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.
Aristotle rejected the extreme views of Plato, though he agreed that the nature of things is to be found in the forms. His reason was that the nature of something is found in its definition, and the definition expresses its form. For example, the definition of 'human being' is 'rational animal.' The human is a species of the genus animal, differentiated from other animal species by its being rational.
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.
Functions are directed at certain ends, so the notion of an end (Greek: telos') is central to Aristotle's account of nature. The end of a thing is its finished, perfected condition. This is another kind of cause (usually called the "final cause"). That for the sake of which a substance changes is a reason for the substance being the kind of thing it is.
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.
To gain knowledge of the universal, we begin with perception of many individuals of a certain sort. Then in a movement called induction, we understand what is common among things of that sort. For example, we might determine by induction that a light's being near and its not twinkling are universally associated. Perception and induction make up what can be called basic knowledge.
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.
Finally, Aristotle inherited a problem from Plato, namely that appeal to universals is not well-suited to explain the dynamic character of nature, how things change. Here, Aristotle was forced to innovation, but as will be seen, the innovation was not a happy one.
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 doctrine of the natural place of things is based on that of the four elements: fire, air, water and earth. Earth tends downward and fire upward, with air tending toward a level between fire and water, and water toward a level between air and earth. Where things move naturally depends on the proportion of their elements, so that a piece of wood floats because it contains air, for example. The earth is a sphere, so the natural place of the element earth is a central sphere, surrounded by a shell of water, which in turn is surrounded by a shell of air, which finally is ringed by a shell of fire.
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.http://hume.ucdavis.edu/mattey/phi022old/arilec.htm
再生核研究所声明296(2016.05.06) ゼロ除算の混乱
ゼロ除算の研究を進めているが、誠に奇妙な状況と言える。簡潔に焦点を述べておきたい。
ゼロ除算はゼロで割ることを考えることであるが、物理学的にはアリストテレス、ニュートン、アンシュタインの相当に深刻な問題として、問題にされてきた。他方、数学界では６２８年にインドで四則演算の算術の法則の確立、記録とともに永年問題とされてきたが、オイラー、アーベル、リーマン達による、不可能であるという考えと、極限値で考えて無限遠点とする定説が永く定着してきている。
ところが数学界の定説には満足せず、今尚熱い話題、問題として、議論されている。理由は、ゼロで割れないという例外がどうして存在するのかという、素朴な疑問とともに、積極的に、計算機がゼロ除算に出会うと混乱を起こす具体的な懸案問題を解消したいという明確な動機があること、他の動機としてはアインシュタインの相対性理論の上手い解釈を求めることである。これにはアインシュタインが直接言及しているように、ゼロ除算はブラックホールに関係していて、ブラックホールの解明を意図している面もある。偶然、アインシュタイン以後１００年 実に面白い事件が起きていると言える。偶然、２０年以上も考えて解明できたとの著書さえ出版された。― これは、初めから、間違いであると理由を付けて質問を送っているが、納得させる回答が無い。実名を上げず、具体的に 状況を客観的に述べたい。尚、ゼロ除算はリーマン仮説に密接に関係があるとの情報があるが 詳しいことは分からない。
１： ゼロ除算回避を目指して、新しい代数的な構造を研究しているグループ、相当な積み重ねのある理論を、体や環の構造で研究している。例えて言うと、ゼロ除算は沢山存在するという、考え方と言える。― そのような抽象的な理論は不要であると主張している。
２：同じくゼロ除算回避を志向して 何と0/0 を想像上の数として導入し、正、負無限大とともに数として導入して、新しい数の体系と演算の法則を考え、展開している。相当なグループを作っているという。BBCでも報じられたが、数学界の評判は良くないようである。― そのような抽象的な理論は不要であると主張している。
