David Berlinski
Author of A Tour of the Calculus
About the Author
David Berlinski is an essayist, philosopher, and mathematician. He holds a Ph.D. from Princeton and has spent many years in various academic positions across America and abroad
Series
Works by David Berlinski
Associated Works
Uncommon Dissent: Intellectuals Who Find Darwinism Unconvincing (2004) — Contributor — 169 copies, 1 review
Poseidon's Gold | Less Than Meets the Eye | The Mamur Zapt and the Girl in the Nile — Contributor — 1 copy
Tagged
Common Knowledge
- Canonical name
- Berlinski, David
- Birthdate
- 1942-02-05
- Gender
- male
- Education
- Princeton University (PhD - Philosophy)
Columbia College (BA) - Occupations
- professor
mathematician
philosopher - Organizations
- Discovery Institute
Center for Science and Culture
Stanford University
Rutgers University
City University of New York
University of Paris - Relationships
- Berlinski, Claire (daughter)
Berlinski, Mischa (son) - Nationality
- USA
- Birthplace
- New York, New York, USA
- Places of residence
- Paris, France
San Francisco, California, USA - Associated Place (for map)
- USA
Members
Reviews
An Ill-Conceived Practical Joke?
At the time that I ordered this book, I had a natural inclination to be sympathetic with its author, since his reputation indicated that he and I had similar views about politics and the philosophy of science. That only increased my disappointment when this ended up being one of the least enlightening and most annoying books I've ever encountered. If Berlinski is as talented as I'd been led to believe, it's hard not to interpret _Infinite Ascent_ as either show more some sort of practical joke or a rush job to fulfill a contract.
In _Infinite Ascent_, Berlinski has a tendency to wax grandiloquent, using metaphors and similes that serve no evident purpose and are sometimes downright bizarre, as when, for example, he likens sets and their elements to the male anatomy (p. 129). Following this up one page later with Berlinski's fantasy about schoolgirls with "their starched shirt fronts covering their gently heaving bosoms" (p. 130) does nothing to ameliorate concern about the author's tendency to get distracted.
One of Berlinski's running themes is the use of "..." in mathematics to represent the continuation of a pattern. He likes to joke about this so much that he starts inserting these dots in his formulas needlessly, just to get to comment on them. For example, instead of just writing down the (extremely short) formula for subtracting complex numbers (p. 69), he leaves an ellipsis and then states that "the crutch of three dots [covers] the transmogrification of a plus to a minus sign and nothing more."
Some of Berlinski's comments are real head-stratchers: "[The Elements] is very clear, succint as a knife blade. And like every good textbook, it is incomprehensible." (p. 14); "[Exponential functions] mount up inexorably, one reason that they are often used to represent doubling processes in biology, as when undergraduates divide uncontrollably within a Petri dish." (p. 71). Huh?
_Infinite Ascent_ has few formulas or other concrete mathematical details, and what there is is often wrong. The formulas for the solutions to quartic equations of quadratic type are botched (p. 93), roots of equations are confused with zeros of functions (p. 80), inscribed rectangles are described while circumscribed rectangles are drawn (p. 56), and g12*du1*du2 is misidentified as a formula for the infinitesimal distance between the points u1 and u2 (p. 120). The sections on logic are the ones Berlinski handles most competently, but even that has been covered better by many others.
Berlinski thinks that Weierstrass's definition of limit is "infinitely wearisome" (p. 145) and is "promptly forgotten" by mathematicians after they have learned it. I think most analysts would disagree strongly with his opinion, and would classify the definition of limit among those things they couldn't forget if they wanted to. (That Berlinski himself very well might have forgotten it is suggested by his unconventional decision to use the letter delta to represent a *large* index (p. 61) in his definition of the limit of a sequence.)
