On the Unity of
by John Ryskamp
In memoriam David Fowler
The twentieth century is dead, and in this essay we view the remains. This is not, of course, to say that that century's influence is gone. Far from it, and that is why we view the remains. How they got that way is the cautionary tale embedded in this brief survey of some of the chief intellectual monuments of the twentieth century. New historical research shows that virtually no discipline has remained immune to the "natural" mathematics developed at the turn of the century in order to cope with the supposed "paradoxes" generated by set theorynot economics, not physics, not biology: apparently no area of inquiry has escaped being made part of the "natural" mathematics project. This mathematics asserts that mathematical formulations are inherently anomalous; the evidence of this is that they generate paradoxes. Therefore, the idea that mathematics is an aspect of human perception, must be made a part of mathematical formulations even though it plays no internally consistent role in any "natural" mathematical formulation.
The role of "natural" mathematics in the disciplines has gone unremarked for the very reason it was influential in the first place. Whether the researcher was the physicist Albert Einstein, the economist Piero Sraffa, the logician Kurt Gödel, the philosopher Ludwig Wittgenstein, or the biologist Motoo Kimura, scientists in non-mathematics disciplines felt they were unable to express their ideas mathematically. This is the chief revelation of the new historical research, and a remarkable and unexpected (given the exalted reputations of these figures) unifying feature of twentieth-century intellectual history. These thinkers had to search for appropriate mathematical terms in the latest mathematics of their day. They were unprepared to cope with the idea that flaws in the mathematics lodged errors in their theories. The current reexamination of the mathematics of the disciplineswhich will turn out to be the chief intellectual enterprise of the early twenty-first centurybegan with the revelation of the faulty approach taken to set theory by some of the chief proponents of "natural" mathematics.
It should be noted that this unification of twentieth-century ideas on the basis of the "natural" mathematics they share, was not the unification sought by twentieth-century thinkers themselves. It has gone pretty much unremarked that twentieth-century thinkers sought to unify the disciplines on the basis of relativity. It has gone unremarked largely because the project was abandoned when physics developed terms of art so recherché that the data and concepts of other disciplines could not be matched to them. The approach was swiftly abandoned, and suppressed out of embarrassment. As we shall see, bringing Einstein's work into alignment with "natural" mathematicssomething which has not been possible until nowallows us to begin asking the kinds of questions which will in the end reveal precisely and in detail, the influence of "natural" mathematics with which we still live and in which we still express our scientific ideas.
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 it was attempted very high up in the western intellectual hierarchy, as Galbraith 1 has shown in his work on Keynes. Ernst Mayr, at one time the doyen of evolutionary studies, claimed during the 1950s that evolution could be seen as a genetic theory of relativity. 2 However, in his later writings that concept went the way of the dinosaur and today he is a figure of fun; no one bothers to investigate what he meant by the term, which is rather too bad, since it may turn out to be one of the few interesting ideas he ever had. 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 until very recently we considered quite impossible.
Keynes did not have a very good grasp of relativity, or, seen through the lens of Sraffa's Production of Commodities By Means of Commodities (1960), even a very good grasp of economics. But it is not altogether fanciful to see internally consistent links between the relativistic world and the biological or economic worlds. After all, light is one of the postulates of relativity, as it is in biology, and humanity is part of biology, and economics the study of one aspect of humanity.
Links like that, however, 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 3 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 argumentabove all, the relativistic "event," what today we would call a spacetime point. To them it was a matter simply of ignoring the subject matterthe materialsof the argument, and just looking at the argument as an internally consistent structure. It was gorgeousit had no flaws. What was even more impressive was that it required Einstein himself to point out the limitations of special relativity. If you could come to terms with his argument, then you could configure the terms of your own discipline so that they mapped to relativity in an internally consistent way. Then you would have a relativity theory of economics, or biologyor even mathematics!
It must be noted that we are still enamored of the explanatory power of the Standard Model. For this reason, historians of ideas pay little attention to the idea that the fundamental ideas of relativity are simply shared by the other disciplines. We are still in an early stage of the examination of the influence of "natural" mathematics. The apparently bad experience of earlier attempts to unify the disciplines, along with disciplinary hubris, still makes us leery of revisiting the settled questions of the various disciplines. And there is nothing wrong with respecting the boundaries these disciplines have set up for themselves. In fact, it allows us to take the chief current ideas of different disciplines one by one, examining them on their own terms in light of the latest mathematical historical research. This examination begins to reveal their shared ideas, and the overarching concerns of twentieth-century thinking. In the course of this examination, we shall see that we have begun to free ourselves of many received ideas.
