Gödel: Proof, Truth & Meta-theory

What is a mathematical theory? We may define a theory as a set of axioms. Of course there needs to be more to a theory than that. What do we do with this set of axioms? Firstly, we need the rules of inference. These rules of inference tell us what we can and can't infer or derive from the set of axioms.

What can we infer or derive from the set of axioms? Theorems. That is, a theorem is any formula which follows from the axioms by repeated application of the rules. The theorems grow out of the axioms (as it were). This must mean that whatever is in the derived or inferred theorems must have already have been in the axioms. Nothing in the theorems which was not first in the axioms – even if hidden or disguised. This makes one ask: How, exactly, do the theorems ‘follow’ from the axioms? What do we mean by ‘follow’, ‘derive’ and ‘infer’?

However, despite the fact that we have a set of axioms for a mathematical theory, this theory "is empty until the axioms have been provided with an interpretation" (394). Intuitively it seems strange that we can ‘interpret’ axioms at all. After all, don’t we create the axioms? Or was Plato right – do we actually discover them? In any case, how do we interpret axioms? We "need to assign values to the primitive terms, and to show how the values of formulae may be derived from the values of their parts" (394). This makes mathematical meta-theory seems like semantics. In semantics too we ‘need to assign values’ to the predicates and phrases found in a truth-valued sentence. Not only that: we also have the other basic Fregean insight that we "show how the values of formulae may be derived from the values of their parts" (394). What are the ‘values of the primitive terms’? What are the primitive terms?

In any case, the interpretation of the axioms, or their primitive terms,

"is not something done by the theory itself, but something done by us, through another theory – the meta-theory" (394).

Clearly the mathematical theory can't interpret itself. It can't even be done by us through the theory itself. We must construct another theory, a meta-theory, in order to offer interpretations of the lower-level theory. That simply means that the theory exists, or can exist, even before an interpretation of its axioms is given. That is why lower-level mathematicians need not concern themselves with meta-theory at all – indeed, they rarely do.

One of the main things mathematical meta-theories do to theories is provides proofs of particular formulas in the theory (or, in Gödel’s case, a proof that a theorem can't in fact be proved). The important point is that the theory itself can't provide a proof of a formula within itself.
 
What would count as a proof of a formula? If
 

"we can show that the axioms generate all truths of arithmetic, and that p is such a truth, then we have proved that p is provable’"(395).

It's hard to imagine that the axioms of a particular theory can ‘generate all truths of arithmetic’; so perhaps this means many or all such theories in mathematics. In addition, if these axioms generate all the truths of arithmetic, then this must mean that the axioms, and perhaps the theory to which they belong, must exist before arithmetic. Or is it the case, instead, that arithmetic exists though it doesn't have its ‘truths’ proved until the meta-theory comes along? However, on a platonistic reading, the truths still exist before the proofs. The proofs are for us (as it were). It isn't that p, in the theory, isn't true until it's proved by the meta-theory. It's just the case that the meta-theory proves that it's true – for us. It would still be true without proof and it was true before proof.

It's often said that a mathematical theory or system can't be complete and fully consistent or fully consistent and complete. This is a question of proof as to consistency and completeness. That is, ‘there can be no proof of the completeness of arithmetic which permits a proof of its consistency and vice versa’ (395). So we either have a proof of arithmetic’s consistency without a proof of its completeness, or a proof of its completeness without a proof of its consistency.

Why is it, precisely, that we can't have both? More to the point, Gödel provided us with a proof that we can't have proofs of both completeness and consistency; though we can have a proof of either one or the other. Again we must stress that this is primarily a question of the set of axioms which generate arithmetic. That is, "we cannot know, of some system of axioms which is sufficient to generate arithmetic, that it is both complete and consistent" (395). Why can’t we show both completeness and consistency? The end result of this is that ‘there may be formulae of arithmetic which are true, but not provable’ (395). So, in this case at least, truth need not come along with proof. Isn’t that what the intuitionists are against? That is, if we have no proof of an equation or formula, then that formula, quite simply, can't be true. The proof brings its truth into existence; just as it's also said that proofs bring the actual numbers of arithmetic into existence (or at least their ‘constructions’).

