A Square Net and a Triangular Net Describe the World With the Same Exponent, and Only the Coefficient Differs ── counting lattice points inside a circle with square, triangular and rectangular lattices of the same density, the error grows as a power of the radius whose exponents 0.525037, 0.521921 and 0.542771 differ by only 0.02 while the coefficients 1.392341, 1.657107 and 1.119127 differ by 0.30; Tractatus 6.341, that the shape of the net is arbitrary and only its fineness says something about the world, is counted as exponent against coefficient ── the separator is whether what was changed is the shape of the net or its fineness ── [Paper 1033]

Abstract

In Tractatus 6.341 Wittgenstein likened mechanics to a net laid over the world: the shape of the mesh (square or triangular) is arbitrary, and all that can be said of the world is that it can be described by a net of this fineness. This paper counts the simile with lattice points. Three lattices of the same density are used to count the lattice points inside a circle against its area, and the exponent and coefficient of the error’s growth are separated. No new theorem or law is claimed, and the exponent is not proved. Scope of this paper (scope note): No new theorem or law is claimed──the Gauss circle problem, that the two-dimensional error is not smaller than the one-half power of the radius (Hardy and Landau 1915), that estimates of the same kind hold for general lattices, and the net simile of Tractatus 6.341 (1922) are all known. The exponent is not proved──a power was fitted to counts up to radius 1000, which is no proof; this paper stands on the same side as Paper 1015. The cause of the difference is not settled──the exponent 0.525037 of the square lattice here and the 0.501657 of Paper 1015 differ, apparently with the sampling of radii, but the cause is not asserted. Interpretation is not entered──what 6.341 claims about the arbitrariness of mechanical description is not read philosophically; the simile is only replaced by one count. Only three lattices are used──square, triangular and 2-to-1 rectangular; no completeness is claimed. Relation to earlier papers: Paper 1015 showed that the error of counting a ball by lattice points comes from the skin in low dimensions and from one shell in high ones──here the same two-dimensional skin fluctuation is counted with three nets, and it is counted that the exponent does not depend on the net while the coefficient does. What is added is separating exponent and coefficient with three lattices of the same density, showing that the spread of exponents is less than a tenth of the spread of coefficients, and giving the reduction when the mesh is made four times finer. First, and this is the core. The exponent does not depend on the net, the coefficient does──for 3000 random radii from 5 to 1000, fitting the root-mean-square error of the square, triangular (hexagonal) and 2-to-1 rectangular lattices (each at 1 point per unit area) to r to the power p gives p = 0.525037, 0.521921 and 0.542771, a spread of 0.020850, with coefficients 1.392341, 1.657107 and 1.119127 and root-mean-square errors near radius 500 of 37.021795, 41.541899 and 31.853360 (spread over minimum 0.304161). The spread of the exponents is less than a tenth of the spread of the coefficients (Section 2). Second, the error vanishes against the area──near radius 500 the error over the area is 4.714x10^-5, 5.289x10^-5 and 4.056x10^-5, about five parts in a hundred thousand for every net (Section 2). Third, a finer mesh reduces the error──quadrupling the density of the square lattice reduces the root-mean-square error near radius 500 from 35.570822 to 14.452407. Fineness is a quantity on the side of the net, the only one that 6.341 allows to say something of the world (Section 3). Fourth, the counting was checked against known values──the square lattice gives 317 points inside a circle of radius 10 and 31417 inside radius 100, matching known values, and the densities of the triangular and rectangular lattices at radius 300 are 1.000048 and 1.000077 (Section 2). Fifth, the exponent here differs from Paper 1015 by 0.02──Paper 1015 obtained 