The Ecstasy of the Proof

Imagine, for a moment, you came across a curious puzzle while reading one day — and indeed you have, this very moment. It goes as follows:

The puzzle is: to what is this sum equivalent? Is it infinite? Does it converge to some particular number and rise no higher? Any article or textbook written since 1735 will follow this cryptic (anachronistic) amalgamation of symbols with ready answers to those questions. But some time earlier it fell on Leonhard Euler's desk, unsolved since it was put forth in 1650 by Pietro Mengoli.4 His solution was quite complex:

"What a marvelous derivation it was. It boasted an all-star cast of transcendental functions: sines, cosines, logs, and exponentials. It ranged from the real to the complex and back again. It featured L'Hôpital's Rule in a starring role. Of course, none of this would have happened without the fluid imagination of Leonhard Euler, symbol manipulator extraordinaire."

Dunham (2009) 6

Euler's solution was marvelous (as were many others10), but let us explore one that you might have stumbled across, given a few key assumptions. Then, after presenting a solution to the Basel problem as an archetypal proof of mathematical insight, I will demonstrate it is an incredible union of reason and imagination. In other words, it is ecstatic by nature. We will expand this idea further, explore some nuances, and proceed to address a myriad of objections.

First, let's suppose that you recall introductory calculus, whether from ambitious high school years, collegiate tedium, or because you love math in all its forms. There is a means, called the Taylor expansion, by which we can express many functions, including basic trigonometry functions, as polynomials:

Then, because you might have been curious to see what happens to the expansion if you substitute \(\pi x\) for \(x\), you find:

This next part is the most improbable. Perhaps flipping through a mathematics textbook, you come across the Weierstrass Factorization Theorem applied, and readily applied it to the sine function without much trouble:

Thinking back to that Taylor expansion, you realize that if you substitute \(\pi x\) for \(x\) again you quickly arrive at:

And then with a few more rearrangements:

But wait, this is the same form as the Taylor expansion! Because both series represent the same analytic function, their power series expansions must agree term by term. If we equate the coefficients of \(x^3\) we find:

Thus, we can conclude:

The purpose of including this concrete, simplified example is not to deter future mathematicians, suggest that Euler's discovery — and many others — are nothing but chance, or even to demonstrate that this proof is difficult and incomprehensible. Rather, Euler had real insight — frankly, far more complex than the insight required for the proof above — and that he was capable of presenting that insight so that anyone with some reasonable mathematical familiarity could be convinced beyond any doubt that \(1 + \tfrac{1}{4} + \tfrac{1}{9} + \cdots \approx 1.645\). This is mathematical proof we have found together.1 Now I shall convince you it is also ecstatic.

I argue the comprehension of a mathematical proof constitutes the ecstatic. Comprehending a proof can be discovering it, as Euler once did; reading another's argument; or reconstructing it from old notes we've rediscovered from ages past. I require complete comprehension; partial comprehension, or memorizing the language, is insufficient. Rather, it is the full understanding of a proof's language and semiotics such that we can be fully convinced by the argument. The full comprehension of proof constitutes what Weiskel and many other thinkers describe as the ecstatic — which, for Weiskel and Kant, I am equating with the sublime.

Prima facie, the anti-structural nature of ecstasy seems to be the opposite of the rigorous certainty of a mathematical proof. The practitioners of Dionysian rites shrug off all Pentheus-like reason and venture off into the woods while those of Turnerian rites give up any technical knowledge or understanding in exchange for mystical faithfulness.823 However, the shaman approaches ecstasy with a purpose, seeking answers to a question, to heal, or to guide the dead. It is ecstasy that enables and accompanies these purposes.19 In the case of the proof, the shaman is the comprehender, seeking answers to the question of what is true. Here the goal is to apply imagination to find reason, and as the imagination reaches toward something that is outside of memory and outside of self, it finds a chain of ideas validated by reason. It is a thing consistent in itself, a sacred idea. Proofs are imbued with a sense of the sacred. For example, Srinivasa Ramanujan claimed that some of his mathematical insights came to him in religious visions while Paul Erdős spoke of "The BOOK," God's collection of the most beautiful proofs.141 One thinker declared "the sacred is equivalent to a power and … to reality."7 He wrote "the religious man desires to be, to participate in reality." A proof makes claims about ideas and what follows from what. Proofs appear in many disciplines, not only math. Given some premises, some ideas — however real they may be — their fixed relation to what follows from them, demonstrated by a proof, could not be more real. Mathematics in particular distinguishes itself from other sciences — except perhaps philosophy — in that it is generally (but not universally) accepted to be a priori: that its ideas follow, in a sense, without experience or empirical observation. Therefore, it remains true in all possible worlds and is the most real. That is to say mathematics is the greatest form of ecstasy along the axis of the real.

