Do we need Mathematics?

The Proposition I.20 of Euclid, nowadays known as The Triangle Inequality, reads as follows:

In any triangle two sides taken together in any manner are greater than the remaining one.

The commentator Proclus wrote:

The Epicureans are wont to ridicule this theorem, saying it is evident even to an ass and needs no prove; it is as much the mark of an ignorant man, they say, to require persuasion of evident truths as to believe what is obscure without question... That the present theorem is known to an ass that make out from the observation that, straw is placed in one extremity of the sides, an ass in quest of provender will make his way along the one side and not by way of the two others.

I do not have enough knowledge to pass a judgement as to whether Epicureans would accept as straight a line drawn by a chalk attached to an ass' tail whenever the latter follows its instincts towards the straw. I consider this a worthy research problem into the history of Mathematics.

In my personal opinion, as a society, we can't do without mathematics. Mathematics is in our culture. It's a vehicle of progress of all other sciences. On an individual level, a small percentage of professions require mathematical knowledge of various degree. Consumer mathematics, however useful, is a misnamed term for numeracy. An average person may be happy without a scruple of mathematical fluency.

But here is what other worthy people had to say with regard to the utility of Mathematics.

  1. J. B. Mencken, De Chralataneria eruditorium, 1715 (quoted in C. Fadiman, The Mathematical Magpie, p. 256):

    Mathematics contains much that will neither hurt one if one does not know it nor help one if one does know it.

  2. Fran Lebowitz (b. 1951), Social Studies, "Tips for Teens", 1981.

    Stand firm in your refusal to remain conscious during algebra. In real life, I assure you, there is no such thing as algebra.

  3. Friedrich Nietzsche (1844-1900). Human, All Too Human, 1878.

    Mathematics ... would certainly have not come into existence if one had known from the beginning that there was in nature no exactly straight line, no actual circle, no absolute magnitude.

  4. H. J. S. Smith (1826-1883), in H. Eves, Mathematical Circles Squared, Boston, Prindle, Weber and Schmidt, 1972.

    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.

  5. G.-C. Rota, Indiscrete Thoughts, Birkhauser, Boston, 1977.

    Stanislaw Ulam: "What makes you so sure that mathematical logic corresponds to the way we think? You are suffering from what French call a deformation professionnelle. Look at the bridge over there. It was built following logical principles. Suppose that a contradiction were to be found in set theory. Do you honestly believe that the bridge might then fall down?"

  6. B. Russell, Autobiography, v1, G.Allen & Unwin, LTD, 1967, p 162

    ... I am glad you abandoned your plan of reading a mathematical book, for any book on the Calculus would have told you lies, and also my book is (I fear) not worth while for you to read, except a few bits. What general value it may have is so buried in technicalities and controversies that it is really only fit for those whose special business it is to go in for such things. The later mathematical volume, which will not be ready for two years or so, will I hope be a work of art; but that will be only for mathematicians. And this volume disgusts me as a whole.

  7. J.W. von Goethe, Faust


    Use well your time, so rapidly it flies;
    Method will teach you time to win;
    Hence, my young friend, I would advise,
    With college logic to begin!

  8. G.-C. Rota, Indiscrete Thoughts, Birkhauser, Boston, 1977.

    The mystery, as well as the glory of mathematics, lies not so much in the fact that abstract theories do turn out to be useful in solving problems, but, wonder of wonders, in the fact that a theory meant for one type of problem is often the only way of solving problems of entirely different kinds, problems for which the theory was not intended. These coincidences occur so frequently, that they must belong to the essence of mathematics. No philosophy of mathematics can be excused from explaining such occurrences.

  9. R. P. Boas, Jr., If This Be Treason..., Amer Math Monthly, 64(1957), 247-249.

    When I was teaching mathematics to future naval officers during the war, I was told that the Navy had found that the men who had studied calculus made better line officers than men who had not studied calculus. Nothing is clearer (it was clear even to the Navy) than that a line officer never has the slightest use for calculus.

  10. H. Eves, Great Moments in Mathematics Before 1650, MAA, 1983

    Bernhard Bolzano (1781-1848) was on a vacation in Prague when he was attacked by an illness that manifested itself in bodily chills and painful weariness. To take his mind from his condition, he picked up Euclid's Elements and for the first time read the masterly exposition of the Eudoxian doctrine of ratio and proportion set out in Book V. The ingenuity of the treatment filled him with such vivid pleasure that, he said, he completely recovered from his illness. Ever after, when any of his friends felt indisposed, he recommended as a cure the reading of Euclid's presentation of the Eudoxian theory.

