What is the meaning of the symbol 1? Gottlob Frege asked at the beginning of the introduction of this book. We may think this is a very simple question. But Frege doesn't think like this. First of all, symbol 1 is not a thing, for if we select something for ourselves to call one, the same proposition about one, then,would mean different things for different people, for such propositions would have no common content. Secondly, 1 is different with the letter a,because in a+a-a=a, We put for a same number, any We please But the same throughout, We get always a true identity. But in the identity 1+1=2, whatever We put for the First 1, We must put something different for the second, even mathematicians can not answer these questions.But questions like these are the First and foremost among science's objects, and are the foundation of the whole struture of arithmetic. Many people will be sure to think this not worth the trouble, they suppose this concept is adequately dealt with in the elementary textbooks, where the subject is settled one for and for all. But that's not a responsible atitude for the First prerequisite to learn anything else.
At this point, John Rawls ,the author of <A Theory of Justice> would agree with Frege. Rawls used to write at the beginning of <A theory of Justice>: truth is the First virtue of systems of thought, a theory however elegant and economical must be rejected or revised if it is unture, so Trege is trying to solve these foundamental questions of mathematics.
From Trege's perspective, the concept of number have a finer stucture than most of the concepts of the other sciences, even although it is still one of the simplest in arithmetic. In orden, then, to dispel this illusion that the pasitive whole numbers really present no difficulies at all, Frege will criticizing some of the views put forward by mathematicians and philosophers on the questions involved, to clear the ground of his own account and settle the question finally.
First of all, Frege thinks the theory that arithmetic founded on muscular sensations are no concern of arithmetic, because all the phases of consciousness are characteristically fluctuating and indefinite, in strong contrast to the definiteness and fixity of the concepts and objects of mathematics. never let us take a description of the originof An idea for a definition ,or An account of the mental and physical conditions on which We become conscious of a proposition for a proof of it.
In achieving knowledge of a concept in its pure form and in stripping off the irrelevant accretions which veil it from the eyes of the mind. immense intellectual effort have continued over centuries. So for those intellectual properties , We should to advance them rather than despite them.
Secondly, We should enter a little into psychology only to reple its invasion of mathematics. Besides, even mathmatical textbooks at times lapse into psychology.
Besides the refusal of all assistance frome the direction of psychology,We need to recognize mathematics' close connexion with logic. when most mathematicians have satisfied their immediate needs by useing a definition, they often regard the definition as sufficiently established. Yet it must still borne in mind that the rigour of the proof remains An illusion, even though no link be missing in chain of our dedutions, so long as the definitions are justified only as An afterthought by our failling to come across any contradiction, for this reason, Frege will go back into the general logical foundations of science.
At the end of the introduction, Frege gives three fundamental principles that will follow in his enquiry:
1:Always to separate sharply the psychological from the logical, the subjective from the objective,
2:Never to ask for the meaning of a work in isolation, but only in the context of a proposition.
3:Never to lose sight of the distinction between concept and object.
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