Letters between a mathematician and a philosopher, Vienna 1929–1931
The letters that follow are fiction. Heinrich Brenner and Elias Wertheimer never existed. But the world they inhabit is real — the cafés, the seminars, the politics, the ideas. Kurt Gödel, Moritz Schlick, Hans Hahn, Rudolf Carnap, and the others appear as they were: members of the Vienna Circle, pursuing one of the twentieth century's great intellectual projects against the backdrop of a civilization beginning to unravel.
The philosophical question at the heart of these letters — is mathematics discovered or invented? — remains unsettled. Gödel believed, with absolute conviction, that mathematical objects exist independently of human minds. The Vienna Circle believed, with equal conviction, that mathematical statements are tautologies devoid of factual content. Gödel's incompleteness theorems did not resolve this debate. They deepened it.
What follows is my attempt to dramatize that deepening.
Mathematician. Privatdozent at the University of Vienna. A Platonist who believes mathematical truths exist independently of human minds. Quiet, precise, more comfortable with proofs than people.
Philosopher. Member of Schlick's Thursday circle at Boltzmanngasse. A logical positivist who holds that mathematical propositions are analytic — true by convention, empty of factual content.
Dear Heinrich,
You left the Herrenhof in such haste last Thursday that I could not press the point. So I shall press it here, on paper, where you cannot retreat behind that maddening silence of yours — that look you give when you believe the philosopher has said something beneath refutation.
You told me that when you work on a problem in analysis, you have the sensation of approaching something — as if the theorem were already there, waiting in some antechamber of reality, and your proof were merely the act of opening the door. I know you meant this literally and not as metaphor. That is precisely what troubles me.
I have just come from Thursday's session at Boltzmanngasse. Schlick was in superb form. We spent two hours on the Tractatus, propositions 6.1 through 6.2, and the discussion has left me more certain than ever that the account Wittgenstein gives is correct: the propositions of logic and mathematics are tautologies. They say nothing about the world. "2 + 2 = 4" is not a discovery about some realm of abstract objects; it is a consequence of the way we have defined "2," "+," "=," and "4." It is true in the same way that "all bachelors are unmarried" is true — by the conventions of our language, nothing more.
I know what you will say. You will say that this cannot be right, because mathematics is surprising. That you cannot explain why the prime numbers behave as they do if they are merely definitional artifacts. That the distribution of primes among the integers has a character, a stubbornness, that resists reduction to convention.
But consider: chess is also surprising. The behavior of the knight on a crowded board can astonish even a grandmaster. Yet no one supposes that chess positions exist in a Platonic heaven, waiting to be discovered. The surprise arises from the complexity of the rules, not from the ontological weight of the pieces.
Mathematics is the richest, the most intricate, the most beautiful game humanity has ever devised. But it remains a game. Its pieces are symbols. Its rules are logical. Its truths are tautological.
Come to the Herrenhof on Saturday and tell me I am wrong. I shall order you a Kapuziner — shade 7, as I recall — and you can explain to me where, exactly, the number seventeen resides.
Your exasperating friend,
Elias
Dear Elias,
I did not leave the Herrenhof in haste. I left because you were beginning to enjoy yourself too much, and I have learned that when you reach that particular pitch of rhetorical pleasure, evidence ceases to matter. You wish to be admired, not answered. On paper, perhaps you will tolerate an answer.
Your chess analogy fails, and it fails instructively. When I study the distribution of prime numbers, I am not exploring the consequences of a game I designed. I am encountering a structure I did not design, that no one designed, that resists my every attempt to impose order upon it from without. The primes do not behave — that is your word, and it is the right word — they do not behave like pieces in a game. They behave like objects in a landscape.
Let me make this concrete. Consider the Goldbach conjecture: every even integer greater than two is the sum of two primes. This proposition has been verified for every even number anyone has ever checked — billions of them. No counterexample exists. And yet we cannot prove it. If mathematics were tautological, if it were simply a matter of unpacking the definitions we ourselves constructed, how can it be that we do not know the answer? We made the definitions. We built the rules. How can the rules keep secrets from their makers?
A tautology cannot surprise you, Elias. Not truly. You may be surprised at how long it takes to verify that a tautology is tautological — that is a matter of computation, of patience. But you cannot be surprised by its content, because it has no content. Your Circle says mathematics has no content. Then how does it continually astonish us with content we did not put there?
