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John Ryskamp
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With the appearance of the general relativity theory, it became increasingly difficult for other disciplines to “map” their own terms of art to those of relativity in an internally consistent fashion. But we know now that, previous to the appearance of the general theory, it was attempted very high up in the western intellectual hierarchy, as Galbraith has shown in his work on Keynes. Today, of course, we say that it’s impossible: there are no quarks in biology, no leptons in economics, and certainly no charm in mathematics. You can’t get, logically, from any concept in any of those disciplines, to any concept of the Standard Model. We smile at the naiveté of Keynes for even attempting what we now consider to have been impossible. But why was it ever considered possible?
Keynes did not have a very good grasp of special relativity, or, seen through the lens of Sraffa’s Production of Commodities By Means of Commodities, even a very good grasp of economics. But it is not altogether fanciful to see internally consistent links between the special relativistic world and the biological or economic worlds. After all, light is one of the postulates of special relativity, as it is in biology, and humanity is part of biology, and economics the study of one aspect of humanity.
But it’s clear that links like that didn’t arouse the competitive instincts of early twentieth-century intellectuals. What did arouse them was the idea that Einstein’s special relativistic argument had wound up at the top of the heap of argumentation. His rhetorical strategy is what proved so seductive. We are starting to take that apart now in the twenty-first century, as I shall show and as Andrea Cerroni has shown. However, at the time of its appearance (although Einstein was frustrated at how long it took to gain recognition even after the publication of the 1905 papers), what impressed intellectuals was the special relativistic argument qua argument---above all, the relativistic “event,” what today we would call a spacetime point. To them it was a matter simply of ignoring the subject matter---the materials---of the argument, and just looking at the argument as an internally consistent structure. It was gorgeous---it had no flaws. What was even more impressive was that it required Einstein himself to point out its limitations.
If you could come to terms with his argument, then you could configure the terms of your own discipline so that they mapped to special relativity in an internally consistent way. Then you would have a relativity theory of economics, or biology---or even mathematics! Is relativity actually lurking somewhere in those disciplines? That is what this comment is about.
Sraffa
We don’t know yet what was in Sraffa’s library throughout his lifetime, or even whether he ever read a single word by Einstein (it appears that Wittgenstein never read a word of Einstein---at least I have seen no documentation of it, although there are comments on relativity in his remarks on the foundations of mathematics and other places). It seems unlikely that a careful, canny, informed intellectual such as Sraffa would never have read anything by Einstein---but we may be in for some surprises in discovering just how uninformed twentieth century intellectuals were about the most famous ideas of their day.
In any event, it is better to look to Sraffa’s work, rather than to that of Keynes, to see the extent to which a serious attempt was made to transform economics into a relativistic discipline. In this regard, I think it is important to take Sraffa seriously when he talks of Production being pulled together out of a mass of old notes. I think it is entirely consistent with his skepticism that he criticized and considered one concept at a time, and was suspicious of links between either concepts or his criticisms of them. That is, perhaps we should regard Production less as a work of synthesis and more as a piece of speculative philosophy. I have no quarrel with Steedman in his assessment of Sraffa’s criticism of Marx, but we seem to be stuck there. That criticism may well turn out to be a very minor sidelight on Sraffa’s achievement, strange as that may seem to say at this point, because currently it is seen as the necessary foundation of his entire economic view. Is it? Let’s ask some special relativistic questions about Production which seem to take the book apart and which seem to take him quite out of economics, perhaps, but also perhaps put him into the mainstream of---at least early---twentieth century thinking:
1. Does he make any assumptions about light?
2. Is there an economic event in the book, and if so, what internally consistent links are there to the special relativistic event as described by Einstein in his book Relativity?
3. Does the approach of Production, and especially the mathematical apparatus, change as Sraffa develops his notes, from 1926 to 1960, to reflect the incorporation of more sophisticated mathematics as general relativity developed? And what, by the way, is in that folder in the Sraffa papers in Cambridge which bears the intriguing label: “D3/12/42 Notes [these notes were gathered by Sraffa in preparation for a work subsequent to Production of Commodities]”?
