Book review: Indiscrete Thoughts by Gian-Carlo Rota

August 14, 2018

First published in 1997, Gian-Carlo Rota’s Indiscrete Thoughts is now a minor classic on the culture of mathematics.1 It is witty and irreverent and difficult not to enjoy. It is also often thoughtful and insightful. Nonetheless, I think its more overtly philosophical parts are seriously flawed. Here I will try to both summarize and critically review this distinctive book.

Indiscrete Thoughts is an anthology of Rota’s essays, philosophy papers, and miscellany, divided into three parts. I will discuss the parts separately because they are quite different in format. But first I will outline the general themes of the book, following Rota’s own Introduction.

The very first sentence of the Introduction is: “The truth offends.” Combined with the title, it creates an accurate impression of the tone of the book, which is terse, blunt, and unapologetic. Rota goes on to explain that the book’s purpose is to explode some persistent “myths” about mathematics and science:2

  1. The myth of the monolithic personality”: Great mathematicians and scientists must be great in all dimensions of their lives.

  2. The myth of reductionism”: All mathematics is reducible to logic and set theory, all science is reducible to physics, and so on.

  3. The zero-one myth”: A concept is either completely precise and universally valid, or else it is completely invalid.3 An example from Rota: “If you don’t believe that everything can be explained in terms of atoms and molecules, you must be an irrationalist.”

The first myth is strongly connected with Part I and the second with Part II, but all three myths are criticized, directly or indirectly, throughout the book.

Part I. Persons and Places

The first Part of Indiscrete Thoughts is its most biographical. Rota reminiscences about fellow mathematicians from throughout his career, first as a math major at Princeton in the early 1950s; next as a graduate student at Yale, advised by functional analyst Jack Schwartz; then as a professor at MIT and a consultant at Los Alamos National Lab, where he became close friends with Stanislaw Ulam. Rota paints portraits of such famous mathematicians as Emil Artin, Alonzo Church, Nelson Dunford, William Feller, Josiah Gibbs, Solomon Lefschetz, Jack Schwartz, and Stanislaw Ulam. The portraits range in size from short vignettes to full essays.

In these biographical sketches, Rota refuses to sanitize his memories in the interest of diplomacy, presenting a far more “human” picture of mathematics than is usually found in the textbooks. The impression created is often, though not always, unflattering. I have not attempted to determine whether what Rota writes about his old professors is accurate or fair. I cannot say whether Alonzo Church possessed almost inhuman detachment and attention to detail, whether Solomon Lefschetz was an arrogant bigot, whether William Feller combined crippling insecurity with occasional outbursts of apoplectic rage, or whether Emil Artin cultivated a cult of personality among his students. What I can say is that Rota’s profiles strike me as being true to life. I expect that anyone acquainted with traditional mathematical culture will agree. From my own undergraduate education, I recall with amusement professors matching Rota’s remarkably well in personality (if not necessarily in stature).

Of the Princeton mathematicians, Alonzo Church gets the most favorable treatment.4 Church is now recognized as a preeminent logician of the 20th century. He invented the lambda calculus and proved (independently of Turing) that first-order logic is undecidable. Rota recounts how Church was abused and disrespected by his Princeton colleagues, who considered mathematical logic a subject unworthy of serious study. He quotes John Kemeny, a contemporary of Church, as saying:

There is no reason why a great mathematician should not also be a great bigot. Look at your teachers in Fine Hall, at how they treat one of the greatest living mathematicians, Alonzo Church.5

This pronouncement made a great impact on the young Rota. Today, the situation seems not much improved, with logic better represented in computer science and philosophy departments than in mathematics departments.

Later in Part I, Rota expands on the curious gradations of prestige in mathematics by contrasting the development of two branches of algebra, which he calls “Algebra One” and “Algebra Two.” Algebra One is motivated by the needs of algebraic geometry and algebraic number theory. It includes commutative algebra, homological algebra, Galois theory, and group cohomology. Algebra Two is more eclectic, being motivated first by invariant theory and then by combinatorics and logic. It encompasses lattice theory, Hopf algebra, tensor algebra, universal algebra, and, more recently, category theory.6 Now, as fifty years ago, Algebra One enjoys far greater prestige in mathematics than Algebra Two. Like Rota, I cannot discern any good reason why this should be the case.

