I am reading the book "The Number-System of Algebra (2nd edition)".
At the starting of page-4 the author writes:
A positive integer is a symbol for the number of things in a group of distinct things$^{1}$.
My question is that, is this the definition of a positive integer? If not then what is the definition of a a positive integer in modern Mathematics.
$^1$Things may refer to either abstract objects or concrete objects.
On
As is often the case, Wikipedia is your friend: see Set-theoretic definition of natural numbers.
Briefly: $0$ is defined as the empty set $\emptyset$; and $n+1$ is defined as $n \cup \{n\}$.
On
A positive integer, or natural number is any object you desire, as long as it belongs to a collection of objects that satisfy the Peano Axioms.
Briefly:
When you have that, you have natural numbers, no matter what your underlying set of objects are, be they ZFC's finite ordinals, strings of only one symbol, pebbles in a bowl (barring another axiom of there being a maximal natural number,) or a category of only a single object.
On
You can see RL Goodstein, Recursive Number Theory: A Development of Recursive Arithmetic in a Logic-Free Equation Calculus (1957), for a modern attempt to "revamp" the so-called "formalist" approach to the definition of number [page 1-on] :
The question 'What is the nature of a mathematical entity?' is one which has interested thinkers for over two thousand years and has proved to be very difficult to answer. Even the first and foremost of these entities, the natural number, has the elusiveness of a will-of-the-wisp when wc try to define it.
One of the sources of the difficulty in saying what numbers are is that there is nothing to which we can point in the world around us when we are looking for a definition of number. The number seven, for instance, is not any particular collection of seven objects, since if it were, then no other collection could be said to have seven members; for if we identify the property of being seven with the property of being a particular collection, then being seven is a property which no other collection can have. A more reasonable attempt at defining the number seven would be to say that the property of being seven is the property which all collections of seven objects have in common. The difficulty about this definition, however, is to say just what it is that all collections of seven objects really do have in common (even if we pretend that we can ever become acquainted with all collections of seven objects). Certainly the number of a collection is not a property of it in the sense that the colour of a door is a property of the door, for we can change the colour of a door but we cannot change the number of a collection without changing the collection itself. It makes perfectly good sense to say that a door which was formerly red, and is now green, is the same door, but it is nonsense to say of some collection of seven beads that it is the same collection as a collection of eight beads. If the number of a collection is a property of a collection then it is a defining property of the collection, an essential characteristic.
This, however, brings us no nearer to an answer to our question 'What is it that all collections of seven objects have in common?' A good way of making progress with a question of this kind is to ask ourselves 'How do we know that a collection has seven members?' because the answer to this question should certainly bring to light something which collections of seven objects share in common. An obvious answer is that we find out the number of a collection by counting the collection but this answer does not seem to help us because, when we count a collection, we appear to do no more than 'label' each member of the collection with a number. (Think of a line of soldiers numbering off.) It clearly does not provide a definition of number to say that number is a property of a collection which is found by assigning numbers to the members of the collection.
To label each member of a collection with a number, as we seem to do in counting, is in effect to set up a correspondence between the members of two collections, the objects to be counted and the natural numbers. In counting, for example, a collection of seven objects, we set up a correspondence between the objects counted and the numbers from one to seven. Each object is assigned a unique number and each number (from one to seven) is assigned to some object of the collection. If we say that two collections are similar when each has a unique associate in the other, then counting a collection may be said to determine a collection of numbers similar to the collection counted.
The weakness in the definition lies in this notion of correspondence. How do we know when two elements correspond? The cups and saucers in a collection of cups standing in their saucers have an obvious correspondence, but what is the correspondence between, say, the planets and the Muses? It is no use saying that even if there is no patent correspondence between the planets and the Muses, we can easily establish one, for how do we know this, and, what is more important, what sort of correspondence do we allow? In defining number in terms of similarity we have merely replaced the elusive concept of number by the equally elusive concept of correspondence.
Some mathematicians have attempted to escape the difficulty in defining numbers, by identifying numbers with numerals [i.e. symbols]. The number one is identified with the numeral 1, the number two with the numeral 11, the number three with 111, and so on. But this attempt fails as soon as one perceives that the properties of numerals are not the properties of numbers. Numerals may be blue or red, printed or handwritten, lost and found, but it makes no sense to ascribe these properties to numbers, and, conversely, numbers may be even or odd, prime or composite but these are not properties of numerals.
The antithesis of "number" and "numeral" is one which is common in language, and perhaps its most familiar instance is to be found in the pair of terms "proposition" and "sentence". The sentence is some physical representation of the proposition, but cannot be identified with the proposition since different sentences (in different languages, for instance) may express the same proposition.
The game of chess, as has often been observed, affords an excellent parallel with mathematics (or, for that matter, with language itself). To the numerals correspond the chess pieces, and to the operations of arithmetic, the moves of the game.
Here at last we find the answer to the problem of the nature of numbers. We see, first, that for an understanding of the meaning of numbers we must look to the 'game' which numbers play, that is to arithmetic. The numbers, one, two, three, and so on, are characters in the game of arithmetic, the pieces which play these characters are the numerals and what makes a sign the numeral of a particular number is the part which it plays, or as we may say in a form of words more suitable to the context, what constitute a sign the sign of a particular number are the transformation rules of the sign. It follows, therefore, that the object of oue study is NOT NUMBER ITSELF BUT THE TRANSFORMATION RULES OF THE NUMBER SIGNS.
This is a book from the 19th century. You should not rely on a book of that age for a definition of any entity in modern mathematics. With that definition, as an example, you would have to know how to count the number of things (in a mathematical precise way), which will get you to the next lacking definition.
Today, most of basic mathematics relies on set theory which is introduced axiomatically. You may want to consider, as an example, Halmos Naive Set Thoery as an introduction, where you will also find a definition of the integers.