Justified True Belief

(●) The logician who proved that no rulebook can ever contain the whole of arithmetic, and who spent the rest of his life working out what that meant for the mind, for the numbers themselves, and for God. + In 1930 there was a plan for the whole of mathematics. David Hilbert, the most respected mathematician alive, had proposed that every mathematical truth be derivable from a short list of starting assumptions by mechanical rules, and that the list be proved free of contradiction using nothing but finite, checkable steps. No question would be permanently closed. Hilbert’s slogan was that in mathematics there is no ignorabimus, no "we shall not know." That September, at a conference in Königsberg, a shy twenty-four-year-old from Vienna mentioned, almost in passing, that the plan could not work, and that he had proved it. He matters to Christian readers for reasons that have nothing to do with the slogans his theorems get wrapped in. Week after week he sat in a room full of philosophers who held that talk about God is not false but meaningless, the way "Tuesday is purple" is meaningless, and he disagreed with every one of them in silence. He was a convinced theist, and he put it in writing where it could be checked. He held that numbers and sets are real things we perceive rather than tokens we invented. He argued to his mother, in letters he never expected anyone else to read, that a rationally ordered world must have another life in it. And in a drawer he kept a formal proof of the existence of God, written out in the symbols of modal logic, which he let a colleague copy in 1970 and never published in his lifetime. One fact about him is worth holding on to, because it explains why so much of him arrived late. His published work would fit in a thin folder, perhaps two dozen papers, one of them a twenty-five-page article in a Vienna journal that ended a thirty-year program. Almost everything else he thought went into notebooks kept in Gabelsberger shorthand, a nineteenth-century German stenography that had gone out of use and that hardly anyone could still read. Scholars are transcribing them to this day.

(●) 1906 AD: Born in Brünn to a German-speaking textile family, a boy the household called Mr. Why. + Kurt Friedrich Gödel was born on 28 April 1906 in Brünn, a manufacturing city in the Austro-Hungarian province of Moravia and today Brno in the Czech Republic. His father Rudolf managed and later part-owned a textile firm; his mother Marianne, educated in France, kept a comfortable German-speaking household in a largely Czech-speaking city. The boy asked questions and kept asking, and the family nickname stuck: der Herr Warum, Mr. Why. At six or seven he came down with rheumatic fever, a joint and heart illness that was common and frightening in the years before antibiotics, and he recovered from it completely. That should have been the end of the matter. Instead, a year or so later, he looked the disease up in the family’s medical books, read that it can damage the heart, and concluded that his own heart had been damaged. Nobody could talk him out of it. He was wrong, and he never got free of it, and a good deal of what went wrong in the last decade of his life is already visible in that eight-year-old with a medical dictionary open on his knees. From 1916 to 1924 he worked his way through the Deutsches Staats-Realgymnasium in Brünn, the German-language state secondary school, and he was good at everything in it: languages and religion quite as much as mathematics. In his whole school career he was marked below the top exactly once, and the subject, of all subjects, was mathematics. His brother Rudolf remembered that by the final years Kurt had already mastered university mathematics, to the astonishment of his teachers and of the other boys.

(●) 1924 AD: He went to Vienna meaning to be a physicist, and mathematics took him instead. + He enrolled at the University of Vienna at eighteen, intending to read theoretical physics, and two things turned him. One was a course of lectures on number theory, the study of the whole numbers and how they behave, given by Philipp Furtwängler, who was paralysed and lectured from a wheelchair while an assistant wrote on the board, and who packed the hall anyway. The other was Hans Hahn, a mathematician with a serious appetite for philosophy, who in 1926 brought his quiet student along to the discussion group that met around Moritz Schlick. That group became famous as the Vienna Circle, and its doctrine was logical positivism: roughly, that a sentence says something only if experience could check it or if it is true by definition alone. "There is a cat in the kitchen" you can settle by walking to the kitchen; "every bachelor is unmarried" is true by what the words mean; and anything that is neither, on their view, is not false but empty, which is where they filed the sentences of metaphysics and theology. Gödel came, sat, and listened. By his own later account he disagreed with nearly all of it from the very beginning, and he said so to almost nobody. Members remembered him as the silent one at the table. In 1929, under Hahn, he finished a doctoral dissertation proving the completeness of first-order logic, which is the everyday logic of "all," "some," "and," "or" and "not" applied to individual things. What he showed is that every statement which comes out true in every structure satisfying a set of axioms can also be derived from those axioms by the rules, or, less formally, that in this part of logic everything true everywhere is provable. It was an elegant result, and it pointed exactly the way Hilbert wanted to go. His father died in the same year.

