Week 2 Reading Quiz

 

Aristotle

Insert 150-200 word answers to each of the following questions. Answers must be specific, well-organized and demonstrate good knowledge of the assigned reading. In text citations and references from assigned readings are required. Avoid inserting direct quotes with more than 10-15 words. Proofread and edit your writing before posting. 

(1)    How does Plato describe relationship between physical and mathematical objects in his theory of Forms? What are Aristotle’s main arguments against Plato’s theory of Forms? What valid arguments can you present in defense of Plato’s positions?

Based on Plato’s theory of form, mathematical objects are in the world of being, and they resemble forms. The descriptions of mathematical objects depend on an individual’s perception. Plato’s theory divides the world of being into two realms, in which the realm of the physical object occurs at the top while the reflection of physical objects exists at the bottom (Shapiro, 2000). Therefore, this theory places the forms on the top and mathematical objects on the bottom, indicating that physical objects are replications of mathematical objects. Furthermore, mathematical objects are the replications of the forms. The divisions are not the same, and forms take the largest space. Mathematically, the relationship between physical objects, forms, and mathematical objects is described by A/B=C/D = A+B/C+D (Shapiro, 2000). In this equation, A represents Forms, B denotes Mathematical Objects, while C and D signify Physical Objects and Reflections, respectively. Aristotle accepted the Forms exist. However, Aristotle believes that Forms are inseparable from individual objects (Shapiro, 2000). Moreover, Aristotle rejects Plato's explanation of infinite magnitudes and infinite sets since people work finitely. Though Aristotle opposes Plato's theory of Forms, there is a problem with his explanations. Rejecting Platonic forms leads to not believing in mathematical objects, the nature in which mathematical objects exist, and the purposes of mathematical objects. Therefore, this implies that Plato's theory of Forms is valid.

(2)   How are numbers and other mathematical objects obtained according to Aristotle?  Compare and contrast abstractionist and fictionalist interpretations of the nature of mathematical objects.

According to Aristotle, mathematical objects and numbers are obtained from the experience of an individual. For example, number six originates from people's experience with sets of six objects. Additionally, the 'triangle' originates from viewing triangular-shaped objects (Shapiro, 2000). In abstractionist interpretation, objects are created or understood by considering physical objects, suggesting that people abstract away from the characteristics of physical objects (Shapiro, 2000). Therefore, geometric objects are like Forms (Aristotelian Forms) (Shapiro, 2000). Again, mathematical objects obtained as a result of abstraction do not occur before or independent of their physical objects. Based on the abstractionist interpretation, natural numbers are determined through abstraction through sets of physical objects (Shapiro, 2000). Contrary to abstractionist interpretation, fictionist interpretation does not abstract from imperfections of physical objects. Fictionist interpretation does not treat physical objects as indivisible (Shapiro, 2000). Fictionist interpretation focuses on the assessments of the consequences of a particular group of characteristics of physical objects. Abstractionist and fictionist interpretations are similar in that geometric objects are like Forms (Platonic Forms or Aristotelian Forms).

(3) How do empiricists and rationalists describe the nature of mathematical truth and applicability of mathematics to the physical world? Is there any common ground between these two opposing schools of thought?  Were there any attempts to synthesize rationalistic and empiricist perspectives?

The rationalists argue that the definitive initial point for all knowledge hinges on the reason but not senses. On the other hand, the empiricists believe that the ultimate origin of all knowledge depends on the individuals’ experience from five senses. The rationalists maintain that people are born with various vital concepts in their minds that give them innate knowledge (Shapiro, 2000). The empiricists assert that any think people know about the world originates from neutral and observation. The rationalism results from reason, while empiricism originates from experience.  The rationalism is responsible for incongruity between objects’ senses and their mathematic similarities (Shapiro, 2000). The common ground between empiricism and rationalism is that rationalists and empiricists consider mathematics as extended objects or physical magnitudes (Shapiro, 2000). There were attempts to elaborate more on the fictionist and rationalist perceptions. The author shows the difference between the concept of rationalist and fictionist. The rationalist perspective exists via reason, while the empiricist viewpoint occurs through experience.

(4) What is the nature of mathematical propositions and how are they knowable according to Kant? Briefly describe two features of Kantian intuition and its role in Kant’s attempt to explain how mathematics is knowable a priori and yet applicable universally.
 

Kant discovered that mathematical prepositions are analytic. Kant adds that mathematical prepositions are knowledgeable priori, suggesting they are independent of sensory experience (Shapiro, 2000). The principles essential for the prospect of experience, this allows endowment of experience with certainty. Moreover, a priori knowledge provides space and time, and cause and substance which function on the multiple of sense impressions, thus making experience possible. Therefore, human’s a priori speculative knowledge hinge on the ampliative or synthetic statements (Shapiro, 2000). Kant’s intuition comprises of two characteristics (singularity and immediacy). The intuition focuses on individual objects, but not general truths, indicating they are singular. Additionally, intuitions produce immediate knowledge. Hence they contain the concept of immediacy. Pure intuitions provide a priori information of essential truths. Consequently, pure intuitions offer forms of empirical intuitions. They also restrict forms of acuity.

Kant shows that mathematics reveals new information via a priori mental construction process. Based on this explanation, mathematics function on and acts on examples (Shapiro, 2000). Kant expounds on his account by using geometry. He applied geometry in describing space and concluded that Euclidean figures form part of the space (Shapiro, 2000).  Euclidean space is the basis of perception. Therefore, it offers forms of understanding.

(5) Compare and contrast Mills and Kant’s views on the nature of mathematical propositions. Provide examples of universal propositions. 
 

Mill refutes Kant's explanation of mathematical nature by stating that the human mind forms the part of nature. So, no substantial knowledge of the ecosphere can be priori (Shapiro, 2000). Mill differs from Kant's explanation by maintaining that mathematical propositions and logic are real, indicating that they are empirical and synthetic (Shapiro, 2000). Mill’s verbal prepositions are factual by definition (Shapiro, 2000). However, they lack genuine content, implying that they do not talk about the world (Shapiro, 2000). Kant’s verbal propositions are opposed to Mill’s verbal propositions (Shapiro, 2000). According to Mill, characteristic mathematical propositions are generalities (Shapiro, 2000). Hence, mathematical propositions summarize and record the experience. Mill directed that generalizations do not have an impact on the force of argument, because all significant inference is from ‘particulars to particulars’ (Shapiro, 2000). As a result, the universal prepositions such as ‘all crows are black’ (Shapiro, 2000) just summarize what we have perceived.

References

Shapiro, S. (2000). Thinking about mathematics: the philosophy of mathematics. Oxford University Press.