Russell's Principia: Paradox, Axiom Systems & Set Theory

Russell's Paradox 
Bertrand Russell’s paradox wasn't taken as some little pleasurable game or puzzle. It "forced logicians to recast set theory and logic in a different way". Why was that? Because it kept ‘creeping in’ to set theory and logic generally. Despite that, logicians wanted to "still retain the bulk of what was useful and descriptive in the original systems" (326).

It was largely as a result of Russell’s Paradox, and the desire to create a solid foundation for mathematics, that Russell himself and Whitehead wrote the Principia Mathematica. At the heart of this book we can say is

"his first full-scale attempt to describe all of mathematics as a formal axiomatic system – an organisation of mathematical ideas based on a small number of statements assumed to be true".

No doubt this too was influenced by the success and structure of Euclid’s geometrical system as well as by the work of Frege, Cantor and others.

Axiomatic Systems

At the core of an axiomatic system is a short list of simple statements called axioms. What do they do? They're combined in specifically defined ways to derive a much larger set of statements called theorems... 

No. No, what are axioms? Not what do they do.

Anyway, forget the axioms themselves, what about the axiom system or systems to which they belong? What are they? More to the point, what did Russell and Whitehead want from their axiomatic system or systems? Take these examples:

i) A system powerful enough to derive sophisticated statements about mathematics as theorems.

ii) A system that avoided all inconsistencies, such as Russell’s Paradox.

iii) A system which could show that all possible mathematical truths could be derived as theorems.

Number iii) is quite amazing. A single system from which all possible mathematical truths could be derived as theorems. Would that have really been just a single axiomatic system rather than a collection of systems?

It can now be said that "their system also eliminated paradoxes of self-reference, such as Russell’s Paradox" (326). Perhaps this isn't surprising since it was his own paradox he was trying to counteract. However, the writer goes onto say

"whether the Principia Mathematica could avoid all inconsistencies and provide a method to derive all of mathematics remained to be seen". (326)

According to Gödel, I thought that this proved to be impossible. However, the writer only talks about eliminating all ‘inconsistencies’. Perhaps that is achievable. After all, Gödel’s proof shows that we can't have both overall consistency and overall completeness. So perhaps in Russell and Whitehead’s case we do have total consistency, without completeness. Clearly Principia Mathematica didn't and couldn't deliver both consistency and completeness!

In retrospect we can now say that even

"though the axiom set of the Principia did solve the problem of Russell’s Paradox, in practice it was awkward, so it didn’t catch on with mathematicians". (327)

Set Theory

This was something that the young Quine realised in those early days. However, a different set of axioms solved the same problem. This well-known and important set of axioms is called the Zermelo-Frankel axioms (ZF axioms). It solved the problem of Russell’s Paradox and other problems of self-reference by "distinguishing sets from more loosely defined objects known as classes" (327).

How did these axioms do that?

I think that primarily it was a case of Frankel’s Foundation Axiom which showed that sets can never have other sets as members. Perhaps Russell’s Theory of Types and Tarski’s ideas about meta-languages and object-languages are also connected to the solution of Russell’s Paradox and other self-referential paradoxes. In any case, today

"the words set theory usually refers to one of several versions of set theory based in the ZF axioms". (327)

E.O. Wilson against Analytic Philosophers

Some philosophers think that some scientists have a naïve and simplistic view of reduction - and indeed of much else - in science. Especially when it comes to the complexities of causation, which they see as their own pet subject. It's not surprising, then, that E.O. Wilson gets a lot of flack from philosophers. Wilson is not, after all, a philosopher. More relevantly when it comes to reduction and causation, he's not an analytic philosopher. It's interesting, then, to see what Wilson himself thinks about these inevitable criticisms from (analytic) philosophers. 

In his book Consilience, he writes:

"'The subject I address they consider their own, to be expressed in their language, their framework of formal thought. They will draw this indictment: conflation, simplism, ontological reductionism, scientism and other sins made official by the hissing suffix. To which I plead guilty, guilty, guilty.’" 

E.O. Wilson cites all the jargon one would expect analytic philosophers to use when criticising not just scientists, but also all non-analytic philosophers. (Sometimes also when criticising other analytic philosophers.)