Even if they happen to be unfair in some respects, Rota’s biographical sketches serve their purpose well enough by demolishing the myth that excellent mathematicians and scientists are necessarily excellent human beings (“the myth of monolothic personality”). Perhaps you think that observation too obvious to need proof. Still, I think there is an unhealthy tendency towards hero worship in scientific culture. It cannot hurt to remind ourselves, from time to time, that some people are better remembered for their sharpness of mind than their generosity of spirit. Gian-Carlo Rota reminds us in a very vivid way.

Readers bored by biography will still find interesting tidbits in Part I. Here is one that resonated with me. While telling the story of Hermann Grassmann and the invention of exterior algebra, Rota draws a useful distinction between definition and description—for example, between defining the real numbers, say by Dedekind cuts or as a complete ordered field, and describing the real numbers, say as a continuum or as a line in the Euclidean plane. Definitions are the official stuff of mathematics, but descriptions are essential because they embody our pre-theoretical intuitions. Historically, descriptions precede definitions. Tensors were described as quantities that transform in certain ways long before they were rigorously defined as multilinear maps or via a universal property. For Rota, descriptions are more fundamental than definitions, even if they are less rigorous, because “mathematics can get by without definitions but not without descriptions” (as proved by physicists and the better part of mathematical history), and because descriptions motivate and ultimately determine definitions, not the other way around.7 On the basis of these arguments, Rota concludes that description is ineliminable from mathematics, or at least from the human understanding of mathematics. We see here our first instance of Rota’s irreductionism (“the myth of reductionism”). We also see a prefigurement of Rota’s hostility to formal analysis, on much greater display in Part II.

Part II. Philosophy: A Minority View

Part II is the most overtly philosophical. In these essays, Rota applies the method of classical phenomenology, as represented by Husserl, Heidegger, and Sartre, to the philosophy of mathematics. He also attacks what he sees as the reductionist tendencies of analytic philosophers, especially in the foundations of mathematics.

One might expect a person of mathematical mind to philosophize in the careful and precise style of mathematics. That would be naive, as Rota’s example shows. Rota not only fails to practice that style of philosophy, but attacks it with great causticness. Of the philosophers I know, I find Rota’s style most reminiscent of Nietzsche’s. Like Nietzsche, he is a shock jock. He often makes statements that cannot be literally true—and that he surely knows to be untrue—but are intended to shock the reader into some realization, in the manner of a Zen koan. Also like Nietzsche, he has a habit of mistaking insults for philosophical arguments.

The first essay in this Part is called “The pernicous influence of mathematics upon philosophy.”8 As the title suggests, it is a protestation against the “mathematization” of philosophy that Rota takes to begin with the early 20th century analytic philosophers. According to Rota, the “mathematizing” philosophers, having grown envious of the definiteness of mathematics and its success at solving problems, try to imitate its method in philosophy. Philosophical problems are, however, intrinsically imprecise. They resist mathematically rigorous formulation, much less rigorous solution. When the mathematizing philosophers fail—inevitably—to solve philosophical problems according to this inappropriate standard of rigor, they conclude that the problems were meaningless all along. As a result, all the great philosophical problems from throughout history are banished from consideration.

This argument is not merely mistaken, it is actively misleading. Although he does not explicitly say so, Rota clearly has in mind the logical positivists of the Vienna Circle, such as Carnap, Hempel, and the early Wittgenstein. It is true that the logical positivists advocated a verificationist theory of meaning, according to which many classical problems of philosophy are meaningless.9 However, this theory was never very popular, and even the positivists eventually weakened it. Some, like Wittgenstein, rejected it entirely. Today virtually no one believes it. But that is not how Rota tells it:

This is not an exaggeration. The classical problems of philosophy have become forbidden topics in many philosophy departments. The mere mention of one such problem by a graduate student or by a junior colleague will result in raised eyebrows followed by severe penalties.10

In fact it is a grotesque exaggeration. Contemporary analytic philosophers are happily engaged with all the classical topics of philosophy, from metaphysics and epistemology to ethics and politics. This can be verified by a cursory glance at the course offerings or CV’s of philosophers at any major American or British university. Rota may not like the style of this work, but that is another matter.