Here is what I believe, and I will say it plainly since you accuse me of hiding behind silence. I believe that when I prove a theorem, I am not constructing something new. I am perceiving something that was already there. Mathematical intuition is a form of perception — imperfect, groping, frequently mistaken, but directed at real objects nonetheless. When Euler discovered that the sum of the reciprocals of the squares converges to π²/6, he did not invent this fact. He found it. The connection between the integers, the squares, and the ratio of a circle's circumference to its diameter exists independently of Euler, independently of humanity, independently of any mind that might contemplate it. If every thinking being in the universe were annihilated, that relationship would persist — not as a sentence, not as a symbol, but as a fact.
You ask me where the number seventeen resides. I confess I do not know its address. But I notice that you cannot tell me where a logical law resides either, and this does not stop you from believing in its validity. The demand for a spatial location is a category error. Not everything that exists occupies a place.
I will come to the Herrenhof on Saturday. But I will have a Schwarzer, not a Kapuziner. You are thinking of someone else. I take my coffee as I take my mathematics: without dilution.
Yours with the patience of a Platonist,
Heinrich
Dear Heinrich,
Four months of silence! I had begun to think my last letter had wounded you mortally, but Menger tells me you have merely been buried in your Habilitation work and have spoken to no human being since Christmas except to request more coffee. This sounds accurate.
I write now because something happened last Thursday at Boltzmanngasse that bears directly on our disagreement, and I want your reaction before the impression fades.
We had a visitor — young Gödel. You know him, of course; he took his doctorate under Hahn last year, the completeness proof for first-order logic. A quiet fellow, almost oppressively polite, sits in the back and says nothing for an hour, then produces a sentence so precise it silences the room. He unnerves Schlick, I think, though Schlick would never say so.
Last Thursday, Hahn was presenting a sketch of the formalist programme — Hilbert's great ambition to establish the consistency of arithmetic by finitary means. You know the idea: formalize all of mathematics, then prove, using only methods everyone accepts, that the formal system generates no contradictions. If this succeeds, the question of whether mathematical objects "really exist" becomes moot. Consistency is enough. We need not quarrel about ontology.
Gödel sat through the entire presentation. At the end, someone — I think it was Waismann — asked whether such a consistency proof was near completion. Hahn said Hilbert's group in Göttingen was optimistic. And then Gödel, very softly, said something I have been turning over in my mind ever since. He said: "One might doubt whether all problems of this type are solvable."
That was all. No elaboration. Schlick moved the discussion along. But I noticed that Hahn looked at Gödel with an expression I can only describe as alert — the way a hound looks when it catches a scent.
I did not know what to make of this then. I am not certain I know what to make of it now. But something in his tone — the absolute calm of it, as if he were reporting a measurement rather than venturing a speculation — stayed with me. It was the voice of someone who knows something he is not yet ready to say.
What do you make of it? You are better placed than I to judge whether Hilbert's programme is as secure as Hahn suggests. Is the young man being cautious, or does he know something?
Meanwhile, the city grows strange around us. I was at the Ringstrasse last week and saw a column of Heimwehr marching in formation, singing. The students at the university grow bolder — there was an incident in the philosophy corridor, pamphlets scattered, names on a list. Schlick's name was among them. He laughed it off. I did not find it amusing.
Come to dinner. Helena asks after you. She believes you are wasting away and wishes to feed you Tafelspitz.
As ever,
Elias
Dear Elias,
Forgive the delay. The Habilitation is submitted. I emerged from the mathematics library yesterday for the first time in weeks and found the chestnut trees in full bloom and the Heimwehr swearing oaths. I scarcely know which disoriented me more.
Your account of Gödel interests me greatly, though I confess no surprise. I have spoken with him at Menger's colloquium — twice, briefly. Each time I came away with the feeling of having been in the presence of someone whose thinking occurs at a depth I cannot access. Not faster than mine — deeper. As if he sees the logical landscape from an elevation that compresses what appears to us as a vast territory into a single clear vista.
You ask whether Hilbert's programme is secure. I will tell you what I think, though it will not comfort you.
I think Hilbert's programme is the noblest failure in the history of human thought. It proceeds from a magnificent ambition: to place all of mathematics on foundations so certain that doubt becomes impossible. Formalize everything. Prove consistency. Close the door on paradox forever. Who would not wish for this? It is the dream of a mind that loves order, and Hilbert's mind loves order as a cathedral loves stone — it is built from it, shaped by it, cannot exist without it.
