An ontological theorist generally begins his discussion with a preconceived notion of what kind of thing an object will turn out to be. Instead, we will here begin with a Thomassonian approach to the ontology of mathematics. First, let us consider what happens when we rst come to determine a mathematical proposition (which I will use synonymously with
'mathematical entitty'). A mathematician does not feel as though he creates mathematical theories. Pythagoras can hardly be thought to have created the claim that a2 + b2 = c2. It becomes clear that a mathematical proposition is a discovered one; that is, we would hardly nd ourselves contending that Pythagoras created his famous theorem. Regardless of who discovers it, the same mathematical
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Already we have found that an ontology must, at the very least, account for the fact that mathematic is about tangible, physical things (even if those things turn out to be merely relations of things). The ctionalist claim seems to address mathematics as a purely linguistic issue, though what mathematics describes is certainly not.
We must then ask whether what math describes is actually there. It seems that the language of mathematics, expressions such as a2 + b2 = c2, are purely constructed terms in the same way that we would be willing to say English is. Perhaps, then, we might lean towards an intuitionist approach like that describes in J.R. Brown [2]. An intuitionist, or constructivist, suggests that mathematical concepts|that is, in our terms, relationships| have no existence until a human mind creates them [4]. However, in suggesting this, we run into some major problems. First, intuitionism looks as though it's going to reject some claims of already accepted mathematics and logic; namely, claims such as the Law of the
Excluded Middle. This is because an intuitionist holds that only until a claim is proven or disprove does it have a true or false truth-value. Propositions such as Goldbach's
I argue that there are three solutions to the question “Under what conditions do objects come into and go out of existence:
Then, he characterizes this rule as something that always and necessarily follows. Also, this rule must make the
I shall also expound Ayer's theory of knowledge, as related in his book. I will show this theory to contain logical errors, making his modified version of the principle flawed from a second angle.
Edmund Husserl’s conception of mathematics was a unique blend of Platonist and formalist ideas. He believed that mathematics had reached a mixed state combining Platonic and formal elements and that both were important for the pursuit of the sciences, as well as for each other. However, he seemed to believe that only the Platonic aspects had significance for his science of phenomenology. Because of the significance of the distinction between these two types of mathematics, I will always use one of the adjectives “material” or “formal” when discussing any branch of mathematics, unless I specifically mean to include both.
... relationship in one problem that doesn’t appear in others. Among all of this, there is such vastness in how one person might approach a problem compared to another, and that’s great. The main understanding that seems essential here is how it all relates. Mathematics is all about relationships between number and methods and models and how they all work in different ways to ideally come to the same solution.
The concept of impossible constructions in mathematics draws in a unique interest by Mathematicians wanting to find answers that none have found before them. For the Greeks, some impossible constructions weren’t actually proven at the time to be impossible, but merely so far unachieved. For them, there was excitement in the idea that they might be the first one to do so, excitement that lay in discovery. There are a few impossible constructions in Greek mathematics that will be examined in this chapter. They all share the same criteria for constructability: that they are to be made using solely a compass and straightedge, and were referred to as the three “classical problems of antiquity”. The requirements of using only a compass and straightedge were believed to have originated from Plato himself. 1
Pythagoras held that an accurate description of reality could only be expressed in mathematical formulae. “Pythagoras is the great-great-grandfather of the view that the totality of reality can be expressed in terms of mathematical laws” (Palmer 25). Based off of his discovery of a correspondence between harmonious sounds and mathematical ratios, Pythagoras deduced “the music of the spheres”. The music of the spheres was his belief that there was a mathematical harmony in the universe. This was based off of his serendipitous discovery of a correspondence between harmonious sounds and mathematical ratios. Pythagoras’ philosophical speculations follow two metaphysical ideals. First, the universe has an underlying mathematical structure. Secondly the force organizing the cosmos is harmony, not chaos or coincidence (Tubbs 2). The founder of a brotherhood of spiritual seekers Pythagoras was the mo...
Geometry, a cornerstone in modern civilization, also had its beginnings in Ancient Greece. Euclid, a mathematician, formed many geometric proofs and theories [Document 5]. He also came to one of the most significant discoveries of math, Pi. This number showed the ratio between the diameter and circumference of a circle.
On closer inspection, however, it is possible to see that these concepts are not in fact contradictory. In the first sentence,
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
is not true either. Therefore, it is not the case that every two objects that are identical are discernible.
The results of study one were the same as the purposed ideas. Yet in study two, their hypothesis was incorrect.
Then in Euclid II, 7, it goes farther to explain that “if a straight line be cut at random, the square on the whole and that on one of the segments both together, are equal to twice the rectangle contained by the whole and said segm...
There are many people that contributed to the discovery of irrational numbers. Some of these people include Hippasus of Metapontum, Leonard Euler, Archimedes, and Phidias. Hippasus found the √2. Leonard Euler found the number e. Archimedes found Π. Phidias found the golden ratio. Hippasus found the first irrational number of √2. In the 5th century, he was trying to find the length of the sides of a pentagon. He successfully found the irrational number when he found the hypotenuse of an isosceles right triangle. He is thought to have found this magnificent finding at sea. However, his work is often discounted or not recognized because he was supposedly thrown overboard by fellow shipmates. His work contradicted the Pythagorean mathematics that was already in place. The fundamentals of the Pythagorean mathematics was that number and geometry were not able to be separated (Irrational Number, 2014).
Euclid, also known as Euclid of Alexandria, lived from 323-283 BC. He was a famous Greek mathematician, often referred to as the ‘Father of Geometry”. The dates of his existence were so long ago that the date and place of Euclid’s birth and the date and circumstances of his death are unknown, and only is roughly estimated in proximity to figures mentioned in references around the world. Alexandria was a broad teacher that taught lessons across the world. He taught at Alexandria in Egypt. Euclid’s most well-known work is his treatise on geometry: The Elements. His Elements is one of the most influential works in the history of mathematics, serving as the source textbook for teaching mathematics on different grade levels. His geometry work was used especially from the time of publication until the late 19th and early 20th century Euclid reasoned the principles of what is now called Euclidean geometry, which came from a small set of axioms on the Elements. Euclid was also famous for writing books using the topic on perspective, conic sections, spherical geometry, number theory, and rigor.