Når jeg starter forsøket om å oppfordre en uinnvidd til å tenke om Gud, pleier jeg å bruke en analogi til matematikk. Tenk deg en immateriell realitet som vi mennesker på en eller annen merkverdig måte har tilgang til gjennom vårt intellekt, og som korresponderer til de fysiske tingene vi opplever omkring oss hver dag. For det virker som om matematikk aldri i prinsippet kan reduseres ned til materielle årsaker, fiktive hjelpemidler eller til å bare være forårsaket av nerveceller i hjernen noen dag.
Matematikk er en objektiv, sikker og bestemmende øvelse av et kaliber som ingen materielle prosesser er. Vår evne til å utføre slike øvelser viser også at menneskesinnet vårt må ha et immaterielt aspekt, som James Ross gir et glitrende argument for i quaddisjonseksempelet her. Likevel virker matematikk til å være uavhengig av både mitt og ditt sinn. Når vi snakker om matematiske enheter som det arabiske numeralet «4» eller det romerske numeralet «IV», og foretar kalkulasjonen «2+2=4» eller «II epler+II epler=IV epler», utfører vi begge nøyaktig det samme. Men det er ikke snakk om materielle årsaker i hjernen min som heldigvis korresponderer til materielle årsaker i hjernen din. Kvantiteter, sett, strukturer, symmetrier, kontinuiteter og perfekte triangler har en ekte relasjon til virkeligheten, og Pytagoras’ teorem, summen av vinklene i en rettvinklet trekant og enkel aritmetikk var sant lenge før det første mennesket kom inn i eksistens, og vil fortsette å være det til universets ende.
Vi oppfinner ikke matematikk – vi oppdager det – som nok et iboende aspekt ved et fabelaktig skaperverk. Denne type platonsk eller aristotelisk realisme til f.eks. matematikk står i kontrast til det vi kaller for nominalisme, som holder til at matematikk bare er et slags språk som vitenskapen benytter, og at alle disse tingene bare eksisterer inne i hjernen din og ikke har noen nødvendig, objektiv relasjon til virkeligheten. Samme argument kan også gis for logiske, nødvendige proposisjoner som det at to gjensidig utelukkende påstander ikke kan være sanne samtidig, eller for universalier som f.eks. rødhet, triangularitet og menneskehet. Jeg har utfordret nominalisme ved en rekke argumenter nederst i denne posten.
Hvis du vil lære mer om en aristotelisk teori for matematikk, kan du lese James Franklins artikkel her.
Så du forstår nå hvordan intellektet vårt kan ha en type direkte tilgang til immateriell matematikk, som igjen kan påvirke hverdagen vår som begrensede, dødelige mennesker? Vel, da er du på vei til å se hvordan mennesket til alle tider også kan ha reell kunnskap om den transcendente, immaterielle og evige Gud. Og det bringer oss til David Berlinski…
«But asking someone like Richard Dawkins about the evidence for God’s existence is a little like asking a quadruple amputee to run the marathon.
The interesting point is elsewhere. There is no argument against religion that is not also an argument against mathematics. Mathematicians are capable of grasping a world of objects that lies beyond space and time.»
Jeg kom over et intervju med denne agnostiske matematikeren og sekulære jøden, som er en skeptiker av darwinisme, samt en selvbeskrevet vennlig kritiker av Intelligent Design. Men sammen med ateisten Thomas Nagel (som jeg har skrevet poster om her og her), er Berlinski også en forfriskende ekte og sjelden fritenker, selv om man ikke trenger å være enig i alt han ytrer. Kontra de mest vanlige «fritenkere» i kulturen vår som tror definisjonen på fritenkning er «avvise alt som ikke passer med virkelighetsforståelsen og kunnskapsteorien jeg allerede har bestemt meg for at jeg vil ha – selv uten gode argumenter i min favør».
Mannen er dessuten vittig til tusen, så du vil nok like intervjuet jeg har gjengitt under mellom ham og Jonathan Witt. Mye læring å hente. Hvis du liker mannen, er det vel verdt å ta en titt på boka hans. Hvis du, som jeg, med diverse poster ute på web de siste ukene (her, her, her, her og noen bidrag her, her og her), himler med øya over ateister som tror at de kan ytre det magiske ordet «naturvitenskap» i troen på at det skal fungere som et get-out-of-jail-free-card i deres favør uten videre spørsmål eller undersøkelse, så er den en fin innføring til å ta diskusjonen litt under dette banalt overfladiske. En annen du trenger, er naturligvis klassikeren til E. A. Burtt.
