Immanuel Kant, Rudolf Carnap And Mary Leng

We use mathematics to our great advantage to explain many things. Although Pythagoras, applied A^2+B^2=C^2, he did not create the substance of the equation, this theorem is timeless, he only brought it to our attention.

The distinction between a priori and a posteriori is made by the two possible modes of knowledge that can be obtained: Experience and intellect. If something is learned through intellect or prior to experience, then the concept is a priori. If it is learned through experience, then it is an a posteriori judgement. Math is an example of a priori knowledge. In the Prolegonema Kant uses the example 7 + 5 = 12 Alone this statement is undeniably true because there’s no augmentation that could alter the truth of the statement. The statement "all bachelorettes are unmarried" is again a priori even though it refers to bachelorettes,

In this paper I will explain the notion of universals, and argue why Platonism is the more correct view, as opposed to Nominalism and Fictionalism. I will also clarify the major differences between Platonism and Nominalism, and explain how they function as philosophical ideologies. Platonists or “realists” in other terms claim that abstract objects are physical; that they exist in some palpable way. Plato, from whom the term is believed to have originated, was with the idea that universals, like “redness”, existed independently from the individual entities (particulars). Platonic realism states that such objects do exist autonomously from the particular. Platonism is the metaphysical opinion that abstract objects exist. Abstract objects are solely non-spatiotemporal, they are also completely non-physical. Abstract objects are extremely central features regarding the context of philosophy. Abstract objects are comprised of all the names and categories of things. These types are abstract. So, for example, a chair is both the token (actual chair) and the type (an abstract classifying as such). This can contain universals, numbers, and ideas like “redness”, concepts like courage and justice, and even individual humans, such as John Smith. Platonists

For example, Kant explains “Geometry is based upon the pure intuition of space…if we omit from the empirical intuitions of bodies and their alterations (motion) everything is empirical, i.e., belonging to sensation, space and time remain.” (Kant Prolegomena, pg. 25). This leads to the fact that space and time for humanity is an a priori part of existence, something everyone is capable of preserving and is surrounding humanity, making it the only way one is able to experiencing reality. Consequently, this comes to an apex in stating pure mathematics is of course possible because of how no matter if a circle is drawn incorrectly within space and time because of the proofs of a circle one versed in geometry will always be able to find its radius. The concept relates back to if the universe can be solved, if mathematics is a form of an absolute derived from synthetic a priori knowledge, then it must be possible to continue this within other fields, including

As a universal language, mathematics allowed for the world to be explained and expressed in ways language cannot. Math cannot only relate to specific numbers and problems on paper, it can be applied to real life situations, which often exists in nature. In this case, is the book of nature written in the language of mathematics and how does perception play a role in mathematics? The book of nature represents the human life, which is made up of a million different patterns that are explained by formulas and equations, to allow the world to be explained in a more comprehensive, simpler state. An example of the use of mathematics in the natural world is Fibonacci's sequence, which states that patterns and sequences exist in the real world and can be applied to almost everything. This explains the patterns found in nature, which includes the symmetry, shapes, lines and color. Another name for Fibonacci sequence is the golden ratio, which has been used to describe many things in history, science and the real world and explain the reasoning behind why the world is the way it is. The most common example of this sequence is the pattern of flower petals. Certain species of flowers have different amount of flower petals, for instance, lilies always has three petals, whereas buttercups have five petals. Another common example of this ratio is with shells or pinecones, the shape or

There have been a lot of debates on the controversy whether mathematics is discovered or invented? Our first discussion was based on one of this controversy and my point of view about mathematics is still the same. Because as stated in the text book, no one quite knows when and how mathematics began. (Berlinghoff, 2015, p.6). I think mathematics is both discovered and invented by humans. Mathematics is an everyday ongoing process. It is all around us, in everything we do and it has past, present and as well as future. (Berlinghoff, 2015, p.1). Most mathematics arise when we try to solve problems. (Berlinghoff, 2015, p.3). Mathematics is a learning process in which we deal with the logic of shape, quantity, and arrangement.

