In the *Convergence* article series “HoM Toolbox, or Historiography and Methodology for Mathematicians,” on the page “Introduction - How Do We Know About the Past?”, Amy Ackerberg-Hastings reminds the reader that sources of historical information can be broken into primary, secondary, and tertiary sources. Articles aimed at an audience of fellow scholars in a given field will be careful to emphasize primary sources. In contrast, it is an unfortunate fact that many of the mathematical quotations circulated on the internet or in publications aimed at a general audience tend to come from tertiary sources; that is, they rarely are drawn from direct examinations of the works of the attributed authors, and in fact such quotations are often shared without any mention of the primary sources in which they originally appeared.

Because mathematical quotations are so often drawn from tertiary sources, it is not surprising that frequently they have been changed in the course of passing through many hands. Sometimes the name of the original author is lost and the quote becomes misattributed to a different author. In other cases, the wording may be changed, becoming a misleading paraphrase of the original author’s intent. Even if the author and quotation are accurate, words stripped from their original setting can appear to mean something very different than when they are viewed in the context of the surrounding material.

Finding the original source of a mathematical quotation can be described as

creating a bibliographic trail, akin to this path around Prescott Fields at

the University of Maine at Farmington. Photo by UMF, supplied to Maine Trail Finder.

In 2006, I started writing a regular column called “Quotations in Context” which appeared in the biannual *CSHPM/SCHPM Bulletin* of the Canadian Society for History and Philosophy of Mathematics. Each column examined one or more related quotations, tracking them to their first appearances in primary sources whenever possible. These past columns, revised and updated, will now be available as a resource here on *Convergence*. These columns contain interesting details that might be included by teachers where appropriate in the mathematics classroom. They also represent a way to prompt students to think about and discuss the potential problems that arise from too much reliance on tertiary sources.

For mathematics educators who wish to involve their students in historical research, these columns could also be viewed as examples/templates of a usefully-bounded student project. A growing number of archives of primary historical materials have become available online, not just to scholars in specialized fields, but as well to teachers, students, and the general public at no cost: the Cambridge Digital Library, e-rara, Google Books, the HathiTrust Digital Library, the Internet Archive, the Linda Hall Library Digital Collection, and the University of Michigan Historical Mathematics Collection, to name only a few. While there still exist some barriers, such as language, to the exploration of these materials, it nonetheless remains quite possible for more advanced students to take quotations they have found in tertiary sources and successfully track that information back to their primary sources; in addition, even if students are unable to find the original source, they could still gain valuable experience by writing a detailed explanation of their work tracing the quotation back as far as they were able to go. Researching a single quotation could also help familiarize students with the vast online archives available to them; in particular, such a task could serve as preparation for more significant research projects.

*“Mathematics is written for mathematicians.”*

In his preface to *De revolutionibus orbium coelestium* (*On the revolutions of the heavenly spheres*), Nicolaus Copernicus described his reasons for hesitating to publish his work. He admitted to fearing the ridicule of others, but finally decided that those mostly likely to heap scorn upon his work were those who are least qualified to judge it. It is at this point in the preface that the phrase “*Mathemata mathematicis scribuntur*” [Copernicus 1543, p. vii] appears.

Painting of Copernicus, 1580. Public domain, *Convergence* Portrait Gallery.

Unfortunately, the Latin here is potentially ambiguous, with the same words sometimes used to refer to astronomy or even astrology, rather than mathematics. Historian Dorothy Stimson translated the preface as part of her 1917 doctoral dissertation at Columbia University, *The Gradual Acceptance of the Copernican Theory of the Universe*. In her text, the words were consistently translated as references to mathematics:

If perchance there should be foolish speakers who, together with those ignorant of all mathematics, will take it upon themselves to decide concerning these things, and because of some place in the Scriptures wickedly distorted to their purpose, should dare to assail this my work, they are of no importance to me, to such an extent do I despise their judgement as rash. For it is not unknown that Lactantius, the writer celebrated in other ways but very little in mathematics, spoke somewhat childishly of the shape of the earth when he derided those who declared the earth had the shape of a ball. So it ought not to surprise students if such should laugh at us also. Mathematics is written for mathematicians [Stimson 1917, p. 115].

