Week 13: Russell, Excerpt from Introduction to Mathematical Philosophy

 

DQ1. Identify Russell's comments about the difference between a number and a plurality. Do you agree? Can you think of a way that someone might argue against him?

 

DQ2. Russell gives the following expression of the logicist approach to mathematics (pg.173):

"So much of modern mathematical work is obviously on the border-line of logic, so much of modern logic is symbolic and formal, that the very close relationship of logic and mathematics has become obvious to every instructed student. The proof of their identity is, of course, a matter of detail: 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. If there are still those who do not admit the identity of logic and mathematics, we may challenge them to indicate at what point, in the successive definitions and deductions of Principia Mathematica, they consider that logic ends and mathematics begins. It will then be obvious that any answer must be quite arbitrary."
  1. Why does it matter (if it does at all) matter whether mathematics 'is just' logic, from the perspective of a philosopher? From the perspective of a mathematician?
  2. What is the logicist definition of a number?
  3. Suppose Russell is right. Is it surprising that most people claim to 'know' that 2+2=4, without knowing the logicist definition of these numbers?

 

(Optional) DQ3. Why should it matter to Russell that the propositions of logic are analytic? Is it possible for him (or anyone) to establish that logical propositions are analytic? How?