… Why do you think the debate about Darwin’s theory of evolution has taken on such a nasty turn?
David Berlinski: Nasty, eh? If so, the nastiness is not entirely ecumenical. As far as I can tell, only one side is now occupying the gutter, even though the gutter is, as gutters generally are, more than spacious enough for two. But you raise a good question. Why are Darwinian biologists so outraged? Like the San Andreas fault, the indignation conspicuous at blogs such as The Panda’s Thumb or Talk Reason is now visible from outer space.
There is a lot at stake, obviously. Money, prestige, power, influence – they all play a role. Darwinism is an ideological system and when such systems come under threat, their supporters react in predictable ways. Freedom of thought very often appears as an inconvenience to those with a position to protect. Look at the attempts made to humiliate Rick Sternberg at the Smithsonian Institute, or the campaign now underway to do the same thing to Guillermo Gonzalez at Iowa State. There is nothing surprising in all this. I myself believe that the world would be suitably improved if those with whom I disagreed were simply to shut up. What is curious is how quickly the Darwinian establishment has begun to appear vulnerable ….
… Not to scientists …
DB: No, perhaps not. But to everyone else. Consider the latest Pew poll. “Two-thirds of Americans,” the New York Times reported, “say that creationism should be taught alongside evolution in public schools.” But even among those quite persuaded of Darwin’s theory, “18 percent said that evolution was ‘guided by a supreme being.’” Now these are astonishing figures. They represent an authentic popular revolt against elite thought. I cannot remember anything like it. The fact that so many Darwinian biologists are utterly tone-deaf when it comes to debate has hardly helped their case. It is no small thing to have appeared before the American public in a way that suggests both illimitable arrogance and scientific insecurity.
… With all due respect, Mr. Berlinski, there are times reading what you have written when it seems that you are right down there in the gutter with the best of them. You did, after all, refer to Richard Dawkins as – and I quote – “a remarkably reptilian character” ….
DB: Did I? Well, mine has been an exercise in defensive slumming.
… I see. What really accounts for your hostility to figures such as Daniel Dennett and Richard Dawkins? …
DB: In the case of Daniel Dennett, I think contempt might be a better word than hostility, and indifference a better word still. There are, of course, lots more where he came from – P.Z. Myers, for example, or Eugenie Scott, or Jason Rosenhouse. Throw in Steven Weinberg, just to reach an even number ….
… The Nobel Laureate? …
DB: None other.
… But Dawkins …
DB: An interesting case, very louche – fascinating and repellant. Fascinating because like Noam Chomsky he has the strange power effortlessly to command attention. Just possibly both men are descended from a line of simian carnival barkers, great apes who adventitiously found employment at a circus. I really should look at this more closely. Repellent because Dawkins is that depressingly familiar figure – the intellectual fanatic. What is it that he has said? “It is absolutely safe to say that, if you meet somebody who claims not to believe in evolution, that person is ignorant, stupid or insane (or wicked, but I’d rather not consider that)”. Substitute ‘Allah’ for ‘evolution,’ and these words might have been uttered by some fanatical Mullah just itching to get busy with a little head-chopping. If he ever gets tired of Oxford, Dawkins could probably find a home at Finsbury Park.
… You do not, I gather, think much of the kind of atheism Dawkins is concerned to promote …
DB: It’s pretty much the sort of stuff Bertrand Russell used to put out when he needed to knock-off a popular best-seller or dazzle one of his mistresses. You see, my dear, belief in god is no better than belief in a teacup orbiting Mars, whereupon my dear would generally begin loosening her undergarments. The fact is that these kinds of arguments have been known to embarrass a wart hog. This has been tested at zoos, by the way, and the experiments widely reported.
… But why should we take seriously religious beliefs that are lacking in evidence?
DB: We shouldn’t. But asking someone like Richard Dawkins about the evidence for God’s existence is a little like asking a quadruple amputee to run the marathon.
The interesting point is elsewhere. There is no argument against religion that is not also an argument against mathematics. Mathematicians are capable of grasping a world of objects that lies beyond space and time ….