For example, "conflation, simplism, ontological reductionism, scientism and other sins". That is, they're accusing all of us (not just scientists) of not being analytic philosophers. Of daring not to talk about ‘conditionals’ and ‘possible worlds’. Of daring not to read at least five papers a month on causation and possible worlds. Of daring not to use "their language, their framework of formal thought" because they truly believe that there's only way of attaining the truth – their way. That there's only one way of being logical – their way. That there's only one way of being rational – their way. And that there's only one way of being philosophical – their way. Anything else is sneered at and criticised for "conflation, simplism, ontological reductionism, scientism and other sins".

The sin, for example, of not reading Synthese, Analysis or Mind. Of not using the sign for the conditional or schematising one’s writing in a pseudo-scientific manner. Of not being up to date with normativity or what Ted Sider said last week. And so on.

So no wonder Wilson "pleads guilty, guilty, guilty". He can't do anything else. No one outside the Analytic Academy can plead anything else but "guilty" to not being an analytic philosopher or writing analytic-philosophy prose.

All I can say is: What’s wrong with ‘ontological reductionism’? What’s wrong with ‘scientism’? Indeed, what’s wrong with simplicity and a bit of ‘conflation’? There may be things wrong with these things. However, in large parts of the Analytic Academy it's simply assumed that reductionism, scientism and the rest are wrong. After all, Wittgenstein and whomever told us that they're wrong.

The real reason - or one main reason - why some analytic philosophers accuse E.O. Wilson of all these things may be because they've "not kept up" with science. Wilson writes:

"It appears to me that professional philosophers have not kept up with the foundational disciplines of neuroscience, behavioural genetics, and evolutionary biology, and as a result have surrendered their franchise to the scientists. The scientists, not the philosophers, now address most effectively the great questions of existence, the mind, and the meaning of the human condition. This surrender seems to be permanent, and professional philosophers have begun a diaspora into other vital and challenging disciplines that include theoretical neuroscience, evolutionary theory, intellectual history and bioethics."

However, in many cases, especially in England rather than America, analytic philosophers were never up to date when it came to science. They probably weren’t even up to date with Newton.

For example, the "ordinary language" and "linguistic" philosophers (as well as some "analytic metaphysicians" today) championed their ignorance of science and said that no scientific findings had any effect on philosophical fundamental problems or truths. That is, philosophy is an essentially a priori discipline which can't be touched by science or its findings. Of course, the Americans and the logical positivists thought otherwise.

All of this, of course, may be a massive generalisation on Wilson’s part. Surely not all philosophers (certainly not philosophers of science) are ignorant of contemporary science. What about Dennett, Churchland, van Fraassen, Putnam, Quine, Fodor and all the rest? They're far from being ignorant of science. Many of them are (or were) mathematicians and logicians. In any case, how up-to-date is up-to-date? After all, philosophers aren't scientists: they're, well, philosophers. Of course, there will be gaps in their knowledge – sometimes large gaps. That’s why they're philosophers and not scientists. If they knew as much as scientists, then they would probably be scientists instead of philosophers.

Wilson also says that scientists

"now address most effectively the great questions of existence, the mind, and the meaning of the human condition".

The "meaning of the human condition" doesn’t sound like a fit subject for science. Perhaps my view is prejudiced or perhaps things have changed in science and its ambit has enlarged somewhat – especially in the advent of "inter-disciplinary research".

And what does Wilson mean by "the great questions of existence"? This sounds like metaphysics or even ontology – surely not a fit subject for any science. And, yes, scientists may well study the mind; though only by reducing it to the physical, or behavioural, or functional, or the computational. Then they'll be studying something that is scientifically respectable. However, will they be studying the mind or consciousness if they leave out, for instance, qualia or the first-person perspective?

Is it true that philosophers 

"have begun a diaspora into… theoretical neuroscience, evolutionary theory, intellectual history and bioethics"? 

Or is it really a case of philosophers becoming more interdisciplinary and therefore using the findings of theoretical neuroscience, evolutionary theory, etc. in their philosophy? That's not the same thing at all.

Steven Stitch and rest are still philosophers who happen to use - and indeed depend upon - the findings and sometimes the methods of science. However, they're still philosophers – interdisciplinary ones!

Philosophers have always been interested in science and indeed up-to-date. Think of Aristotle, Descartes, Leibniz, J.S. Mill, Russell, Quine, Carnap and all the rest. Indeed, science even provided the philosophers with some of their own problems, as in the case of scepticism about the external world in Descartes’ case.