What has all this to do with the failure of the logicist programme? We can now say that "no logical system, however refined, will suffice to generate the full range of mathematical truths" (395). Isn’t it the case that the logicists tried to reduce mathematics to logic? Here it is said that ‘no logical system will suffice to generate mathematics’? Does this amount to the same thing? That is, is the generation of mathematics from logic the same as the reduction of mathematics to logic? That is, if mathematics is genuinely generated from logical principles, even if that is not fully known, then clearly it will also be the case that mathematics can be reduced to the logic. If logic generates maths, then maths can be reduced to logic.

There also seems to be a connection being made here between the logicist programme, or between logic itself, and mathematical provability. It ‘follows too that we cannot treat mathematics as Hilbert had wished, merely as strings of provable formulae: the theory of “formalism” is false’ (395). That is, if mathematics is reducible to logic, then all mathematical formulae must be capable of being proved. This means that proof is essential in logic. However, as Gödel and others discovered, not all mathematical statements can be proved. Thus, because proof isn’t everything in mathematics, then maths must be something above and beyond logic, in which, we can say, proof is everything. This lack of (complete) provability in mathematics, in fact, makes it very different from logic.

Does this make mathematics non-a priori and therefore a posteriori? Or is it non-a priori without also being a posteriori – if that is possible? Whatever the truth is, mathematics, or its propositions, outstrip provability and, one must conclude, the principles of logic too!

This conclusion has a very platonistic ring to it. That is, ‘if there can be unprovable truths of mathematics, then mathematics cannot be reduced to the proofs whereby we construct it’ (395). This conclusion not only goes against Hilbert’s attempted reduction, as well as the logicist programme, but also against the intuitionists who believe that truth and proof are intimately connected. Indeed that one can't have truth without proof.

Not only that: the intuitionists also believed that the truths of mathematics were generated by the proofs whereby we construct them. As for the platonistic conclusion to all this, we can say that if truth is genuinely divorced from proof, and that mathematical truths can be true without thereby being proved to be true, then they must be true regardless of our means of proving or even knowing them. Thus they must be mind-independent in some substantial sense.

We can conclude that there ‘is a realm of mathematical truth, whether or not we can gain access to it through our own intellectual procedures’ (395). This is not just a question of the rejection of the truth = proof principle; but of the necessity of mathematical truth being necessarily connected to any of ‘our own intellectual procedures’ (395). Thus it isn't necessary that we have any kind of access to the ‘realm of mathematical truth’! We certainly don’t need causal or epistemic access to it. We may have access to this mathematical realm; though it isn't necessary that we do so. It would still exist, mind-independently, even if we didn't know that it in fact exists. This platonic realm doesn't even care about being known by us. It exists happily without being known by us.

Not only did Gödel prove that certain truths within mathematics can't be proved to be true, he also proved that this platonic realm of unprovable and provable mathematical truths exists. He proved that p can't be proved. However, p is still true; though not provably true.

The question remains as to whether or not Gödel needed this platonic realm in order to believe, or prove, what he said about the nature of mathematics. Can Godelianism be non-platonic or must it be platonic? Can we accept the notion of mathematics’ essential incomplete provability and yet not be a Platonist as well? Or do all these conclusions about mathematics come along with a commitment to Platonism?

Platonic Mathematics

Roger Scruton argues that the platonic position on the true nature of mathematics is basically based on four fundamental points. He lists them thus:


i) We know many arithmetical truths, and know them without a shadow of a doubt.
ii) Arithmetical truths are about numbers.
iii) Numbers are the subject-matter of identities, and indeed identity of number is one of the primary mathematical concepts.
iv) Truth means correspondence to the facts.


I suppose that I can provisionally agree with 1) above. We do seem to know arithmetical truths without a shadow of a doubt. We also intuitively believe that arithmetical truths are about numbers. I suppose if arithmetic is about numbers, then this aboutness implies that such numbers are not our own invention.