(●) 1930 AD: At twenty-four he showed that arithmetic can never be finished, and one man in the room understood him. + The Second Conference on the Epistemology of the Exact Sciences met at Königsberg from 5 to 7 September 1930, and the programme was a three-cornered debate about the foundations of mathematics, one school of thought to a corner. Gödel read a short paper on his completeness theorem, which was welcome news for the plan everyone in the hall had come to advance. Then, on the final day, in the middle of a roundtable discussion, he remarked that one could give examples of statements that are true and yet not provable in the usual formal systems. You would expect a room like that to stop dead. It did not: the published record of the discussion shows almost no reaction at all, and the conversation moved on. One man had heard it. John von Neumann, who missed nothing, sought him out afterwards and saw at once what had been done, and within weeks he had worked out the unprovability of consistency for himself. He wrote in November to tell Gödel about it, only to learn that Gödel had it already and had put it in the paper. The paper came out the following year in the Monatshefte für Mathematik und Physik under the title "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I," which is to say "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I." It ran to twenty-five pages, and the "I" promised a second part that never appeared, for the simplest of reasons: nobody disputed the first. What it contains (T1) did not damage Hilbert’s program. It closed it. Vienna accepted the paper on 1 December 1932 as his habilitation, the second thesis a German-speaking academic must produce before he is allowed to teach, and in March 1933 he became a Privatdozent, which is a lecturer with no salary who lives on whatever fees his students bring in.

(●) 1936 AD: Moritz Schlick was shot dead on the university steps, Gödel broke down for the second time, and Adele Porkert, whom his family had opposed for eleven years, married him in 1938 and spent the next forty years keeping him alive. + The decade took him apart. He lectured at Princeton over the winter of 1933 to 1934, met Albert Einstein there for the first time, and collapsed on the way home, spending months in a sanatorium being treated for depression. Then, on 22 June 1936, Moritz Schlick, the philosopher who had built the Vienna Circle and had made room in it for him, was shot dead on the steps of the university by Johann Nelböck, a former doctoral student of his. Gödel broke down a second time. The fear of being poisoned that would eventually kill him dates from about these years. One thing in his life was steady, and it was the thing his relatives thought least of. He had met Adele Porkert in 1927, when she was working at a Vienna nightclub called Der Nachtfalter, across the street from where he lived. She was six years older than he was, divorced, a trained dancer, and not at all the wife a Gödel was expected to bring home, and his family opposed the match for more than a decade. The two married in a civil ceremony in September 1938 anyway. She looked after him for forty years and tasted his food in front of him when he was afraid of it, and there is a strong case that she is the reason he lived as long as he did. In those same years, with all of this going on around him, he did the second great piece of mathematics of his life, announced at the end of 1938, and it is worth slowing down for. Set theory is the study of collections, and by the 1930s it had two famous unsettled questions. The axiom of choice says that from any family of non-empty boxes you can pick one item out of each, even when there are infinitely many boxes and no rule that tells you which item to take. Cantor’s continuum hypothesis says that there is no size of infinity in between the counting numbers, 1, 2, 3 and onward, and the points on a line. Mathematicians had argued over both for decades and settled neither. What Gödel did was settle half of each. He built a model of the standard axioms of set theory, meaning a miniature mathematical world in which all of those axioms come out true, and he built it out of "constructible" sets, sets that can be spelled out stage by stage rather than merely declared to exist. In that world both disputed statements hold, so neither of them can be disproved from the standard axioms. Paul Cohen finished the story in 1963, by a quite different method, showing that neither can be proved from them either. The axioms, in other words, do not decide the question at all, which is exactly the sort of gap (T1) says to expect.

