More bad philosophy

Another excellent post by the Maverick on a topic in which he excels, namely the hamfisted and sophomoric way in which scientists deal with philosophical questions.

One would think that a scientist, trained in exact modes of thought and research, would not fall into such a blatant confusion. Or if he is not confused 'in his own mind' why is he writing like a sloppy sophomore? Scientific American is not a technical journal, but it is certainly a cut or two above National Enquirer.

The problem is that philosophy, unlike 'science' is a subject which most people feel qualified to talk about. Unlike physics or maths or chemistry, it is not taught in school, so most people have no idea of the difficulty of acquiring expertise in it. Philosophy is also rather like drama, and unlike music, in that it is difficult for non experts to spot lack of expertise. Let me explain. Bad playing of a musical instrument is immedately obvious to anyone who has no training in music. All parents will remember those primary school concerts when the young ones play violins, trumpets, pianos and so on, where the pain of listening whose only just counterbalanced by the love we all bear towards our progeny. In contrast, bad acting is less obvious to those without training in the dramatic arts.

Supposedly this is why membership of the actor's union Equity is so difficult to obtain, whereas membership of the Musicians Union is not. Being a good actor is something that good actors have to judge, being a good musician is obvious to the world. No one can pretend to be a good musician, everyone can pretend to be an actor, and so rules must be drawn up. Now in an ideal world, there would be a philosopher's union, and no one would be allowed to write or even speak about it unless they were a member. But that is not so, and the best we have is Bill's occasional entertaining ranting - which is good enough, to be sure, and lightens the darkness of our days.

Essex on dreaming

Joey Essex (My London, Evening Standard, 27th April 2012) says

I had a nightmare the other day. It felt like I was still asleep but I was awake, but it was weird because I was actually asleep.  When I woke up I was like 'Wow'.  It was so weird.

I'm not sure what he is going on about here.  Is the point that, when you are dreaming, you are usually dreaming that you are awake, i.e. dreaming that you are walking, reading, talking to people, doing all the things that you are doing when you are awake. But sometimes you might be dreaming that you are dreaming, or in this case, dreaming that you are awake, but in one of those waking states where it seems as though you might be dreaming.  And then you actually wake up. Weird, eh?

[See also Alarm clock dreams, posted six years ago]

Lectura (Scotus)

Now in the Logic Museum, book I of the Lectura by Duns Scotus. These are Scotus' notes for lectures he gave on Books 1 and 2 of the Sentences as a bachelor theologian at Oxford. It is the only material from his Oxford lectures that were available for some parts of the Sentences, as the Ordinatio (the revised and edited version of the lectures) was never completed.  The work is also useful for presenting an earlier and generally simpler and less 'subtle' version of his thought.

As with much of the material in the museum, it is untranslated, I am afraid.

Apples and oranges (and cherries)

"You are comparing apples with oranges". How many times have you heard that argument. Is it valid? Depends. Clearly apples and oranges are different species, and insofar as they are different, it is wrong to make a comparison. Yet they are of the same genus. If the objection is about a fruit-related claim, without regard to species of fruit, it is invalid. For example, you claim that all fruit is completely safe to eat. I object that cherries contain hydrogen cyanide, which is not completely safe to eat. You reply "You are comparing apples with oranges (or perhaps cherries)". Invalid. If cherries are fruit, and if cherries are not completely safe to eat, then it is false that all fruit is completely safe to eat. It is no good claiming that I am comparing apples with oranges, if I am talking about fruit.

The fallacy is obvious in this case, but it is frequently abused, and there some wonderful examples on Jimmy Wales talk page here, in a discussion about the 'paid editing' problem. Wikipedians don't like paid editing, because they see it as a 'conflict of interest'. If you are paid, your interest is to write an article satisfying the interests of the person making the payment, not the interests of a comprehensive, reliable and neutral reference work. That is bad, so paid editing is bad. Now someone has (rightly, in my view) objected that the same principle applies to any case where there is a conflict of interest.

I really, truly don’t understand why someone who gets $1000 to edit an article is more inclined to violate our policies (WP:NPOV, etc.) someone who is a “true fan” committed to showing the world how great their favorite team/singer/restaurant, or someone who is absolutely certain that Ethnic group X is better than Ethnic group Y. I don’t understand why we want or need a special policy to deal with one specific form of motivation, when we have a perfectly good set of policies that govern all forms of biased and improper editing behavior.