As for 3), numbers are essentially about identities. 1+ 1 = 2; thus 2 = 2. Or, more complexly, 2 + 2 = 4 itself equals 4 = 4.


Poincare said that the whole of mathematics boils down to a gigantic A = A – though only if one thinks that mathematics deals with tautologies!
 
Now 4). We say that the statement ‘snow is white’ is made true by the fact of snow’s being white. Thus the inscription ‘2 + 2 = 4’ is made true by the mathematical fact that 2 + 2 = 4. Mathematical statements correspond with mathematical facts. Indeed, each number has its own reference. The inscription ‘4’ refers to the number 4,just as ‘snow’ or ‘Tony Blair’ has a reference when embedded in a true sentence. (Though does ‘4’ also have a Fregean ‘sense’? Indeed does ‘2 + 2 = 4’ have a Fregean ‘sense’ or does it express a ‘Thought’?)


The upshot of all this is simple. We can now say that it is hard to accept 1) to 4) and still "deny that numbers are objects" (383). Of course many philosophers do reject 1) to 4) – especially 2) and 4). Wittgenstein, primarily, rejected the view that mathematics is about objects, correspondence and mathematical facts. This is a mistake. It is to conflate what is true about world-directed or empirical facts with what is true about mathematical propositions. They are not, in fact, the same, and for many reasons which Wittgenstein forcibly gives.


There is one final and often noted problem with the platonic position on numbers or mathematics generally. If numbers really are as Plato thought they are, then numbers "take no part in any change or process; they are causally inert" (383). That is why Plato put them in a transcendent realm. Though if they are causally inert, how do we gain access to them in the first place? Of course Plato had an answer to this. We ‘intuit’ them with our intellectual faculty. Is this really an answer or a bunch of bullshit?
 

Externalism in Broad Outline

The thing we need to get out of the way is the idea that externalism has nothing to do with the "boring reason that we are causally affected by our surroundings, so we are likely to notice when those surroundings changed" (421). No one would deny that truism. Of course we are causally affected by our surroundings. Externalism, instead, argues that the "contents of our thoughts are 'fixed' or 'determined' by the context in which they occur" (421). That means that the contents of our thoughts are fixed or determined even if we don't know that this is so. Perhaps it's the case that we never really know that our thoughts are fixed or determined in these ways by our environment. The fact that we don't know is, I suppose, the whole point of the externalist thesis.

We can now ask how they are so determined or fixed without our knowing it.

The incredible thesis of externalism is that

"two indistinguishable thinkers might entertain thoughts with very different contents – thoughts about very different things – if the thinkers are embedded in very different environments" (421).

In what way, then, are the thinkers indistinguishable? This means that their psychological access to their thoughts (and even the thoughts themselves) are indistinguishable even though they have different contents. The fact that they have different contents occurs because they are in different environments; even though they are psychologically indistinguishable. This must mean that the different environments must determine or fix the different contents even if the thinkers are thinking the same things in the same ways. This also means that their minds, when taken apart from their environments, are indistinguishable.

What makes the contents of the thoughts different are the different environments. John Heil argues that this "suggests that there is no way to infer from intrinsic features of a thought to its content: what it is about" (421). We can't analyse their minds, and even what they say about their psychological states, to the real content of those states. The environment, along with internal states, determines and fixes the content. What the states ‘are about’ determine and fix the content of the states. This means that these thoughts can be about something which the thinkers don't effectively know they're about. What they are about, and their environments, fix and determine the contents of such thoughts.

This thesis goes squarely against Cartesian internalism in which all there is to know is what the subject himself knows about his own mind. On that picture, the environment simply doesn't matter – at least not once it has fixed or determined such internal mental states. The Cartesian thinker can break free (as it were) of his environment and still know all there is to know about his own mind and the contents of his own thoughts. This Cartesian vision, however, creates the sceptical dilemma. It means that "agents [are] faced with the problem of 'matching' their thoughts to the world" (422). If minds can exist in splendid isolation from the world, and if agents can know all there is to know, and all they need to know, about their thoughts, then how do they get back to the world and match their thoughts to the world? The radical separation of mind and world has created the sceptical problem of whether or not we do in fact actually correctly match the world when we think about it.