(●) 1940 AD: He crossed Siberia to reach Princeton, where Albert Einstein became the friend of his life. + Austria was absorbed into Germany in March 1938, and one by one the props went out from under him. The rank of Privatdozent was abolished, his applications for the post that replaced it dragged on without an answer, and a medical board looked him over and found him fit for military service. He was not Jewish himself, but his whole intellectual world had been, and by the winter of 1939 the roads west were shut. So in January 1940 he and Adele went east instead, which meant the Trans-Siberian Railway across the whole width of Russia, a ship out of Japan, San Francisco on 4 March 1940, and then a train the other way across America to New Jersey. He never saw Europe again. Princeton gave him an ordinary membership at the Institute for Advanced Study in 1940, a permanent membership in 1946, and, remarkably late for a man of his standing, a full professorship in 1953. What Princeton also gave him was Einstein. The two walked to and from the Institute together almost daily for fifteen years, talking German the whole way, and the economist Oskar Morgenstern recorded Einstein saying that his own work no longer meant much to him and that he came in to the Institute merely to have the privilege of walking home with Gödel. That is worth sitting with for a moment, because of who is saying it. Two episodes from those years belong side by side, and they show the two sides of him. On 5 December 1947, at his citizenship hearing in Trenton, with Einstein and Morgenstern beside him as witnesses, Gödel told Judge Phillip Forman that he had found an internal inconsistency in the Constitution of the United States by which the country could lawfully be turned into a dictatorship. Forman did not take the bait, Einstein steered the conversation elsewhere, and Gödel became an American. And for Einstein’s seventieth birthday in 1949 he produced the sort of present only he would think of: an exact solution to the equations of general relativity describing a rotating universe in which a traveller can, in principle, follow a path around into his own past.

(●) 1961 AD: His mother asked whether they would see each other again, and he sat down and wrote out his reasons. + Marianne Gödel had stayed in Europe, and mother and son wrote to each other for decades. In 1961 she put a direct question to him: did he believe in a Wiedersehen, a seeing-again? Between July and October he answered her in a series of letters, patiently, in plain German, with none of the guardedness he used at the Institute (T5). The world is rationally ordered, he told her. A rational order does not build a creature capable of so much and then let it finish almost none of what it was built for. So he expected another existence, in which what was begun here gets completed. At the same time he was pressing these convictions in his professional work and getting nowhere with publishing them. Six separate drafts survive of an essay called "Is mathematics syntax of language?", written for a volume on Rudolf Carnap, one of the leaders of the old Vienna Circle, and he submitted none of the six. A lecture from 1961 that sorts philosophies by whether they lean toward metaphysics and theology or away from them stayed in his papers as well. Then, in February 1970, believing himself to be dying, he let the logician Dana Scott copy out a page he had been carrying since about 1941: a formal proof of the existence of God (T4). Morgenstern noted in his diary that August that Gödel was holding back from publication because he was afraid people would conclude that he actually believed in God, whereas he was only, as he put it, carrying out a logical investigation.

(●) 1978 AD: He starved to death at seventy-one, certain that his food was poisoned. + The fear he had carried since the 1930s finally lost its counterweight, and the counterweight had always been Adele. For years he had eaten only what she prepared and tasted in front of him, and that arrangement held for as long as she was well. In the summer of 1977 she had a stroke, then major surgery, and was in hospital for months. During those months he stopped eating. He was admitted to Princeton Hospital and died there, sitting in a chair, on 14 January 1978. He weighed sixty-five pounds, under thirty kilograms. The death certificate gave the cause as malnutrition and inanition caused by personality disturbance. Inanition is the old medical word for the exhaustion of a body that has had no food put into it. He is buried in Princeton Cemetery, and Adele, who died in 1981, lies beside him. It was she who gave his papers to the Institute for Advanced Study, thousands of pages of a shorthand that almost nobody living could read, and the reading of them has taken decades. The first volume of his collected works appeared in 1986, eight years after his death, and the last two in 2003.

