1. A Former Smoking Room

The Chicago Board Options Exchange opened on April 26, 1973, in what had been the Chicago Board of Trade’s smoking room. The parent exchange was celebrating its 125th anniversary. A new market was beginning in a space previously used for a break from the old one.

A few weeks later, Fischer Black and Myron Scholes published their paper on option pricing. Robert Merton’s treatment appeared that spring too; a working version dated from 1971. Black later recalled that the Black–Scholes paper had first been rejected and gained another hearing with help from Chicago economists. What subsequently became an emblem of mathematical finance did not arrive wearing that emblem.

Its central move was not simply to offer a clever formula. Under specified assumptions, a portfolio of stock and borrowing could replicate the payoff of an option. An option price inconsistent with the replicating portfolio would create an arbitrage opportunity. This offered a way to price the option without first measuring each investor’s aversion to risk. The assumptions mattered: a mathematical replication argument was not a promise that actual trading would be frictionless or that prices would follow the chosen stochastic process.

Black’s career traced the movement that followed. He graduated in physics from Harvard in 1959 and earned an applied-mathematics doctorate there in 1964, working on an artificial-intelligence question-answering system. He worked on AI in industry, moved into consulting in 1965, and encountered finance. Chicago’s business school followed in 1971, MIT in 1975, and Goldman Sachs in 1984. He became a partner two years later. The route ran from mathematics through consulting and universities into an investment bank.

The equations travelled too. Black sold printed tables of model prices to traders. Programmable calculators and then computers made calculation available during trading rather than only before it. By the later 1970s, the model was part of the market’s working language.

Donald MacKenzie’s historical and sociological studies add a further possibility: using a model can help to create the conditions under which prices resemble its predictions. Traders learning the same relationships and exploiting the same discrepancies may bring the market closer to the model. This is a historically situated claim about feedback between knowledge and practice, not a theorem that markets must eventually obey equations.

Mathematics had always worked for money and power. The scribes at the beginning of this series accounted for grain and taxes. Navigation needed astronomical tables; insurance supplied problems in probability; surveying and artillery required calculation. Bletchley Park was hardly a retreat from practical affairs. If something changed after the 1970s, usefulness alone cannot explain it.

2. Strangers on an Open Line

In November 1976, Whitfield Diffie and Martin Hellman published “New Directions in Cryptography.” Its opening located the problem in an emerging material world: inexpensive digital equipment, remote terminals, and automated banking. Strong cryptography was becoming technically affordable far beyond the institutions that had traditionally possessed it.

Could two people who had never met, and had exchanged no prior secret, establish a shared secret over a channel others could overhear? Their paper developed the public-key framework, digital signatures, and a key-agreement method. There is an important qualification to the familiar story: unauthenticated key agreement protects against passive listening under its computational assumptions, but not by itself against an active intermediary impersonating each participant to the other. Authentication is a separate part of making strangers’ communication secure.

This was not the first public academic research on cryptography. Shannon’s published work alone rules out that description. It was, however, a decisive expansion of what an open research community could attempt. Cryptographic design became a field in which university researchers could publish mechanisms, expose them to attack, and build on one another’s results.

That openness met a practical threat. In July 1977, Joseph Meyer, an employee of the US National Security Agency, wrote to the IEEE suggesting that publication of cryptographic research might violate arms-export regulations. An information-theory symposium was approaching at Cornell. Hellman’s students Ralph Merkle and Stephen Pohlig were due to present.

Merkle had already experienced a different barrier to recognition. As a Berkeley undergraduate in 1974, he proposed a computer-security course project containing the germ of his public-key work. His instructor did not accept it. He simplified the proposal, met continued disinterest, left the course, and pursued the idea himself. A paper submitted in August 1975 then faced an adverse anonymous assessment and appeared only after a long delay.

