6.1 Mathematical Objects and Truths
Even though Aristotle’s contributions to mathematics are significantly important and lay a strong foundation in the study and view of the science, it is imperative to mention that Aristotle, in actuality, “never devoted a treatise to philosophy of mathematics” [5]. As aforementioned, even his books never truly leaned toward a specific philosophy on mathematics, but rather a form or manner in which to attempt to understand mathematics through certain truths.
To better attempt to understand Aristotle’s view on mathematical truths, further inquiry will be made in regards to a fictionalist versus a literalist view point of mathematical objects. Both literalism and fictionalism have been attributed to Aristotle
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However, his greatest contribution to mathematics is considered to be logic, for without logic there would be no reasoning and therefore no true valid rules to the science of mathematics.
Aristotle saw logic as a tool that led to probing and eventually to explanations through argumentation rather than deductions alone [6]. In Aristotle’s view, deductions were not sufficient to lead to any type of validity, and most certainly not in the sciences, where arguments should “feature premises which are necessary” in order to avoid false suppositions [6]. He insisted that because science “extends to fields of inquiry like mathematics and metaphysics,” it is essential that not only facts had to be reported, but also explained through their “priority relations” [6].
Aristotle’s method of reasoning involved syllogisms, which are pairs of propositions that when combined gave a further explanation or connection between the pair, leading to a valid conclusion. His logic was based on “inclusion and exclusion relations” as seen in the following example provided by the Stanford Encyclopedia of Philosophy,
Aristotle, . Metaphysics. Aristotle: Selections. Edited by Terence Irwin and Gail Fine. Indianapolis: Hackett , 1995.
The subject-matter that I would like to discuss today is a sample of how these commentators can still contribute to understand Aristotle. I would like to warn, however, that the theme of the indefinite terms is especially illustrative of what I indicate, for the modern comments on this topic have been made without a profounder consideration of the ancient teaching. In my opinion, however, a more reliable and complete explanation about this difficult subject is to be found in the analysis of the ancient view.
The influence of Aristotle can be seen in almost every era of history that followed his death over 2300 years ago. In the Middle Ages thinkers used Aristotle’s work as a sort of “final authority on all sorts of issues” (Patterns, 141). In the 16th and 17th centuries philosophers had to first tackle...
Aristotle is considered by many to be one of the most influential philosophers in history. As a student of Plato, he built on his mentor’s metaphysical teachings of things like The Theory of Forms and his views on the soul. However, he also challenged them, introducing his own metaphysical ideas such as act and potency, hylemorphism, and the four causes. He used these ideas to explain his account of the soul and the immateriality of intellect.
Aristotle is regarded by many as one of the most important thinkers of the ancient era. Although many of his theories regarding the physics of the natural world were later disproved by Galileo, Aristotle nevertheless offered the world at that time a relevant and consistent explanation of physics of impressive breadth and explanatory ability. Many of his theories endured for up to 1200 years, and helped to form the basis of the midieval christian perspective of the natural world. Much of his physics, when combined with Ptolemy's mathematical model of planetary motions, was used by midieval thinkers to describe the behavior of the cosmos.
It is interesting that despite the fame he achieved because of his mechanical inventions, he believed that pure mathematics was the more worthwhile pursuit. Plutarch describes his attitude:
On the European continent, rationalists like Descartes, Spinoza and Leibniz were plato’s natural heirs. They emphasised the importance of reason, as opposed to sensory experience in obtaining knowledge. The rationalist model for knowledge-gathering is mathematics - mathematical demonstration in particular. Demonstrating this, Spinoza’s Ethics has the same format as Euclid’s Elements, containing ‘propositions’ and ‘demonstrations’. It is thought science is supposed to be founded on philosophical first-principles.
Many have scrutinized and compared the dissimilarities and similarities of Aristotle's doctrine of categories and Plato's theory of forms. The observations found are of an interesting nature.
To know a thing, says Aristotle, one must know the thing’s causes. For Aristotle the knowledge of causes provides an explanation. It is a way to understand something. Because of the importance of causality to knowledge and understanding, Aristotle developed something like the complete doctrine of causality, distinguishing efficient, material, formal, and final causes, and later concepts of causality have been derived from his analysis by omission. Aristotle’s four causes gives answers to the questions related to the thing to help ascertain knowledge of it, such as what the thing is made of, where the thing comes from, what the thing actually is, and what the thing’s purpose is. The thing’s purpose is used to determine the former three, in addition to the purpose being basically the same thing as what the thing actually is, as the purpose of the thing is used to determine whether or not a thing is what it is.
yet we know relatively few facts of his life. We are not exactly sure of his birth
Leonardo da Vinci was one of the greatest mathematicians to ever live, which is displayed in all of his inventions. His main pursuit through mathematics was to better the understanding and exploration of the world. He preferred drawing geographical shapes to calculate equations and create his inventions, which enlisted his very profound artistic ability to articulate his blueprints. Leonardo Da Vinci believed that math is used to produce an outcome and thus Da Vinci thought that through his drawings he could execute his studies of proportional and spatial awareness demonstrated in his engineering designs and inventions.
Many of Aristotle’s teachings have shown remarkable insight into the human mind, especially considering the time in which he lived. Just as some of his teachings on physics were held as true for nearly 2000 years, many of his teachings on the human mind were well ahead of his time. His method of study and experimentation, followed by logical deduction are the basis for all sciences now, something which was completely new when he wrote of this approach.
Burton, D. (2011). The History of Mathematics: An Introduction. (Seventh Ed.) New York, NY. McGraw-Hill Companies, Inc.
The history of math has become an important study, from ancient to modern times it has been fundamental to advances in science, engineering, and philosophy. Mathematics started with counting. In Babylonia mathematics developed from 2000B.C. A place value notation system had evolved over a lengthy time with a number base of 60. Number problems were studied from at least 1700B.C. Systems of linear equations were studied in the context of solving number problems.
Aristotle made contributions to logic, physics, biology, medicine, and agriculture. He redesigned most, if not all, areas of knowledge he studied. Later in life he became the “Father of logic” and was the first to develop a formalized way of reasoning. Aristotle was a greek philosopher who founded formal logic, pioneered zoology, founded his own school, and classified the various branches of philosophy.