Background to Seventeenth Century Philosophy
Aristotelian Philosophy
The Seventeenth Century philosophers to a large extent overthrew the prevailing philosophy of the Middle Ages (medieval philosophy). Medieval philosophy most characteristically combined Roman Catholic theology with the natural philosophy of Aristotle. Thomas Aquinas was responsible for this synthesis.
Much of the thought of the ancient philosophers was lost to the West with the end of the Roman empire, though it was preserved in Arab lands. The works of Aristotle were well-known and commented upon. Contact between Christians and Arabs eventually led to the re-introduction of Aristotle's thought to Western thinkers.
The Renaissance ("rebirth") brought to light much of the rest of ancient philosophy, including the work of Plato. This created a problem for orthodox medieval philosophy, and though some tried to incorporate the newly-discovered elements of ancient thought into medieval philosophy, the seeds of its downfall had now been planted. The Protestant Reformation presented another severe challenge to orthodoxy. In this context, the assault on Aristotle's natural philosophy began in earnest.
We may begin our approach to Aristotle's philosophy by posing the fundamental question: what is the nature of things? Some early answers equated everything with some element of the world, such as water. Aristotle classified the possible answers to this question as emphasizing either the form or the matter of things.
Aristotle understood the ancient atomists as claiming that the nature of things we observe is found in their constituents, the atoms. Atoms are tiny, hard particles which move about in otherwise empty or void space. Insofar as the atoms are the matter making up things in the world, the atomists are materialists. Aristotle rejected atomism entirely. He advanced arguments against the void, but more generally he held that the nature of things is their form.
At the other extreme were philosophers who emphasized form to the extent that matter becomes almost an afterthought. The Pythagorean philosophers found numerical forms in all things. Their esoteric doctrine did not find much favor, though it spurred the development of Greek mathematics.
Plato held that the forms of things are separate from the material world. These things, which we perceive by the senses, are pale imitations of the forms themselves. We are like prisoners in a cave, viewing the shadows on the wall but unable to see the real objects casting them. Only the cultivation of the intellect, trained through the study of mathematics, allows us to rise above sense perception and apprehend the true nature of things.
The break with Plato comes over whether the forms are more basic than the individuals which are instances of them. Aristotle makes individuals the "primary substances," while forms are only secondary. The form is that which makes the thing the kind of thing that it is, but it has no separate existence without the thing. Because it is responsible in a sense for a thing's being the kind of thing it is, Aristotle called the form a cause (often referred to as the "formal cause" of a substance). This ranking of individual over form was commonly accepted at the beginning of the seventeenth century.
One might ask whether Aristotle begs the question against partisans of matter when he declares that the definition expresses the form. After all, why should the definition be stated in terms of formal characteristics? Perhaps 'human being' should be defined as a group of atoms organized as a body with a specific biological structure.
But Aristotle's model for the form of a thing is the mature biological individual. The form is at bottom to be understood as the function of the thing. Thus Aristotle might answer the charge of question-begging by claiming that the definition of a human as a biological structure already incorporates the notion of form, and it is only the form which could differentiate different kinds of groups of atoms.
The Greek philosophers were not content with stating what a thing is; they also pondered the question of what a thing is not. For example, both Socrates and Plato are human beings, but Plato is not Socrates. Though they agree in their form, they differ in their matter. For Aristotle, it is the matter of things which "individuates" them, or distinguishes different members of the same species. The respects in which the things differ are called "accidents," as opposed to the "essence" which is the nature of the thing.
Once again, the matter of a thing is part of what is responsible for its being the thing that it is, so Aristotle deemed matter a cause (usually called the "material cause"). We must be careful to recognize that the material cause is understood as the specific type of matter of which a thing is made, so that bronze is the material cause of a statue, etc.
Forms are universals, characteristics shared by many individuals. Human beings are by definition one and all rational. One can explain human behavior by appealing to humans' rationality. Socrates drank the hemlock because he believed it was the best thing to do, that is, insofar as he was exercising his rationality.
But there is still more involved: what Aristotle called demonstration, which makes connections between what has been perceived or got through induction. There are two types of knowledge that can be gained in demonstration.
The first is demonstration of the fact, whereby we gain new information from old. Suppose we know through perception that planets do not twinkle and by induction that what does not twinkle is near. Then we may make this demonstration.
