Wittgenstein: ‘+’ and Rule-following

We assume that if we use the symbol ‘+’ then that symbol must have a meaning. Not only that, but also that meaning must be in the mind somehow before its use. That is, we may assume that certain “conscious states or events” must have “accompanied” (372) the use of ‘+’. Perhaps, however, we can use ‘+’ without any meaning or different mental events or states. That is, without things in addition to the cognition of the actual use of ‘+’.




It could be the case that the use of ‘+’ has a normative dimension. That is, it is not a case of what ‘+’ means, but what it should mean. Or, more correctly, how it should be used. Of course, we can now ask if there can be examples of use, of any kind, that can function without meaning or separate mental events or processes. However, Heidegger talked, in his Being and Time, of a particular use of a hammer without its users thinking at all about the nature of the hammer or hammers generally or what function it fulfils. That is, he simply uses the hammer. He knows how the hammer should be used, in the Wittgensteinian normative sense, but he does not rely on an intellectual attitude to the hammer.




Back to the normative status of ‘+’. We know what it should do, but this knowledge doesn’t depend on mental meaning or meaning of any description. What is determinate is not the meaning of ‘+’, but the rules of its usage. So if there are any meanings to understand, it is not the meaning of ‘+’, but the meanings of the rules that determine its usage.




What all this means is that




‘+’ does not mean +




If we accept meaning, then ‘+’ could mean slinmp. It is not the meaning of ‘+’ that determines that 2 + 2 = 4, but the use of the symbols in it.




It may therefore follow that if the symbol ‘+’ doesn’t depend on meanings, then it doesn’t depend on private mental events or processes either. Therefore the use of ‘+’ may depended, instead, on what the community takes the symbol ‘+’ to be, not what they think it means. Indeed, it is precisely because ‘+’ doesn’t depend on meanings, which are essentially private, that it can be used in a public or communal manner. If it depended on private meanings, then it couldn’t be captured in a public discourse and it couldn’t be communicated intersubjectively from a person’s private meaning to other people who cannot be aware of that person’s private mental states and processes. If the use of ‘+’ depended on private meanings, then, as with the ‘private language argument’, there would be no going right or wrong with an individual’s use of ‘+’. That is, no one could tell him, and he couldn’t even tell himself, whether or not he is going right or going wrong.




However, if the individual cannot know if he is going right or wrong, what happens when we take the community as a whole? That is, perhaps the community as a whole can’t know if it’s going right or wrong. The community itself would need another community, or a larger community, to guarantee its correct use of ‘+’. Each member of the community can’t depended on other members of the community to determine whether or not he is using ‘+’ correctly, or even on a larger subsection of the community, because other members of the community are by definition using ‘+’ in the same way, just as the private individual outside the community cannot compare his present uses of ‘+’ to other of his past uses and see if his present use is correct. So just as the private individual doesn’t rely on external sources of justification in his use of ‘+’, so too the community doesn’t rely on external sources to justify its uses.




*) Meanings can’t be facts or be factual because the things that determine facts or explain the facts must themselves contain meanings. Meanings are not facts, but what they mean can have a factual status. That is, it is a fact that the word ‘cat’ is invariably used to refer to cats. But then the fact is not the word’s meaning, but the situation that people use that word, ‘cat’, to refer to cats. Indeed, in that case the word ‘cat’ may not need some kind of internal and abstract meaning. That is, ‘cat’ means cat because the word ‘cat’ is used to refer to cats. Nothing more is needed. No mental item and no abstract meaning, nothing but the usage and nothing but the communal rules that determine the use of the word ‘cat’. Why do philosophers cry out for something more than this? Because communal rules change and they are not shared between different communities. And what if there are communities within communities (as there are)? What the realist wants is something determinate. Something timeless or out of time. Something essential and non-contingent. Something that can settle disputes once and for all. Something that can firmly answer, say, the question: What does ‘justice’ mean?

