The peculiar service which was rendered at this juncture by the ` Cambridge School' was that, instead of opposing a mere dogmatic opposition to the Tubingen critics, they met them frankly on their own ground; and instead of arguing that their conclusions ought not to be and could not be true, they simply proved that their facts and their premisses were wrong.
With these definitions it is now possible to prove the following six premisses applying to finite cardinal numbers, from which Peano 2 has shown that all arithmetic can be deduced i.
The proof of the six premisses requires an elaborate investigation into the general properties of classes and relations which can be deduced by the strictest reasoning from our ultimate logical principles.
They are excellent principles of the highest value, but they are in no sense the necessary premisses which must be proved before any other propositions of cardinal numbers can be established.
But on the assumption that "mathematics" is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ "mathematics" in the general sense' of the "science concerned with the logical deduction of consequences from the general premisses of all reasoning."