Our earlier discussion of Aristotle's argument that the continuum is not composed solely of points ('indivisibles') neatly illustrated an important philosophical principle: that the only adequate reply to a philosophical argument is to show what is wrong with it. It is no good simply saying that the conclusion is false. Nor claiming that some respected authority says it is false. Nor even stating an arguments against it (which simply shows that there are two arguments with conflicting conclusions). No: the only suitable way is to show what is wrong with the argument. And there are only two ways of doing that: either show that the argument is not valid, i.e. that the premisses can be true with the conclusion false. Or show that the argument is not sound, i.e. one or more of the premisses is false.
Now Aristotle's argument is this.
(1) If a continuum is composed solely of points, these points must be either continuous with one another, or touch, or be in succession.
(2) The points are not continuous
(3) They do not touch
(4) They are not in succession
(5) Therefore the continuum is not composed solely of points.
Clearly the argument is valid. If the four premises are true, the conclusion cannot be false. For the consequent of the implication in (1) is a disjunction. A disjunction is false when all of the disjuncts are false. Premisses (2)-(4) assert the falsity of each of the disjuncts, so if they are true, the consequent is false. If the consequent is false (and if the implication is good) the antecedent is false – consequens falsum ergo antecedens. And if the antecedent is false, the conclusion is true, for the conclusion is the opposite of the antecedent. Therefore the argument is valid.
Is the argument sound? I don't see anything wrong with premises (2)-(4), given the definitions that Aristotle supplies in the text, i.e. the definitions of continuity, contact, succession etc. (A common problem with replies to philosophical arguments is that they ignore careful definitions given in the preliminaries, and focus on something else). So the culprit is clearly (1), as is obvious after a little thought. Take any two points on the continuum you like. Then it does not have to be true that they are either 'in contact', or 'continuous' or 'in succession'. Of any two different points, it can – and indeed always is - true that there are further points in between them.
Why is the argument convincing at all? Probably because it has the appearance of rigour, and because it starts with the covert and natural assumption that each point has a successor. If it has a successor, then there can't be a point in between them (otherwise the point between would be the successor). But there must be something else in between, something that is not a point, and therefore the continuum can't consist wholly of points. But only if there is a successor, which is not necessarily true.
So Aristotle's argument is flawed. But not because Cantor was right, nor because modern mathematics is better, or clearer, or because mathematics is different from the real world (whatever the real world is). It is flawed because it has a flaw, a flaw which we can clearly demonstrate. That is where we philosophers are coming from.