Broadly speaking, a set is a collection of objects, and in mathematical discourse these objects are mathematical ones such as numbers, points in space or other sets. If we wish to rewrite sentence(3) symbolically, another way to do it is to define P to be the collection, or set, of all prime numbers.
Then (3) can be rewritten, “5 belongs to the set P”.
This notion of belonging to a set is sufficiently basic to deserve its own symbol, and the symbol used is 2. So a fully symbolic way of writing the sentence is 5 2 P.
The members of a set are usually called its elements, and the symbol 2 is usually read “is an element of”. So the “is” of sentence (3) is more like 2 than =. Although one cannot directly substitute the phrase “is an element of” for “is”, one can do so if one is prepared to modify the rest of the sentence a little.
There are three common ways to denote a specific set. One is to list its elements inside curly brackets: {2, 3, 5, 7, 11, 13, 17, 19}, for example, is the set whose elements are the eight numbers 2, 3, 5, 7, 11, 13, 17 and 19.
The majority of sets considered by mathematicians are too large for this to be feasible - indeed, they are often infinite - so a second way to denote sets is to use dots to imply a list that is too long to write down: for example, the expressions {1, 2, 3, . . . , 100} and {2, 4, 6, 8, . . . } represent the set of all positive integers up to 100 and the set of all positive even numbers respectively.
A third way, and the way that is most important, is to define a set via a property: an example that shows how this is done is the expression {x : x is prime and x < 20}. To read an expression such as this, one first says, “The set of”, because of the curly brackets. Next, one reads the symbol that occurs before the colon. The colon itself one reads as “such that”. Finally, one reads
what comes after the colon, which is the property that determines the elements of the set. In this instance, we end up saying, “The set of x such that x is prime and x is less than 20,” which is in fact equal to the set {2, 3, 5, 7, 11, 13, 17, 19} considered earlier.
Many sentences of mathematics can be rewritten in set-theoretic terms. For example, sentence (2) earlier could be written as 5 2 {n : n < 10}. Often there is no point in doing this - as here where it is much easier to write 5 < 10 - but there are circumstances where it becomes extremely convenient.
For example, one of the great advances in mathematics was the use of Cartesian coordinates to translate geometry into algebra, [Cross references, such as Kollar and my remarks on geometry later in this section.] and the way this was done was to define geometrical objects as sets of points, where points were themselves defined as pairs or triples of numbers.
So, for example, the set {(x, y) : x2 + y2 = 1} is (or represents) a circle of radius 1 about the origin (0, 0). That is because, by Pythagoras’s theorem [CR?], the distance from (0, 0) to (x, y) is p x2 + y2, so the sentence “x2 + y2 = 1” can be reexpressed geometrically as “the distance from (0, 0) to (x, y) is 1”.
if all we ever cared about was which points were in the circle, then we could make do with sentences such as “x2 + y2 = 1”, but in geometry one commonly wants to consider the entire circle as a single object (rather than as a multiplicity of points, or as a property that points might have), and then set-theoretic language is indispensable.
A second circumstance where it is hard to do without sets is when one is defining new mathematical objects - unless they are exceedingly simple. Very often such an object is a set together with a mathematical structure imposed on it, which takes the form of certain relationships amongst the elements of the set. For examples of this use of settheoretic
language, see the later sections on number systems and algebraic structures. Sets are also very useful if one is trying to do metamathematics, that is, to prove statements not about mathematical objects but about the process of mathematical reasoning itself.
For this it helps a lot if one can devise a very simple language - with a small vocabulary and an uncomplicated grammar - into which it is in principle possible to translate all mathematical arguments. Sets allow one to reduce greatly the number of parts of speech that one needs, turning almost all of them into nouns. For example, with the help of the membership symbol 2 one can do without adjectives, as the translation of “5 is a prime number” (where “prime” functions as an adjective) into “5 2 P”has already suggested.
This is of course an artificial process - imagine replacing “roses are red” by “roses belong to the set R” - but in this context it is not important for the formal language to be natural and easy to understand. [Cross-reference to Ellenberg’s article in Section IV for further discussion of adjectives.]
Saturday, July 10, 2010
Four basic concepts
Another word, which famously has three quite distinct meanings, is “is”. The three meanings are illustrated in the following three sentences.
(1) 5 is the square root of 25.
(2) 5 is less than 10.
(3) 5 is a prime number.
In the first of these sentences, “is” could be replaced by “equals”: it says that two objects, 5 and the square root of 25, are in fact one and the same object, just as it does in the English sentence “London is the capital of the United Kingdom.”
