Pythagoras is certainly not noting the existence of the formula, but, rather, he is noticing the relation between a hypoteneuse and its sides. This relationship comes to be expressed in his formula. So we already see that while a genuine relationship exists between a hypoteneuse and its sides, a genuine theorem is contingent on language; the language in this case is that of mathematics. We are met, then, with two questions. The rst is whether we should consider the terms of mathematics, such as wo" or four," to abstract or concrete.
Starting with Descartes’ vision of a philosophy with a mathematical certainty, rationalists claimed to have grasped a rather large portion of reality, including the world, God, consciousness, and whatever falls in-between. As empiricists argued, most of this "knowledge" was in effect assumed, a habit, as it had no representation in the real world. The rationalists’ notorious abstractness and their disregard for the seeming discrepancy between their proofs and the real world have been the main reasons for the fearsome opposition and caricature they faced: even Voltaire, though influenced to a great extent by Leibniz’s philosophy, ridicules it in his masterpiece Candide in the form of ludicrously optimistic Pangloss. . Kant, especially, has put a rather impressive dent in the hull of rationalist philosophy, branding it dogmatic metaphysics.
While agreeing with Popper's falsifiability criteria, I question his initial assumptions of the nature of science. He suggests that all scientific thought is purely logical and scientific theories are rigorous, mathematical and precise. While true for most modern theories, this assumption is not true for ancient scientific theories. Modern science is a product of Hellenistic thought, which evolved from Alexandrine culture. Modern theories, as well as those which follow the Hellenistic tradition, are characterized by their narrow focus of logic and mathematics -- they explain how something works (Kuhn 104).
The key word there was abstract. The meaning of abstract is “existing in thought or as an idea but not having a physical or concrete existence”, which helps the theory of the non-Platonists. They argue that mathematical statements definitely do not exist physically, hence the word abstract. Following the logic of the non-Platonists, math is therefore: an invented logic exercise with no existence outside of mankind’s conscious thought. The purpose of math, they argue, is to use patterns to discerned by brain, to create useful, but artificial order from
Certainly the effort should be made. Perhaps, through Pythagorean ideas, logicism and Platonism, a firmer understanding can be known of the grasp that mathematics has on the world. Due to the secrecy of the society in which Pythagoras, it is difficult to distinguish between any original works of Pythagoras from those of his followers. However, it is not the author that is important, but rather the notions presented. According to the view of the Pythagoreans that "all is number," the first four numbers have a special significance in that their sum accounts for all possible... ... middle of paper ... ...l proofs for someone who accepts the axioms from which they begin."
In addition to this, Leibniz supports the claim that all necessary truths are demonstrable within a finite series of steps. He does not allow for infinite non-recurring decimal numbers such as pie to be necessary truths because of the infinite step-process involved in the demonstration. The essay will also emphasize the function of Leibniz’s account in the possible world context. It will finally evaluate the extent to which contingent truths can be adequately distinguished from necessity. The analysis of infinite series in mathematical propositions is Leibniz’s source of inspiration for the acc... ... middle of paper ... ...lanation for contingency.
Pythagoras felt that all things could be explained and represented by mathematical formulae. Plato, Socrate’s most important disciple, believed that the world was divided into two realms, the visible and the intelligible. Part of the world, the visible, we could grasp with the five senses, but the intelligible we could only grasp with our minds. In their own way they both sought to explain the nature of reality and how we could know what is real. Pythagoras held that an accurate description of reality could only be expressed in mathematical formulae.
Instead, this school of thought maintains that because the world that we experience through our sense is in a state of constant change it can, therefore, not be relied upon in deriving distinct and reliable truths, also known as absolute truths. Rene Descartes, a seventeenth-century mathematician, was one of the most influential philosophers in rationalism. Descartes, like all rationalists, rely on the absolute truths found only in mathematics and logic, and place ultimate value in analytic statements. "An analytic statement attributes a property to something, and that property is already implicit in the definition of that object or concept". (White & Rauhut, pg.72) Descartes introduced the idea of "radical doubt", as we... ... middle of paper ... ...lank state", provide us with a logical explanatory argument against rationalism.
The Cartesian Doubt Experiment and Mathematics ABSTRACT: The view that Descartes called mathematical propositions into doubt as he impugned all beliefs concerning common-sense ontology by assuming that all beliefs derive from perception seems to rest on the presupposition that the Cartesian problem of doubt concerning mathematics is an instance of the problem of doubt concerning existence of substances. I argue that the problem is not 'whether I am counting actual objects or empty images,' but 'whether I am counting what I count correctly.' Considering Descartes's early works, it is possible to see that for him, the proposition '2+3=5' and the argument 'I think, therefore I am,' were equally evident. But Descartes does not found his epistemology upon the evidence of mathematical propositions. The doubt experiment does not seem to give positive results for mathematical operations.
(1) The product of the two distinctions yields three kinds of knowledge: synthetic à priori, analytic à priori and synthetic à posteriori; analytic à posteriori being impossible. For Kant propositions like; "7+5=12," "all bodies have mass" and "every event has a cause." were synthetic and known à priorily. (2) Post-Kantian philosophy witnessed an attack on the possibility of synthetic à priori knowledge such as the rejections of analysis, geometry and arithmetic as synthetic à priori by Bolzano, Helmholtz and Frege respectively. (3) These were motivated by a fear that Kant's conceptualism, of the mind imposing space and time on the world, may lead to anti-realism, such as that of Husserl's bracketing the existence of the world based on his extensions of Descartes and Kant.