To what extent has Philosophy shaped Mathematics?
Kant vs Mill
Preface
There was of course considerable philosophical action after Aristotle and through the Middle Ages, but there was very little focused on mathematics. The seventeenth century saw major revolutions in science and mathematics, through the work of Descartes, Newton and Leibniz. This meant that when Kant began his philosophy in the eighteenth century, he was in a position to build on philosophy after the scientific developments. In particular, physics began to grow in importance and popularity. This lead to new branches in mathematics and new conceptions in the traditional branches. These mathematical revolutions included new ways of linking geometry with algebra and arithmetic (Fermat and Descartes), and the development of the calculus, through the work of Newton and Leibniz, which was needed for the study of gravitation and motion.
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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. Descartes attempted a mathematical-style derivation of the laws of motion.
Empiricism, the main opposition to rationalism, is an attempt to base knowledge on experiences from the five senses. During this period, the major writers were Locke, Berkeley, Hume and Reid, all of whom lived in the British Isles. A common theme in empiricism is that ultimately anything we know about the world around us must come from our only access to the world - through our eyes, ears, and so
Another motive for action is when something is done in accordance to duty, and actually wants to do it – this is also called immediate inclination. An example of this principle would be a man who is happily married. However, at the office, there is an attractive new intern that constantly hits on him. He does find the intern to be physically attractive but does not actually desire to be with her. He reflects that he could indeed have an affair with this intern if he wanted to but he wont in a million years because he is extremely happy with his wife. He wouldn’t risk that relationship for a chance at a fling. According to Kant, this would not have moral worth because it comes from immediate inclination, not from the motive of duty.
John Stuart Mill famously criticized Immanuel Kant and his theory of the Categorical Imperative by arguing that,
1650, some European thinkers began to analyze nature in order to determine the laws governing the universe. They employed experimentation and abstract reasoning to discover general principles behind phenomena such as the motions of planets and stars, the behavior of falling objects, and the characteristics of light and sound. Above, all Enlightenment philosophers emphasized acquiring knowledge through reason, taking particular delight challenging previously unquestioned assumptions. John Lockes Essay
Immanuel Kant Versus John Stuart Mill Immanuel Kant and John Stuart Mill; two opposing philosophers of their time. Even though they were living in different countries, their works have been against each other. In his book, Grounding for the Metaphysics of Morals, Kant argues that there is nothing better than wanting goodwill by itself. He emphasizes the importance of goodwill over and over again and tries to show how effective moral philosophy can be if goodwill is used as the key element. Therefore, for Kant, the sole foundation of philosophy rests on goodwill.
John Stuart Mill (1808-73) believed in an ethical theory known as utilitarianism. There are many formulation of this theory. One such is, "Everyone should act in such a way to bring the largest possibly balance of good over evil for everyone involved." However, good is a relative term. What is good? Utilitarians disagreed on this subject.
Unlike rationalists, empiricists believe that sense perception is the main source of knowledge. John Locke explained this by dividing ideas into 2 parts: 1) simple, and 2) complex. Simple ideas are based only on perception, like color, size, shape, etc. Complex ideas are formed when simple ideas are combined.
Although philosophy rarely alters its direction and mood with sudden swings, there are times when its new concerns and emphases clearly separate it from its immediate past. Such was the case with seventeenth-century Continental rationalism, whose founder was Rene Descartes and whose new program initiated what is called modern philosophy. In a sense, much of what the Continental rationalists set out to do had already been attempted by the medieval philosophers and by Bacon and Hobbes. But Descartes and Leibniz fashioned a new ideal for philosophy. Influenced by the progress and success of science and mathematics, their new program was an attempt to provide philosophy with the exactness of mathematics. They set out to formulate clear and rational principles that could be organized into a system of truths from which accurate information about the world could be deduced. Their emphasis was upon the rational ability of the human mind, which they now considered the source of truth both about man and about the world. Even though they did not reject the claims of religion, they did consider philosophical reasoning something different than supernatural revelation. They saw little value in feeling and enthusiasm as means for discovering truth, but they did believe that the mind of an individual is structured in such a way that simply by operating according to the appropriate method it can discover the nature of the universe. The rationalists assumed that what they could think clearly with their minds did in fact exist in the world outside their minds. Descartes and Leibniz even argued that certain ideas are innate in the human mind, that, given the proper occasion, experience would cause...