What He Taught

(T1) No fixed list of rules can prove every truth about the whole numbers. + Suppose you set out to write the complete rulebook for arithmetic. You would need a short list of starting assumptions, called axioms, and a list of permitted moves for getting new statements out of old ones, called rules of inference. Put those two lists together and you have what logicians call a formal system: a machine for grinding out theorems, meaning the statements the system can prove, in which no step anywhere appeals to insight, or to meaning, or to a mathematician’s taste. A clerk who knew no mathematics could check every line of it. That is what Whitehead and Russell had attempted in Principia Mathematica, three volumes of symbols meant to derive mathematics from logic alone, and it is what Hilbert asked the world’s mathematicians to finish. Gödel’s method was a piece of bookkeeping so plain that it is easy to miss how strange it is. He gave every symbol, every formula and every proof in the system a number of its own, the way a library gives every book on the shelf a call number. Once that is done, a statement about proofs turns into a statement about numbers, and statements about numbers are exactly what the system was built to handle. The system can now talk about itself. And so Gödel built a sentence which, decoded, says of itself: I have no proof in this system. Follow what happens. If the system proves that sentence, the system has proved something false, and a rulebook that proves falsehoods is worthless. If the system cannot prove it, then what the sentence says is exactly the case, and there is a true statement of arithmetic that the rulebook never reaches. Either the rulebook contradicts itself or it is incomplete. You might think the repair is obvious, that you could simply bolt the missing sentence on as one more axiom and carry on, and that is the first thing everybody says. It does not work. The same trick runs again on the enlarged rulebook and hands you a fresh sentence it cannot prove, and it will keep doing so for as long as you keep patching. That is the first incompleteness theorem. The second theorem follows from the first, and it is the one that finished Hilbert’s program. Call a system consistent if it never proves both a statement and its denial, which is the least anyone would ask of a rulebook. Gödel showed that a consistent system strong enough to handle ordinary arithmetic cannot prove its own consistency. The rulebook cannot certify itself, any more than a company can settle doubts about its books by having its own bookkeepers audit them. If you want to show that arithmetic is free of contradiction, you have to stand outside it and lean on assumptions at least as strong as the ones you were trying to make safe. Two things all of this does not say, worth saying plainly because they get claimed so often. It does not say that mathematics is unreliable, or that mathematical truth is beyond us: the sentence Gödel built is true, and we can see that it is true, which is rather the point. And it says nothing whatever about subjects that are not formal systems in the required sense, a list that includes ethics, history and the reading of Scripture. The theorems are about what fixed rulebooks can do. That is where their force lies, and it is enormous there.

(Q) "It may therefore be surmised that these axioms and rules of inference are also sufficient to decide all mathematical questions which can in any way at all be expressed formally in the systems concerned. It is shown below that this is not the case, and that in both the systems mentioned there are in fact relatively simple problems in the theory of ordinary whole numbers which cannot be decided from the axioms." + Source: Gödel, On Formally Undecidable Propositions of Principia Mathematica and Related Systems I, section 1 (Meltzer translation). The opening page of the 1931 paper. He states the reasonable expectation everyone held, and then, in the flattest sentence in twentieth-century mathematics, says that it is false.

(T2) Numbers and sets are discovered, not invented. + Ask a mathematician whether 2,147,483,647 is a prime number, meaning a number that nothing divides evenly except itself and one, and you will get an answer rather than a show of hands. It is prime. It was prime before anyone checked, and it would have been prime if the human race had never appeared at all. Something in ordinary mathematical practice quietly resists the idea that we are making all this up as we go. Gödel took that resistance at face value, and the position has a name: mathematical Platonism, after Plato, or mathematical realism, which is the view that mathematical objects exist in their own right and that mathematical statements are true or false independently of us. Ranged against it were the views that filled the room he sat in every week. Formalism says mathematics is a game played with marks on paper under rules we chose. Conventionalism says mathematical truths are true only because of decisions we made about how to use words. On either of those, asking what the numbers are really like is a confusion. His argument from the theorems runs like this. If mathematics were nothing but the consequences of rules we laid down, then the rules would have to settle every mathematical question, because on that account there is nothing else for an answer to depend on. The theorems show that the rules do not settle every question (T1). So there is something else. And we plainly have some access to it, because mathematicians do go on settling questions by adopting new axioms, and when they do they are not taking a vote and they do not feel free to choose either way. They are recognising something. Gödel called that recognition mathematical intuition and insisted, provocatively, that it is a kind of perception: less vivid than eyesight, certainly, and no less entitled to be trusted. A Christian reader will hear something familiar in this and should also hear what is missing. An order of truth we did not make and cannot alter sits very comfortably in a world spoken into being by a God in whom all things hold together (Colossians 1:17), and Augustine argued precisely that, placing the eternal truths in the mind of God rather than leaving them hanging in mid-air. Gödel never takes that step in print. His mathematical objects are simply there, unexplained and unowned.