As we shall see with Weiskel, this realness is not sufficient for his "sublime moment,"24 but novelty is also essential, for this truly distinguishes this new reality from the reality that a mind already knows. Novelty is implicitly essential to thinkers such as Eliade and Turner because we cannot find the sacred by participating in reality that we are already participating in. It is nonsensical to physically participate in a sacred rite twice simultaneously. In the sense of what is known, the equivalent of participation emerges in changing a mind's notion of what is true. This can't be done for the same rite — truth — more than once, and so novelty is essential. There must be some doubt to be satisfied by reason. Comprehending a proof we have previously comprehended fails to be novel to the extent that we do not modify our understanding of what is true; there is no new certainty. In this way we may readily imagine the forgetful to be the most ecstatic of all — not because ignorance is ecstatic, but because (re)discovery is. If mathematics is a practice of creation, the mind uncovers what is truly new. If it is better taken as discovery or revelation, all the better; in that way, it is more real and more absolute, and by partaking in discovering or finding the universally certain, the participant partakes in the real and therefore the ecstatic. Altogether, these various lenses upon the comprehension of a proof demonstrate its potential to be ecstasy is not so far-fetched.

The Nature of Ecstasy

Up to this point, we have been informal. Now, we shall derive a definition of the ecstatic from the sublime. Kant's notion of the sublime is of "an object (of nature) the representation of which determines the mind to think the unattainability of nature as a presentation of [reason's] ideas."242 The "unattainability" is twofold: first, of the imagination to comprehend the object and second, via the former, of reaching reason's ideas. In the case of the proof, it is the unattainability of the conception of an idea (or ordered sequence of ideas, what mathematicians might call a tuple) that emerges from the imagination while still conforming to natural order; that is, while still validated by reason because it is logically sound and comprehensible. Weiskel reformulates Kant's definition in a clearer, more generalized way to define the sublime as any object, such as a proof, "if the attempt to represent it determines the mind to regard its inability to grasp wholly the object [proof] as a symbol of the mind's relation to a transcendent order." The mind and proof "become radically indeterminate" because the mind contains what it knows to be true and the proof becomes true, collapsing the border between them. There is a "signifier," the proof, "which finds no reflected signified in our minds" because this conviction did not exist before there was evidence for its existence. The "mind's new relation to a transcendent order" emerges from its reconstitution of what it understands to be true. If there is no recharacterization of the true, likely due to incomplete or previous comprehension, the lack of change is described by Weiskel's "homologous relation of habitual perception," which is the ordinary: the unchanged notion of the true. This distinction is common to many thinkers discussed above. In other words, we have demonstrated the comprehension of a proof constitutes a moment of the sublime in Weiskel and Kant's sense, where the mind's relation to truth changes.

To also briefly attach the sublime to the ecstatic, we see from both Kant and Weiskel how readily the sublime brings about the ecstatic as a conduit.

To spell this idea out further, representing or comprehending the proof compels the mind to look back upon itself and its own limitations. This act inhibits it from fully comprehending how the act that brought this all about — representing the proof — ennobles the mind in terms of its relation to Weiskel's "transcendent order." Mathematics, in particular, best exists in this transcendental state — as opposed to some rite or natural awe-inspiring phenomenon — because it transcends the incidental facts of existence. Math would be true in any world; it transcends more than just this one. We could go so far as to say that mathematics proofs are the purest form of ecstasy, but we will be satisfied with a form of ecstasy.3

To better connect Weiskel's "sublime moment" to what we take to be ecstatic we shall identify the ecstasy of the proof by framing it in terms of, first, the familiar and, second, infinite potential. The more well-defined something is, the more it is commonplace, the more formulaic and applicable it is, and the further it is from the ecstatic. The sublime — the ecstatic — fights "the familiar"; in fact, "the sublime [is] an antidote to the boredom." It does this by "tamper[ing] with the received 'natural' meanings of signs" in what Weiskel identifies to be two ways. A proof matches Weiskel's first way exactly by putting ideas to the words that formerly lacked proof and collapsing the "gap between word and thing." However, a proof also violates the outdated meaning of signs in a novel way. Weiskel's second way is to "violate decorum," to misuse words. The words of mathematical language are separate from belief; mathematical equality is a logical certainty (with caveats beyond the scope of this paper). To use mathematical language to assert \(\sum \frac{1}{n^2} = \frac{\pi^2}{6}\) is to misuse language until we have reason — proof — to use language in that way.4 This is how language is misused and decorum violated. The comprehension of a proof brings about a breakdown in the mind between the relation of what we hold to be true and the use of language to claim something different is true.