Photograph of Dean Dorothy Stimson, 1926. Donnybrook Fair Yearbook Collection,

Goucher College Digital Library, Goucher College Archives, Towson, MD*.*

On the other hand, a more recent translation by historian Edward Rosen consistently interpreted the Latin in terms of astronomy instead:

Perhaps there will be babblers who claim to be judges of astronomy although completely ignorant of the subject and, badly distorting some passage of Scripture to their purpose, will dare to find fault with my undertaking and censure it. I disregard them even to the extent of despising their criticism as unfounded. For it is not unknown that Lactantius, otherwise an illustrious writer but hardly an astronomer, speaks quite childishly about the earth’s shape, when he mocks those who declared that the earth has the form of a globe. Hence scholars need not be surprised if any such persons will likewise ridicule me. Astronomy is written for astronomers [Copernicus 1978, p. 5].

Which of these translations interprets “*Mathemata mathematicis scribuntur*” correctly? Interestingly, that very issue seemed to be addressed by Copernicus in the introduction of Book I. Copernicus began by arguing that, of all things that can be studied, nothing is more important than the study of the heavens and the movements of the stars. Copernicus explicitly recognized the ambiguity of how this study has been named:

If then the value of the arts is judged by the subject matter which they treat, that art will be by far the foremost which is labeled astronomy by some, astrology by others, but by many of the ancients, the consummation of mathematics. Unquestionably the summit of the liberal arts and most worthy of a free man, it is supported by almost all the branches of mathematics. Arithmetic, geometry, optics, surveying, mechanics and whatever others there are all contribute to it [Copernicus 1978, p. 7].

If the study of the stars is truly to be viewed as the “consummation of mathematics,” arguing about the distinction between “Mathematics is written for mathematicians” and “Astronomy is written for astronomers” appears somewhat unimportant.

Copernicus, Nicholas. 1543. *De revolutionibus orbium coelestium*. Nuremberg: Johann Petreius.

Copernicus, Nicholas. 1978. *On the Revolutions*. Translation and commentary by Edward Rosen. Edited by Jerzy Dobrzycki. Vol. 2. London: Palgrave Macmillan.

Stimson, Dorothy. 1917. *The Gradual Acceptance of the Copernican Theory of the Universe*. New York: Baker & Taylor.

“Quotations in Context” is a regular column written by Michael Molinsky that has appeared in the *CSHPM/SCHPM Bulletin* of the Canadian Society for History and Philosophy of Mathematics since 2006 (this installment was first published in May 2008). In the modern world, quotations by mathematicians or about mathematics frequently appear in works written for a general audience, but often these quotations are provided without listing a primary source or providing any information about the surrounding context in which the quotation appeared. These columns provide interesting information on selected statements related to mathematics, but more importantly, the columns highlight the fact that students today can do the same legwork, using online databases of original sources to track down and examine quotations in their original context.

*“But there is another reason for the high repute of mathematics: it is mathematics that offers the exact natural sciences a certain measure of security which, without mathematics, they could not attain.”*

*“As far as the laws of mathematics refer to reality, they are not certain;
and as far as they are certain, they do not refer to reality.”*

The two Albert Einstein quotations above frequently appear without citation. For example, the first quotation is printed in the “They Say, What They Say, Let Them Say” pages at the beginning of E. T. Bell’s *Men of Mathematics*, and the second can be found in the third volume of James R. Newman’s *The World of Mathematics*. The quotations, placed side-by-side, might look somewhat contradictory at first glance, since one seems to indicate that mathematics is a boon to natural science, while the other appears to say that mathematics is of no use in describing the real world; however, both of these quotations actually originate from the same source and occur only a few sentences apart.

Photograph of Albert Einstein. *Convergence* Portrait Gallery.

On January 27, 1921, Albert Einstein presented a lecture to the Prussian Academy of Sciences entitled “Geometry and Experience.” His complete paper was printed later that year and appeared in English translation a year later. In his paper, Einstein discussed the geometric model of the universe and how to best determine what that model must be. He also considered ways to visualize non-Euclidean concepts such as sets that are finite, but unbounded. Both of the quotations above, however, appeared at the beginning of the paper, when Einstein sets out to distinguish between “axiomatic” and “practical” mathematics.