Planets are untwinkling
Untwinkling objects are near
Therefore, Planets are near
The conclusion is not something which is accessible to perception or induction, so it constitutes new information, or demonstration of the fact that the planets are near.
The second type of knowledge is the explanation of the known fact. In our example, Aristotle's example, there is a closely related demonstration of the "why" of the first premise, why the planets do not twinkle.
Planets are near
Near objects are untwinkling
Therefore, Planets are untwinkling
The conclusion does not give us any new information, but instead gives us an explanation for what is known through perception or induction.
Aristotle's three-part picture of knowledge has several short-comings. First, there is the problem of induction. How can we know when we have observed enough cases to draw a universal conclusion? Why do our observations about near torch lights apply to the light of planets? This problem was brought out dramatically by David Hume in the eighteenth century. It is not the demonstration, but the induction, which packs the real power in Aristotelian explanation. Aristotle defended his account of induction only by analogy. It is like the rallying of a fleeing army around an individual who takes a stand.
Second, the demonstrations are purely qualitative in character. The properties involved (near, untwinkling) are imprecise. In the seventeenth century, the emphasis was on the mathematical formula as the "middle term" giving the real "why" of the observed phenomenon. We will see below how deficient was Aristotle's use of quantitative demonstrations.
In the course of nature things become what they were not. An acorn becomes a mature oak tree, with a trunk, branches, leaves and its own acorns. (Note that after reaching its peak of vigor, the perfection of its form, the tree declines.) The transition is from potential to actual.
There are several ways in which things change, becoming what they were not. Aristotle called three ways of change "motion." The three are: quantitative change, increase and diminution (e.g. the growth and decline of the oak), qualitative change (e.g. the changing of the color of the leaves), and change of place or motion proper.
These kinds of change may result from internal or external factors. Change of place, in particular, can have an internal or external source in the case of animals, which move themselves or can be moved (e.g. by the flow of a river). Inanimate objects also have an internal source of motion: the tendency to move toward their natural place.
The outer shell of fire is where the mixing of the elements stops. Outside this shell is that of a fifth element (quintessence) which is the home of the stars, planets, sun and moon. The planets, sun and moon move in concentric shells independently of the outermost shell, that of the fixed stars.
There would be no motion at all if things were all in their natural places. So Aristotle posits a first or prime mover, responsible for mixing up the elements into their present unsettled state. The motion of the first mover applies to the outermost sphere, which communicates motion by contact to all the others. Aristotle believed that all motion is passed on by contact, there being no void, on his view.When change is brought about by an agent, the agent can be said to be the "efficient cause" of the change.
The theory of contact motion explains the violent (unnatural) motion of projectiles, objects which are hurled or shot through another medium. When contact with the shooting or hurling body is lost, air (for example) which has been displaced from the front of the body rushes to the rear to push it forward. Aristotle believed that this explanation shows the theory of the void to be incorrect, for a projectile would have nothing to keep it moving in a vacuum.
Aristotle presented little by way of a theory of the motion of bodies. He claimed that velocity in a medium is proportional to its density. But this runs the risk of being an empty rule, since the definition of density seems to be based on the velocity which it permits. The atomists had a proper notion of density based on the quantity of atoms in the given part of space. But Aristotle claimed that the velocity of a body in the void makes no sense.
Since a void has zero density, the determination of the proportion of the velocity of an object moving in a void to that of an object moving in a medium would require division by zero, which cannot be done.
Aristotle's few quantitative descriptions of motion illustrate the weakeness of his science. He describes the application of the "rule of proportion" to cases of motion. Suppose object A moves object B over space C during time D. He claims that a moved object with half the quantity of B (.5B) will be moved over twice the space (2C) by A in time D. Algebraically, if A/B = C/D, then A/.5B = 2C/D. However, it is also the case that A/2B = .5C/D, so that if the formula holds generally, a moved object with twice the quantity will be moved half the distance. Aristotle then notes that this is not the case, observationally. The object might not be moved at all. So at best, the rule of proportion holds for only some A, B, C, D.