Berlinski opines that the Fundamental Theorem of Calculus (connecting differentiation to definite integration) is something that "no one at all would expect". On the contrary, I consider it to be eminently plausible. Berlinski also describes the classic math book _Counterexamples in Analysis_ as consisting of "a series of misleading proofs supporting theorems that are not theorems." _Counterexamples in Analysis_ actually contains nothing of the sort. Rather than containing fallacious "proofs" of non-theorems, it contains exactly what its title says it does: Counterexamples (i.e., examples that show why the hypotheses of (true) theorems are necessary and why stronger conclusions are unwarranted). show less
At the time that I ordered this book, I had a natural inclination to be sympathetic with its author, since his reputation indicated that he and I had similar views about politics and the philosophy of science. That only increased my disappointment when this ended up being one of the least enlightening and most annoying books I've ever encountered. If Berlinski is as talented as I'd been led to believe, it's hard not to interpret _Infinite Ascent_ as either show more some sort of practical joke or a rush job to fulfill a contract.
In _Infinite Ascent_, Berlinski has a tendency to wax grandiloquent, using metaphors and similes that serve no evident purpose and are sometimes downright bizarre, as when, for example, he likens sets and their elements to the male anatomy (p. 129). Following this up one page later with Berlinski's fantasy about schoolgirls with "their starched shirt fronts covering their gently heaving bosoms" (p. 130) does nothing to ameliorate concern about the author's tendency to get distracted.
One of Berlinski's running themes is the use of "..." in mathematics to represent the continuation of a pattern. He likes to joke about this so much that he starts inserting these dots in his formulas needlessly, just to get to comment on them. For example, instead of just writing down the (extremely short) formula for subtracting complex numbers (p. 69), he leaves an ellipsis and then states that "the crutch of three dots [covers] the transmogrification of a plus to a minus sign and nothing more."
Some of Berlinski's comments are real head-stratchers: "[The Elements] is very clear, succint as a knife blade. And like every good textbook, it is incomprehensible." (p. 14); "[Exponential functions] mount up inexorably, one reason that they are often used to represent doubling processes in biology, as when undergraduates divide uncontrollably within a Petri dish." (p. 71). Huh?
_Infinite Ascent_ has few formulas or other concrete mathematical details, and what there is is often wrong. The formulas for the solutions to quartic equations of quadratic type are botched (p. 93), roots of equations are confused with zeros of functions (p. 80), inscribed rectangles are described while circumscribed rectangles are drawn (p. 56), and g12*du1*du2 is misidentified as a formula for the infinitesimal distance between the points u1 and u2 (p. 120). The sections on logic are the ones Berlinski handles most competently, but even that has been covered better by many others.
Berlinski thinks that Weierstrass's definition of limit is "infinitely wearisome" (p. 145) and is "promptly forgotten" by mathematicians after they have learned it. I think most analysts would disagree strongly with his opinion, and would classify the definition of limit among those things they couldn't forget if they wanted to. (That Berlinski himself very well might have forgotten it is suggested by his unconventional decision to use the letter delta to represent a *large* index (p. 61) in his definition of the limit of a sequence.)
Berlinski opines that the Fundamental Theorem of Calculus (connecting differentiation to definite integration) is something that "no one at all would expect". On the contrary, I consider it to be eminently plausible. Berlinski also describes the classic math book _Counterexamples in Analysis_ as consisting of "a series of misleading proofs supporting theorems that are not theorems." _Counterexamples in Analysis_ actually contains nothing of the sort. Rather than containing fallacious "proofs" of non-theorems, it contains exactly what its title says it does: Counterexamples (i.e., examples that show why the hypotheses of (true) theorems are necessary and why stronger conclusions are unwarranted). show less
I wanted to like this more than I did... Berlinski weaves together linguistics, computer science, mathematics, statistics, and molecular genetics to come to reveal enlightening connections between them on the nature of life and success of evolution. For instance:
For me a key point:
This conservative navigation through the vast space of possibilities can be modeled probabilistically using techniques applicable to AI-based linguistic modelling and depict a process benefitting from apparent direction as well as partial successes.
What detracts from this is Berlinski is enamored of his airy, peripatetic academic life and insists on inserting vignettes of his career ranging from name dropping to trite scenes to a recurring focus on breasts ("curiously bustless coeds", "humungous knockers", etc.). Chomsky gets special focus for his work in linguistics. Apparently, many of his peers are not impressed and more so since he only knowns one language. These criticisms are presented in a way that seems catty. show less
“Let us contemplate the following experiment,” Schutzenberger said, “the Weizenbaum Experiment.”show more
Joseph Weizenbaum of MIT had just published a book on the misuse of computer theory. Largely because he wished that he had said
many of the things that Weizenbaum had said, Schutzenberger had taken an immense boisterous dislike to Weizenbaum and hence thought it appropriate that our experiment be named after him.