(Q) "But, despite their remoteness from sense experience, we do have something like a perception also of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true." + Source: Gödel, "What is Cantor’s Continuum Problem?", supplement to the 1964 revised version (Collected Works II, p. 268). The most quoted sentence he ever wrote about the philosophy of mathematics, and the one that made him an embarrassment to his positivist friends. He goes straight on to say that he sees no reason to trust this kind of perception less than the sense perception we use to build physical theories.

(T3) Either the human mind outruns every machine, or there are mathematical questions nobody will ever answer. + At Christmas 1951 the American Mathematical Society gave Gödel its Gibbs Lecture, one of the society’s most prestigious invitations, and he used it to say out loud what he thought his theorems meant for the mind. Characteristically, he never published it, and it appeared only in 1995. One term first. A Diophantine problem, named after Diophantus of Alexandria, is a question about whether an equation has solutions in whole numbers. Whether there are whole numbers with x squared minus two times y squared equal to one is a Diophantine problem. So is Fermat’s famous question about whether x to the power n plus y to the power n can equal z to the power n for n greater than two, which took three and a half centuries to settle. They look like schoolroom questions and some of them are bottomless. Now put the two together. The brain, as Gödel said, is to all appearances a finite machine with a finite number of parts, the neurons and the connections between them. If the human mind is no more than that machine, then everything mathematical that human beings could ever establish is whatever some fixed rulebook can generate, and incompleteness (T1) then guarantees that there are arithmetical questions permanently closed to us. His conclusion is a fork, and he was careful to leave both prongs standing: either the mind is not that machine, or such absolutely unsolvable problems exist. He believed the first. What is easy to miss is that he thought the second prong, if it turned out to be the true one, would hand him something he wanted anyway. A question that nobody could ever settle either way would still have an answer, and that answer could not be coming from us or from our rules, since our rules are exactly what fail to reach it. It would have to be true in its own right. That is realism about mathematics (T2) arriving by a second road. Where later writers have pressed this into a proof that the mind is not a machine, the pressing runs into a genuine difficulty, and it is worth naming rather than hiding. To turn the theorems against a proposed machine model of your own mind, you need to know that your own reasoning is consistent, and consistency is exactly what (T1) says a system cannot establish about itself. Gödel understood that perfectly well. It is why he stated a disjunction, an either/or with both branches left standing, rather than a proof.

(Q) "Either mathematics is incompletable in this sense, that its evident axioms can never be comprised in a finite rule, that is to say, the human mind (even within the realm of pure mathematics) infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable diophantine problems of the type specified (where the case that both terms of the disjunction are true is not excluded, so that there are, strictly speaking, three alternatives)." + Source: Gödel, "Some basic theorems on the foundations of mathematics and their implications," the Gibbs Lecture of 1951 (Collected Works III, p. 310). He calls the result a mathematically established fact, and adds that its philosophical implications, whichever way the fork falls, are very decidedly opposed to materialistic philosophy. Materialism is the view that physical facts are the only facts there are, so that a mind is a brain and nothing besides. The lecture was written out in full, delivered, and then filed away.

