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Whittaker-ET: Mathematician review (STEP 2)

Reviewer: PhD mathematician. Scope: derivations, notation, units, hidden assumptions, and the Investigator's findings as they bear on the mathematics. Tags: [read] document, [web] URL, [memory] recalled (confidence), [calc] my own computation (SymPy/mpmath/NumPy in the project venv).

Source gap carried over from claims.md: the 1903 Mathematische Annalen scan has no text layer, so equations were read from page images; journal pp. 347-353 (Helmholtz equation, generalized Bessel functions) rest on a digest alone and are not judged here. The 1910 book is digest-only. Not obtained: History Vol. 2 (1953), the 1927 paper, the 1944 Endeavour article, From Euclid to Eddington. Where a conclusion below rests on [memory] it says so.

I re-read the 1903 gravity section myself on page images (PDF pp. 23-24 = journal pp. 354-355) and re-ran the key checks.

Reader's glossary (used below)

  • Laplace's equation: del-squared V = 0 (V = potential; del-squared = sum of second derivatives in x, y, z). Describes static Newtonian gravity and electrostatics away from sources.
  • Wave equation: del-squared V = k^2 d^2V/dt^2, where 1/k is the wave speed.
  • Plane wave of arbitrary profile: V = F(n.r + t/k) where n is a unit vector (direction of travel) and F any function; it keeps its shape and moves at speed 1/k.
  • Null vector: a vector whose "length squared" is zero. The whole trick of the 1903 paper is that (i cos u, i sin u, 1) is such a vector for Laplace's equation and (sin u cos v, sin u sin v, cos u, 1) is one for the wave equation (space part length 1, time part 1, opposite signs in the equation).

Claim 1. General solutions of Laplace's and the wave equation by superposed functions of one variable (1903)

a. Internally consistent? Yes, with stated limits. - Why the forms solve the equations [calc]: with s = z + ix cos u + iy sin u, the gradient of s is (i cos u, i sin u, 1), and (i cos u)^2 + (i sin u)^2 + 1 = 0, so del-squared f(s,u) = f'' x 0 = 0. For the wave case w = x sin u cos v + y sin u sin v + z cos u + t/k has space-gradient a real unit vector and time derivative 1/k, so del-squared f(w) - k^2 d^2f/dt^2 = f'' (1 - 1) = 0. Residuals 0 in SymPy. Note the Laplace case needs complex arguments; the wave case is real. - Coefficient counts (2n+1) for Laplace and (n+1)^2 for the wave equation: arithmetic correct; independence of the relations is assumed by Whittaker and is true (ranks 3,5,7,9 and 4,9,16,25,36 reproduced by the deep reader; I confirm the identities (n+1)(n+2)/2 - n(n-1)/2 = 2n+1 and C(n+3,3) - C(n+1,3) = (n+1)^2 by hand). The (2n+1) independent solutions come from the Fourier coefficients of (Z + iX cos u + iY sin u)^n; the argument (highest power of Z, parity in Y) is sound. - What is proved and what is asserted (agrees with the deep read): proved, degree by degree, that each homogeneous harmonic polynomial has the integral form. Asserted: that the sum over all degrees has it too. That step needs the per-degree functions f_n to sum to a convergent f and the sum and integral to be interchanged; neither is argued. The result is therefore local (near a point where V is analytic, i.e. equals its convergent Taylor series), and for real-analytic V. In modern terms this is the Whittaker/Bateman representation of harmonic functions [memory, high: it appears as "Whittaker's integral" in Whittaker and Watson, A Course of Modern Analysis]. It is not a statement about arbitrary smooth, non-analytic, or singular solutions (e.g. the 1/r potential at the source). - Hidden condition on the "1/r" identity (journal p.334): 1/r = (1/2 pi) times the integral of du / [(z-c) + i(x-a) cos u + i(y-b) sin u] holds only for z > c. For z < c it gives -1/r. [calc] At (0.3, 0.4, 1.2) the integral is +0.769231 = 1/r; at (0.3, 0.4, -1.2) it is -0.769231. So the "potential of any distribution of matter" has this form only above all the masses (or after rotating axes). Whittaker does not state it on p.334 (he says "z supposed positive" later, p.10 of the PDF). - Misprints confirmed by the deep reader and not repeated here: m = 0 Fourier factor 1/(2 pi) not 1/pi; sign and P-index error in the Bernoulli/Legendre series (journal p.342); missing x in the integrand (journal p.346); "= 0" printed for "= 1" in the spheroid. None affects the theorem. - Small gap the deep read passes over: the measure is du dv, not the solid-angle measure sin u du dv. Since f is arbitrary this is harmless in the interior, but to reproduce a function with a smooth behaviour at the poles u = 0, pi, f must be allowed to carry a factor 1/sin u. Whittaker himself inserts sin u in §5, part 3 (journal p.348-ish; deep read). Minor. - "f periodic in u, v without loss of generality": only values on the integration range matter, so periodicity imposes nothing on the integral. It matters only if one also Fourier-analyses f in its first argument (§5, part 1), which needs f finite on a range; Whittaker says "general finite solution". Múnera's objection (quoted by Titleman 2023 [web: https://www.qeios.com/read/Y3D1HM, via Investigator]) that the claim of generality is "unacceptable due to the assumption of periodicity" is therefore correct only if "general" is read as "every solution on all space and time"; it is not a defect in the local theorem.