According to externalism, however, there is no such sceptical problem because

"if your thoughts are fixed by the world there can be no question of their 'matching' or failing to match an 'external reality': what your thoughts are about automatically matches the world around you" (422).

If one’s thoughts are fixed by the world in the first place, then the problem of matching or not matching the world ceases to be a real problem. We wouldn't even have these thoughts if they weren't already fixed or determined by the world. We would have no thoughts about the world if the world itself hadn't fixed those thoughts which are about the world. There would be no aboutness if the world had not fixed or determined that aboutness in the first place. The Cartesian wouldn't have thoughts about this or that aspect of the world in the first place if this or that aspect of the world had not already fixed or determined the content of his thoughts about the world (or his thoughts about this or that aspect of the world).

All this lead Putnam to say that "meanings ain’t in the head". Does that mean that meanings are literally in the world? As Heil puts it:

"What we mean by the sentences we utter is partly a matter of how we are situated in the world." (423)

Instead of meanings being parts of the world, so to speak, meaning is a matter of how we are situated in the world. Our place in the world will determine or fix what it is that we mean by our sentences (or words?). What does Heil mean by ‘partly’ a matter of how we are situated in the world? What are the other parts of meaning which we need to know about?

We can think of the Cartesian and externalist pictures in the following ways. The Cartesian thinks that

"words are connected to things… by 'outgoing' chains of significance guided by the agents’ thoughts ('noetic rays') (423)."

The externalist, on the other hand, thinks that words are connected to things by

incoming causal chains.

In the Cartesian picture, the relation is one of mind-to-world. In the externalist picture, the relation is world-to-mind. In the former case, the agent’s thoughts determine the meanings of our words and sentences about the world. In the externalist case, the meanings of our words and sentences are determined and fixed by incoming causal chains.

What are these incoming causal chains? Putnam "emphasises causal connections; Tyler Burge discusses social factors affecting the meaning of what we say" (423). Putnam’s version seems to be a strictly scientific account of meaning (if there can be such a thing). Burge, on the other hand, seems to go back to Wittgenstein’s idea that ‘meaning is use’ – perhaps with an equal emphasis on the causal chains which come from social practices into the minds of agents within those social practices.

David Lewis’s ‘Possible Worlds’ (1986)

i) Quantifying Over Anything

ii) Quine, Lewis and Meinong's Jungle

iii) Necessity

iv) Possible-worlds Realism

v) Existence and Actuality

vi) The Point of Possible Worlds

This is an account of the 'Possible Worlds' chapter of David Lewis's book Counterfactuals (1973).

Quantifying Over Anything

It can be assumed that most people will accept that

“there are many ways things could have been besides the way they actually are”.

However, the American philosopher David Lewis (1941 - 2001) believed that this sentence involves an existential quantification. Why did he believe that? Surely you can only existentially quantify over that which exists. That’s why it’s existential. “Ways things that could have been” don't actually exist. There could have been three-headed snakes; though there aren’t. Therefore we can’t quantify over three-headed snakes. As W.V.O. Quine put it: “To be is to be the value of a bound variable.” Things that could be can’t be the values of variables... Actually, if the philosopher or logician wants to, he can quantify over everything and anything – over literally everything in some cases (as well as nothing, in the case of the dialetheist philosopher Graham Priest).

Lewis qualified his argument by saying that people who believe in possibilities “believe in the existence of entities”. What is Lewis’s argument for this move from belief (in possibilities) to existence? Believing in ways things could have been doesn't entail (or imply) their existence. So if these ways things could have been are possible worlds (or parts thereof), then possible worlds exist?