In Wittgenstein’s Tractatus, he discusses and develops his view on the nature and relation of the world, fact, atomic fact, object, simplicity and complexity. Wittgenstein starts with asserting what the world is, and then builds each concept. In this paper, I will expound upon each concept and what I believe he is expressing with each one.

Kant uses two ways to distinguishes whether a judgement is known to be true. Those which are completely true is a priori and is independent of the senses. On the other hand, there are those known to be a true posterior which is based on the senses. A priori judgements are not necessary according to Kant because they cannot be

Immanuel Kant was born Emmanuel in eastern Prussia into a poor family in 1724. His family had originally emigrated from Scotland where they spelled their last name Cant. Emmanuel’s family was quite large by today’s standards consisting of nine children, he was the fourth born and only one of four to make it to adulthood. He and his siblings grew up in a Pietist house, the focus of Pietism was the literal study of the bible. Their childhood was a strict and punitive one that stressed religious teachings over the sciences and mathematics. After learning Hebrew, Emmanuel changed his name to Immanuel. While attending the Collegium Fridericianum was a good student, Immanuel improved his studies and eventually enrolled at the University of Kongsberg by age 16 in 1740.

For Kant, there are two sources of human knowledge: sensibility (Sinnlichkeit) and understanding (Verstand.) The former is given to us and the latter is thought. Sensibility imposes pure a priori forms on the objects that are given to us in sense-experience. What is given to us, however,

All knowledge is derived from experience according to Kant’s piece the conclusion drawn from this premise is that there is no a priori knowledge.This conclusion can be drawn from this since experience is part of a posteriori knowledge, then all knowledge begins with a posteriori knowledge. This is supported by a quote from Kant’s piece in which he states, “That all our knowledge begins with experience there can be no doubt.”Therefore a priori knowledge which is said to be derived from reason cannot exist according to Kant's theory that all knowledge begins with experience. The premise of a piece is a statement from which a conclusion can be drawn, the premise in this piece is that all knowledge begins with

David Hume and Immanuel Kant argue about the origin of something nearly everyone agrees on, our actions are a result of reason. Hume holds the idea that actions, when cut down to their core, are a result of the universal feelings that a species shares. Kant’s counterargument is that actions, when not done as a means to an end or faculty of desire, are done from duty and only such have true moral worth. He goes on to define good will with and without limitations, and separates the duties towards oneself and to others. Both arguments seem to be a bit incomplete to me, though I believe Hume to be more on the right track. David Hume does a better job arguing for the source of actions of moral worth than Kant because of his simpler approach and more easily applicable standard.

However, this position calls into question the validity of mathematics. Fiction, if assumed completely truthful, becomes a lie. So if mathematics are purely a human made language created to describe phenomena that cannot be described in another language, how can we ascertain its precision and how much power should we give it in a discussion? Mathematical entities do not exist concretely, yet we give them characteristics and magnitudes that allow them to be ranked, ordered and even to interact amongst

The knowledge of mathematics results from proofs that consist of valid and certain conclusions. Verification of mathematical statements is not through experiments or social agreements but logical deductions from basic assumptions. The method assures that the knowledge has universal application since mapping a mathematical statement into physical reality holds

Mathematics is different from some other disciplines such as art and music in that it builds upon itself. It has traditionally been based on a series of axioms and theorems derived from these axioms, from which more theorems can be derived. Therefore, when an essay asks us to eliminate a concept fundamental to mathematics, it is difficult to make a choice since each concept has its merits and is necessary for understanding concepts that build upon it. This essay, although not feasible to implement in the real world, accomplishes the goals of a Writing in Mathematics seminar, which are to encourage students to think creatively about mathematics, increase the students’ skills in writing effective arguments, and show the students that there