  11. P. E. B. Jourdain, The Nature of Mathematics, from The World of Mathematics by J.R.Newman, Simon and Schuster, NY, 1956

    I remember reading a speech made by an eminent surgeon, who wished, laudably enough, to spread the cause of elementary surgical instruction. "The higher mathematics," said he with great satisfaction to himself, "do not help you to bind up a broken leg!" Obviously they do not; but it is equally obvious that surgery does not help us to add up accounts; ... or even to think logically, or to accomplish the closely allied feat of seeing a joke.

  12. Robert Recorde, The whetstone of witte, London, 1557

    It is confessed emongste all men, that knowe what learnyng meaneth, that besides the Mathematicalle artes, there is noe vnfallible knowledge, excepte it bee borowed of them.

  13. B. Russell, Autobiography, v1, G.Allen & Unwin, LTD, 1967, p 43

    There was a footpath leading across fields to New Southgate, and I used to go there alone to watch the sunset and contemplate suicide. I did not, however, commit suicide, because I wished to know more of mathematics.

  14. J.Napier, Mirifici logarithmorum canonis descriptio, Edinburgh, 1614.

    The use of this book is quite large, my dear friend,
    No matter how modest it looks,
    You study it carefully and find that it gives
    As much as a thousand big books.

  15. Jane Muir, Of Men & Numbers, Dover, 1996.

    At eighteen, Euler published his first mathematical paper, a treatise on the masting of ships, which he submitted in the annual contest held by the French Academy of Science. Although he was competing against Europe's top mathematicians and scientists, many of them two or three times his age, he won second prize. ... Coming from landlocked Switzerland, he knew next to nothing about ships or their sails. But this lack of firsthand experience did not bother him, for since his conclusions on the height and thickness of masts were "deduced from the surest foundations in mechanics; their truth or correctness could not be questioned."

  16. L. Hogben, Mathematics for the Million, W.W.Norton & Co, 1993 (H. Eves in his In Mathematical Circles questions the veracity and even the plausibility of this story.)

    There is a story about Diderot, the Encyclopædist and materialist, a foremost figure in the intellectual awakening which immediately preceded the French Revolution. Diderot was staying at the Russian court, where his elegant flippancy was entertaining the nobility. Fearing that the faith of her retainers was at stake, the Tsaritsa commissioned Euler, the most distinguished mathematician of the time, to debate with Diderot in public. Diderot was informed that a mathematician has established a proof of the existence of God. He was summoned to court without being told the name of his opponent. Before the assembled court, Euler accosted him with the following pronouncement, which was uttered with due gravity: "(a + bn)/n = x, donc Dieu existe, repondez." Algebra was Arabic to Diderot. ... He left the court abruptly amid the titters of the assembly, confined himself to his chambers, demanded a safe conduct, and promptly returned to France.

  17. S. K. Stein, Strength in Numbers, John Wile & Sons, 1996

    A physician, Arthur Sadden, who had majored in mathematics, wrote, "Mathematics opened the doors to the very best medical schools. The discipline of analytical thought processes prepared me extremely well for medical school. In medicine one is faced with a problem which must be thoroughly analyzed before a solution can be found. The process is similar to doing mathematics."

    Another mathematics major, Jonathan Battiness, who went on to become a lawyer, had a similar view. "Although I had no background in the law - not even one political science course - I did well at one of the best law schools. I attribute much of my success there to having learned, through the study of mathematics, and, in particular, theorems, how to analyze complicated principles. Lawyers who have studied mathematics can master the legal principles in a way that most others cannot."

  18. A.H.Beiler, Recreations in the Theory of Numbers, Dover, 1966

    Rabbi Joseph ben Jehuda Ankin, in the twelfth century, recommended the study of perfect numbers in his book Healing of Souls.

  19. B. Bollobás, Littlewood's Miscellany, Cambridge University Press, 1990

    There was a rent act after 1914, and the definition of when a house was subject to it was as follows (my notations in brackets). The 'standard rent' (R) was defined to be the rent in 1914 (R0), unless this was less than the rateable value (V), in which case it was to be the rateable value. 'The house is subject to the act if either the standard rent or the rateable value is less than £105.' There were many law suits, argued ad hoc in each case. The subject is governed by a fundamental theorem, unknown to the Law:

    The house is subject to the act iff V < 105.

    This follows from Lemma: Min{Max{R0, V}, V} = V.