Rationalism and empiricism have always been on opposite sides of the philosophic spectrum, Rene Descartes and David Hume are the best representative of each school of thought. Descartes’ rationalism posits that deduction, reason and thus innate ideas are the only way to get to true knowledge. Empiricism on the other hand, posits that by induction, and sense perception, we may find that there are in fact no innate ideas, but that truths must be carefully observed to be true.
In the Prolegomena, Kant states that reading David Hume, "awakened him from his dogmatic slumber." It was Hume's An Inquiry Concerning Human Understanding that made Kant aware of issues and prejudices in his life that he had previously been unaware of. This further prompted Kant to respond to Hume with his own analysis on the theory of metaphysics. Kant did not feel that Hume dealt with these matters adequately and resolved to pick up where Hume had left off, specifically addressing the question of whether metaphysics as a science is possible.
Rationality from the Latin ‘rationari’ meaning to ‘think’ or ‘calculate’ is a significant concept in Western philosophy born out of the Enlightenment. During the 17th and 18th centuries many philosophers began to emphasise the use of reason as the best method of learning objective truth. Pioneers in this field include Descartes and Locke.
To begin with the question of rationalism versus empiricism, it is important to understand, first, what it is that rationalists argue. This school of thought infers that all knowledge comes from within, an innate source that arrives with us at birth. Rationalists "suggest that only the truths we arrive at through our minds alone can count as knowledge".(White & Rauhut, pg.64) They argue that the conclusions that we arrive at through our senses are not adequate enough to count as legitimate knowledge. Instead, this school of thought maintains that because the world that we experience through our sense is in a state of constant change it can, therefore, not be relied upon in deriving distinct and reliable truths, also known as absolute truths.
It is interesting to note that the ongoing controversy concerning the so-called conflict between Wilhelm Gottfried Leibniz and Isaac Newton is one that does not bare much merit. Whether one came up with the concepts of calculus are insignificant since the outcome was that future generations benefited. However, the logic of their clash does bear merit.
To what degree is a rational agent allowed to pursue his own goals or to choose one action over another? Both Immanuel Kant and John Stuart Mill answer the question of what makes a person free. Two different conceptions of individual freedom and autonomy are present by them and for this reason these philosopher differ on why it is that freedom and self-governance should be valued. In Grounding for the Metaphysics of Morals Kant puts forward a normative conception of freedom and autonomy where by one has the capacity to deliberate and give himself laws. It is based on this claim that he makes his argument that autonomy should be valued because it is the sole principle of our moral law. In On Liberty, Mill propounded that freedom was doing as one pleases, and unlike Kant promoted a personal account of autonomy wherein an individual is encouraged to decide for one’s self one what ever course of action they desired- often regardless of a particular moral. The good consequence of progress was the core reason that Mill felt that one should value this type of autonomy.
...ocity. On the other hand, Leibniz had taken a geometrical approach, basing his discoveries on the work of previous thinkers like Fermat and Pascal. Though Newton had been the first to derive calculus as a mathematical approach, Leibniz was the first one to widely disseminate the concept throughout Europe. This was perhaps the most conclusive evidence that Newton and Leibniz were both independent developers of calculus. Newton’s timeline displays more evidence of inventing calculus because of his refusal to use theories or concepts to prove his answers, while Leibniz furthered other mathematician’s ideas to collaborate and bring together theorems for the application of calculus. The history of calculus developed as a result of sequential events, including many inventions and innovations, which led to forward thinking in the development of the mathematical system.
The basic of mathematics was inherited by the Greeks and independent by the Greeks beg the major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress continued in Islamic countries Unlike the Babylonians, the Egyptians did not develop fully their understanding of mathematics. Instead, they concerned themselves with practical applications of mathematics. Mathematics flourished in particular in Iran, Syria and India from 450B.C. Major progress in mathematics in Europe began again at the beginning of the 16th Century.