Begin with reflected paths in Alexandria and the fastest curve in Groningen, then minimize functions and errors in Lausanne and Paris. Compute constraints and plans in Chicago, Leningrad, Washington, Berkeley, and Santa Monica before reaching interior points in Murray Hill and learning in Toronto—separating ‘the calculation is correct’ from ‘the objective is right.’
QUESTION FOR THE ROUTE
How does optimization turn ‘what is feasible?’ and ‘what is better?’ into mathematics, and who must question the people, costs, and values omitted from that mathematics?
WHAT THIS RIVER DOES NOT CLAIM
The river is not geography. Distance downstream stands for time passing, and the light turns from dawn to dusk as the centuries go by. The objects by each stele are symbols of the kind of event and of how each century band wrote and calculated; they do not reconstruct any real artefact. The land around each stop sketches the natural geography of the scene’s real place, and an iconic building appears only if it already stood in that year. Each scene keeps its real place and evidence basis; open it on the map to read where it happened. This route does not equate optimization with universal efficiency, automated justice, or one engine of AI. It separates shortest distance from shortest time, local descent from global optimality, necessary from sufficient conditions, worst-case theory from practical performance, and Pareto efficiency from justice. The map line is an edited comparison across distinct problems and institutions, not one text’s proven transmission.
WHAT YOU SEE ON THIS RIVER
- Diagram in the sky
- The shortest path across a network
- Emblem at the source
- Stakes lighting the shortest path
- The real place around each stop
- Around each stele the land takes on the natural geography of that scene’s real place — sea or lake, plain, hills or mountains, the colour of the ground and its common trees — and, where one defines the place, its landform: a volcano, snow peaks, granite domes, a mesa, dunes, a fjord, islands, a rock hill, a gorge or loess terraces. The water near the stop takes the colour of the real river or sea, and the haze the place’s climate. A small globe on the stele marks where it is, with the route from the previous place. Where a city has an iconic building that already stood in the scene’s year, its schematic silhouette rises behind the stop and is named on the card. The land follows today’s terrain and climate as a sketch and the silhouettes are not measured reconstructions. Between stops the river itself stays symbolic.
- A figure board at every stop
- Each board draws the mathematics of that scene. When the boat arrives, the construction is drawn in and the key result rises in red. The drawings are schematic reconstructions, not historical manuscripts.
- Century bands along the banks
- to 499 · Sandstone stele · braziers · earthen villages and beacons · rafts · flocks of birds
- 1450–1749 · Marble stele · iron lanterns · three-arch bridge · windmills and clock-tower towns · sailing ships
- 1750–1899 · Cast-iron plaque · gas lamps · iron truss bridge · factory chimneys, railway and steam train · steamboats
- 1900–1969 · Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft
- 1970 onward · Glass marker · LED lights · cable-stayed bridge · glass towers, wind turbines and data centres · ferries · satellites
Where the century band changes, the boat passes under a bridge of the new band. Villages, mills, factories, pylons and towers stand for the technology of each century, not for any real place or architectural style.
01·c. 60 CE·Alexandria(basis: Composition)
Seeing a Shortest Route in a Broken Path of Light — the Catoptrica
Unfold a reflected path at a plane mirror and the equal-angle route becomes the shortest broken line. This geometric argument is an important shoulder for later variational principles. Yet the work survives only in a medieval Latin translation, was misattributed to Ptolemy, and is only doubtfully assigned to Heron. It does not say that light minimizes geometric distance in every medium and setting.
- Pause and ask
- Why is the equal-angle reflected path the shortest as the reflection point moves?
- How thinking changed
- Move from measuring one path to comparing a family of possible paths and selecting an extremum.
- What we cannot claim
- The Catoptrica is usually but doubtfully attributed to Heron. Its plane-mirror geometry is not projected backward into a universal claim that light minimizes distance in every medium or into a modern optimization algorithm.
- This place
- The optical and geometric tradition associated with Alexandria organized mirrors, vision, and reflection as diagrammatic arguments, but the surviving Latin translation cannot fix an exact writing room. (Mediterranean coast · date palms · flat sand · 31.2°N 29.9°E · landmark: Lighthouse of Alexandria (Pharos) (280 BCE))
- Figure board
- Comparing paths as the reflection point moves along a plane mirror: the broken path with equal angles is the shortest.
- On the river
- A desk holding a written record · to 499 (Sandstone stele · braziers · earthen villages and beacons · rafts · flocks of birds)
02·1696 CE·Groningen(basis: Publication)
Asking for a Curve Faster Than the Shortest Line — Bernoulli’s Challenge
Johann Bernoulli challenged mathematicians to find the quickest descent under gravity between two points. The cycloid is longer than the straight segment but drops steeply first and gains speed earlier. The 1696 Acta Eruditorum challenge and a 1697 Groningen reprint moved the problem through European correspondence; the result assumes an ideal point mass, no friction, and uniform gravity.
