You shouldn’t believe everything you read

One of the things that I have been reading recently is a very interesting paper by John N. Crossley, the Anglo-Australian logician and historian of mathematics, about the reception and adoption of the Hindu-Arabic numbers in medieval Europe.[1]Here I came across this wonderful footnote:[2]


It is interesting to note that Richard Lemay in his entry “Arabic Numerals,” in Joseph Reese Strayer, ed., Dictionary of the Middle Ages(New York, 1982–89) 1:382–98, at 398 reports that in the University of Padua in the mid-fifteenth century, prices of books should be marked “non per cifras sed per literas claras.” He gives a reference to George Gibson Neill Wright, The Writing of Arabic Numerals(London, 1952), 126. Neill Wright in turn gives a reference to a footnote of Susan Cunnigton, The Story of Arithmetic: A Short History of Its Origin and Development(London, 1904), 42, n. 2. She refers to Rouse Ball’s Short History of Mathematics, in fact this work is: Walter William Rouse Ball, A Short Account of the History of Mathematics, 3rded. (London, 1901), and there one finds on p. 192: “…in 1348 the authorities of the university of Padua directed that a list should be kept of books for sale with the prices marked ‘non per cifras sed per literas claras’ [not by cyphers but by clear letters].” I am yet to find an exact reference for this prohibition. (There is none in Rouse Ball.) Chrisomalis Numerical Notations, p. 124, cites J. Lennart Berggren, “Medieval Arithmetic: Arabic Texts and European Motivations,” in Word, Image, Number: Communication in the Middle Ages, ed. John J. Contreni and Santa Casciani (Florence, 2002), 351–65, at 361, who does not give a reference.

Here we have Crossley the historian following a trail of quotes, references and footnotes; his hunt doesn’t so much terminate in a dead-end as fizzle out in the void, leaving the reader unsure whether the university of Padua really did insist on its book prices being written in Roman numerals rather than Hindu-Arabic ones or not. What we have here is a succession of authors writing up something from a secondary, tertiary, quaternary source with out bothering to check if the claim it makes is actually true or correct by looking for and going back to the original source, which in this case would have been difficult as the trail peters out by Rouse Ball, who doesn’t give a source at all.

This habit of writing up without checking original sources is unfortunately not confined to this wonderful example investigated by John Crossley but is seemingly a widespread bad habit under historians and others who write historical texts.

I have often commented that I served my apprenticeship as a historian of science in a DFG[3]financed research project on Case Studies into a Social History of Formal Logic under the direction of Professor Christian Thiel. Christian Thiel was inspired to launch this research project by a similar story to the one described by Crossley above.

Christian Thiel’s doctoral thesis was Sinn und Bedeutung in der Logik Gottlob Freges(Sense and Reference in Gottlob Frege’s Logic); a work that lifted him into the elite circle of Frege experts and led him to devote his academic life largely to the study of logic and its history. One of those who corresponded with Frege, and thus attracted Thiel interest, was the German meta-logician Leopold Löwenheim, known to students of logic and meta-logic through the Löwenheim-Skolem theorem or paradox. (Don’t ask!) Being a thorough German scholar, one might even say being pedantic, Thiel wished to know Löwenheim’s dates of birth and death. His date of birth was no problem but his date of death turned out to be less simple. In an encyclopaedia article Thiel came across a reference to c.1940; the assumption being that Löwenheim, being a quarter Jewish and as a result having been dismissed from his position as a school teacher in 1933, had somehow perished during the holocaust. In another encyclopaedia article obviously copied from the first the ‘circa 1940’ had become a ‘died 1940’.

Thiel, being the man he is, was not satisfied with this uncertainty and invested a lot of effort in trying to get more precise details of the cause and date of Löwenheim’s death. The Red Cross information service set up after the Second World War in Germany to help trace people who had died or gone missing during the war proved to be a dead end with no information on Löwenheim. Thiel, however, kept on digging and was very surprised when he finally discovered that Löwenheim had not perished in the holocaust after all but had survived the war and had even gone back to teaching in Berlin in the 1950s, where he died 5. May 1957 almost eighty years old. Thiel then did the same as Crossley, tracing back who had written up from whom and was able to show that Löwenheim’s death had already been assumed to have fallen during WWII, as he was still alive and kicking in Berlin in the early 1950s!

