Certainty Quotes - Page 20

• Infinity is a fathomless gulf, into which all things vanish.

  Marcus Aurelius

  Ontology, Infinity, Logic

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• The slicing technique from Flatland still remains one of the most powerful tools for dealing with aggregates in higher dimensions.

  Thomas Banchoff

  Powerful, Technique, Tools

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• 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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• Science asks no questions about the ontological pedigree or a priori character of a theory, but is content to judge it by its performance; and it is thus that a knowledge of nature, having all the certainty which the senses are competent to inspire, has been attained--a knowledge which maintains a strict neutrality toward all philosophical systems and concerns itself not with the genesis or a priori grounds of ideas.

  Chauncey Wright

  Philosophical, Character, Ideas

  Chauncey Wright (2003). “Philosphical Discussions”, p.47, A&C Black

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• One certainty when you travel is the moment you arrive in a foreign country, the American dollar will fall like a stone.

  Erma Bombeck

  Country, Travel, Fall

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• The offer of certainty, the offer of complete security, the offer of an impermeable faith that can’t give way, is an offer of something not worth having. I want to live my life taking the risk all the time that I don’t know anything like enough yet; that I haven’t understood enough; that I can’t know enough; that I’m always hungrily operating on the margins of a potentially great harvest of future knowledge and wisdom. I wouldn’t have it any other way.

  Christopher Hitchens

  Giving, Risk, Want

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• Still, being fragile creatures, humans always try to hide from themselves the certainty that they will die. They do not see that it is death itself that motivates them to do the best things in their lives. They are afraid to step into the dark, afraid of the unknown, and their only way of conquering that fear is to ignore the fact that their days are numbered. They do not see that with an awareness of death, they would be able to be even more daring, to go much further in their daily conquests, because then they would have nothing to lose- for death itself is inevitable.

  Paulo Coelho

  Fear, Dark, Trying

  Paulo Coelho (1992). “The diary of a magus”, Harpercollins

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• Discovering, for example, that as witnesses to your life diminish, there is less corroboration, and therefore less certainty, as to what you are or have been. [p. 65]

  Julian Barnes

  Example, Witness, Discovering

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• Mitchell Sanders was right. For the common soldier, at least, war has the feel-the spiritual texture-of a great ghostly fog, thick and permanent. There is no clarity. Everything swirls. The old rules are no longer binding, the old truths no longer true. Right spills over into wrong. Order blends into chaos, love into hate, ugliness into beauty, law into anarchy, civility into savagery. The vapors suck you in. You can't tell where you are, or why you're there, and the only certainty is overwhelming ambiguity.

  Tim O'Brien

  Spiritual, Hate, War

  Tim O'Brien (2009). “The Things They Carried”, p.78, Houghton Mifflin Harcourt

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• An expectation was there, mixed in with so many other emotions - excitement, resignation, hesitation, confusion, fear - that would well up then wither on the vine. You're optimistic one moment, only to be racked the next by the certainty that it will all fall to pieces. And in the end it does.

  Haruki Murakami

  Fall, Optimistic, Expectations

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• The greatest weapon any warrior can carry into battle is absolute certainty of her eternal soul.

  Chuck Palahniuk

  Warrior, Soul, Battle

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• There was the single abiding certainty that they would never be at a loss for things to carry.

  Tim O'Brien

  Loss, Abiding

  Tim O'Brien (2009). “The Things They Carried”, p.15, Houghton Mifflin Harcourt

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• What is history? Any thoughts, Webster?' 'History is the lies of the victors,' I replied, a little too quickly. 'Yes, I was rather afraid you'd say that. Well, as long as you remember that it is also the self-delusions of the defeated. ... 'Finn?' '"History is that certainty produced at the point where the imperfections of memory meet the inadequacies of documentation." (quoting Patrick Lagrange)

  Julian Barnes

  Memories, Lying, Self

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