4. What are Sraffa’s mathematical assumptions in Production? Are they entirely Euclidean, or Euclidean at all? Remember that Einstein adopts strict Euclidean ideas as the assumptions of special relativity, along with the constancy of the speed of light.
5. Does the train experiment in Relativity map logically to the Production “event”?
Gödel
It is clear now that Garciadiego’s book on the set-theoretical “paradoxes” is a dagger pointed straight at the heart of Gödel’s theorem. Above all, this devastating book demolishes not only Richard’s paradox, but also, the rest of the book shows that the various paradoxes which so entranced Russell and his contemporaries, weren’t paradoxes at all. This is disconcerting enough. It is doubly disconcerting to note that Gödel approvingly cites Richard’s paradox in his 1931 paper. So the first thing to establish is the internally consistent links, if there are any, between Gödel’s theorem and Richard’s paradox. All arguments are wrong, that is, eventually they have all been proved internally inconsistent, but so far Gödel’s theorem has been sustained in its logic. Perhaps understanding an internal link with Richard’s paradox will open the door to finding the internal inconsistency in the theorem.
And that is probably what it will have to be: something sitting there on the page, like an assumption that 1 does not equal 0 leading to a conclusion that 1 does equal 0 or a 1/0 stuck somewhere it is easily overlooked, which was simply too close to prejudices we have about language for us to be able to spot it. Gödel, like Sraffa, is a very careful, clever rhetorician, much too aware of presentational strategy for us NOT to have difficulty locating the internal inconsistency in his paper. Doubtless not careful enough, but we don’t yet know where he slipped up.
And special relativity? In fact, we know very little about Gödel’s study of relativity through the years, apart from his rather uninteresting later relativistic studies, and Feferman in his editorial notes to Gödel’s Works is quite dismissive of some of Gödel’s restatements of relativistic ideas. When did Gödel first read the 1905 papers, or did he ever read them? What exactly did he read by Einstein? What was his first reaction on hearing of special, or general, relativity? We just don’t know. On this crucial subject, there is very little documentation for the cases of many important twentieth-century intellectuals (except, perhaps, Duchamp, who freely confessed that much of what he learned about science he gathered from conversation—apparently he never read a word by Einstein).
This leads us to ask the same sorts of questions about Gödel’s paper as we do about Sraffa’s book. Is there an assumption about light in that paper? This seems a very odd question, even an inappropriate one, to ask about a mathematical argument. However, Gödel provokes it with this remarkable statement, which appears in his paper: “Numbers cannot in fact be put into a spatial order.” What does he mean by a fact? by space? What are the Euclidean assumptions, if any, of the paper? What, in special relativistic terms, is a Gödelian event? Is Gödel’s theorem an argument at all, and if so, is it, not a metamathematical argument or even a piece of formal logic, but in fact a straightforward physical theory? Is the paper nothing more than a retelling of Einstein’s train experiment?
Kimura
It may well turn out that it was not Einstein who developed the special relativity theory, but instead, Mendel and Darwin, because the rhetoric of geometry in both Mendel’s paper and Darwin’s Origin (in the latter case, in the only figure in the book which can be called geometrical) is what we now recognize as demonstrably similar to the geometry Einstein sets forward in the train experiment in Relativity, and it serves a similar purpose: to articulate the biological event. It is in biology, of course, that we are most justified in asking for an internally consistent discussion of light. Do Darwin and Mendel, and later Kimura, have light as an assumption in their arguments, and what is that assumption? Are their assumptions Euclidean? Or better yet, if Einstein were to posit a relativistic biological event, how would he express it? Or is he expressing it? Is selection the relativistic event?
These are not questions necessarily restricted to special relativity. This is because Kimura is a statistician. His increasingly sophisticated use of statistical concepts led him to a mathematical apparatus which, in The Neutral Theory of Molecular Evolution, looks remarkably similar to the mathematical apparatus of, say, Feynman’s QED. Are the similarities internally consistent? Is Kimura’s random drift an exception to selection, or is it an exception to relativity? Of all twentieth-century researchers, it appears to be Kimura who took his discipline closest to relativity. Is that true?