This episode convinced Thiel to set up his research project Case Studies into a Social History of Formal Logic in order, in the first instance to provide solid, verified biographical information on all of the logicians listed in Church’s bibliography of logic volume of the Journal of Symbolic Logic, which we then proceeded to do; a lot of very hard work in the pre-Internet age. Our project, however, was not confined to this biographical work, we also undertook other research into the history of formal logic.

As I said above this habit of writing ‘facts’ up from non-primary sources is unfortunately very widespread in #histSTM, particularly in popular books, which of course sell much better and are much more widely read than academic volumes, although academics are themselves not immune to this bad habit. This is, of course, the primary reason for the continued propagation of the myths of science that notoriously bring out the HISTSCI_HULK in yours truly. For example I’ve lost count of the number of times I’ve read that Galileo’s telescopic discoveries proved the truth of Copernicus’ heliocentric hypothesis. People are basically to lazy to do the legwork and check their claims and facts and are much too prepared to follow the maxim: if X said it and it’s in print, then it must be true!

[1]John N. Crossley, Old-fashioned versus newfangled: Reading and writing numbers, 1200–1500, Studies in medieval and Renaissance History, Vol. 10, 2013, pp.79–109

[2]Crossley p. 92 n. 42

[3]DFG = Deutsche Forschungsgemeinschaft = German Research Foundation




Filed under History of Logic, History of Mathematics, Myths of Science

16 responses to “You shouldn’t believe everything you read

  1. Fernando

    I wrote a short article in MAA Focus some years ago about trying to chase down the sources from Rouse Ball’s version of the story of the Delian problem of doubling the cube. I concluded that he had made the story “better” by supplying details that none of the Ancient sources provide. It’s certainly possible that he “improved” the Padua story as well…

    • I read Rouse Ball many years ago and as far as I can remember it was quite for the time it was written. Whereas I think it’s possible that he, like many others, might embellish a story I don’t seriously think that he would invent an entire quote.

  2. I love tracking sources back through notes like this. On Rouse Ball, if anyone was inclined, they could check Monumenti della Università di Padova (Venice, 1884–8) which sets out all the statutes in order and, if I was a betting man, I’d expect where he got the quote from, assuming it is genuine. As we have a year, it might be easy to look. There’s a copy in the Warburg Institute, alas no longer somewhere I have access to.

    The split personality between Roman and Arabic numerals lasted well into the sixteenth century. At the Queens’ College, Cambridge in the 1500s, the accounts were still being produced with Roman numerals (Cantab, UL, QCA 1-4) at the same time as the university lecturer on maths, Roger Collingwood, was a fellow of the college teaching students how to do arithmetic in Arabic numbers (his lecture notes Oxford, CCC, MS 102). As the resident mathematician, I do wonder if he was also required to do the accounts. (I do provide my references to the primary sources ;).)

  3. Ray

    Not to discount the importance of checking one’s sources, but the Galileo example seems quite different from the other two, in that the error cannot be resolved by simply appealing to primary sources ( or their absence.)

    I don’t see what sort of primary source you could possibly show someone to demonstrate that Galileo’s observations did or didn’t constitute proof for the theory of Copernicus. At best you could show that some contemporary scientist claimed they constituted proof (I wouldn’t be surprised if Galileo or one of his followers did in fact make such a claim.) Rather, I see no substitute for actually understanding the science.