The Necessary Nature of Numbers

We can say a world in which there are no animals is also a world in which there is no horses. That appears to be a necessary truth in that at no world at which there are no animals can there be horses. Is it, however, only a conceptual truth and not a metaphysical truth? Is it a truth about our concepts, or concept-kinds, in that contained within the concept [horse] is the concept [animal]? Thus a horse must be an animal, conceptually speaking. Or is it a truth about kinds as they are in themselves – independently of minds and concepts? However, we can only get at horses and animals through our concepts or through our classifications and categories. And they are mind-dependent.


Some have argued that there may well be a world at which 2 + 1 equals 4. Is that possible? Where would the number 3 come at this world? Could it come after, say, 4? In that case, 4 would be 4 + 1. If that number shifted, then so too would all the others. If 4 were 4 + 1, then 5 would be 4 + 2 or 5 + 1. Alternatively, perhaps there is a world in which 3 is simply missing and 4 is the immediate successor of 2. Wouldn’t that pattern need to be repeated? Not necessarily. If it were, then 6 may also be the immediate successor of 4. Would that mean that this is effectively a different arithmetic to our own, or perhaps not arithmetic at all? Can there actually be alternative arithmetics in the way that there are alternative geometries (despite the fact that ‘alternative’ geometries do not necessarily contradict each other).


Despite all that, we can say that what makes a natural number the number it is, is its position in the number series. This seems to unequivocally rule out the possibilities so far discussed. That is, 3 is 3 precisely because it comes after 2 and before 4. Change the series and you change everything. Thus you couldn't even call 3 ‘3’ at this possible world. 3 is its positions in the natural number series. It gains its identity or meaning through its position in the natural number series (which is the basis for arithmetic).


Alternatively, we can simply say that "if per impossible 3 and 4 switched places, 4 would now just be 3" (133). This world may use the inscription ‘4’; though that inscription would still actually be the number 3. Even if it this isn’t a case of inscriptions versus real numbers, their 4, not their ‘4’, would still be 3 (or do we mean ‘our’ 3?).

Frank Ramsey’s Redundancy Theory of Truth

F.P. Ramsey claimed that the statement

p is true

is logically equivalent to 

p

Both have the same truth-value. However, surely if that were true, then we wouldn't even require the concept of truth at all. The predicate "is true" is a useless or meaningless appendage to p. In fact, "we can say everything we want to say without it".

What is lost when we say p rather than "p is true" besides the linguistic predicate itself? Do the two words "is true" actually add anything to what's already there? However, if it were already there, then perhaps we haven't done away with truth after all. We can say that contained within p is the meaning that p is actually true. Alternatively, perhaps the predicate "is true" is simply a "reaffirmation" of the p which comes before it.

We can ask: What makes p true? Is it its truth-condition? Then we can pair the proposition with its truth-condition. The problem is that this will give "a different result for each proposition". The truth-condition of one p may be snow's being white and another may be ravens' being black. Why not ask, then, what it is that makes any proposition true? What do all true propositions share?

Perhaps they don’t share anything. 

Perhaps the question is "over-generalised". ("How much does anything weigh?") We can compare each proposition with its truth-condition. However, "there is no general truth about truth in the sense required by the traditional theories".

For example, are truths about the past the same as truths in mathematics or truths about what’s happening now? What about negative truths like the truth that Gordon Brown is not in my room now or the truth (if it is a truth) that God doesn't exist?

Instead, we should see truth as part of our language or as part of a language. As S puts it, truth

"does not have the magic property which nothing could have, of leading us out of language, into some direct of transcendental encounter with the world".

What would such a direct or transcendental encounter be like? It would be languages-less. Thus, we can't have such an encounter because everything we say about the world is said in some language or other.

And the predicate "is true" is unequivocally part of that language. If it is part of our language, then it is part of us. It belongs to our concepts, senses, categories, classifications and the like. It is polluted by minds and by language.

The problem has been that many metaphysicians have wanted something that is somehow language-less or even mind-less that can make truth something deep and more profound than just about everything else. They have wanted truth to be, then, a non-natural property or thing both in and out of the world. Something non-observable, non-concrete than can somehow belong to the world or be applied to the world. They wanted the best of both worlds. They wanted something transcendent to be applied to the immanent. Something abstract (or just non-spatiotemporal) to be part of - or be applied to - the concrete and spatiotemporal.

The problem is that we have become used to the predicate "is true". We use it everyday of our lives. Is it any wonder, then, that it would be hard to do without it in our everyday discourse? 

For example, take the locution

"The truth about Mozart’s death."