Now mention thus of an experiment may suggest that something or other is about to be executed or performed; in fact, the Weizenbaum Experiment is one of those purely imaginary affairs in which trains of thought are allowed to meander and then merge by anastomosis.
“In life,” Schutzenberger went on thoughtfully, “there are two mathematical structures or spaces. There is a space made up of DNA or the proteins. This is an alphabetic space. Its objects are words. And there is a zoological space. Its objects are organisms. This is a space of alphabetic representatives. We say that in both spaces a natural metric exists – the natural metric – and that evolution proceeds in both spaces according to this natural metric. What is more, there is a mapping between the two spaces. It is this mapping that establishes that my DNA serves to express me.
I nodded and began to take notes.
“Furthermore,” Schuztenberger said, “there is in the space of the nucleic acids or the proteins a probability transition system.”
“This probability transition system – do you have in mind a finite-state Markov process?”
“Yes,” said Schutzenberger dreamily, “a finite-state Markov process.”
Schutzenberger stopped pacing and folded his hands in front of him, the long, curled, tobacco-stained fingers locked.
“We now observe,” Schutzenberger said, “that the probability transition system is roughly in accord with the natural metric. We are speaking only of the alphabetic space, remember.”
I looked up, for the moment unconvinced. “Why?”
“We say that the probability transition system is in accord with the natural metric because the most likely changes in the system are those that transform strings into nearby strings.”
“I don’t see this principle follows from the idea that alphabetic changes are independent.”
“It does not,” Schutzenberger agreed. “It follows from the observation that the probabilistic structure of the alphabetic space is not uniform. Indeed, it is this observation that shows ultimately that life does not comprise an ergodic system.”
This was the sort of lovely lunatic leap that Schutzenberger was forever taking in conversation. I must have looked up with an expression of radiant confusion; Schutzenberger directed a warm, beaming smile into the ambient atmosphere, and went on, untroubled by my lack of confidence.
“Life,” he said, “is conservative. Not everything that can change does change. For the most part, biological strings do not change at all. When they do change, they change in only one position. It is highly unlikely that a given string will change in respect to every position. We do not in life see a strand of DNA change its character at every possible codon in one sudden mutation.”
I caught Schutzenberger’s point.
“Now we need something more,” he said, with the air of a man constructing a wonderful instrument, “a mirror, so to speak. We wish to say with regard to arbitrary strings not only how far apart they are under the natural metric, but how far apart their representatives in the real world are. For this we require an induced metric. Very common in mathematics.”
I stopped writing to look up, and shook my hand to release its cramp.
“So when we talk of strings of DNA or strands of protein,” Schutzenberger went on, “we can talk of the natural distance between them or their induced distance. Two sets of strings may be close under the natural metric and far apart under the induced metric. You know, there is the famous experiment in which chimpanzee and human polypeptides were compared. Simply considered as strings there is virtually no difference between them. Evidently there is some difference between a chimpanzee and a human being.”
We were for the moment both quiet.
“In fact, zoologists often assume that the chimpanzee and the human being are closer than they really are.” Schutzenberger held his own hand in the air, palm outward.
Having grown impatient with his own digressions, Schutzenberger finally said, “Let us now perform the Weizenbaum Experiment. We suppose that we have certain strings of alphabetic objects, and that there is some initial probability distribution defined upon them. That is to say, at the beginning of the experiment, the strings are most likely to be in a certain initial configuration. We also have – are you writing? Good – a probability transition system, one that tells which changes in the strings are probable and which are not.”
For me a key point:
“Life,” he said, “is conservative. Not everything that can change does change. For the most part, biological strings do not change at all. When they do change, they change in only one position. It is highly unlikely that a given string will change in respect to every position. We do not in life see a strand of DNA change its character at every possible codon in one sudden mutation.”