In any case, Rota’s criticism of formal philosophy is wrong in a deeper, more important way. I grant that the most slippery concepts of philosophy will probably never admit mathematical formalizations that are fully satisfactory. But it does not follow that formal analysis can shed no light on philosophical concepts. Ironically, in prosecuting his case against the myth of reductionism, Rota falls victim to his own zero-one myth. A formal theory need not be perfectly accurate to be useful. In science and statistics this statement is a banality. It is equally true in philosophy.

Consider the concept of causality, among the most vexing in the history of philosophy. More progress has been made in understanding causality in the last several decades than in the whole preceding two thousand years. How is that possible? It is possible because a group of computer scientists and philosophers11 have subjected to formal analysis a concept once regarded as unanalyzable. They have not resolved every philosophical problem about causality, but they have clarified many issues. Their insights are now trickling into practical scientific and statistical procedures. The brand of philosophy that Rota advocates is incapable of producing such knowledge.

After a short essay on “Philosophy and computer science,” Rota turns to his main topic, the phenomenology of mathematics, in a series of three essays entitled “The phenomenology of mathematical truth,”12 “The phenomenology of mathematical beauty,”13 and “The phenomenology of mathematical proof.”14

In the first of the series, “The phenomenology of mathematical truth,” Rota contrasts two conceptions of mathematical truth. On the first conception, mathematical truths are like natural facts. A mathematical theorem records a regularity of the mathematical world, just as a law of nature records a regularity of the physical world. Mathematical truths are mind-independent. They are discovered by observation and experimentation, even if they are later proved by formal reasoning. On the second conception, mathematical theorems are proofs. Though they may be very formidable to lowly humans, they are analytic trivialities, logically equivalent to any tautology, like . As Rota memorably puts it:

Every mathematical theorem is eventually proved trivial. The mathematician’s ideal of truth is triviality, and the community of mathematicians will not cease its beaver-like work on a newly discovered result until it has shown to everyone’s satisfaction that all difficulties in the early proofs were spurious, and only an analytic triviality is to be found at the end of the road.15

To Rota, it is a paradox that mathematical theorems are, on the one hand, mind-independent truths, apprehended by humans only with great difficulty, if at all, but are, on the other hand, analytic statements, trivial in the strict logical sense. His solution to this problem is to expand the paradox from mathematics to all of science, declaring that “science may be defined as the transformation of synthetic facts of nature into analytic statements of reason.”16 He calls this “the ex universali argument” and attributes it to Husserl.

I cannot see how drastically expanding the scope of a paradox should be counted as a philosophical success. More importantly, I cannot see what is paradoxical about Rota’s picture of mathematical truth in the first place. Most theories of truth, mathematical or scientific, distinguish between truth and verification. A statement may be true, in the analytic or synthetic sense, but very difficult to verify, because the proof is too long, or your computer is too slow, or the experimental apparatus is too intricate to build, or whatever else. This is a banality, not a paradox. There are now whole fields of mathematics, such as computability theory and complexity theory, dedicated to understanding the many ways an analytic statement can be true but difficult to verify. Of course, this would not be news to Rota. That makes it all the harder for me to understand why he does not anticipate and respond to this obvious objection.

I could continue to critique Rota’s philosophical articles along these lines, but I do not have the energy. Instead, I will try to sum up what I see as the strengths and weaknesses of his approach. First, Rota is always attentive to the psychological and sociological dimensions of mathematics. That is admirable, because these aspects are often neglected in writing about mathematics. But he then goes too far, by extending these insights to philosophical conclusions through careless reasoning. So, for example, he moves sneakily from a psychological observation about the difficulty of apprehending mathematics to an epistemological problem about mathematical truth, without bothering to explain how this is justified. Second, Rota attacks formal philosophy in general and, within the philosophy of mathematics, the emphasis on foundations of math in particular. I have already argued that this is unfair. However, one important point may be found underneath all the invective. Rota rightly observes that by focusing almost exclusively on foundations (logic, set theory, and elementary number theory), philosophers of mathematics have neglected the remarkable and diverse accomplishments of contemporary mathematics. More recently, this cause has been taken up in several books, like David Corfield’s Towards a Philosophy of Real Mathematics17 and Fernando Zalamea’s Synthetic Philosophy of Contemporary Mathematics.18 I hope to review at least one of these books in a future post.

