[W]e shall be concerned with the general nature of pure mathematics, and how it is distinguished from other sciences. Here there are... two distinct … - Frank P. Ramsey

The first problem I propose to tackle is this: how much of its income should a nation save? To answer this a simple rule is obtained valid under conditions of surprising generality; the rule, which will be further elucidated later, runs as follows. The rate of saving multiplied by the marginal utility of money should always be equal to the amount by which the total net rate of enjoyment of utility falls short of the maximum possible rate of enjoyment.

[T]he formalist school, of whom the most eminent representative is Hilbert, have concentrated on the propositions of mathematics, such as '2 + 2 = 4'. They have pronounced these to be meaningless formulae to be manipulated according to arbitrary rules, and they hold that mathematical knowledge consists in knowing what formulae can be derived from what others consistently with the rules. ...for example...'2' is a meaningless mark occurring in these meaningless formulae. But... '2' occurs not only in '2 + 2 = 4', but also in 'It is 2 miles to the station', which is not a meaningless formulae, but a significant proposition, in which '2' cannot conceivably be a meaningless mark.

Tautologies and contradictions are not real propositions, but degenerate cases. ...Clearly, by negating a contradiction we get a tautology, and by negating a tautology a contradiction. ...A genuine proposition asserts something about reality, and it is true if reality is as it is asserted to be. But a tautology is a symbol constructed so as to say nothing whatever about reality, but to express total ignorance by agreeing with every possibility.