This conservative navigation through the vast space of possibilities can be modeled probabilistically using techniques applicable to AI-based linguistic modelling and depict a process benefitting from apparent direction as well as partial successes.
What detracts from this is Berlinski is enamored of his airy, peripatetic academic life and insists on inserting vignettes of his career ranging from name dropping to trite scenes to a recurring focus on breasts ("curiously bustless coeds", "humungous knockers", etc.). Chomsky gets special focus for his work in linguistics. Apparently, many of his peers are not impressed and more so since he only knowns one language. These criticisms are presented in a way that seems catty. show less
Whose side are you on, anyway?
As someone with a degree in mathematics, I of course know something of Euclid, but mostly for his contributions to number theory (an area of mathematics that Euclid didn't even have a name for). As far as geometry was concerned, I knew of his work, but didn't know much about it. And, of course, it is for geometry that Euclid is primarily known.
So let it be said, for starters, that this is not an introduction to Euclid. You'll learn some things about Euclid's show more system of postulates, but very little about the rules of logic that Euclid used, and only a tiny handful of results are covered.
The first part of the book, in particular, seems to have a peculiar love-hate relationship with its material. Author Berlinski is correct in stating that Euclid made more assumptions than he states -- in effect, that his system of postulates and axioms is incomplete. So what is the solution? The solution is to bring in the additional postulates that are needed. This is never done; Berlinski eventually gets around to David Hilbert and his formalization of Euclid, but we don't learn much beyond that Hilbert did so. We neither learn what Euclid left out nor what we need instead.
I'm also a little bothered that Berlinski just lays into Euclid without considering the matter of language. Euclid wrote in classical Greek, not English, and words in Greek do not automatically mean the same in Greek as in English. Take, as a very elementary example, the verb "to be" (is, are, am, was, were, etc.). In English, the use of this verb it is mandatory, even in the present indicative. In Greek, it is not. So in English, you say "This is geometry." In Greek, you can say "This is geometry" or merely "This geometry" -- or, possibly, depending on context, "is geometry" without "this." And which form you use affects, if not the meaning, at least the emphasis. "Is," in Greek, is a much stronger verb than in English. And "is" can mean an equality, so the exact use of "is" can result in different meanings in Greek and English. There are many, many other places where Greek and English words do not precisely align. It is not fair to go after Euclid's failures based on English translations of Euclid without reference to the Greek. How much this matters I don't know (I assuredly don't have enough Greek to understand Euclid in that language!), but it irked me a lot.
Oh, and by the way, Euclid did not know that the interior angles of a triangle add up to π. Euclid didn't measure angles in radians -- and most people don't, either; we measure angles in degrees. Euclid didn't even know that π was an irrational number, or call it by that name. Measurement in radians, as I understand it, arises from the unit circle, not geometry. Please, Dr. Berlinsky, if you're going to talk about the interior angles of a triangle, either measure them in degrees (which everyone knows) or at least say you're doing measurements in radians.
The latter part of the book is devoted to non-Euclidean geometry -- that is, the geometries that emerge if one decides that Euclid's infamous fifth postulate (the "parallel postulate") is not correct, or at least is not axiomatic. This is a vital result -- the universe is non-Euclidean -- but I felt that this was a bit weak, too. There are at least four ways to address the parallel postulate, which basically says that if you have a line and a point not on the line, there is only one parallel line that passes through the point. One can assume that the parallel postulate is correct, and there is only one parallel line (Euclid's approach). One can assume that parallel lines are more than one (probably infinitely many). One can assume there are none. Or one can just say, "I won't assume any of these; I don't know what a parallel line is." These produce different non-Euclidean geometries. I really wish this had been clearer.
If Berlinski had just done a history of gripes about Euclid, it would have been a useful book. If he had done a real job of giving the information Euclid lacked or didn't understand the need for, it would have been a very useful book. As it is, I feel as if this falls between two stools. I spent far too much time saying, "Tell me more" (and, once in a while, "Tell me less"). Almost never did I feel like what I read was just the right amount. show less
As someone with a degree in mathematics, I of course know something of Euclid, but mostly for his contributions to number theory (an area of mathematics that Euclid didn't even have a name for). As far as geometry was concerned, I knew of his work, but didn't know much about it. And, of course, it is for geometry that Euclid is primarily known.