In the second sentence, “is” plays a completely different role. The words “less than 10” form an adjectival phrase, specifying a property that numbers may or may not have, and “is” in this sentence is like “is” in the English sentence “grass is green.” As for the third sentence, the word “is” there means “is an example of”, as it does in the English sentence “Mercury is a planet.”
These differences are reflected in the fact that the sentences cease to resemble each other when they are written in a more symbolic way. An obvious way to write :
(1) is 5 = p25. As for
(2) it would usually be written 5 < 10, where the symbol < means “is less than”.
The third sentence would normally not be written symbolically because the concept of a prime number is not quite basic enough to have universally recognised symbols associated with it. However, it is sometimes useful to do so, and then one must invent a suitable symbol. One way to do it would be to adopt the convention that if n is a positive integer, then P(n) stands for the sentence “n is prime”.
Another way, which doesn’t hide the word “is”, is to use the language of sets.
(1) 5 is the square root of 25.
(2) 5 is less than 10.
(3) 5 is a prime number.
In the first of these sentences, “is” could be replaced by “equals”: it says that two objects, 5 and the square root of 25, are in fact one and the same object, just as it does in the English sentence “London is the capital of the United Kingdom.”
In the second sentence, “is” plays a completely different role. The words “less than 10” form an adjectival phrase, specifying a property that numbers may or may not have, and “is” in this sentence is like “is” in the English sentence “grass is green.” As for the third sentence, the word “is” there means “is an example of”, as it does in the English sentence “Mercury is a planet.”
These differences are reflected in the fact that the sentences cease to resemble each other when they are written in a more symbolic way. An obvious way to write :
(1) is 5 = p25. As for
(2) it would usually be written 5 < 10, where the symbol < means “is less than”.
The third sentence would normally not be written symbolically because the concept of a prime number is not quite basic enough to have universally recognised symbols associated with it. However, it is sometimes useful to do so, and then one must invent a suitable symbol. One way to do it would be to adopt the convention that if n is a positive integer, then P(n) stands for the sentence “n is prime”.
Another way, which doesn’t hide the word “is”, is to use the language of sets.
Tuesday, July 6, 2010
The language and grammar of mathematics
It is a remarkable phenomenon that children can learn to speak without ever being consciously aware of the sophisticated grammar they are using. Indeed, adults too can live a perfectly satisfactory life without ever thinking about ideas such as parts of speech, subjects, predicates or subordinate clauses.
Both children and adults can easily recognise ungrammatical sentences, at least if the mistake is not too subtle, and to do this it is not necessary to be able to explain the rules that have been violated. Nevertheless, there is no doubt that one’s understanding of language is hugely enhanced by a knowledge of basic grammar - it is almost tautologous to say so - and this understanding is essential for anybody who wants to do more with language than use it unreflectingly as a means to a non-linguistic end.
The same is true of mathematical language. Up to a point, one can do and speak mathematics without knowing how to classify the different sorts of words one is using, but many of the sentences of advanced mathematics have a complicated structure that is much easier to understand if one knows a few basic terms of mathematical grammar.
The object of this section is to explain the most important mathematical “parts of speech”, some of which are similar to those of natural languages and others quite different. These are normally taught right at the beginning of a university course in mathematics. Much of the Companion can be understood without a precise knowledge of mathematical grammar, but a careful reading of this section will help the reader who wishes to follow some of the more advanced parts of the book.
The main reason for the importance of mathematical grammar is that the statements of mathematics are supposed to be precise, and it is not possible to achieve a high level of precision unless the language one uses is free of many of the vaguenesses and ambiguities of ordinary speech. Mathematical sentences can also be highly complex: if the parts that made them up were not clear and simple, then the unclarities would rapidly propagate and multiply and render the sentences unintelligible.
To illustrate the sort of clarity and simplicity that is needed in mathematical discourse, let us consider the famous mathematical sentence “Two plus two equals four” as a sentence of English rather than of mathematics, and try to analyse it grammatically.
On the face of it, it contains three nouns (“two”, “two” and “four”), a ver(“equals”) and a conjunction (“plus”). However, looking more carefully we may begin to notice some oddities. For example, although the word “plus” resembles the word “and”, the paradigm example of a conjunction, it doesn’t behave in quite the same way, as is shown by the sentence, “Mary and Peter love Paris”.
The verb in this sentence, “love”, is plural, whereas the verb in the previous sentence, “equals” was singular. So the word “plus” seems to take two objects(which happen to be numbers) and produce out of them a new, single object, while “and” conjoins “Mary” and “Peter” in a looser way, leaving them as distinct people.