- Pause and ask
- Why is the fastest descent between two points not the shortest straight line?
- How thinking changed
- See the power of the objective: replace distance with time and the optimal curve changes.
- What we cannot claim
- The cycloid is fastest under ideal assumptions such as a frictionless point mass and uniform gravity. The problem was neither Johann’s isolated invention nor a set of mathematically identical solutions.
- This place
- While teaching in Groningen, Bernoulli used a Leipzig journal and a separate reprint to send the challenge across Europe and create a public arena for comparing solutions. (Open fields · broadleaf trees · flat land · 53.2°N 6.6°E · landmark: Martinitoren (1482))
- Figure board
- Beads marked at equal time steps: the one on the cycloid reaches B first. Swap length for time as the goal and the best curve changes.
- On the river
- A stack of books · 1450–1749 (Marble stele · iron lanterns · three-arch bridge · windmills and clock-tower towns · sailing ships)
03·1744 CE·Lausanne(basis: Publication)
Choosing an Entire Function Instead of a Number — Euler’s Calculus of Variations
Euler’s Methodus inveniendi systematically studied curves that maximize or minimize quantities such as length, time, or action. The unknown was no longer just a finite list of numbers but an entire function. Lausanne and Geneva mark the publication network rather than Euler’s main workplace, and one book did not single-handedly invent every part of the calculus of variations.
- Pause and ask
- How can one choose the best object from an entire family of functions rather than a few numbers?
- How thinking changed
- Expand the unknown from numbers to curves and functions, then extremize a total quantity attached to each.
- What we cannot claim
- The pin marks publication, not Euler’s sole workplace. The book is not compressed into one moment that invented every variational idea or completed the physical principle of least action.
- This place
- The Bousquet publishing network in Lausanne and Geneva turned Euler’s long work across St Petersburg and Berlin into a systematic Latin book for an international readership. (Lake Geneva · Alpine snow peaks · vineyard slopes · 46.5°N 6.6°E)
- Figure board
- Every curve between the same endpoints carries one integral value; the best curve is chosen from the whole family of functions.
- On the river
- A stack of books · 1450–1749 (Marble stele · iron lanterns · three-arch bridge · windmills and clock-tower towns · sailing ships)
04·1805 CE·Paris(basis: Publication)
Squaring Errors to Choose the Best-Fitting Orbit — Legendre
In a work on comet orbits, Legendre first published least squares: minimize the sum of squared differences between observations and a model. Many residuals became one objective for selecting parameters. Gauss later claimed earlier private use and published a theory in 1809, so first publication and claimed prior practice remain distinct. Squared error is not automatically the right loss for every dataset.
- Pause and ask
- How can ‘best fit’ among conflicting observations be defined as one number?
- How thinking changed
- Replace separate explanations of each error with one sum-of-squared-residuals objective for choosing model parameters.
- What we cannot claim
- Legendre’s first publication is separated from Gauss’s claim of earlier private use. Squared loss is sensitive to large errors and is not a natural or neutral criterion for every dataset and purpose.
- This place
- Parisian astronomy, geodesy, printing, and academy networks made a procedure for reconciling observations public and comparable inside a work on comet orbits. (Seine banks · broadleaf trees · flat basin · 48.9°N 2.4°E · landmark: Notre-Dame de Paris (1250), Dôme des Invalides (1706))
- Figure board
- Squares are raised on the gaps between observations and a line; the chosen line makes the total area of the squares smallest.
- On the river
- A stack of books · 1750–1899 (Cast-iron plaque · gas lamps · iron truss bridge · factory chimneys, railway and steam train · steamboats)
05·1847 CE·Paris(basis: Presentation)
Walking in the Steepest Local Downhill Direction — Cauchy
Cauchy proposed following the negative gradient while solving simultaneous equations, an ancestor of gradient descent. Direction alone is not enough: a tiny step can be slow, a large one can oscillate or diverge, and a nonconvex landscape can contain local minima, saddle points, and flat regions.
- Pause and ask
- Once a downhill direction is known, how far should one move?
- How thinking changed
- Replace one closed-form solution with repeated local gradient calculations approaching an approximation.
- What we cannot claim
- The negative gradient is a local descent direction, not a guarantee of a global optimum for an arbitrary nonconvex function. Step size, initialization, stopping, and numerical error change the result.