Why does belief in possibilities entail (or imply) existential quantification? Again, David Lewis himself might have been a bricklayer; though David Lewis the bricklayer didn't and still doesn’t exist. That is, we can’t quantify over a bricklaying David Lewis (unless it’s just someone with the same name).

Lewis, however, preempts these problems by asking the following question:

“If our modal idioms are not quantifiers over possible worlds, then what else are they?”

Was Lewis asking us where the bricklaying David Lewis is if he isn’t at a possible world? Who is it, precisely, that I’m talking about? Is this Plato’s Beard all over again? That is, if we talk about a bricklaying David Lewis, then he must exist in some shape or form. Thus:

Are there different modes of existence (or being), including a mode of existence at possible worlds?

Quine, Lewis and Meinong's Jungle

Quine (in his ‘What There Is’) has provided us with strong arguments against such extravagant Meinongism. (Though was Lewis really an extravagant Meinongian?) This problem itself brings in a whole host of accompanying problems about the references of words and the names of entities that seemingly don’t actually exist.

Yet we can quite happily talk about a round square: does that fact somehow bring about the round square’s existence? I can talk about a Possible Murphy who has three thousand girlfriends. Does my talk alone bring this Possible Murphy into existence? Indeed who is this God, for example who “doesn’t exist”? (This is something Bertrand Russell grappled with way back in 1918 in his 'Existence and Description'.)

However, this is still a fair question:

What is it we're talking about when we talk about “way things could have been”?

Necessity

The same is true, according to Lewis, when we talk about any given x being necessary (rather than merely possible). What are we talking about when we talk about this or that being necessary? What are we referring to? What makes this or that necessary?

Necessity can’t be seen (as it were) in one world – in our own world. Therefore it must be something about (as it were) every possible world. That is one possible expression of Lewis's position.

Thus when we say that

2 + 2 = 4 is necessarily true.

what are we saying? We're saying that this equation is true at every possible world - even in a world made of alcohol seas or one without our own physics. We can only make sense of necessity - in this and in all instances - by believing in the possible worlds that make our statements of necessity true. Without possible worlds, what is it that makes 2 + 2 = 4 necessarily true? After all, it may be true in our world; though how do we know that it's true at all other possible worlds? We know by imagining other possible worlds (of all shapes and forms) and then we quickly realise that 2 + 2 = 4 must be necessarily true at these worlds too. If 2 + 2 = 4 were true only in our world, then it wouldn’t be necessarily true.

Again, why are logical or mathematical truths necessarily true? Because they're true at all possible worlds. Their necessity comes from their being true at all possible worlds. Thus, in order to guarantee (or insure) necessity and possibility, we need possible worlds. That, anyway, is part of Lewis's argument (in this paper at least).

Possible-worlds Realism

So let’s be clear what Lewis believed about possible worlds. Thus:

i) Are possible worlds simply theoretical constructs?

ii) Are they convenient posits which somehow solve a whole host of problematic modal issues?

iii) Are they fictions-for-a-purpose?

iv) Or, in Lewis’s own words, are they “linguistic entities”?

The answer in all cases is: Absolutely not! Lewis was a realist when it comes to possible worlds. That’s what he’s famous for. He wanted to “be taken literally”.

Though precisely what should we take literally?

Well, for a start, “possible worlds are like our world”, according to Lewis. They are, in fact, (often?) very similar to our world. So what’s different about them? Well, different things “go on in them” than go on in our world. Therefore it can be said that - departing a little from Lewis - possible worlds have exactly the same constituents as our world; though those constituents differently configured. (This was D.M. Armstrong's position in his 'The Nature of Possibility'.) This means that possible worlds have legs, buses, atoms, trees, tables, etc; and, presumably, explosions, orgasms, car chases and so on. They also have David Lewises, Houses of Parliaments and so on. However, at one – or more - possible world, David Lewis (his “counterpart” - who's not literally our David Lewis) is a bus conductor. At another world, David Lewis is Prime Minister. In addition, at other possible worlds there are different configurations of atoms, molecules, etc., as well as different laws, constants of nature, etc.

