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Combinators: A Centennial View
Combinators have inspired ideas about computation ever since they were first invented in 1920, and in this innovative book, Stephen Wolfram provides a modern view of combinators and their significance. Informed by his work on the computational universe of possible programs and on computational language design, Wolfram explains new and existing ideas about combinators with unique clarity and stunning visualizations, as well as provides insights on their historical connections and the curious story of Moses Schönfinkel, inventor of combinators. Though invented well before Turing machines, combinators have often been viewed as an inaccessibly abstract approach to computation. This book brings them to life as never before in a thought-provoking and broadly accessible exposition of interest across mathematics and computer science, as well as to those concerned with the foundations of formal and computational thinking, and with the history of ideas.
370 pages, Hardcover
First published June 17, 2021
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About the author
Stephen Wolfram
43Ìýbooks441ÌýfollowersStephen Wolfram is the founder & CEO of Wolfram Research, creator of Mathematica, Wolfram|Alpha & Wolfram Language, author of A New Kind of Science and other books, and the originator of Wolfram Physics Project.
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Displaying 1 - 5 of 5 reviews
August 29, 2021
I had heard of Moses Schönfinkel, the inventor of combinators, but he was just a name, and I was pretty vague about what combinators were too. Having read this book, a product of the inimitable Stephen Wolfram, I am better informed. Schönfinkel was born a Russian Jew towards the end of the 19th century. He did well in school, straight As, got into mathematics, and set his sights on studying with Hilbert, then the undisputed king of mathematicians. When he was offered a place at Göttingen, his letters say he thought he'd died and gone to heaven. He went off to Germany shortly before the outbreak of WW I... and then no one seems to be very sure about what happened.
Schönfinkel definitely hung around with Hilbert and his circle between 1914 and 1924. Wolfram has put a good deal of energy into tracking down records. He appears in photographs and is mentioned from time to time. But in these ten years, all he actually seems to have done was give a couple of talks in 1920 and 1921, both on topics in mathematical logic. Just before he left Göttingen for good, to return to what was now the Soviet Union, he submitted a paper based on his 1920 talk to a journal; a colleague assisted him in writing it up. A second paper, based on his 1921 talk, was submitted to another journal by his coauthor Paul Bernays in 1927. Wolfram believes that Bernays put it together based on notes Schönfinkel had left.
Schönfinkel never completed a thesis - as far as Wolfram knows, he never even started one - and he never received a doctorate. After he got back to the Soviet Union, his life collapsed. He was committed to a "sanatorium" (the actual nature of the institution is apparently uncertain), and former colleagues were told that he had had a mental breakdown. He never again published anything, never married, never had any children, and died in miserable circumstances. Given the above, you would be forgiven for thinking that his life was the most abject failure.
The curious thing is that this may not be true. Schönfinkel's 1920 paper "Über die Bausteine der mathematischen Logik" ("On the building-blocks of mathematical logic") was not immediately recognised as being of any importance. It describes what seems to be a perverse, weirdly minimalist approach to defining the fundamental logical structure of mathematics, where everything is reduced to the interaction between two operators called S and K. They are defined by the cryptic equations:
K[x][y] = x
S[x][y][z] = x[z][y[z]]
Unless you have a certain amount of background in mathematical logic, it's not obvious that these equations say anything meaningful, and even if you do it's not obvious that it's interesting. But Wolfram goes through the background and links it up to other things in logic; after a while, you see that it's at least possible that Schönfinkel's idea may be important. Many people have now tried to develop it, the best-known being Haskell Curry. Wolfram says he's been thinking about it, on and off, for decades. A substantial chunk of the book shows you graphs summarising trillions of automatically executed calculations with K's and S's, carried out by software that Wolfram has put together. The jury's still out on what they might tell us. It's a strange story. Schönfinkel's whole academic career essentially consisted of two lines of mathematics, and we still don't know if they give us anything worthwhile.
The book contains all the details, though I found the order of presentation frustrating: it starts with the simulations, then it gives you the mathematical background, and finally it tells you about Schönfinkel's life. I would recommend reading the sections in the opposite order. If you're curious about the history of ideas, definitely worth a look.
Schönfinkel definitely hung around with Hilbert and his circle between 1914 and 1924. Wolfram has put a good deal of energy into tracking down records. He appears in photographs and is mentioned from time to time. But in these ten years, all he actually seems to have done was give a couple of talks in 1920 and 1921, both on topics in mathematical logic. Just before he left Göttingen for good, to return to what was now the Soviet Union, he submitted a paper based on his 1920 talk to a journal; a colleague assisted him in writing it up. A second paper, based on his 1921 talk, was submitted to another journal by his coauthor Paul Bernays in 1927. Wolfram believes that Bernays put it together based on notes Schönfinkel had left.