A proof, though it employs language in a specific and rigorous way, also has to do with infinite potential. Hyper-specificity can be taken as the opposite of the ecstatic because it is in the immensity, the belief of the transcendence of comprehensibility, that triggers the ecstatic. Weiskel states "the sublime moment establishes depth because the presentation of unattainability is phenomenologically a negation, a falling away from what might be seized." Kant states this "sublime is [a] feeling of displeasure that arises from the imagination's inadequacy."15 The depth is this immensity, but how are we to reconcile the negative with the increased understanding of truth a proof provides? I argue the ecstasy of the proof is in the conception of some logical argument (the tuple of ideas) from the vastness of possibility. It is from the awe of this sliver of conviction — emerging by way of imagination and validated by reason — from the complete unattainability of this same practice applied ad infinitum to the whole space of possibility. By comprehending ("seizing") what could have just as easily been any other sound argument, or every other sound argument — which is truly unattainable — we find ecstasy. Whether Euler's ecstasy, with his own imagination conceiving a proof for the first time, or our ecstasy, comprehending his work in awe of that imagination which could be ours and which could create such an argument — they both emerge from the comprehension of a proof. In this way, the ecstasy of proof hinges upon the infinite potential of the imagination.

The Second Proof

Euler's proof, though vastly different from ours, reaches the same conclusion. To us, after having found our own way there, is it still ecstatic? I argue in the affirmative, and a brief digression onto more informal ground will help us comprehend how the mathematical community esteems proof itself, rather than what is proven verifiably.

There is a deep and enduring reverence for proofs, both old and new, within the mathematical community. This is not an isolated case — mathematicians have uncovered proofs addressing the Basel problem through Fourier analysis, combinatorics, and probability theory, each offering its own distinct revelations.2 This pursuit of new proofs for old theorems is not merely an exercise in redundancy but is frequently celebrated as a means of deepening understanding; improving accessibility and pedagogical clarity; achieving elegance and simplicity; and uncovering unexpected connections between mathematical domains. A popular example is the work of G. H. Hardy and J. E. Littlewood, who uncovered powerful techniques for analytic number theory while working on alternative approaches to proofs in analysis.13

Despite the fact that the classic Pythagorean theorem is one of the most well-known results in mathematics and there exist numerous simple, elegant proofs, it continues to be proven in new and interesting ways. In fact, there are hundreds of distinct proofs and the number continues to grow; in 2023, a group of high school students discovered a new trigonometric proof of the Pythagorean theorem, igniting widespread public interest.20 While many journals in contemporary academic publishing reasonably prioritize original results over novel proofs for existing theorems, some journals, such as the American Mathematical Monthly, explicitly encourage new proofs of known results, recognizing their didactic and heuristic value.16 All this is to say, there is a deep recognition not only for novel discovery of new theorems, but also of new ways to be convinced of these pre-proven theorems.

A new proof for a proven theorem is ecstatic via the infinite imagination. Our earlier exposition around the resolution of a proof from the whole space of possibility does not depend on the final claim being novel. Regardless of whether we happen to know the final line, the umpteenth proof demonstrates how each idea necessarily follows from one to the next. This is in itself novel and reshapes the mind's notion of the true. We may even be tempted to save ourselves the trouble of writing in the final line, but there is novelty there too. That this way may have been the first is itself a possibility and impossibility all at once.

Novelty is essential, yes, but the final declaration need not be the ecstatic part. Anyone may posit \(\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}\). This is indeed a misuse of symbols, and to claim it is true is exactly that: a claim or proposition. The Basel assertion alone — \(\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}\) — while a part of the proof, and an avenue unto it (the misuse of symbols, etc.), is, alone, not ecstatic. The doubt does not morph into certainty by the final claim; the proof entailing it causes the reconfiguration of the truth. The proposition becoming a theorem is ecstatic, yes, but not without the whole of the proof behind it. Therefore, we encounter no issue with a second novelty — a distinct proof itself ecstatic — to realize the justified use of symbols: \(\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}\).

Would you call the quip in the first line of this essay proof? I incontrovertibly convince you that you have come across a curious puzzle while reading one day. Setting aside the self-fulfilling issue, this raises a deeper question: what constitutes proof?