This is where things get complicated.

Existence and Actuality

David Lewis said that all these other possible worlds exist; though they aren't “actual”... What the hell does that mean? Well, for a start, the word “actual” is indexical (like “here”, “there” and “now”). That is, what is and what isn’t actual is dependent upon (or contextual to) the circumstances of utterance. That is, our world is actual to us; and other possible worlds are merely, well, possible. However, at w, it's the case that w is actual. And other worlds, to w, are merely possible. So every possible world is actual according to itself (even if this is a personification); though only possible according to every other possible world.

Can we make sense of this distinction between actual and existent? At a prima facie level, “actual” and “existent” seem to be virtual synonyms. However, as stated, actual and existent aren't synonyms in Lewis’s scheme.

Strangely enough, Lewis actually says that the "unactualised inhabitants [of possible worlds] do not actually exist". That is according to us (i.e., not them), these inhabitants don't exist. Again, actuality is indexical.

Can we make sense of this strange ontology?

Lewis himself was explicit:

"To actually exist is to exist and to be located at our actual world…"

Here Lewis seems to be conflating existence and actuality. That is, surely we can we say that other-worldly persons aren't actual because they don't exist. What “mode of existence” (or being) do they have? (Being and existence aren't the same thing in philosophical literature.) If they "don't exist according to our world”, then what kind of existence do they have? There seems to be a logical contradiction looming here. Other-worldly persons both exist and don't exist. They don't exist (or aren't actual) according to us; though they do exist (or are actual) according to their own worlds.

What's going on here?

To some philosophers, the conclusion can only be that possible worlds don't exist. So what was Lewis's reply to this? In Lewis's own words, it "does not follow that realism about possible worlds is false". He also came out with this Zen-like statement:

“[T]here are more things than actually exist."

So some things that don't exist do actually…well, what, have being?

Again, the word “unactualised” seems to be synonymous with “non-existent”; whereas earlier Lewis offered us a distinction here. However, Lewis - in this paper a least - doesn't offer us a precise account of his ontological position; which would, hopefully, clear away some of these problems. Again, do non-actuals have some kind/mode of, well, existence? If not existence, then some mode of being?

The extent of Lewis's realism about possible worlds can be seen in the following passage. In it he stated:

"[T]here is much about them [possible worlds] that I do not know…"

So possible worlds certainly weren't imaginative creations to Lewis. If they were, then he, presumably, would have known everything about them. Possible worlds are therefore like unknown planets. Thus there are indeed a vast amount of planets out there; though we know precisely nothing about the vast majority of them.

The Point of Possible Worlds

Lewis didn't just believe in possible worlds because he thought that they exist or have being. He also thought that their existence solves various philosophical problems. So his interest (or belief) in possible worlds wasn't entirely contextless (if that's the right word to use).

So what did possible worlds do for Lewis and other possible-worldists? Firstly, they "systematize [our] pre-existing modal opinions". That is, they serve a philosophical purpose over and above the mere fact of their existence... or being.

What are these other worlds like, according to Lewis?

As stated above, although earlier in this paper Lewis wrote that possible worlds are very much like our world (only reconfigured - at least that's a word one can use), he also argued that the physics of some of these possible worlds will be – or are - different to our own. Indeed it follows from a belief in possible worlds that certain possible worlds must have alternative physics.

However, Lewis didn’t accept that any possible world can have an alternative logic or alternative mathematics. And isn't that the primary point (if they need a primary point) of possible worlds? Of course Lewis wasn't talking about specific (or generally-accepted) logical or mathematical systems; only that logical and mathematical truths and realities will be true regardless of our efforts to codify them. Thus Lewis was also a realist (as it were) about logic and mathematics. That is, there may be some logical or mathematical truths (or realities) that we human beings can never - or will never - know or be able to formulate (e.g., Goldbach's conjecture).

