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.

Intentional being

Maverick has a nice post on intentional being.  He mentions Aquinas, but here is the magnificient Ockham on the very same subject.

Reference and supposition

I made a start here on an account of medieval 'supposition theory', although there is a lot more to do. I was getting irritated with the Wikipedia article, although it is not bad by Wikipedia standards, having been written in 2006 by someone who is actually a professor of philosophy. Brian is not an expert though (few people are in this arcane branch of logic) and the article has a few problems. It's sad too that he could not continue his work on Wikipedia because work for inaccessible peer-reviewed journals was more important, according to his Dean of studies. (I know him from the days I was naively trying to recruit academics to Wikipedia).

Anyway, one of the problems with the article is the way it compares supposition theory to modern theories of reference. That's actually not a problem for Wikipedia – most standard reference works say exactly the same. But Catarina Dutilh Novaes has objected, in a number of published works including her Ph.D. thesis, that this is an anachronism. She doesn't think the concepts of supposition and reference are comparable at all, and part of the problem, she says, is that modern philosophical logicians do not understand medieval theories of supposition, and medieval scholars do not understand the concepts of modern philosophical logic such as reference. Each is comparing something they do understand to something they don't understand, and it's all wrong.

I used to think she was right about this, but now I don't. The problem is that modern philosophical logicians don't really agree on what 'reference' is, and the definitions they give are extremely vague. Consider the definition from the much-better-than-Wikipedia SEP:

Reference is a relation that obtains between expressions and what speakers use expressions to talk about. When I assert ‘George W. Bush is a Republican’, I use the proper name ‘George W. Bush’ to refer to a particular individual, an individual about whom I go on to speak. [...] More picturesquely, we are able to use language to talk about the world because words, at least certain types of words, somehow ‘hook on to’ things in the world — things like George W. Bush. Proper names — expressions like ‘George W. Bush’ — are widely regarded as paradigmatic referring expressions.

What does all that mean? One of Catarina's objections to comparing supposition to reference is that in classical supposition theory, common terms as well as proper names have supposition. 'Man' supposits for all men, 'Socrates' supposits for just Socrates. But according to the SEP definition above, it seems like common terms can refer. If reference is 'talking about', then we can use the expression 'every man' to talk about every man, and when we assert 'every man is an animal' we use the quantified expression 'every man' to talk about a particular group of individuals, individuals about whom we go on to speak.

Now Frege had an objection to this (and Frege is the source of all our modern ideas about reference).

If I utter a sentence with the grammatical subject 'all men', I do not wish to say something about some Central African chief wholly unknown to me. It is thus utterly false that I am in any way designating [my emphasis] this chief when I use the word 'man', or that this chief belongs in any way to what the word 'man' means.

Which is not correct at all. When you use the expression 'every man' or 'all men', you certainly are saying something about this individual, even though he is unknown to you. And why is it false that you are designating him? It depends what mean by designation, but then you are back to the problem of defining 'reference'. And the chief does belong in some way to what the name means. If it meant something different, e.g. if the English word 'man' meant only European men, then the sentence wouldn't be about this chief, would it?

In summary, there is nothing about the definition of reference as given by at least two authoritative sources (the SEP and Frege) that distinguishes it from the medieval concept of supposition. More later.

Is any man's parent male?

Nice to know these medieval problems are still a difficulty. Anthony argues that 'every man's parent is male' is false because (if I am understanding him) every man has two parents. OK, then if Aristotelian theory is correct, and if that sentence really does have the logical form 'every A is B', it follows that its contradictory 'some man's parent is not male' is true. But what makes that true? Well, take any man you like, say John. Jean is a parent of his, and she is not male, so 'some man's parent is not male' is true because of John and Jean. But John was any man you like. It therefore follows that 'every man's parent is not male' is true.

But that causes a problem. Assuming that every parent is either male or female, 'every man's parent is not male' implies 'every man's parent is female'. But the whole argument began with the assumption that 'every man's parent is male' is false because every man has two parents. Why, by equal reasoning, isn't 'every man's parent is female' false also?

On donkeys and deformed thinking

I found the Wittgenstein quotation I was thinking of, which is in the Logic Museum here.

"Mathematical logic" has completely deformed the thinking of mathematicians and of philosophers, by setting up a superficial interpretation of the forms of our everyday language as an analysis of the structures of facts. Of course in this it has only continued to build on the Aristotelian logic.

It's what he says about Aristotelian logic which is the interesting one. There's a school of thought in the medievalist world according to which Aristotelian (scholastic) logic is somehow more faithful to ordinary language than modern mathematical logic. Wittgenstein would clearly have disagreed. I have also been looking at the donkey sophism in Worcester 13 again. The problem is that 'every man's donkey is running' has the form 'every A is B', where A = man's-donkey and B = running. According to Aristotelian logic 'every A is B' and 'every A is non-B' are contraries, they can't both be true at once. But clearly the ordinary language sentences 'every man's donkey is running' and 'every man's donkey is not running' can both be true at the same time, namely in the case where every man has two donkeys, one of which is running and the other of which isn't. It's not a problem for ordinary language at all. But it is a problem for the Aristotelian formalism of the sentence. In that formalism every sentence has two terms, joined by a copula and a quantifier attached to the subject term. It is a procrustean bed which fits our actual thinking very badly, in some cases.