Non-Deductive & Deductive Arguments

Deductive reasoning forms - if they're correctly formulated - can't be defeated. In that sense, they're like larger-scale versions of analytical statements. That is, they're guaranteed to be true and correct. Induction, on the other hand, is defeasible. It's defeasible because the possibility is always left open that “new knowledge” may come along and defeat either the premises or the conclusion (or both). With deductive arguments, on the other hand, new knowledge would be irrelevant. Nothing could possibly invalidate a deductive conclusion if the way to that conclusion correctly followed the rules and forms of deductive logic.

Flach gives as an example of non-deductive (therefore defeasible) logic this:

Birds typically fly.

Tweety is a bird.

Therefore Tweety flies.

I assume that this is a non-deductive argument quite simply because in the initial premise we have the word “typically”. That would make it a non-categorical statement. Therefore quite clearly it's defeasible. Though surely it could still be taken as a deductive argument simply by taking out the word “typically”. That is, this argument clearly leaves the possibility that there are certain non-typical birds. And if they aren't typical birds, then such birds may not in fact fly.

The first premise is effectively an inductive generalisation about birds. And if we call it a “generalisation”, then evidently there may well be exceptions to this rule.

For example, that two plus two equals four couldn't be deemed a generalisation of any kind. It is, however, a “plausible” piece of reasoning. That is, if it's indeed the case that birds typically fly, then there's a good chance that if Tweety is a bird, that Tweety can fly. However, because of the qualified first premise which implies that it's not logically necessary that it flies because that very premise left the possibility open that there are birds that can't fly. It's that qualificatory premise that really makes the above argument a non-deductive argument.

In these non-deductive inductive arguments, we need to distinguish whether or not it's the conclusion that can be defeated, or the premises (or both).

For example

i) Everyday during my life the sun rose.

ii) I don’t know of any trustworthy report of the sun not rising one day in the past.

iii) Therefore, the sun will rise every day in the future.

This too is a qualified argument based on a qualified initial premise. It doesn't say that the sun must rise everyday. It simply says that the sun has risen everyday. This simply means that it leaves open the possibility that there could be an occasion in which the sun doesn't rise. Indeed it's that which makes it a non-deductive argument

Inference & Sense-data: Davidson against Russell

Bertrand Russell (in his Problems of Philosophy) took a similar view to C.S. Peirce which smacks of the latter’s idea that ‘opinions will converge in the long run’[see Hookway, 1985]. Take the example of the ‘reality’ of a table:

"… the ‘real’ shape is not what we see; it is something inferred from what we see."

This is why the early Russell distinguished between ‘physical objects’ and ‘sense-data’:

"… if the physical sun had cased to exist within the last eight minutes, that would make no difference to the sense-data which we call ‘seeing the sun’. This affords a fresh illustration of the necessity of distinguishing between sense-data and physical objects."

There is a simple reason given here why sense-data and object are not, and can’t be, identical. Is this position that different to Berkeley’s point that "an idea can only be like another idea"? For a start, if we did not accept sense-data we should believe that a stick in the water really is actually bent. We may also think that the earth is flat (if we lived in Holland, that is). That the stars and the moon are much closer than they actually are. That train tracks converge in the distance. That the half moon is in fact, well, a half moon.

Even in these instances only, we do in fact infer from sense-data, Russell would have argued, to the reality of the object. But these are all examples of visual distortions. What about a plain and simple cognition of a table (as in Russell’s own example)? Do we infer from sense-data in this case?

