What is the significance of Gödel’s Incompleteness Theorems for the philosophy of Mathematics?
Gödel’s incompleteness theorems, established in the first half of the twentieth century, have transformed the way many mathematicians, philosophers and even computer scientists have thought about mathematics. Although throughout the entirety of his work, he is neither concise nor always clear; it is obvious Gödel's theorems unearth a series of restrictions of an axiomatic and mechanical view of mathematics. He is understood to claim that any complete axiomatic system cannot be consistent. His theorems changed the understanding of various fields of philosophy, particularly to the philosophy of mathematics; they pose prima facie problems for Hilbert's program and directly to logic, to intuitionism and also invites controversial comparisons between the scope of mathematics and the human mind. The extent of the first will be the focus of this essay. I will discuss the efforts of Gödel to unveil a new era of mathematics, in doing so he successfully discovered a flaw in mathematicians reasoning, but whilst his theorems were non-the-less significant, a physical change in mathematics has not been dramatic; the theorems did not over-rule the astounding perfection mathematics has already established.
Before we consider such significance, I feel it’s important to confirm a mutual understanding of the theorems and its foundation. Considered to be one of the greatest philosophical mathematicians of a generation David Hilbert published his pursuit of an ideology for a systematic basis for arithmetic; ‘turning every mathematical proposition into a formula…thus recasting mathematical definitions and inferences in such a way that they are unshakeable and...
... middle of paper ...
...ems and to arithmetic. A mathematician says 2+2=4, and Gödel says prove it. Some things require no more proof than the thought of its negation. Some non-axiomatic truths do not require proof; Gödel’s threat is in instance rather than reality.
Lecture
David Hilbert; The Foundations of Mathematics, The modern development of the foundations of mathematics in the light of philosophy 1927
Journal
Kurt Gödel, Collected Works, Volume III (1961) publ. Oxford University Press, 1981
Essay
Michael Detlefsen, Hilbert’s Program: An Essay on Mathematical Instrumentalism, Kluwer Academic Publishers 1986
Journal
Daniel Isaacson, Arithmetical truth and hidden higher-order concepts in W.D. Hart, ed., The Philosophy of Mathematics (OUP, 1996)
Journal
David Hilbert, ‘On the infinite’, in Benacerraf and Putnam, eds., Philosophy of Mathematics: Selected Readings, 2nd ed. (CUP, 1983)
The Mathematical Principles of Natural Philosophy (1729) Newton's Principles of Natural Philosophy, Dawsons of Pall Mall, 1968
Severe as it is, this level of doubt is not utterly comprehensive, since the truths of mathematics and the content of simple natures remain unaffected. Even if there is no material world (and thus, even in my dreams) two plus three makes five and red looks red to me. In order to doubt the veracity of such fundamental beliefs, I must extend the method of doubting even more hyperbolically.
Not every great writer can be correct in what he or she is saying. This is the idea that Gaunilo had in mind when he wrote his criticism to St. Anselm’s Ontological Argument which states that if something greater than anything else that could be thought of is conceived in the understanding then it must exist. Gaunilo says it is foolish to believe in the existence of something just because it is understood. He says there must be some kind of other explanation. In this paper, I will try to explain both Anselm’s theory and Gaunilo’s argument by first breaking each of them down in simpler terms. I will attempt to show what Gaunilo is trying to discredit with his objection.
Ball, Rouse. “Sir Isaac Newton.” A Short Account of the History of Mathematics. 4th ed. Print.
Infinity has long been an idea surrounded with mystery and confusion. Aristotle ridiculed the idea, Galileo threw aside in disgust, and Newton tried to step-side the issue completely. However, Georg Cantor changed what mathematicians thought about infinity in a series of radical ideas. While you really should read my full report if you want to learn about infinity, this paper is simply gets your toes wet in Cantor’s concepts.
Wigner, Eugene P. 1960. The Unreasonable Effectiveness of Mathematics. Communications on Pure and Applied Mathematics 13: 1-14.
1) Oxford Readings in Philosophy. The Concept of God. New York: Oxford University press 1987
Immanuel Kant’s (1724-1804) Critique of Pure Reason is held universally as a watershed regarding epistemology and metaphysics. There have been anticipations regarding the notion of the analytic especially in Hume. The specific terms analytic and synthetic were first introduced by Kant at the beginning of his Critique of Pure Reason book. The mistake that metaphysicians made was viewing mathematical judgments as being “analytic”. Kant came up with a description for analytic judgments as one that is merely elucidatory, that is, what is implicit is transformed into explicit. Kant’s examples utilize the judgments of subjects or rather predicates, for instance the square has four sides. The predicates content is always already accounted for in
Furthermore, during 1619 he invented analytic geometry which was a method of solving geometric problems and algebraic geometrically problems. After, Rene worked on his method of Discourse of Mindand Rules for the Directions of th...
In Martin Hollis and Steven Lukes editors Rationality and Relativism (Cambridge Press, 1982).
Pythagoras held that an accurate description of reality could only be expressed in mathematical formulae. “Pythagoras is the great-great-grandfather of the view that the totality of reality can be expressed in terms of mathematical laws” (Palmer 25). Based off of his discovery of a correspondence between harmonious sounds and mathematical ratios, Pythagoras deduced “the music of the spheres”. The music of the spheres was his belief that there was a mathematical harmony in the universe. This was based off of his serendipitous discovery of a correspondence between harmonious sounds and mathematical ratios. Pythagoras’ philosophical speculations follow two metaphysical ideals. First, the universe has an underlying mathematical structure. Secondly the force organizing the cosmos is harmony, not chaos or coincidence (Tubbs 2). The founder of a brotherhood of spiritual seekers Pythagoras was the mo...
The concept of impossible constructions in mathematics draws in a unique interest by Mathematicians wanting to find answers that none have found before them. For the Greeks, some impossible constructions weren’t actually proven at the time to be impossible, but merely so far unachieved. For them, there was excitement in the idea that they might be the first one to do so, excitement that lay in discovery. There are a few impossible constructions in Greek mathematics that will be examined in this chapter. They all share the same criteria for constructability: that they are to be made using solely a compass and straightedge, and were referred to as the three “classical problems of antiquity”. The requirements of using only a compass and straightedge were believed to have originated from Plato himself. 1
Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculator science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry. Euclid, who lived around 300 BC in Alexandria, first stated his five postulates in his book The Elements that forms the base for all of his later Abu Abd-Allah ibn Musa al’Khwarizmi, was born abo...
Many mathematicians established the theories found in The Elements; one of Euclid’s accomplishments was to present them in a single, sensibly clear framework, making elements easy to use and easy to reference, including mathematical evidences that remain the basis of mathematics many centuries later. The majority of the theorem that appears in The Elements were not discovered by Euclid himself, but were the work of earlier Greek mathematician such as Hippocrates of Chios, Theaetetus of Athens, Pythagoras, and Eudoxus of Cnidos. Conversely, Euclid is generally recognized with ordering these theorems in a logical ...