In taking a retrospective glance at the works of these three in relation to relativity, we are free to put ourselves very far in the future, at a time when an internal inconsistency has been found in relativity itself and that theory is an historical artifact. Then the three look to be, not attempting to map their work to relativity, but rather, using the inherited concepts of their respective disciplines to critique relativity, looking for that internal inconsistency. Then we will be in a position to ask: did one or more of them actually find the flaw, and were they expressing that discovery in the terms of art of their disciplines?
For example, consider this passage from Lawson’s accurate translation of Einstein’s Relativity:
Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to be embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length AB of the embankment. But the events A and B also correspond to positions A and B on the train. Let M1 be the mid-point of the distance AB on the traveling train. Just when the flashes (as judged from the embankment) of lightning occur, this point M1 naturally coincides with the point M but it moves towards the right in the diagram with the velocity v of the train.
I cite at length because this passage contains an anomaly which is very easy to miss. If you examine the Italian and French translations of the text (which no one has ever bothered to do), you will see that the anomaly has always presented problems for those who had to look very closely at the words. The oddity is, of course, the term “naturally coincides,” which leaps out at us because we are looking at it with twenty-first century eyes, not twentieth-century eyes; indeed, perhaps the most difficult cultural task now before us is simply to realize that we are not living in the twentieth century. Einstein has already spent thirty-odd pages of this very brief book laying out the assumptions which underlie this train experiment. He is very careful about being consistent with them, and he is a devoted and very strict Euclidean; he wished never to deviate from Euclid, a stance which reminds us that Sraffa wished never to deviate from Marx. But Einstein was not, it appears, quite careful enough. We know that he is assuming, along with Euclid, that the definition of the coincidence of two points is a point. However, we have never gotten a definition of a “natural” coincidence of two points. This alone prevents us from going on.
We also have a problem if we try to resolve the issue ourselves. If we simply drop the “naturally” we run into a situation in which Einstein has told us to assume two Cartesian coordinate systems, but now leaves us with one, since, following from the definition of the coincidence of two points, if two parallel coordinate systems coincide at one point, they coincide at all points and are one coordinate system, not two. We have been led to a contradiction. There is nothing to be done about that contradiction. It cannot be resolved, and we have to confront what it is about our changed situation that we see it, as it were, as an object in an intellectual landscape we have never seen before.
If this is the pea under the mattress of the Standard Model, then we are bound to look at the works of Sraffa, Kimura and Gödel with new eyes. We can’t go back and see them as they were seen in the twentieth century, that point of view is no longer an option for us. For example, Einstein also has an algebraic formulation of special relativity; however, this formulation simply begs the question of the “natural” coincidence of points, it covers it up with a metric, a gap-filler. Is this the cause of the formulation of a metric in Gödel’s theorem? Assuming Kimura’s mathematics can be mapped to any degree to the Standard Model, what is it saying about that Model? Is Production not a synthesis, and if it is not---if it lies in pieces of speculation---what does that tell us about Sraffa’s attitude to relativity? And as for Wittgenstein’s remarks on relativity, which always seemed to be useless to any professional physicist, are they also now telling us that something is wrong? Is Duchamp also telling us something is wrong? Einstein said that he hoped his work would provide a few hours’ diversion, and Duchamp said a work of art lasted twenty years. Perhaps we should have taken them at their word.
In short, we need a much more dynamic approach to what we consider the principal monuments of the twentieth century. Every educated person, during the nineteenth century, was presumed to read widely and be up to date in the research of all areas of inquiry, including art. With the advent of specialization---that is, with the development of terms of art within the disciplines---intellectual life lost that character, because, to the extent there was internal consistency within any two given disciplines, it became increasingly difficult to build logical bridges between concepts in the two disciplines. There aren’t twenty people in the world who have read both Production of Commodities and The Neutral Theory. Have you? And yet no highly educated person in the latter eighteenth century could have claimed to be so without having read both Newton and Smith.
