PDF Indiscrete Thoughts Modern Birkhäuser Classics GianCarlo Rota Fabrizio Palombi 9780817647803 Books
Download As PDF : Indiscrete Thoughts Modern Birkhäuser Classics GianCarlo Rota Fabrizio Palombi 9780817647803 Books
Indiscrete Thoughts gives a glimpse into a world that has seldom been described - that of science and technology as seen through the eyes of a mathematician. The era covered by this book, 1950 to 1990, was surely one of the golden ages of science and of the American university. Cherished myths are debunked along the way as Gian-Carlo Rota takes pleasure in portraying, warts and all, some of the great scientific personalities of the period. Rota is not afraid of controversy. Some readers may even consider these essays indiscreet. This beautifully written book is destined to become an instant classic and the subject of debate for decades to come.
PDF Indiscrete Thoughts Modern Birkhäuser Classics GianCarlo Rota Fabrizio Palombi 9780817647803 Books
"I enjoyed it just for the account of Prof. Rota's encounters with other mathematicians. It is a very good autobiography of a mathematician for those who are inclined to these."
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Indiscrete Thoughts Modern Birkhäuser Classics GianCarlo Rota Fabrizio Palombi 9780817647803 Books Reviews :
Indiscrete Thoughts Modern Birkhäuser Classics GianCarlo Rota Fabrizio Palombi 9780817647803 Books Reviews
- Rota is a specialist in a branch of mathematics called combinatorics and he is also a philosopher of the Husserl school (phenomenology). The book has three main parts. In the first he tells us about the life of famous mathematicians in a candid way their greatness as professionals and some of their failures as human beings. It reads as easily as a novel. The second part is more dense and the reading is more difficult, but there are some interesting ideas. For example he equates beauty in mathematics with enlightment. Enlightment is what distinguishes mathematics from puzzles. He is also critical of how math is frequently taught and presented in professional papers. Take, for example, the ergodic theorem. See its description in Wikipedia. It is hard to make any sense of it until you click the external link at the end of the page What is Ergodicity? I would call that an example of enlightment. Rota also warns mathematicians that if they don't make the effort to make themselves more understood by the educated public (much as cosmologists, for example, do), mathematics could have trouble in finding public funds. No doubt some potentially good mathematicians never became one because of the way math has been taught (remember that fashion called modern math, i.e, set theory).
The third part is a collection of "lessons" and short thoughts (one chapter is called "A Mathematician's Gossip") and the book ends with reviews of some books.
To sum up Rota reveals us how mathematics is carried out by the professionals and discusses some philosophical aspects of mathematics a subject that has not yet been explored too deeply and that is also the subject of another book "What is Mathematics Really" , by his friend Reuben Hersh who is, by the way, the author of the very good foreword of Rota's book. - He's quite a character, and one with a lot of interesting stories.
- I enjoyed it just for the account of Prof. Rota's encounters with other mathematicians. It is a very good autobiography of a mathematician for those who are inclined to these.
- great
- Among the many beautiful articles collected here, I am restricting my comments to the one on "The Phenomenology of Mathematical Beauty," for I think it is a pity that Rota's valuable contribution to this neglected topic has been left unmined.
The purpose of a phenomenology is to stimulate a synthesis. So first I present my synthesis and then I test it against Rota's phenomenology.
My definition of mathematical beauty
A beautiful proof is one which the mind can play its way through with a natural grace, as if it was created for this very purpose. We grasp a beautiful proof as a whole, yet see the rôle of every detail; it is vivid and transparent; we are its masters and its connoisseurs, like a conductor directing a symphony. I shall call this type of proof "cognisable" for short.
An ugly proof resorts to computations, algorithms, symbol manipulation, ad hoc steps, trial-and-error, enumeration of cases, and various other forms of technicalities. The mind can neither predict the course nor grasp the whole; it is forced to cope with extra-cognitive contingencies. The mind's task is menial it can only grasp one step at a time, checking it for logical adequacy. It can become convinced of the results but it is not happy since all the work was being done outside of it. Our memory is strained, our mind distorted to accommodate some artificial logic, like a student struggling with German grammar. I shall call this type of proof "noncognisable."
I found it easiest to express this delineation in terms of beautiful and ugly _proofs_, but little importance attaches to this fact. For cognitively speaking there is no sharp distinction between theorem and proof, or the between proof and interplay of ideas, or between mathematics and science. What brings us aesthetic satisfaction is an idea that enlarges the cognisable domain, whether it be a theorem, a proof technique, or an empirical discovery. Some common classes of such ideas are cognisable theorems and proofs (e.g., infinitude of primes, Desargues's theorem); new cognisable entities (e.g., ideal numbers, points at infinity); connections between cognisable entities (e.g., the prime number theorem; Euler's formula e^i(pi)'=-1; Bernoulli's theorems on the logarithmic spiral; fundamental theorem of calculus); new cognisable ways of thinking about cognisable entities (e.g., classification of conics, cubics, surfaces); ways of making the noncognisable cognisable (e.g., Riemann surfaces, Galois theory).