    • Galileo’s own publications on his telescopic observations clearly show that they neither refute geocentrism nor, more importantly, do they in anyway confirm a heliocentric model of the cosmos. This is something that Galileo was well aware of. Of course you have to understand the science to realise this but if you don’t understand the science you shouldn’t be making claims about it anyway

      • Ray

        The impression I get from reading Galileo’s publications is that he is deliberately trying to make the reader think he is claiming his observations confirm the Copernican system. He typically does this by (rightly) demonstrating that his observations rule out the Ptolemaic system as well as a handful of other incorrect Aristotelian claims, and then neglecting to mention that there is a popular non-Copernican alternative to Ptolemy (i.e. that of Tycho) which does not make any of the refuted claims. It is for this reason that I feel going back to primary sources is not the most effective prescription for this particular disease. Moreover, in my experience, even those who claim Galileo “proved” heliocentricity tend not to credit him with any concrete astronomical observations beyond those which he actually made (moons of Jupiter, sunspots, phases of Venus, planetary disks much larger than first magnitude stars through the telescope, Saturn’s “ears”.) This is the sort of thing you’d expect people to get wrong if primary sources were the main problem.

        There is also some weirdness about what the claim about Galileo’s observations even is supposed to mean. It is generally bad form to describe anything outside of mathematics (in the modern sense) as having been “proven,” but if we take a more colloquial “beyond a reasonable doubt” sense of proof, it’s not clear Galileo was right to stop short of taking his observations as proof of heliocentrism. After all, it seems likely that if we were to transport the theoretical framework of Newton back to the time of Galileo, it could easily stand based on the observations made thus far (periods and orbital radii for the moons of Jupiter, Galileo’s observations of falling bodies and pendulums, Kepler’s periods and orbital radii for the planets based on the data of Tycho, known correlations of tides with lunar position.)

        But, I suppose the fact remains, neither Galileo nor any of his contemporaries were able to develop a theoretical framework strong enough to make his observations as convincing a demonstration of heliocentricity as they should have been in hindsight, and here I admit primary sources are of some help. But even there, I think the failure mode we see most often is not just a matter of not sourcing one’s claims, but not understanding what the sources would need to have said in order for the claims to be true. I don’t think even the worst Galileo hagiographies credit Galileo with proposing universal gravitation or making accurate photometric estimates of stellar distances, even though it would have been feasible with the observations available to him.

      • As usual your comment is less historical accuracy and more distorted hindsight and wishful thinking. In neither of his reports on his telescopic observations the Sidereus Nuncius and The Letters on Sunspots does Galileo, to use you phraseology, “deliberately try to make his reader think he is claiming his observations confirm the Copernican system.”

        He was very wise to do so. Being no fool, Galileo knew full well that there was nothing in Sidereus Nuncius to either refute a geocentric system or to confirm a heliocentric one. The Letters on Sunspots contain his observations of the phases of Venus, which of course refute a pure geocentric system, but Galileo, like his readers, knew full well that that were also explained by the Capellan and Tychonic systems both of which stood prominently in the middle of the contemporary debate on cosmological systems, for which see Pietro Daniel Omodeo Copernicus in the Cultural Debates of the Renaissance: Reception, Legacy, Transformation.

      • Ray

        Here’s an example of a passages where I think Galileo is trying to be deliberately misleading, from the Letter to the Grand Duchess Christina:

        “They know that as to the arrangement of the parts of the universe, I hold the sun to be situated motionless in the center of the revolution of the celestial orbs while the earth rotates on its axis and revolves about the sun. They know also that I support this position not only by refuting the arguments of Ptolemy and Aristotle, but by producing many counter-arguments; in particular, some which relate to physical effects whose causes can perhaps be assigned in no other way. In addition there are astronomical arguments derived from many things in my new celestial discoveries that plainly confute the Ptolemaic system while admirably agreeing with and confirming the contrary hypothesis.

        I think Galileo wants the reader to think “the contrary hypothesis” refers to that of Copernicus specifically, as he has mentioned no third option. He has plausible deniability I guess to claim it refers to a broader range of alternatives, but if that’s all he means, the italicized clause is unnecessary.

      • Ray writes:

        it seems likely that if we were to transport the theoretical framework of Newton back to the time of Galileo …

        which strikes me as a rather strange hypothetical. I mean, woulda shoulda coulda! Search for “Bending Spacetime in the Basement”, and you will find this remark:

        It seems plausible, then, given the knowledge at hand and a chain of inference which, in retrospect at least, appears straightforward, that Archimedes could have suspected the universality of gravitation. But could he have demonstrated it?

        accompanied by video showing that Archimedes could have performed the Cavendish experiment with the equipment available to him! Do I need to comment on how ahistorical this is?