This topic is expansive, subject to extensive philosophical debate. First, different schools admit different conditions on what a proof might be. Intuitionists demand constructive proof; Platonists are extremely open-minded. But whatever it takes to satisfactorily move a mind from doubt to certainty, meeting those conditions has ecstatic potential.

Implicitly, so far we have assumed a Cartesian approach, whereby proofs can be totally understood "all at once."12 This feels intuitive and familiar, until one is reminded that proofs can be gargantuan — Gödel and others have published proofs on the lengths of proofs, in particular that however large you want your proof, there exist theorems that must take at least that long to prove.1121 Leibniz would disagree that proofs must be understood altogether and instead defined a "more logical" proof, one with numbered lines which each receives a tick mark after being systematically verified from the previous lines according to the rules of logic. The Cartesian method may seem more suited to our argument because its proofs appear more analytic; rather than constructing a logical structure from premises or uncovering complex implications through symbolic manipulation, they clarify the structure of truth by making explicit what was already implicit. A critic might argue that it is precisely because of their analytic nature that such proofs can be grasped in their entirety by the mind. But the Leibnizian understanding of proofs is not incompatible — and, indeed, is more agreeable to both Euler's proof of the Basel problem and our own than the Cartesian viewpoint. The justification builds systemically, relying on a sequence of convictions which are themselves abstractions of smaller lemmas (possibly proofs themselves, but we shall distinguish them for clarity), to achieve the ecstatic. From a modular perspective, the objection falls apart.

Unfortunately, there are bigger obstacles. To claim the Four-Color Theorem — which was proved in 1976 only through the help of computers which exhaustively checked nearly 2,000 cases3 — conveys ecstasy may lead us to restricting ourselves to non-computer aided proofs. The issue arises from the inability to hold the whole tuple of ideas in the mind, thereby inhibiting the mind from fully justifying its transcendental transformation of the truth.5 However, we would like to say that if someone could prove it without aid, it would be ecstatic. Indeed, this may be possible, in the same way we dealt with the Leibnizian viewpoint. It is simply so inconvenient no one may bother, so neither do they bother to experience the ecstatic.

The first line of this essay does something akin to proof. It does not derive its claim from axioms, nor does it proceed through formal logical inference, yet it leaves no room for doubt. Its success is immediate: in reading it, you find yourself necessarily convinced. It's similar to ideas like: "this sentence is true." This is a different kind of necessity from proofs, one not grounded in syntactic derivation but in the transparency of the idea itself. As such we will not call it proof, but an analytic truth. One way to intuit this distinction is to observe that ecstasy arises not merely from observation, but from the process of traversing the unknown — of constructing a path from uncertainty to necessity. It is in this traversal that the ecstatic resides. This perspective strikes to the heart: the ecstatic dares to convince the mind, either of mathematical certainty or of the mind's inability to fathom the reality of nature. Even if this reality is out of grasp, the recognition of its impossibility takes the ecstatic mind from ambivalence and skepticism to certainty. Whether the awe is grasping nothing or something, finally and for real, the experience is certainly not boring.

Now is time to address the non-ecstatic proof. Should the rationalist sincerely trade their poetry for a math textbook? Admitting textbook study is readily accommodated as a rite, as fair as any other, but proofs can be constructed systematically. For example, for any novel true phrase you like, called A, we can infinitely conjunct them (i.e. A and A and A ...). This is problematic because it conflicts with our novelty condition. Lost on the formalism is the triviality we immediately recognize. No one would deny it is proof, but any particular instance of these (setting aside the recognition of this triviality) we would be hard-pressed to even admit to an ecstatic potential. Thus, have we found a non-ecstatic proof? Unfortunately, it's more complicated; there exists the domain of nonclassical logics, dedicated to formulating non-classic systems and identifying how strong they are, where often such simple premises — such as the nth conjunction of A — may not be redundant at all. Therefore, we are left with the compromise of ennobling proofs with ecstatic potential.