Now publication carried a possible criminal risk. After consulting their families, the students decided not to deliver their talks. According to Hellman’s participant account, Stanford offered to defend him if prosecuted, but could not pay a criminal penalty or serve a sentence on his behalf. He presented the work with the students beside him. The scene belongs to a recollected institutional conflict, not a transcript of every conversation.

Meanwhile, Ronald Rivest, Adi Shamir, and Leonard Adleman developed the system known by their initials. Martin Gardner introduced it to a popular readership in August 1977, with an encrypted challenge and a hundred-dollar reward. The formal paper followed in February 1978. A company appeared in 1982, and a US patent in 1983. Public mathematics could become a published method, a commercial enterprise, and a legally controlled implementation at the same time.

3. Four Years Earlier

On December 18, 1997, Clifford Cocks publicly described a history that had been unavailable to the public participants themselves.

Around 1969, James Ellis at Britain’s Government Communications Headquarters conceived what he called non-secret encryption. His 1970 internal report argued that encryption need not require a previously shared secret. What it lacked was a workable construction.

Cocks arrived in communications-security work in September 1973, aged twenty-two. He had studied mathematics at King’s College, Cambridge, and begun research in number theory with Bryan Birch at Oxford before leaving for GCHQ. A colleague, Nick Patterson, told him about Ellis’s problem. Cocks found a construction closely related to the RSA system independently published four years later. His internal note was dated November 20, 1973. In 1974, Malcolm Williamson developed a key-agreement method closely related to what Diffie and Hellman would publish.

Calling this merely “large primes multiplied together” leaves out much of the mathematics. The construction uses modular arithmetic and an asymmetry between public operations and information retained by the recipient. Its significance here is also institutional: none of the three men could submit the work to a journal. These were not rejected papers. They were discoveries outside the category of publishable knowledge.

The IEEE’s historical milestone recognizes that secret sequence. Declassification came in 1997. Ellis had written an account in 1987, but it remained unavailable for another decade. He died on November 25, 1997, aged seventy-three. His account was released the following month, shortly before Cocks’s public presentation.

Recognition then arrived on a different timetable. Cocks received the Companion of the Order of the Bath in 2008. The IEEE commemorated the three researchers in Cheltenham in 2010. Cocks became a Fellow of the Royal Society in 2015, after a career that included serving as GCHQ’s chief mathematician.

Two systems of priority meet here. Dated internal documents establish an earlier discovery. A responsible history must include it. But Diffie, Hellman, Merkle, Rivest, Shamir, and Adleman did not borrow from a history they could not read. Their independent public work created an accessible body of methods that other people could examine, criticize, implement, and extend. An archive can revise the chronology of discovery without retrospectively cancelling the achievement of making knowledge public.

Intelligence agencies have also employed mathematicians on a substantial scale in the United States. Claims that the NSA is among the largest mathematical employers are not a substitute for an independently auditable annual headcount. Secrecy obscures both the workforce and its work.

At least three effects need separating: research never published, research published decades later, and research whose ideas may be discussed while operational details remain restricted. Each changes how credit accumulates and how much of a researcher’s expertise can travel with them.

4. Easy in One Direction

In 1940, G. H. Hardy defended mathematics partly through its distance from practical harm. He suggested that number theory and relativity had not been found useful in war and were unlikely to become so soon. Whatever one makes of the accuracy of that historical judgment, the moral attraction is clear: mathematical beauty seemed to offer a refuge from destructive usefulness.

Public-key cryptography unsettled that refuge. Multiplying three by five gives fifteen. Factoring fifteen is easy too. With 143, one may need a little trial before finding eleven and thirteen. As the integers grow, multiplication remains relatively inexpensive, while the best known general classical factoring methods become much more demanding.

But the popular explanation needs several brakes. A product of two hundred-digit primes is not guaranteed to require any fixed number of years to factor. Difficulty depends on the size and structure of the number, the algorithm, the hardware, and the available resources. Nor has mathematics proved that general integer factorization has no efficient classical algorithm.