Now to compare with Rota. First some quotations from Rota that fits the above very well.
"A beautiful theorem may not be blessed with an equally beautiful proof ... It is however probably impossible to find instances of beautiful proofs of theorems that are not thought to be beautiful."
This follows immediately from my definition on reading "cognisable" for "beautiful."
"We think back to instances of mathematical beauty as if they had been perceived by an instantaneous realization, in a moment of truth, like a light-bulb suddenly being lit. All the effort that went in understanding the proof of a beautiful theorem, all the background material that is needed if the statement is to make any sense, all the difficulties we met in following an intricate sequence of logical inferences, all these features disappear once we become aware of the beauty of a mathematical theorem, and what will remain in our memory of our process of learning is the image of an instant flash of insight, of a sudden light in the darkness."
Beauty is precisely the elevation of a proof from "an intricate sequence of logical inferences" to a cognitively coherent whole. It is precisely when our cognitive faculties are capable of performing this transformations that we experience beauty.
"The most common instance of beauty in mathematics is a brilliant step in an otherwise undistinguished proof. ... [But] Most frequently, the _word_ 'beautiful' is applied to theorems. In the second place we find proofs; ... Axiom systems can also be beautiful. ... [There can also be] beauty in a definition."
That beauty does not attach to a specific type of object is explained by the fact that the distinctions between them are, while formally strong, cognitively weak.
"The beauty of a mathematical theory is independent of the aesthetic qualities, or the lack of them, of any of the theory's rigorous expositions." "Mathematical beauty and mathematical elegance are distinct phenomena ... Mathematical elegance has to do with the presentation of mathematics, and only tangentially does it relate to content."
This is because printed proofs are different from cognitive processes.
Now I turn to some more specific topics. The first concerns inevitability as a possible defining property of beauty.
"We say that a proof is beautiful when such a proof finally gives away the secret of the theorem, when it leads us to perceive the actual, not the logical inevitability of the statement that is being proved."
This is exactly what my theory says if one reads "cognisable" and "noncognisable" for "actual" and "logical" respectively. Could these terms mean something else?
One possible alternative sense of the terms could be that "actual" means something like "straight to the conclusion," whereas "logical" means that the result follows as a corollary of seemingly unrelated considerations. But I do not think "actuality" in this sense confers any aesthetic advantage. Consider for example the standard algebraic derivation of the quadratic equation formula, which shows the "actual inevitability" in this sense (i.e., the proof does not go on a detour) but is nevertheless entirely void of beauty. The sense in which this proof fails to be inevitable, I say, is this its steps do not inevitably suggest themselves to the mind. Which brings us back to the view that the interesting dichotomy is that in terms of cognisability, not that in terms of inevitability.
Another possibility is that the distinction between "logical" and "actual" is that between verification and discovery. The idea that mere verification is unattractive is surely sound (nobody likes to verify a solution formula for the general quadratic equation or a particular type of differential equation by "plugging it in") but this just what my theory says. Likewise, I have said that beauty is the enlargement of the cognitive domain, which in a sense seems synonymous with discovery. So something sharper is needed if we are to differentiate Rota's claim from mine along these lines. This can be done by taking "logical" to mean that the proof is designed on the assumption that the theorem is true. On this view, to establish the "actual" inevitability of the theorem the proof must start with an open mind, as it were, and lead us to discover the theorem. But I do not think that this is a credible aesthetic criterion. For a counterexample we may refer to Rota "An example of mathematical beauty upon which all mathematicians agree is Picard's theorem, asserting that an entire function of a complex variable takes all values with at most two exceptions. The limpid statement of this theorem is fully matched by the beauty of the five-line proof provided by Picard himself." The proof proceeds by showing that a function that omits three values is a constant. Since we would never have thought of considering such a function if we did not know the theorem, the proof is undoubtedly closer to verification than discovery.
Another possible defining property of beauty is enlightenment. Rota himself suggests this characterisation of beauty
"Mathematicians may say that a theorem is beautiful when they really mean to say that the theorem is enlightening. ... All talk of mathematical beauty is a copout from confronting the logic of enlightenment."
Again this is precisely my view, provided that one understands enlightenment as cognisability. Rota does not define enlightenment; its meaning must be inferred from a number of different statements. Many of these I have already quoted above and we have seen that Rota's views in this regard appear similar to mine. Once, however, he offers a possibly different characterisation of enlightenment
"We acknowledge a theorem's beauty when we see how the theorem 'fits' in its place, how it sheds light around itself, like a Lichtung, a clearing in the woods."