- This place
- The Paris Academy’s short proceedings format quickly recorded a numerical procedure for simultaneous equations and circulated it among analysts and calculators. (Seine banks · broadleaf trees · flat basin · 48.9°N 2.4°E · landmark: Notre-Dame de Paris (1250), Dôme des Invalides (1706))
- Figure board
- Repeated steps along the steepest local downhill direction, perpendicular to the contours, zigzag toward the bottom depending on the step size.
- On the river
- A desk holding a written record · 1750–1899 (Cast-iron plaque · gas lamps · iron truss bridge · factory chimneys, railway and steam train · steamboats)
06·1896 CE·Lausanne(basis: Teaching / position)
Finding the Boundary Where Helping One Person Hurts Another — Pareto
In political-economy teaching and books produced at Lausanne, Pareto analyzed allocations involving several preferences. A state later called Pareto efficient cannot improve one person without worsening another. That property guarantees neither fairness, justice, equality, nor a unique solution. The chosen preferences, resources, and measures come first.
- Pause and ask
- Is a boundary where one improvement requires another loss automatically a fair answer?
- How thinking changed
- Replace one objective value with a frontier of trade-offs among several agents and multiple nondominated solutions.
- What we cannot claim
- Pareto efficiency means no further unanimous improvement under given preferences and resources. It guarantees neither justice, equality, nor a unique social optimum; a highly unequal allocation can be efficient.
- This place
- Political-economy teaching and publication at Lausanne linked mathematical choice and allocation to institutions, without turning the period’s economic and class assumptions into universal values. (Lake Geneva · Alpine snow peaks · vineyard slopes · 46.5°N 6.6°E)
- Figure board
- With two people’s shares on two axes, there are many points on the frontier where helping one means hurting the other.
- On the river
- A lectern and a board · 1750–1899 (Cast-iron plaque · gas lamps · iron truss bridge · factory chimneys, railway and steam train · steamboats)
07·1939 CE·Chicago(basis: Composition)
Writing Conditions for Minima under Inequality Constraints — Karush
William Karush’s University of Chicago master’s thesis treated multiplier conditions for optimization with inequality constraints. It remained little known until its precedence was recovered after Kuhn and Tucker’s 1950 work, producing the name KKT. The conditions do not automatically guarantee a global optimum: necessity needs regularity assumptions, and sufficiency needs additional structure such as convexity.
- Pause and ask
- At an inequality boundary, which conditions identify candidates for a minimum?
- How thinking changed
- Replace the zero-gradient test with a joint calculation of active constraints, multipliers, and complementarity.
- What we cannot claim
- KKT is not an unconditional solver. Necessity needs regularity such as a constraint qualification, global sufficiency needs further assumptions such as convexity, and a nonconvex stationary point need not be optimal.
- This place
- The University of Chicago master’s-thesis system recorded the result, while limited circulation and later wartime and disciplinary paths kept it outside the standard story for years. (Lake Michigan · broadleaf trees · flat prairie · 41.9°N 87.6°W · landmark: Chicago Loop skyscrapers (1892))
- Figure board
- An interior minimum has multiplier zero; a minimum on the boundary makes −∇f point along ∇g — one of the two factors is always zero.
- On the river
- A desk holding a written record · 1900–1969 (Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft)
08·1939 CE·Saint Petersburg(basis: Publication)
Giving Scarce Factory Resources Shadow Values — Kantorovich
In Leningrad, Kantorovich published a booklet using linear inequalities and solution methods to plan production with limited machines, labor, and materials. Multipliers on scarce resources later supported the interpretation of shadow prices. The work remained largely unknown in the West for years, so it is not merged into one direct transmission line with later American linear programming.
- Pause and ask
- How should scarce machines, labor, and materials be allocated among production plans?
- How thinking changed
- Translate factory rules of thumb into a planning problem with a linear objective, inequalities, and marginal values for scarce resources.
- What we cannot claim
- Kantorovich and Dantzig reached core structures independently in different institutions. The Soviet work is not asserted to have directly produced the 1947 American simplex method, and the planning model did not capture every real-world value.
- This place
- Leningrad University, industrial consulting, and Soviet planning made production allocation urgent and publishable, while linguistic and institutional barriers delayed international movement. (Neva delta and gulf · birch and pine · flat · 59.9°N 30.3°E · landmark: Peter and Paul Cathedral (1733), Saint Isaac’s Cathedral (1858))
- Figure board
- In the production polygon cut by machine, labour and material limits, the gain from one more unit of a scarce resource is its shadow price.