I also noticed this other comment by Wittgenstein in the same place:

The curse of the invasion of mathematics by mathematical logic is that now any proposition can be represented in a mathematical symbolism, and this makes us feel obliged to understand it. Although of course this method of writing is nothing but the translation of vague ordinary prose.

What on earth does he mean by that?

Burley on being

There is an interesting post from the Maverick today on the being-essence distinction. About this question "you have no idea how much ink, and vitriol too, has flooded the scholastic backwaters". Very true. He mentions Aquinas, but there were many others. Here is a link to my translation of Walter Burley's discussion of the question, with parallel Latin text. Walter provides a nice summary of the origins of the question in Boethius, and Avicenna and Al-Ghazali, as well as the different positions held by the main thirteenth century philosophers (Albertus Magnus, Thomas Aquinas, Giles of Rome, Henry of Ghent and Godfrey of Fontaines). Walter agrees with Godfrey that existence (i.e. being) and essence are the same:

Concerning this [Godfrey's] position, it should be understood that nothing is actually in a real genus unless it actually exists. Which is clear from this: the Philosopher (Metaphysics VI at the end), divides being into true being outside the soul, and diminished being that only has being in the soul, and that being he excludes from his consideration. Next, he divides true being outside the soul into the ten categories, and thus every category is true being outside the soul, and nothing is actually in a category unless it actually exists outside the soul.

Maverick's discussion has more in common with the question of individuation, however, over which much ink was also spilled.

[Edit] I am right.  The question of whether it is existence that determines numerical distinction is discussed by Scotus in Question 3 of the third distinction of Book II of his Ordinatio.  The third distinction is all about the problem of numerical individuation.  Angels are immaterial, they have no matter in which their form is embedded.  But angels are numerically distinct, and if so, the distinction cannot be grounded in different material of which they are made, for they aren't made of material at all. In the six questions of this distinction, Scotus considers different answers to the problem, before settling on his own answer: numerical distinction and hence individuation must be grounded in some positive feature, some intrinsic 'thisness' or haecceity (from the Latin haec meaning 'this').  I never got round to working on questions 4-6 because they are so difficult, and not on account of the Latin.  Scotus is one of those writers - Sartre is another - whose prose becomes more obscure and more difficult in proportion to the difficulty of the question.  Why not the other way round?

The price of knowledge

At least one commenter has spotted the problem about the application of Hayek's paper and Wikipedia. Hayek's central argument is that organisation is best made by people who are familiar with local circumstances, who know directly of the relevant change in circumstances and have the resources immediately available to meet them. No central planner can have this knowledge of circumstance. Thus organisation is best performed – and is being performed in many cases – by a decentralised price system.

Now if there were an analogy between price-based decentralisation and Wikipedia's decentralisation, there would have to be a way for consumers to signal demand for knowledge by paying for it, and a way for the 'miners' of that knowledge to be compensated by digging deep into the earth of information, trying to find veins of wisdom running through the slag of trivia, or by refining the crude ore containing knowledge thinly spread, into the pure metal of subtle and profound wisdom.

There is no such mechanism in Wikipedia. It is paid for by an annual fundraiser which appeals for donations to collect 'the sum of human knowledge', without any mechanism for donors to sponsor chosen articles containing parts of that sum. Even if such targeting were possible, there is no way of directing donations to individual 'knowledge miners'. All editors of Wikipedia are unpaid volunteers*. This fact has already been noted by Harvard researcher Andreea D. Gorbatai in 2011, who questioned whether collective production such as in Wikipedia creates social utility.

Now there is a sort of reward system on Wikipedia whereby editors can achieve non-financial status similar to kindergarten 'stars'. But follow-ups on Wikipedia to Gorbatai's study concluded that this reward system was perverse, with greater recognition being given to editors producing large numbers of low importance articles than to editors producing small numbers of high importance articles.  An article in the Wikipedia signpost concluded -

- It is interesting to compare the most prominent author of high importance articles at low production rates, Garrondo, with the most prominent author of low importance articles at high production rates, Ucucha. Garrondo has written one FA, Parkinson’s disease in 2011. Ucucha has produced 14 FAs on rare, Latin-named, mammal species. Garrondo has a lousy strategy for climbing up the WBFAN. However, when we look at the impact of the two editors' articles for the readers, there is little question. Because the single Parkinson’s disease article has 180 times the views as Ucucha’s average article, Garrondo achieved 13 times the total contribution to reader-viewed FA content. The problem is all our systems of rewards, all our tracking systems, all our unconscious assumptions, talk page remarks, and so on simply talk about number of stars…instead of the importance of them. We are incentivising the high production of low importance articles and discouraging the opposite. Yet the latter strategy is the more efficient way to serve the readers. [My emphasis]

This provoked fury from Wikipedia's established editors. The discussion is here - beware the heated and often incoherent ranting.