Part III. Readings and Comments

Part III, the shortest and most eclectic, returns to more lighthearted fare. The first two chapters, “Ten lessons I wish I had been taught”19 and “Ten lessons for the survival of a mathematics department,”20 offer sensible advice to young mathematicians. The last chapter is a collection of book reviews, short and long. Of these I especially enjoyed “Professor Neanderthal’s World,” a warm review of T. Y. Lam’s First Course in Noncommutative Rings21 that Rota hijacks to mock the proverbial Professor Neanderthal who reflexively opposes all expository work in mathematics. The importance of mathematical exposition is a theme throughout Part III. I wholeheartedly endorse it.

The heart of Part III is a collection of aphorisms by Rota, called “A mathematician’s gossip.” The aphorisms are mainly, but not exclusively, about the content and culture of mathematics. This chapter is the ultimate distillation of Rota’s style: gnomic, pithy, and provocative. I am reminded again of Nietzsche, now of the “Epigrams and Interludes” in Beyond Good and Evil. If you read nothing else in Indiscrete Thoughts, read the aphorisms. You will surely find something worthwhile and you will get a good sense of whether you will enjoy reading the rest of the book.

Rather than try to summarize the aphorisms, whose topics range widely, I will just list a few of my favorites. On the difficulty of mathematical exposition:

A common error of judgment among mathematicians is the confusion between telling the truth and giving a logically correct presentation. The two objectives are antithetical and hard to reconcile… The truth of any piece of mathematical writing consists of realizing what the author is “up to”; it is the tradition of mathematics to do whatever it takes to avoid giving away this secret.22

On the psychological need for foundations of mathematics:

When we ask for foundations of mathematics, we must first look for the unstated wishes that motivate our questions. When you search into the Western mind, you discover the craving that all things should be reduced to one, that the laws of nature should all be consequences of one law, that all principles should be reduced to one principle. It is a great Jewish idea. One God, one this, one that, one everything. We want foundations because we want oneness.23

Rota is not afraid to crack a joke:

Textbooks in algebraic geometry should be written by Italians and corrected by Germans.24

Nor is he afraid to make pronouncements on the status or the future of whole mathematical subjects:

Nowadays, if you wish to learn logic, you go to the computer science shelves at the bookstore. What lately goes by the name of logic has painted itself into a number of unpleasant corners—set theory, large cardinals, independence proofs—far removed from the logic that will endure.25

Twenty years after the publication of Indiscrete Thoughts, it would be interesting to evaluate Rota’s many predictions. And it would be interesting to evaluate them again after fifty years. I am mostly not competent to do so, although I am inclined to agree with Rota about logic. The most important developments in logic today are not happening within pure mathematics.

Rota does not confine his aphorisms to mathematics, sometimes venturing into adjacent disciplines like physics, philosophy, and computer science. He even makes a few comments about artificial intelligence (or “pattern recognition,” as it was then commonly called).26 As a researcher in this subject, I found these very enjoyable. First, Rota says:

Of some subjects we may say they exist, of others, that we wish they existed. Cluster analysis is one of the latter.27

This statement is as true today as it was twenty years ago. Although there have been attempts to put cluster analysis on a firmer footing (say by reducing it to density estimation), it remains a poorly understood area of machine learning. The same is true of unsupervised learning generally. Developing a convincing conceptual foundation for unsupervised learning—making a “subject” out of it—is a problem for the future. I suspect Rota was unoptimistic about this prospect, because in the next aphorism he says:

Pattern recognition is big business today. Too bad that none of the self-styled specialists in the subject—let us charitably admit it is a subject—know mathematics, even those who know how to read and write.27

On the accuracy of this statement, then or now, I will not dare to comment.


Indiscrete Thoughts is often fascinating, sometimes frustrating, and always entertaining. I would recommend it, though not without reservations, to anyone seeking a glimpse into the peculiar world of mathematicians. Rota is at his best when describing the psychological, social, and cultural aspects of doing mathematics. His attempts to transform these observations into a coherent philosophy of mathematics are not very successful. The same temperament that makes him an engaging essayist—bold, aphoristic, impatient, caustic—makes him a poor philosopher. But that does not detract too much from the book, which is mainly very enjoyable.