Reflecting on the word “and” a bit more, one finds that it has two very different uses. One, as above, is to link two nouns whereas the other is to join two whole sentences together, as in “Mary likes Paris and Peter likes New York”. If we want the basics of our language to be absolutely clear then it will be important to be aware of this distinction. (When mathematicians are at their most formal, they simply outlaw the noun-linking use of “and” - a sentence such as “3 and 5 are prime numbers” is then paraphrased as “3 is a prime number and 5 is a prime number”.)
This is but one of many similar questions: anybody who has tried to classify all words into the standard eight parts of speech will know that the classification is hopelessly inadequate. What, for example, is the role of the word “six” in the sentence, “This section has six subsections”? Unlike, “two” and “four” earlier, it is certainly not a noun.
Since it modifies the noun “subsection” it is somewhat adjectival, but it does not behave like an ordinary adjective: the sentences “My car is red” and “Look at that tall building” are perfectly grammatical, whereas the sentences “My car is six” and “Look at that six building” are not just nonsense but ungrammatical nonsense.
So do we invent a new part of speech called a “numeral”? Perhaps we do, but then our troubles will only just be beginning:
the more one tries to refine the classification of English words, the more one realizes just how great is the variety of different ways we use them.
Both children and adults can easily recognise ungrammatical sentences, at least if the mistake is not too subtle, and to do this it is not necessary to be able to explain the rules that have been violated. Nevertheless, there is no doubt that one’s understanding of language is hugely enhanced by a knowledge of basic grammar - it is almost tautologous to say so - and this understanding is essential for anybody who wants to do more with language than use it unreflectingly as a means to a non-linguistic end.
The same is true of mathematical language. Up to a point, one can do and speak mathematics without knowing how to classify the different sorts of words one is using, but many of the sentences of advanced mathematics have a complicated structure that is much easier to understand if one knows a few basic terms of mathematical grammar.
The object of this section is to explain the most important mathematical “parts of speech”, some of which are similar to those of natural languages and others quite different. These are normally taught right at the beginning of a university course in mathematics. Much of the Companion can be understood without a precise knowledge of mathematical grammar, but a careful reading of this section will help the reader who wishes to follow some of the more advanced parts of the book.
The main reason for the importance of mathematical grammar is that the statements of mathematics are supposed to be precise, and it is not possible to achieve a high level of precision unless the language one uses is free of many of the vaguenesses and ambiguities of ordinary speech. Mathematical sentences can also be highly complex: if the parts that made them up were not clear and simple, then the unclarities would rapidly propagate and multiply and render the sentences unintelligible.
To illustrate the sort of clarity and simplicity that is needed in mathematical discourse, let us consider the famous mathematical sentence “Two plus two equals four” as a sentence of English rather than of mathematics, and try to analyse it grammatically.
On the face of it, it contains three nouns (“two”, “two” and “four”), a ver(“equals”) and a conjunction (“plus”). However, looking more carefully we may begin to notice some oddities. For example, although the word “plus” resembles the word “and”, the paradigm example of a conjunction, it doesn’t behave in quite the same way, as is shown by the sentence, “Mary and Peter love Paris”.
The verb in this sentence, “love”, is plural, whereas the verb in the previous sentence, “equals” was singular. So the word “plus” seems to take two objects(which happen to be numbers) and produce out of them a new, single object, while “and” conjoins “Mary” and “Peter” in a looser way, leaving them as distinct people.
Reflecting on the word “and” a bit more, one finds that it has two very different uses. One, as above, is to link two nouns whereas the other is to join two whole sentences together, as in “Mary likes Paris and Peter likes New York”. If we want the basics of our language to be absolutely clear then it will be important to be aware of this distinction. (When mathematicians are at their most formal, they simply outlaw the noun-linking use of “and” - a sentence such as “3 and 5 are prime numbers” is then paraphrased as “3 is a prime number and 5 is a prime number”.)
This is but one of many similar questions: anybody who has tried to classify all words into the standard eight parts of speech will know that the classification is hopelessly inadequate. What, for example, is the role of the word “six” in the sentence, “This section has six subsections”? Unlike, “two” and “four” earlier, it is certainly not a noun.
Since it modifies the noun “subsection” it is somewhat adjectival, but it does not behave like an ordinary adjective: the sentences “My car is red” and “Look at that tall building” are perfectly grammatical, whereas the sentences “My car is six” and “Look at that six building” are not just nonsense but ungrammatical nonsense.
So do we invent a new part of speech called a “numeral”? Perhaps we do, but then our troubles will only just be beginning:
the more one tries to refine the classification of English words, the more one realizes just how great is the variety of different ways we use them.
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