One of the most important goals of the discussion which follows, is to briefly introduce specialists to major monuments outside their disciplines and to provide reasons for specialists to familiarize themselves with these works which, initially, may seem to be remote from their concerns. Why should a chemist read Sraffa, or an economist read Kimura? Hopefully, the linkage of these writers through "natural" mathematics, will provide, above all, the stimulus for specialists to reexamine ideas in their own fields which they take too much for granted.
Piero Sraffa's Economics of "Natural" Mathematics
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." 5 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 (that the labor theory of value is untenable), 6 but we seem to be stuck there; the Cambridge controversies (the close examination of the latter part of Production) have never quite gone away, but they have stalled. Sraffa's criticism of Marx 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? Since the Standard Model is currently regarded as being in every way an argument superior to Sraffa's in Production, 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 ofat least earlytwentieth century thinking. This frees us to open up the discussion later to links between relativity and Sraffa's work on the basis of their shared "natural" mathematics:
Now, how did Sraffa choose to express these ideas mathematically? We shall begin to answer this question by noting below the path he took along with other twentieth-century thinkers, the path to "natural" mathematics.
Kurt Gödel's Insufficient Examination of "Natural" Mathematics
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 Solomon 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? There were discussions of relativity in the Vienna Circle, but he seems to have shied away from 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 conversationapparently 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 in his paper: "Numbers cannot in fact be put into a spatial order"this is the infamous footnote 8. 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?
Motoo Kimura's Search for a "Natural" Mathematics
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, Richard Feynman's QED (1985). Are the similarities internally consistent? Is Kimura's random drift an exception to selection, or is it an exception to relativity? What is his biological event: substitution? mutation? selection? something else? Is the neutral theory a biological theory, or a physical theory? This latter question arises in considering a comment drawn from Kimura by a critic. In response, Kimura says: "Just as synonyms are not 'noise' in language, it is not proper to regard the substitution of neutral alleles simply as noise or loss of genetic information….It seems to me to be more appropriate to say that strictly neutral alleles are absolutely noiseless." 9 These metaphors are physical ideas. Of what?
The basis for unfolding the context of the terms of art of these different disciplines, is the understanding that they emerge from a shared "natural" mathematics. 10 Neither Kimura nor Sraffa came to their disciplines from mathematics, and they felt they needed a mathematical expression for their ideas. Kimura learned French rather late just so he could read Gustave Malécotwho pioneered the use of "natural" mathematics in biologyand Sraffa went, like Diogenes, through mathematician after mathematician searching for the mathematical expression of his ideas. So they inheritedmore or less unawaresthe increasingly problematic nature of the "natural" mathematical ideas in which both of their arguments are expressed. Historical research is revealing the difficulties in the chief ideas of "natural" mathematics. For example, L.E.J. Brouwer promulgated what he called an "infinite ordinal number." Supposedly this notion had been ratified by Georg Cantor's well-ordering of the ordinal numbers. But it turns out that Cantor never did so, never claimed he had done so, and never used the term "infinite ordinal number." As Garciadiego says: "[G. G.] Berry maintained that Cantor had virtually proved the existence of the well-ordering of the ordinal numbers by showing that ordinals of the second class are well-ordered….but Cantor simply indicated that 'we shall show that the transfinite cardinal numbers can be arranged according to their magnitude, and, in this order, [they] form, like the finite numbers, a 'well-ordered aggregate' in an extended sense of the words.'" 11 Nevertheless, Brouwer's term worked its way into the discourses of Émile Borel (the mentor of Malécot), Andrei Kolmogorov, Haskell Curry and John von Neumann, and is, regrettably, at the heart of contemporary probability and computational theory; computer science is replete with "natural" mathematicswhat false results is it thereby giving us? 12 It is likely that we can put most twentieth-century disciplines in the form of Richard's "paradox," see how they partook of "natural" mathematics, and reveal their flaws. Now that we are more familiar with the idea that the project of the twentieth centuryregardless of disciplineis "natural" mathematics, it is probably best to approach any idea in a twentieth-century discipline with two questions: what "paradox" is it trying to avoid? what "paradox" is it trying to express?