… Come again …
DB: No need to come again: I got to where I was going the first time. The number four, after all, did not come into existence at a particular time, and it is not going to go out of existence at another time. It is neither here nor there. Nonetheless we are in some sense able to grasp the number by a faculty of our minds. Mathematical intuition is utterly mysterious. So for that matter is the fact that mathematical objects such as a Lie Group or a differentiable manifold have the power to interact with elementary particles or accelerating forces. But these are precisely the claims that theologians have always made as well – that human beings are capable by an exercise of their devotional abilities to come to some understanding of the deity; and the deity, although beyond space and time, is capable of interacting with material objects.
… And this is something that you, a secular Jew, believe? …
DB: What a question! I feel like I’m being interviewed by the Dean at some horrible community college. Do you believe in the university’s mission – that sort of thing. Look, I have no religious convictions and no religious beliefs. What I do believe is that theology is no more an impossible achievement than mathematics. The same rational standards apply. Does the system make sense; does it explain something? Are there deep principles at work. Is it productive?
… You know, Dawkins, at least, is quite clear that insofar as religion is expressed as a sense of wonder, he counts himself a religious man ….
DB: … Sure. But that’s because he has found it remarkably convenient to associate his views with those of Albert Einstein – you know, the standard starry sky at night, my goodness the universe is wonderful routine. Why should Dawkins, of all people, find the universe wonderful if he also believes it is largely a self-sustaining material object, something bigger than a head of cabbage but not appreciably different in kind? The whole place supposedly has no meaning, no point, no purpose, and no reason for its existence beyond itself. Sounds horrible to me. Wonder is the last reaction I’d expect. It’s like being thrilled by Newark, New Jersey. A universe that is nothing more than a collection of atoms whizzing around in the void is a material slum …
…How would you react to the argument that Dawkins has made that any form of religion that goes beyond the scientific facts about the universe really represents a form of brainwashing …
DB: He’s probably right. Most education is a form of brainwashing – so much better in French, by the way, lavage de cerveau. Give a child to the Jesuits, they say, and ten years later the man will cringe when he spots the Cross. But look, ten years or so spent studying physics is a pretty effective form of brainwashing as well. You emerge into the daylight blinking weakly and talking about an endless number of universes stacked on top of one another like an old-fashioned Maine pancake breakfast. Or you start babbling inanely about how meaningless the universe is. But if you ask me just who is the more credulous, the more suggestible, the dopier, the more perfectly prepared to convey absurdity to an almost inconceivable pitch of personal enthusiasm – a well-trained Jesuit or a Ph.D. in quantum physics, I’ll go with the physicist every time. There is nothing these people won’t believe. No wonder used-car salesmen love them. Biologists are, of course, worse. Tell them that in the future Richard Dawkins is going to conduct a personal invasion of Hell in order to roust the creationists, and The Panda’s Thumb will at once start vibrating with ticket sales.
… Perhaps this isn’t the most productive of topics to pursue …
DB: That’s fine. You lead, I’ll follow …
…Can you say a little bit more by what you mean by an ideological system?…
DB: Marxism is an ideological system, or was, and Darwinism is like Marxism. Darwinism, I must stress, the sibilant distinguishing the man from his message. By itself, Darwin’s theory of random variation and natural selection would simply be a hopelessly premature 19th century thought experiment, vastly less important than Clerk Maxwell’s theory of the electromagnetic field, which was completed at roughly the same time. But like confined quarks (or any number of quacks), Darwin’s theory never appears by itself in contemporary thought …
… Let me interrupt you. Can you be a little clearer on the difference, as you see it, between Darwin’s theory and Darwinism? …
DB: It is a matter of attitude and sentiment, Look, for thousands of intellectuals, becoming a Marxist was an experience of disturbing intensity. The decision having been made, the world became simpler, brighter, cleaner, clearer. A number of contemporary intellectuals react in the same way when it comes to the Old Boy – Darwin, I mean. Having renounced Freud and all his wiles, the literary critic Frederick Crews – a man of some taste and sophistication – has recently reported seeing in random variations and natural selection the same light he once saw in castration anxiety or penis envy. He has accordingly immersed himself in the emollient of his own enthusiasm. Every now and then he contributes an essay to The New York Review of Books revealing that his ignorance of any conceivable scientific issue has not been an impediment to his satisfaction.