We are given no principle to explain this exception, and it violates Aristotle's view that to understand we must grasp the universal. The exceptions to the rule are ad hoc deviations (i.e., they are brought in for this special purpose and are unmotivated by any general principle). The modern philosophers sought to exclude ad hoc exceptions from their theories.
Another difficulty with the universality of Aristotle's science stems from the irregular motion of some of the planets. He claimed that the planets lie in concentric spheres surrounding the sphere of fire. Like others of his time, he also believed that the natural motion of heavenly bodies is circular. It was impossible for the ancient astronomers to reconcile these principles with observations.
A key problem was how to explain the retrograde motion of the outer planets, e.g., Jupiter. At some times, they do not move in the normal direction, but stop and loop back on themselves, before resuming normal motion. Astronomers modeled this behavior mathematically using circles on circles. The basic circle around the earth (the deferent) has another circle centered on a point on its circumference. The compound motion of the two circles would account for the looping of the planets. However, this motion is incompatible with the concentric sphere theory. All points on the circumference of the sphere are equally distant from the center, while the points on the compound system vary in their distance from the center.
Other anomalies in observation led to even more deviation from the basic system of concentric spheres, so that by the time of Copernicus, it could justly be asserted of the astronomer Ptolemey's system that it "seemed neither sufficiently absolute nor sufficiently pleasing to the mind."
In summary, Aristotle's system sought to explain the "why" of things by appeal to the universal element in them. Change was explained by identifying the universal with a form, which, in turn, was identified as the end or telos toward which an object moves from a state of mere potentiality, as an acorn becomes a mature oak. Aristotle's science was fundamentally teleological and qualitative: two elements which would be expunged on methodological grounds from scientific explanation in the seventeenth century. Moreover, Aristotelian science was unable to provide a comprehensive explanation of the phenomena of motion, both celestial and terrestial. It was the problem of celestial motion which captured the imagination of the scientists of the sixteenth century. Galileo and others made numerous advances in the seventeenth, which culminated in the grand synthesis effected by Newton.
とても興味深く読みました：
再生核研究所声明353（2017.2.2） ゼロ除算 記念日
2014.2.2 に 一般の方から100/0 の意味を問われていた頃、偶然に執筆中の論文原稿にそれがゼロとなっているのを発見した。直ぐに結果に驚いて友人にメールしたり、同僚に話した。それ以来、ちょうど３年、相当詳しい記録と経過が記録されている。重要なものは再生核研究所声明として英文と和文で公表されている。最初のものは
再生核研究所声明 148（2014.2.12）: 100/0=0, 0/0=0 － 割り算の考えを自然に拡張すると ― 神の意志
で、最新のは
Announcement 352 (2017.2.2): On the third birthday of the division by zero z/0=0
である。
アリストテレス、ブラーマグプタ、ニュートン、オイラー、アインシュタインなどが深く関与する ゼロ除算の神秘的な永い歴史上の発見であるから、その日をゼロ除算記念日として定めて、世界史を進化させる決意の日としたい。ゼロ除算は、ユークリッド幾何学の変更といわゆるリーマン球面の無限遠点の考え方の変更を求めている。― 実際、ゼロ除算の歴史は人類の闘争の歴史と共に 人類の愚かさの象徴であるとしている。
心すべき要点を纏めて置きたい。
１） ゼロの明確な発見と算術の確立者Brahmagupta (598 - 668 ?) は 既にそこで、0/0=0 と定義していたにも関わらず、言わば創業者の深い考察を理解できず、それは間違いであるとして、１３００年以上も間違いを繰り返してきた。
２） 予断と偏見、慣習、習慣、思い込み、権威に盲従する人間の精神の弱さ、愚かさを自戒したい。我々は何時もそのように囚われていて、虚像を見ていると 真智を愛する心を大事にして行きたい。絶えず、それは真かと 問うていかなければならない。
３） ピタゴラス派では 無理数の発見をしていたが、なんと、無理数の存在は自分たちの世界観に合わないからという理由で、― その発見は都合が悪いので ― 、弟子を処刑にしてしまったという。真智への愛より、面子、権力争い、勢力争い、利害が大事という人間の浅ましさの典型的な例である。
４） この辺は、２０００年以上も前に、既に世の聖人、賢人が諭されてきたのに いまだ人間は生物の本能レベルを越えておらず、愚かな世界史を続けている。人間が人間として生きる意義は 真智への愛にある と言える。
５） いわば創業者の偉大な精神が正確に、上手く伝えられず、ピタゴラス派のような対応をとっているのは、本末転倒で、そのようなことが世に溢れていると警戒していきたい。本来あるべきものが逆になっていて、社会をおかしくしている。
６） ゼロ除算の発見記念日に 繰り返し、人類の愚かさを反省して、明るい世界史を切り拓いて行きたい。
以 上
追記：
The division by zero is uniquely and reasonably determined as 1/0=0/0=z/0=0 in the natural extensions of fractions. We have to change our basic ideas for our space and world:
Division by Zero z/0 = 0 in Euclidean Spaces
Hiroshi Michiwaki, Hiroshi Okumura and Saburou Saitoh
International Journal of Mathematics and Computation Vol. 28(2017); Issue 1, 2017), 1-16.