(T4) God is more than a person, not less, and his existence is a question reason can take up. + Gödel was a Lutheran by baptism who belonged to no congregation and told almost nobody what he believed. The record is clear enough all the same, because he set it down twice. The first time was for a sociologist. Burke Grandjean sent him a questionnaire in 1974 asking about his background and his convictions, then sent it again, and again, with letters asking him please to answer. Gödel filled it in carefully in 1975 and drafted a covering letter, and then posted neither. Both were found among his papers. He described his belief as theistic rather than pantheistic, and he named the philosopher he was following. Theism here means what it usually means, one God who is distinct from the world and made it. Pantheism, the view that God and the universe are two names for one thing, was Spinoza’s; Leibniz’s God is a being distinct from the world who chose to create it and had his reasons for choosing as he did. To Hao Wang, the logician who spent years in conversation with him, Gödel put the difference in a line: Spinoza’s god is less than a person, and his own is more than a person, because God can play the role of a person. The second time was in symbols. Around 1941 he worked out his own version of the ontological argument, the argument that runs from what God is to the conclusion that God is, first pressed by Anselm of Canterbury in the eleventh century and reworked by Leibniz. Gödel’s version fills about two dozen lines of modal logic, which is the logic of necessity and possibility, of what must be so as against what merely happens to be so. It turns on the notion of a positive property, meaning a genuine perfection rather than a lack or a limit: knowledge counts as one, blindness does not. A God-like being, on his definition, is a being that has every positive property there is. From there the steps are short, and they are worth taking one at a time. Such a being is at least possible. A being with every perfection has those perfections essentially rather than by accident, which is to say it could not have lacked them and still been itself. And necessary existence, existing in such a way that failing to exist is not an option, is itself one of the perfections. Put the three together and the conclusion follows: if a God-like being is possible at all, then a God-like being exists, and exists necessarily. He handed a copy of all this to Dana Scott in February 1970 and let it circulate in typescript, and it reached print only after his death. Logicians have worked it over hard since, and the honest summary is that it is unfinished business rather than a knockdown. Jordan Howard Sobel showed in 1987 that Gödel’s axioms as they stand imply modal collapse, the awkward consequence that whatever is true is necessarily true. Follow that out and God had no free choice about creating anything, nobody has a free choice about anything else either, and the world could not have been in any respect other than it is, which is a very high price to pay for a proof. C. Anthony Anderson proposed an amended set of axioms in 1990 to block it. Then, in 2013, Christoph Benzmüller and Bruno Woltzenlogel Paleo did something nobody had tried before: they fed the whole argument to automated theorem provers, the programs that check a piece of formal reasoning line by line, and let the machines grind. Two findings came back. The first is that Gödel’s axioms exactly as he left them turn out to be inconsistent, which is a harsher verdict than Sobel’s, because a set of assumptions that contradicts itself will prove anything you like, the existence of God and the non-existence of God with equal ease. The second is that the slightly amended set Dana Scott wrote down fares better: it is consistent, it really does entail that a God-like being exists necessarily, and it really does collapse the modalities. So everything rests where it rests in every version of this argument, on the single question of whether a being with every perfection is possible.

(Q) "My belief is theistic, not pantheistic, following Leibniz rather than Spinoza." + Source: Gödel, answer to the Grandjean questionnaire, 1975, written out and never mailed (Gödel papers). The sentence directly before it, giving his religion, reads: baptized Lutheran (but not member of any religious congregation).

(T5) Death cannot be the end of a person, because a rational world does not dig a foundation and then walk away from the house. + The question that drew out his fullest statement about God and the soul did not come from a philosopher. It came from his mother, who was old, living in Europe, and wondering whether she would ever really see her son again. He answered her in four letters written between July and October 1961, and because the exchange was private he wrote what he thought. The argument has three steps and he set them out for her in order. First, the world is rationally ordered. It is not chaotic and it is not arbitrary; natural science keeps turning up regularity everywhere it looks, and finds it in places nobody expected to find it, and order, he wrote, is itself a form of rationality. Second, a rationally ordered world is a world in which things have meaning, and he took this to be the exact counterpart of the assumption every scientist works by, that everything has a cause. Philosophers call that the principle of sufficient reason: Leibniz’s rule that nothing is the case without a reason why it is so and not otherwise. Third, hold a human life up against that standard, which is where the argument stops being abstract. A person is built for learning, for work, for friendship and love, for a hundred kinds of development, and then dies having reached, as Gödel put it to his mother, not a thousandth of it. Think of a builder who digs the foundation, lays out the rooms, orders the timber and then walks off the site for good. The labour is wasted, and anyone watching would say so. A rational world does not waste like that. So the building goes on somewhere else. He also held that materialism, the view that physical facts are the only facts, could not be true, and he told her why in the same letters: on that view the world is an unordered heap of atoms with no meaning in it and death is complete annihilation. His own theorems, he thought, made materialism about the mind unlikely into the bargain (T3). It is worth being exact about what this argument delivers. It is an argument from the rational structure of the world to the continuation of persons, and it stands in a long line of such arguments; Kant reached for something similar, and so does anybody who feels in his bones that a world of unfinished lives cannot be the last word. What it does not contain is a resurrection, a judgment or a Redeemer.