  1. Gian-Carlo Rota, 1997. Indiscrete Thoughts. Springer Science and Business Media. DOI

  2. Ibid., pp. xx–xxi. 

  3. In his defense of “critical scientific realism,” Ilkka Niinuluoto attributes a similar fallacy, which he calls the all-or-nothing fallacy, to scientific antirealists. See: Niinuluoto, 1999. Critical Scientific Realism. Oxford University Press. pp. 81, 93, 293. 

  4. Although Rota cannot resist saying of Church that “he looked like a cross between a panda and a large owl,” a comment I find inexplicably funny, if not exactly charitable (Rota, 1997, p. 4). 

  5. Ibid., p. 7. 

  6. Rota locates categories and topoi within Algebra One, which may be true historically, inasmuch as they were originally motivated by algebraic geometry and algebraic topology, but it appears to me that the modern study of categories and topoi in their own right is much closer in spirit (and in prestige) to Algebra Two. Consider, for example, that universal algebra has been swallowed whole by category theory and now goes by the name “categorical logic.” Or that Hopf algebras and tensor categories are closely connected and often presented together, as in: Ross Street, 2007. Quantum Groups: A Path to Current Algebra. Cambridge University Press. 

  7. Rota, 1997. p. 49. 

  8. First published as: Gian-Carlo Rota, 1991. The pernicious influence of mathematics upon philosophy. Synthese, Vol. 88, No. 2, pp. 165-178. DOI

  9. Even so, Rota’s treatment of the positivists is unfair. Although the verificationist criterion of meaning is too strong, and although it is fashionable nowadays to ridicule it, the logical positivists made important contributions to the philosophy of science, paving the way for our modern understanding of scientific explanations and theories. The spirit of positivism is alive and well today. 

  10. Rota, 1997. p. 98. 

  11. Prominent members of this group include Judea Pearl, at UCLA, and Clark Glymour, Richard Scheines, and Peter Spirtes, at the CMU philosophy department. For an introduction to their work, see: Pearl, Glymour, and Jewell, 2016. Causal Inference in Statistics: A Primer. John Wiley & Sons. 

  12. First published as: Gian-Carlo Rota, 1991. The concept of mathematical truth. The Review of Metaphysics. Vol. 44, No. 3. pp. 483-494. JSTOR

  13. Also published as: Gian-Carlo Rota, 1997. The phenomenology of mathematical beauty. Synthese. Vol. 111, No. 2. pp. 172-182. DOI

  14. Also published as: Gian-Carlo Rota, 1997. The phenomenology of mathematical proof. Synthese. Vol. 111, No. 2. pp. 183-196. DOI

  15. Rota, 1997. p. 118. 

  16. Ibid., p. 119. 

  17. David Corfield, 2003. Towards a Philosophy of Real Mathematics. Cambrige University Press. DOI

  18. Fernando Zalamea, 2012. Synthetic Philosophy of Contemporary Mathematics. Translated by Zachary Luke Fraser. Urbanomic & Sequence Press. 

  19. Also published as: Gian-Carlo Rota, 1997. Ten lessons I wish I had been taught. Notices of the American Mathematical Society. Vol. 44, No. 1. pp. 22-25. Online

  20. First published as: Gian-Carlo Rota, 1992. Ten rules for the survival of a mathematics department. MAA Focus. Vol. 12, No. 6. pp. 13-14. Online

  21. T.Y. Lam, 1991. A First Course in Noncommutative Rings. Springer. DOI. Second edition published in 2001. 

  22. Rota, 1997. p. 215. 

  23. Ibid., p. 218. 

  24. Ibid., p. 232. 

  25. Ibid., p. 218. 

  26. The incessant rebranding of AI is an important aspect of the hype cycle. Within the statistical paradigm of AI, beginning with the rediscovery of statistics by AI researchers in 1980s, statistical reasoning in AI has been successively rebranded as “pattern recognition,” then “machine learning,” then “deep learning,” and, now that the phrase’s taint has worn off, simply “artificial intelligence.” 

  27. Ibid., p. 229.