But I believe the dream contains a hidden impossibility. Not a technical obstacle — something deeper. Consider: Hilbert asks us to prove, within a finitary framework, that our formal system is consistent. But a proof is itself a mathematical object. To prove the consistency of mathematics, you must use mathematics. You are asking the system to vouch for itself. It is as if a man were asked to provide a letter of reference for his own character — and the letter were to be accepted only if it could be shown, by the man's own testimony, that his testimony is reliable.
I do not say this constitutes a proof that the programme must fail. I am not Gödel; I have no proof. I have only an intuition — which I know you despise as a philosophical category. But I believe that mathematical truth outruns any formal system that attempts to contain it. We can build larger and larger nets, but the ocean of mathematical reality is not the sort of thing that can be netted. This is not a deficiency in our nets. It is a feature of the ocean.
And here our disagreement sharpens. If mathematics is a game, as you insist — a formal manipulation of symbols according to rules we invented — then Hilbert's programme ought to succeed. A game is a closed system. Its rules define its reality. There is nothing in chess that chess does not contain. If Hilbert's programme fails, you must explain how a game can outrun its own rules. A game cannot do this. Which means that mathematics is not a game.
I heard about the pamphlets. I heard about the list. Schlick should not laugh it off. One of the students in my seminar last term — I will not name him — gave a presentation on a perfectly routine topic in real analysis and managed to insert a remark about "the Jewish corruption of mathematical thought." I stared at him. He stared back. He was not embarrassed. He was proud.
Thank Helena for the invitation. I will come on Sunday if the offer stands. I am indeed wasting away, though I attribute this to Boltzmann's Vorlesungen rather than to insufficient Tafelspitz.
Yours in mathematical realism,
Heinrich
Dear Heinrich,
I have just returned from Königsberg and I am writing this in a state of — I do not know what to call it. Not confusion exactly. Something closer to vertigo.
You must hear what happened. I will try to be precise, though precision is difficult when the ground is shifting.
The conference was on the Epistemology of the Exact Sciences. Three days. On the first day, Carnap presented the logicist position — mathematics reducible to logic — with his customary rigour. Heyting spoke for the intuitionists. Von Neumann defended formalism on behalf of Hilbert, who was in Königsberg but attending a different meeting. Waismann read a paper on Wittgenstein's views. It was all very civilized, very orderly, very much what one expected.
On the last day, there was a roundtable discussion. Hahn chaired it. Gödel was present — sitting quietly, as always. Near the end, almost as an afterthought, he spoke. He said — I am paraphrasing, but the sense is exact — that assuming the consistency of classical mathematics, one can construct propositions in the language of mathematics that are true but unprovable within the system.
Heinrich, I want you to understand what happened in that room. Nothing happened. The discussion moved on. Carnap made a remark. Someone asked a technical question. The session ended. People went to dinner. It was as if a man had announced the death of God and the room had responded by discussing the weather.
Only von Neumann reacted. I saw him lean forward when Gödel spoke — the way Hahn leaned forward in March, that same alertness. After the session he went directly to Gödel and they spoke for a long time in a corridor. I could not hear what was said. But I saw von Neumann's face, and I have never seen such an expression on any man. It was the face of someone who has just been told a secret he both dreads and recognizes as true.
The next day — this is almost too perfect to bear — Hilbert gave his retirement address to the Society of Natural Scientists and Physicians. He was 68 years old, full of fire, and he closed with the words: Wir müssen wissen — wir werden wissen. We must know — we shall know. It was broadcast on the radio. The old lion, roaring his defiance at the limits of human knowledge, one day after a 24-year-old had quietly announced that those limits are real.
I do not yet understand the full technical import of what Gödel claims. I am not a mathematician; I am a philosopher, and I need the proof in front of me before I can judge. But I understand enough to know that if he is right — if there are true propositions that cannot be proved — then something I believed about mathematics must be wrong.
Not necessarily what you think is wrong, mind you. I am not ready to surrender. But I feel the ground shifting, and I wanted you to know.
Shaken but still standing,
Elias
P.S. — They recorded Hilbert's radio address. The irony of the timing will be lost on everyone who was not at both sessions. I think it may be the most dramatic juxtaposition in the history of ideas, and perhaps four people in the world are aware of it.
Dear Elias,
I have read Gödel's paper. Hahn lent me the manuscript before publication — it will appear in the Monatshefte in January. I have spent two weeks with it. I have checked the argument three times. It is correct.
