— Alleged to have appeared over the door of Plato’s Academy.
In 1552, books of
astronomy and geometry were destroyed under the reign of Edward VI, because
they were "infected with magic".
— Scientific American, 5/11/1859,
303.
I enclose an
attempt to devise a numerical system that may prove to be entirely new.
Briefly, here is what it is … By using a binary system based on the number 2
instead of the decimal system based on the number 10, I am able to write all of
the numbers in terms of 0 and 1. I have done this not for mere practical
reasons, but rather to allow new discoveries to be made … This system can lead
to new information that would be difficult to obtain in any other way …
— Gottfried Wilhelm Leibniz (1646 – 1716), letter written to the French Academy
of Sciences in 1701.
Will no one tell
me what she sings? —
Perhaps the plaintive numbers flow
For old, unhappy, far-off things,
And battles long ago.
— William Wordsworth (1770 – 1850), The
Solitary Reaper.
Let nobody who is
not a mathematician read my works
— Leonardo da Vinci, Trattato della
Pittura.
Also he made a
molten sea of ten cubits from brim to brim, round in compass, and five cubits
the height thereof; and a line of thirty cubits did compass it round about.
— Holy Bible, Chronicles II, 4:2.
It is sometimes
suggested that pure mathematicians glory in the uselessness of their work, and
make it a boast that it has no practical applications. The imputation is
usually based on an incautious saying attributed to Gauss…I’ve never been able
to find the exact quotation. I’m sure that Gauss’s saying (if indeed it be his)
has been rather crudely misinterpreted.
— Godfrey Harold (G. H.) Hardy (1877 – 1947)
2 + 2 = 5 (for
sufficiently large values of 2)
— graffito, Princeton, 1990, reported in several journals.
If we take in
hand any volume; of divinity or school metaphysics, for instance; let us ask, Does it contain any abstract reasoning
concerning quantity or number? No. Does
it contain any experimental reasoning, concerning matter of fact or existence?
No. Commit it then to the flames, for it can contain nothing but sophistry and
illusion.
— David Hume (1711-1776), An Enquiry
Concerning Human Understanding (1748).
In mathematics he
was greater
Than Tycho Brahe, or Erra Pater:
For he by geometric scale
Could take the size of pots of ale;
Resolve by sines and tangents straight
If bread or butter wanted weight;
And wisely tell what hour o’ day
The clock doth strike, by algebra.
— Samuel Butler (1612-1680) Hudibras.
Whoever thinks
algebra is a trick in obtaining unknowns has thought in vain. No attention
should be paid to the fact that algebra and geometry are different in
appearance. Algebras are geometric facts which are proved.
— Omar Khayyam, quoted in Boyer, A
History of Mathematics, 242.
It is a common
belief that mathematics is the hallmark of Science, and some people are apt to
imagine that the introduction of a little mathematics into subjects like
economics entitles them to rank as genuine science. The truth is that science
rests on the painstaking recognition of uniformities in nature.
— Lancelot Hogben (1895 – 1975), Science
for the Citizen.
Mathematics,
rightly viewed, possesses not only truth, but supreme beauty — a beauty cold
and austere, like that of sculpture.
— Bertrand Russell (1872 – 1970), Mysticism
and Logic.
Histories make
men wise; poets, witty; the mathematics, subtile; natural philosophy, deep;
moral, grave; logic and rhetoric, able to contend.
— Francis Bacon (1561 – 1626).
But the age of
chivalry is gone. That of sophisters, economists, and calculators, has
succeeded; and the glory of Europe is extinguished forever.
— Edmund Burke (1729 – 1797).
— Sir Peter Medawar, Introduction to Pluto’s Republic, Oxford University Press, 1984, 1.
Perhaps we’d
better distinguish between the two types of space by spelling one of them with
a capital. Small s space — the variety that Euclid got hung up on — is what
stops everything from being in the same place.
— Arthur C. Clarke, The View from
Serendip, Gollancz, 1978, 42.
