Quotes: Mind, Judgment,
Decision, Uncertainty, Chance, Probability, Analysis, Intuition, Coherence,
Many of these quotes are from the Furman
Math Quotation Server.
Contributions and comments welcome
Darry LePatner or Rita Mae Brown (depending on the source)
Good judgment comes from experience, and experience comes from bad judgment.
Judgment, or assent to probability, supplies our want of knowledge. The faculty which God has given man to supply the want of clear and certain knowledge, in cases where that cannot be had, is judgment: whereby the mind takes its ideas to agree or disagree; or, which is the same, any proposition to be true or false, without perceiving a demonstrative evidence in the proofs. The mind sometimes exercises this judgment out of necessity, where demonstrative proofs and certain knowledge are not to be had; and sometimes out of laziness, unskilfulness, or haste, even where demonstrative and certain proofs are to be had.
Descartes, René (1596-1650)
I thought the following four [rules] would be enough, provided that
I made a firm and constant resolution not to fail even once in the observance
of them. The first was never to accept anything as true if I had not evident
knowledge of its being so; that is, carefully to avoid precipitancy and
prejudice, and to embrace in my judgment only what presented itself to
my mind so clearly and distinctly that I had no occasion to doubt it. The
second, to divide each problem I examined into as many parts as was feasible,
and as was requisite for its better solution. The third, to direct my thoughts
in an orderly way; beginning with the simplest objects, those most apt
to be known, and ascending little by little, in steps as it were, to the
knowledge of the most complex; and establishing an order in thought even
when the objects had no natural priority one to another. And the last,
to make throughout such complete enumerations and such general surveys
that I might be sure of leaving nothing out.
Discours de la Méthode. 1637.
Galbraith, John Kenneth
There can be no question, however, that prolonged commitment to mathematical
exercises in economics can be damaging. It leads to the atrophy of judgement
Economics, Peace, and Laughter.
Milton, John (1608-1674)
Chaos umpire sits
And by decision more
embroils the fray
by which he reigns: next
him high arbiter
Chance governs all.
Ibn Khaldun (1332-1406)
Geometry enlightens the intellect and sets one's mind right. All of
its proofs are very clear and orderly. It is hardly possible for errors
to enter into geometrical reasoning, because it is well arranged and orderly.
Thus, the mind that constantly applies itself to geometry is not likely
to fall into error. In this convenient way, the person who knows geometry
The Muqaddimah. An Introduction to History.
Brown, George Spencer (1923 - )
To arrive at the simplest truth, as Newton knew and practiced, requires
years of contemplation. Not activity Not reasoning. Not calculating. Not
busy behaviour of any kind. Not reading. Not talking. Not making an effort.
Not thinking. Simply bearing in mind what it is one needs to know. And
yet those with the courage to tread this path to real discovery are not
only offered practically no guidance on how to do so, they are actively
discouraged and have to set abut it in secret, pretending meanwhile to
be diligently engaged in the frantic diversions and to conform with the
deadening personal opinions which are continually being thrust upon them.
Einstein, Albert (1879-1955)
The human mind has first to construct forms, independently, before
we can find them in things.
Einstein, Albert (1879-1955)
The truth of a theory is in your mind, not in your eyes.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
The effort of the economist is to "see," to picture the interplay of
economic elements. The more clearly cut these elements appear in his vision,
the better; the more elements he can grasp and hold in his mind at once,
the better. The economic world is a misty region. The first explorers used
unaided vision. Mathematics is the lantern by which what before was dimly
visible now looms up in firm, bold outlines. The old phantasmagoria disappear.
We see better. We also see further.
Transactions of Conn. Academy, 1892.
Gauss, Karl Friedrich (1777-1855)
I have had my results for a long time: but I do not yet know how I
am to arrive at them.
In A. Arber The Mind and the Eye 1954.
La Touche, Mrs.
I do hate sums. There is no greater mistake than to call arithmetic
an exact science. There are permutations and aberrations discernible to
minds entirely noble like mine; subtle variations which ordinary accountants
fail to discover; hidden laws of number which it requires a mind like mine
to perceive. For instance, if you add a sum from the bottom up, and then
from the top down, the result is always different.
Mathematical Gzette, v. 12.
Minsky, Marvin Lee (1927 -)
Logic doesn't apply to the real world.