I will not attempt a full technical exposition here; you are a philosopher and I must translate. But I want you to understand what he has done, because I believe it bears on our argument in a way that neither of us anticipated.
Gödel has shown two things. The first: any formal system powerful enough to express arithmetic, if it is consistent, must contain propositions that are true but cannot be proved within the system. The second — and this is the coup de grâce — such a system cannot prove its own consistency.
The method is diabolical in its elegance. He assigns numbers to every symbol, every formula, every proof in the system — so that the system can, in effect, talk about itself. And then he constructs a sentence that says, in the language of arithmetic: "I am not provable." If this sentence is provable, then the system proves something false, and is inconsistent. If it is not provable, then it is true — a true sentence the system cannot prove. The system is caught in a trap of its own making.
Do you see what this means for Hilbert? The consistency of arithmetic cannot be established by arithmetic's own methods. The cathedral cannot certify its own foundations. The formal system that was supposed to contain all mathematical truth contains, instead, a permanent gap — a window through which we can see truths it cannot reach.
But Elias — and here I must be careful, because I am about to make a philosophical claim, not a mathematical one — I believe this result means something more. Not just that Hilbert's programme fails. That the formalist picture of mathematics is wrong.
Your Circle tells me that mathematics is tautological — that its truths are consequences of definitions, empty of factual content. But Gödel has shown that mathematical truth exceeds mathematical provability. There are truths — genuine, determinate truths — that no formal system can capture. If mathematics were merely a game, merely a manipulation of symbols according to human-made rules, this would be impossible. A game cannot outrun its own rules. The rules define the game. Everything that is true of chess follows from the rules of chess.
But mathematics does outrun its rules. There are facts about numbers that are true independently of any formal system in which we might try to express them. This is not a gap in our knowledge. It is a feature of mathematical reality. The truths are there — really there, objectively there — and our formal systems are windows onto a landscape they can never fully contain.
I told you in my last letter that I believed this. Now Gödel has proved it. Not proved Platonism — that would be too much to claim. But proved something that Platonism predicts and that formalism and conventionalism cannot explain: that mathematical truth has a life of its own, beyond and independent of the systems we build to capture it.
You say the ground is shifting. I say the ground was always there. You are only now noticing that you are standing on it.
With something approaching exhilaration,
Heinrich
P.S. — I spoke to Gödel after Menger's colloquium last week. I asked him, cautiously, what he believed about the nature of mathematical objects. He looked at me for a long time — that disconcerting stare of his — and said: "I have held, since about 1925, that mathematical objects have an existence of their own." Since 1925! He was nineteen. He has been a Platonist since before any of us knew his name, sitting in those Thursday sessions at Boltzmanngasse, listening to your Circle declare mathematics tautological, knowing — knowing — they were wrong, and saying nothing.
Dear Heinrich,
I have not written in four months, and I owe you an explanation. It is not that I have been idle. I have been reading Gödel's paper — slowly, with Carnap's help on the technical passages — and thinking. Much thinking. The sort of thinking that does not produce letters because it does not yet know what to say.
I will now try to say it.
You are right that Gödel's result damages the formalist position. This I concede entirely. If a consistent formal system cannot prove its own consistency, then the dream of mathematics as a self-certifying formal structure is finished. Hilbert aimed to show that the building stands firm on its own foundations; Gödel has shown that no building can inspect its own foundations from within. This is devastating, and anyone who pretends otherwise is not paying attention.
You are also right that the result creates difficulties for my position — for the view that mathematical truths are analytic, tautological, conventional. If there are truths of arithmetic that no formal system can prove, then those truths are not consequences of any set of rules. They cannot be tautological in any straightforward sense. I feel the force of this, and I will not pretend I do not.
But I am not ready to follow you into your Platonic heaven. Let me tell you why.
Gödel's sentence — the sentence that says "I am not provable" — is undeniably true. But what kind of truth is it? It is true in the way that "this sentence has five words" is true: it is true because of what it says about itself. It is a self-referential construction, extraordinary in its ingenuity, but its truth does not require us to posit a realm of abstract objects existing beyond space and time. It requires only that we understand self-reference and the relationship between a formal system and its own expressive capacity.