It should not be surprising if biology turns out to be a branch of physics. Most of Gregor Mendel's published papers are in meteorology. Charles Darwin began as a physicist seeking to describe reality and that concern is recurrent. He first sought to do so in the context of cosmology and geology and only later turned to biology, as we see when he presents his physical ideas in a book no one reads anymore, The Structure and Distribution of Coral Reefs (1842). 13 For Darwin, the identity of physics and biology is due to the progressivism of reality. Natureencompassing all the disciplinesis the continuum of that progressivism; paradox supposedly flowed from the tension between perfection as an assumption and progressivism as a conclusion. Both Mendel and Darwin seem to have turned to biology because it offered more, and more internally continuous, physical data than cosmology or geology. Of all twentieth-century researchers, it appears to be Kimura who took his discipline closest to relativity. Is that true? Both Darwin and Kimura set their work in the context of physics. Darwin says "that, whilst this planet has gone cycling on according to the fixed law of gravity, from so simple a beginning endless forms most beautiful and most wonderful have been, and are being evolved." Kimura's gloss on this passage is to remind us that although mutational "random processes are slow and insignificant for our ephemeral existence, in the span of geological times, they become colossal."14 We in turn hunt among concepts such as "fixed law," "gravity," "random" and "geological times" for the necessary internal links between geology, physics and biologybut perhaps these words have fallen apart and we cannot use them anymore.
Another idea is also beginning to take shape: there are no "paradoxes," at least as far as we know. Researchers have resisted looking into the set-theoretical paradoxes because it leads us further and further back in time and so implicates more and more important ideas. If the set-theoretic "paradoxes" are not paradoxes, are the earlier paradoxes (for example, the liar paradox) really paradoxes? And more importantly, to what extent are the earlier mathematical expressions in the various disciplines, simply projects to "avoid" or "solve" these paradoxes, which in turn may not be paradoxes at all? To what extent is the history of objective discourse, a falsely based "natural" mathematics having no logical object? To what extent can we say to everything we currently consider to be internally consistent: what is your argument?
And relativity? In taking even a retrospective glance at the works of only three twentieth-century figures 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 an internal inconsistency which actually lies in the "natural" mathematics Einstein shares with them. Consider this passage from Lawson's accurate translation of Einstein's Relativity:
I cite at length because this passage contains an anomaly, which of course is the term "naturally coincides." This term ("fällt zwar…zusammen" in the original German) 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.
"Natural" coincidence is otherwise known as a spacetime point. Einstein has already spent twenty-odd pages of this very brief book laying out the assumptions which underlie the 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 (and never get, in any of Einstein's writings) a definition of a "natural" coincidence of two points. This alone prevents us from going on and this argument, which defined the twentieth century, abruptly ends.
We also have a problem if we try to resolve the issue ourselves. If we simply drop the term "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. A spacetime point is no longer a physical fact, it is an outmoded doctrine. This is the first occasion we have to note a logical mistake in Einstein's fundamental ideas. At it happens, we know how he came to make it. As pointed out recently,16 Einstein was enormously impressed by Poincaré's Science and Hypothesis (1902). What he was totally unprepared for was the "natural" mathematical point of view Poincaré was trying so hard to sell him. As Garciadiego points out, Poincaré used the book to set out "numerous inconsistencies arising from set theory....Poincaré was hunting for 'paradoxes' because he was trying to discredit both Cantor's theory of sets and Russell's logicism." 17 Thus, the young Einstein faces both a well-developed mathematical debate and a polemic. Note that at no time did Einstein ever question the status of the set theory, or other paradoxes, or the historical approach developed to deal with them (neither did Kimura or Sraffa). Instead, he felt comfortable expressing the relativity of simultaneity through "natural" mathematics without ever examining it, with disturbing consequences for his theory. In Poincaré he read and accepted the idea that "the mind has a direct intuition of this power ["proof by recurrence" or "mathematical induction"], and experiment can only be for [the mind] an opportunity of using it, and thereby of becoming conscious of it." In geometry "we are brought to [the concept of space] solely by studying the laws by which…[muscular] sensations succeed one another." 18 We now understand why we never find "natural" coincidence among Einstein's postulates or definitions: neither of those is its job. Its job is to serve as a facilitator of arguments which cannot be carried out logically, and so we see exactly why it occurs where it does in the argument for the relativity of simultaneity: it "allows" one point to "succeed" another, in conformity with the demands of "natural" mathematics. Here we also see emerging in Einstein's thought a tenet of "natural" mathematics not usually associated with him: he believed that reality is progressive. That, of course, is not an acceptable stand today in any scientific argument, but it required perspective on "natural" mathematics in order to realize that it is part of Einstein's thought. Examination of "natural" mathematics allows us to integrate in a logical way, the prejudices of thinkers into their ideas: ideas currently considered to have some internal consistency, become the history of ideas.