Schönfinkel never completed a thesis - as far as Wolfram knows, he never even started one - and he never received a doctorate. After he got back to the Soviet Union, his life collapsed. He was committed to a "sanatorium" (the actual nature of the institution is apparently uncertain), and former colleagues were told that he had had a mental breakdown. He never again published anything, never married, never had any children, and died in miserable circumstances. Given the above, you would be forgiven for thinking that his life was the most abject failure.
The curious thing is that this may not be true. Schönfinkel's 1920 paper "Über die Bausteine der mathematischen Logik" ("On the building-blocks of mathematical logic") was not immediately recognised as being of any importance. It describes what seems to be a perverse, weirdly minimalist approach to defining the fundamental logical structure of mathematics, where everything is reduced to the interaction between two operators called S and K. They are defined by the cryptic equations:
K[x][y] = x
S[x][y][z] = x[z][y[z]]
Unless you have a certain amount of background in mathematical logic, it's not obvious that these equations say anything meaningful, and even if you do it's not obvious that it's interesting. But Wolfram goes through the background and links it up to other things in logic; after a while, you see that it's at least possible that Schönfinkel's idea may be important. Many people have now tried to develop it, the best-known being Haskell Curry. Wolfram says he's been thinking about it, on and off, for decades. A substantial chunk of the book shows you graphs summarising trillions of automatically executed calculations with K's and S's, carried out by software that Wolfram has put together. The jury's still out on what they might tell us. It's a strange story. Schönfinkel's whole academic career essentially consisted of two lines of mathematics, and we still don't know if they give us anything worthwhile.
The book contains all the details, though I found the order of presentation frustrating: it starts with the simulations, then it gives you the mathematical background, and finally it tells you about Schönfinkel's life. I would recommend reading the sections in the opposite order. If you're curious about the history of ideas, definitely worth a look.
July 31, 2021
I was sent a copy of this book in exchange for an honest review. It's a nicely bound edition, full of full-color diagrams and printed on good quality paper. I know it sounds silly when I compliment a book on its physical attributes, but if you're going to read a "sit up straight and pay attention" book, it's good to have visual and tactile positives to help keep you motivated. Plus, for some odd reason, it smells good (I think this is the first time I've ever complimented a book on its smell). But now on to the contents.
The topic of this book is not, in my opinion, combinators. It's not even what the author was primarily thinking about (although he doubtless did spend a good amount of time on that). If combinators were the actual topic, we would have heard a lot more about Haskell Curry (who we do hear about very late in this book), and a lot less about the actual topic of the book, which is a man named Moses Schönfinkel. Wolfram has clearly developed a fascination with this individual, in much the same way that others have with Évariste Galois (who did important early work in mathematical group theory, and died in a duel at age 20). What we basically know about Moses Schönfinkel is that he was born in Russia, moved to Germany, published the first paper on combinators, moved back to Russia, and died (perhaps, but not for certain, after spending some time in an insane asylum).
The first half of the book appeared as if it was going to be something along the lines of Wolfram's earlier "A New Kind Of Science", or "A Project To Find The Fundamental Theory of Physics". We are talking about some abstractions which I'll call "mathematical", although perhaps they are better thought of as "computational", and Wolfram is using the graphical abilities of his Wolfram language to make five bajillion graphical representations of them. It's pretty cool, if you like the visuals, and are content to only sometimes understand what he's talking about. I'm not saying he doesn't make sense, just that I am only sometimes able to follow what he's talking about in detail. I have a Master's Degree in Electrical Engineering (thesis topic: "The Uses Of Statistical Measures of Correlation in Artificial Neural Networks"), I'm not a complete doofus (usually), but I'll admit that some of the formulas did not mean much to me. For example: "s[s][s[s][s[s[s]]][k]][k[s[s][s[s[s]]][k]]]". That's by no means the longest one, it's just the longest I was willing to type here.
But, Wolfram doesn't actually expect or intend for the reader to parse the combinator expressions, no doubt he's just putting them there to give you the idea of how much complexity we're talking about. It is, at times, rather like reading about chemistry by spelling out all of the atoms of each chemical compound, never saying "ethanol" but rather listing all the C's and H's involved. But then, Wolfram is involved in a field with not nearly so much work done in it yet, so naming conventions have not been developed. In any case, for the first quarter of the book I tried hard to keep up with what he was saying, but for the second quarter of the book I just read it and looked at the pretty pictures, without worrying too much if I had followed the details or not.
