditionally associated with Cambridge or Oxford i.
The implication here is that a certain kind of physical space is necessary in order for numbers and their related operations to exist. His was an approach that accepted the conventional nature of mathematical concepts, but only in so far as those conventions were the product of empirical observations and inferences about the space within which motion is possible. While it is the case that empirical evidence is the best judge of what does and does not exist in this world, direct empirical knowledge is limited.
It is circumscribed by the finite reach of our sensory capacities. Because of this, Clifford accepted that our mathematical descriptions can only ever be conventional expressions of what we believe to be true about the world. In order to advance mathematical and scientific knowledge, experimentalists and mathematicians had to be willing to speculate upon, and use, concepts related to the nature of the world as they knew it indirectly. Accepting such an interpretation required that mathematicians revise their beliefs regarding the universality of certain mathematical principles, in particular the principle of commutativity.
It also required that mathematicians quench any desire to ideologically preserve commutativity in light of the useful tools that could emerge when adherence to such traditional principles were abandoned. In his presentation, Clifford posited that continuity was a workable hypothesis, which enjoined scientists and mathematicians to adhere to a fundamentally material view of the universe such that the interaction between physical entities say, of atoms at the most basic level of material reality would dominate descriptions of physical phenomena.
Clifford wrote that if scientists were to adopt the Riemannian hypothesis that space might not be flat i. In other words, continuous geometrical displacements in non-flat space might manifest themselves in the form of seemingly discontinuous phenomena perceived by humans in Euclidean space i. Clifford combined his views of geometry with his overarching belief in the existence of spatial and temporal continuity in order to make the following claims:.
In so doing, he aimed to expand upon both Hamiltonian quaternions as well as Grassmannian algebra.
Throughout the s, Clifford developed his bi-quaternions in the hopes of linking the seemingly empirical reality of Euclidean space to a geometric model that would be able to describe rigid body motion in non-flat spaces. Lauded in a posthumous review by his biographer H. As Smith wrote,.
Some men who have an ardent love for new knowledge find it difficult to maintain an unflagging interest in geometry, because they regard it as a purely deductive science of which the first principles axioms, postulates, and definitions , whether derived from experience or not, are unquestionable and contain implicitly in themselves all possible propositions concerning space. Those two mathematicians defended the usefulness and justifiability of non-Euclidean geometrical models in mathematical physics.
As Tucker recounted:. Upon the view put forward by Riemann and adopted by Clifford, the essential properties of space have to be regarded as things still unknown, which we may one day hope to find out by closer observation and more patient reflection, and not as axioms to be accepted on the authority of universal experience, or of the inner consciousness. To adopt either of the two opposite hypotheses that space is continuous or that it is discontinuous, while admitting fully that no phenomena have yet been observed which point to its discontinuity. The idea that objects may not stay the same shape as they move through space suggests that space might not be homogeneous.
Clifford thought the topological structure of space might vary from one section of space to another. Thus, our perceptions of an object in one section of space compared to our perceptions of that same object in another section of space might yield very different empirical results. The true definitions appear in the forms of axioms. The relation of these assumptions remains consequently in darkness. We neither perceive whether and how far their connection is necessary, nor, a priori, whether it is possible.
Different spatial structures require different metric systems. Triply extended magnitudes space of three dimensions that can be measured using Euclidean lines and line segments on a Cartesian graph, constitute only one particular case [Riemann , 56]. That is to say the system Euclid had established constitutes only one system of measure-relations in space. Many other systems could, in theory, be created. Riemann, therefore, suggested that mathematicians extend their systems to include models that could describe and grasp that which is not directly observable.
This distinction is one that Clifford would later reproduce in his own mathematical papers and in the Common Sense ofthe Exact Sciences. Examples of discrete manifolds, Riemann wrote, are common in everyday language. Any group of distinct things, such as the letters in the alphabet, can be considered to be a discrete manifold. Continuous manifolds, on the other hand, are more difficult to imagine. One common example is the position of perceived objects, when definite portions of manifoldness are distinguished by boundaries, which themselves take up no space 11 [Riemann , 57].
The number of directions of movement determines the type of space being defined.
Measure-relations can be studied in the abstract, because their dependence upon one another can be represented by formulae. However, geometric representations always lay at the bases of these formulae. If the position of an object changes, its displacement can be represented by a relative change in quantities measure-relations in a given number of directional dimensions. In cases in which the manifold contains no curvature, the equivalent mathematical model at play is a Euclidean one.
However, Riemann contended that such a model should be considered to be the exception and not the norm, because in such spaces figures could be moved about with no change in their size or shape. The entire structure of good science, therefore, depends upon recognition of this fact. Riemann concluded:. Researches starting from general notions, like the investigation we have just made, can only be useful in preventing this work from being hampered by too narrow views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices.
The idea of a multiply extended magnitude was, roughly, something that could be measured by a given number of coordinates. Riemann concluded that geometry could be thought of as dealing with various spaces n -fold extended magnitudes in which distances were to be measured using an infinitesimal ruler [Gray , ]. If the infinitesimal measuring rod could be placed anywhere and still offer the same scale of measurement, it would mean that the space being measured is constant in curvature.
Thus, the sum of the angles in any one triangle in such a spatial model would be equivalent to the sum of triangles at all places. However, Riemann ultimately wanted to discuss the measurement of distances between points without having to rely on Euclidean space. He was not able to work this up into a coherent theory, but it seems clear that this idea of conceptually rethinking the nature of space in a direct physical context accounts for many features of his lecture.
In some sense, real analysis is a pearl formed around the grain of sand provided by paradoxical sets. Aksoy, M.
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