Embedded Files
In everyday life, we use context and our general knowledge of the world and of our lives to fill in missing information in what is written or said, and to eliminate the false interpretations that can result from ambiguities. For example, we would need to know something about the context in order to correctly understand the statement
The man saw the woman with a telescope.
The man saw the woman with a telescope.
Who had the telescope, the man or the woman? Ambiguities in newspaper headlines—which are generally written in great haste— can sometimes result in unintended but amusing second readings. Some funny ones that have appeared over the years are:
Sisters reunited after ten years in checkout line at Safeway.
Large hole appears in High Street. City authorities are looking into it.
Mayor says bus passengers should be belted.
To take another example, in the phrase “small rodent” the word “small” means something different (in terms of size) than it does in the phrase “small elephant.” Most people would agree that a small rodent is a small animal, but a small elephant is definitely not a small animal. The size range referred to by the word “small” can vary depending on the entity to which it is applied.
To systematically make the English language precise (by defining exactly what each word is to mean) would be an impossible task. It would also be unnecessary, since people generally do just fine by relying on context and background knowledge. But in mathematics, things are different.
Precision is crucial, and it cannot be assumed that all parties have the same contextual and background knowledge in order to remove ambiguities. Moreover, since mathematical results are regularly used in science and engineering, the cost of miscommunication through an ambiguity can be high, possibly fatal. As a result, when mathematicians use language in doing mathematics, they rely on the literal meaning.
This is why beginning students of mathematics in college are generally offered a crash course in the precise use of language. This may sound like a huge undertaking, given the enormous richness and breadth of everyday language. But the language used in mathematics is so constrained that the task actually turns out to be relatively small. The only thing that makes it difficult is that the student has to learn to eliminate the sloppiness of expression that we are familiar with in everyday life, and instead master a highly constrained, precise (and somewhat stylized) way of writing and speaking
Modern, pure mathematics is primarily concerned with statements about mathematical objects. Mathematical objects are things such as: integers, real numbers, sets, functions, etc. Examples of mathematical statements are:
There are infinitely many prime numbers.
For every real number a, the equation x squared + a = 0 has a real root.
Square root of 2 is irrational
Not only are mathematicians interested in statements of the above kind, they are, above all, interested in knowing which statements are true and which are false. For instance, in the above examples, (1), (3) are true and (2) is false. The truth or falsity in each case is demonstrated not by observation, measurement, or experiment, as in the sciences, but by a proof, about which you can read here
At first, it might seem like a herculean task to make the use of language in mathematics sufficiently precise. Fortunately, it is possible because of the special, highly restricted nature of mathematical statements. Every key statement of mathematics (the axioms, conjectures, hypotheses, and theorems) is a positive or negative version of one of four linguistic forms: (1) Object a has property P (2) Every object of type T has property P (3) There is an object of type T having property P (4) If STATEMENT A, then STATEMENTB
Report abuse