RSA adds another distinction. Its public operation is modular exponentiation, not multiplication alone. Knowing the factorization of the modulus provides an efficient route to recovering the private exponent in the usual construction. It has not been proved in full generality that every way of inverting the RSA operation must factor the modulus. “Factoring is hard” and “RSA is secure” are related claims, not interchangeable theorems. Secure implementations additionally require appropriate padding and protection against attacks on the surrounding system.

The language of one-way functions is therefore conditional here. Such a function should be easy to compute and computationally difficult to invert in the specified sense. The existence of suitable one-way functions is not an established unconditional theorem. Cryptography turns carefully investigated hardness assumptions into useful mechanisms; it does not abolish the assumptions.

The challenge Gardner publicized used a 129-digit modulus. In April 1994, seventeen years after its appearance, an international effort completed the factorization. Roughly six hundred volunteers contributed access to about sixteen hundred machines over several months. The recovered message was “THE MAGIC WORDS ARE SQUEAMISH OSSIFRAGE.” The hundred dollars went to the Free Software Foundation.

RSA-250 fell in 2020. The researchers estimated the computation at approximately 2,700 core-years, a measure of aggregate work, not a claim that one machine had actually run for twenty-seven centuries. Their published account of related record computations records the advance. A 2,048-bit RSA modulus, a familiar deployed size, has roughly 617 decimal digits; that observation is not a blanket recommendation for every application or an assurance of indefinite security.

The branch of mathematics Hardy prized for its apparent uselessness had become valuable enough to classify, patent, attack, and deploy. Usefulness changed. The need to distinguish proof from confidence did not.

5. Is Source Code Speech?

Daniel Bernstein developed his Snuffle encryption work while a graduate student at Berkeley. In 1995, supported by the Electronic Frontier Foundation, he sued over export restrictions affecting publication of his paper and source code. Communicating code to foreign colleagues could trigger requirements that made ordinary scholarly circulation a regulated act.

In April 1996, a federal district court accepted that source code had expressive qualities protected by the First Amendment. A Ninth Circuit panel in May 1999 went further, finding an unconstitutional prior restraint in the licensing regime it considered. For researchers, source code was not merely an executable object. It was also a precise language for communicating an algorithm to another person.

Yet that appellate ruling cannot be presented as an uncomplicated final precedent. The panel opinion was withdrawn in September 1999 when rehearing by the full court was granted. The litigation subsequently changed course as the regulations changed; it did not culminate in a Supreme Court ruling establishing a universal exemption for encryption software. The case archive preserves the procedural sequence as well as the more famous opinion.

The practical environment did change. Policy announcements in 1999 and rules in 2000 substantially liberalized parts of the export regime, including provision for publication of certain source code subject to conditions such as notification. That is a historical description, not a statement that all encryption exports became unrestricted.

Another struggle concerned government access rather than publication. The Clinton administration proposed the Clipper chip in 1993. Its key-escrow arrangement was meant to preserve a route for authorized government decryption. In 1994, Bell Labs researcher Matt Blaze identified a way to defeat the escrow mechanism by exploiting its short, sixteen-bit authentication check.

The distinction matters here too. Blaze did not thereby break the underlying Skipjack cipher or show that ordinary eavesdroppers could read every protected conversation. He showed how a user could evade the intended law-enforcement access mechanism while continuing to encrypt. NIST’s contemporary response explicitly distinguished those issues. By the middle of the decade, the initiative had lost momentum; saying that nobody mentioned it again would erase the continuing policy debate.

The coalition resisting restrictions did not consist solely of university mathematicians. Cryptographers, computer scientists, engineers, lawyers, civil-liberties organizations, and software businesses participated for overlapping reasons. Calling them all mathematicians would obscure precisely what was changing: mathematically intensive work no longer fitted neatly inside an academic department’s occupational boundary.

