The Neo-Kantians and the 'Logicist' Definition of Number

This enterprise yields some powerful ideas. (1) Some of the relationships studied have great interest, numerical identity in particular. Indeed, seeing Kant discuss it here, one wonders why he did not include it in the Table of Categories. (2) Kant gives a solid argument for the necessity of a sensible element in representations, something not found elsewhere in the Transcendental Analytic.The Transcendental Analytic of Kant's Critique of Pure Reason ends with a little appendix on what Kant calls the Amphiboly of the Concepts of Reflection. As an appendix, the passage is more than a little curious.

Mathematics In an attempt to express certain basic concepts of mathematics precisely, one should consider a handful of different accepted and developed conceptions. Pythagoras, in the Fifth Century B.C., believed that the ultimate elements of reality were numbers; therefore the explanation for the existence of any object could only be explained in number. Gottlob Frege stated, in an idea referred to as logicism, that mathematics could in some sense be reduced to logic. The views of Plato state that we "know" these rules of mathematics at the intuitive level rather than the conscious level. Plato also believed that these forms existed previously in their perfect forms; humans know them in their imperfect forms through concept and imagination.

Comparing Spinoza’s Ethics and Dostoyevsky’s Notes from the Underground Perhaps my choice of the subject may come across as a little eccentric, to say the least. To appear quaint and whimsical, however, is not my intention, so I figured as an introduction, I would explain my choice. From so far as I can tell, philosophy, or the search for truth, has all too often been equated with certainty. This quality of certainty has been especially magnified in the rationalist branch of philosophy. 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.

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.

In this essay, published in 1738, Voltaire explains the philosophies of not only Newton, but in a large part Descartes because of his contributions in the fields of geometry. In Voltaire's concise explanation of Newton's and other philosophers' paradigms related in the fields of astronomy and physics, he employs geometry through diagrams and pictures and proves his statements with calculus. Voltaire in fact mentions that this essay is for the people who have the desire to teach themselves, and makes the intent of the book as a textbook. In 25 chapters, and every bit of 357 pages, as well as six pages of definitions, Voltaire explains Newton's discoveries in the field of optics, the rainbow spectrum and colors, musical notes, the Laws of Attraction, disproving the philosophy of Descarte's cause of gravity and structure of light, and proving Newton's new paradigm, or Philosophy as Voltaire would have called it. Voltaire in a sense created the idea that Newton's principles were a new philosophy and acknowledged the possibility for errors.

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.

This doctrine is vague and misconceived. In this essay, I will show that it is vague and misconceived and, consequently, why it does not cure his dilemma. Wittgenstein stated in the preface of his book that he had solved the problems of philosophy. That these problems had been formulated by the misuse of the logic of our language by philosophers. What philosophers had been saying could simply not be said.

(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. (4) Nominalism and idealism are anti-realist but conceptualism and conventionalism need not be. I extend the typology of knowledge by adding knowledge by invention. Many fundamental propositions of mathematics, science and metaphysics hence shift from the realm of synthetic à priori to the realm of knowledge by invention. For Poincaré fundamental definitions of mathematics are neither à priori nor à posteriori, but conventional.

Metaphysics as Addressed by Kant and Hume In the Prolegomena, Kant states that reading David Hume, "awakened him from his dogmatic slumber." It was Hume's An Inquiry Concerning Human Understanding that made Kant aware of issues and prejudices in his life that he had previously been unaware of. This further prompted Kant to respond to Hume with his own analysis on the theory of metaphysics. Kant did not feel that Hume dealt with these matters adequately and resolved to pick up where Hume had left off, specifically addressing the question of whether metaphysics as a science is possible. Hume basically asserted in his writings that metaphysics, as a science, is not possible.

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).