The paper began with separating mathematics from science by means of certainty: mathematical propositions are not subject to debate, whereas scientific propositions, even when supported by all current facts, may be falsified by later data. Einstein pointed out that, while the natural sciences may not be completely certain, their use of mathematics introduces at least some level of certainty to the mix. It was at this point that the first quotation appeared:

Die Mathematik genießt vor allen anderen Wissenschaften aus einem Grunde ein besonderes Ansehen; ihre Sätze sind absolut sicher und unbestreitbar, während die aller andern Wissenschaften bis zu einem gewissen Grad umstritten und stets in Gefahr sind, durch neu entdeckte Tatsachen umgestoßen zu werden. Trotzdem brauchte der auf einem anderen Gebiete Forschende den Mathematiker noch nicht zu beneiden, wenn sich seine Sätze nicht auf Gegenstände der Wirklichkeit, sondern nur auf solche unserer bloßen Einbildung bezögen. Denn es kann nicht wundernehmen, wenn man zu übereinstimmenden logischen Folgerungen kommt, nachdem man sich über die fundamentalen Sätze (Axiome) sowie über die Methoden geeinigt hat, vermittels welcher aus diesen fundamentalen Sätzen andere Sätze abgeleitet werden sollen. Aber jenes große Ansehen der Mathematik ruht andererseits darauf, daß die Mathematik es auch ist, die den exakten Naturwissenschaften ein gewisses Maß von Sicherheit gibt, das sie ohne Mathematik nicht erreichen könnten [Einstein 1921, p. 3].

One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of all other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts. In spite of this, the investigator in another department of science would not need to envy the mathematician if the laws of mathematics referred to objects of our mere imagination, and not to objects of reality. For it cannot occasion surprise that different persons should arrive at the same logical conclusions when they have already agreed upon the fundamental laws (axioms), as well as the methods by which other laws are to be deduced therefrom. But there is another reason for the high repute of mathematics, in that it is mathematics which affords the exact sciences a certain measure of security, to which without mathematics they could not attain [Einstein 1922, pp. 27–28].

Einstein then asked a familiar question: why should mathematics, based on axioms and independent of experience, be so useful as a tool for modeling reality? It is in response to the idea that human beings might be able to determine the properties of the world by thought and reason alone that the second quotation appears.

An dieser Stelle nun taucht ein Rätsel auf, das Forscher aller Zeiten so viel beunruhigt hat. Wie ist es möglich, daß die Mathematik, die doch ein von aller Erfahrung unabhängiges Produkt des menschlichen Denkens ist, auf die Gegenstände der Wirklichkeit so vortrefflich paßt? Kann denn die menschliche Vernunft ohne Erfahrung durch bloßes Denken Eigenschaften der wirklichen Dinge ergründen?

Hierauf ist nach meiner Ansicht kurz zu antworten: Insofern sich die Sätze der Mathematik auf die Wirklichkeit beziehen, sind sie nicht sicher, und insofern sie sicher sind, beziehen sie sich nicht auf die Wirklichkeit [Einstein 1921, p. 3].

At this point an enigma presents itself which in all ages has agitated inquiring minds. How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? Is human reason, then, without experience, merely by taking thought, able to fathom the properties of real things?

In my opinion the answer to this question is, briefly, this:—As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality [Einstein 1922, p. 28].

Although the quotation itself only says “mathematics,” the following sentences make clear that, in this context, the mathematics in question is “axiomatic” mathematics. Using geometry as an example, Einstein proceeded through the familiar argument that, in its purely axiomatic form, the expressions “straight line” and “point” have no connections to real objects or properties of the real world. Any properties they have proceed from the axioms and only from the axioms.

Einstein then defined “practical geometry,” which takes axiomatic geometry and adds a proposition about how it relates to reality; for example, that solid bodies in the real world behave just like solid bodies in Euclidean Geometry. He pointed out that this proposition is not accepted as an axiom, but instead as a scientific premise, to be tested through experience and experimentation. This is clearly the intended context of the first quotation: that while science benefits from a degree of certainty through the use of “practical” mathematics, it cannot provide full certainty, because future experience may demonstrate that a proposed correspondence between a mathematical object and a property of the universe will be found to be invalid.

Einstein, Albert. 1921. *Geometrie und Erfahrung*. Berlin: Julius Springer.

Einstein, Albert. 1922. *Sidelights on Relativity*. Translated by G. B. Jeffery and W. Perrett. London: Methuen & Co., Ltd.