0.501657 for the square lattice from counts up to radius 1000; the 0.525037 here differs by 0.023380, the radii being sampled differently (random real radii and bins). The cause of the difference is not settled (Section 2). Sixth, the separator is whether what was changed is the shape of the net or its fineness──changing the shape moves the coefficient and not the exponent; changing the fineness moves the size of the error. The exponent belongs to the smoothness of the boundary, that is, to the side of the world (Section 4). A square net and a triangular net describe the world with the same exponent, and only the coefficient differs. Counting lattice points inside a circle with square, triangular and rectangular lattices of the same density, the exponents 0.525037, 0.521921 and 0.542771 differ by only 0.020850, the coefficients are 1.392341, 1.657107 and 1.119127, and the spread over minimum of the errors near radius 500 is 0.304161. Making the mesh four times finer reduces the error from 35.570822 to 14.452407──"the shape of the net is arbitrary" of 6.341 is the side of the exponent, and "only the fineness says something" the side of the coefficient. The separator is whether what was changed is the shape of the net or its fineness. Placed among the earlier papers──Paper 1015 showed that the error of counting by lattice points comes from the skin fluctuation in low dimensions. Here that skin fluctuation was counted with three nets, and the exponent was counted not to depend on the net. To be honest──the exponent has only been fitted, not proved. On the making of this work: The ideas and content of this work stem from the author's own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author's. Feedback and corrections are sincerely appreciated. Keywords: lattice points, Gauss circle problem, square lattice, triangular lattice, Tractatus Logico-Philosophicus, exponent. ----- ウィトゲンシュタインは『論理哲学論考』6.341 で、力学を世界に被せる網にたとえた。網の目の形(正方形でも三角形でも)は任意で、世界について言えるのは「この細かさの網で書ける」ということだけだ、と。本稿は、このたとえを格子点で数える。同じ点密度の三つの格子で円の中の格子点の数と面積の差を取り、誤差の伸び方の冪と係数を分ける。新しい定理も法則も主張せず、冪を証明しない。 本稿の射程(射程注記):新しい定理も法則も主張しない──ガウスの円問題、2 次元のずれが半径の 1/2 乗より小さくならないこと(ハーディとランダウ 1915)、一般の格子でも同じ型の評価が成り立つこと、『論理哲学論考』6.341 の網のたとえ(1922)は、いずれも既知である。冪を証明しない──半径 1000 までの数え上げに冪を当てはめただけで、証明にはならない。論文1015 と同じ側に立つ。差の原因を決めない──本稿の正方格子の冪 0.525037 と論文1015 の 0.501657 の差は、半径の取り方の違いに伴うものと見られるが、原因は断定しない。解釈に入らない──6.341 が力学の記述の任意性について何を主張しているかの哲学的な読みには立ち入らず、たとえを一つの数え上げに置き換えるだけである。三つの格子に限る──正方・三角・縦横比 2 の長方だけで、網羅ではない。既刊との関係:論文1015 は、格子点で数えた中身の誤差が低い次元では皮の揺らぎ、高い次元では一つの殻の段差で決まることを示した──ここでは、同じ 2 次元の皮の揺らぎを三つの網で数え、冪が網によらず係数が網によることを数える。加えたのは、同じ密度の三つの格子で冪と係数を分けたこと、冪の幅が係数の幅の 10 分の 1 より小さいことを出したこと、網の目を 4 倍細かくしたときの減り方を出したことである。 第一に、これが本稿の芯である。冪は網によらず、係数は網による──半径 5 から 1000 の乱数 3000 個で、正方格子・三角(六角)格子・縦横比 2 の長方格子(いずれも 1 点/単位面積)の二乗平均誤差を r の p 乗に当てると、p は 0.525037・0.521921・0.542771 で幅 0.020850、係数は 1.392341・1.657107・1.119127 で、半径 500 付近の二乗平均誤差は 37.021795・41.541899・31.853360(幅÷最小 0.304161)である。冪の幅は係数の幅の 10 分の 1 より小さい(第2節)。 第二に、誤差は面積に比べれば消えていく──半径 500 付近の誤差÷面積は 4.714x10^-5・5.289x10^-5・4.056x10^-5 で、どの網でも面積の 10 万分の 5 程度である(第2節)。 第三に、網の目を細かくすると誤差は減る──正方格子の密度を 4 倍にすると、半径 500 付近の二乗平均誤差は 35.570822 から 14.452407 に減る。細かさは世界の側ではなく網の側の量で、6.341 が「世界について言う」とした唯一のものである(第3節)。 第四に、数え上げは既知の値で検算した──正方格子で半径 10 の円の中の点は 317、半径 100 で 31417 で既知の値に一致し、三角格子と長方格子の密度は半径 300 で 1.000048・1.000077 である(第2節)。 第五に、本稿の冪は論文1015 の値と 0.02 違う──論文1015 は正方格子で半径 1000 までの数え上げから冪 0.501657 を得た。本稿の正方格子の 0.525037 との差は 0.023380 で、半径の取り方(乱数の実数の半径と刻み)が違う。差の原因は決めない(第2節)。 第六に、分離子は、動かしたのが網の形か、網の目の細かさか──形を動かせば係数が動き、冪は動かない。細かさを動かせば誤差の大きさが動く。冪は境界の滑らかさ、すなわち世界の側にある(第4節)。 正方形の網と三角形の網は同じ冪で世界を写し、違うのは係数だけである。同じ点密度の正方・三角・長方の格子で円の中の格子点を数えた誤差の冪は 0.525037・0.521921・0.542771 と 0.020850 しか違わず、係数は 1.392341・1.657107・1.119127 で、半径 500 付近の二乗平均誤差の幅÷最小は 0.304161 である。網の目を 4 倍細かくすると誤差は 35.570822 から 14.452407 に減る──6.341 の「網の形は任意」は冪の側、「細かさだけを言う」は係数の側である。分離子は、動かしたのが網の形か、網の目の細かさかである。既刊との位置──論文1015 は格子点で数えた中身の誤差が低い次元では皮の揺らぎで決まることを示した。ここでは、その皮の揺らぎを三つの網で数え、冪が網によらないことを数えた。正直に言えば──冪は当てはめただけで、証明していない。 作成にあたって:本稿の着想と内容は、著者自身の考察に基づくものです。文章の構成整理や英訳、数式の確認には AI(大規模言語モデル)の助力を得ました。最終的な内容の解釈や誤りがあれば、それらはすべて著者の責に帰します。お気づきの点があれば、ご教示いただければ幸いです。 キーワード:格子点、ガウスの円問題、正方格子、三角格子、論理哲学論考、冪。