Gottlob Frege was a German mathematician, logician and philosopher who influenced the shape of 20th century analytical philosophy. The main philosophical problem investigated by him was the question: What is a number? He maintained that the question is a common problem for mathematicians and philosophers. Frege worked out some ways of defining a number. Although Frege’s conception of number has already been described quite well I would like to show his achievement form other point of view, and pay special attention to the following problems:

1. The development of Frege's conceptions of number.

2. The reasons why he kept looking for a better conception.

3.His logical, semantic and philosophical tools used to present the definition of number.

I shall stress the considerable differences between some expressions on number based on differences in:

1. using primitive terms.

2. accepting assumed philosophical background.

I will try to highlight the differences and work on assumption that there are three original and different conceptions of number in his papers:

1. The inductive definition: Begriffsschrift 1879, Anwendung der Begriffsschrift 1879; Die Grundlagen der Arithmetik 1884.

"a" is a positive whole number when “a" belongs to the sequence which begins with 0 and arises from a constant increase by 1 (Anwendung der Begriffsschrift);

the number [Zahl] (n+1) belongs to a concept F, if there is an object a falling under F and such that the number [Zahl] n belongs to the concept “falling under F, but not a” (Die Grundlagen der Arithmetik, § 55).

2. The definition by equinumerosity: Die Grundlagen der Arithmetik 1884; Grundgesetze der Arithmetik, vol. I, II 1893, 1903; Nachwort added to II vo. of Grundgesetze der Arithmetik 1903.

The Number which belongs to the concept F is the extension of the concept “concept equal to the concept F” where a concept F is called equal to a concept G if there exists the possibility of one-one correlation referred to above Die Grundlagen der Arithmetik, § 107).

3. The definition based on geometry (unfinished): Zahlen und Arithmetik 1924/1925; Neuer Versuch der Grundlegung der Arithmetik 1924/1925; Erkenntnisquellen der Mathematik und Naturwissenschaften,1924/1925.

The number that in this way defines the size of angle is the number yielded by measuring the arc of C included by its sides with the radius of C. […] The number 1 defines an angle for which the length b [arc] is equal to r [radius] (Erkenntnisquellen...,1924/1925).