- On the river
- A stack of books · 1900–1969 (Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft)
09·1947 CE·Washington DC(basis: Main activity)
Turning Military Plans into Moves between Vertices — Dantzig
While working on US Air Force planning, George Dantzig developed a linear-planning model and the simplex method, which moves among vertices of a feasible polytope. ‘Programming’ meant organizing plans, not primarily writing computer code. The method is powerful but not neutral with respect to its military and administrative objectives, and it cannot value harms or benefits omitted from the objective.
- Pause and ask
- When there are too many feasible plans, how can a good vertex be found without checking every point?
- How thinking changed
- Read a planning table as a high-dimensional polytope and improve the objective by moving among adjacent vertices.
- What we cannot claim
- Programming here means planning rather than coding. Simplex is often powerful in practice, while some pivot rules have exponential worst cases, and neither its military objective nor omitted social costs are neutral.
- This place
- Postwar US Air Force planning organizations in Washington assembled deployment, training, logistics problems, and computing labor to test a general planning model. (Potomac banks · broadleaf trees · low hills · 38.9°N 77.0°W · landmark: United States Capitol (1866))
- Figure board
- A planning table read as a polytope; the method walks along edges to adjacent vertices, improving the objective at every move.
- On the river
- A place of ongoing work, marked only by the route’s emblem · 1900–1969 (Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft)
10·1950 CE·Berkeley(basis: Presentation)
Publishing Stationarity Conditions for Constrained Nonlinear Problems — Kuhn and Tucker
At the Berkeley symposium on nonlinear programming, Kuhn and Tucker combined objective and constraint gradients, multipliers, and complementary slackness. Published in the 1951 proceedings, their work is read with Karush’s 1939 predecessor as the KKT conditions. Satisfying them does not prove global optimality for an arbitrary nonconvex problem, and constraint qualifications matter.
- Pause and ask
- How can one equation express the balance between objective and constraint gradients at a boundary?
- How thinking changed
- Translate feasible-region geometry into multipliers, active constraints, and complementarity, creating a common language for nonlinear programming.
- What we cannot claim
- Kuhn and Tucker’s public influence is recognized without erasing Karush’s predecessor. Satisfying KKT and being the unique global optimum of an arbitrary problem are different claims.
- This place
- The 1950 Berkeley symposium gathered distinct optimization results in one venue and proceedings, giving nonlinear programming a name and an inspectable research community. (San Francisco Bay · oaks and eucalyptus · hills · 37.9°N 122.3°W · landmark: Sather Tower (1914))
- Figure board
- At a corner where two constraints meet, −∇f lies in the cone of the constraint gradients, and nonnegative multipliers λ record the balance.
- On the river
- A desk holding a written record · 1900–1969 (Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft)
11·1951 CE·Chapel Hill(basis: Publication)
Reducing Step Size while Estimating through Noise — Robbins and Monro
Robbins and Monro published a stochastic-approximation process for locating a root from noisy observations of an otherwise inaccessible function. Properly diminishing steps can average random disturbance while continuing toward the target. It is an important ancestor of stochastic gradient methods, not the identical mini-batch neural-network algorithm used today.
- Pause and ask
- Can a target be found when exact function values are unavailable and only noisy observations remain?
- How thinking changed
- Replace the demand for a complete function table with repeated observations and diminishing steps that average uncertainty.
- What we cannot claim
- The original problem is noisy root finding, not identical to modern mini-batch SGD. Convergence needs conditions on steps, noise, and the function and does not guarantee success after any finite run.
- This place
- The mathematical-statistics community at the University of North Carolina linked probability with iterative computation and supported publication in the Annals of Mathematical Statistics. (Woods · oak and pine · rolling hills · 35.9°N 79.1°W)
- Figure board
- Seeing the function only through noisy observations, ever smaller correction steps home in on the root θ of the target level α.
- On the river
- A stack of books · 1900–1969 (Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft)
12·1953 CE·Santa Monica(basis: Publication)
Folding a Long Decision into the Values of Remaining Problems — Bellman
At RAND, Bellman organized multistage decisions through dynamic programming: after a current choice, the remaining tail must itself be optimal from the new state. ‘Programming’ again meant planning, and Cold War military systems analysis supplied institutional support. As the number of states grows, the curse of dimensionality can overwhelm the recursion.
- Pause and ask
- Can a long decision be folded into values of remaining subproblems instead of being solved afresh each time?
- How thinking changed
- Replace comparison of complete paths with reusable optimal values for each state, computed backward.
- What we cannot claim
- Programming again means planning. The state must model all information relevant to the future, and exact computation can become infeasible under the curse of dimensionality.