There is another, more subtle, question here. The 'market mechanism' assumes that what consumers want is what they actually need (or rather, it makes no distinction but wanting and needing). There is an older tradition, dating from at least Plato, that the ordinary mass of human beings don't really know what they need, or what is really good for them. The most recent proponent of this view was Lord Reith of the BBC, who believed that broadcasting should be 'an authoritarian system with a conscience', carrying to the greatest number of people everything that is 'best' in every department of human knowledge, endeavour and achievement, and to avoid whatever is 'harmful'. He was criticised for not giving the public what it wanted but he replied that few people know what they want, and fewer still know what they need. I discuss that in an old post here.

*Except for paid editors, of course, who are employed by public relations agencies or rich individuals to embellish articles about their clients, or themselves. But that's another problem.

Are Meinongian objects incomplete objects?

The Maverick asks whether Meinongian objects are 'incomplete'? According to his definition, these are objects which have all and only the properties specified in their descriptions. For example, The Yellow Brick Road is described (in the story) as being yellow, but is not described as being coloured. So The Yellow Brick Road is yellow, because the property of being yellow is ascribed to it, but does have the property of being coloured. So Meinongian objects are 'incomplete'.

Which makes no sense to me. Every yellow object is coloured. Therefore there are no yellow objects which are not coloured. Therefore there is no 'incomplete object' such as Maverick has described. You object: Bill has described such an object, therefore there is such an object as Bill has described. I reply, the verb 'described' is logically intransitive. It takes a grammatical object, i.e. an accusative, but no logical object. 'S describes A' does not entail 'for some x, S describes x'. I can describe a yellow brick road, and perhaps I can even describe a road that is yellow but not coloured. But it does not follow that there is some yellow non-coloured object described by me, for no object is yellow but not coloured.

Kilwardby on the usefulness of logic

As I argued before, Ockham's nominalism is not simply about whether universals exist or not.  It is more radical than that: problems in philosophy and theology arise from misunderstandings and disagreements about logic.  This is not an idea that originated with him, as the following quote* from the mid thirteenth-century theologian Robert Kilwardby shows.

The origin of this science [i.e. logic] ... was as follows.  Since in connection with philosophical matters there were many contrary opinions and thus many errors (because contraries are not true at the same time regarding the same thing), thoughtful people saw that this stemmed from a lack of training in reasoning, and that there could be no certainty in knowledge without training in reasoning.  And so they studied the process of reasoning in order to reduce it to an art, and they established this science by means of which they completed and organised both this [science] itself and all others; and it is the science of the method of reasoning on all [subject] matters.

Note that 'logic' in the medieval period covered more than formal logic, and covered metaphysics, semantics, informal and demonstrative reasoning as well. Nor did symbolic logic exist.  Medieval logic is the logic of natural language, as ordinary people use it in argumentation.

*De ortu scientarum, ed. Albert G. Judy, London, The British Academy, 1976, chapter 53.

Budget special

As it's budget day here in London I will make a rare concession to things that are happening now, rather than the late thirteenth century.  Philosopher Stephen Law, whose debate with William Lane Craig I discussed in a few posts last year, has an excellent and entertaining rant about taxation and the rich on his blog here.  The comments section is especially funny.  Law's premiss is that the Conservative government has a lot of rich mates and that their entire policy is designed to make them richer, and the poor poorer.

This is presumably why today's budget will include a rise in the 'personal allowance', namely the bit you substract from your taxable income, and which therefore isn't taxed at all.  The rise will take some people out of the tax net entirely.  And presumably why the London Times today has a headline saying 'Super rich to pay for Osborne's tax cuts'.

I could make a whole post and more about the logic of taxation, but not today.  Perhaps later.  (Actually our friend Anthony who occasionally posts here knows something about this, but he has been a bit quiet lately. Where are you, Anthony?)

Wikipedia, Hayek and the marketplace

'Belette' comments here that "there is no marketplace on wiki". Well perhaps there isn't, but its founders certainly thought there is. Hayek's paper about market mechanisms "The Use of Knowledge in Society" is supposed by some, not least by Jimmy Wales himself, to be the model for Wikipedia. You can read the paper for yourself, but Hayek's central argument is that there are two types of knowledge, namely knowledge of the eternally true, scientific sort of stuff that you read about in books, and knowledge of "circumstance", i.e. of little facts peculiar to a time or place, knowledge of which "never exists in concentrated or integrated form but solely as the dispersed bits of incomplete and frequently contradictory knowledge which all the separate individuals possess." Hayek argues that all economic activity is planning, but all planning must be based on knowledge which is not initially available to the planner but to somebody else, which somehow will have to be conveyed to the planner. Which is the best way of utilising this knowledge: central planning or the use of a price mechanism?