b. Consistent with experiment? The mathematics makes no physical prediction by itself; it is a representation theorem. Its mainstream descendants are in use: twistor theory and the Penrose transform (via Bateman) [memory, medium; the Investigator found 26 zbMATH citing documents, many in twistor theory: research/2026-10-07-zbmath.md], and "scalar Whittaker potentials" in plasmonics (Berthel et al., Phys. Rev. E 92, 033202, 2015 [web: via research/2026-10-07-qeios.md]).

c. Tested? A theorem is proved, not tested. Independent confirmation: the deep reader's numerical rank checks and my identity checks. Stoney (MNRAS 63, 1903) praised it as "striking and most useful" and "a singularly elegant proof" [web: Investigator, research/2026-10-07-oup-mnras.md].

d. If it were true? Nothing in mainstream mathematics changes; it is already part of it.

Verdict: Established (for real-analytic solutions, locally). Split: the strong reading "every solution, globally" is Not testable as stated / not proved; Whittaker did not claim it beyond "finite solution".

Claim 2. Every finite wave-equation solution is a superposition of plane waves in all directions

a. Consistent. Each term F(n.r + t/k) is a plane wave of arbitrary profile; on a range where F is finite and continuous it is represented by Fourier's integral, so it is a sum of sinusoidal plane waves. [read: PDF pp.15-16] Directional detail: the phase lambda(x + t/k) moves toward negative x in the rotated axes (the deep reader's direction note is right); irrelevant since all directions are included. The Fourier-integral step needs more than "finite and continuous" by modern standards (bounded variation, integrability on the range); that is 1903 rigour, not an error. b. This is textbook Fourier/plane-wave analysis (Stoney 1897 had it by physical reasoning, cited by Whittaker [read: PDF p.16]). c. Used daily in acoustics, optics and radio; not in dispute. d. Nothing. Verdict: Established.

Claim 3. Two scalar potentials F and G give the full electromagnetic field of moving electrons (1904)

a. Consistent? Yes. I re-verified independently [calc, SymPy]: for f = F(s) z-hat, g = G(s) z-hat with s = n.r - ct (n any unit vector), d = curl curl f + (1/c) curl g-dot and h = (1/c) curl f-dot - curl curl g satisfy div d = 0, div h = 0, curl h = (1/c) d_t and curl d = -(1/c) h_t exactly, and d.n = h.n = 0: every plane-wave term gives a transverse electromagnetic wave, with no longitudinal part. This agrees with the deep reader's numerical Lienard-Wiechert check (12 digits) and the claims.md statement. - What the construction is, in modern words [memory, high]: f and g are the electric and magnetic Hertz vectors (Debye potentials), restricted to a fixed direction. The count "two" is not an accident: source-free Maxwell fields have two independent polarizations, and four potentials (phi and A) carry a gauge redundancy. Two scalars is the minimum, at the price of tying the description to an axis (Whittaker says no symmetrical choice exists). - Whittaker's own gloss, which the Investigator flagged for checking against the Vol. 1 PDF: it IS there. The 1951 History Vol. 1 (PDF p.212, "The Two Scalar Potentials", Chapter XIII) says: "an electrostatic field can be described by F alone, and a magnetostatic field by G alone; again, if the field consists of a plane wave of light, then the functions F and G correspond respectively to two plane-polarised components into which it can be resolved." [read: PDF p.212 text layer]. He also explains that different choices of axes give different pairs (F, G). So Whittaker himself reads F and G as the two polarizations (transverse), not as two "longitudinal waves". This is direct evidence against the reading of Titleman 2023 that the 1904 paper makes EM waves "from the interference of two longitudinal waves" [web: Qeios, via research/2026-10-07-qeios.md]. The 1904 text never mentions gravity or longitudinal waves (deep read; agreed). b. Reproduces Maxwell-Lorentz, hence all the experiments Maxwell-Lorentz theory reproduces. c. Verified by the deep reader numerically and by me analytically for plane waves; used in plasmonics. d. Nothing. Verdict: Established (as a reformulation of Maxwell-Lorentz theory). The extra claims attached to it by later writers (hidden longitudinal "scalar waves") are not in the paper; as a reading of the paper they are Internally inconsistent with Whittaker's own gloss (he calls F, G plane-polarised components); as physics they belong to the Bearden project.