Another example – I’ve got hundreds. Daniel Dennett has in Darwin’s Dangerous Idea written about natural selection as the single greatest idea in human intellectual history. Anyone reading Dennett understands, of course, that his acquaintance with great ideas has been remarkably fastidious. Mais, je divague …
In the case of both Crews and Dennett, it’s that God-awful eagerness to explain everythingthat is the give-away. The eagerness is entirely academic or even literary. But, you know, what sociologists call prole-drift is present even in a world without proles. Look at Christopher Hitchens – very bright, very able. Just recently he felt compelled to release his views on evolution to a public not known eagerly to be waiting for them. What does he have to say? Pretty much that he doesn’t know anything about art but he knows what he likes. The truth of the matter, however, is that he pretty much likes what he knows, and what he knows is what he has heard smart scientists say. Were smart scientists to say that a form of yeast is intermediate between the great apes and human beings, Hitchens would, no doubt, conceive an increased respect for yeast. But that’s a journalist for you: all zeal and no content. No, no, not you, of course. You’re not like the others.
… Thank you, I’m sure. I am still not sure what you are getting at when you refer to Darwinism as an ideological system? Many biologists such as Paul Gross simply reject the term altogether …
DB: Yes, I know. The term – Darwinism, I mean – has been a long standing banana peel for poor Gross. No matter how often he swears not to slip, he can inevitably be spotted straddling that banana and about to slip-up all over again. Ah, there he goes – vawhoomp. I have a service that lets me know every time Gross topples.
But enough about Gross. Let’s get back to me. It’s not that easy to say what Darwinism amounts to, but then again, it was never easy to say what Marxism amounted to either. If you look at Marxists journals from the 1930s, the party line shifted all the time, so much so that in the 1940s, Stalin had to sit down and write an account of the principles of socialism. It reads very much like a high-school textbook in biology – a very sophisticated high-school textbook, of course. The real mark of an ideological system is its presumptuousness. There is nothing it cannot explain by means of a few trite ideas. Why is romantic love a sign of bourgeois decadence, Comrade? Because, Comrade, it represents a form of false consciousness. In Darwinism, natural selection has displaced such old standbys as false consciousness or the class struggle, Comrade. You don’t mind if I call you Comrade? It’s the least I can do ….
… But …
DB: Take the short essay in a most recent issue of The London Review by Thomas Jones, one of the review’s editors – no dope, by the way. “Since we use our brains to make up stories, and to make sense of the stories of others,” Jones says, “it is hard to disagree with the idea that the capacity for storytelling is the result of evolution.”
And here’s something Stephen Pinker said, it’s even better …
… But look, someone like Jones is simply stating the obvious – like everything else, literature must be understood in evolutionary terms. What other terms are there? …
DB: Why must literature be understood in any terms beyond the literary? Just recently someone named David Barash – an evolutionary psychologist, it goes without saying – published a book together with his wife called Madame Bovary’s Ovaries. Her ovaries? Look, set aside the appalling vulgarity of the book and its title, its almost unfathomable literary and intellectual crudeness. To talk about Madame Bovary’s ovaries is a little like looking at one of Rembrandt’s late self-portraits of his face and wondering whether the man suffered from bunions. What we know of the man is right there on the canvass. Nothing else. To imagine that somehow there is a real woman to be found in Flaubert’s nacreous masterpiece is to regard art the way an infant or a primitive regards art.
If you think you can take the story of Anna Karenina and connect its meaning to any anatomical, physiological, neurological, or biochemical feature of the human brain as it is now understood, by all means go ahead. We do not know how the human brain establishes that the word ‘cat’ designates a cat. Or that it does. Or that it has. Or that it can.
But even setting that aside, what reason do we have for supposing that differential reproduction tells us anything more about the anatomy of criticism than the class struggle tells us about the anatomy of love? That’s a learned reference, by the way …
… I have read Northrop Frye, Mr. Berlinski …
DB: Glad to hear it. Then you understand how pointless it is to coordinate our remarkable human powers with a filter so crude as the biological desire to promote oneself into the main chance.
You wouldn’t argue that the capacity for carpet-weaving is the result of evolution, would you?
… Yes, I would …
DB: Well, you would be wrong. Men and women make up stories, wander around foreign cities, take up sky-diving, invent financial swindles, learn to speak Mandarin, or weave carpets out of silk pretty much because they feel like it. Evolution has nothing to do with it.
… But how they feel and the decisions they make are shaped by evolution …
DB: That’s trivially true. If human beings did not have the kinds of brains they do, they wouldn’t make the choices they do. Different livers would probably lead to different choices, too, and who knows what a man shaped by evolution to have six sexual organs might contemplate.