You leap from "there are truths that outrun any formal system" to "therefore mathematical objects exist independently." But this leap is not licensed by the theorem. One might equally conclude that our notion of "truth" is more complex than our notion of "proof" — that the concept of mathematical truth has a richer structure than any formal proof procedure can exhaust — without committing oneself to Platonism. The truths may be objective without being objects.
I am not certain this is the right response. I am groping. But I refuse to abandon the principle of parsimony simply because a brilliant young man has shown that formal systems have limits. Of course they have limits. Everything human has limits. This does not mean that what lies beyond those limits constitutes an independent reality. It may mean only that the tools of mathematical reason, like all tools, have edges.
There is something else I want to say, and I am uncertain how to say it. The atmosphere at the university has become unbearable. Last week a group of students disrupted a lecture by Wirtinger — Wirtinger, who is 65 and has done nothing to anyone — with catcalls and thrown papers. There are lists circulating. The faculty meeting last month dissolved into shouting. I look at the faces of my colleagues and I see, in some of them, a calculation being made — not an intellectual calculation but a political one: which side to be on when the reckoning comes.
I do not say this to change the subject. I say it because I wonder whether our debate — this beautiful, intricate, passionately conducted argument about the nature of mathematical truth — belongs to a world that is ceasing to exist. Gödel has shown that formal systems cannot prove their own consistency. I am beginning to suspect that civilizations cannot prove their own consistency either.
Yours, with diminishing confidence in all foundations,
Elias
Dear Elias,
Von Neumann has written to Carnap. Hahn showed me the letter. It says: "Gödel has shown the unrealizability of Hilbert's programme. There is no more reason to reject intuitionism." Von Neumann — who stood in Königsberg nine months ago and defended formalism — has abandoned his position. The sharpest mathematical mind of our generation encountered Gödel's result and simply turned around. There was no gradual retreat, no face-saving compromise. He saw the truth and accepted it.
I mention this not to gloat — though I confess the temptation — but because it bears on the question you raised in your last letter: the question of what kind of truth Gödel's sentence expresses.
You say the incompleteness theorem shows only that formal systems have limits, not that mathematical objects exist independently. You say the concept of truth may be richer than the concept of proof without requiring Platonism. This is subtle, and I have spent weeks thinking about it. I will tell you what I have concluded.
You are right that the theorem alone does not prove Platonism. No single result could prove or refute a philosophical position; philosophy is not mathematics. But I think Gödel's result does something more interesting than proving a position. It shifts the burden.
Before Gödel, the burden was on the Platonist. You could say: "Why posit abstract objects? Mathematics is just a formal game, and we can explain everything — truth, proof, consistency — within the formalist framework." The Platonist had to explain why his additional ontological commitment was necessary.
After Gödel, the burden shifts. Now you must explain how there can be mathematical truths that no formal system captures. If mathematics is merely formal, merely conventional, merely tautological, then its truths should be exhausted by its formalisms. They are not. Something exceeds the formalism. What is it? If you refuse to call it mathematical reality, you must tell me what to call it instead. And your answer — that "truth is richer than proof" — is, if you will forgive me, a description of the problem, not a solution to it. Why is truth richer than proof? What makes it so? What is this truth about?
A Platonist has an answer: truth is richer than proof because mathematical reality is richer than any formal system. The truths are about objects — numbers, sets, structures — that exist independently and cannot be fully captured by any finite set of axioms. The incompleteness of formal systems is a consequence of the richness of mathematical reality, just as the incompleteness of any map is a consequence of the richness of the territory it represents.
You say this is extravagant. I say it is the simplest explanation available. The alternatives — that truth somehow floats free of both formal systems and mathematical objects, that it is neither tautological nor about anything — seem to me far more mysterious than the position they hope to avoid.
But I want to turn, in this last part, to what you said about the university, and about Vienna, and about whether our argument belongs to a world that is ceasing to exist.
I think you are right to be frightened. I am frightened too, though I have told no one. Last month one of my students — the same young man who spoke of "Jewish corruption" in my seminar — came to my office to ask, with perfect politeness, whether I intended to remain in my position. The question was not a question. It was a notification.
And yet. When I sit in the library and work through a proof — when I follow an argument from premise to conclusion and feel the click of logical necessity — I am in a place that no political movement can reach. The Heimwehr cannot march into the realm of prime numbers. The thugs in the corridor cannot burn the truths of analysis. This is not escapism, Elias. This is the deepest argument for Platonism I know.