“Quotations in Context” is a regular column written by Michael Molinsky that has appeared in the *CSHPM/SCHPM Bulletin* of the Canadian Society for History and Philosophy of Mathematics since 2006 (this installment was first published in May 2007). In the modern world, quotations by mathematicians or about mathematics frequently appear in works written for a general audience, but often these quotations are provided without listing a primary source or providing any information about the surrounding context in which the quotation appeared. These columns provide interesting information on selected statements related to mathematics, but more importantly, the columns highlight the fact that students today can do the same legwork, using online databases of original sources to track down and examine quotations in their original context.

*“I have hardly ever known a mathematician who was capable of reasoning.”*

In 1888, Benjamin Jowett (1817–1893), the master of Balliol College, Oxford, published the third edition of his English translation of Plato’s *Republic*. This edition is the source of the quotation above. The quotation certainly appears to claim that mathematicians are incapable of any kind of reasoning. Paul Shorey (1857–1934), a Greek scholar at the University of Chicago, referred to this line as “Jowett’s wicked jest” [Plato 1969, p. 531]. When the quotation is placed within the context of Plato’s arguments in the sixth and seventh books of the *Republic*, it is clear that Plato’s intended meaning was very different.

Photograph of Benjamin Jowett. Public domain, from the book *Oxford Men
& Their Colleges* (Oxford, 1893), as digitized by the Internet Archive.

In the sixth book, Plato argued that the world can be divided into the visible and the intellectual. In the dialogue, Socrates tells Glaucon that the intellectual world can be broken further into two categories:

There are two subdivisions, in the lower of which the soul uses the figures given by the former division as images; the enquiry can only be hypothetical, and instead of going upwards to a principle descends to the other end; in the higher of the two, the soul passes out of hypotheses, and goes up to a principle which is above hypotheses, making no use of images as in the former case, but proceeding only in and through the ideas themselves [Plato 1888, p. 211].

The “lower” subdivision of the intellectual world uses hypotheses to engage in deductive reasoning. This subdivision is specifically identified with arithmetic, geometry and other mathematical sciences, which make use of the visible world as models for understanding the invisible world of thoughts and ideas. This is separated from the dialectic, the sphere of the intellect that uses tools such as discussion and debate to delve into the realm of ideas without making any use of the world of the visible.

The seventh book of the *Republic* explored the best way to train minds to understand the visible and the intellectual. For the subdivision of the intellectual that is concerned with deductive reasoning, the appropriate training is rigorous preparation in arithmetic, plane and solid geometry, astronomy and harmonics (i.e., the quadrivium). But Plato suggested that this training is not enough to fully prepare a student, and the quotation that is the topic of this column appears in Glaucon’s reply to Socrates:

Do you not know that all this is but the prelude to the actual strain which we have to learn? For you surely would not regard the skilled mathematician as a dialectician?

Assuredly not, he said; I have hardly ever known a mathematician who was capable of reasoning [Plato 1888, p. 235].

Glaucon’s statement, in the context of the surrounding material, clearly was not intended to mean that mathematicians are incapable of all kinds of reasoning; instead, the implication was that expertise in deductive reasoning does not necessarily translate into mastery of the form of reasoning identified as dialectic.

Photograph of Paul Shorey. Public domain, from the article “The Spirit of the University of Chicago”

in *The University of Chicago Magazine* 1, no. 6 (April 1909): 228, as reproduced in Wikimedia Commons.

Other translators seem to have avoided Jowett’s witty but potentially misleading phrasing. For example, in Shorey’s translation of the *Republic* (first published in two volumes in 1930 and 1935), the same text is translated into English as:

Or do we not know that all this is but the preamble of the law itself, the prelude of the strain that we have to apprehend? For you surely do not suppose that experts in these matters are reasoners and dialecticians?

No, by Zeus,” he said, “except a very few whom I have met [Plato 1969, p. 531].

Even Jowett himself gave a clearer translation of the original Greek in his first edition of the *Republic*, back in 1871:

Are we not advised that this is but the prelude of the actual strain which we have to learn? For I imagine that you would not regard the skilled mathematician as a dialectician?

No indeed, he said; very few mathematicians whom I have ever known are reasoners in that sense [Plato 1871, p. 359].