(Q) "If the world is rationally organised and has meaning, then it must be the case. For what sort of a meaning would it have to bring about a being (the human being) with such a wide field of possibilities for personal development and relationships to others, only then to let him achieve not even 1/1,000th of it?" + Source: Gödel to his mother Marianne, letter of 23 July 1961 (translated by Alexander T. Englert; German original in the Wienbibliothek im Rathaus, Vienna). "It" is the seeing-again she had asked about. Her own letters have not survived, so the correspondence reads like a dialogue with one speaker’s lines removed.

(Q) "The idea that everything in the world has meaning is, by the way, the exact analogue of the principle that everything has a cause on which the whole of science is based." + Source: Gödel to his mother Marianne, letter of 6 October 1961 (translated by Alexander T. Englert). The whole of his natural theology in one sentence. Natural theology is the attempt to reach God by reason and by looking hard at the world, rather than by revelation, and this is his: the search for meaning and the search for causes are the same habit of mind, so a scientist who trusts the second has no principled reason to abandon the first.

What Christian Thinkers Made of Him

(†) Christian philosophers took up his ontological proof, and are still working on it. + Anselm of Canterbury put the ontological argument into circulation in about 1078 and it has been dying and reviving ever since. Kant delivered what most philosophers took for the death blow, that existence is not a property a thing can have or lack in the way redness is. Then, in the middle of the twentieth century, the argument came back in modal dress, because the modern logic of necessity and possibility gave it a form Kant had not been arguing with: the claim is not that God has the extra property of existing but that God, if possible at all, cannot fail to exist. Norman Malcolm set out such a version in 1960, Charles Hartshorne another in The Logic of Perfection in 1962, and Alvin Plantinga gave the best-known one in The Nature of Necessity in 1974, arguing that if a being of maximal greatness is possible, then that being exists in every possible world, this one included. A possible world, in this way of talking, is not a planet somewhere; it is a complete way things could have been, the world as it would be if you had taken the other job or if the dinosaurs had never died out. Gödel’s own version, let out of the drawer in 1970 and printed only after his death, reached the public after all of these and turned out to be the most rigorously worked of the family, which is why it is the one logicians reach for when they want to test the argument properly. Robert Maydole, Alexander Pruss and Joshua Rasmussen have carried the work forward since, and the machine-checked version by Benzmüller and Woltzenlogel Paleo (T4) put a piece of natural theology through the same tools used to verify aircraft software. What every version of the argument shares is a single load-bearing premise: that a maximally perfect being is possible. Prove that and the rest is bookkeeping. Nobody has yet found a way of establishing it that convinces those not already convinced, and the best defenders of the argument say so themselves.