Prima facie, it is certainly not obvious that we do. But surely sometimes we don’t cognise enough of an object to know that it is the object that we think it is. For example, when I see a table I do not see all four legs in one go, but I still know that I am looking at a table. Did Russell simply mean that we apply concepts and categories to what is, in fact, our insufficient sensory data? In that case, the inference to the existence of a table is carried out via the application of prior concepts to the table’s sense-data. These concepts, as it were, fill in the gaps left by insufficient sensory material. But what if we do have sufficient sensory data to work on? Perhaps then we would still need to apply concepts in order to come to a cognitive decision that it is in fact a table that we are looking at. Sense-data do not speak for themselves. They do not tell us what they are sense-data of. Of course we get to work on the sense-data; but they, in and of themselves, are not enough. Put that way, it seems obvious that sense-data cannot tell us what to think. As Davidson might have said: causal contact with the table is not enough to tell us that it is a table. We infer from the sense-data that we receive, care of concepts, etc., that it is, in fact, a table. Sense-data alone are not enough. After all, a native who is unfamiliar with tables may receive exactly the same sense-data as us, and have the same kind of causal contact with the table; but no amount of sensory material will tell him that it is a table in front of him. However, even if we do know about tables, we still infer from the sense-data, according to Russell, to the reality of the table. According to the half-full/half empty bottle example, it is our concepts and attitudes that help us decide what it is that we perceive. One person says the bottle is half empty. The other says that it is half full. Yet both persons are experiencing exactly the same sense-data. In this case, neither person is in the right or in the wrong. They are, in effect, letting their concepts and attitudes get in the way of a pure description of the bottle with water in it. Similarly, when we see the facade of a building - we automatically assume that it is a four-dimensional structure. Yet the sense-data themselves only ‘tell us’ that it is a facade – that is, something one-dimensional. We could, after all, be looking at the set for a western film in which the facades of the ‘buildings’ have nothing behind them. The well-known example in this context is the ‘elliptical penny’. The sense-data tell us that the penny is elliptical, whereas in fact it is round. It only appears to be elliptical, in this instance, because of the angle of our perception of the penny. If we were above the penny, looking directly down, we would indeed see a round rather than elliptical penny. Similarly, seen from a great distance, the penny would be neither round nor elliptical; it would be simply a dimensionless dot in the distance.

According to Davidson, even though we do not get sufficient sensory data, we still do not infer the existence of objects [Davidson, 1989]. We see objects. Indeed we will see tables even with a possible dearth of sensory data. This simply means that there are no inferential processes carried out by the percipient at all, even in the case of the elliptical penny. Concepts are applied instantaneously. There is no epistemic gap between the perception and the cognisance of the object in question. Davidson quite happily accepts the fact that there is a ‘causal gap’. That is, sensory data must come before the application of concepts. But the cognitive mind is not aware of this causal gap. In that sense, it is incorrect to use the word ‘data’ at all in that data is commonly perceived to be something that we work - or infer - from. It is the basis for cognitive work, as it were. If there is no cognitive work actually involved, then there is no data. Sense-data are in fact a myth. If anything we have object-data and event-data, not sense- data.

References and Further Reading

Davidson, D. – (1989/2000) ‘A Coherence Theory of Truth and Knowledge’, in Epistemology: An Anthology, eds. E. Sosa and J. Kim, Blackwell Publishers
- (1974/1984) ‘On the very idea of a conceptual scheme’, reprinted in his Inquiries into Truth and Interpretation, Oxford University Press
Hookway, C. – (1985) Peirce, London
Russell, B. – (1912) The Problems of Philosophy, Oxford University Press

Derrida on the Metaphysics of Presence

I interpret (or “read”) the term “metaphysics of presence” as referring to the (philosophical) belief that there's a strict and determinate one-to-one relation between between a word/concept and our knowledge (or cognisance) of its meaning. I also believe that Jacques Derrida stretched this out to include many - or all - aspects of Western philosophy and religion (dating back to the ancient Greeks).

The Western fixation (or belief), so Derrida claimed, is that we could gain a precise and exact grasp of what a word/concept means. (Or, more broadly and concretely, what “truth”, “justice”, “knowledge”, etc. mean.) But in the case of words/concepts, that's impossible because they bear a stronger relation to other words/concepts than they do to their (abstract) meanings. (Or, in Derrida's jargon, “signifiers” bear a stronger relation to other signifiers than they do to what they “signify” – or to the “signified”) What you get, then, is a kind of semantic holism in which a word's meaning ripples out (as it were) to all the other words in the given language (langue or system) - or at least to those words to which it is directly connected. That means that no one person (by virtue of semantic holism) can fully grasp the meaning - or determine the complete meaning - of a word/concept.

This lack of “presence” (see also “trace”) also applies (as stated) to all philosophical concepts: such as truth, freedom, knowledge, God, etc. In other words, there is no metaphysics of presence.

So, yes, there is no fixed center to consciousness or the self either; though Derrida, again, didn't put it that way. However, if this is just a denial of the self (or a critique of essentialism), then Derrida was hardly original in this respect. That's because such a critque can be found in, for example, David Hume (who both was and wasn't an essentialist). In addition, philosophical anti/non-essentialism dates back to the ancient Greeks.

Derrida’s views on the self and his anti-essentialism are (somewhat) separable from his views on the metaphysics of presence (which I've commented upon); though there are obvious connections.

Derrida's Prose Style

It's difficult to understand Derrida. However, I don't think that this has anything to do with the complexity or deepness of his thought. Often what he says (if after you've unpacked it) is often pretty obvious and sometimes truistic. Either that or it's been said before; which isn't to say that no other philosopher has repeated what previous philosophers have said.

What is different with Derrida is, specifically, his (“deconstructive”) prose style; which largely belongs to a particular Continental (specifically French) tradition. Partly because of that you'll find endless neologisms; which, to those outside Derrida's fan-base, will prove to be flummoxing. In addition, there's a strong sense of one-upmanship, outflanking, pretentiousness, being deliberately outré, self-conscious philosophical radicalism, philosophical exhibitionism, point-scoring against the last (usually French) "radical philosopher" who's just died or gone, excess, etc. which also makes Derrida's work hard to understand - at least as far as I'm concerned. However, if you'd said all that to Derrida himself (or to a fan of Derrida), you probably wouldn't be able to understand his reply (a scholar - or follower - of Derrida would understand it). And that, no doubt, would have been precisely Derrida's intention.

Final Note on the Opening Meme

It's quite possible that Derrida would have taken the claim that he fucked minds as some kind of compliment. Though, of course, he would never have said, “I take that as a compliment.” He would, instead, have said the same thing in pretentious deconstructionese; perhaps to hide the simplicity and possible banality of his words.

The Infinities of Cantor & Dedekind

One way if which you can defend (or explain) the reality of infinite numbers is the way Richard Dedekind did.

It's claimed that an infinite set has the “property” of having a proper subset that is as large as itself. Now how does that generate numerical infinity? Well, if the infinite set itself has a subset which is equal to itself, that in itself seems to imply some kind of infinity. After all, wouldn't you expect an infinite set to include subsets which are themselves infinite in nature?...

I'm not sure if that's the argument or reality though. I suppose it could mean that if the infinite set is truly infinite in nature, then it must have room (as it were) for a subset which is itself infinite. That is, if a set is infinitely large, it has space (as it were) for something which is also infinitely large.

Nonetheless, how can infinities be compared in such a way? How can one infinity be compared to another infinity? More to the point: how can we use such spatial metaphors as “as large as itself” when we compare an infinite set to one of its subsets? Can infinite sets be compared at all – such as the set of natural numbers and the set of even numbers?

Well, in a sense (or even not in a sense) they were compared in this way by George Cantor. He compared one infinite set with another via what's called the “diagonal argument”. That is, he compared the set of real numbers with the set of natural numbers. More specifically, he matched each number from the set of real numbers diagonally (in terms of how he represented it graphically) with one number from the set of natural numbers. At each end of the diagonal line a real number is matched with a natural number.

Actually, Cantor showed that the set of real numbers is larger than the set of natural numbers. Nonetheless, as I said earlier, how can this be the case when we're talking about rival infinite sets? How can one infinity be larger or smaller (or even equal) to another infinite set? More specifically, why did the set of real numbers win in this battle against the set of natural numbers? In other words, why, exactly, are there more real numbers than natural numbers?

Wittgenstein, for one, rejected Cantor's infinite sets. As some kind of constructivist, he thought that Cantor had simply magicked them out of a hat. In other words, Wittgenstein believed that they aren't real or actual.