A Thumbnail History of the Philosophy of Mathematics
"It is beyond a doubt that all our knowledge begins with experience."  Imannuel Kant ( 1724 – 1804 )
( However naïve realism is no substitute for truth^{ [1] } )
^{[1] }" ... concepts have reference to sensible experience, but they are never, in a logical sense, deducible from them. For this reason I have never been able to comprehend the problem of the á priori as posed by Kant", from "The Problem of Space, Ether, and the Field in Physics" ( "Das Raum, Äether und FeldProblem der Physik." ), by Albert Einstein, 1934  source: "Beyond Geometry: Classic Papers from Riemann to Einstein", Dover Publications, by Peter Pesic, St. John's College, Sante Fe, New Mexico
Mathematics does not have a universally accepted definition during any period of its development throughout the history of human thought.
However for the last 2,500 years, beginning first with the pre  Hellenic Egyptians and Babylonians, mathematics encompasses possible deductive relationships concerned solely with logical truths derived by accepted philosophic methods of logic which the classical Greek thinkers of antiquity pioneered. Although it is normally associated with formulaic algorithms ( i.e., mechanical methods ), mathematics somehow arises in the human mind by the correspondence of observation and inductive experiential thinking together with its practical predictive powers in interpreting future as well as "seemingly" ephemeral phenomena. Why all of this is true in human progress, no one can answer faithfully.
In other words, human experiences and intuitive thinking first suggest to the human mind the abstract symbols for which the economy of human thinking mathematics is well known; but it is those parts of mathematics most disconnected from observation and experience and therefore relying almost wholly upon its internal, self  consistent, deductive logics giving mathematics an independent reified, almost ontological, reality, that mathematics most powerfully interprets the ultimate hidden mysteries of nature. It is this connected symbiosis of experience and á priori logic, neither of which is sufficient alone, which exists at the very heart of the history of human thought and thereby the history of mathematics.
The Greeks^{∗}
Nature to the Hellenics, although seemingly variegated and even heterogeneous, possessed underlying and inherently universal elements of concreteness and continuity. Nature to the classical Greeks, in other words, was essentially reasonable in that there is a total identification of mathmatics and nature; and, therefore, nature becomes completely available to human understanding by means of intellectual abstraction such as shown by Euclid's and Pythagoras's geometry of line and extant distance and Archimedes's geometry and mechanics.
On the other hand even though the Greeks "discovered" the human mind as distinct from the human body as well as distinct of the external world, their naïve realism derived of innocent intuition confused human thought with the external world and, hence, Greek rejection of other imaginative mathematical postulates or axioms which did not conform to immediate and personal sensate experience of the external world, became the greatest weakness in Greek logic and mathematics!
Stlll, this way of thinking differed considerably from the antipodal pre  Hellenics of Egypt and Babylon who sought no such valid elements of [ geometric, etc. ] invariance in nature, notwithstanding their vast accumulation of numerical relationships by means of incomplete inductions; and, therefore, throughout their vast body of empirical investigations to which the Hellenic Greeks were indeed indebted, neither the Egyptians nor the Babylonians of antiquity exhibited the Greek freedom of imagination but rather exhibited a sort of psychological and even philosophical nominalism in their perception of nature. Classical Greeks were the original inventors of secular humanism who went beyond and transcended the metaphysics of religious dogma where everything in human life and external nature is capable of submission to the human mind!
For ourselves, classical Greek thinking is an essential precursor to both modern mathematics and the concomitant scientific method  i.e., definitions, hypothesis, and logical methods for empirical investigations.
Greek mathematical thought, however, excluded the number zero; excluded the number one thus only beginning with the number two for their discrete system of numbers ( Physica. IV. 220a, Aristotle ); never invented a system for irrational numbers, although they did invent a theory of irrational continuous magnitudes of lines, areas, and volume; in fact, excluded the infinitely large and infinitely small from their mathematical reasoning owing to their sense of naïve realism. Rather, the Greeks thought primarily in terms of congruent geometric magnitudes of continuous forms as well as proportions and ratios of extant lines, areas, and volumes, and forever maintained a philosophical dichotomy or disconnect between obvious notions of continuous motions but yet discrete, irreducible, finite numbers.
Because the hypothenuse of right triangles ( the diagonal of the square ) is not reducible to either an irreducible whole integral number nor reducible down to a whole multiple of either side ( i.e., discovering the least common integral divisor ), the Greeks invented a theory of comparable irrational continuous magnitudes from which they disassociated their system of discrete numbers as was just previously mentioned and from this they invented a cumbersome method of geometric exhaustion, the precursor to the modern calculus of integration for determining comparative areas and volumes. Archimedes ( c. 287 – c. 212 B.C. ), the world's greatest mathematician of classical antiquity, nearly perfected this heuristic method of geometric exhaustion, first rigorously introduced by Eudoxus of Cnidus ( c. 410 or 408 – c. 355 or 347 B.C. ) from Antiphon ( c. 480 – c. 411 B.C. )'s own suggestive mathematics, by employing upper and lower bounded geometries surrounding areas and volumes and thereby applying the original Greek concept of geometric magnitudes, proportions and ratios but without the use of irrational numbers which they instinctively dismissed as not necessarily congruent with line magnitudes and hence continuous motion.