Assume that somewhere in the world a new opportunity for the use of some raw material, say, tin, has arisen, or that one of the sources of supply of tin has been eliminated. It does not matter for our purpose—and it is very significant that it does not matter—which of these two causes has made tin more scarce. All that the users of tin need to know is that some of the tin they used to consume is now more profitably employed elsewhere and that, in consequence, they must economize tin. There is no need for the great majority of them even to know where the more urgent need has arisen, or in favor of what other needs they ought to husband the supply.

Wales claims (in an email to me) that this idea underpins his whole thinking about Wikipedia. Larry Sanger also had some similar ideas which he expressed in a mini-essay here in 2001.

Academia is sometimes compared to the marketplace of ideas. That's also an apt description of Wikipedia at present: it's unregulated (except for some basic ground rules), and anyone can come in and "set up shop" (write an article), but other "business owners" (contributors) can "compete" (improve the article) according to their understanding of what the facts are, what the best way to express them is, etc. Competition improves articles. Regulation tends to stifle free competition."

Why Wikipedia's Fans Shouldn't Gloat

A wonderful wonderful article in The Atlantic. It speaks for itself, no further comment required.

The 'one ring' theory of cranks

I'm still working on the Wikipedia book, looking today at the history of the 'three revert rule', introduced in November 2004. This says that “An editor must not perform more than three reverts on a single page within a 24-hour period”, theoretically puts an end to the endless edit/revert cycle between warring editors.  In theory, it is a numbers game which gives the edge to subject matter experts. As I understand the rule (I'm sure Belette will correct me if I am wrong) it applies per player, not per group of players.  Two players working together can revert a single opponent up to six times a day, but the opponent, if on their own, can revert only three times.  Thus whoever has the largest number on their side, wins the edit war.

This seems to give the edge to subject matter experts because generally (though not always) they tend to agree, at least on the kind of slightly out of date knowledge that is appropriate for reference works.  Cranks, by contrast, have their own theory of everything which is peculiar to itself and inconsistent with every other crank theory of everything. The theory that the earth is flat and the theory that it is a cube are both opposed to the mainstream theory that it is roughly spherical, but they contradict each other too. Likewise for the theories that the moon is made of cheese, and that it is made of candy floss. Thus 3 experts can beat any number of cranks, so long as the cranks don't agree on anything.

However, neurologist Steven Novella makes an insightful observation here that brings this idea into question.

... cranks around the world have been able to form their own “alternative” community, publish their own journals, and have their own meetings. There is just one requirement in this alternative community – acceptance. All ideas are accepted (there is no chaff, all is wheat), that is except for one. Whatever is accepted by mainstream science is wrong [my emphasis]. That is “the one ring” of crank mythology, that brings all crank theories together and in the darkness of their community binds them together. Otherwise they are largely mutually incompatible. Each crank’s “theory of everything” is a notion unto itself, and is mutually exclusive to every other crank’s own theory of everything (unless there is some incidental overlap). So they get together, present their theories without criticism, and all agree that the evil conspiracy of mainstream science must be taken down. Of course, if any of them got their way and their ideas became accepted, they would instantly become rejected by the rest of the crank community as mainstream physics.

Correct.  My enemy's enemy is my friend, whatever my enemy believes.  I have seen this effect in Wikipedia a number of times.  Cranks unite to defeat the mainstream, orthodox view.  Orthodox editors get blocked or banned.  Cranks then war with each other, and get banned themselves. The orthodox editors mount appeals to the powers that be - the arbitration committee, none of whom have any expert credentials as far as I can see, and get unbanned.  Or they just open 'sockpuppet' accounts and start editing again under a different name. So do the cranks, and the whole nightmare begins again.  Another difficulty that Novella omits is 'mainstream' crankery.  That is, bad science or quackery that unites its practitioners by financial interest. Homeopathy and 'Neurolinguistic programming' are good examples of this.

This would not matter at all, if Wikipedia were not increasingly used as a 'reliable source' by students, and even some medical researchers, as I noted in an earlier post.

New Aristotelians according to Connolley

A piece here by William Connolley gives me a chance to write about something today.  He says that 'with the success of science' the idea of 'abstracting problems', i.e. abstraction, isn't difficult. He considers the example of a ball moving on a surface, which requires the idea of a perfect sphere on a perfectly flat surface, 'ignoring friction'. This takes us straight away to Newton's laws of motion, such as the law that the ball moves in a straight line unless deflected or stopped by some force, and so on.

He claims that the ancients, such as Aristotle, were unable to make that abstraction. "They were still trying to understand the whole world as it was".