So let it be said, for starters, that this is not an introduction to Euclid. You'll learn some things about Euclid's show more system of postulates, but very little about the rules of logic that Euclid used, and only a tiny handful of results are covered.
The first part of the book, in particular, seems to have a peculiar love-hate relationship with its material. Author Berlinski is correct in stating that Euclid made more assumptions than he states -- in effect, that his system of postulates and axioms is incomplete. So what is the solution? The solution is to bring in the additional postulates that are needed. This is never done; Berlinski eventually gets around to David Hilbert and his formalization of Euclid, but we don't learn much beyond that Hilbert did so. We neither learn what Euclid left out nor what we need instead.
I'm also a little bothered that Berlinski just lays into Euclid without considering the matter of language. Euclid wrote in classical Greek, not English, and words in Greek do not automatically mean the same in Greek as in English. Take, as a very elementary example, the verb "to be" (is, are, am, was, were, etc.). In English, the use of this verb it is mandatory, even in the present indicative. In Greek, it is not. So in English, you say "This is geometry." In Greek, you can say "This is geometry" or merely "This geometry" -- or, possibly, depending on context, "is geometry" without "this." And which form you use affects, if not the meaning, at least the emphasis. "Is," in Greek, is a much stronger verb than in English. And "is" can mean an equality, so the exact use of "is" can result in different meanings in Greek and English. There are many, many other places where Greek and English words do not precisely align. It is not fair to go after Euclid's failures based on English translations of Euclid without reference to the Greek. How much this matters I don't know (I assuredly don't have enough Greek to understand Euclid in that language!), but it irked me a lot.
Oh, and by the way, Euclid did not know that the interior angles of a triangle add up to π. Euclid didn't measure angles in radians -- and most people don't, either; we measure angles in degrees. Euclid didn't even know that π was an irrational number, or call it by that name. Measurement in radians, as I understand it, arises from the unit circle, not geometry. Please, Dr. Berlinsky, if you're going to talk about the interior angles of a triangle, either measure them in degrees (which everyone knows) or at least say you're doing measurements in radians.
The latter part of the book is devoted to non-Euclidean geometry -- that is, the geometries that emerge if one decides that Euclid's infamous fifth postulate (the "parallel postulate") is not correct, or at least is not axiomatic. This is a vital result -- the universe is non-Euclidean -- but I felt that this was a bit weak, too. There are at least four ways to address the parallel postulate, which basically says that if you have a line and a point not on the line, there is only one parallel line that passes through the point. One can assume that the parallel postulate is correct, and there is only one parallel line (Euclid's approach). One can assume that parallel lines are more than one (probably infinitely many). One can assume there are none. Or one can just say, "I won't assume any of these; I don't know what a parallel line is." These produce different non-Euclidean geometries. I really wish this had been clearer.
If Berlinski had just done a history of gripes about Euclid, it would have been a useful book. If he had done a real job of giving the information Euclid lacked or didn't understand the need for, it would have been a very useful book. As it is, I feel as if this falls between two stools. I spent far too much time saying, "Tell me more" (and, once in a while, "Tell me less"). Almost never did I feel like what I read was just the right amount. show less
Berlinski recapitulates most of the familiar arguments between atheists and (mostly Christian) believers; I don't think he treads any new ground.
The ten chapters are more like related essays rather than being steps in developing an overall thesis.
It's an accessible introduction to the subject, but not a scholarly exegesis (although Berlinski is a fairly prominent Public Intellectual).
The book lacks a bibliography, footnotes or citations. The index is incomplete and IMO somewhat haphazard show more (for instance, the Penrose quote on page xv is neither sourced nor indexed). show less
The ten chapters are more like related essays rather than being steps in developing an overall thesis.
It's an accessible introduction to the subject, but not a scholarly exegesis (although Berlinski is a fairly prominent Public Intellectual).
The book lacks a bibliography, footnotes or citations. The index is incomplete and IMO somewhat haphazard show more (for instance, the Penrose quote on page xv is neither sourced nor indexed). show less
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