Wales replies. "You are positing a competition between two things that are completely different. Apples and oranges." Fallacy. Of course paid editing is different from "favourite band" editing. One is motivated by money, the other is motivated by a fanatical obsession with some stupid 'band'. These are different species of editing, just as apples and oranges are different species of fruit. But they are the same genus for all that. The genus is 'conflicted editing', and one case of conflicted editing is not 'completely different', qua genus than any other case. If it is wrong in the one case, why is it not wrong in the other, at least if the wrongness flows from the genus, rather then the species.

Wales goes on: "the incentives before them and the motivations are different". Certainly, as species of editing. But the question is whether the different incentives and motivations involve a conflict or not, and surely they do. Then he says that one size does not fit all, which is another, frequently encountered, version of the fallacy. Of course one size does not fit all, if the cases are different sizes. But the original claim may have had nothing to do with 'size'. Another editor objects that it is 'self evident' that money is special, and that it is "ridiculous to posit otherwise". How so? It is self evident that oranges are special qua species, and not the same as cherries, apples, bananas etc. But they are all the same insofar as they are fruit. The question is whether the different cases (favourite band editing, nationalistic editing) involve a conflict of interest, and surely they all do. Payment is just one instance of a general, problematic case.

Later on the real reason comes out: unpaid editors would be demoralised if paid editing. That is a valid objection: conflicted editing that is demoralising is specifically different from conflicted editing that is not. But that had to be spelled out.

Wikipedia, Britannica and Wales

Lest anyone accuse me of bias against the internet encyclopedia Wikipedia, here is something about the inadequacy of its arch-rival Britannica, and its articles about Wikipedia.

Britannica on the origin and growth of Wikipedia: " In 1996 Jimmy Wales, a successful bond trader, moved to San Diego, Calif., to establish Bomis, Inc., a Web portal company."

On Jimmy Wales: "From 1994 to 2000 he was an options trader in Chicago, amassing enough money to allow him to quit and start his own Internet company."

There are almost too many errors to count. First mistake, if you are going to be wrong, at least be consistently wrong.  Elementary logic suggests it is unlikely that he was both in Chicago and San Diego at the same time, unless he was commuting frequently.  Was he a bond trader or an options trader?  Did he start Bomis in 1996 or 2000?

Second, the facts, as far as I can establish.  Wales never traded bonds but rather futures and options, which are derivatives of bonds and other interest rate instruments.  He was probably not trading to begin with, as he joined Chicago Options Associates in June 1994 as a research associate.  He probably stopped trading in 1998, although he gives conflicting accounts of his time as a trader.  He moved to San Diego around August 1998, not 2000.  He had already established Bomis in 1996, but not as a Web portal company, that came slightly later, probably in October 1997.  Wales and the other founder, Tim Shell, explored many other ideas, including an online takeaway service, before they settled on the idea of the 'portal'.

What else?  There is no compelling evidence that Wales was a 'successful' trader.  In early interviews such as this, he claimed that he had made enough to support himself and his wife for the rest of their lives. In later interviews, such as with Andrew Lih (author of the Wikipedia Revolution, which contains a mostly accurate account of the Wales's early years), he said he simply made 'enough'.

Bursty traffic

Two records broken today. First, today's page views exploded to over 1,000 for the first time in the history of Beyond Necessity. Usually it chugs along at around 150. Second – a direct cause of the burst – the number of monthly page views exceeded 5,000 for the first time.

All because of a post last year on Tolkien's Two Towers, spotted today by someone on Reddit. The interest was because of the five towers mentioned in the book, which came as a surprise to some (even though the Wikipedia entry also mentions this). This was not the object of the post, however, which concerned reference to non existent things, and the possibility of counting them. Never mind. I forget this blog is a humble cottage on the shores of the dark sea of the internet, from which the occasional storm is bound to blow in.

Learning negation

I have a further question about this discussion and generally about any argument that we learn logical laws by experience, observation or induction or anything like that. Suppose it is argued that I learn that no x is white and not white by observing particular x's and noting of each one that it is either white, or not white. I then generalise this to 'no x is white and not white', and further generalise (by substituting other predicates like 'round', 'soft', 'large' etc) to 'no x is F and not F'.

