What is logicism?

Bertrand Russell, answered from the texts and cited to the page.

Bertrand Russell answers

Logicism is the thesis that mathematics and logic are not two subjects but one. Historically they appeared entirely distinct — mathematics connected with science and calculation, logic with the Greek tradition of the syllogism — but both have developed in modern times until it has become wholly impossible to draw a line between them.¹ The proof of identity is not a slogan but a matter of detail.

Starting from premisses which any instructed student would admit belong to logic, and proceeding by strict deduction, one arrives at results which as obviously belong to mathematics; and there is no point in that progression at which a sharp line can be drawn, with logic to the left and mathematics to the right.² Any answer to the question "where does logic end and mathematics begin?" must be quite arbitrary, and I have invited those who doubt this to indicate the point in the successive definitions and deductions of Principia Mathematica — the work Whitehead and I published in three volumes beginning in 1910 — where they would draw it.

The Kantian picture is the main obstacle to seeing this clearly. Kant held that mathematical reasoning always requires an appeal to intuition, something beyond strict logic, because the geometers of his day could not prove their theorems by unaided argument and had to appeal to the figure.³ What that observation actually showed, though not what Kant concluded from it, is that the mathematical proofs of his day were simply unsound — defective as logical arguments.⁴

Once the proofs are made rigorous, the supposed peculiarity of mathematical reasoning disappears entirely.⁵ The positive consequence is this: what can be known by mathematical methods is precisely what can be deduced from pure logic, and what cannot be so deduced cannot be known a priori at all.⁶ All mathematics, in short, is deducible from the primitive propositions of formal logic; these being admitted, no further assumptions are required.⁷

Sources

1. Mathematics and logic, historically speaking, have been entirely distinct studies... But both have developed in modern times: logic has become more mathematical and mathematics has become more logical. The consequence is that it has now become wholly impossible to draw a line between the two; in fact, the two are one.

   Introduction to Mathematical Philosophy

2. starting with premisses which would be universally admitted to belong to logic, and arriving by deduction at results which as obviously belong to mathematics, we find that there is no point at which a sharp line can be drawn, with logic to the left and mathematics to the right.

   Introduction to Mathematical Philosophy

3. Kant, having observed that the geometers of his day could not prove their theorems by unaided argument, but required an appeal to the figure, invented a theory of mathematical reasoning according to which the inference is never strictly logical, but always requires the support of what is called 'intuition.'

   Introduction to Mathematical Philosophy

4. the fact is, that the whole difference lay in the fact that mathematical proofs were simply unsound.

   The Principles of Mathematics

5. Thus the supposed peculiarity of mathematical reasoning has disappeared.

   The Principles of Mathematics

6. What can be known, in mathematics and by mathematical methods, is what can be deduced from pure logic. What else is to belong to human knowledge must be ascertained otherwise—empirically, through the senses or through experience in some form, but not a priori.

   Introduction to Mathematical Philosophy

7. All mathematics, we may say — and in proof of our assertion we have the actual development of the subject — is deducible from the primitive propositions of formal logic: these being admitted, no further assumptions are required.

   The Principles of Mathematics