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!’.