Claim 4. Gravity as longitudinal waves in the light medium; a steady field built of waves (1902 note, 1903 §5 part 5); the "longitudinal standing wave" hypothesis put to the review (Q07)

This is the claim the owner asked about. I treat it in six parts.

4.1 What the mathematics actually is [read: PDF p.23-24, journal pp.354-355; calc]

Whittaker's example: outgoing spherical waves in the band mu to mu + d mu contribute (2 d mu / (pi mu)) sin(mu(Vt - r))/r. Summing over mu from 0 to infinity: (2/(pi r)) integral of sin y / y dy = (2/(pi r))(pi/2) = 1/r, "after the waves have reached the point r, so that Vt - r is positive". I confirm the page reads exactly this (not "4/r", as the digest had).

Three things Whittaker does not say, all verifiable: 1. The sum is -1/r for r > Vt (the integral of sin(a mu)/mu over mu is (pi/2) sgn(a)). [calc, mpmath with a large frequency cutoff M: field x r = +1 at r < Vt, 0 exactly at r = Vt, -1 at r > Vt]. The "steady 1/r" holds only inside the sphere r < Vt. 2. The source strength is a step. Each term sin(mu(Vt - r))/r is an outgoing wave from a point source oscillating as sin(mu V t); the sum over mu with weights 2/(pi mu) is the Fourier series of the sign function, so the point source strength is -1 for t < 0 and +1 for t > 0. The "steady gravitational potential" is the retarded field of a source that switches from -1 to +1 at t = 0, seen inside the expanding sphere of radius Vt. The outside (r > Vt) still carries the field of the earlier, opposite source, i.e. -1/r. This is a plain consequence of the formula and needs no physics. 3. The leftover at the wavefront. Differentiating the field with its band cutoff, force = -dV/dr = +1/r^2 inside, -1/r^2 outside, and a 1/r-weighted spike at r = Vt (derivative of the step: -(2/r) delta(r - Vt)). With a finite bandwidth M the step is smoothed with Gibbs ringing of overshoot (2/pi) Si(pi) = 1.179 [calc], i.e. 18 percent. So Whittaker's "constituent fields" (each with a 1/r flicker, which Stoney correctly computed: force component = (R m cos(m(R - Vt)) - sin(m(R - Vt)))/R^2 [calc]) do cancel exactly in the interior, but only because the whole bandwidth is present; the cancelled 1/r radiative parts are not destroyed, they are concentrated at the moving front.

Consequence for Stoney's objection ("flickering" forces, "overwhelming improbability" [web: MNRAS 63 (1903) 424-428, via research/2026-10-07-oup-mnras.md]): his arithmetic is right (I re-derived his force components), and his point that only one special spectrum (amplitude proportional to 1/mu) works is right. The reply Whittaker could make is that this special spectrum is simply the spectrum of a step source, which is not exotic. The reply Stoney could make back is that the model then needs a gravitating particle that has been "switched on", with a front that expands at speed V. Whittaker's answer, if any, has not been found by the Investigator. Both sides are kept here.