Of course I could not agree more that this is one possible way to bring about beauty; and I think this is how Rota meant it too, though this is not entirely clear from the context. But let us consider the possibility that what is meant here is rather that it is a theorems connectivity and usefulness that confers beauty. But it seems to me that there are enough beautiful theorems unexceptional in this regard (e.g., Bernoulli's theorems on the logarithmic spiral, the theorem that there are five regular polyhedra) that a criterion weak enough to include them would be too weak to have any teeth. But perhaps Rota meant instead that it is the ability of a theorem to lighten the burden of everyday research that confers beauty. But this does not seem very plausible. Rota notes for example that "the theory of differential equations, both ordinary and partial, is fraught with distinctively ugly theorems and with awkward arguments," whose ugliness surely no one would attribute to lack of usefulness in this sense.
Another topic to consider is whether beauty is objective or subjective. My theory predicts that aesthetic judgements should be universal among everyone to the (presumably very considerable) extent that human cognitive endowments are so, except insofar as judgement is precluded by lack of study and training. "Appreciation of mathematical beauty requires thorough familiarity with a mathematical theory," as Rota says (e.g., one cannot appreciate the prime number theorem without knowing something about logarithms). But my theory predicts that there should be no great differences in aesthetic judgement that cannot be so explained. This at first seems to agree with Rota's description
"Given the historical period and the context, one finds substantial agreement among mathematicians as to which mathematics is to be regarded as beautiful. ... In other words, the beauty of a piece of mathematics does not consist merely in subjective feelings experienced by an observing mathematician. The beauty of a theorem is a property of the theorem, on a par with its truth or falsehood. Mathematical beauty and mathematical truth are not to be distinguished by labeling the first as a subjective phenomenon and the second as an objective phenomenon. Both the truth of a theorem and its beauty are equally objective qualities, equally observable characteristics of a piece of mathematics which are equally shared and agreed upon by the community of mathematicians."
However, Rota's restriction to "the historical period and the context" separates his view from mine
"The rise and fall of synthetic geometry ... shows that the beauty of a piece of mathematics is strongly dependent upon schools and periods of history. ... Nowadays, synthetic geometry is a field largely cultivated by historians, and an average mathematicians ignores even the main results of this once flourishing branch of mathematics. ... In retrospect, one wonders what all the fuss was about."
I do not think this disproves my theory. The reason why is suggested in a related passage where Rota uses the same example to make a point about the difference in aesthetic judgement between mathematicians and non-mathematicians
"Theories that mathematicians consider to be beautiful seldom agree with the mathematics thought to be beautiful by the educated public. For example, classical Euclidean geometry is often proposed by non-mathematicians as a paradigm of a beautiful mathematical theory, but I have not heard it classified as such by professional mathematicians, despite occasional pressure upon mathematicians to do so on the part of historians of mathematics."
Thus Rota has established only that aesthetic judgements _among mathematicians_ change over time, not aesthetic judgements _among humans_, which is the relevant class for my theory. Such a change can be due to either a change in the definition of "beauty," or a change in the definition of "mathematician." All evidence plainly points to the latter. A person enticed by the beauty of synthetic geometry was a mathematician in antiquity and part of the "educated public" today. The only thing that has changed is his professional affiliation. - `Funny stories about mathematicians!' An oxymoron, you might counter. Need I say that the title of the book is a pun. If you aren't from math, you might say that it is an inside joke. Pick up the book! If you are anything like me, you will not be able to put it down!
And I think you will not be disappointed; even if you might initially have misgivings.
The book is funny. If you don't believe me, give it a try, and judge for yourself. I had one of the best laughs of the year. The book is also unique in several ways; autobiographical in many ways, and written by an outstanding scientist; one with a rare talent for writing, for making observations about human nature, and for interpersonal skills. Had Rota not turned to math, he might well have become a novelist.
A number of the protagonists in the book are the famous math professors Rota encountered when he was an undergraduate in Princeton in the early fifties; that was also the period of another illustrious mathematician, John Nash [later to become a Nobel Laureate, and the subject of a bestseller, and a movie; `A Beautiful Mind'].
The stories I enjoyed the most in Rota's little book was those about Alonzo Church, a pioneer in logic; William Feller, one of the founders of modern probability theory; Solomon Lefschetz (of topology), to mention only some. But you will likely select your own favorites from Rota's illustrious gallery. Rota paints his subjects with a mix of colors humor, respect, love, insight in the human soul, wisdom, and personal reflection. What is charming and amusing is to observe thru the eyes of the then young and impressionable undergraduate student Gian-Carlo Rota, that the famous scientists shared personal weaknesses, and failed human relationships, with the rest of us.
Reviewed by Palle Jorgensen, November, 2004.
