Playing with Infinity (Dover Books on Mathematics)

This point of view does not mean that infinity cannot be used in physics. For convenience's sake, calculations, equations, theories and approximations often use infinite series , unbounded functions , etc. Physicists however require that the end result be physically meaningful. In quantum field theory infinities arise which need to be interpreted in such a way as to lead to a physically meaningful result, a process called renormalization. However, there are some theoretical circumstances where the end result is infinity.

One example is the singularity in the description of black holes. Some solutions of the equations of the general theory of relativity allow for finite mass distributions of zero size, and thus infinite density. This is an example of what is called a mathematical singularity , or a point where a physical theory breaks down. This does not necessarily mean that physical infinities exist; it may mean simply that the theory is incapable of describing the situation properly.

Two other examples occur in inverse-square force laws of the gravitational force equation of Newtonian gravity and Coulomb's law of electrostatics. The first published proposal that the universe is infinite came from Thomas Digges in Living beings inhabit these worlds. Cosmologists have long sought to discover whether infinity exists in our physical universe: Are there an infinite number of stars? Does the universe have infinite volume? Does space "go on forever"?

This is an open question of cosmology. The question of being infinite is logically separate from the question of having boundaries. The two-dimensional surface of the Earth, for example, is finite, yet has no edge. By travelling in a straight line with respect to the Earth's curvature one will eventually return to the exact spot one started from. The universe, at least in principle, might have a similar topology.

If so, one might eventually return to one's starting point after travelling in a straight line through the universe for long enough. The curvature of the universe can be measured through multipole moments in the spectrum of the cosmic background radiation.

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As to date, analysis of the radiation patterns recorded by the WMAP spacecraft hints that the universe has a flat topology. This would be consistent with an infinite physical universe. However, the universe could be finite, even if its curvature is flat. An easy way to understand this is to consider two-dimensional examples, such as video games where items that leave one edge of the screen reappear on the other.

The topology of such games is toroidal and the geometry is flat. Many possible bounded, flat possibilities also exist for three-dimensional space. The concept of infinity also extends to the multiverse hypothesis, which, when explained by astrophysicists such as Michio Kaku , posits that there are an infinite number and variety of universes.

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These are defined as the result of arithmetic overflow , division by zero , and other exceptional operations. Some programming languages , such as Java [41] and J , [42] allow the programmer an explicit access to the positive and negative infinity values as language constants. These can be used as greatest and least elements , as they compare respectively greater than or less than all other values.

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They have uses as sentinel values in algorithms involving sorting , searching , or windowing. In languages that do not have greatest and least elements, but do allow overloading of relational operators , it is possible for a programmer to create the greatest and least elements. In languages that do not provide explicit access to such values from the initial state of the program, but do implement the floating-point data type , the infinity values may still be accessible and usable as the result of certain operations.

Perspective artwork utilizes the concept of vanishing points , roughly corresponding to mathematical points at infinity , located at an infinite distance from the observer. This allows artists to create paintings that realistically render space, distances, and forms. Escher is specifically known for employing the concept of infinity in his work in this and other ways. Variations of chess played on an unbounded board are called infinite chess. Cognitive scientist George Lakoff considers the concept of infinity in mathematics and the sciences as a metaphor. The symbol is often used romantically to represent eternal love.

Several types of jewelry are fashioned into the infinity shape for this purpose.

Enjoyable Books, Department of Mathematics, Texas A&M University

From Wikipedia, the free encyclopedia. For other uses of "Infinity" and "Infinite", see Infinity disambiguation. Cardinality and Ordinal number. Cardinality of the continuum. The Princeton Companion to Mathematics. Archived from the original on Extract of page Archived at the Wayback Machine. Retrieved April 3, Haddad; VijaySekhar Chellaboina February 17, Nonlinear Dynamical Systems and Control: Archived from the original on April 4, Archived from the original on April 3, Annals of Library and Information Studies. Retrieved 16 May Landmark Writings in Western Mathematics In Zalta, Edward N.

Stanford Encyclopedia of Philosophy. The Question of the Reality of Infinitesimal Magnitudes".

60 second math book review(60SMBR): Intro to Topology

Archived from the original on 15 February Retrieved 16 February Retrieved 7 April Communicating the Science and Culture of Astrobiology illustrated ed. In Quest of the Universe illustrated ed. Archived from the original on 1 June Retrieved 16 March Archived from the original on 10 April The Shape of Space. Knopf Doubleday Publishing Group. Archived from the original on 9 June Retrieved 6 September Archived from the original on 25 March Mathematics for the nonmathematician.

An infinite chess scheme.

One, Two, Three...Infinity: Facts and Speculations of Science

A; Joel David Hamkins Controversy over Cantor's theory. Internal set theory Non-standard analysis Set theory Synthetic differential geometry. Beginning mathematics students and people in the humanities and other fields will find the book particularly outstanding for their purposes. Paperback , pages. Published August 19th by Dover Publications first published January 1st To see what your friends thought of this book, please sign up. To ask other readers questions about Playing with Infinity , please sign up. Be the first to ask a question about Playing with Infinity.

Lists with This Book. Apr 09, Abbas Mehrabian rated it it was amazing. I read this book when a kid, and it was such a wonderful introduction to mathematics, that led me to become a mathematician! La cosa si vede molto bene nella parte iniziale, dove si comincia con le operazioni elementari: Ci deve essere stata una grande fiducia nella traduzione originale, insomma!

Jun 18, Erika RS added it Shelves: This is a introduction to mathematics for non-mathematicians, and therefore only enjoyable to people who don't know math or who really know math. Pretty good overall; the bit on calculus was a bit tedious, but she warned that it might be and advised people to just skip it if they found it to be so.

Of course, the bit on logic was highly annoying since she essentially treated constructive logic like a heretic sect of a well established religion. Leonard rated it liked it Mar 10, Kevin rated it liked it Feb 25, Chantile rated it really liked it May 13, Tom Reid rated it really liked it Feb 03, Sandra rated it really liked it Dec 10, Renee rated it liked it Jul 14, Kaiser rated it it was amazing Apr 26, May 29, Sky Nguyen rated it liked it Shelves: Bill rated it it was amazing May 27, Amy rated it liked it Apr 30, Vdy rated it liked it Dec 12, Carolina rated it liked it Feb 01, Ata rated it really liked it May 30, Manish rated it really liked it Jul 17,