Why is it that America has an unequalled excellence in literature and yet Americans are advised to study maths? Is there anywhere in the world literature as inspiring as the Nebraskan novels of Willa Cather, the Tennessee novels of Mary Noailles Murfee or the Sonoran novels of Conrad Richter or Carlos Castaneda? Is there any literature in the owrld which describes and extolls the harmony of humans with nature and with each than these texts? Surely in these times of ecolological destruction and human violence we need to read this literature as a priority.
It is clear that all over the Western world the most students who obtain mediocre standards are those who are compelled to study mathematics. In each Western country the number of students who fail or barely pass last year high school maths is over 50%. What is wrong here? After six years of intensive study students sometimes fail to answer on question correctly.
The "success" of maths when applied to the world appears to prove mathematics. But this is wrong. Despite what leading physicists and mathematicians say mathematics is imposed on the world and can be "proved" only by means within mathematics. In the following few examples I will demonstrate that this is most important to concede if we are to understand maths fully.
1. The irrationality of root 2.
This problem has interested mathematicians since it was discovered by Pythagoras who lectured to farm animals. It is often used to show students the "power" of proof, there are even you tube presentations. But the proof is incorrect. In the classical proof the reductio is achieved by by alluding to the "evenness" of 2. But a close analysis of the proof shows that if it were true it should be extendable to other even numbers. But, of course, it can't (for example, the root of 4 is not irrational). To make the proof correct and generalizable it must be noted that root 2 is irrational because it is prime. The the proof can be extended to other primes. Here it must be conceded that a precise proof can only be made after a careful analysis within maths.
2. Achilles and the turtle.
When Zeno proposed this proof he was laughing. He knew that when scholars attempt to apply maths to the world they get into a horrible mess. To understand the paradox we must state precisely the initial conditions, this is the most important stage of any mathematical or scientific proof. But these initial conditions are somewhat bizarre. On a football field Achilles starts at the end zone and must catch the turtle who starts at the 50 yard mark. After each unit of time both Achilles and the turtle can only run half as far as they did in the previous unit of time (they get tired at the same rate, of course). Also the turtle and Achilles must get smaller at each time (to avoid Heisen berg Uncertainty problems). Then achilles can never catch the turtle and neither can make it to the other end zone. But the exact proof of this is obtained by using the limit theorems of convegent sequences, not by thinking about the competitors.
3. The Prime Numbers
All high school students know that no-one (even the most powerful computer) can predict the next prime number. It seems that the prime number sequence is anarchic. But what few know is that the argument can be extended to all non narchic sequences. Even the sequence 1,2,3,4,.... No one can "predict" the next number in this sequence. You know what it is (5) but only because you know the sequence and it's rule. Like all other non-anarchic sequences it is an imposed rule. No one can predict the next number because if they could they would have the means to precict the next prime, which they do not have.
4. What are numbers?
The previous example underlines the importance of knowing what numbers are. According to mathematicians a natural number has the essential property of being made up of a unique factorixation of primes. Perhaps this is an "Aristotelian" essential property. If it is it is essential as a metaphysical construct, it is the property without which the number would not exist. This problem of the nature of numbers has been studied by Badiou. Interestingly, he uses set theory inorder to overcome the "schoolboy" notions of numbers as the ordinal sequence (which as the previous example shows is imposed). But his argument seems incorrect. When he distiguishes an element of a set and a part of a set he equivocates to such an extent that it can't be accepted that is "part" and "his "element" can be one and the same thing (see chapter7 of Number). But Badiou is onto something in using set theory. Our "democracy" is based on the overturning of nearly every axiom of set theory. From the axiom that voters should be clearly "distinguishable" to the axiom of choice where the voting should not be reducible to other constructs (such as geographical position or self-interested sub-group.
5. The free rider problem
Professor Fodor of Rutgers has used the free rider problem to expose the logical problems within the scinece of evolution. Biologists who choose to focus on the importance of a property in reproductive success fail to see that the property is always co-extensive with other properties no so focysed on. When this happens it leads to the imposition of a logical order on the natural world which amounts only to tautological definitions whose only validity is semantic inter-definability. But the free rider problem is not exclusive to biology, it occurs everywhere in math ( the irrationality of root 2 is a good example) and in physical science.
6. The BInomial coefficent.
Picking a basketball team form a group o players can be done in many different ways. But we never end up with a fraction of a person. This is often given as a "proof" that the coefficient is always an integer. But the real proof can only be mathematical. It concens the presence of k consecutive numbers in the numerator and k factorial in the denominator. It is purely a mathematical proof.
Many other examples can demonstrate that if math is studied we must not see it as something that gathers its validity form the world. Infact the application of maths often turns out to be very destructive.