...It was "obvious" to them that the first and most obvious property of moving objects was that they stopped moving once you stopped pushing them. An ox-cart rumbling down some rutted muddy path stopped when the oxen stopped pulling it; that was obvious, and the study of such was so mired in the nitty-gritty reality of the world that precious little progress was made until Galileo abstracted (I know, I know, I simplify: Oresme etc worked on the problem too and got some of the way there; but again, only by picking on simpler examples).

Sorry, but that's quite horrible.  The ancients such as Aristotle were well used to abstraction.  Aristotle, as is well known, was profoundly influenced by Euclid and the abstract world of geometry, you know, perfect circles, spheres, lines, planes and so on.  As was Plato before him, of course, who was so impressed with abstraction that he constructed a quasi-religious theory about it.

To get Newton's laws abstraction may be necessary, but not sufficient. The crucial part of Connolley's example is the 'friction' bit. We have to 'ignore friction'. Yes, but what is friction?  Well, as we observe it with our own eyes in the natural world, it simply is the tendency of bodies in the sublunary world to slow down and stop.  An ancient philosopher could easily have abstracted away friction, after all, that's exactly what Euclid did.  There is no friction in Euclid's geometry, nor is there gravitation.  But there's nothing in that abstraction that tells you what friction really is.  The ancients thought that bodies slow down because that's what they naturally do, just as we think that bodies are naturally attracted to each other by gravitational force (which we still don't understand, except as some Aristotelian essential characteristic of matter).

The capacity to abstract is nothing to do with ancient science.  In fact, the problem was too much abstraction. By contrast, a little more observation and attention to the actual world would have done the trick. As I observed here and elsewhere, it was Buridan's observation of milwheels and ships, plus a spell in an armchair, that led him to reject Aristotle's theory of impetus.  Forget abstraction.

Jimmy Wales, civil servant

So Jimmy Wales is to become an advisor to the British government. Is he going to make the workings of government more like Wikipedia? Here are a few suggestions.

Remove all that stuff about qualifications, Oxbridge, classics education and all that fuddy duddy old world elitist nonsense. The first thing is that everyone should have an equal say in the way that the health service, the financial system, tax collection and so on are run. And why stop at the people who work there? Why not open the doors of government completely, on the lines of an 'open source' software collective? Remove those turnstiles, guards, doors, just let anyone walk in off the street and work at anyone's desk. Give them access to any kind of file they want. The only requirement is that they get a numbered ticket at the front door, which they must use to get in again. (Don't worry if they lose their tickets, they can simply ask for a new one).

Obviously this may result in the destruction or loss of confidential or important information. But that doesn't matter. Wikipedia managed with an elite corps of patrollers who can check instantly whether information, whatever it is, has been changed or destroyed. They will back up all the information and as soon as some common vandal comes in from the street and messes something up, they will restore things to the way they were before, and take away the entrance ticket of the vandal. (Of course, the vandal can easily get another ticket, but don't worry about that).

Angry disputes about how the country should be run are inevitable, so there should be a 'civility code' against shouting, politically incorrect insults and so forth. This is so important that it should be enforced by an elite hierarchy of administrators. In fact, the only role to promotion in the new Civil Service should be through this hierarchy. Again, forget that nasty old elitist thing about Oxbridge and education. The only thing that matters is the ability to break up fights and to punish those who seem to have started the fight. This does not require any knowledge or understanding of what the fight was about. You could go round asking the people who work at the doors of pubs or clubs, in case they are interested in a permanent job.

This way, the British Civil Service will become much more efficient.

Every man's donkey is running

Suppose that every man has two donkeys, one running and one not running. Then every man's donkey is running, for the donkey of this man runs, the donkey of that man runs, and so on for each individual man. But on the other hand, a donkey of this man does not run, (namely the other one of his which is not running), a donkey of that man does not run, and so on for each individual man. Therefore every man's donkey is not running. Therefore it is not the case that every man's donkey is running.

This is one of those puzzles which caused medieval logicians all sorts of mental strain, but which is completely resolved by translation to modern predicate logic. It can easily be shown that the scenario of each man having two donkeys, one running and one not running, implies the following two propositions of predicate logic

(1) (x) Ey x owns y and y runs
(2) (x) Ey x owns y and not (y runs)

where x ranges over men and y over donkeys. Obviously the two propositions are not contraries: they can both be true at the same time. Yet the English sentences which they translate ('every man's donkey runs'and 'every man's donkey does not run') do appear contraries. This is clearly a problem for English, not for logic.

The program of modern analytic philosophy was to resolve all philosophical puzzles by means of the same kind of translation into modern predicate logic. I think this has failed, but that does not imply there can't be some way of formalising paradoxical or aporetic sets of English sentences in a way that dissolves the aporia.

