Are finite numbers an assumption in mathematics?
Well this is a philosophy forum, not a math forum, so we need to try to stick to the basics of the philosophy of mathematics. Stick with me and read the following:
"It is thus not surprising that Hegel's book began with a devastating, even if very ironical, critique of Jacobi's position against Kantianism (and all the forms of post-Kantianism), namely that we are in possession of a kind of 'sense-certainty' about individual objects in the world that could not be undermined by anything else and which showed that there was an element of 'certainty' about our experience of the world that philosophy was powerless to undermine. Hegel called this a thesis about 'consciousness.' If we begin with our consciousness of singular objects and present to our senses (an awareness of 'things' that is supposedly prior to fully fledged judgments), and hold that what makes those awarenesses true are in fact the singular objects themselves, then we take objects to be the 'truth-makers' of our judgments about them; however in taking these objects to be the truth-makers of our awareness of them, we find that our grasp on them simply dissolves and the impetus for such a dissolution lies in the way we are taking them to play a role in consciousness. The result, Hegel argued, is that in the process of working out these tensions, we discover that it could not be the singular objects of sense-certainty that had been playing the normative role in making those judgments of sense-certainty true, but the objects of a more developed, more mediated perceptual experience had to have been playing that role all along... The dialectic inherent in Jacobi's sense-certainty thus turns on our being required to see the truth-making of even simpler judgments about the existence of singular things of experience as consisting of more complex unities of individual things-possessing-general-properties of which we are perceptually and not directly aware. That is, we can legitimate judgments about sigfular objects only be referring them to our awareness of them as sigfular objects possessing general properties, which in turn requires us to legitimate them in terms of our take on the world in which they appear as perceptual objects... We must acknowledge, as Kant put it, that it must be possible for an 'I think; to accompany all our consciousness of things." Terry Pinkard
So I am not trashing mathematics but, instead, probing it's assumptions. The view of many philosophers is that infinities alone exist and I'm wondering what happens to the rest of mathematics when this is accepted by someone. Infinities have cardinality and density, the former perhaps being bound to the latter and perhaps geometry as well. Hegel in particular thought mathematics eventually turns into theories of the infinite and pure logic and I thought it would be interesting to see other peoples' take on these questions