  20. E. T. Bell, MATHEMATICS: Queen & Servant of Science, MAA Spectrum, 1987

    Only one person in hundreds ever actually uses the common algebra he learned.

  21. A 7 grader Amanda C. Xiques sent me the following verse

    My Feelings About Math
    by Amanda Xiques

    Roses are Red
    Violets are Blue
    Math is a subject
    I'll always use

    From cooking to dancing
    Hard work to pleasure
    Life is full
    Of values we measure

    Math is a skill
    We need throughout life
    We use it at work,
    As a husband, or wife

    To balance a checkbook,
    Buy groceries, teach school
    Most life experiences
    Use mathematical rules

    Math is objective
    There's no in-between
    It's logical, explainable,
    Practical, and clean

    The fact that it's perfect
    Intimidates me
    Sometimes solutions
    Are hard to see

    I have one major problem
    Related to testing
    Seems I always know the answer
    To the question no one's asking

    But I've learned more than I realized
    When I recently tested
    My SAT Math scores
    Were really impressive

    Math isn't easy
    Every puzzle's a test
    But I respect it, I need it
    And I always do my best

  22. D. Niederman and D. Boyum, (What The Numbers Say, Broadway Books, 2003)

    ... large amounts of quantitative information are a feature of many such jobs, and good quantitative thinking is crucial to doing the jobs well. But matrix algebra is not required. Nor are high school staples such as quadratic equations, analytic geometry, and imaginary numbers (p. 2).

    Lest there be any doubt, and lest your grade school experiences suggest otherwise, percentages exist in order to make our lives easier (p. 80).

  23. Ron Aharoni, a math professor at Technion, Israel, in the Forward to his recent Arithmetic for Parents, admits (unwittingly I believe - AB) that most of the adults make do without mathematics:

    Most adults have long buried their memories of studying mathematics. All they really want is to forget the trauma. They accept their past incomprehension as a tolerable, albeit painful, fact. "You don't really need to know mathematics," they console themselves. Until one day the need does arise and the old anxieties resurface. This happens when their child begins dealing with the same experiences.

  24. A lovely advice is found in Mary Everest Boole's Philosophy And Fun Of Algebra (London: C. W. Daniel, LTD, 1909).

    Algebra can be made about anything which any human wants to know about (but does not - AB). Everybody ought to be able to make Algebras; and the sooner we begin the better. It is best to begin before we can talk; because, until we can talk, no one get us into illogical habits; and it is advisable that good logic should get the start of bad.

  25. Sir John Herschel (1792-1871), who named seven moons of Saturn and four moons of Uranus, has been quoted by A. de Morgan (A Budget of Paradoxes, p. 81):

    Admission to its sanctuary, and to the privileges and feelings of a votary, is only to be gained by one means--sound and sufficient knowledge of mathematics, the great instrument of all exact inquiry, without which no man can ever make such advances in this or any other of the higher departments of science as can entitle him to form an independent opinion on any subject of discussion within their range.

  26. Among many other things, Benjamin Franklin was an amateur mathematician who drew much enjoyment in inventing and solving mathematical problems, see, for example, P. C. Pasles' Benjamin Franklin's Numbers from Princeton University Press (2008). In a letter to his sister Jane Mecom (Sept. 20, 1787) he wrote

    It seems to me, that if statesmen had a little more arithmetic, or were more accustomed to calculation, wars would be much less frequent.

  27. In autobiographical recollections that he wrote in 1876 for his children, Charles Darwin expressed himself (Dr. Euler's Fabulous Formula, p. 11) as follows:

    During the three years which I spent at Cambridge my time was wasted, as far as academical studies were concerned ... I attempted mathematics, and even went during the summer of 1828 with a private tutor ... but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense.


  1. W.Dunham, Journey through Genius, Penguin Books, 1991
  2. W.Dunham, The Mathematical Universe, John Wile & Sons, 1994


H. Eves questions the veracity and even the plausibility of the Euler-Diderot story. According to Eves, the story was first told by Thiébault in his Mes Souvenirs de vinght ans de séjour à Berlin of 1801, and later retold, with significant additions, by Augustus De Morgan in his Budget of Paradoxes in 1878. Since then, the story has been repeated by many authors, always in De Morgan's highly colored version.

In fact, ... Diderot was a very good mathematician and had published five creditable memoirs on the subject prior to his trip to Russia. ... One even wonders if the Thiébault version is true. ... the story hardly fits Euler's character; Euler did not indulge in thoughtless and asinine behavior.


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