„We are told … statements that are not mathematical or logical formulae may look as if they were necessarily or certainly true, but they only look like that. They cannot really be either necessary or certain.“

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— Robert Maynard Hutchins

Great Books: The Foundation of a Liberal Education (1954)

Estratto da Wikiquote. Ultimo aggiornamento 03 Giugno 2021.

Robert Maynard Hutchins38

philosopher and university president 1899 - 1977

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„We had principles in mathematics that were granted to be absolute in mathematics for over 800 years, but new science has gotten rid of those absolutism, gotten — forward other different logics of looking at mathematics, and sort of turned the way we look at it as a science altogether after 800 years.“

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— Mahmoud Ahmadinejad 6th President of the Islamic Republic of Iran 1956

Columbia University speech, 24 September 2007
[24 September 2007, http://www.azstarnet.com/sn/hourlyupdate/202820.php, "Iran's president at Columbia University - a transcript", azstarnet.com, 2007-09-25]
2007

„[Oddly enough, Putnam believes part of the attraction of some of these formalisms is their obscurity]. "I think part of the appeal of mathematical logic is that the formulas look mysterious - you write backward Es!"“

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— Hilary Putnam American philosopher 1926 - 2016

Putnam as quoted in: Julian Baggini, Jeremy Stangroom (2005) What Philosophers Think. p. 233

„Necessary truth is merely the subject-matter of mathematics, not the reward we get for doing mathematics. The object of mathematics is not, and cannot be, mathematical certainty. It is not even mathematical truth, certain or otherwise. It is, and must be, mathematical explanation.“

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— David Deutsch, libro The Fabric of Reality

The Fabric of Reality (1997)

„I will ask of you only the ability to read English and to think logically—no high school mathematics, and certainly no higher mathematics.“

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— Edmund Landau German Jewish mathematician 1877 - 1938

Grundlagen der Analysis [Foundations of Analysis] (1930) Preface for the Student, as quoted by Eli Maor, Trigonometric Delights (2013)

„After defining semantical concepts like logical truth and related ones, I proposed to interpret the modalities as those properties of propositions which correspond to certain semantical properties of the sentences expressing the propositions. For example, a proposition is logically necessary if and only if a sentence expressing it is logically true.“

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— Rudolf Carnap German philosopher 1891 - 1970

Origine: Carnap’s intellectual biography (1963), p. 62

„The viewpoint of the formalist must lead to the conviction that if other symbolic formulas should be substituted for the ones that now represent the fundamental mathematical relations and the mathematical-logical laws, the absence of the sensation of delight, called "consciousness of legitimacy," which might be the result of such substitution would not in the least invalidate its mathematical exactness. To the philosopher or to the anthropologist, but not to the mathematician, belongs the task of investigating why certain systems of symbolic logic rather than others may be effectively projected upon nature. Not to the mathematician, but to the psychologist, belongs the task of explaining why we believe in certain systems of symbolic logic and not in others, in particular why we are averse to the so-called contradictory systems in which the negative as well as the positive of certain propositions are valid.“

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— L. E. J. Brouwer Dutch mathematician and logician 1881 - 1966

as translated by Arnold Dresden from: Brouwer, L. E. J. (1913). Intuitionism and formalism. Bulletin of the American Mathematical Society, 20(2), 81–96. (quote on p. 84)

„The object of mathematics is to discover "true" theorems. We shall use the term "valid" to describe statements formed according to certain rules and then shall discuss how this notion compares with the intuitive idea of "true."“

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— Paul Cohen American mathematician 1934 - 2007

Set theory and the continuum hypothesis, p. 8. https://books.google.com/books?id=Z4NCAwAAQBAJ&pg=PA8
Set Theory and the Continuum Hypothesis (1966)

„Mathematical knowledge may, just like our scientific knowledge, be deep and broad, it may be subtle and wonderfully explanatory, it may be uncontroversially accepted; but it cannot be certain.“

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— David Deutsch, libro The Fabric of Reality

Origine: The Fabric of Reality (1997), Ch. 10

„Even measured by Wiener's standards Cybernetics is a badly organised work — a collection of misprints, wrong mathematical statements, mistaken formulas, splendid but unrelated ideas, and logical absurdities. It is sad that this work earned Wiener the greater part of his public renown, but this is an afterthought. At that time mathematical readers were more fascinated by the richness of its ideas than by its shortcomings.“

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— Hans Freudenthal Dutch mathematician 1905 - 1990

Norbert Wiener (2008)

„Cubism is not a formula, it is not a school. Cubism is a philosophy, a point of view in the universe. It is like standing at a certain point on a mountain and looking around. If you go higher, things will look different; if you go lower, again they will look different. It is a point of view.“

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— Jacques Lipchitz American and French sculptor 1891 - 1973