How happy the lot
of the mathematician! He is judged solely by his peers, and the standard is so
high that no colleague can ever win a reputation he does not deserve. No
cashier writes a letter to the press complaining about the incomprehensibility
of Modern Mathematics and comparing it unfavourably with the good old days when
mathematicians were content to paper irregularly shaped rooms and fill bathtubs
without closing the waste pipe.
— W. H. Auden (1907 – 1973), ‘Writing’, in The
Dyer’s Hand, 1963, 15.
In every respect
but one, in fact, the old Mathematical Tripos seemed perfect. The one
exception, however, appeared to some to be rather important. It was simply — so
the young creative mathematicians, such as Hardy and Littlewood kept saying —
that the training had no intellectual merit at all. They went a little further,
and said that the Tripos had killed serious mathematics in England stone dead
for a hundred years. Well, even in academic controversy, that took some
skirting around, and they got their way.
— C. P. Snow (1905 – 1980), The Two
Cultures and the Scientific Revolution, Rede Lecture, 1959.
It is India that
gave us the ingenious method of expressing all numbers by means of ten symbols,
each symbol receiving a value of position as well as an absolute value; a
profound and important idea which appears so simple to us now that we ignore
its true merit. But its very simplicity and the great ease which it has lent to
computations put our arithmetic in the first rank of useful inventions; and we
shall appreciate the grandeur of the achievement the more when we remember that
it escaped the genius of Archimedes and Apollonius, two of the greatest men
produced by antiquity.
— Pierre-Simon de Laplace (1749 – 1827).
The long chains
of simple and easy reasonings by means of which geometers are accustomed to
reach the conclusions of their most difficult demonstrations had led me to
imagine that all things, to the knowledge of which man is competent, are
mutually connected in the same way and that there is nothing so far removed
from us as to be beyond our reach, or so hidden that we cannot discover it,
provided only that we abstain from accepting the false for the true and always
preserve in our thoughts the order necessary for the deduction of one truth
from another.
— René Descartes (1596 – 1650), Discourse
on Method, 16.
To divide a cube
into two other cubes, a fourth power or in general any power whatever into two
powers of the same denomination above the second is impossible, and I have
assuredly found an admirable proof of this, but the margin is too narrow to
contain it.
— Pierre de Fermat (1601 – 1665), in the margin of his copy of Diophantus’ Arithmetica.
In order
therefore to appreciate the requirements of the science, the student must make
himself familiar with a considerable body of most intricate mathematics, the
mere retention of which in the memory materially interferes with further
progress. The first process therefore in the effectual study of the science, must
be one of simplification and reduction of the results of previous investigation
to a form in which the mind can grasp them. The results of this simplification
may take the form of a purely mathematical formula or of a physical hypothesis.
— James Clerk Maxwell (1831 – 1879), “On Faraday’s Lines of Force”, Transactions of the Cambridge Philosophical
Society, 1856.
… I left Caen,
where I was living, to go on a geologic excursion under the auspices of the
School of Mines. The incidents of the travel made me forget my mathematical
work. Having reached Coutances, we entered an omnibus to go to some place or
other. At the moment when I put my foot on the step, the idea came to me,
without anything in my former thoughts seeming to have paved the way for it,
that the transformations I had used to define the Fuchsian functions were
identical with those in non-Euclidean geometry. I did not verify the idea; I
should not have had time to do so, as, upon taking my seat on the omnibus, I
went on with a conversation already commenced, but I felt a perfect certainty.
On my return to Caen, for conscience’s sake, I verified the results at my
leisure.
— Jules Henri Poincaré (1854 – 1912), quoted by David Well, Curious and Interesting Mathematics,
Penguin 1997. Originally from Poincaré’s 1914 Science and Method.
Six is a number
perfect in itself, and not because God created the world in six days; rather
the contrary is true. God created the world in six days because this number is
perfect, and it would remain perfect, even if the work of the six days did not
exist.
— St Augustine of Hippo (354 – 430).
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