With the notable but tentative exceptions advanced by the Pythagoreans, who were primarily concerned with permanence of form but not conditions of variable change in nature, and the Greek mathematical atomists, foremost of whom was Democritus, Greek mathematics did not incorporate concepts of the infinitesimal nor the infinitely extant, either of which would have required the invention of the modern concept of the limit. Given this inherent weakness ( naïve realism ) in classical Greek mathematics, Zeno ( c. 490? – c. 430 B.C.? )'s 4 Arguments against continuous motion gained philosophical ascendance which, however, modern differential calculus recognizes as mere sophistry. That is, owing to the lack of accepted concepts by the classical Greeks of the space  time continuum and aggregate limits going either to the infinitesimal or to the infinitely large for converging series of terms equivalent to a finite sum with an infinitely small remainder term, the Classical Greeks could never progress to the modern era of mathematical physics.
computerized tomography of Fragment A slices ( 2  negative )
source: http://www.antikytheramechanism.gr/data/ct
copyright: the Antikythera Mechanism Research Project
Besides inventing the algorithmic calculus during the respective periods 1665  66 for Isaac Newton and 1673  76 for Gottfried Wilhelm von Leibnitz ( also: Leibniz ), Newton also in his Methodus Fluxionum and De Quadratura effectively created tables of integrals, these two mathematical geniuses contemporaneously made the amazing discovery of the anti  derivative of the integral, viz., The Fundamental Theorem of The Calculus, where
Also, the Leibniz notational symbol
is used widely  in fact, used exclusively  to represent
of an infinitely converging sequence of terms equivalent to a finite sum with an infinitely small remainder term and thus fulfilling the ancient Greek desire for a theory of aggregation or method of exhaustion.
They thus invented the algorithms of the modern calculus and discovered the anti  derivative of the integral, although both without having given a rigorous mathematical analysis of the infinitesimal ( as being either the aggregate of discrete infinitesimal magnitudes of irreducible monads vs. that of the continuous limit of motion of points, tangent lines and planes best represented by infinite series and infinite aggregates ), for the foundation of the modern calculus for which only later 19th century mathematicians ( Lagrange, Bolzano, Euler, Cauchy, et al. ) successfully accomplished this vitally important analytic task as follows:
For a deeper understanding of Newton's persona, science, Christian religious philosophy and deeply felt attachment to the ancient Hebrew texts, go to:
Beyond the calculus algorithms of Newton and Leibnitz
Calculus mathematics involves concepts of a transcendent infinitely small and infinitely large, initially inspired by primitive sensory intuitions at the very edge of maximally sensible human experience, but yet understood nevertheless by human minds by means of thought  reasoning.
Calculus mathematics therefore admits to the idea of limits where either the limit goes to zero or goes to infinity for converging series of terms. And because of the mathematically self  consistent logical constructs for limits and space  time continuities suggested by nature, the human mind is not insulted by any sort of contradiction with human experience. Thus, the inherent power of predictive mathematics is allowed by the human mind to transcend any minimal ( or maximal ) threshold of direct and immediate human experience. In fact, mathematics is readily accepted by the human mind beyond being merely allowed!!
Therefore, the atomic theory in quantum science was predicted by the infinitely small differentials [ to be considered as continuous variables in and of themselves ] constructed by the infinitesimal analysis of the calculus where the derivative is the quotient of two related differentials such as time and space, thereby giving the "point properties" of a curvilinear function. However, in the modern calculus the ratio of two related but somewhat distinct differentials is more regarded as a single quantity owing finally to the mathematical acceptance by means of rigorous logic and analysis of the concept of continuous limit of a converging infinite series of real number ratio terms.
Likewise at the macro  level for the universe, relativity theory is made possible by the modern calculus.
In summary, all of mathematics is philosophy, but not all of philosophy is mathematics, where the modern calculus is truly a magnificent humanist achievement in the history of human thought !
Nature's Power of Abstraction Goes Way Beyond Mathematics
There is nothing more beautifully abstract than those contained in the images of nature for which the entire body of present mathematical knowledge is not yet capable of capturing these aspects of nature's beauty as can only the eye of an artist!
§ Further admiring nature up on the Golan Heights, Israel:
§ Further reading: "The History of The Calculus and its Conceptual Development", by Carl B. Boyer, 1949 and reprinted 1959
"Mathematics, rightly viewed, possesses not only truth, but supreme beauty  a beauty cold and austere, like that of sculpture."  Bertrand Russell ( 1872  1970 )
^{∗}click here to view a map depicting the origins of Greek mathematics
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