D. R. Hofstadter and D. C. Dennett (eds.) The Mind's I, 1981.
Pascal, Blaise (1623-1662)
There are two types of mind ... the mathematical, and what might be
called the intuitive. The former arrives at its views slowly, but they
are firm and rigid; the latter is endowed with greater flexibility and
applies itself simultaneously to the diverse lovable parts of that which
Discours sur les passions de l'amour. 1653.
Poincaré, Jules Henri (1854-1912)
...by natural selection our mind has adapted itself to the conditions
of the external world. It has adopted the geometry most advantageous to
the species or, in other words, the most convenient. Geometry is not true,
it is advantageous.
(Contrast below with other quotes on the purity of math.)
Sanford, T. H.
The modern, and to my mind true, theory is that mathematics is the
abstract form of the natural sciences; and that it is valuable as a training
of the reasoning powers not because it is abstract, but because it is a
representation of actual things.
In N. Rose Mathematical Maxims and Minims, Raleigh NC:Rome Press Inc.,
Franklin, Benjamin (1706-1790)
To Joseph Priestley
Dear Sir, London Sept. 19. 1772 In the Affair of so much Importance to you, wherein you ask my Advice, I cannot for want of sufficient Premises, advise you what to determine, but if you please I will tell you how. When these difficult Cases occur, they are difficult chiefly because while we have them under Consideration all the Reasons pro and con are not present to the Mind at the same time; but sometimes one Set present themselves, and at other times another, the first being out of Sight. Hence the various Purposes or Inclinations that alternately prevail, and the Uncertainty that perplexes us. To get over this, my Way is, to divide half a Sheet of Paper by a Line into two Columns, writing over the one Pro, and over the other Con. Then during three or four Days Consideration I put down under the different Heads short Hints of the different Motives that at different Times occur to me for or against the Measure. When I have thus got them all together in one View, I endeavour to estimate their respective Weights; and where I find two, one on each side, that seem equal, I strike them both out: If I find a Reason pro equal to some two Reasons con, I strike out the three. If I judge some two Reasons con equal to some three Reasons pro, I strike out the five; and thus proceeding I find at length where the Ballance lies; and if after a Day or two of farther Consideration nothing new that is of Importance occurs on either side, I come to a Determination accordingly. And tho' the Weight of Reasons cannot be taken with the Precision of Algebraic Quantities, yet when each is thus considered separately and comparatively, and the whole lies before me, I think I can judge better, and am less likely to make a rash Step; and in fact I have found great Advantage from this kind of Equation, in what may be called Moral or Prudential Algebra. Wishing sincerely that you may determine for the best, I am ever, my dear Friend, Yours most affectionately
Smith, Henry John Stephen (1826-1883)
It is the peculiar beauty of this method, gentlemen, and one which
endears it to the really scientific mind, that under no circumstance can
it be of the smallest possible utility.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and
I have often pondered over the roles of knowledge or experience, on
the one hand, and imagination or intuition, on the other, in the process
of discovery. I believe that there is a certain fundamental conflict between
the two, and knowledge, by advocating caution, tends to inhibit the flight
of imagination. Therefore, a certain naivete, unburdened by conventional
wisdom, can sometimes be a positive asset.
R. Langlands, "Harish-Chandra," Biographical Memoirs of Fellows of
the Royal Society 31 (1985) 197 - 225.
Kasner, E. and Newman, J.
Mathematics is often erroneously referred to as the science of common
sense. Actually, it may transcend common sense and go beyond either imagination
or intuition. It has become a very strange and perhaps frightening subject
from the ordinary point of view, but anyone who penetrates into it will
find a veritable fairyland, a fairyland which is strange, but makes sense,
if not common sense.
Mathematics and the Imagination, New York: Simon and Schuster, 1940.
Everyone complains of his lack of memory, but nobody of his
want of judgment.
You can use all the quantitative data you can get, but
you still have to distrust it and use your own intelligence and judgment.
Logic: The art of thinking and reasoning in strict accordance
with the limitations and incapacities of the human misunderstanding.
The Devil's Dictionary
Robert E. Ornstein
This duality has been reflected in classical as well as modern literature as reason versus passion, or mind versus intuition. The split between the conscious mind and the unconscious. There are moments in each of our lives when our verbal-intellect suggests one course, and our hearts, or intuition, another.
State University of New York
A. Rockefeller College of Public Affairs and Policy