Today, advances in understanding the rhetoric of the twentieth century have led us to be much more cautious about the caution of twentieth-century thinkers, and hopefully much more direct than our own twentieth-century selves. Those selves are no longer with us, we left them at the door of this century. We understand more of the prejudices which went into the thinking of people in the twentieth century, and that is part and parcel of the endless process of building up and tearing down ideas. We will go much further in this direction, and much faster, if we try to understand how---regardless of the barriers which specialists felt surrounded their disciplines---they nevertheless communicated in internally consistent ways across those barriers: and built bridges over them!
Keynes did not have a very good grasp of special relativity, or, seen through the lens of Sraffa’s Production of Commodities By Means of Commodities, even a very good grasp of economics. But it is not altogether fanciful to see internally consistent links between the special relativistic world and the biological or economic worlds. After all, light is one of the postulates of special relativity, as it is in biology, and humanity is part of biology, and economics the study of one aspect of humanity.
But it’s clear that links like that didn’t arouse the competitive instincts of early twentieth-century intellectuals. What did arouse them was the idea that Einstein’s special relativistic argument had wound up at the top of the heap of argumentation. His rhetorical strategy is what proved so seductive. We are starting to take that apart now in the twenty-first century, as I shall show and as Andrea Cerroni has shown. However, at the time of its appearance (although Einstein was frustrated at how long it took to gain recognition even after the publication of the 1905 papers), what impressed intellectuals was the special relativistic argument qua argument---above all, the relativistic “event,” what today we would call a spacetime point. To them it was a matter simply of ignoring the subject matter---the materials---of the argument, and just looking at the argument as an internally consistent structure. It was gorgeous---it had no flaws. What was even more impressive was that it required Einstein himself to point out its limitations.
If you could come to terms with his argument, then you could configure the terms of your own discipline so that they mapped to special relativity in an internally consistent way. Then you would have a relativity theory of economics, or biology---or even mathematics! Is relativity actually lurking somewhere in those disciplines? That is what this comment is about.
Sraffa
We don’t know yet what was in Sraffa’s library throughout his lifetime, or even whether he ever read a single word by Einstein (it appears that Wittgenstein never read a word of Einstein---at least I have seen no documentation of it, although there are comments on relativity in his remarks on the foundations of mathematics and other places). It seems unlikely that a careful, canny, informed intellectual such as Sraffa would never have read anything by Einstein---but we may be in for some surprises in discovering just how uninformed twentieth century intellectuals were about the most famous ideas of their day.
In any event, it is better to look to Sraffa’s work, rather than to that of Keynes, to see the extent to which a serious attempt was made to transform economics into a relativistic discipline. In this regard, I think it is important to take Sraffa seriously when he talks of Production being pulled together out of a mass of old notes. I think it is entirely consistent with his skepticism that he criticized and considered one concept at a time, and was suspicious of links between either concepts or his criticisms of them. That is, perhaps we should regard Production less as a work of synthesis and more as a piece of speculative philosophy. I have no quarrel with Steedman in his assessment of Sraffa’s criticism of Marx, but we seem to be stuck there. That criticism may well turn out to be a very minor sidelight on Sraffa’s achievement, strange as that may seem to say at this point, because currently it is seen as the necessary foundation of his entire economic view. Is it? Let’s ask some special relativistic questions about Production which seem to take the book apart and which seem to take him quite out of economics, perhaps, but also perhaps put him into the mainstream of---at least early---twentieth century thinking:
1. Does he make any assumptions about light?
2. Is there an economic event in the book, and if so, what internally consistent links are there to the special relativistic event as described by Einstein in his book Relativity?
3. Does the approach of Production, and especially the mathematical apparatus, change as Sraffa develops his notes, from 1926 to 1960, to reflect the incorporation of more sophisticated mathematics as general relativity developed? And what, by the way, is in that folder in the Sraffa papers in Cambridge which bears the intriguing label: “D3/12/42 Notes [these notes were gathered by Sraffa in preparation for a work subsequent to Production of Commodities]”?
4. What are Sraffa’s mathematical assumptions in Production? Are they entirely Euclidean, or Euclidean at all? Remember that Einstein adopts strict Euclidean ideas as the assumptions of special relativity, along with the constancy of the speed of light.