http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
再生核研究所声明359（2017.3.20） ゼロ除算とは何か ― 本質、意義
ゼロ除算の理解を進めるために ゼロ除算とは何か の題名で、簡潔に表現して置きたい。 構想と情念、想いが湧いてきたためである。
基本的な関数y=1/x を考える。 これは直角双曲線関数で、原点以外は勿論、値、関数が定義されている。問題はこの関数が、x=0 で どうなっているかである。結論は、この関数の原点での値を ゼロと定義する ということである。 定義するのである。定義であるから勝手であり、従来の定義や理論に反しない限り、定義は勝手であると言える。原点での値を明確に定義した理論はないから、この定義は良いと考えられる。それを、y=1/0=0 と記述する。ゼロ除算は不可能であるという、数学の永い定説に従って、1/0 の表記は学術書、教科書にもないから、1/0=0 の記法は 形式不変の原理、原則 にも反しないと言える。― 多くの数学者は注意深いから、1/0=\infty の表記を避けてきたが、想像上では x が 0 に近づいたとき、限りなく 絶対値が大きくなるので、複素解析学では、表現1/0=\infty は避けても、1/0=\infty と考えている事は多い。（無限大の記号がない時代、アーベルなどもそのような記号を用いていて、オイラーは1/0=\inftyと述べ、それは間違いであると指摘されてきた。 しかしながら、無限大とは何か、数かとの疑問は 続いている。）。ここが大事な論点である。近づいていった極限値がそこでの値であろうと考えるのは、極めて自然な発想であるが、現代では、不連続性の概念 が十分確立されていて、極限値がそこでの値と違う例は、既にありふれている。― アリストテレスは 連続性の世界観をもち、特にアリストテレスの影響を深く受けている欧米の方は、この強力な不連続性を中々受け入れられないようである。無限にいくと考えられてきたのが突然、ゼロになるという定義になるからである。 しかしながら、関数y=1/xのグラフを書いて見れば、原点は双曲線のグラフの中心の点であり、美しい点で、この定義は魅力的に見えてくるだろう。
定義したことには、それに至るいろいろな考察、経過、動機、理由がある。― 分数、割り算の意味、意義、一意性問題、代数的な意味づけなどであるが、それらは既に数学的に確立しているので、ここでは触れない。
すると、定義したからには、それがどのような意味が存在して、世の中に、数学にどのような影響があるかが、問題になる。これについて、現在、初等数学の学部レベルの数学をゼロ除算の定義に従って、眺めると、ゼロ除算、すなわち、 分母がゼロになる場合が表現上現れる広範な場合に 新しい現象が発見され、ゼロ除算が関係する広範な場合に大きな影響が出て、数学は美しく統一的に補充，完全化されることが分かった。それらは現在、３８０件以上のメモにまとめられている。しかしながら、世界観の変更は特に重要であると考えられる：
複素解析学で無限遠点は その意味で1/0=0で、複素数0で表されること、アリストテレスの連続性の概念に反し、ユークリッド空間とも異なる新しい空間が 現れている。直線のコンパクト化の理想点は原点で、全ての直線が原点を含むと、超古典的な結果に反する。更に、ゼロと無限の関係が明らかにされてきた。
ゼロ除算は、現代数学の初等部分の相当な変革を要求していると考えられる。
以 上
付記： The division by zero is uniquely and reasonably determined as 1/0=0/0=z/0=0 in the natural extensions of fractions. We have to change our basic ideas for our space and world
Division by Zero z/0 = 0 in Euclidean Spaces
Hiroshi Michiwaki, Hiroshi Okumura and Saburou Saitoh International Journal of Mathematics and Computation Vol. 28(2017); Issue 1, 2017), 1 -16.