… Oh please, isn’t that just clever word play? If the human brain did not arise by evolution, how did it arise? …
DB: I have no idea. It’s not my problem.
… That is an awfully convenient out for you …
DB: Sure. It’s the same out that Darwinian biologists take when it comes to the origins of life. Not our problem. What’s good enough for Richard Dawkins is good enough for me.
… How do you see Darwinism in the larger context of social or academic attitudes …
DB: A congeries of sentimental attitudes are at work in the humanities – atheism, moral relativism, materialism. They are incarnated locally in the United States by Richard Rorty, a philosopher, I must say, who while espousing irony as an antidote to anomie (and anything else that ails you) seems to me, at least, to exhibit an almost elephantine earnestness in everything he writes. The man could paralyze an infantry battalion just by beginning a lecture. I may have to consult with my spies in the Pentagon about this. Within the sciences, the governing attitude is often designated by the word ‘naturalism,’ especially by those sophisticated enough to know that adverting to the Temple of Reason after Robespierre might not be a good idea.
… Meaning? …
DB: Hard to say – again. Naturalism is sometimes taken to mean that there is only one body of human knowledge, and that is contemporary science; at other times, it is taken to mean that there is only one method by which knowledge can be acquired, and that is the scientific method. This is a little like arguing that cabbage is the only food and that prayer is the only way to get it.
… Why? …
DB: Mathematics is a counter-example to the first thesis, and the law, a counter-example to the second. In any case, science has no more method than golf …
… You don’t believe that …
DB: You mean about the scientific method? Certainly I do. Where science has a method, it is trivial – look carefully, cut the cards, weigh the evidence, don’t let yourself be fooled, do an experiment if you can. These are principles of kennel management as well as quantum theory. Where science isn’t trivial, it has no method. What method did Einstein follow, or Pauli, or Kekulé? Kekulé saw the ring structure of benzene in what he called a waking dream. Some method.
… I wonder whether we could get back to naturalism …
DB: A vos ordres. Carl Sagan seems to have captured the emotional content of naturalism when he remarked that the universe is all that there is, was, or would ever be. A curious sentence, don’t you think, and one that embodies a curious claim? Its denial is a contradiction, and so the claim is itself a logical triviality. This has not discouraged any number of commentators from embracing it warmly. Eugenie Scott is a small squirrel-like creature who is often sent out to defend Darwin. Whenever doubts are raised, she withdraws a naturalistic nut from her cache and flaunts it proudly. And if naturalism won’t do, there is always methodological naturalism. One nut is, after all, pretty much as good as another.
… What is the connection between Darwinism and naturalism? …
DB: There is none – at least if by a connection, you mean a logical connection. There is, however, a sentimental connection. A commitment to naturalism, however defined, very often makes Darwin’s theory seem more plausible than it otherwise might be. Naturalism is sentimentally a sufficient condition for Darwinism. By the same token, Darwinism is sentimentally a necessary condition for naturalism. Richard Lewontin has made this point explicitly, by the way. The point is elementary but it explains a good deal, as so many elementary points do. Biologists persuaded that there is nothing out there but atoms and the void are naturally made apprehensive by the thought that Darwin’s theory might be false, for in that case, it follows by contraposition that naturalism might be false as well.
… What do you think accounts for these sentimental connections, as you put it …
DB: Fashion, for one thing. It’s what everyone seems to be saying in the faculty dining room at Mongaheela State Community College, or at The New York Review of Books, much the same environment, now that I think about it. A good deal of this is changing, I should hasten to add, as academics prepared to sneer at religious experience or moral absolutes remember just who happens to pay their salaries. This consideration alone has a wonderfully clarifying effect on one’s theoretical commitments.
… If Darwinism is so unworthy of respect, what is the appeal of Darwinism? After all, a great many scientists disagree with you. They can’t all be fools, after all…
DB: I’m not sure why not.
… I’d like better to understand your views on science. You talk very often of, and I quote, “the serious sciences.” I take it you mean to exclude biology altogether. Is that your view? …
DB: To a certain extent. My real view is that there is only one science, and that is mathematics, and that the physical sciences are really forms of experimental mathematics. The idea that there is out there a physical world which just happens to lend itself to mathematical description has always seemed to me to be incoherent. There is only one world – the universe, in fact, and it has the essential properties of a mathematical model. For reasons that we cannot even begin to understand, that model interacts with out senses, and so without measuring devices, allowing us to pretty much confirm conclusions antecedently reached by pure thought.