- This place
- RAND in Santa Monica supplied sustained salaries, military systems problems, and computing resources for turning sequential-decision theory into reports, books, and computation. (Pacific coast · palms · coastal plain · 34.0°N 118.5°W)
- Figure board
- Stages of states are filled backward with the optimal value V of the remaining problem; each choice then follows the optimal path.
- On the river
- A stack of books · 1900–1969 (Concrete marker · electric streetlights · concrete bridge · pylons, apartment blocks and radio masts · barges · aircraft)
13·1972 CE·Berkeley(basis: Publication)
Grouping Problems Whose Optima May Be Hard to Find Quickly — Karp
Richard Karp connected many combinatorial problems by polynomial-time reductions, greatly expanding the family known to be NP-complete. This does not mean no optimum exists or every input is impossible to solve. It is a worst-case, conditional hardness claim while P versus NP remains open; approximation, heuristics, special structure, and small instances still matter.
- Pause and ask
- Can an exact optimum exist while the time needed to find it becomes unmanageable as input grows?
- How thinking changed
- Separate existence and verification from fast search, linking distinct problems by reductions to compare worst-case difficulty.
- What we cannot claim
- NP-complete does not mean absolutely unsolvable. P versus NP remains open, the claim concerns worst cases and input growth, and special structure, approximation, and heuristics can work well on large real instances.
- This place
- Berkeley’s theoretical-computer-science community and expanding conference and journal networks made computational complexity a shared question in algorithm design. (San Francisco Bay · oaks and eucalyptus · hills · 37.9°N 122.3°W · landmark: Sather Tower (1914))
- Figure board
- Polynomial-time reductions chain satisfiability to clique, colouring, tour and subset-sum problems; whether P = NP stays open.
- On the river
- A stack of books · 1970 onward (Glass marker · LED lights · cable-stayed bridge · glass towers, wind turbines and data centres · ferries · satellites)
14·1984 CE·Murray Hill(basis: Publication)
Crossing the Interior Instead of Following the Polytope’s Edges — Karmarkar
At Bell Labs, Karmarkar published a polynomial-time linear-programming method that traveled through the feasible region’s interior. It opened a major era for interior-point methods, but Khachiyan’s 1979 ellipsoid method was the first polynomial-time LP algorithm. Worst-case theory and practical speed are different comparisons, and simplex remains valuable on many problems.
- Pause and ask
- Can one reach an optimum quickly by crossing a polytope’s interior instead of following its vertices?
- How thinking changed
- Use projective and barrier geometry distinct from simplex boundary moves to build a polynomial-time path for linear programming.
- What we cannot claim
- Karmarkar’s method was not the first polynomial-time LP algorithm. Theoretical complexity and practical performance differ, and interior-point methods did not replace simplex on every problem.
- This place
- Bell Labs in Murray Hill combined long-term basic research, telecommunications optimization, and large machines, allowing a new algorithm to be tested in theory and implementation. (Woods · broadleaf trees · low ridges · 40.7°N 74.4°W)
- Figure board
- Instead of stepping along the boundary vertices of the feasible polytope, an interior path cuts across the inside toward the optimal vertex.
- On the river
- A stack of books · 1970 onward (Glass marker · LED lights · cable-stayed bridge · glass towers, wind turbines and data centres · ferries · satellites)
15·2012 CE·Toronto(basis: Publication)
Learning Features by Lowering Loss across Millions of Examples — AlexNet
Krizhevsky, Sutskever, and Hinton combined GPUs, large-scale ImageNet data, convolutional networks, backpropagation, and stochastic optimization to sharply reduce image-classification error. AlexNet did not single-handedly invent gradient descent, backpropagation, or deep learning. Minimizing benchmark loss measures performance on chosen data and metrics, not general intelligence, fairness, or social value.
- Pause and ask
- Does lowering loss on millions of examples mean a system understands the world best?
- How thinking changed
- Move from hand-designed features to a computational regime where large data, GPUs, and backpropagation jointly learn representations and parameters.
- What we cannot claim
- AlexNet did not invent gradient descent, backpropagation, or deep learning; it combined three authors’ work with multiple infrastructures. Minimizing benchmark error is not the same objective as general intelligence, fairness, truth, or social good.
- This place
- Long-running neural-network research at Toronto, public ImageNet data, NVIDIA GPUs, and an international conference created a stage for testing a combination of algorithm, data, and hardware. (Lake Ontario · maples · flat shore · 43.7°N 79.4°W)
- Figure board
- Loss from predictions through stacked convolution layers is sent back by backpropagation, slowly lowering the loss over many examples.
- On the river
- A stack of books · 1970 onward (Glass marker · LED lights · cable-stayed bridge · glass towers, wind turbines and data centres · ferries · satellites)