4.2 Is it a standing wave?

  • A standing wave is a time-dependent pattern with fixed nodes: V = R(r) cos(omega t) (two counter-propagating waves of equal amplitude). Whittaker's steady field has omega = 0 in total; its parts are travelling waves. Neither the 1902 note nor the 1903 paper uses "standing" [read; web: MNRAS 1902 via Investigator]. The word comes from Barrett (1993) and the Bearden literature, repeated by Titleman 2023 [web, via research/2026-10-07-qeios.md]. At best "static field" is the omega -> 0 limit of a standing wave, which is a trivial member of the family.
  • If one wants a genuine standing longitudinal wave that looks like Newton's law, the scalar wave equation gives a concrete test. The spherical solution regular away from the origin is V = cos(kappa r)/r times cos(omega t), with omega = c kappa (c for the wave speed). The force is -dV/dr, and [calc, SymPy series] r^2 F = 1 + (kappa r)^2/2 + O(kappa^4). So the inverse-square law holds to relative accuracy (kappa r)^2/2 over a distance r. Worked example [calc]: to keep the deviation below 1 part in 10^8 (kappa r = 1e-4) at r = 1 AU = 1.496e11 m requires kappa < 6.7e-16 per metre, wavelength 2 pi/kappa > 9.4e15 m (about 1 light year), and omega = c kappa = 2.0e-7 rad/s, period 3.1e7 s (about 363 days). So "static over the solar system" and "a wave with c-speed" together force periods of about a year or longer; a faster standing wave would produce a different force law at AU scale. (This is my own illustration, not from any source, and assumes the wave speed is c; with V >> c the periods shorten by c/V.)
  • The same calculation shows what a standing wave predicts that a static one does not: the field strength oscillates in time at one frequency everywhere (V proportional to cos(omega t)), unless a steady part is added. Any such oscillation of g is a measurable quantity in principle (gravimeters, pulsar timing, planetary ranging) and in A07 is listed as not yet tested against a specific model. I do not state limits here without a source.

4.3 "Longitudinal" is a consequence of the choice of a scalar potential

For any plane-wave term V = F(n.r - ct), the force -grad V = -n F' lies along n, the direction of travel. So every scalar-potential wave is automatically longitudinal; this is not an extra physical property. Maxwell's transverse EM waves arise because their fields are curls of vector (or Hertz) potentials (claim 3 check above). The two statements Whittaker makes (gravity force ~ longitudinal; EM ~ transverse) therefore differ by what kind of field is waving (a gradient of a scalar versus a curl), a fact the 1903 text compares as "similar" without noticing; it is the reason the "identical property of the medium" does not carry over. This is a mathematical observation; the physics (mainstream gravitational waves are tensor) is the Physicist's.

4.4 Why the speed is unconstrained by the mathematics

Because dV/dt = 0 inside, k drops out of every static equation: "k is any constant" [read: p.354]. This is exactly A07's inference and is right. It is also why Laplace's old aberration argument matters: a moving source breaks the degeneracy. Laplace's bound on the speed of "Newtonian" gravity is c_g > 2e10 c, and in GR aberration is almost exactly cancelled by velocity-dependent interactions, which allows c_g = c [web: https://arxiv.org/abs/gr-qc/9909087, Carlip, "Aberration and the Speed of Gravity"; abstract read]. Whittaker's "may be enormously greater" fits the old Laplace constraint on a scalar, force-along-the-line-to-the-source model; this is a consistency note, not a verdict on whether his specific model satisfies it (that needs the moving-source field of the scalar wave equation, which Whittaker did not compute). The Investigator's lead on the 1902 note shows the same sentences appeared a year earlier [web: research/2026-10-07-oup-mnras.md].

4.5 Dimensions and units

Whittaker's V in section 5 is "the potential (or force-component)"; the wave amplitude has no stated units and the weights 2 dmu/(pi mu) are dimensionless in his example, so the example is a statement about functional form (1/r), not about G, masses or magnitudes. No numerical prediction can be extracted from it without adding a coupling the paper does not give. The sin y / y integral is dimensionless as written (y = mu(Vt - r)); fine.

