Mathematics Proof Philosophy
Mathematics have the concept of rigorous proof, which leads to knowing something with complete certainty. Consider the extent to which complete certainty might be achievable in mathematics and at least one other area of knowledge.
Mathematics has played a significant part in the development of past and present civilizations. It is likely that the Egyptian Pyramids and construction on the Aztec and Mayan sites in South America were build by mathematically sophisticated architects. Similarly, cultures in China, India and throughout the Middle East developed mathematics a very long time ago. It is also the case that there have been successful cultures, example Rome, Australian Aborigines which have had little need for mathematical methods.
Mathematics has developed two distinct branches. Pure mathematics, which is studied for its own sake and applied mathematics which is studied for its usefulness. “Proof” has a very special meaning in mathematics. We use the word generally to mean “proof beyond reasonable doubt” in situations such as law court when we accept some doubt in a verdict. For mathematics, “proof” is an argument that has NO doubt at all. When a new proof is published, it is scrutinised and criticized by other mathematicians and is accepted when it is established that every step in the argument is legitimate. Only when this has happened a “proof” becomes accepted.
Textbooks of mathematical logic say that a “proof” is a sequence of statements, each of which either follows from previous statements in the sequence or from agreed conjectures unproved but explicitly stated assumptions that define the area of mathematics. Achieving complete certainty in mathematics is possible but is limited by its own theories.
Mathematics is often described as the queen of the sciences. I would guess that mathematics arose long ago when two cave women couldn't visualize the intuitive concept of “many”. One cave women said, “let's pick this tree over here because it has more bananas.” Without a number system to visualize their concept of “many”, the two cave women could never prove to each other which tree offered better pickings.
We, as human beings, are faced with questions every day. Then often pertain to our most basic need to understand ourselves which mathematics attempts to do. We know that certain drugs can alter our mood. We also have calculated and mapped out those billions of individual cells making up any organism in order to solve life's puzzles. The fundamental question of how those billions of cells with no consciousness of their own are able to form one cohesive personality still eludes an answer. The reason for mathematics inability to explain how our minds work lies in its underlying philosophy, which is to study and define the unknown in understandable terms.
While Mathematics is, of course, the homeland of geometrical explanations, it turns out that intuitive understanding even has a role to play in this discipline. For as Pascal observes, while we may use deduction to reason from axioms to theorems, the axioms themselves are not subject to proof. All proof must end some way, and ends with first principles that we can justify only on the basis of intuitive understanding. Furthermore, even when it comes to proving theorems, the French mathematician Henri Poincare (1854-1912) claimed that great mathematicians are guided as much by the intuition of beauty as by mechanical calculation:
“Mathematical work is not a simple mechanical work, and it could not be entrusted to any machine, wherever the degree of perfection we suppose it to have been brought too. It is not merely a question of applying certain rules, of manufacturing as many combinations as possible according to certain fixed laws. The combinations so obtained would be extremely numerous, useless and encumbering. The real work of the discoverer consists in choosing between these combinations with a view to eliminating those that are useless, or rather not giving himself the trouble of making them at all. The useful combinations are precisely the most beautiful, I mean those that can most charm that special sensibility that all mathematicians know, but of which Laymen are so ignorant”.
At a more specific level, Roger Penrose, in his book The Emperor's New Mind, contends that one of the implications of Godel's Incompleteness Theorem is that the way in which we decide whether or not certain mathematical statements are true is intuitive rather than rule governed. Be that as it may, we have all, at some level, had the “ah-ha” experienced of suddenly being able to see how to solve a mathematical problem and this is surely a faint echo of the higher level intuitions of the great mathematicians.
History is a good example of a subject area in which intuitive understanding rather than geometrical explanation is the appropriate intellectual mode. Since history trades in concepts that resist precise mathematical specification, it cannot be reduced to a measurable science. For example, an important factor in determining the outcome of a battle might be the open “morale” of the troops; but “morale” is an inherently vague, qualitative term; and while you may be able to measure the troops' height or weight, you cannot measure their morale. Certainly, statistics have a role to play in history; but any attempt to reduce historical explanations to numbers can only result in absurdity, as the following example from the American historian Barbara Tuchman shows:
“In a quantification study of the origins of World War I which I have seen, the operatives have divided all the diplomatic documents, messages, and utterances of the July crisis into categories labelled “hostility”, “friendship”, “frustration”, “satisfaction” and so on, with each statement rated for intensity on a scale from 1 to 9, including fractions. But no pre-established categories could match all the private character traits and public pressures variously operating on the nervous monarchs and ministers who were involved. The massive effort that went into this study brought forth a mouse-the less than startling conclusion that the likely hood of war increases in proportion to the rise in hostility of the messages.”
The motivation behind such quantification studies is to find repeatable patterns in history; but this remains an empty drain. For history is by its very nature concerned, not with the abstract and general, but with the concrete and particular. Moreover, the meaning and significance of particular events cannot be understood in isolation, but only in the overall context in which they take place. Since historical events are unique and unrepeatable, the most for which we can hope when we enquire into, say, the cause of revolution or the consequences of neutrality, are similarities rather than identities, and tendencies rather than laws. Given that history is a matter of intuitive understanding and judgement rather than geometrical explanations and proof, it is not surprising that historians rarely achieve consensus. However, this does not exclude history as a discipline, but simply reflects the nature of the subject matter with which it deals.
If the historian is limited to what he can observe, what will he report when, for example, he sees that you don't vote (is it because of laziness? or disgust? or rebelliousness? or a bribe?) or when you stand still on Armistice Day? When the mathematician propounds a theory to explain phenomena, he introduces precise rules that coordinate these unobservable rules with something that can be observed. The historian does not know what motive to coordinate with your not voting; he must refer to his own motives in order to formulate the conditions under which such events occur. When you bomb your enemy in war time, do you predict his submission because you empathize with the terror, or do you predict his resistance because you empathize with the challenge? or with Hitler? or with believers in witch craft?
In Mathematics, phenomena may be studied without regard to their past (Third Degree Polynomials is just what it is), whereas human beings and societies are only what they have come to be. This is a problem for the social sciences (history) which may find their predictions falsified because of unobservable and unverifiable past histories. Only a burnt child still dreads a fire. Living creatures have memories, dispositions, and expectations. Behaviour is altered by habits and conditioning. Thus a person's past history influences his present reactions. Rocks do not remember. But this constraint does not preclude the search for generalisations about behavioural phenomena (for example, one might investigate whether all burned children dread fire equally) and in mathematics the influence of the past is not always irrelevant (Fermat's Last Theorem, subsequently proved by Andrew Wiles). Everything is what it has come to be… In evolution every living species is what it is as the result of a long history of natural selection; but only the history which is incorporated into its present structure is of any scientific significance.
Bibliography
Damasio, Antonio B. Descartes Error: Emotion, Reason and the Human Brain
Pinker, Steven. How the Mind Works
Lagemaat, Richard. Theory of Knowledge
Krytatas, Vassilis. Theory of Knowledge
Davis, Philip and Reuben Hersh. Mathematical Experience
Vicki Strid, Mathematical Studies, Fabio Cirrito, published 1998
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