INTRODUCTION  

§1. Recent work in mathematics has shown a tendency towards rigour of proof and sharp definition of concepts.  
§2. This critical examination must ultimately extend to the concept of number itself. The aim of proof.  
§3. Philosophical motives for such an enquiry: the controversies as to whether the laws of arithmetic are analytic or synthetic, a priori or a poteriori. Sense of these expressions.  
§4. Task of the present work.  
I.  VIEWS of CERTAIN WRITERS on the NATURE of ARITHMETICAL PROPOSITIONS  
Are numerical formulae provable?  §5.  
§6.  
§7.  
§8.  
Are the laws of arithmetic inductive truths?  §9.  
§10.  
§11.  
Are the laws of arithmetic synthetic a priori or analytic?  §12.  
§13.  
§14.  
§15.  
§16.  
§17.  
II.  VIEWS of CERTAIN WRITERS on the CONCEPT of NUMBER  
§18.  
§19.  
§20.  
Is number a property of external things?  §21.  
§22.  
§23.  
§24.  
§25.  
Is number something subjective?  §26.  
§27.  
Numbers as sets.  §28.  
III.  VIEWS on UNITY and ONE  
Does the number word "one" express a property of objects?  §.29  
§30.  
§31.  
§32.  
§33.  
Are units identical with one another?  §34.  
§35.  
§36.  
§37.  
§38.  
§39.  
Attempts to overcom the difficulty.  §40.  
§41.  
§42.  
§43.  
§44.  
Solution of the difficulty.  §45.  
§46.  
§47.  
§48.  
§49.  
§50.  
§51.  
§52.  
§53.  
§54.  
IV.  The CONCEPT of NUMBER  
Any individual number is a selfsubsistent object.  §55.  
§56.  
§57.  
§58.  
§59.  
§60.  
§61.  
To obtain the concept of number, we must fix the sense of a numerical identity.  §62.  
§63.  
§64.  
§65.  
§66.  
§67.  
§68.  
§69.  
One definition completed and its worth proved.  §70.  
§71.  
§72.  
§73.  
§74.  
§75.  
§76.  
§77.  
§78.  
§79.  
§80.  
§81.  
§82.  
§83.  
Infinite numbers.  §84.  
§85.  
§86.  
V.  CONCLUSIONS  
§87.  
§88.  
§89.  
§90.  
§91.  
Other numbers.  §92.  
§93.  
§94.  
§95.  
§96.  
§97.  
§98.  
§99.  
§100.  
§101.  
§102.  
§103.  
§104.  
§105.  
§1069. 
"In the enquiry that follows, I have kept to three fundamental principles:
