On touching and feeling

I have another difficulty with Aristotle's argument against the continuum that he sets out here.  He distinguishes between two things are continuous, i.e. such that their extremities are one, i.e. are identical, and two things which are contiguous or in contact, i.e. such that their extremities are together.  What is this notion of together?  It's a bit like touching, which is at once natural and philosophically difficult.  I put my hand on the desk.  I have no glove, and so I touch the desk. There is nothing between my hand and it.  How so?

The surface of my hand is clearly not 'one' with the surface of the desk.  I can feel them as quite separate. Well, sort of. When I do it (for I am typing right now), it's rather like the touchingness were a single sensation.  So perhaps they are one. But logic says they cannot be one. They must be separate.  But how can they be separate, when they are in continuous space, and when there is nothing in between? Impossible.

Points and indivisibles

Following my post yesterday, William has updated his post. He writes

So if you're A[ristotle], then given a line segment between two points, you can keep cutting it and keep finding points, none of with (of course) touch. And in your mind, therefore, you have a series of line segments separated by points. What you can't do is consider all possible cuts, because that kind of realised infinity is foreign to his way of thinking.

In which case, the final step is to go back and say, given that definition / idea, is his original proof valid? I think that, given that, his original result is valid, but vacuously so: he refuses to consider completed infinities, and a line, to be made of points, needs an infinite number of points, which he has ruled out, therefore a line isn't made of an infinite number of points. But only because of his artifical restriction on the meaning of infinity.

I think the idea of points 'appearing' when you divide the continuous is foreign to Aristotle's intention (at least at Physics 231a21). Rather, you divide the continuous and you get more continuous, period. You don't 'find' any points after a finite number of division, for the 'points' could only appear when the process of division is complete, which (for Aristotle) can never happen.

Remember that Aristotle doesn't talk about 'points'. He talks about the 'indivisible'. You start with the idea of a continuous thing as something which when divided gives two continuous things. It follows logically from this that the continuous is not indivisible, since it is part of of its definition that it can always be divided. It also follows that no finite process of division will yield anything that cannot be divided further.

If we then define 'composed of' as the relation between the continuous and any set of parts that result from any process of division, it follows that the continuous is composed solely of parts which are continuous when the process is finite. I.e. no points, no 'indivisibles' at all. Just many bits of continuous. Now add the assumption, which Aristotle thinks is impossible, that the process of division can be completed, and by definition (a) the process cannot be finite, from our original definition (b) what is left over will be indivisible, otherwise the process would not be complete and (c) the original continuous thing will be 'composed' of these indivisible thingies, from our definition of 'composed'.

That is, it’s not that the points start appearing as soon as you start splitting the marble. Rather, you only get more bits of marble. But if you keep bashing away hard enough so as to get millions of tiny grains of marble, a heap of fine sand, you can visualise where the process is heading – do this infinitely many times and those little grainy atoms as it were turn into real atom which cannot be further subdivided. Then, and only then, do the points appear. For points are indivisible.

On William's claim that Aristotle has an 'artifical restriction on the meaning of infinity' that's completely wrong. Aristotle understands the same as we do: an infinite process is one that cannot be completed in a finite number of steps. But he also holds that such a process cannot be completed at all, because it is infinite.

Connolley on the continuum

Bill Connolley has post at Stoat about Aristotle and the continuum, and I think I finally see what his problem is.  (and it's also my problem). Is Aristotle's notion of the continuum roughly congruous with the modern notion, and did Aristotle simply get it wrong? In which case, how on earth could he have got it so wrong?

.. the problem I'm having now is to see how his argument can ever have been believed, by him or by anyone else

Or was Aristotle's notion something quite different, such that his view that 'it' is not composed of indivisibles is perfectly consistent. In which case,  what on earth was his notion?

I think I see a way out (noting carefully that I am not a mathematician, and this is just my two cents).  Connolley starts with the idea that the continuum is just the real numbers between two points (say 0 and 1).  If that's what the continuum is, i.e. if it is just those numbers, then it's surreal to ask whether it is composed solely of indivisibles, i.e. composed of numbers. If that's how you define it, it's an absurd question. And even more absurd to argue that is isn't composed of numbers at all. That would be like concluding that bachelors are married men.

But we don't have to start with that idea at all. Suppose we characterise what-is-continuous as that which is divisible into parts, and which after any finite number of such divisions leaves parts which are continuous themselves.  Then it is an open question whether such divisibility could be completed or not.  Clearly there would have to be an infinite number of such divisions, since by definition any finite division leaves continuous parts which can be further subdivided. And if that were possible, i.e. if it were possible to complete the process, then by definition of 'complete', what was left over would be indivisible.

So perhaps it is coherent to hold that  Aristotle agrees with the moderns in a defining characteristic of the continuum (i.e. infinite divisibility into parts), but disagrees over the accidental property of whether the process of divisibility can be completed.  And disagrees, of course, that it is an accidental property at all, for he holds that the impossibility of completion can be proved by logical means, and is thus an essential property.

What is the continuum?

There's a discussion going on here about how Aristotle defined 'the continuum'. The problem, of course, is that he didn't, and couldn't, define the English word 'continuum', since he wrote in ancient Greek. A further problem is the English word is imported from medieval Latin. Is the English imported sense the same as the medieval sense? Even assuming that the medieval Latin was an accurate translation of Aristotle's Greek, how far does the modern usage of the word reflect the medieval usage?

The modern use, as I understand it, is as an abstract noun referring to an abstract non-physical entity with certain idealised properties. This contrasts with the medieval use we find, e.g., in Aquinas here which retains the sense of the adjective, namely as signifying that (physical thing) which has continuity, rather than the abstract feature of contuinity itself (whatever that means). 'Continuum' in Latin, like 'vacuum' is an adjective in the neuter which (in that usage) has a noun-like sense, meaning 'the continuous', or 'that which is continuous' or 'that which is unbroken'. E.g. when he says that it is impossible that "aliquod continuum componi ex indivisibilibus" he is not saying that it is impossible for some abstract object called 'the continuum' to be composed of indivisibles. Rather, he is saying that it is impossible for any real object possessing the property of continuity or unbrokenness to be composed of indivisibles.

Dicit ergo primo quod si definitiones prius positae continui, et eius quod tangitur, et eius quod est consequenter, sunt convenientes (scilicet quod continua sint, quorum ultima sunt unum: contacta, quorum ultima sunt simul: consequenter autem sint, quorum nihil est medium sui generis), ex his sequitur quod impossibile sit aliquod continuum componi ex indivisibilibus, ut lineam ex punctis; si tamen linea dicatur aliquid continuum, et punctum aliquid indivisibile.

Now immediately, hearing this, there will be those who cry that Aristotle was thinking too hard about the 'real world' or the 'physical world' or something like that. As opposed to the 'mathematical world' or some abstract world of abstract things. To which I confess: I don't understand. If there is a mathematical world, in what sense is it not real? As for abstraction, I commented earlier (somewhere) that abstraction is considering normal, real things without considering the features which we are abstracting from. For example, while there is no such thing as a frictionless surface, I can still consider surfaces in respect of their shape and form, without considering properties such as friction. That is all that abstraction is. Or I can consider a triangle without considering whether it has (A) all three sides equal, or (B) two sides only, or (C) none. Now any triangle I consider must be one of (A), (B) or (C). Yet I can consider any one of them without considering whether it is such, i.e. in abstraction from whether it is any one of those three types. That is all 'abstraction' means. It doesn't mean there are any such things as 'abstract objects', as though, absurdly and impossibly, there could be a frictionless surface, or a triangle which does not have three sides equal, nor two side, or none.

Another argument against indivisibles

Here's another argument* against the continuum being composed of indivisibles.  An indivisible has a magnitude of zero.  Thus adding the magnitudes of indivisibles will always result in a magnitude of zero.  For, obviously, zero plus zero is zero. But anything which does have a magnitude, can only be composed of things which have magnitude when added.

Someone objects that this is only true when there are finite additions, or merely countably infinite additions.  I don't understand enough of the subject to reply.

*Philosophers always refer to their arguments as 'arguments' and never as 'proofs'.  This is because there is nothing in the entire, nearly three thousand year history of philosophy that would count as a proof of anything. Nothing.

The history of the continuum

Belette asks about the history of the continuum problem. I'm not an expert, and the subject is huge, but there are a couple of interesting books I recommend. One is Paolo Mancusu's Philosophy of Mathematics and Mathematical Practice in the Seventeenth Century, which covers a lot of the history of the 'indivisibles' question in the seventeenth century and before. The other is Ewald's excellent source book From Kant to Hilbert which covers the period in the nineteenth century when a lot of the advances were made, both in the theory of the continuum and in mathematical logic (although the two subjects overlap considerably at this point).

In the fourteenth century and afterwards the main debate was not so much about whether the continuum could be composed of indivisibles (points), but whether indivisibiles could exist at all. Was the continuum composed of indefinitely divisible lines alone, or a mixture of lines and points? Ockham's discussion of the continuum is here in chapter 45 of part I of the Summa, where he argues against the existence of points, lines etc.

On Cantor's contribution, the idea of transfinite number is often mentioned, but I believe Frege predates him with (The Foundations of Arithmetic). Cantor's main contribution was the idea that the number of the reals was different from the number of the natural number. His argument for this, as I commented here, is unusual and remarkable – possibly the most unusual and remarkable thing in all logic and mathematics - in that nothing appears to predate it.

Aristotle against the continuum

Belette ponders how we could show how Aristotle's argument that the continuum can't be composed of indivisibles is wrong. For reference, the argument is in Physics book 6 at 231 a2. Thomas Aquinas' discussion of it is in his lectures on Physics 6, lecture 1 n2.

Aristotle says that two thing are 'continuous' if their extremities are one, 'in contact' if the extremities are together, and 'in succession' if there is nothing of their own kind in between them. An 'indivisible' is that which has no parts.

Thus a continuum cannot be composed of indivisibles. For such indivisibles are either continuous, or in contact, or in succession. Not continuous, for no point can have separate extremities. Not in contact, for one thing can be in contact with another only if whole is in contact with whole or part with part or part with whole. But since indivisibles have no parts, they must be in contact with one another as whole with whole. And if they are in contact with one another as whole with whole, they will not be continuous: for what is continuous has distinct parts: and these parts into which it is divisible are spatially separate. Not in succession, for things are in succession if there is nothing of their own kind intermediate between them. But there is always a line between two points. And (supplementing his argument) a line is either composed of points, in which case the points are not in succession, by definition, or it is not, in which case the continuum is not composed of indivisibles alone.

What is wrong with the argument?

Ockham on the continuum

Newly translated, for the first time on the Internet etc., here is chapter 45 of part I of Ockham's Summa Logicae.  Here, Ockham's applies his nominalism to the age-old question of the continuum.  Is a point something separate and indivisible from the line of which it is a point?  Is number something different from the things which are numbered?

Ockham says no. Aristotle’s intention, according to Ockham, was to deny that there is anything indivisible 'in this world below' (in istis inferioribus).  Continuous quantity is nothing other than a single thing having one part at a distance from another part, and discrete quantity (number) is nothing other than the numbered things themselves. The difference between continuous and discrete quantity is simply that the parts of continuous quantity mutally protude onto one another [ad se protensae mutuo], whereas the parts of discrete quantity (i.e. two men) can be as near or as far as you like, with no 'medium' between them.

... in the case of discrete quantity it does not matter whether or not the items which constitute the discrete quantity are distinct in place and situation or not, or whether there is a medium between them. Hence, for two men to be 'two', it does not matter whether there is a medium between those two men or not. For they are two when there is no medium between them, just as when they are distant from each other by a hundred leagues, nor does the predication 'two' of those men vary because of anything to do with nearness or distance. On the contrary, if they were in the same place at the same time they would be two, just as if they were not in the same place.

 The translation is new, and has not been through any of the review stages required in the Logic Museum, so all suggestions welcome.