Abstract particulars

Anthony asks about abstract particulars here. In reply, I am not sure what an abstract particular is, so I am not entirely sure I can answer the question. I don't think Ockham ever discusses such things, and in any case, beginning with distinctions in reality rather than distinctions in language is foreign to his whole project. When he begins to talk about 'concrete' and 'abstract' in Chapter 5 of Part I of the Summa, he is talking about types of term. He says that ‘concrete’ and ‘abstract’ are names that have a similar beginning verbally, but not similar endings. Thus 'human' and 'humanity', 'just', 'justice', 'wise', 'wisdom'and so on. This is a linguistic distinction, and Ockham's whole argument is that we confuse distinction in language with a distinction in reality. If Socrates is wise, then 'wise' denotes him, but so also does 'wisdom'. 'Wisdom' does not denote any common substance or nature, a singular thing common to all wise things. Rather, like 'wise', it denotes all wise things.

What about the distinction between particulars that exist in space and time (trees, chairs, people) and particulars which do not (triangles, numbers, etc)? Ockham does have something to say about this, but it takes up all of chapters 40-62 of the Summa, in his discussion of Aristotle's categories. So, more later.

Stromboli and flying horses

I replied to this Maverick post, but will copy it here as well.  Maverick says that the following argument is valid but not sound:

Pegasus exists.
Pegasus is a flying horse.
Ergo
A flying horse exists.

And he claims that the first premise [Pegasus exists], though false, is needed for the argument to be valid. Do I agree?  Certainly not. On my view, which is not a standard view, 'Pegasus is a flying horse' has two meanings, a literal and a non-literal. The literal meaning is such that its truth implies that something is a flying horse. I also hold the Brentano-Quine view that 'something' is always 'existing something'. Thus the argument is valid with the second premise alone. "Pegasus is a flying horse" is equivalent (literally understood) to "Something is Pegasus and it is a flying horse" which implies "something is a flying horse" without the help of the first existential premise.  Literally understood, "A is B" is always existential. 

The second meaning, the non-literal one, is 'In Greek mythology, Pegasus is a flying horse', which means the same as 'Greek mythology says that Pegasus is a flying horse'. This is not existential, at least in respect of the that-clause.  It can be true that I say that some horses fly, even though it is false that any horse flies (as far as I know).  Thus, on this non-literal interpretation, Bill's argument is not valid, for it really reads:

Pegasus exists.
Greek mythology says that Pegasus is a flying horse
Ergo
A flying horse exists.

which is not valid. At most, it would imply 'some [existing] creature is said by Greek mythology to be a flying horse'.

This is not a standard view, of course. The standard view would be that 'Pegasus' is meaningless because it fails to refer. Therefore Bill's argument is invalid anyway.

Pinterest

I found this website a bit troublesome.  It's a website, apparently used mostly by women.  It is mostly about cooking, clothes and children.  Why?  Is there some fundamental biological distinction between men and women?  Or is it that patriarchal domination, or the black magic of false consciousness forces women into such things?

Or do they like them at all?  Most women I know don't really like cooking.  My wife doesn't, nor does a friend of mine who goes home and lies on the sofa with a glass of wine while her husband operates in the kitchen.  Clothes are for people under 30, of both sexes.  As for children, these are a biological necessity.

I was more disturbed by the horrible cute kitsch of the whole site.

What concept does 'exists' instantiate?

Maverick has a great post here. If 'some horse exists' says that the concept horse is instantiated, what concept does 'something exists' instantiate? (I have simplified his argument somewhat, without missing its force, I hope).

Actually this is all too difficult for a Saturday afternoon, so instead here is Happy Go Lively by Laurie Johnson, who also wrote the Avengers theme music. Happy Go Lively to me sounds identical to Holiday for Strings by David Rose, who also wrote The Stripper.

I first went to the States in 1971, when inside absolutely every lift and in every shopping mall they played music like Holiday for Strings. To the teenager I was then, Jimi Hendrix represented all that was true, and shopping mall music everything that was horrible. I don't know if they are playing Hendrix in shopping malls, but music has moved on since then, in some sense of 'moved on'. For example, here is Bring Me The Horizon playing something completely horrid at the Reading Festival last year. The singer has so many tattoos it seems he is infected by gangrene.

Is a universal one or many?

Following on from my earlier discussion of Maverick's posts. Assume that a universal is either one thing or many things (an assumption which may be false).  If it is many things, then we have an infinite regress.  If Socrates' wisdom is numerically different from Plato's wisdom, and if all the wisdoms of many philosophers are many different things, just as the philosophers themselves are many different things, how do we explain the underlying common nature of all these wisdoms? Don't we need another universal to explain this common factor?  And then I ask whether this common factor is one or many.  If one, then eadem ratione standum fuit in primo: by the same logic we should have stopped with the original case and gone no further.  If many, procedetur in infinitum, there is an infinite regress.