One of the tenets of "natural" mathematics is that it either is one of the natural sciences, or is intimately related to them. Cantor expressed his devotion to "natural" mathematics in his belief that chemistry and mathematics are the same thing. Of all his formulae, the most important is: chemical valence=cardinal number. Einstein himself seems to regard the shift to Darwinian biologyeffected by a substitution of assumptionsas paradigmatic of the shift from Newtonian to relativistic mechanics: "[a]s an example, a case of general interest is available in the province of biology, in the Darwinian theory of the development of species by selection in the struggle for existence, and in the theory of development which is based on the hypothesis of the hereditary transmission of acquired characteristics." 19 Is relativity a biological theory? Interestingly, and not surprisingly given his general inquisitiveness, Feynman visited "natural" coincidence when, apparently apropos of renormalization, he remarked: "perhaps the idea that two points can be infinitely close together is wrongthe assumption that we can use geometry down to the last notch is false. If we make the minimum possible distance…the smallest distance involved in any experiment today…, the infinites disappear, all rightbut other inconsistencies arise…." 20 Today we would have to say that that's rather good, and very nearly hits the mark. But not quite. In the event there is nothing to be done about the contradiction in which, due to this "natural" coincidence of points, we are led from an assumption of two Cartesian coordinate systems to a conclusion of one such system. 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. We might begin by asking: what "paradox" is the relativity of simultaneity designed to avoid, and what is the "paradox" it tries to express? It may well turn out that natural selection=natural coincidence, thus unifying biology and physics on the basis of an error. These new questions and formulation indicate a change in the direction of science.
"Natural" coincidence is the pea under the mattress of the Standard Model, and it makes us uncomfortable looking at the works of Sraffa, Kimura and Gödel in the old way. We can't go back and see them as they were seen in the twentieth centurythat point of view is no longer an option for us, it's over. For example, Einstein also has an algebraic formulation of special relativity; however, we see now that that formulation simply begs the question of the "natural" coincidence of points, it covers it up with a metric, a gap-filler. Indeed, Einstein probably found sanction for the algebraic formulation in the "natural" mathematics of the algebraist and "natural" mathematics advocate Ernst Schröder. 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 notif it lies in pieces of speculationwhat does that tell us about Sraffa's attitude to relativity? It's been a long time since anyone took Wittgenstein seriously, but perhaps his remarks on relativity, which always seemed to be useless to any professional physicist, 21 are now telling us that something is wrong. Is Duchamp also telling us something is wrong?
Like Kimura and Sraffa, Einstein suffered from an Achilles' heel: mathematics. Like them, he needed a mathematical expression for his ideas, and was seduced by the same intuitionist-style mathematics. The idea of a "natural" mathematics as a part of perception, reflects doubt that geometry or other forms of math express propositions, and a belief that perception and expression are one. We find it not only in economics, biology, mathematics and physicsand more evidence of it in chemistry than simply Cantor's yearning. The foundation of contemporary chemical theory is the "natural" mathematics of Condillac which found its way to chemistry, through Lavoisier, where it expresses Condillac's idea of "'analysis' as originating in simple sensory experiences, followed by the process of 'synthesis' in which the ideas were reconstructed in such a way that the relations between them were clearly revealed."22 What "paradox" is Condillac attempting to "avoid" as well as to "express," by this formulation?
Whatever the independent validity of the notion of "natural" mathematics, it is not logically incorporated in any of the arguments it seeks to express. It may well be that Richard's set E becomes Wittgenstein's "private language," Einstein's "natural coincidence," Kimura's "population," Sraffa's "product," and that we have a basis for unifying much of the work of the twentieth century through this anomalous concept. This understanding would mark the clearest break with twentieth-century civilization. At the very least, this exposure of "natural" mathematics has begun a revolution in chemistry, physics, economics and biology. As for mathematics, the Pythagorean theorem is itself probably an attempt to "avoid" as well as to "express" a "paradox." Which one?