Plato. 1871. *The Dialogues of Plato*. Translated by Benjamin Jowett. Vol. 2. New York: Charles Scribner and Company.

Plato. 1888. *The Republic of Plato*. Translated by Benjamin Jowett. Oxford: Clarendon Press.

Plato. 1969. *Plato in Twelve Volumes*. Translated by Paul Shorey. 2nd ed. Vols 5–6. Cambridge: Harvard University Press.

“Quotations in Context” is a regular column written by Michael Molinsky that has appeared in the *CSHPM/SCHPM Bulletin* of the Canadian Society for History and Philosophy of Mathematics since 2006 (this installment was first published in November 2006). In the modern world, quotations by mathematicians or about mathematics frequently appear in works written for a general audience, but often these quotations are provided without listing a primary source or providing any information about the surrounding context in which the quotation appeared. These columns provide interesting information on selected statements related to mathematics, but more importantly, the columns highlight the fact that students today can do the same legwork, using online databases of original sources to track down and examine quotations in their original context.

*“Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”*

This quotation from the logician Bertrand Russell originally appeared in an article, “Recent Work on the Principles of Mathematics,” published in 1901 by *The International Monthly* [Russell 1901]. Russell later included the work in his book *Mysticism and Logic* in 1918, where the essay in question appeared as the chapter “Mathematics and the Metaphysicians” [Russell 1918]. In this later work, he amended the original article with some corrective footnotes, reflecting changes in his understanding of some of the topics over the intervening years. Russell also admitted in the preface to *Mysticism and Logic* that the tone of the article was not entirely his idea, and that the editor of the magazine had urged him to make it “as romantic as possible” [Russell 1918, p. vi].

Photograph of Bertrand Russell in 1924. Public domain, Wikimedia Commons.

The article “Recent Work in the Philosophy of Mathematics” was intended for a general audience, and covered topics such as Zeno’s paradoxes of motion, infinitesimals, infinite cardinals, and the differences between Euclid’s work and that of modern geometers. The overall theme of the work concerned the definition of mathematics.

Russell opened the essay with the claim that “pure” mathematics was born in the 19th century, specifically in the work of George Boole. Pure mathematics, in this context, was defined by Russell as being identical to formal, symbolic logic. Russell argued that “pure” mathematical work is concerned only with discovering rules of valid inference; for example, given that property \(A\) is true of anything, can you safely conclude that property \(B\) is true of anything? He stated that questions about whether or not property \(A\) really is true, or what the “thing” is that we are talking about, are questions that belong to “applied” mathematics.

It was at this point in the article that the quotation appeared:

We start, in pure mathematics, from certain rules of inference, by which we can infer that if one proposition is true, then so is some other proposition. These rules of inference constitute the principles of formal logic. We then take any hypothesis that seems assuring, and deduce its consequences. If our hypothesis is about anything, and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true [Russell 1901, p. 84].

Since Russell was considering mathematics as nothing but formal logic, and since he required that mathematics be distinguished from the “applied” world by concentrating on the validity of rules of inference and not the validity of the hypotheses on which those rules are used, he summed this argument up with the given quotation. In the context of the article, the quotation appears to be a somewhat tongue-in-cheek, but accurate, summation of the idea that logical rules of inference are not concerned with what the hypotheses of an argument may be or whether they are true.

In closing, I’d like to include the equally witty line that follows the quotation from the article:

People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate [Russell 1901, p. 84].

Russell, Bertrand. 1901, July. Recent Work on the Principles of Mathematics. *The International Monthly* 4(1): 83–101.

Russell, Bertrand. 1918. *Mysticism and Logic and Other Essays*. London: Longmans, Green and Co.

“Quotations in Context” is a regular column written by Michael Molinsky that has appeared in the *CSHPM/SCHPM Bulletin* of the Canadian Society for History and Philosophy of Mathematics since 2006 (this installment was first published in May 2006). In the modern world, quotations by mathematicians or about mathematics frequently appear in works written for a general audience, but often these quotations are provided without listing a primary source or providing any information about the surrounding context in which the quotation appeared. These columns provide interesting information on selected statements related to mathematics, but more importantly, the columns highlight the fact that students today can do the same legwork, using online databases of original sources to track down and examine quotations in their original context.