If by contrast it is one thing, we must suppose some relation between individual objects which are instances of the universal, and the universal itself. Socrates, who is wise, bears this relation to the universal wisdom.  So does Plato.  But Joe Stupid, who is not wise, does not.  And so we can construct a predicate, e.g. '- has wisdom' which is satisfied by all wise people, and only wise people. But what is the common feature that explains why some individuals bear this relation to the universal, and some do not?  Is this one or many? And so we have an infinite regress again.

I conclude that, if there are universals, they are  neither one nor many.  More later, and this may involve bringing Avicenna into it.

I was amused by this post from the Maverick which attempts to classify the various forms of nominalism. I particularly liked the idea of 'mad dog nominalism' – a form of the genre that collapses into linguistic idealism.

I pointed out in the comments box that there is a further ingredient in Ockham's nominalism, namely the thesis that by ignorance of logic we are led astray into certain false and fantastical beliefs. This is not a million miles from Wittgenstein.

[Philosophical problems] are, of course, not empirical problems; they are solved, rather, by looking into the workings of our language, and that in such a way as to make us recognise those workings: in despite of an urge to misunderstand them. The problems are solved, not by giving new information, but by arranging what we have always known. Philosophy is a battle against the bewitchment of our intelligence by means of language. [Philosophical Investigations ~109]

The Encyclopedia Game

There's an fascinating new documentary in the making. The website is here with some promotional clips here.

"Wikipedia is a massive online free encyclopedia. It's one of the most visited sites on the web, with two billion hits every month. With the stated goal of compiling the sum of all human knowledge, it's the most used reference source in the world. Yet, at any given moment, it may contain information that is completely untrue and misleading. How can this be?

"The world of Wikipedia, which many assume to be a collegial community of experts collaborating to write an encyclopedia, often more closely resembles an arena in which personal conflict and gang warfare are standard operating procedures. Wikipedians are collaborating to write the world’s largest and most used encyclopedia, but their behaviors often more closely resemble those of a large group of anonymous characters playing a vast and intricate online game in which “writing an encyclopedia” merely provides the basic scenario and context within which the game is played".

That seems about right. I have identified the accounts in question, plus quite a few more that the site doesn't mention, and this is probably the most skilled 'sockmaster' I have ever seen. But to understand what a 'sockmaster' is I will have to explain one of the fundamental design flaws of Wikipedia.

Wikipedia was set up in 2001 by free culture types, who believe that knowledge should be both free and open. Free means, well, free, and I can deal with that, and open means user-generated, which is more problematic. Having anyone edit means that anyone can open an account without restriction. Indeed, the same person can open as many accounts as they like without restriction. This is a fundamental principle of Wikipedia, and is unlikely ever to change, given the politics and ideology of the people who run the site. Add to this the premise that people generally edit for selfish reasons, as I argued here, and you have two nicely toxic ingredients. If you have some passionate belief in some piece of nonsense – it doesn't matter what the nonsense is – then you want the nonsense in Wikipedia. Given that other people may oppose the nonsense, then you will want to stack the argument with as many multiple accounts or 'sockpuppets' as you can create, defending the nonsense against all comers.

This takes some skill. Using sockpuppets to stack the vote is a (theoretically) a banning offence on Wikipedia, unless you are a highly influential member of the administration. So ideally at least one of your socks is an administrator, which this guy was*. In addition, the rest of the sockpuppets should not be 'single purpose'. If they all start to edit just the one article, you will normally be caught out pretty quickly. So you edit in many different and non-overlapping subject areas, occasionally editing the subject in dispute. At crucial moments you bring the socks together, usually to gain consensus, or stack a vote, or to bypass the '3RR'** rule. This all takes time and effort, but young people with an agenda and time on their hands can manage that. The achievement of Cognition was to add significant content in various contentious areas without being caught out. I haven't yet checked to see how much of the nonsense is still there, but it is likely to be significant. The main point, as the character says, is that Wikipedia is not an encyclopedia, and the skill required to participate in it are more aligned to John le Carre, than Diderot.

And he is still editing.

* I can imagine Belette's dorsal fins flapping here. No, not with the Cognition account.
** 3RR = three revert. If you revert someone's edit more than three times, you can be blocked.  So you get your socks to do some of the reverting.  Of course your opponent will be using socks too.  It is not uncommon to see what looks like a whole crowd of people arguing with each other, when in reality there are just two.  Sometimes there is only one!  Really skilled sockmasters will create good/bad hand accounts to increase the appearance of reality, often putting silly arguments in the mouths of the bad hand account in order to refute them more surely, and to make the opposition look foolish.

Ockham and the London friaries

Three additions to the Logic Museum. First, a summary of Ockham's Summa of Logic. The material on part I is pretty much complete, but parts II and III, and particularly part III, need further work.  Part III has been little studied - even Boehner only got as far as chapter 2.  I wrote the Wikipedia article on the same subject some years ago, but that is a mere stub.

