In Kant’s time and in the nineteenth century the tremendous difficulties involved in this problem could not be clearly seen, and the older thinkers arrived at their results in rather a superficial manner, but when during the last two or three decades the problem of the nature of mathematical propositions presented itself to the mathematicians within their own territory and called for a definite solution for the sake of pure mathematics alone, it was taken up again by logicians empirically-minded philosophers, and mathematicians, under the leadership of the most subtle and critical of the last.
The work of all these thinkers has not yet been entirely completed — a few little gaps still have to be filled out — but there is not the slightest doubt any more what the final solution looks like. It can be expressed briefly in the following way: mathematical and logical propositions are so entirely different in nature from ordinary empirical propositions that perhaps it is even unwise to call them both by the same name. It is so difficult to realize this difference, and it has never been clearly seen before, because both are expressed in language by sentences of the same form. When I say: “The latitude of San Francisco is 38°,” and when I say: “The fifth power of 2 is 32,” both sentences not only seem to have a similar grammatical construction, but they also seem to impart some real information. The one speaks of a certain city in California in a similar way as the other seems to speak about certain numbers: how can there be any intrinsic difference between them?
Strict logical analysis has shown that there is the greatest imaginable difference. It is simply this, that empirical propositions really deal with something in the universe, they communicate actual facts, while mathematical propositions do not deal with anything real. “Numbers” cannot be found anywhere in the real world in the same way in which San Francisco can be found there. It is also not true that numbers are simply imaginations of the human mind, like dragons or angels, and it is not true that, together with other “unreal” objects, they belong, to a world of their own, which Plato called the realm of Ideas, and in which many present day philosophers also believe, giving different names to it. The fact is that numbers or other logical entities cannot be regarded as “objects” in any sense at all. The propositions apparently dealing with them do not communicate any “facts” about them and therefore are no proper “propositions” at all. What are they?