But to tell you the truth, I’m not at all sure I understand my own views, remarkable as they are.
… I’m sure that in this you are not alone, Mr. Berlinski …
DB: No doubt. But it is odd, isn’t it, that we really have no good views about science itself. Its existence is as much of a mystery as the phenomena that it explains. I know of nothing like an imagined overall theory that even begins to explain the role of science in the universe. No theory explains itself, after all, even if it could explain everything else.
… I’m not sure what you mean …
DB: Suppose one had a fabulous final theory. The universe is made up ultimately of wriggling strings – or whatever. The theory would not explain itself in the simple sense that unless the theory is in some odd and perverse sense self-referential, it would leave something out – the reasons why it just happens to be true. For that, one would have to deduce the theory from something else, and so far as we know or understand, deduction is itself a relationship between theories.
… But how is this connected …
DB: Not to worry. It’s probably not.
… Mr. Berlinski, you have frequently been accused of being a crank, someone more generally participating in what has come to be called crank science. I know that …
DB: So?
… Well, is the accusation one that you accept? …
DB: Sure. It’s obviously true in essence, although I prefer to describe myself as an iconoclast, one whom history will vindicate …
… No doubt …
DB: But the point is the same, whatever the terms. But speaking of terms, maybe I spoke too soon. Look, it’s one thing to say that someone like me is a crank. That’s fine because it’s true. It’s quite another thing to talk about crank science.
… Surely crank science is what cranks do? …
DB: Surely. But that is not how the term crank science has come to be used. Look at someone like Jeremy Bernstein – a good physicist and a very good writer about physics. He means something quite specific by the term crank science, and that is a willingness to deny the cumulative structure of modern physics, the fact that each great physical theory represents an enlargement of its predecessors. This is terrifically important as a rhetorical strategy because it means that the burden of skepticism becomes impossibly high with each new theory. This is just another way of protecting the sciences from criticism. To go on the attack, it is not enough to say, hey look, this particular theory is wrong, or absurd, or preposterous. You must instead take on the entire history of a tradition. Not quite sporting, I say.
… Yes, but isn’t it true? Science is cumulative and the more it accumulates the greater the weight of evidence in its favor …
DB: Yes, this is the claim. Steven Weinberg has made it explicitly. He at least knows of no advance in physical theory that has really overturned previous developments.
… How could you possibly object to that? …
DB: How? By remarking that it’s just nuts, that’s how. Weinberg is a very good physicist, but as an intellectual historian he rather resembles a horse put to work in a glass factory. He can’t help it, of course, it’s just not his métier. He gives that pompadour of his a shake, and a dozen fragile figurines just topple. Far from being cumulative, it’s the reverse that’s more really true. Let’s try and be just a little bit more precise. What’s a theory, for example? Now I’m an old logic hand and the only answer I know is that a theory in the physical sciences is just like a theory in mathematical logic – a consistent set of sentences satisfied in a model. Not the best way of putting things, but so far as I know, the only good way. Now take Newtonian mechanics and compare it to general relativity. Is it true that GR is a consistent extension of Newtonian mechanics?
… Surely many physicists would say so …
DB: Yes, and they would be wrong. Newtonian mechanics is committed to the view that the spatial structure of the universe is classically Euclidean. Not so GR. Newtonian mechanics holds that if you accelerate a rigid rod, neither its length nor certain temporal intervals will change. GR holds the opposite. But why am I telling you all this. It’s obvious.
… But Mr. Berlinski, no one would deny these points? GR is an extension of Newtonian mechanics. It goes further and because it does, we see better …
DB: An extension, maybe, but a consistent extension? Never. Consistent? If so, then Newtonian mechanics and GR must be satisfied in the same model by the compactness theorem. But how can a single mathematical model satisfy the postulates of both theories? It just can’t be done. No, no, I’m not appealing to anything like a paradigm shift. It’s perfectly possible to compare Newtonian mechanics and GR. One theory is better than the other. It explains more. It reaches for deeper principles. It is more elegant. I’m talking about Newtonian mechanics, of course. But the intersection of the set of sentences in both theories is inconsistent and so satisfied in no model whatsoever. If this is so, then the whole image of science as a cumulative structure breaks down. What one really has is a collection of cathedrals on a kind of fruited plane [sic!]. Some are taller and grander than others, others are smaller and more elegant. No one cathedral is really built on top of the other.