4.6 Other constructions of the same type

  • The Investigator's Brush (1911), See (1917), Múnera (2019) and Titleman (2022-2023) items build on the same gradient-of-scalar-wave idea; their mathematics is not in the folder and was not checked. Titleman's "standing waves" sentence is quoted from Barrett (1993) and is not Whittaker's; his list of implications (lensing, Aharonov-Bohm, MOND and others) are claims I could not check from the project files. I do not discard them; they stay in the record as the work of a living author developing claim 4 [web: arXiv:2205.08309 via research/2026-10-07-qeios.md].

a. Internally consistent? The 1/r result is correct for r < Vt (and -1/r for r > Vt, unstated). The decomposition is valid. The physical reading ("a medium that carries light carries gravity") is offered as "it seems probable" and is not derived from the equations. The speed claim is consistent (degenerate for static fields). b. Consistent with established experiment? Mathematically consistent with Newtonian statics. Dynamical consequences for moving sources were not worked out by Whittaker; the A07 follow-up lists the experimental constraints (GW170817, polarizations, Hulse-Taylor, MICROSCOPE) and these are for the Physicist to weigh. The one mathematical contact: any scalar-wave model radiates monopole/dipole-type waves for changing sources in a way tensor GR does not; the pulsar-decay comparison in A07 (Ibison, 2/3 of GR) is the quantitative form of this. c. Tested? The mathematics was criticised on physical grounds by Stoney in 1903; no replication or neutral test of the specific "longitudinal waves in the light medium" model is on record in the folder or in the Investigator's material. Whittaker's later 1927 work works inside GR [web: A07]. No standing-wave model with a specified frequency has been tested against oscillating-g searches (A07, open). d. If true? Gravity would need a medium with a scalar longitudinal wave mode and a finite speed; GR's spin-2 radiation, the equality of gravitational and inertial mass (the field couples to mass), and the observed absence of scalar polarization would all need to be reconciled; and the cancellation of aberration would need a replacement mechanism.

Verdict, split: - The mathematical statement (a 1/r potential is a superposition of waves whose sum is steady inside the front r < Vt): Established. - The physical statement that gravity actually is such a superposition in the medium of light (strong form, with speed possibly much greater than c): Never properly tested as a specific model (dismissed on physical grounds by Stoney in 1903 without a controlled test; no experimental test of Whittaker's version was run). Where specific scalar-wave versions have been compared with data (A07), the Physicist's and Arbiter's reading governs. - "Standing wave" version (oscillating, with nodes): Not testable as stated until a frequency and amplitude are given; with them it becomes Never properly tested (A07 open item on oscillating-g searches). It is not what Whittaker wrote.

Claim 5. Whittaker's 1951 history: the vacuum may be called the aether; Riemann, Maxwell and gravity; Maxwell's waves transverse

Not a mathematical claim; I checked the mathematical content of the passages. - Maxwell 1864's energy argument, which I read in the Vol. 1 text: if gravitational potential energy sits in the medium its energy per cubic centimetre must be a - beta R^2 (R = gravitational force per gram, a and beta positive constants); with positive energy, a must exceed beta R^2 everywhere, so where gravity vanishes "the intrinsic energy in the medium must have an enormously great value"; Maxwell: "I cannot go any further in this direction in searching for the cause of gravitation" [read: 1951 PDF, Chapter VIII; also deep read]. The sign comes from the fact that like masses attract while like charges repel: the field energy of Newtonian gravity is negative where the electrostatic energy is positive. This is a correct and sharp piece of mathematics, and it is the same obstacle any "medium carries gravity" model must pass; Whittaker's 1902-03 waves do not address it (he says the investigation "does not explain the cause of gravity"). [memory, high: modern statement that the Newtonian field energy density is -g^2/(8 pi G)]. - Riemann 1853 (resistance to compression gives gravity and electrostatics, resistance to rotation gives light and magnetism) is a model, not a derivation; I checked only that the summary in facts.json matches the deep read. - The history never gives a derivation of a longitudinal EM wave; Kelvin's 1896 suggestion is reported as a speculation contested by FitzGerald (one party's private letter, as the deep read notes). - The preface proposal to call the quantum vacuum "aether" is a terminological choice. Verdict: Historically superseded but instructive (aether-as-medium gravity models); the preface remark is Not testable as stated (terminology).