*“The early study of Euclid made me a hater of geometry.”*

In 1869, James Joseph Sylvester was selected to serve as President of the Mathematical and Physical Sciences Section of the British Association for the Advancement of Science. Although his Presidential Address to the section meeting touched on a variety of issues, the main body of the speech was a rebuttal of a sequence of characterizations of mathematics published by Thomas Huxley that same year. In an article entitled “Scientific Education: Notes of an After-Dinner Speech,” published in *MacMillian’s Magazine*, Huxley emphasized the importance of observation in scientific training. By way of contrast, Huxley briefly described his view of mathematical training:

Mathematical training is almost purely deductive. The mathematician starts with a few simple propositions, the proof of which is so obvious that they are called self-evident, and the rest of his work consists of subtle deductions from them [Huxley 1869b, p. 182].

Similarly, Sylvester highlighted Huxley’s statement from “The Scientific Aspects of Positivism,” published in *The* *Fortnightly Review*, that mathematics is “that which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation” [Huxley 1869a, p. 667].

Photograph of Huxley by Elliott & Fry. Public domain, courtesy of Wellcome Collection.

Sylvester argued that Huxley could not be further from the truth. He pointed out that great mathematicians such as Lagrange and Gauss had emphasized the importance of observation in mathematics. He provided a list of mathematical ideas that originated from experiment; for example, how the motion of compound pendulums led to Sturm’s theorem regarding the number of unique real roots of a polynomial. Sylvester argued that “observation, divination, induction, experimental trial, and verification” [Sylvester 1908, p. 657] are all important aspects of mathematical training and discovery.

Stipple engraving of Sylvester by G. J. Stodart. Public domain, courtesy of Wellcome Collection.

Despite their difference of opinions on the nature of mathematics, Sylvester agreed with Huxley that experimental science should be introduced in schools, and he expressed a belief that mathematics could benefit from being taught in connection with science:

I think that that study [of natural and experimental science] and mathematical culture should go on hand in hand together, and that they would greatly influence each other for their mutual good. I should rejoice to see mathematics taught with that life and animation which the presence and example of her young and buoyant sister could not fail to impart, short roads preferred to long ones, Euclid honourably shelved or buried “deeper than did ever plummet sound” out of the schoolboy’s reach [Sylvester 1908, p. 657].

Following the response to Huxley, Sylvester touched on a variety of topics, but toward the end of the address he discussed the connections between arithmetic, algebra and geometry, saying that they “are constantly becoming more and more intimately related and connected by a thousand fresh ties” [Sylvester 1908, p. 659]. Sylvester particularly emphasized the importance of geometry, proclaiming the “wonderful influence geometry has exercised” [Sylvester 1908, p. 659] over modern mathematics. Having praised geometry as a general topic, Sylvester responded to those who may have been dismayed by his dismissal of Euclid from the curriculum, and it is in this context that the quotation at the head of this column appeared:

The early study of Euclid made me a hater of Geometry, which I hope may plead my excuse if I have shocked the opinions of any in this room (and I know there are some who rank Euclid as second in sacredness to the Bible alone, and as one of the advanced outposts to the British Constitution) by the tone in which I have previously alluded to it as a schoolbook; and yet, in spite of this repugnance, which had become a second nature to me, whenever I went far enough into any mathematical question, I found I touched, at last, a geometrical bottom [Sylvester 1908, p. 660].

From the material above, it can be seen that the isolated quotation for this column is both accurate and misleading. It seems quite accurate in portraying Sylvester’s actual feelings toward Euclid’s *Elements*, in particular to the use of that work as a geometry textbook. However, whatever his early feelings about geometry may have been, the context of the address makes it apparent that in later life he was clearly no longer a “hater” of geometry.

Huxley, Thomas. 1869a, June 1. The Scientific Aspects of Positivism. *The Fortnightly Review* 11(30): 653–670.

Huxley, Thomas. 1869b, June. Scientific Education: Notes of an After-Dinner Speech. *Macmillan’s Magazine* 20(116): 177–184.

Sylvester, James Joseph. 1908. *The Collected Mathematical Papers of James Joseph Sylvester*. Vol. 2. Cambridge: Cambridge University Press.