Origine: Jacques Lipchitz: The Artist at Work, 1966, p. 199

„I cannot see why it is necessary that every deduction from algebra should be bound to certain conventions incident to an earlier stage of mathematical learning, even supposing them to have been consistently used up to the point in question. I should not care if any one thought this treatise unalgebraical, but should only ask whether the premises were admissible and the conclusions logical.“

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— Augustus De Morgan British mathematician, philosopher and university teacher (1806-1871) 1806 - 1871

Advertisement, p.3
The Differential and Integral Calculus (1836)

„Pure mathematics is, in its way, the poetry of logical ideas. One seeks the most general ideas of operation which will bring together in simple, logical and unified form the largest possible circle of formal relationships. In this effort toward logical beauty spiritual formulas are discovered necessary for the deeper penetration into the laws of nature.“

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— Albert Einstein German-born physicist and founder of the theory of relativity 1879 - 1955

1930s, Obituary for Emmy Noether (1935)

„Every philosophical problem, when it is subjected to the necessary analysis and justification, is found either to be not really philosophical at all, or else to be, in the sense in which we are using the word, logical.“

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— Bertrand Russell logician, one of the first analytic philosophers and political activist 1872 - 1970

Variante: Every philosophical problem, when it is subjected to the necessary analysis and purification, is found either to be not really philosophical at all, or else to be, in the sense in which we are using the word, logical.
Origine: 1910s, Our Knowledge of the External World (1914), p. 33

„[M]athematics is not a popular subject... The reason for this is to be found in the common superstition that [it] is but a continuation... of the fine art of arithmetic, of juggling with numbers. [We] combat that superstition, by offering, instead of formulas, figures that may be looked at and that may easily be supplemented by models which the reader may construct. This book... bring[s] about a greater enjoyment of mathematics, by making it easier... to penetrate the essence of mathematics without... a laborious course of studies.“

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— David Hilbert, Geometry and the Imagination

Preface
Geometry and the Imagination (1952)

„To get at the meaning of a statement the logical positivist asks, "What would the world be like if it were true?" The operationist asks, "What would we have to do to come to believe it?" For the pragmatist the question is, "What would we do if did believe it?"“

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— Abraham Kaplan American philosopher 1918 - 1993

Origine: "The Conduct of Inquiry", p. 43.

„One thing I have to say about Japanese anime is that there’s a certain sort of tone there that bypasses the Asian part. And the characters really turn into something more Western. I’d like to see a true Japanese character. We don’t need to make the eyes look round, we don’t need the light in her hair. We can have dark hair and the eyes look like mine. They can be speaking English. We have Asian-Americans. And certainly there are plenty of people from Hawaii that are very Asian and totally local. It’s part of America…“

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— Cary-Hiroyuki Tagawa American actor 1950

On Japanese anime in “Comic-Con 2001: An Interview With Cary-Hiroyuki Tagawa” http://fanboyplanet.com/comic-con-2001-an-interview-with-cary-hiroyuki-tagawa/ in Fanboy Planet (2001 Jul 27)

„The final truth about a phenomenon resides in the mathematical description of it; so long as there is no imperfection in this, our knowledge of the phenomenon is complete. We go beyond the mathematical formula at our own risk; we may find a model or a picture which helps us understand it, but we have no right to expect this, and our failure to find such a model or picture need not indicate that either our reasoning or our knowledge is at fault. The making of models or pictures to explain mathematical formulas and the phenomena they describe is not a step towards, but a step away from reality; it is like making a graven image of a spirit.“

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— James Jeans, libro The Mysterious Universe

The Mysterious Universe (1930)

„In many ways it is true to say that syntax is mathematical logic, semantics is philosophy or philosophy of science, and pragmatics is psychology, but these fields are not really all distinct.“

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— Frank Honywill George British psychologist 1921 - 1997

Origine: The Brain As A Computer (1962), p.42 as cited in: Sica Pettigiani (1996) La comunicazione interumana. p.48

„The principles of logic and mathematics are true simply because we never allow them to be anything else.“

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— Alfred Jules Ayer, libro Language, Truth, and Logic

Origine: Language, Truth, and Logic (1936), p. 77.
Contesto: The principles of logic and mathematics are true simply because we never allow them to be anything else. And the reason for this is that we cannot abandon them without contradicting ourselves, without sinning against the rules which govern the use of language, and so making our utterances self-stultifying. In other words, the truths of logic and mathematics are analytic propositions or tautologies.

„A physicist looks for causes; that does not necessarily imply that there are causes everywhere. A man may look for gold without assuming that there is gold everywhere; if he finds gold, well and good, if he doesn't he's had bad luck. The same is true when the physicists look for causes.“

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— Bertrand Russell logician, one of the first analytic philosophers and political activist 1872 - 1970

BBC Radio Debate on the Existence of God, Russell vs. Copleston (1948)
1940s

„We are told … statements that are not mathematical or logical formulae may look as if they were necessarily or certainly true, but they only look like that. They cannot really be either necessary or certain.“

Robert Maynard Hutchins38

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