Corrections to Step 1

  1. facts.json (1903 entry) and claims.md claim 4: incomplete, not wrong. They report that the sum of Whittaker's spherical waves is 1/r "once Vt > r". It is -1/r for r > Vt, and the superposition is the retarded field of a source that steps from -1 to +1 at t = 0 [calc; page image PDF p.23]. So the "steady field" exists only inside the expanding front, and the "flickering" 1/r parts that Stoney computed sit at the front. A07's table row ("1/r for r < Vt") is correct and should gain the r > Vt line.
  2. Title of the idea. claims.md already says "not a textbook standing wave"; I agree and add the quantitative distinction in 4.2 (a genuine standing wave must oscillate; Whittaker's has omega = 0).
  3. claims.md claim 3 / facts.json 1904 entry: confirm and strengthen. The statement that each scalar term gives a transverse wave is confirmed analytically (not only numerically), and the Investigator's lead to check Titleman's quotation against Vol. 1 is resolved: the passage is on PDF p.212 of the project's 1951 Vol. 1 PDF (Chapter XIII). Whittaker there calls F and G "two plane-polarised components". The facts.json entry for the 1951 book should list this passage (only Chapter VIII was deep-read).
  4. facts.json 1903 entry: "f periodic ... asserted". Add that periodicity in u, v is vacuous for the integral (only the range [0,2 pi] x [0,pi] is used); the real restriction is analyticity at a point and local validity. This answers Múnera's objection as relayed by Titleman: it bites only if "general" means global.
  5. Pole measure. The du dv measure (no sin u) needs f to be allowed a 1/sin u factor for functions smooth at the poles; the deep reader said "loses nothing". Minor.
  6. README (project note): "secondary source". Fine. The Vol. 2 lending note was already corrected by the Investigator (sites.md); nothing to add.
  7. No discrepancy found with A07's quantitative statements that are mathematical (the 1/r integral, "any speed fits a static field").

Investigator's findings: how they bear on the mathematics (kept whatever the verdict)

  • 1902 MNRAS note (the earlier, shorter statement; sentences on "identical property of the medium" and "enormously greater" appear word for word; "force perpendicular to the wave-front"): consistent with 4.1-4.3 [web, Investigator].
  • Stoney 1903 critique: his force formula checks; his "special spectrum" point is mathematically right (it is the step-source spectrum); his Mercury postscript is a different issue (motion of bodies) and is exactly the moving-source degeneracy of 4.4. Whittaker's reply not found.
  • Wangerin's 1903 Jahrbuch review and McCrea's 1957 obituary: contemporary and later readings. McCrea's sympathetic reading ("how a static gravitational field can be regarded as resulting from effects that are being propagated through the field") matches the mathematics in 4.1. Interested-party note: McCrea was a colleague and writer of Whittaker's obituary.
  • Twistor lineage (Penrose, Hitchin, Ward; Thompson, Swearngin, Bouwmeester 2014 build linearised gravitational radiation from the "Whittaker solution"): mainstream use of the same decomposition for tensor waves; supports claims 1-2, and shows the mathematics does not itself select a longitudinal scalar reading [web/memory as tagged in sites.md].
  • Barrett/Bearden/Titleman "standing wave" lineage; Múnera 2019; Brush 1911-29; See 1917; Darmos 2022: related work, not checked mathematically beyond what is stated in 4.6. Brush's free-fall claim and its Bureau of Standards check (result not located) remain open leads.
  • Absence of evidence: Whittaker's reply to Stoney, the 1902-03 daily-press coverage, and the 1944 Endeavour article have not been found; they are records that would show whether he ever worked out the moving-source field (4.4) or the front (4.1). Competing explanations (he considered it answered; he moved to GR by 1927; the exchange was dropped) are all consistent with the record so far.

Summary of verdicts

  1. Whittaker's integral solutions of Laplace and the wave equation: Established (local, real-analytic); global generality not proved.
  2. Plane-wave analysis of finite wave solutions: Established.
  3. Two scalar potentials F, G: Established as a reformulation of Maxwell-Lorentz; the "longitudinal scalar wave" reading is not in the paper and contradicts Whittaker's own gloss (two plane-polarised components).
  4. Gravity as superposed longitudinal waves in the light medium: mathematics Established (steady 1/r inside r < Vt only; -1/r outside; source is a step); physical model Never properly tested (dismissed on physical grounds by Stoney, 1903); "standing wave" version Not testable as stated, and not Whittaker's wording.
  5. 1951 aether/vacuum and Maxwell/Riemann history: Historically superseded but instructive; terminology Not testable as stated.

Files: review/Whittaker-ET/mathematician.md