I am a huge fan of Gian-Carlo Rota, who has been said to be the founding father of modern algebraic combinatorics. (He is also my. Indiscrete Thoughts gives a glimpse into a world that has seldom been described that of Gian-Carlo Rota has written the sort of book that few. Indiscrete thoughts. Front Cover. Gian-Carlo Rota. Birkhäuser, – Mathematics – pages Bibliographic information. QR code for Indiscrete thoughts.
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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, towas 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 thoughrs indiscreet.
This beautifully written book is destined to become an instant classic and the subject of debate for decades to come. It is pages of Rota calling it like he sees it Readers are bound to find his observations amusing if not insightful. Gian-Carlo Rota has written the sort of book that few mathematicians could write. What will appeal immediately to anyone with an interest in research mathematics are the stories he tells about the practice of modern mathematics.
It has aged very well, and richly deserves its inclusion in this series. Indiscrete Thoughts gives a glimpse into a world that has seldom been described, that of science rhoughts technology as seen through the eyes of a mathematician.
The era covered by this book, towas surely one of the golden ages of science as well as of the American university. After the publication of the essay “The Pernicious Influence of Mathematics upon Philosophy” reprinted six times in five languages the author was blacklisted in analytical philosophy circles. Indiscrete Thoughts should become an instant classic and the subject of debate for decades to come. Would you like to tell us about a lower price?
Indiscrete thoughts – Gian-Carlo Rota – Google Books
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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: 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.
Enlightment is what distinguishes mathematics from puzzles. He is also critical of roota 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: I would call that an example of enlightment.
Rota also warns mathematicians that if they don’t make the effort to make indidcrete more understood by the educated public much as cosmologists, for example, domathematics 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. 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. 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: He’s quite a character, and one with a lot of interesting stories.
Kindle Edition Verified Purchase. 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. One person found this helpful. 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 indicsrete 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. 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 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. For cognitively speaking there is no sharp distinction between theorem and inidscrete, 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: Now to compare with Rota. First some quotations from Rota that fits the above very well.
It is however probably impossible to find instances of beautiful proofs of theorems that are not thought to be beautiful. 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 ihdiscrete an instant flash of insight, of a sudden light in the darkness.
It is precisely when our cognitive faculties are capable of performing this transformations that we indiscreet beauty. In the second place we find proofs; Axiom systems can also be beautiful.
Mathematical elegance has to do with the presentation of mathematics, and only tangentially does it relate to content. Now I turn to some more specific topics. The first concerns inevitability as a possible defining property of beauty. 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 indjscrete the quadratic equation formula, which indisdrete the “actual inevitability” in this sense i. The sense in which this proof fails to be inevitable, I say, is this: 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 lndiscrete must start with an open mind, as it were, and lead us to discover the theorem.
Rota’s Indiscrete Thoughts
But I do not think indkscrete this is a credible aesthetic criterion. For a counterexample we may refer to Rota: The limpid statement of this theorem is fully matched by the beauty of the five-line proof provided by Picard himself.
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: All talk of mathematical beauty is a copout from confronting the logic of enlightenment.