Then, unexpectedly, we stop talking (as much) about combinators, and turn instead to the subject of Moses Schönfinkel. Wolfram has done, it seems, a great deal of work to try to rescue him from the oblivion into which he had sunk, pulled down by a combination of war, anti-semitism, and the poverty which those two forces had pushed him into. It seems likely that a great deal more work by Schönfinkel was burned after his death in Russia in 1942, probably for tinder during the winter. Wolfram is persistent, however, and by combing through German and Russian records still available from the first half of the 20th century, he is able to piece together some of his life.
To a certain extent, Schönfinkel was lost to the "Urkatastrophe", that series of cascading catastrophes which was begun by World War 1. But, in some ways, he was also simply born too early. Combinators are an idea which cries out for computing to explore, and this was simply not available to Schönfinkel (or anyone else, during his lifetime). Like Ada Lovelace and Alan Turing and Haskell Curry, Schönfinkel was one of those thinkers whose lot it was to be born just barely too early for their greatest talents to be put to best use. What if Shakespeare had been born before theater, or J.S.Bach before the invention of musical instruments? No doubt they would have found other ways to put their talents to use, but perhaps Wolfram is aware (at some level, at least) that he is fortunate to have been born in an age when he is able to play the equivalent (for him) of J.S.Bach's church organ. Schönfinkel never got the chance, but in this book, you can get a glimpse of the landscapes that existed in his mind, waiting for a chance to get out.
The topic of this book is not, in my opinion, combinators. It's not even what the author was primarily thinking about (although he doubtless did spend a good amount of time on that). If combinators were the actual topic, we would have heard a lot more about Haskell Curry (who we do hear about very late in this book), and a lot less about the actual topic of the book, which is a man named Moses Schönfinkel. Wolfram has clearly developed a fascination with this individual, in much the same way that others have with Évariste Galois (who did important early work in mathematical group theory, and died in a duel at age 20). What we basically know about Moses Schönfinkel is that he was born in Russia, moved to Germany, published the first paper on combinators, moved back to Russia, and died (perhaps, but not for certain, after spending some time in an insane asylum).
The first half of the book appeared as if it was going to be something along the lines of Wolfram's earlier "A New Kind Of Science", or "A Project To Find The Fundamental Theory of Physics". We are talking about some abstractions which I'll call "mathematical", although perhaps they are better thought of as "computational", and Wolfram is using the graphical abilities of his Wolfram language to make five bajillion graphical representations of them. It's pretty cool, if you like the visuals, and are content to only sometimes understand what he's talking about. I'm not saying he doesn't make sense, just that I am only sometimes able to follow what he's talking about in detail. I have a Master's Degree in Electrical Engineering (thesis topic: "The Uses Of Statistical Measures of Correlation in Artificial Neural Networks"), I'm not a complete doofus (usually), but I'll admit that some of the formulas did not mean much to me. For example: "s[s][s[s][s[s[s]]][k]][k[s[s][s[s[s]]][k]]]". That's by no means the longest one, it's just the longest I was willing to type here.
But, Wolfram doesn't actually expect or intend for the reader to parse the combinator expressions, no doubt he's just putting them there to give you the idea of how much complexity we're talking about. It is, at times, rather like reading about chemistry by spelling out all of the atoms of each chemical compound, never saying "ethanol" but rather listing all the C's and H's involved. But then, Wolfram is involved in a field with not nearly so much work done in it yet, so naming conventions have not been developed. In any case, for the first quarter of the book I tried hard to keep up with what he was saying, but for the second quarter of the book I just read it and looked at the pretty pictures, without worrying too much if I had followed the details or not.
Then, unexpectedly, we stop talking (as much) about combinators, and turn instead to the subject of Moses Schönfinkel. Wolfram has done, it seems, a great deal of work to try to rescue him from the oblivion into which he had sunk, pulled down by a combination of war, anti-semitism, and the poverty which those two forces had pushed him into. It seems likely that a great deal more work by Schönfinkel was burned after his death in Russia in 1942, probably for tinder during the winter. Wolfram is persistent, however, and by combing through German and Russian records still available from the first half of the 20th century, he is able to piece together some of his life.