      • Ray

        Michael Weiss,

        In order to evaluate claims about historical causality, we pretty much have to consider counterfactual scenarios. I understand my dispute with Thony as follows: I think he is claiming that Galileo and his contemporaries were uncertain of the truth of heliocentrism because the available observations were insufficient to distinguish between heliocentrism and geocentrism. I instead argue that the cause for confusion was not the absence of observations, but the failure of Galileo to consider (or more properly invent) the version of heliocentrism that most economically acounted for the observations he already had.

        Now counterfactuals are often very hard to test, but I think in this case we have a very good natural experiment in Newton. The observations Newton used to develop his theory of universal gravitation in the 1660s really weren’t different in any important way from what Galileo already had when he wrote the letter I referenced in my last comment. ( Here I include at the very least the results of his 1603 inclined plane experiments, his telescopic observations, and the data referenced in Kepler’s Astronomia Nova.) if you dispute this, I’d like to know what experiment or observation you think made the difference. Rather, the main advantage Newton had over Galileo was that he was exposed to more good ideas and fewer bad ideas over the course of his education.

        So in sum. Just like primary sources in history of science need a thorouogh grounding in the science of the day to be as useful as they should be, so too scientific observations need to work in concert with the development of theory. Neither science nor history is or should be mere stamp collecting.

  4. Thony, I’m sorry to say I no longer have the copies I took and can’t quote details of the letters, but some time ago I had reason to read a lot of Athanasius Kirchers’ correspondence and that of people associated with him. I was surprised to learn that his circle of acquaintances were embarrassed by the ferocity with which he insisted Galileo be condemned for heresy, when most of his circle found nothing particularly outrageous about G’s. work. But Kircher persisted, and succeeded in getting the Pope to be less hospitable than he had been to G. The tone of the letters was along the lines of ‘Yes, Kircher’s behaviour is excessive and embarrassing – but what can you do, he’s Kircher.” It was all the more strange given the interest K. obviously felt in astronomy, but he found the idea of an earth which moved through space to be irreconcilable with his own interpretation of the Bible.

  5. Postscript – I’ve had a quick look online for references to the above, but find only a hint in the introduction (p.xx) to ‘Athanasius Kircher (1602-1680), Jesuit Scholar: An Exhibition of His Works …’ published by the Harold B. Lee Library in 2003.

  6. I remember running across, in the 1970s, erroneous statements about Löwenheim’s death in the holocaust. Only later did I learn about his survival. One fact that lent plausibility to the erroneous assumption: Löwenheim is known almost entirely for the LS theorem, admittedly one of the most important results ever discovered in mathematical logic. Skolem on the other hand had quite a few arrows in his quiver, for example Skolem functions, the Skolem-Noether theorem in ring theory, and his role in the formulation of the ZF axioms of set theory, It was natural to assume Löwenheim’s life had been cut short, like Lindenbaum’s.

    A curiousity: the Löwenheim-Skolem theorem is always referred to as such; the term is even commonly used for related theorems (like the upwards LS theorem) that were not due to L or to S. But the paradox is, in my experience, almost always called Skolem’s paradox. (Although the MacTutor bio of L calls it the LS paradox.) I know that Skolem discussed the paradox in detail in one of his papers. Did Löwenheim do the same?

    • Anyone who has studied the cosmology of the twentieth century has run across many references to “Milne and McCrea”. However, the paper in question is:

      Newtonian Universes and the curvature of space
      McCrea, William Hunter; Milne, Edward Arthur
      Quart. J. Math. Oxford 5, 73-80 (1934)
      Full Publication:
      The Quarterly Journal of Mathematics, Volume os-5, Issue 1,
      January 1934, Pages 73–80

      This is not just in casual speech; people even cite it wrongly.

  7. Every time I start asking questions about a footnote, I get this horrible feeling, because I know –I know– that if I dig I’ll lose a shit ton of time, and will probably only be more cynical in the end.

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