5. Does the train experiment in Relativity map logically to the Production “event”?
Gödel
It is clear now that Garciadiego’s book on the set-theoretical “paradoxes” is a dagger pointed straight at the heart of Gödel’s theorem. Above all, this devastating book demolishes not only Richard’s paradox, but also, the rest of the book shows that the various paradoxes which so entranced Russell and his contemporaries, weren’t paradoxes at all. This is disconcerting enough. It is doubly disconcerting to note that Gödel approvingly cites Richard’s paradox in his 1931 paper. So the first thing to establish is the internally consistent links, if there are any, between Gödel’s theorem and Richard’s paradox. All arguments are wrong, that is, eventually they have all been proved internally inconsistent, but so far Gödel’s theorem has been sustained in its logic. Perhaps understanding an internal link with Richard’s paradox will open the door to finding the internal inconsistency in the theorem.
And that is probably what it will have to be: something sitting there on the page, like an assumption that 1 does not equal 0 leading to a conclusion that 1 does equal 0 or a 1/0 stuck somewhere it is easily overlooked, which was simply too close to prejudices we have about language for us to be able to spot it. Gödel, like Sraffa, is a very careful, clever rhetorician, much too aware of presentational strategy for us NOT to have difficulty locating the internal inconsistency in his paper. Doubtless not careful enough, but we don’t yet know where he slipped up.
And special relativity? In fact, we know very little about Gödel’s study of relativity through the years, apart from his rather uninteresting later relativistic studies, and Feferman in his editorial notes to Gödel’s Works is quite dismissive of some of Gödel’s restatements of relativistic ideas. When did Gödel first read the 1905 papers, or did he ever read them? What exactly did he read by Einstein? What was his first reaction on hearing of special, or general, relativity? We just don’t know. On this crucial subject, there is very little documentation for the cases of many important twentieth-century intellectuals (except, perhaps, Duchamp, who freely confessed that much of what he learned about science he gathered from conversation—apparently he never read a word by Einstein).
This leads us to ask the same sorts of questions about Gödel’s paper as we do about Sraffa’s book. Is there an assumption about light in that paper? This seems a very odd question, even an inappropriate one, to ask about a mathematical argument. However, Gödel provokes it with this remarkable statement, which appears in his paper: “Numbers cannot in fact be put into a spatial order.” What does he mean by a fact? by space? What are the Euclidean assumptions, if any, of the paper? What, in special relativistic terms, is a Gödelian event? Is Gödel’s theorem an argument at all, and if so, is it, not a metamathematical argument or even a piece of formal logic, but in fact a straightforward physical theory? Is the paper nothing more than a retelling of Einstein’s train experiment?
Kimura
It may well turn out that it was not Einstein who developed the special relativity theory, but instead, Mendel and Darwin, because the rhetoric of geometry in both Mendel’s paper and Darwin’s Origin (in the latter case, in the only figure in the book which can be called geometrical) is what we now recognize as demonstrably similar to the geometry Einstein sets forward in the train experiment in Relativity, and it serves a similar purpose: to articulate the biological event. It is in biology, of course, that we are most justified in asking for an internally consistent discussion of light. Do Darwin and Mendel, and later Kimura, have light as an assumption in their arguments, and what is that assumption? Are their assumptions Euclidean? Or better yet, if Einstein were to posit a relativistic biological event, how would he express it? Or is he expressing it? Is selection the relativistic event?
These are not questions necessarily restricted to special relativity. This is because Kimura is a statistician. His increasingly sophisticated use of statistical concepts led him to a mathematical apparatus which, in The Neutral Theory of Molecular Evolution, looks remarkably similar to the mathematical apparatus of, say, Feynman’s QED. Are the similarities internally consistent? Is Kimura’s random drift an exception to selection, or is it an exception to relativity? Of all twentieth-century researchers, it appears to be Kimura who took his discipline closest to relativity. Is that true?
In taking a retrospective glance at the works of these three in relation to relativity, we are free to put ourselves very far in the future, at a time when an internal inconsistency has been found in relativity itself and that theory is an historical artifact. Then the three look to be, not attempting to map their work to relativity, but rather, using the inherited concepts of their respective disciplines to critique relativity, looking for that internal inconsistency. Then we will be in a position to ask: did one or more of them actually find the flaw, and were they expressing that discovery in the terms of art of their disciplines?