To a certain extent, Schönfinkel was lost to the "Urkatastrophe", that series of cascading catastrophes which was begun by World War 1. But, in some ways, he was also simply born too early. Combinators are an idea which cries out for computing to explore, and this was simply not available to Schönfinkel (or anyone else, during his lifetime). Like Ada Lovelace and Alan Turing and Haskell Curry, Schönfinkel was one of those thinkers whose lot it was to be born just barely too early for their greatest talents to be put to best use. What if Shakespeare had been born before theater, or J.S.Bach before the invention of musical instruments? No doubt they would have found other ways to put their talents to use, but perhaps Wolfram is aware (at some level, at least) that he is fortunate to have been born in an age when he is able to play the equivalent (for him) of J.S.Bach's church organ. Schönfinkel never got the chance, but in this book, you can get a glimpse of the landscapes that existed in his mind, waiting for a chance to get out.
September 4, 2021
Driven by a love for people who come up with great ideas this book shows a desire to explain that greatness and tell their stories.
Combinators are very abstract and it is hard to develop an intuition for the significance of a particular formulation. But they do seem to be useful for exploring the fundamental question computation.
It seems like very simple building blocks can support universal computation. I was amazed that a simple-looking construction could take a hundred and thirty million steps to stabilize.
This story is a unique and little-understood episode in the development of the modern understanding of computation.
Combinators are very abstract and it is hard to develop an intuition for the significance of a particular formulation. But they do seem to be useful for exploring the fundamental question computation.
It seems like very simple building blocks can support universal computation. I was amazed that a simple-looking construction could take a hundred and thirty million steps to stabilize.
This story is a unique and little-understood episode in the development of the modern understanding of computation.
March 25, 2022
This is a weird book. Not the topic (well, that's debatable), but the way it's structured and presented. The first part is autobiographical: Wolfram's experiments with the S and K combinators over the years. It's a printed playground of exploring a multitude of ways to visualize executing sequences of combinators. It examines combinators as values, information, and as a universal system of computation.
You can enjoy this book in much the same way that anyone can enjoy looking at fractals. It's fun and fascinating to look at the little universes of complexity that are contained in surprisingly compact sequences of combinators.
It's really quite beautiful even if I didn't follow every twist and turn of Wolfram's thinking (following along is made especially difficult if you're not intimately familiar with the Wolfram Language...and I'm not, though I'm certainly aware of it). I appreciated the candid discussions about the "nuts and bolts" of his attempts to make combinators easier to visualize and reason about.
By the way, I had much better luck following the definitions of K and S as given on the Wikipedia page for "Combinatory logic", which I'll reformat for brevity here:
The second part of the book is a biography of the little-known Moses Schönfinkel, who published an astounding paper introducing combinatory logic and later disappeared (perhaps into madness and/or poverty). Wolfram has clearly put effort into digging up whatever can be found of Schönfinkel to celebrate his life and work.
I don't know if this book would appeal to either the average person on the street on one extreme or a mathemetician on the other extreme. But for someone like me, with an interest in combinatorics, logic, lambda calculus, or grammars (all as they apply to computation and computer programming), this book is a fun and thought-provoking meander through the topic.
You can enjoy this book in much the same way that anyone can enjoy looking at fractals. It's fun and fascinating to look at the little universes of complexity that are contained in surprisingly compact sequences of combinators.
It's really quite beautiful even if I didn't follow every twist and turn of Wolfram's thinking (following along is made especially difficult if you're not intimately familiar with the Wolfram Language...and I'm not, though I'm certainly aware of it). I appreciated the candid discussions about the "nuts and bolts" of his attempts to make combinators easier to visualize and reason about.
By the way, I had much better luck following the definitions of K and S as given on the Wikipedia page for "Combinatory logic", which I'll reformat for brevity here:
Another simple combinator is K, which manufactures constant functions: (K x) is the function which, for any argument, returns x, so we say ((K x) y) = x for all terms x and y.
A third combinator is S, which is a generalized version of application: (S x y z) = (x z (y z)). S applies x to y after first substituting z into each of them.
The second part of the book is a biography of the little-known Moses Schönfinkel, who published an astounding paper introducing combinatory logic and later disappeared (perhaps into madness and/or poverty). Wolfram has clearly put effort into digging up whatever can be found of Schönfinkel to celebrate his life and work.
I don't know if this book would appeal to either the average person on the street on one extreme or a mathemetician on the other extreme. But for someone like me, with an interest in combinatorics, logic, lambda calculus, or grammars (all as they apply to computation and computer programming), this book is a fun and thought-provoking meander through the topic.
December 26, 2021
A beautiful book. The layout choices and color, are thoughtful. The topic, reflects a modern view, of an appreciation, for mathematics; in particular, the world of Haskell Curry and Moss Schonfinkel. It is clear, Mr. Wolfram is literate in multiple languages, like most good mathematicians. Recommended.
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