May 23, at I don't see any evidence whatsoever of your deity. Having said that, I'm more closely aligned with the agnostic point of view which rejects any unsupported belief.
Ancient Greek mathematics While the practice of mathematics had previously developed in other civilizations, special interest in its theoretical and foundational aspects was clearly evident in the work of the Ancient Greeks.
Early Greek philosophers disputed as to which is more basic, arithmetic or geometry. The Pythagorean school of mathematics originally insisted that only natural and rational numbers exist.
Aristotle took a majority of his examples for this from arithmetic and from geometry. Euclid justifies each proposition by a demonstration in the form of chains of syllogisms though they do not always conform strictly to Aristotelian templates.
Aristotle's syllogistic logic, together with the axiomatic method exemplified by Euclid's Elements, are recognized as scientific achievements of ancient Greece. Platonism as a traditional philosophy of mathematics[ edit ] This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources.
Unsourced material may be challenged and removed. October Further information: Platonism mathematics Starting from the end of the 19th century, a Platonist view of mathematics became common among practicing mathematicians.
The concepts or, as Platonists would have it, the objects of mathematics are abstract and remote from everyday perceptual experience: Their existence and nature present special philosophical challenges: How do mathematical objects differ from their concrete representation?
Are they located in their representation, or in our minds, or somewhere else? How can we know them? The ancient Greek philosophers took such questions very seriously.
Indeed, many of their general philosophical discussions were carried on with extensive reference to geometry and arithmetic. He believed that the truths about these objects also exist independently of the human mind, but is discovered by humans.
In the Meno Plato's teacher Socrates asserts that it is possible to come to know this truth by a process akin to memory retrieval. Above the gateway to Plato's academy appeared a famous inscription: In this way Plato indicated his high opinion of geometry.
He regarded geometry as "the first essential in the training of philosophers", because of its abstract character. This philosophy of Platonist mathematical realism is shared by many mathematicians. It can be argued that Platonism somehow comes as a necessary assumption underlying any mathematical work.
Not our axioms, but the very real world of mathematical objects forms the foundation. Aristotle dissected and rejected this view in his Metaphysics.
These questions provide much fuel for philosophical analysis and debate. Middle Ages and Renaissance[ edit ] For over 2, years, Euclid's Elements stood as a perfectly solid foundation for mathematics, as its methodology of rational exploration guided mathematicians, philosophers, and scientists well into the 19th century.
The Middle Ages saw a dispute over the ontological status of the universals platonic Ideas: Realism asserted their existence independently of perception; conceptualism asserted their existence within the mind only; nominalism denied either, only seeing universals as names of collections of individual objects following older speculations that they are words, "logoi".
Descartes' book became famous after and paved the way to infinitesimal calculus. Isaac Newton — in England and Leibniz — in Germany independently developed the infinitesimal calculus based on heuristic methods greatly efficient, but direly lacking rigorous justifications.
Leibniz even went on to explicitly describe infinitesimals as actual infinitely small numbers close to zero. Leibniz also worked on formal logic but most of his writings on it remained unpublished until The Protestant philosopher George Berkeley —in his campaign against the religious implications of Newtonian mechanics, wrote a pamphlet on the lack of rational justifications of infinitesimal calculus: May we not call them the ghosts of departed quantities?
Concerns about logical gaps and inconsistencies in different fields led to the development of axiomatic systems. In his work Cours d'Analyse he defines infinitely small quantities in terms of decreasing sequences that converge to 0, which he then used to define continuity.
But he did not formalize his notion of convergence. It gives a rigorous foundation of infinitesimal calculus based on the set of real numbers, arguably resolving the Zeno paradoxes and Berkeley's arguments.
Mathematicians such as Karl Weierstrass — discovered pathological functions such as continuous, nowhere-differentiable functions.Are ghosts real? Well, there's as much scientific evidence to support their existence as there is for the existence of black holes.
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