(†) His theorems became the sharpest tool in the argument that the mind is not a machine. + C. S. Lewis had argued in Miracles in 1947 that naturalism, the view that nature is all there is and that nothing stands behind it, saws off the branch it is sitting on. If all reasoning is the product of non-rational causes, atoms bumping in a skull, then so is the reasoning that produced naturalism, and there is no reason left to believe it. The argument was philosophical and it wanted teeth. Gödel gave it some. In 1961 the Oxford philosopher J. R. Lucas published "Minds, Machines and Gödel," arguing that for any machine proposed as a model of a human mathematician we can produce the sentence that machine cannot prove and see that it is true, so no machine models us. Roger Penrose, who is not a Christian, pressed a version of the same case in The Emperor’s New Mind in 1989 and Shadows of the Mind in 1994. Victor Reppert brought the two strands together in C. S. Lewis’s Dangerous Idea in 2003, where incompleteness serves the argument from reason rather than standing in for it. The replies have been heavy, and they have to be faced rather than waved past. Hilary Putnam, George Boolos and Solomon Feferman each pressed the same point: the argument needs us to know two things about ourselves, that our own reasoning is consistent and what our own program is, and we do not know either one (T3). So the state of the question is roughly this. The strong claim, that Gödel refuted mechanism, mechanism being the view that a human mind just is a machine of some kind, does not command assent among philosophers of mind. Gödel’s own careful fork, on the other hand, has never been refuted at all, and a materialist is uncomfortable whichever prong of it he takes.

(†) The rotating universe he gave Einstein turned into an argument about time that Christian philosophers still use. + The birthday present of 1949 came with a philosophical essay attached, written for the Schilpp volume of tributes to Einstein. Gödel’s reasoning went beyond the mathematics. In his rotating universe there is no way to slice spacetime into a single objective sequence of moments, a universal "now" ticking forward. If such a universe is genuinely possible, he argued, then the passing of time cannot be something built into the physical world, since whether time passes could hardly depend on how fast the universe happens to be spinning. He concluded that change is an appearance produced by our way of perceiving, and that time, in the sense that matters to us, is not real. Christian philosophers of time have taken the argument seriously while refusing the conclusion. William Lane Craig worked through it at length in Time and the Metaphysics of Relativity in 2001, arguing that what Gödel really showed is that relativity by itself cannot settle whether time passes, which reopens a question physics had seemed to close. That matters more than it sounds. A faith whose centre is a set of events, a birth under Augustus, a death under Pilate and an empty tomb on the third day, needs those events to have actually happened, and happened in that order, rather than sitting timelessly in what philosophers call a block universe, a four-dimensional loaf in which every moment is equally there and nothing ever really comes to pass. Gödel, who wanted time unreal, handed the other side one of its better arguments for keeping the question open.

(†) His religion reached a God who explains the world, and stopped short of the God who was raised on the third day. + The ledger here is his own, and it is short, and it is real. - A God reached by proof and not by revelation. The God of the ontological argument and of the letters to his mother is Leibniz’s God: the reason there is order rather than chaos, the guarantee that a rational world will not waste a human life. Every word of it could have been written before Bethlehem. Paul’s verdict on that road is not that it is foolish but that it does not arrive: "For since, in the wisdom of God, the world did not know God through wisdom, it pleased God through the folly of what we preach to save those who believe" (1 Corinthians 1:21). - A hope with no resurrection in it. What he expected after death was the completion of unused potential, a second existence in which a person finishes what he began. Christian hope is a different thing and rests on a different kind of fact: a body raised because one particular body was raised on one particular morning, with a judgment on the way to it (Hebrews 9:27; 1 Corinthians 15:19). - A conviction kept in a drawer. He was baptized Lutheran and joined no congregation. By Adele’s account he read the Bible in bed on Sunday mornings, by himself. He said that religions are for the most part bad although religion is not, and he withheld his proof, on Morgenstern’s evidence, because he did not want to be taken for a believer. Whatever else the New Testament is, it is not a set of conclusions to be held privately (Hebrews 10:24-25), and the eternal life it offers is knowing a person rather than securing a result (John 17:3). None of this makes him a hostile witness, and it would be ungrateful to read him that way. What theologians call general revelation, meaning the knowledge of God that is available to anyone through the created world and the conscience, as distinct from the special revelation given in Scripture, is exactly what Gödel spent his life running into: an order he did not make, a mind that will not reduce to matter, a world whose regularity looks like the work of reason, and a life too big for the years allotted to it. He kept meeting it in his own workplace, which is not where a man goes looking for God. Paul says that much is clearly perceived in the things that have been made, and that it leaves us without excuse (Romans 1:20). Gödel perceived it about as clearly as anyone has. What he did not have, and never claimed to have, was the road down from the mountain.