Next, London Greyfriars,  one of five proposed articles on London friaries.  This is an obvious continuation of the version I wrote in December 2010 for Wikipedia.  Unfortunately I was blocked by administrators, halfway through writing this section, which as you can see is still incomplete.  One of the administrators actually deleted the article, although it was soon restored.  This was all part of an ongoing feud with Wikipedia whose origins have long since been forgotten, as with most feuds.  The new article completes the section on the buildings and - very important - has a list of some of the books which John Leland found in the Greyfriars library before it was dispersed by King Henry's henchmen.  I am still fascinated by the idea that Ockham stayed at Greyfriars while he wrote Summa Logicae.  Many of his biographers think so, but I had a correspondence with William Courtenay who persuaded me that this is still not certain.  It is pretty certain that Ockham, Wodeham and Chatton were all together at some time.  What we are not certain about is where this was.  Note, however, that the Greyfriars library contained quite a few works by the venerable inceptor.

Finally, a stub on the Carmelite friary, which also had an extensive library.  The crypt of the friary is still visible under the offices of Freshfields in Whitefriars street, and there is a link to some photos.

On defining nominalism

"According to Ockham, everything in external reality is singular".  This is not a good way of characterising Ockham's nominalism, for it could equally apply to some of the brands of realism which he criticises and caricatures.  For example, a realist who holds that a universal is a singular thing, inhering in some way in many things, also holds that everything (including universals) is singular. But they would clearly not be a nominalist.

A similar observation applies to the so-called  Ockham's razor.  "Do not multiply entities beyond what is necessary". Sure, but realists agree with that too. The disagreement is over what counts as necessary.  Realists would hold that universals are necessary, of course.

The medical condition known as glucojasinogen

Here is the lovely example, possibly the best yet, of what I have called Wikipedia faction.  This is where some nonsense information gets added to Wikipedia and stays there long enough for 'reliable sources' to pick it up, so that Wikipedia can then cite the reliable sources for the nonsense.

In 2007 an anonymous IP adds this entry to the Wikipedia article on diabetic neuropathy.

It is important to note that people with diabetes are more likely to develop symptoms relating to peripheral neuropathy as the excess glucose in the blood results in a condition known as Glucojasinogen. This condition is affiliated with erectile dysfunction and epigastric tenderness which in turn results in lack of blood flow to the peripheral intrapectine nerves which govern the movement of the arms and legs.

It's nonsense of course (we spent some time going through a medical dictionary to check).  The nonsense then got picked up by two journals: "Influence of Murraya koenigii on experimental model of diabetes and progression of neuropathic pain" by S.V. Tembhurne and D.M. Sakarkar, Journal of Research in Pharmaceutical Sciences, 2010, and African Journals Online (which cites the same paper).  It is now visible in Google scholar.

It would have been more amusing if the sources actually had been added, to complete the circle.  But then it's not amusing at all, is it.  It's one thing to get most of the key facts about medieval philosophers completely wrong.  That just damages learning.  It's another to slander someone in public, under the umbrella of a supposedly comprehensive and reliable reference work.  That merely damages someone's reputation or even destroys their career.  But getting important medical information wrong can damage someone's health or life.  That's not amusing.  In fact none of it is amusing. 

The Wikipedians, by contrast, are having a bit of a laugh about it.  Another Wikipedia hoax. The IP editor even got a special 'barnstar'.  This is part of the frustration of the place.  If something goes desperately wrong, it's a bit of a giggle.  Challenge any of this from the outside, however, and you are immediately pointed in the direction of the famous 'Nature' article in 2005. "Wikipedia articles come close to the level of accuracy in Encyclopedia Britannica".

Ockham on induction

I have made a start on book III-3 of Ockham's Summa Logicae by translating some of the chapters on induction. The part 3 of the third part is received little attention from philosophers, and as far as I know has never been translated. The chapters I have looked at come across as terribly weak, unlike the powerful and insightful work of the first two parts.

Ockham says that induction "is a progression from singulars to the universal", which is pretty much the modern understanding of the term. He gives the example "this [man] runs, and that one, and so on, therefore every man runs". What does the 'and so on' mean? If it means 'every other man apart from this one and that one', then the argument amounts to 'Man a and man b and every other man apart from a and b run, therefore every man runs', which is trivial, though admittedly valid. Or does he mean that the 'and so on' is a placeholder for a longer proposition which enumerates every man there is, and so does not include the word 'every', but concludes with the word 'every'? But that would not work, because the enumeration would have to conclude 'and there are no more men'. Mathematical induction, of course, is different from any of these. As I understand it, it is an inference from the properties of the successor of every number, to the properties of all numbers of a particular type (the naturals). Thus the antecedent and the consequent of mathematical induction both contain the word 'every'. I may be wrong, I'm sure my mathematically minded commenters will leap in to correct me if so.