For example, consider this passage from Lawson’s accurate translation of Einstein’s Relativity:
Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to be embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length AB of the embankment. But the events A and B also correspond to positions A and B on the train. Let M1 be the mid-point of the distance AB on the traveling train. Just when the flashes (as judged from the embankment) of lightning occur, this point M1 naturally coincides with the point M but it moves towards the right in the diagram with the velocity v of the train.
I cite at length because this passage contains an anomaly which is very easy to miss. If you examine the Italian and French translations of the text (which no one has ever bothered to do), you will see that the anomaly has always presented problems for those who had to look very closely at the words. The oddity is, of course, the term “naturally coincides,” which leaps out at us because we are looking at it with twenty-first century eyes, not twentieth-century eyes; indeed, perhaps the most difficult cultural task now before us is simply to realize that we are not living in the twentieth century. Einstein has already spent thirty-odd pages of this very brief book laying out the assumptions which underlie this train experiment. He is very careful about being consistent with them, and he is a devoted and very strict Euclidean; he wished never to deviate from Euclid, a stance which reminds us that Sraffa wished never to deviate from Marx. But Einstein was not, it appears, quite careful enough. We know that he is assuming, along with Euclid, that the definition of the coincidence of two points is a point. However, we have never gotten a definition of a “natural” coincidence of two points. This alone prevents us from going on.
We also have a problem if we try to resolve the issue ourselves. If we simply drop the “naturally” we run into a situation in which Einstein has told us to assume two Cartesian coordinate systems, but now leaves us with one, since, following from the definition of the coincidence of two points, if two parallel coordinate systems coincide at one point, they coincide at all points and are one coordinate system, not two. We have been led to a contradiction. There is nothing to be done about that contradiction. It cannot be resolved, and we have to confront what it is about our changed situation that we see it, as it were, as an object in an intellectual landscape we have never seen before.
If this is the pea under the mattress of the Standard Model, then we are bound to look at the works of Sraffa, Kimura and Gödel with new eyes. We can’t go back and see them as they were seen in the twentieth century, that point of view is no longer an option for us. For example, Einstein also has an algebraic formulation of special relativity; however, this formulation simply begs the question of the “natural” coincidence of points, it covers it up with a metric, a gap-filler. Is this the cause of the formulation of a metric in Gödel’s theorem? Assuming Kimura’s mathematics can be mapped to any degree to the Standard Model, what is it saying about that Model? Is Production not a synthesis, and if it is not---if it lies in pieces of speculation---what does that tell us about Sraffa’s attitude to relativity? And as for Wittgenstein’s remarks on relativity, which always seemed to be useless to any professional physicist, are they also now telling us that something is wrong? Is Duchamp also telling us something is wrong? Einstein said that he hoped his work would provide a few hours’ diversion, and Duchamp said a work of art lasted twenty years. Perhaps we should have taken them at their word.
In short, we need a much more dynamic approach to what we consider the principal monuments of the twentieth century. Every educated person, during the nineteenth century, was presumed to read widely and be up to date in the research of all areas of inquiry, including art. With the advent of specialization---that is, with the development of terms of art within the disciplines---intellectual life lost that character, because, to the extent there was internal consistency within any two given disciplines, it became increasingly difficult to build logical bridges between concepts in the two disciplines. There aren’t twenty people in the world who have read both Production of Commodities and The Neutral Theory. Have you? And yet no highly educated person in the latter eighteenth century could have claimed to be so without having read both Newton and Smith.
Today, advances in understanding the rhetoric of the twentieth century have led us to be much more cautious about the caution of twentieth-century thinkers, and hopefully much more direct than our own twentieth-century selves. Those selves are no longer with us, we left them at the door of this century. We understand more of the prejudices which went into the thinking of people in the twentieth century, and that is part and parcel of the endless process of building up and tearing down ideas. We will go much further in this direction, and much faster, if we try to understand how---regardless of the barriers which specialists felt surrounded their disciplines---they nevertheless communicated in internally consistent ways across those barriers: and built bridges over them!