In logic, the question of the relation between mathematics and logic is one of the most crucial questions. If we conceive of logic as the structure of correct reasoning whereby truth can be inferentially related through a formal series of premises that flow into a conclusion filling it with life, truth, and necessity, and we also see the same sort of occurrence in mathematical modes of calculation, then we might well ask whether or not logic a species of mathematics or vice versa? And while this question is hardly new, the impact of how we answer this question is hardly to be ignored. For if logic is a species of calculation then we might well say that the highest philosophy is nothing other than abstract mathematics, that Pythagoras is the true innovator and not Plato. On the other hand, if mathematics is a subset of logic, as it were, then we might well affirm with Plato that mathematics is simply the first step towards enlightenment and not the last. So the question of course boils down to the historical distinction between Thomas Hobbes who claimed the former and Frege who argued for the latter. Ironically it was Frege, who in saying that all arithmetic can be deduced from logical concepts who needed Cantor’s insight of the continuum hypothesis. As they say, one sits upon the shoulders of giants: and just as Plato sat atop Pythagors, so too Frege peers from atop the great work of a mathematician. So what does Frege say, but “It is only recently that infinite numbers have been introduced in a remarkable work by G. Cantor. I heartily share his contempt for the view that in principle only finite Numbers ought to be admitted as actual. Perceptible by the senses they are not, nor are they spatial – any more than fractions are, or negative numbers, or irrational or complex numbers; and if we restrict the actual to what acts on our senses or at least produces effects which may cause sense-perceptions as near or remote consequences, then naturally no number of this kind is actual… For us, because our concept of NUmber has from the outset covered infinite numbers as well, no extension of its meaning has been necessary at all.” (p. 443 The Development of Logic) There is certainly more to say here. But we can at least begin to see the ways in which Frege felt that Cantor was correct in his postulation of cardinality. Cantor’s set theory eventually set the groundwork for Frege to make the great innovation of his theory of number in which numbers can be defined in terms of cardinal sets. For an excellent video to introduce you to both Cantor and Frege please see:
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