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. Perhaps individuals we marginalizedand ideas we thought had been synthesized out of the argumentare now waiting to contribute something relevant. If "noise" matters, perhaps we should bring Bartók, Schoenberg and Webern into the discussion (but how?). Perhaps we can finally bring into alignment two concepts which rattle around in the twentieth century like two peas: chance and infinity. Einstein famously said that God does not play dice with the universe. What does he mean by chance (assuming he thinks dice is an example of chance) and God?
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 specializationthat is, with the development of terms of art within the disciplinesintellectual 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. During the twentieth century, we "couldn't" or "shouldn't" read both booksit would have been like professing two religions. Perhaps this essay will make possible an ecumenical approach.
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 and demanding 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 also freely grant influence within certain groups such as the Vienna Circle or through such well-connected figures as Frank Ramsey, whose ideas found expression in works as apparently diverse as those of Gödel, Wittgenstein and Sraffa. And then there is the ubiquitous Poincaré. We will go much further in this direction, and much faster, if we try to understand howregardless of the barriers which specialists felt surrounded their disciplinesthey nevertheless communicated in internally consistent ways across those barriers: and built bridges over them!
1 James K. Galbraith, "Keynes, Einstein and Scientific Revolution," The American Prospect, Volume 5, Number 16 (1994), pp. 62-67.
2 Motoo Kimura, The Neutral Theory of Molecular Evolution (1981), pp. 20-21.
3 "Discovering relativity beliefs: towards a socio-cognitive model for Einstein's Relativity Theory formation," Mind & Society, Volume 3, Number 5 (2002), pp. 93-109.
4 I take his remarks on experiments to be criticisms of relativity, but perhaps I am assuming too much about his knowledge.
5 Production, vi.
6 Ian Steedman, Marx after Sraffa, 1977.
7Alejandro Garciadiego, Bertrand Russell and the Origins of the Set-Theoretic 'Paradoxes', 1992. It is worth examining Richard's own formulation of his "paradox"even though it is rather lengthybecause it provides a sort of template for other paradoxes both ancient and modern. Are other paradoxes simply this argument and do they suffer from its flaw? "I am going to define a certain set of numbers, which I shall call set E, through the following considerations. Let us write all permutations of the twenty-six letters of the French alphabet taken two at a time, putting these permutations in alphabetical order; then, after them, all permutations taken three at a time, in alphabetical order; then, after them, all permutations taken four at a time, and so forth. These permutations may contain the same letter repeated several times; they are permutations with repetitions. For any integer p, any permutation of the twenty-six letters taken p at a time will be in the table; and, since everything that can be written with finitely many words is a permutation of letters, everything that can be written will be in the table formed as we have just indicated. The definition of a number being made up of words, and these words of letters, some of these permutations will be definitions of numbers. Let us cross out from our permutations all those that are not definitions of numbers. Let u1 be the first number definied by a permutation, u2 the second, u3 the third, and so on. We thus have, written in a definite order, all numbers that are defined by finitely many words. Therefore, the numbers that can be defined by finitely many words form a denumerable infinite set. Now, here comes the contradiction. We can form a number not belonging to this set. 'Let p be the digit in the nth decimal place of the nth number of the set E; let us form a number having 0 for its integral part and, its nth decimal place, p + 1 if p is not 8 or 9, and 1 otherwise.' This number N does not belong to the set E. If it were the nth number of the set E, the digit in its nth place would be the same as the one in the nth decimal place of that number which is not the case. I denote by G the collection of letters between quotations marks. The number N is defined by the words of the collection G, that is, by finitely many words; whence it should belong to the set E. But we have seen that it does not. Such is the contradiction." Pp. 141-142. As Garciadiego notes, Richard called this a "contradiction," not a paradox, and said that "the collection G had meaning only if the set E was defined in totality; this could not be done except with infinitely many words." This is also reproduced in van Heijenoort, Jean, ed., From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931, 2002.
For an equally devastating investigation, this time of the way in which physics renormalized itself out of any logical content whatsoever, see David Pickering, Constructing Quarks: A Sociological History of Particle Theory, 1984. Pickering's book is a good beginning for a study of the process by which the intellectual world defiled itself during the twentieth century. Feynman, by the way, had considerable contempt for renormalization: a "shell game," he called it. QED, 128. Nevertheless, it is a part of the "natural" mathematics gameand Feynman played it. Poincaré made sure to enumerate a series of laws for enforcing it, laws which he says are summed up as a "group." Science and Hypothesis (1952 edition), p. 64.