http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
Relations of 0 and infinity
Hiroshi Okumura, Saburou Saitoh and Tsutomu Matsuura：
http://www.e-jikei.org/…/Camera%20ready%20manuscript_JTSS_A…
http://www.e-jikei.org/…/Camera%20ready%20manuscript_JTSS_A…
再生核研究所声明357（2017.2.17）Brahmagupta の名誉回復と賞賛を求める。
再生核研究所声明 ３３９で 次のように述べている：
世界史と人類の精神の基礎に想いを致したい。ピタゴラスは 万物は数で出来ている、表されるとして、数学の重要性を述べているが、数学は科学の基礎的な言語である。ユークリッド幾何学の大きな意味にも触れている（再生核研究所声明315（2016.08.08） 世界観を大きく変えた、ユークリッドと幾何学）。しかしながら、数体系がなければ、空間も幾何学も厳密には 表現することもできないであろう。この数体系の基礎はブラーマグプタ（Brahmagupta、598年 – 668年?）インドの数学者・天文学者によって、628年に、総合的な数理天文書『ブラーマ・スプタ・シッダーンタ』（ब्राह्मस्फुटसिद्धान्त Brāhmasphuṭasiddhānta）の中で与えられ、ゼロの導入と共に四則演算が確立されていた。ゼロの導入、負の数の導入は数学の基礎中の基礎で、西欧世界がゼロの導入を永い間嫌っていた状況を見れば、これらは世界史上でも顕著な事実であると考えられる。最近ゼロ除算は、拡張された割り算、分数の意味で可能で、ゼロで割ればゼロであることが、その大きな影響とともに明らかにされてきた。しかしながら、 ブラーマグプタは その中で 0 ÷ 0 ＝ 0 と定義していたが、奇妙にも１３００年を越えて、現在に至っても 永く間違いであるとされている。現在でも0 ÷ 0について、幾つかの説が存在していて、現代数学でもそれは、定説として 不定であるとしている。最近の研究の成果で、ブラーマグプタの考えは 実は正しかった ということになる。 しかしながら、一般の ゼロ除算については触れられておらず、永い間の懸案の問題として、世界を賑わしてきた。現在でも議論されている。ゼロ除算の永い歴史と問題は、次のアインシュタインの言葉に象徴される：
Blackholes are where God divided by zero. I don't believe in mathematics. George Gamow (1904-1968) Russian-born American nuclear physicist and cosmologist re-
marked that "it is well known to students of high school algebra" that division by zero is not valid; and Einstein admitted it as the biggest blunder of his life [1] 1. Gamow, G., My World Line (Viking, New York). p 44, 1970.
物理学や計算機科学で ゼロ除算は大事な課題であるにも関わらず、創始者の考えを無視し、割り算は 掛け算の逆との 貧しい発想で 間違いを１３００年以上も、繰り返してきたのは 実に残念で、不名誉なことである。創始者は ゼロの深い意味、ゼロが 単純な算数・数学における意味を越えて、ゼロが基準を表す、不可能性を表現する、神が最も簡単なものを選択する、神の最小エネルギーの原理、すなわち、神もできれば横着したいなどの世界観を感じていて、0/0=0 を自明なもの と捉えていたものと考えられる。実際、巷で、ゼロ除算の結果や、適用例を語ると 結構な 素人の人々が 率直に理解されることが多い。
１３００年間も 創始者の結果が間違いであるとする 世界史は修正されるべきである、間違いであるとの不名誉を回復、数学の基礎の基礎である算術の確立者として、世界史上でも高く評価されるべきである。 真智の愛、良心から、厚い想いが湧いてくる。
以 上
追記
The division by zero is uniquely and reasonably determined as 1/0=0/0=z/0=0 in the natural extensions of fractions. We have to change our basic ideas for our space and world:
http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007
http://www.ijapm.org/show-63-504-1.html
http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf
0 件のコメント:
コメントを投稿