Mathematics Numerical Analysis From Rs

Five methods, one count

Numerical analysis is the branch of mathematics that designs and studies algorithms for approximating solutions to continuous problems, such as finding roots of equations, integrating functions, solving linear systems, or transforming signals. Its history runs from Newton's method in the 17th century to the Fast Fourier Transform (FFT) of Cooley and Tukey in 1965. The classical toolkit is broad, but a small set of methods appears everywhere: Newton's method for root finding, Euler integration and Runge-Kutta for ordinary differential equations, Gaussian elimination for linear systems, and the FFT for discrete Fourier transforms. These five are the workhorses of scientific computing.

In Recognition Science (RS), the framework's machine-checked library of formal theorems takes these five canonical methods as a single object. The library defines an inductive type with exactly five constructors: newton, eulerIntegration, rungeKutta, gaussElimination, and fft. A theorem proves that the cardinality of this type is 5. This is not a claim about the relative merits of the methods; it is a structural statement: the framework's recognition cycle forces a dimension count of 5, and these five methods are the canonical representatives of that count.

The module then connects the FFT to the framework's dimension count. The discrete Fourier transform with 8 modes (DFT-8) is declared as 2^3, and a theorem proves 2^3 = 8. The FFT's operation count per tick is defined as 8 * 3, and a theorem proves 8 * 3 = 24. The number 8 is not arbitrary: it is 2^D where D = 3, the framework's forced spatial dimension. The number 24 is 3 * 8, the product of the dimension and the mode count. These equalities are proved by computation, with no axioms beyond the standard three.

The module assembles these three facts, five methods, eight modes, and 24 operations, into a certificate structure. The certificate is a record with three fields, each a proof of one equality. The library provides a single definition that constructs the certificate from the three theorems. In plain language, the module says: the five canonical numerical methods, the 8-point DFT, and the 24 operations of the FFT are all consistent with the framework's forced dimension count of 5 and its spatial dimension of 3. It does not derive the methods' convergence or stability; it establishes a counting correspondence.

What this changes for a reader: numerical analysis is not just a bag of tricks but a set that the framework counts exactly. The five methods are not five arbitrary choices; they are the complete set of canonical methods in this account. The FFT's 24 operations per tick are not a performance detail; they are a structural echo of 3 times 8. The module is a small piece of a larger mosaic where recognition costs force dimensions, and here the dimensions show up in the algorithm count.

theorem numericalMethodCount : Fintype.card NumericalMethod = 5 := by decide

theorem dft8Modes_8 : dft8Modes = 8 := by decide

theorem fftOps_24 : fftOps = 24 := by decide

def numericalAnalysisCert : NumericalAnalysisCert where
  five_methods := numericalMethodCount
  eight_modes := dft8Modes_8
  fft_ops := fftOps_24

What this page does not claim

This module does not prove that Newton's method converges or that Runge-Kutta is stable. The five methods are not claimed to be the only numerical methods that exist in classical mathematics. The number 8 is not claimed to be the only possible DFT size in the framework.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Mathematics/NumericalAnalysisFromRS.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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