Certainty Quotes - Page 10

• Today the major reason for our interest in Flatland is that for the first time we can achieve some of the dreams of our ancestors a century ago and obtain direct visual experience of phenomena in a dimension higher than our own.

  Thomas Banchoff

  Dream, Ontology, Today

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• My theory stands as firm as a rock; every arrow directed against it will return quickly to its archer. How do I know this? Because I have studied it from all sides for many years; because I have examined all objections which have ever been made against the infinite numbers; and above all because I have followed its roots, so to speak, to the first infallible cause of all created things.

  Georg Cantor

  Archer, Years, Roots

  "Journey Through Genius". Book by William Dunham, 1990.

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• This view [of the infinite], which I consider to be the sole correct one, is held by only a few. While possibly I am the very first in history to take this position so explicitly, with all of its logical consequences, I know for sure that I shall not be the last!

  Georg Cantor

  Views, Ontology, Lasts

  "Journey Through Genius". Book by William Dunham, 1990.

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• The actual infinite arises in three contexts: first when it is realized in the most complete form, in a fully independent otherworldly being, in Deo, where I call it the Absolute Infinite or simply Absolute; second when it occurs in the contingent, created world; third when the mind grasps it in abstracto as a mathematical magnitude, number or order type.

  Georg Cantor

  Independent, Order, Numbers

  "Mind Tools: The Five Levels of Mathematical Reality". Book by Rudy Rucker, 1987.

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• No human investigation can claim to be scientific if it doesn't pass the test of mathematical proof.

  Leonardo da Vinci

  Ontology, Tests, Logic

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• My opinion of mankind is founded upon the mournful fact that, so far as I can see, they find within themselves the means of believing in a thousand times as much as there is to believe in.

  Augustus De Morgan

  Believe, Mean, Ontology

  Augustus De Morgan (1872). “A Budget of Paradoxes Reprinted, with the Author's Additions, from the Athenaeum Augustus De Morgan”, p.70

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• Although my knowledge grows more and more, nevertheless I do not for that reason believe that it can ever be actually infinite, since it can never reach a point so high that it will be unable to attain any greater increase.

  Rene Descartes

  Believe, Ontology, Logic

  Rene Descartes, Elizabeth Sanderson Haldane, G. R. T. Ross (2003). “Discourse on Method and Meditations”, p.89, Courier Corporation

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• Neither the true nor the false roots are always real; sometimes they are imaginary; that is, while we can always imagine as many roots for each equation as I have assigned, yet there is not always a definite quantity corresponding to each root we have imagined.

  Rene Descartes

  Real, Roots, Ontology

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• We call infinite that thing whose limits we have not perceived, and so by that word we do not signify what we understand about a thing, but rather what we do not understand.

  Rene Descartes

  Ontology, Limits, Logic

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• Calculus works by making visible the infinitesimally small.

  Keith Devlin

  Ontology, Logic, Calculus

  Keith Devlin (2003). “Sets, Functions, and Logic: An Introduction to Abstract Mathematics, Third Edition”, p.7, CRC Press

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• The increased abstraction in mathematics that took place during the early part of this century was paralleled by a similar trend in the arts. In both cases, the increased level of abstraction demands greater effort on the part of anyone who wants to understand the work.

  Keith Devlin

  Art, Effort, Levels

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• Since I found that one could make a case shadow from a three-dimensional thing, any object whatsoever - just as the projecting of the sun on the earth makes two dimensions - I thought that by simple intellectual analogy, the fourth dimension could project an object of three dimensions, or, to put it another way, any three-dimensional object, which we see dispassionately, is a projection of something four-dimensional, something we are not familiar with.

  Marcel Duchamp

  Simple, Two, Intellectual

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• Complete knowledge of the nature of an analytic function must also include insight into its behavior for imaginary values of the arguments. Often the latter is indispensable even for a proper appreciation of the behavior of the function for real arguments. It is therefore essential that the original determination of the function concept be broadened to a domain of magnitudes which includes both the real and the imaginary quantities, on an equal footing, under the single designation complex numbers.

  Carl Friedrich Gauss

  Appreciation, Determination, Real

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• When the boy begins to understand that the visible point is preceded by an invisible point, that the shortest distance between two points is conceived as a straight line before it is ever drawn with pencil and paper...the fountain of all thought has been opened to him...the philosopher can reveal him nothing new, as a geometrician he has discovered the basis of all thought.

  Johann Wolfgang von Goethe

  Distance, Boys, Two

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• In my paper the fact the XY was not equal to YX was very disagreeable to me. I felt this was the only point of difficulty in the whole scheme...and I was not able to solve it.

  Werner Heisenberg

  Paper, Ontology, Able

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• Many persons entertain a prejudice against mathematical language, arising out of a confusion between the ideas of a mathematical science and an exact science. ...in reality, there is no such thing as an exact science.

  William Stanley Jevons

  Reality, Ideas, Confusion

  William Stanley Jevons (1879). “Political Economy”

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• It is well known that the central problem of the whole of modern mathematics is the study of transcendental functions defined by differential equations.

  Felix Klein

  Ontology, Logic, Study

  Felix Klein (1893). “The Evanston Colloquium: Lectures on Mathematics Delivered from Aug. 28 to Sept. 9, 1893 Before Members of the Congress of Mathematics Held in Connection with the World's Fair in Chicago at Northwestern University, Evanston, Ill”

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• ALGEBRA is a general Method of Computation by certain Signs and Symbols which have been contrived for this Purpose, and found convenient.

  Colin Maclaurin

  Purpose, Ontology, Logic

  Colin MacLaurin (1756). “A Treatise of Algebra, in Three Parts: Containing I. The Fundamental Rules and Operations; II. The Composition and Resolution of Equations of All Degrees, and the Different Affections of Their Roots; III. The Application of Algebra and Geometry to Each Other. To which is Added an Appendix Concerning the General Properties of Geometrical Lines”, p.1

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• A good proof is one that makes us wiser.

  IU?. I. Manin

  Ontology, Logic, Proof

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• Thought that accepts reality as given is no thought at all.

  Herbert Marcuse

  Reality, Ontology, Logic

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• Infinity is a dark illimitable ocean, without bound.

  John Milton

  Ocean, Dark, Infinity

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• Heaven is angered by my arrogance; my proof [of the four-color theorem] is also defective.

  Hermann Minkowski

  Color, Heaven, Arrogance

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• If we base our belief systems on the humble assumption that the complexities of the world are ontologically beyond our understanding, then maybe our belief systems will make more sense and end up causing less suffering.

  Moby

  Humble, Understanding, Suffering

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• The world is too big and too intricate to conform to our ideas of what it should be like... Just because we invent myths and theories to explain away the chaos we're still going to live in a world that's older and more complicated than we'll ever understand.

  Moby

  Ideas, World, Ontology

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• How many pizzas are consumed each year in the United States? How many words have you spoken in your life? How many different peoples names appear in the New York Times each year? How many watermelons would fit inside the U.S. Capital building? What is the volume of all the human blood in the world?

  John Allen Paulos

  New York, Blood, Years

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