Within the foundations of mathematics it is traditional to see the irreconcilability of realism and constructivism as emerging, at least in its starkest form, at the level of the classical continuum or the real line. However I argue here that there is a sense in which the dichotomy between classical mathematics and anti-realism cannot really be confined to the level of the continuum, for there is a strong argument that it must also penetrate to the level of the theory of the natural numbers. This can be brought out most effectively by looking at some recent work on the consistency of the formal system implicit in Frege's Grundlagen (Boolos [1], [2], [3] and Wright [23]) and, in particular, a defence of a version of number theoretic logicism by Wright. This will be contrasted with a recent attack also by Wright ([24], pp. 131-37), but in the spirit of Dummett's anti-realism on the coherence of Cantor's famous diagonal argument ([6], [7]) for the existence of an uncountable set, viz. the continuum, which is surely the primary distinctive result of set theory. What Wright attacks here is the notion of an arbitrary subset of natural numbers, a notion that seems to be needed for us to be able to claim that there is in fact an uncountable set. Yet it is precisely this notion which is crucial to the non-contradictory reconstruction of Fregean logicism which Wright wants to defend as being plausible and thus at least coherent. Thus it seems that, either the argument against Cantor shows that the modern logicist account of arithmetic is incoherent along with the Cantorian conception of the classical continuum, incomplete as it may be; or the attack on the latter fails. The point here is of general interest and is not just an ad hominem argument against Wright, for his position is unusual and challenging in that it tries to combine logicism (admittedly of a sophisticated form) about the natural numbers with constructivism of a very radical kind about the real numbers. This is very much against the general tradition in the philosophy of mathematics which sees as the only viable position constructivism for both or neither. I shall argue that this middle way is incoherent and this for rather deep and general reasons.