“Quotations in Context” is a regular column written by Michael Molinsky that has appeared in the *CSHPM/SCHPM Bulletin* of the Canadian Society for History and Philosophy of Mathematics since 2006 (this installment was first published in November 2007). In the modern world, quotations by mathematicians or about mathematics frequently appear in works written for a general audience, but often these quotations are provided without listing a primary source or providing any information about the surrounding context in which the quotation appeared. These columns provide interesting information on selected statements related to mathematics, but more importantly, the columns highlight the fact that students today can do the same legwork, using online databases of original sources to track down and examine quotations in their original context.

*“Everything of importance has been said before
by somebody who did not discover it.”*

Another Quotations in Context column addresses a quotation from James Joseph Sylvester’s Presidential Address in 1869 to the Mathematical and Physical Sciences Section of the British Association for the Advancement of Science. In 1916, Alfred North Whitehead gave his own Presidential Address to this same institution. The topic of his speech was “The Organisation of Thought.”

Photograph of Whitehead. Public domain, courtesy of Wellcome Collection.

The first half of his address was about the nature of science. Whitehead discussed the practical and theoretical sides of science, the discipline’s use of inductive reasoning and experience, and its foundation in common sense. In the end, he concluded:

Science is essentially logical. The nexus between its concepts is a logical nexus, and the grounds for its detailed assertions are logical grounds. King James said, 'No bishops, no king.' With greater confidence we can say, 'No logic, no science' [Whitehead 2007a].

The speech briefly turned to those who viewed logical thought as barren or sterile. He mentioned the belief that deductive reasoning could not create new ideas, since any conclusions came part and parcel with the premises with which one began. Whitehead offered a brief rebuttal to this characterization, but he then moved on, although he would address this topic again later in the speech.

Whitehead gave a brief overview of modern logic, dividing it into four general sections, which he termed arithmetic, algebra, general-function theory and analytic. The arithmetic section involved general propositions, while the algebra section introduced propositional functions. The general-function theory section then considered classes of propositional functions. The final section, the analytic stage, Whitehead concluded as being the “whole of mathematics . . . neither more nor less” [Whitehead 2007b].

The quotation in question for this column appeared immediately following this summary and categorization of logical thought, when Whitehead returned to the idea of the barrenness of logic:

The question arises, How many forms of propositions are there? The answer is, an unending number. The reason for the supposed sterility of logical science can thus be discerned. Aristotle founded the science by conceiving the idea of the form of a proposition, and by conceiving deduction as taking place in virtue of the forms. But he confined propositions to four forms, now named A, I, E, O. So long as logicians were obsessed by this unfortunate restriction, real progress was impossible. Again, in their theory of form, both Aristotle and subsequent logicians came very near to the theory of the logical variable. But to come very near to a true theory, and to grasp its precise application, are two very different things, as the history of science teaches us. Everything of importance has been said before by somebody who did not discover it [Whitehead 2007b].

I have occasionally found Whitehead’s quotation being misinterpreted as equivalent to Stephen Stigler’s “law of eponymy.” But in the context of the original address, Whitehead clearly seems to be saying the exact opposite: that the first person to state a result or make a claim is rarely the first person to truly understand the idea precisely and completely.

Whitehead, Alfred North. 2007a, April. A. N. Whitehead addresses the British Association in 1916, Part 1. MacTutor History of Mathematics Archive.

Whitehead, Alfred North. 2007b, April. A. N. Whitehead addresses the British Association in 1916, Part 2. MacTutor History of Mathematics Archive.

“Quotations in Context” is a regular column written by Michael Molinsky that has appeared in the *CSHPM/SCHPM Bulletin* of the Canadian Society for History and Philosophy of Mathematics since 2006 (this installment was first published in November 2008). In the modern world, quotations by mathematicians or about mathematics frequently appear in works written for a general audience, but often these quotations are provided without listing a primary source or providing any information about the surrounding context in which the quotation appeared. These columns provide interesting information on selected statements related to mathematics, but more importantly, the columns highlight the fact that students today can do the same legwork, using online databases of original sources to track down and examine quotations in their original context.

Michael Molinsky is a Professor of Mathematics at the University of Maine at Farmington, the oldest public institution of higher education in Maine, where he encourages his students (especially those preparing to be K–12 teachers) to explore the history of mathematics. He is an active member of the Canadian Society for History and Philosophy of Mathematics, and he served as an Associate Editor of *Convergence* from 2017 through 2022.