３：最近、アインシュタインの理論の専門家達が アインシュタインの理論から、0/0=1, 1/0=無限 が出て、ゼロ除算は解決したと報告している。― しかし、これについては、論理的な間違いがあると具体的に指摘している。結果も我々の結果と違っている。
４：数学界の永い定説では、1/0 は不可能もしくは、極限の考え方で、無限遠点を対応させる. 0/0 は不定、解は何でも良いとなっている。― 数学に基本的な欠落があって、ゼロ除算を導入しなければ数学は不完全であると主張し、新しい世界観を提起している。
ここ２年間の研究で、ゼロ除算は 何時でもゼロz/0=0であるとして、 上記の全ての立場を否定して、新しい理論の建設を進めている。z/0 は 普通の分数ではなく、拡張された意味でと初期から説明しているが、今でも誤解していて、混乱している人は多い、これは真面目に論文を読まず、初めから、問題にしていない証拠であると言える。
上記、関係者たちと交流、討論しているが、中々理解されず、自分たちの建設している理論に固執しているさまがよく現れていて、数学なのに、心情の問題のように感じられる微妙で、奇妙な状況である。
我々のゼロ除算の理論的な簡潔な説明、それを裏付ける具体的な証拠に当たる結果を沢山提示しているが、中々理解されない状況である。
数学界でも永い間の定説で、初めから、問題にしない人は多い状況である。ゼロ除算は算数、ユークリッド幾何学、解析幾何学など、数学の基本に関わることなので、この問題を究明、明確にして頂きたいと要請している：
再生核研究所声明 277(2016.01.26)：アインシュタインの数学不信 ― 数学の欠陥
再生核研究所声明 278(2016.01.27)： 面白いゼロ除算の混乱と話題
再生核研究所声明279(2016.01.28) ： ゼロ除算の意義
再生核研究所声明280(2016.01.29) : ゼロ除算の公認、認知を求める
我々のゼロ除算について８歳の少女が３週間くらいで、当たり前であると理解し、高校の先生たちも、簡単に理解されている数学、それを数学の専門家や、ゼロ除算の専門家が２年を超えても、誤解したり、受け入れられない状況は誠に奇妙で、アリストテレスの２０００年を超える世の連続性についての固定した世界観や、上記天才数学者たちの足跡、数学界の定説に まるで全く嵌っている状況に感じられる。
以 上
考えてはいけないことが、考えられるようになった。
説明できないことが説明できることになった。
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.
Aristotle rejected the extreme views of Plato, though he agreed that the nature of things is to be found in the forms. His reason was that the nature of something is found in its definition, and the definition expresses its form. For example, the definition of 'human being' is 'rational animal.' The human is a species of the genus animal, differentiated from other animal species by its being rational.
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.
Functions are directed at certain ends, so the notion of an end (Greek: telos') is central to Aristotle's account of nature. The end of a thing is its finished, perfected condition. This is another kind of cause (usually called the "final cause"). That for the sake of which a substance changes is a reason for the substance being the kind of thing it is.
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.
To gain knowledge of the universal, we begin with perception of many individuals of a certain sort. Then in a movement called induction, we understand what is common among things of that sort. For example, we might determine by induction that a light's being near and its not twinkling are universally associated. Perception and induction make up what can be called basic knowledge.
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.
Finally, Aristotle inherited a problem from Plato, namely that appeal to universals is not well-suited to explain the dynamic character of nature, how things change. Here, Aristotle was forced to innovation, but as will be seen, the innovation was not a happy one.
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 doctrine of the natural place of things is based on that of the four elements: fire, air, water and earth. Earth tends downward and fire upward, with air tending toward a level between fire and water, and water toward a level between air and earth. Where things move naturally depends on the proportion of their elements, so that a piece of wood floats because it contains air, for example. The earth is a sphere, so the natural place of the element earth is a central sphere, surrounded by a shell of water, which in turn is surrounded by a shell of air, which finally is ringed by a shell of fire.