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Philosophical Notes (1)

1) ‘Light-waves and the different mixtures of lengths of wave emitted’ are the causes of colour; though not themselves colour. Are ‘differences in mean kinetic energy’ therefore separable from heat? Not according to Kripke: heat = mean kinetic energy. Can we say the same about, say, the colour blue? That is: blue = … lengths of wave…

Blue cannot exist without sensory receptors, according to Kripke. This does make sense. After all, the sun may boil, heat up, water without sensory receptors being there to check, as it were, this fact. If there were no sensory receptors, there would be no blue (as Berkeley made clear nearly three hundred years ago).

2) Why should Paul and Twin Paul ‘mean different things by the same word’? Water may be XYZ on Twin Earth and H2 O on earth; though they are phenomenally and epistemically identical. Why should their scientific constituents determine mental content and not their phenomenal features? Do H2 O molecules, or XYZ molecules, enter the brain and therefore determine mental content?

3) So many new theories of mind simply ignore the mind-body problem. I'm suspicious, but perhaps there is nothing more to say about it. Kim or Davidson doesn’t seem to have solved the problem.

4) Some theories of mind are so neat and tidy. It’s as if the mind-brain was designed by a logician or a computer designer. It’s all there in the books on logic. Similarly with Fodor’s LOT – it is language-centric. Churchland’s theories are messier because it is science-based. That is, they take into account evolution, etc. A philosopher of mind, e.g. Fodor, sees language-like computations and structures in the mind. A computer nerd sees computational events and structures. A logician… What comes first? The chicken or the egg? (Cheap ‘psychologism’?)

5) Does mathematics have meanings or references, or is it just the manipulation of symbols? Symbols of what? Abstract numbers in a platonic heaven?

6) What a strange juxtaposition – Ryle and Heidegger. However, they certainly share anti-Cartesianism. How deep does the resemblance go?

Ryle’s aggressive style is very refreshing when it's compared to the boring bullshit of J.L. Austin and other linguistic philosophers (although Ryle took some things from them). Ryle is largely ignored today. It is, of course, a shame. Such a lively writer, to say the least. It was surprising, therefore, that there is a chapter on behaviourism. Surely the question of whether or not Ryle was a behaviourist is answered there. (Or at least if he thought he was one, even if others thought differently.) This is not to say that we should accept his denials or affirmations; though at least the water will be made a little less muddy in this area. More precisely, was behaviourism an instrumentalist or pragmatic theory on his part, or did it squarely and ontologically deny mentality? Indeed could it have rejected mentality (bearing in mind the possibility that behaviourists must have had mental states themselves!)?

What I note, idiosyncratically, is how much Ryle knows about non-philosophical subjects. This is consciously missing in most Anglo-American analytic philosophy. He was, perhaps, a Renaissance man, unlike so many other analytic philosophers. (Excepting, perhaps, Roger Scruton! However, he’s sometimes an arse!)

7) Despite the popular-science approach of Daniel Dennett’s Brainstorms, there is still the use of logical notation:

(x) (Mx≡Px)

This is a very simple notation, involving symbols for the biconditional, predicates and a variable:

For every x such that each x is mental if and only if x is physical.

‘x’ could stand for Tony Blair. Then it would be:

Tony Blair is a politician if and only if he is corrupt.

8) Of course Kant wasn’t a Pietist is the obvious sense. He was a philosopher who built a huge edifice of moral philosophy. Calvin didn’t argue his case. He stated it. Kant argued his case with powerful philosophy. Powerful enough to convince a Puritan or even an atheist. But he was still a pietist with a small ‘p’!

10) The more you read of a particular philosopher, the more you understand and the more connections you can make. The broad concerns become apparent and it is realised that, in a sense, all his works are tackling the same issues or have the same concerns and presuppositions.

Take Quine. In nearly all of his papers and books he refers to ‘Neurath’s boat’. This is a story about philosophy’s dependence and necessary relation to science. Hume’s ‘mitigated’ scepticism resounds through all his work. And Kant’s a priori is as applicable to metaphysics as to moral philosophy.