8 Garciadiego, op. cit., p. 24.
9 Kimura, op. cit., p. 50.
10 The latest expression of this point of view is self-confessedly ad hominem: "humans are so constructed as to conceptualize the world in terms of some simple fundamental categories (e.g., as comprised of individual objects standing in various relations; that the world, to a large extent, is properly described as so constructed (up to the point of quantum mechanics, at least); and that a rudimentary logic is implicit in these shared structures…." Maddy, Penelope, "Three Forms of Naturalism," in Shapiro, ed., The Oxford Handbook of Philosophy and Mathematics(2005), p. 450.
11 Garciadiego, op. cit., p. 134 and note 3.
12 The project of "avoiding" or "solving" the "paradoxes," comes almost immediately to dominate twentieth-century mathematics itself, with all the problems inherent in addressing issues which do not exist. It is worth noting that neither Ramsey nor Church nor Turing—nor later figures such as Gödel, Carnap or Tarski—ever considered whether the "paradoxes" might be simply meaningless. They all believed that these arguments had at least some logical content, and that that content had implications with which they had to deal. From this initial error, many other errors followed. As Garciadiego makes abundantly clear, the "problems" of the "paradoxes" proceeded in no way from logic, but instead, from Russell’s megalomania. See Feferman, A. and Feferman, S., Alfred Tarski: Life and Logic, 2004; Dokic, Jérôme and Engel, Pascal, Frank Ramsey: Truth and Success (2002) Alonzo Church had problems with definitions: "A function is a rule of correspondence by which when anything is given (as argument) another thing (the value of the function for that argument) may be obtained." The problem is the word "thing." This is Richard’s set E, which cannot be defined. The Calculus of Lambda-Conversion (1951), pp. 1-2. Church subscribes to the "definition of simple order in terms of the relation precedes," which he attributes to Cantor. However, this attribution is in the context of Cantor’s formulation of the notion of a set, a notion, as Garciadiego says, comprising "properties…so unsound that the theory seems to be the product of a charlatan." Introduction to Mathematical Logic (1958), note 541; Garciadiego, p. 9. Many of Alan Turing’s ideas are still highly respectable in mathematical circles, but this statement in "Systems of Logic Based on Ordinals" would hardly pass muster in today’s math world: "Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two faculties, which we may call intuition and ingenuity. The activity of intuition consists in making spontaneous judgments which are not the result of conscious trains of reasoning. These judgments are often but by no means invariably correct (leaving aside the question what is meant by ‘correct’). Often it is possible to find some other way of verifying the correctness of an intuitive judgment. We may, for instance, judge that all positive integers are uniquely factorizable into primes; a detailed mathematical argument leads to the same result. This argument will also involve intuitive judgments, but they will be less open to criticism than the original judgment about factorization. I shall not attempt to explain this idea of ‘intuition’ any more explicitly." Copeland, B. Jack, ed., The Essential Turing (2004), p. 192. Note that he regards mathematical reasoning as having only two components—not one, or four, or some other number. This base 2 system is a metaphor which traces itself back through Turing’s own bifurcation of the mathematical process to Brouwer’s own bifurcation of the operation of the human mind ("the connected and the separate, the continuous and the discrete")—all in an attempt to "avoid" the "paradoxes." Turing’s entire apparatus of calculability is designed to "restore" "distinctions."
13 Darwin's physical ideas should be looked for, in part, among his theological ideas. See Dov Ospovat, The Development of Darwin's Theory: Natural History, Natural Theology, and Natural Selection, 1838-1859, 1981.
14 Kimura, op. cit., p. 327.
15 Fifth edition (1952), pp. 19-20.
16 Howard, D. and Stachel, J., eds., Einstein the Formative Years 1879-1900, 2000.
17 Garciadiego, op. cit., p. 140.
18 Poincaré, op. cit., pp. 13, 58.
19 Einstein, op. cit., p. 142.
20 Feynman, op. cit., p. 129.
21 His grasp of Gödel's theorem was also feeble. See Timothy Bays, "On Floyd and Putnam on Wittgenstein on Gödel," Journal of Philosophy, Volume CI, Number 4 (2004), pp. 199-210.
22 Grattan-Guinness, Ivor, The Search for Mathematical Roots 1870-1940 (2000), p. 15.
The final version of John Ryskamp's poem The Twenty-First Century also appears in this issue.
An earlier version of The Twenty-First Century appeared in FlashPoint 7.