Bernstein later introduced Curve25519 in 2005. Related mechanisms entered widely used secure communications, including remote-login tools, transport protocols, and messaging systems. The researcher who had needed litigation to defend the circulation of code helped create code that could circulate and operate at enormous scale. Whether code counted as expression had become a question about the infrastructure of everyday communication.

6. Giving a Model Power

Black died of throat cancer in August 1995, aged fifty-seven. Scholes and Merton received the 1997 economics prize; the announcement recognized Black’s contribution, but the prize was not awarded posthumously.

Both recipients were partners in Long-Term Capital Management. Founded by John Meriwether in 1994, the fund combined trading experience with academic finance. It pursued heavily leveraged relative-value positions: identify prices expected to converge and borrow enough that small discrepancies could produce large returns.

After Russia’s default in August 1998, investors rushed toward safety. Relationships that had looked dependable moved against the fund. Markets in which positions needed to be unwound became illiquid. By the end of August, LTCM had lost approximately half its capital.

On September 23, a consortium of private institutions agreed to a $3.6 billion recapitalization, coordinated through discussions at the Federal Reserve Bank of New York. The Federal Reserve facilitated the arrangement but did not supply the rescue capital. The Federal Reserve’s historical account is explicit about that distinction.

Alan Greenspan’s October 1 congressional testimony did not describe a single equation suddenly becoming false. He discussed models drawing on historical relationships, rising leverage, extraordinary market conditions, liquidity, and the failure of counterparties to constrain exposures through adequate scrutiny and stress testing.

“Nobel formula fails” is consequently too compressed a story. The Black–Scholes–Merton option-pricing result is not identical to every strategy pursued by a fund employing Scholes and Merton. The dangerous system combined confidence in models, borrowed money, crowded positions, disappearing liquidity, and institutions willing to finance the whole arrangement.

The financial crisis a decade later did not make models disappear. It helped produce an expanded professional apparatus for supervising them. US banking guidance in 2011 described model-risk management through inventories, independent validation, effective challenge, continuing monitoring, and limits on use. Mathematics was being employed to construct models, test models, and govern the authority granted to models.

Emanuel Derman had taken another route from physics to finance: several years in particle physics, Bell Labs in 1980, Goldman Sachs in 1985, and collaboration with Black on an interest-rate model. In January 2009, he and Paul Wilmott published a financial modelers’ manifesto.

Its proposed commitments were epistemic and ethical. The world is not the modeler’s creation and need not obey the equations. A model can be used boldly without its mathematical sophistication becoming an excuse for misplaced confidence. Assumptions and omissions should be disclosed; precision should not be promised where it cannot be justified. The social consequences may exceed the modeler’s understanding.

Their manifesto did not renounce mathematics. It challenged the conversion of a useful representation into an unquestionable decision-maker.

7. Away from the Desk

Corporate laboratories offered another institutional home. Bell Labs could sustain research far from immediate products through the resources of the regulated AT&T system. The 1984 breakup and subsequent reorganizations changed that arrangement without simply erasing the laboratory.

Xerox established its Palo Alto Research Center in 1970. Microsoft Research followed in 1991. Such organizations could employ researchers inside companies while maintaining substantial connections to papers, conferences, and universities. Corporate employment did not automatically imply secret research.

Search engines and large digital platforms added a different arrangement: mathematical research embedded directly in an operating product. Recommendation, advertising auctions, dynamic prices, fraud detection, and search ranking require collaboration among researchers, engineers, economists, and product teams. The algorithm is attached to logs, experiments, computing infrastructure, and a stream of real decisions.

Access to these resources can determine the questions a researcher is able to investigate. Books, papers, and colleagues remain important, but so do proprietary data, computing capacity, users, and permission to experiment. Leaving an employer can mean losing not merely income but the research object itself.

The outputs also take different forms: publications, patents, production code, products, and trade secrets. Some are visible in a bibliography; others are not. A classified state archive may eventually open. A failed trading strategy or discarded recommendation system may disappear with its organization and leave no comparable public record. Neither outcome is inevitable, but they expose the inadequacy of treating a publication list as a complete inventory of mathematical work.