That certainly makes sense. We can say: "The facts about Mozart’s death." However, facts are almost as metaphysically controversial as truths. Not only that: there may be a strong relation between fact and truth.

What is a fact? 

It is something that is true? 

What makes something a fact? 

That something is true of the world and that makes it a fact? 

In addition, facts are said to correspond with true propositions or statements. 

Thus, there doesn’t seem to be a way out of this semantic circle. 

The same is true about "a story which is largely true". We can say: "A story that is largely factual." Here we have the same problems again. What makes this story factual? The fact that it is largely true!

According to Frank Ramsey’s theory, "how could you remove these words from these phrases"? However, the fact that we can’t remove the word ‘true’ from these phrases doesn't automatically mean that truth is a metaphysical property - or any property at all. We also use the word ‘Superman’. However, Superman doesn't exist. We can even use the phrase ‘"the round square"; though the round square doesn't exist (not even at a possible world or abstractly).

Similarly, we can say that ‘not’, ‘or’ and ‘and’ don't refer to anything. However, such words do have a use and they can be implicitly defined. Perhaps we can say that ‘true’ has a use and we can define its use in our discourse. However, it may have a use and also a definition which does not entail that truth is also a metaphysical property of some kind. It may function like ‘or’ or ‘not’ in our discourse. Alternatively, it may be closer to the word ‘yes’ or the word ‘stop!’.

Constructivism & Intuitionism

Constructivism

The primary position of constructivism is simple. The constructivist "believes that we have no conception of mathematical truth apart from the idea of proof".

Simply, 

proof = truth

Or:

mathematical truth = proof

It follows that if truth = proof, then truth and proof are (despite Platonism) inventions of the human mind. Proof is all there is. More specifically, we can say the same about numbers. Numbers "do not exist until constructed, by operations which generate them in a finite number of steps". Mathematical operations don't just use numbers: they also construct them.

This leads us to a question: 

What did these mathematical constructions use before they constructed the numbers? 

What constituted the mathematical constructions before the numbers were actually created? Were numbers there from the beginning? In that case, who or what created them? Or, if they were there from the start, perhaps they weren't constructed (or created) at all and Plato was right after all.

The stark conclusion of constructivism is the ‘anti-realist’ idea that "all existing numbers are contained in the books and papers of the mathematicians" (384). Numbers aren't discovered or intuited by mathematicians. They're constructed or created. Thus if a number hasn't been constructed or proved, then it quite simply doesn't exist to be discovered or intuited. In addition, only numerals (not numbers) really exist. And to say "that numbers exist is to say that there are valid proofs involving numerals" (384). (This appears to be very like Hartry Field’s position.)

This position is very similar to that endorsed by Kant over a hundred and forty years earlier. Kant believed that mathematical propositions "are known a priori since we ourselves are the authors of them" (385). Is this mathematical idealism? Their a priori status is guaranteed simply because we don't need to look outside of our own minds to the empirical world (or even to a platonic realm) to discover numbers and their nature.

Intuitionism

Now we arrive at intuitionism, a variant on constructivism.

Here too proof is everything. However, there's a surprising conclusion to this emphasis on mathematical proof. We've already said that a 

"mathematical proposition is true only if there is a proof of it; similarly, it is false only if there is a proof of its negation" (385). 

But what if there is proof of neither? Does that mean that the proposition is neither true nor false? Perhaps it simply means that the proposition is "meaningless" or that it's not a genuine example of a mathematical proposition.

However, the intuitionists accepted one of these conclusions. The proposition may well be neither true nor false. It's still, however, a bona fide proposition. We must, therefore, deny the law of the excluded middle for such mathematical propositions. That is, we must deny the principle: either p or not-p. This means that such mathematical propositions must have a "third value". This third (truth?) value is often called "indeterminate".

There are more surprising conclusions one must accept if one is an intuitionist. For example,

"as Heyting demonstrated, we shall need an entirely new system of logic – which he called intuitionistic logic – in order to accommodate the constructivist vision of mathematical truth" (385).

The logicists tried to reduce mathematics to logic. Now we find that a discovery in mathematics will have a profound effect on logic itself. If mathematics requires a third truth-value (indeterminate), then so too will logic (which, of course, also deals with truth). Indeed a logical vision or system must ‘accommodate’ the new findings of constructivism or intuitionism. Does this in itself show us that logic is part of mathematics, rather than that mathematics is part of logic? Perhaps not in all cases.