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.http://hume.ucdavis.edu/mattey/phi022old/arilec.htm
再生核研究所声明296(2016.05.06) ゼロ除算の混乱
ゼロ除算の研究を進めているが、誠に奇妙な状況と言える。簡潔に焦点を述べておきたい。
ゼロ除算はゼロで割ることを考えることであるが、物理学的にはアリストテレス、ニュートン、アンシュタインの相当に深刻な問題として、問題にされてきた。他方、数学界では６２８年にインドで四則演算の算術の法則の確立、記録とともに永年問題とされてきたが、オイラー、アーベル、リーマン達による、不可能であるという考えと、極限値で考えて無限遠点とする定説が永く定着してきている。
ところが数学界の定説には満足せず、今尚熱い話題、問題として、議論されている。理由は、ゼロで割れないという例外がどうして存在するのかという、素朴な疑問とともに、積極的に、計算機がゼロ除算に出会うと混乱を起こす具体的な懸案問題を解消したいという明確な動機があること、他の動機としてはアインシュタインの相対性理論の上手い解釈を求めることである。これにはアインシュタインが直接言及しているように、ゼロ除算はブラックホールに関係していて、ブラックホールの解明を意図している面もある。偶然、アインシュタイン以後１００年 実に面白い事件が起きていると言える。偶然、２０年以上も考えて解明できたとの著書さえ出版された。― これは、初めから、間違いであると理由を付けて質問を送っているが、納得させる回答が無い。実名を上げず、具体的に 状況を客観的に述べたい。尚、ゼロ除算はリーマン仮説に密接に関係があるとの情報があるが 詳しいことは分からない。
１： ゼロ除算回避を目指して、新しい代数的な構造を研究しているグループ、相当な積み重ねのある理論を、体や環の構造で研究している。例えて言うと、ゼロ除算は沢山存在するという、考え方と言える。― そのような抽象的な理論は不要であると主張している。
２：同じくゼロ除算回避を志向して 何と0/0 を想像上の数として導入し、正、負無限大とともに数として導入して、新しい数の体系と演算の法則を考え、展開している。相当なグループを作っているという。BBCでも報じられたが、数学界の評判は良くないようである。― そのような抽象的な理論は不要であると主張している。
３：最近、アインシュタインの理論の専門家達が アインシュタインの理論から、0/0=1, 1/0=無限 が出て、ゼロ除算は解決したと報告している。― しかし、これについては、論理的な間違いがあると具体的に指摘している。結果も我々の結果と違っている。
４：数学界の永い定説では、1/0 は不可能もしくは、極限の考え方で、無限遠点を対応させる. 0/0 は不定、解は何でも良いとなっている。― 数学に基本的な欠落があって、ゼロ除算を導入しなければ数学は不完全であると主張し、新しい世界観を提起している。
ここ２年間の研究で、ゼロ除算は 何時でもゼロz/0=0であるとして、 上記の全ての立場を否定して、新しい理論の建設を進めている。z/0 は 普通の分数ではなく、拡張された意味でと初期から説明しているが、今でも誤解していて、混乱している人は多い、これは真面目に論文を読まず、初めから、問題にしていない証拠であると言える。
上記、関係者たちと交流、討論しているが、中々理解されず、自分たちの建設している理論に固執しているさまがよく現れていて、数学なのに、心情の問題のように感じられる微妙で、奇妙な状況である。
我々のゼロ除算の理論的な簡潔な説明、それを裏付ける具体的な証拠に当たる結果を沢山提示しているが、中々理解されない状況である。
数学界でも永い間の定説で、初めから、問題にしない人は多い状況である。ゼロ除算は算数、ユークリッド幾何学、解析幾何学など、数学の基本に関わることなので、この問題を究明、明確にして頂きたいと要請している：
再生核研究所声明 277(2016.01.26)：アインシュタインの数学不信 ― 数学の欠陥
再生核研究所声明 278(2016.01.27)： 面白いゼロ除算の混乱と話題
再生核研究所声明279(2016.01.28) ： ゼロ除算の意義
再生核研究所声明280(2016.01.29) : ゼロ除算の公認、認知を求める
我々のゼロ除算について８歳の少女が３週間くらいで、当たり前であると理解し、高校の先生たちも、簡単に理解されている数学、それを数学の専門家や、ゼロ除算の専門家が２年を超えても、誤解したり、受け入れられない状況は誠に奇妙で、アリストテレスの２０００年を超える世の連続性についての固定した世界観や、上記天才数学者たちの足跡、数学界の定説に まるで全く嵌っている状況に感じられる。
以 上
考えてはいけないことが、考えられるようになった。
説明できないことが説明できることになった。
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