I shouldn’t rely on interpreters or explicators of philosophers’ work. I should trust my own understanding of them. Why not? I will, of course, study interpretations and commentaries. I just won’t rely on them. This, surely, is a healthy position to take. After all, certain interpretations and commentaries are more difficult to understand than the original works themselves. Why did I ever believe that philosophers themselves would be harder to read? Think of all the great philosophers who were great writers too: Augustine, Plato, Hume, Schopenhauer, Nietzsche, Wittgenstein (?), Quine, Dennett, Fodor (?), Russell, Descartes, Plotinus…

11) Most analytic philosophers have a rudimentary knowledge of science; though most scientists know next-to-nothing about philosophy. (Lewis Wolpert has a positive distaste for philosophy and philosophers.) Interestingly enough, I have a compendium of science and mathematics which has a philosophy section - all the articles are written by scientists.

12) I was surprised to read that figure, solidity, extension, size, etc. were all deemed subject-relative in the Humean system. This just proves that, in these respects at least, he took more from Berkeley than he did from Locke.

13) I don’t think that there are total empiricists or total rationalists nowadays. The brain must structure experience in some way and to some degree. Similarly, experience generates nearly all conscious or brain processes.

14) Causation doesn’t particularly interest me; but Humean scepticism does. Perhaps my position is more psychological and political than strictly philosophical. I certainly don’t have the arguments for philosophical or analytical scepticism. On the other hand, Hume’s general position seems entirely commendable.

15) Why is causality so important? Because it is the ‘glue of the universe’? Because without necessary connections, science would lose much of its power? The same goes for the inexistence of substance. Lack of a self is fine and dandy. If Hume is right, change, even radical change, wouldn’t be a problem for anyone. Why can’t people change? That’s the fundamental question. There must be something that is as hard as concrete stopping people changing. What is it? It must be physical – that is, neurochemicals and/or possibly anatomical. Do people have a misguided loyalty to their past selves? This is a new possibility. Is it the case?

Hume says that the mind partly determines our experience of the world in contingent and psychological ways. Kant, on the other hand, claims that these determinations are necessary and a priori. Our acquired experiences are passed on, as it were, to the world, according to Hume. Kant claims that we are born with the concepts and categories which partly determine the nature of the world which we experience.

 I’m beginning to understand what Hume is all about. Primarily, his work is an onslaught against rationalist metaphysics. Hume uses the word ‘cause’ uncritically alongside a philosophical critique of our and scientific notions of causation. In this a contradiction? No. He didn’t reject causation in itself, only our conceptions of it. However, Russell, in his History of Western Philosophy, did detect certain contradictions and inconsistencies in Hume’s position.

16) On one hand the Catholic Church proclaims its absolutism and anti-relativism. On the other hand it tells us how it accepted Copernicus and Darwin. Can Catholics have it both ways, or are past adaptations acceptable but contemporary ones unacceptable?

17) 18th century philosophers were cultured men. Hume was an expert on Cicero and Ovid. Kant, likewise, on Horace.

18) Despite the technical detail and the power of the arguments, I still can’t intuitively accept that causation, space and time are not ‘out there’ in the world. Why are my intuitions relevant? Why do they matter? Why do I care at all about intuitions and common-sense? Perhaps the truth is counterintuitive. Is quantum mechanics intuitive or commonsensical? Of course not. Virtually all scientific theories were counterintuitive to the common sense of the lay person.

Kant’s work is extremely complex and difficult, and yet, in places, it isn’t ‘dry’ and ‘obscure’.

I simply assume that counterintuitive things can’t be true. What kind of reasoning lies behind this? Nothing springs immediately to mind. Perhaps it’s just a prejudice and nothing more. I’ve accepted other initially counterintuitive things in the past, so why should I stop now?

19) One reading of Kant’s own work, rather than another interpretation or commentary, both changed my view of Kant and made me understand him better.

20) Quine’s juxtaposition of pragmatism and empiricism, with tiny platonic assumptions about numbers and mathematics in general. He needs to tackle obscure ontologies in order to legitimise his more empiricist assumptions.