I ask, how did I learn the meaning of the negation 'not'? Is this a sign whose meaning I understood correctly before all these observations? Or as part of the process of observation that led to the general conclusion? Surely not the first. Could anyone who thought it was possible that 'Socrates is white and Socrates is not white' was true, really understand the meaning of the word 'not'? It means negation, and negation means denial, and how could you assert and deny the same thing at the same time? So not the first.

But if the second, that means we learn the concept of negation by observation. Perhaps by your teachers pointing to different things and saying 'not white' when they were not white, and 'white' if the things were white. But that doesn't tell me whether the predicate 'not white' also applies to the white things. To do that, my teachers would have to say 'not not white' when pointing to the white things. And that still doesn't of itself tell me how to use the negation operator for I still haven't been taught that 'not not not white' applies to the not white things, and so on ad infinitum. To understand negation properly, I would have to understand its basic properties before all this took place. But if I understood that, the first point would apply, i.e. I would have to understand that 'x is white and x is not white' can never be true, on account of the meaning of the negation sign.

On the point attributed to Tim Crane, namely that one can perceive something 'as A and not-A' but rejects it through giving greater weight to the principle of contradiction, I'm not sure we can perceive something as A and not-A. Rather, it may seem that it is A and not A, but our understanding of the meaning of the word 'not' assures us that it is not the case that it is A and not A.

Objectivist epistemology

Researching the early history of Wikipedia, and in particular the effect of Ayn Rand's 'Objectivism' on the early development of Wikipedia, I came across this Usenet post from April 1994 by Jimmy Wales, one of the founders of Wikipedia.

Essentially, in Peikoff's presentation, the process goes like this: I perceive (directly, via observation) that "this man is not both white and nonwhite" (at the same time and in the same respect, of course). I see that this pail of water is not both wet and non-wet. At a later point in time, I abstract from the particulars that I've observed and note that "No being is both A and non-A." This holds no matter what being and what attribute is being considered. 

Peikoff is a leading exponent of 'Objectivism'.  Thoughts?  My initial question is how one can perceive, and 'directly, via observation' that this man is not both white and not white.  How exactly do we perceive this?

Completing the infinite

Anthony writes "An infinite procedure is, by definition, a procedure which can never be completed". Well no. By definition, an infinite procedure is one that can never be completed in a finite amount of time. To say it can never be completed at all begs the question, for it presumes that all processes are finite.

Cantor: If one considers the arguments which Aristotle presented against the real existence of the infinite (vid. his Metaphysics, Book XI, Chap. 10), it will be found that they refer back to an assumption, which involves a petitio principii, the assumption, namely, that there are only finite numbers, from which he concluded that to him only enumerations of finite sets were recognizable. [Punktmannigfaltigkeiten § 4 p. 104-5]

You ask what 'finite' means, which Russell answers as follows.

§ 183 Of all the philosophers who have inveighed against infinite number, I doubt whether there is one who has known the difference between finite and infinite numbers. The difference is simply this. Finite numbers obey the law of mathematical induction; infinite numbers do not… It is in this alone, and in its consequences, that finite and infinite numbers differ.

The principle may otherwise be stated thus: If every proposition which holds concerning 0, and also holds concerning the immediate successor of every number of which it holds, holds concerning the number n, then n is finite; if not, not. This is the precise sense of what may be popularly expressed by saying that every finite number can be reached from 0 by successive steps, or by successive additions of 1. This the principle which the philosopher must be held to lay down as obviously applicable to all numbers, though he will have to admit that the more precisely his principle is stated, the less obvious it becomes. [The Principles of Mathematics]

Mathematical existence

There's a tendency to get into all sorts of philosophical quicksand when talking about existence – does it mean physical existence, or space-time existence, is there a sense in which Harry Potter or Frodo have a fictional existence, etc. But let's keep it simple. When I ask whether the square root of 2 exists, all I am asking is whether there is a positive number such that when you multiply it by itself, the product is the number 2. In that way I haven't used the word 'existence' or 'exist' at all. Of course, I used the expression 'there is', but people don't seem to find the verb 'is' problematic in the same way as the verb 'exists'. So, is there a square root of 2?

This brought us to a discussion of sequences of decimals. We agreed that such sequences exist, i.e. that some things are sequences of decimals, or some sequences are of decimals, and we agreed that these sequences can be multiplied. Then we agreed that these sequences can be infinite, and that infinite sequences can also be multiplied. Finally, we agreed that (if we bought the other stuff), there is at least one infinite sequence such that when multiplied by itself, the product is 2 (or rather, the product is 1.999999… which you can verify on an Excel spreadsheet). So, there is a square root of 2. Or, if you like, the square root of 2 'exists'.

Is that it? Of course, we had to buy a couple of ideas. First, that some things are numbers. Some things are chairs, some things are tables, some things are stars or planets, some things are or may be angels (pace Anthony, who does not believe that any things are angels). And some things are numbers. Note my avoidance of the word 'exists'. Second, some numbers correspond to finite sequences of decimals, others to infinite sequences. Do we buy that? Time to read some more Ockham. More later.

On the misunderstanding of logic

Maverick claims a great misunderstanding in my earlier post about the usefulness of logic. I don't think so. Maverick's original post, as I understand it, was an objection to the principle of addressing the small questions of philosophy because the 'big questions' are just too difficult, and probably without resolution. His objection is that the small questions are also "widely and vigorously contested", and so this supposed advantage of pulling in our horns is lost, "and we may as well concentrate on the questions that really matter, which are most assuredly not questions of logic and language"

My reply was twofold. First, the 'little problems' are assuredly not without resolution, at least not if Ockham is right. Now Bill's point may not be that there is no uncontroversial resolution of the little problems. I.e. he is not claiming that they are insoluble, but rather that there is no widespread agreement or consensus on how to resolve them. I reply, Ockham's point is about the understanding of logic. There may be disagreement about his resolution, but that is because those who disagree fail to understand logic.

Second, Ockham's point is that the little problems and the big ones are connected. Resolution of the simple, logical problems opens a door to the larger problems (for example, the problem of the Holy Trinity).

… the gateway to wisdom is open to no one not educated in logic. [...] For it resolves all doubts, dissolves and penetrates all the difficulties of Scripture

This addresses Bill's point that the big questions "are assuredly not questions of logic and language". Ockham says that they are, or rather, that the big questions can be resolved by addressing questions of logic and language.

Whether Ockham is right about the understanding of logic is of course widely and vigorously disputed. See, e.g. "The Failure of Ockham's Nominalism".  But that is beside the point. The fact that the both the big and the little questions are widely disputed is not in itself a good reason to go for the big questions.  If you truly believe that the little questions can be resolved and, even better, are a doorway to understanding and resolving the big ones, then that is a good reason to take the little ones first.  The existence of popular disputes, misunderstandings and confusion is not in itself a good reason not to tackle their root cause.

On the great usefulness of logic

The Maverick fires off a broadside today against the usefulness of logic.

If by 'pulling in our horns' and confining ourselves to problems of language and logic we were able to attain sure and incontrovertible results, then there might well be justification for setting metaphysics aside and working on problems amenable to solution. But if it turns out that logical, linguistic, phenomenological, epistemological and all other such preliminary inquiries arrive at results that are also widely and vigorously contested, then the advantage of 'pulling in our horns' is lost and we may as well concentrate on the questions that really matter, which are most assuredly not questions of logic and language — fascinating as these may be.

Against, I cite the venerable Ockham in his prologue to the Summa , as well as my humbly intended summary of the Summa. As Ockham, following the blessed Augustine, rightly asserts, the study of logic (and semantics) "resolves all doubts, dissolves and penetrates all the difficulties of Scripture, as the distinguished teacher Augustine testifies". "The gateway to wisdom is open to no one not educated in logic". For

It often happens that younger students of theology and other faculties overlay their study with subtleties, before they have much experience in logic, and through this fall into difficulties that are inexplicable to them - difficulties which are nonetheless little or nothing to others - and slip into manifold errors…

So if Maverick has a pipe, as I believe he does, he knows what to put in it.

On the existence of root two

Mathematicians are fond of defining things into existence. "If you say it exists, then it exists". This infuriates philosophers, who see existence as something that has to be proved.  You can say that p, and it can thus be true that you say that p. But that does not entail p.

Nonetheless, mathematician (and Fields medal winner) Tim Gower has an entertaining dialogue on whether you can prove the existence of the square root of two.

I like the idea that for any non-rational number you can define an ever-decreasing interval that appears to converge on the number.  For example, take any finite decimal expansion of root 2.  This will be slightly lower than the fully expanded number. Then add 1 to the digit at the very right hand side.  This will give a number slightly higher than the fully expanded number. Thus for any finite expansion whatever, you can define an interval which contains root 2.  For example, the interval

[1.4142135623, 1.4142135624]

Furthermore, for any such interval you can find the next digit in the expansion, and define in interval which lies completely inside the first interval, and which also contains root 2. For example, we know that the next digit in the expansion above is 7.  Thus  the interval

[1.41421356237, 1.41421356238]

contains root 2, and lies inside the first interval.

Thus we can prove the existence of an infinite set of intervals each of which contains root 2, and each of which can be further subdivided into another such subinterval inside that, and so on and so on.  But does that prove the existence of root 2 itself, which is not an interval, and which is 'indivisible'?  That is precisely where I lost Gowers' argument (the bit right at the end, where the disputant appears to accept the idea that you can define something into existence).

[Added as an afterthought.  I have a paper somewhere by Bolzano, written in 1917, where he attempts to prove by logic the intermediate value theorem.  It is blatantly fallacious, although a part of it eventually turned into the Bolzano-Weierstrass theorem.  This must be connected in some way, but it's so long ago I've forgotten most of it.  Could we not say, for example, that there are two sequences implied in the example above, one of increasing numbers that approach root 2 'from below', and the other of decreasing numbers which approach if 'from below'? ]

Not proven, not guilty

I have been leafing through Ueberweg's System of Logic, which is an interesting nineteenth-century and Teutonic look at that subject.  Very few logic text books would now mention Hegel's logic, for example.  He has an interesting discussion of the principle of Excluded Middle, the one that says any sentence, or its negation must be true.  He claims (p. 263) that the principle may be invalid in certain instances. For example, 'not proven' fills an obvious gap between 'guilty' and 'not guilty'.

Surely not.  What does 'not proven' mean?  It means not proven to be guilty.  'It is proved that' is an operator on the proposition 'x is guilty', not a third truth value filling the gap between sentence and negation. What an elementary mistake, or have I missed something?

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.

Circular reference

David Brightly has a post I just noticed here, replying to a post by Maverick that I also discussed here. DB says “Is Man subordinate to Species? No If it were then some branch of the Porphyrean tree would be labelled 'Species', which isn’t the case.” Very true, and Ockham himself (the medieval one) would have relished it.

Maverick is productive today and has a further post about a throwaway remark in my post earlier today. (An obiter dictum is posh for ‘throwaway remark’, by the way). I am dismayed he calls me “cantankerous and contrary” and even suggests I am known for this. Only my wife knows that, or thinks she does.

But there is an little gem in his post: while nothing of any real substance has been ‘proved’ in philosophy, it is certainly true – as anyone who has taught philosophy to undergraduate students, or argued on the internet knows – that people unschooled in the basics of philosophy make all sorts of idiotic and silly claims which are refuted like lightning by their tutors. Bill mentions the ‘sophomoric relativist’. This reminds me of Adam Morton, who once told me of a student who said to him ‘that’s just your opinion’. Adam fixed her with his X-ray vision and replied ‘Well of course it is’.

This is what makes philosophy interesting. It has established absolutely nothing, no fact you could cite in Wikipedia. Yet it has refuted thousands, or tens of thousands of nonsensical claims. It’s as though it were sort of tailor-made for them.

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.

Subsumption and subordination

Maverick dedicates his post here to me, saying that I like “sophisms and scholastic arcana”. Yes, and Fregean arcana too, Bill, heh heh. His post is about the odd-looking inference

Man is a species; Socrates is a man; ergo, Socrates is a species.

Note that the scholastics would have resolved this by treating the conclusion as a reduplicative proposition: Socrates insofar as he is a man, is a species, but never mind that. Maverick goes on to discuss the ‘modern’ Fregean treatment of propositions like ‘man is a species’. According to Frege, a universal proposition like ‘every man is an animal’ has a fundamentally different form from ‘Socrates is an animal’. In the former, one concept is subordinated to another: the concept ‘man’ is said to be subordinate to the concept ‘animal’. In the latter, the individual Socratres is subsumed under the concept ‘animal’.

Fregegives some very bad arguments for this in his famous essay ‘On Concept and Object’, but I won’t discuss those now. For the moment, here are two arguments against his view.

1. Argument from obviousness. It is obvious that ‘every man is animal’ does not say that one concept is subordinate to another, for the simple reason that it does not say anything about concepts at all. What it says is that every man is an animal. Thus, Socrates is an animal, Plato is an animal. It is talking about every man, not about some concept.

2. Frege’s position requires taking on the absurd idea of ‘object dependence’, i.e. that the meaning of a proper name depends on the existence of some object referred to. I discuss this at length here, with reference to another less well-known essay of Frege's*. Briefly, if we allow that a proper name N can be meaningful and empty, there must be some relation which holds between the name and its referent when its referent exists, and which fails to hold when there is no referent. But then a proper name is not essentially different from a common name like ‘man’. We can say ‘there are no men’ if nothing falls under the meaning of ‘man’, and we can say ‘there is no Socrates’ if nothing falls under the meaning of ‘Socrates’. Frege correctly rules this out as inconsistent with the concept-object distinction. In summary: the distinction between ‘subordination’ and ‘subsumption’ implies and is implied by the object-concept distinction. And the object-concept distinction implies and is implied by the position that the meaning of a proper name is object-dependent.

*Though I note that Maverick mentions this essay, at second-hand, in his book here.

Aristotle against the continuum - reply

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.

Attention to detail

I'm not one for attention to detail, as any casual reader of this blog will appreciate (spelling, sentences without endings etc).  I like to be roughly right rather than exactly wrong.  I've just spent a few hours linking the proposed Logic Museum version of the Quaracchi edition of the Ordinatio to an online pdf I found. A sample  Logic Museum page is here, containing a link to the corresponding page in the pdf.  Now click on the link to take you to the pdf.  You notice the button at the bottom right, inviting you to move to the next page.  You do so, and you notice the special effect that makes the page turn as though by an invisible hand. All very nice, and must have taken a long time to code all that up. (The work was funded by a grant from the University of Toronto).  But look again at the right page, particularly the bottom half. Anyone familiar with digitisation will recognise the problem this.  The contrast is poorly set, or it is overlit.  The poor digitiser will give up in disgust.

And that is precisely what happened. Go to the digitised, pure text version, you see that while the even-numbered pages are not too bad, the odd-numbered pages (that's the pages on the right, mathematicians) are unintelligible.  A bit of thought at the beginning would have saved a lot of effort.

This sort of thing is everywhere in 'digital medievalism'.  Too much focus on software, not enough on the bleeding obvious.  Same applies to Wikipedia, by the way.

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?

History of the infinite

Scotus' discussion of the infinite reminded me of a (very old) Logic Museum page on the same subject.  Many of the links, particularly to those involving Aquinas, are to material in external sites that is now in the Logic Museum itself.  There is other material that ought to be there but isn't (including the passages from Scotus that I quoted recently).  The picture of the corridor at the top was taken in Brompton Cemetery.

More Scotus

Some more Scotus: the Prologue and the first seven distinctions of Book I of the immense Ordinatio.  Following the post the other day about actual infinity, note Distinction 2, Question 1 on whether there is an actual infinite, and Question 2 whether some infinite being is per se known to us, such as the being of God.

Scotus on the actual infinite

Cantor thought that the Scholastics strictly followed Aristotle in rejecting the actual infinite. Aquinas certainly did, but Scotus seemed to have come pretty close to accepting. Here is a passage from Quodlibet 5, where he argues that we can conceive of an actually infinite being. He says that Aristotle ( Physics book III 207a8) defines the infinite as that which for those taking any quantity (i.e. any quantity however large) there always remains something else to take*.

But why can’t we imagine all the parts that could be taken to be actually taken together, so that then we would have an actually infinite quantity, something as great in actuality, as it was potentially? Furthermore, if we can imagine the actual infinite in respect of mere quantity, why not something actually infinite in respect of being? This would be absolutely perfect. For, while an actually infinite quantity has parts which are imperfect (for example, the series of even numbers is imperfect because it lacks the series of odd numbers to make the whole number series), a being that was infinite in being would have perfect parts. According to Scotus anyway, but I didn’t follow the argument. Why would any part of anything be perfect, given that it lacked precisely the remaining bits of the whole in virtue of which it is a part. Isn’t that in the very meaning of the word ‘part’?

Nonetheless, the passage is historically interesting.

*The Latin formulation is “cuius quantitatem accipientibus”, which you can find in at least four places in Aquinas’ Summa: Part I q14 a12 and q86 a2, part IIa q30 a4 and part III q10 a3.

Scotus Quodlibetal Questions

I've been a bit quiet for a few days, which is because I located  a source for Scotus' difficult to obtain Quodlibetal Questions. Now in the Logic Museum.