Besides proof, there is the realm of mathematical explanation. Within this domain is Euler's proof as he first presented it and effectively the proof as presented herein. The critical reason is the Weierstrass Factorization Theorem was not proven at the time of Euler's original announcement.6 To Euler's credit, he conducted extensive empirical validation, it was proven not long after, and his goal was primarily finding the value it converged to with solid justification, as opposed to specifically proving it. There is another twist to Euler's proof of the Basel problem: it is a popular demonstration of how plausible reasoning conveys mathematical confidence, or even nigh-certainty; while not called proof, it holds a revered place in the domain of mathematical explanation.12172218

Kant's notion of the sublime refers to an experience where the imagination is overwhelmed by an object of nature (e.g. the vastness of the ocean). A mathematical proof, however, may not be an object of nature. The practice of mathematics has been described as "the enigmatic matching of nature with mathematics and of mathematics by nature,"5 or roughly per Galileo, "nature is written in the language of mathematics,"26 and therefore distinct from nature itself. Many pure mathematicians would prefer a more abstract definition. There is a longstanding debate between the Platonists, formalists, and intuitionists, to name a few, about to what extent mathematical truths are objective or natural. Ultimately, though, what Weiskel and Kant both identify for ecstasy is necessarily phenomenological; it is an experience within the mind. Regardless of how we realize multivalent mathematical "intuition," the nature of a proof as described precedes the senses used to perceive nature; as it exists wholly in the mind, its effect should be all the stronger for it, and certainly constitute the ecstatic.

Kant noted that the sublime involves a paradox: while our imagination is displeased by its own inadequacy to fully grasp an object, it simultaneously experiences pleasure because this very inadequacy is in harmony with our rational aspirations.15 At first glance, there is a major problem in locating the displeasure of the proof. I contend the displeasure is in the complexity. If all necessary truths were analytically self-evident, there would be no ecstasy. This displeasure (and the challenge) is the necessity of finding a coherent proof from the vast space of possibility. There are some cases which especially highlight this conundrum. Renowned mathematician William Browder, writing on behalf of the National Academy of Sciences in a brief report titled Progress in Theoretical Mathematics, wrote:

"Herein lies the agony as well as the ecstasy of Langlands' program. To merely state the conjectures correctly requires much of the machinery of class field theory, the structure theory of algebraic groups, the representation theory of real and p-adic groups, and (at least) the language of algebraic geometry. In other words, though the promised rewards are great, the initiation process is forbidding."

Gelbart (1984) 9

We may readily suppose that Euler's experience may have had greater effect on him than on us. After all, he was able to publish it and resolve a longstanding problem — and quite early in his career, too. Is he, then, more entitled to ecstasy? No. The transformative nature of an ecstatic experience is not solely a function of originality or historical impact but rather of the intrinsic structure of the understanding itself. Ecstasy is not apportioned by merit or precedence; it has the potential to arise wherever comprehension reflects upon itself, irrespective of who experiences it or why.

A contrarian might object that by insisting proofs must evoke an element of ecstasy, we risk shifting their purpose from establishing mathematical truths to celebrating aesthetics. This view suggests that proofs could be judged more on their beauty and insight than their rigor — a concern underscored by computer-aided methods and self-verifying languages like Coq, which, while analytically sound (setting aside engineering issues), often receive criticism for lack of those very properties: aesthetics and insight. If a proof is solely a tool for establishing truth, then its presentation and elegance should be secondary. Yet, if the insight quality is deemed essential, the boundary between logic and aesthetics becomes blurred, reducing what was once rigor to a potentially subjective experience. We can counter that proofs are primarily practical instruments for uncovering mathematical truth. While elegance and simplicity can enhance their appeal, these qualities naturally emerge from a proof's fundamental function: transforming our understanding of what is true, rather than serving as its main criterion. By transforming our understanding of the truth, proofs, by their nature, must have the potential to be ecstatic. We might even wonder if there exists a bijection between the two.

A proof is a comprehension of reality; it is a particular instance of what must be true from the relation between other ideas within the mind. As the mind changes to reconstitute these ideas, it is that moment we would call ecstatic. Comprehending a new proof is the opposite of profane; it differentiates itself from the ordinary with its novelty, its change, and its impact on the mind. While a proof may have been discovered before, comprehending it has the potential to be ecstatic because it can change what the mind constitutes as true. At the same time, it presents the imagination with a proof (an ordered set of ideas validated by reason) inextricably linked to the incomprehensibly infinite possibility of any and every other proof that can be conceived through imagination and validated by reason, all self-contained within a mind.

I have argued that, consistent with the definitions of Weiskel and Kant, but more broadly put: the sublime is the recognition of an impossibility in the representational power of the mind; the ecstatic moment is the transfiguration from ambivalence and skepticism to conviction. If that happens to be in awe at the impossibility of comprehension or the pure reduction to certainty, both reconfigure what the mind calls truth and both are valid. In conclusion, this proof within a proof explores the relation between reason and thinking and imagination and ecstasy, working to show they are not so dichotomous as they appear.