Occupational statistics create a related illusion. The US Bureau of Labor Statistics reports about 2,200 jobs in the narrowly defined occupation “mathematicians” for 2025, with 53 percent in the federal government. These are not counts of everyone holding a mathematics doctorate or doing mathematical work. Research scientists, applied scientists, quantitative researchers, faculty, and many others can sit in different categories. The BLS classification describes a job label, not the entire population of mathematical thinkers.

Three institutional pictures can help, provided they are treated as contrasts rather than exhaustive histories. Ancient Greek mathematical writing survived through a small, dispersed community of texts and commentators. Göttingen assembled chairs, seminars, reading rooms, and editorial networks into a university centre. The later twentieth century expanded durable careers in intelligence, finance, corporate laboratories, and platforms.

Ancient scholars also depended on patrons; university mathematicians never possessed unlimited independence. Still, the portability of work can differ sharply. Someone changing universities may take the central problems with them. Someone leaving a platform or trading firm may lose the data, code, transaction records, computing cluster, and publication permission that made a problem researchable. Institutional location becomes part of the mathematics that can actually be done.

8. Repricing a Career

The US Survey of Earned Doctorates makes one change unusually concrete. Among mathematics and statistics doctorate recipients with definite US employment commitments outside postdoctoral study, 226 of 325 entered academia in 2003: approximately 70 percent. In 2023, the corresponding figures were 220 of 814, or 27 percent.

The academic numerator barely changed; the denominator more than doubled. These figures do not describe every doctorate recipient, every country, or an entire career. They describe a specified group at graduation, with classification changes also requiring care in comparisons. The NCSES methodological notes supply the underlying counts.

For the 2024 cohort, reported median expected salaries in mathematics and statistics were $150,000 for industry or business, $68,000 for academic employment, and $70,000 for postdoctoral positions. “Industry or business” is not a synonym for every nonacademic sector: government and nonprofit employment are distinct. These are expected starting salaries, not lifetime earnings. Even with those limits, the survey’s salary comparison shows a labour market placing markedly different prices on available paths.

Other questions concern the source of the money. In 1970, Alexander Grothendieck left the IHÉS amid disagreements that included military funding. He helped establish Survivre, concerned with armaments, ecology, and scientific responsibility. His institutional break did not end his mathematical work: substantial research and writing continued afterward, including in the 1980s.

Grothendieck asked where support came from. Bernstein asked who could prevent research from circulating. Financial modelers asked how much power an organization should give their constructions. These are different questions, but all locate mathematics inside arrangements of authority.

What changed after the 1970s was not the invention of applied mathematics. A much larger doctorate-producing system fed a more extensive employment market outside universities. Mathematical models increasingly operated inside decisions rather than merely advising the people who made them: generating quotes, executing trades, allocating advertisements, and ranking what users would see. Data, computation, market access, and experimental permissions became essential assets. Classification was joined by patents, commercial confidentiality, and proprietary datasets as a constraint on circulation.

This is not a simple story of capital capturing formerly innocent knowledge. Companies have supported genuinely open foundational work; Shannon’s 1948 information theory appeared in the Bell System Technical Journal. Universities themselves depend on public budgets, defense funding, foundations, and tuition. The Cocks case shows how thoroughly a state can suppress circulation, not a universal theorem that states always conceal more than firms.

Applications also return problems to mathematics. Cryptography draws on number theory, finite fields, and elliptic curves; finance generates work in stochastic analysis, partial differential equations, numerical methods, and optimization. Platform systems interact with learning theory, combinatorial optimization, and mechanism design. Some results travel back into public knowledge.

One way to understand a researcher’s position is to ask what they can take with them when they leave.

On August 22, 2006, the International Congress of Mathematicians in Madrid announced four Fields medalists. One was absent and refused the medal. The question of what mathematics was worth was about to take another form.