If mathematics is a human invention — a game, a convention, a social construction — then it is as vulnerable as every other human institution. It can be corrupted, rewritten, instrumentalized, destroyed. If the mob decides that certain mathematical truths are degenerate, then on your account they are as entitled to that judgment as any other, because there is no fact of the matter to correct them.
But if mathematics is discovered — if its truths exist independently of any human mind, any human society, any human politics — then they are safe. Not safe in our keeping; we may well lose access to them. But safe in themselves. The theorems will be true whether anyone proves them or not. The primes will be distributed as they are distributed whether anyone studies them or not. The ratio of a circle's circumference to its diameter will remain what it is when the last university closes and the last library burns.
In times like these, Elias, I find that I need this to be true. Not as comfort — or not only as comfort. As conviction. The truth of mathematics does not depend on the survival of mathematicians. It does not depend on the survival of civilizations. It is the one thing we can participate in that is genuinely, irrevocably beyond the reach of human failure.
You say civilizations cannot prove their own consistency. I am afraid you may be right. But the integers can survive anything, because they were never ours to begin with.
Come to dinner. I insist. Bring Helena. Bring wine. We will eat, and argue, and be human beings together, for as long as being a human being in Vienna remains possible.
Yours, now and in whatever comes next,
Heinrich
There are no further letters.
What happened next is a matter of history, not fiction, and it must be recorded plainly.
Hans Hahn, Gödel's doctoral supervisor and a pillar of the Vienna Circle, died of cancer in 1934. He was 55.
Moritz Schlick, the founder of the Vienna Circle, was shot and killed on the steps of the University of Vienna on June 22, 1936, by a former student. The murderer's defense argued that Schlick's anti-metaphysical philosophy had undermined his moral restraints. Austrian Nazi sympathizers exploited the killing, emphasizing Schlick's connections to Jewish intellectuals. He was not Jewish.
The Ernst Mach Society, the Vienna Circle's public face, had been forcibly dissolved in 1934 by the Austro-Fascist government. One by one, the members scattered. Carnap left for Prague, then Chicago. Feigl went to Iowa. Neurath fled to The Hague, then Britain. Menger departed for Notre Dame. Waismann escaped to Oxford.
Kurt Gödel left Vienna in January 1940, traveling by rail through the Soviet Union and then by ship from Yokohama to San Francisco, eventually reaching Princeton, where he joined the Institute for Advanced Study. He became an American citizen in 1948. He remained a mathematical Platonist for the rest of his life, growing more explicit about it as the decades passed. In his 1951 Gibbs Lecture, he argued that the incompleteness theorems support mathematical realism — that mathematical truth outstrips formal provability precisely because it is about something real. He died in Princeton in 1978, of self-starvation brought on by paranoia, convinced that someone was trying to poison his food. He weighed 65 pounds.
David Hilbert died in Göttingen on February 14, 1943, in a Germany that had driven out nearly all of its great mathematicians. When a Nazi official asked him whether mathematics at Göttingen had suffered from the departure of Jewish scholars, Hilbert replied: "Suffered? It hasn't suffered. It simply doesn't exist anymore." Only ten people attended his funeral.
The question at the heart of these letters — is mathematics discovered or invented? — has not been settled. It may be unsettleable. But the incompleteness theorems ensure that it cannot be dismissed. There is something in mathematics that exceeds our formal grasp of it. Whether we call that something a Platonic reality, or an inexhaustible structure, or simply the depth of mathematical truth, it remains — as Heinrich might have said — independent of us. The theorems are true. The primes are distributed as they are distributed. The ratio holds.
Whatever else has been lost, this has not.
A note on historical accuracy: All historical events, dates, quotations, and biographical details in this piece are accurate to the best of available scholarship. The August 26, 1930 meeting at Café Reichsrat, the Königsberg conference, Hilbert's radio address, von Neumann's letters, Gödel's Platonist confession, and the fates of the Vienna Circle members all happened as described. Heinrich Brenner and Elias Wertheimer are fictional. Their arguments are drawn from real philosophical positions held by real people, compressed and dramatized for narrative purposes.
Principal sources: Rebecca Newberger Goldstein, Incompleteness: The Proof and Paradox of Kurt Gödel (2005). Karl Sigmund, Exact Thinking in Demented Times: The Vienna Circle and the Epic Quest for the Foundations of Science (2017). John W. Dawson, Jr., Logical Dilemmas: The Life and Work of Kurt Gödel (1997). David Edmonds, The Murder of Professor Schlick (2020).