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.

Tarski's Convention T

Even though many philosophers believe that Alfred Tarski’s ‘theory’ of truth is not about correspondence, "he suggested [that it] captured the idea" of correspondence. This makes intuitive sense. In any case, he saw the notion of truth as foundational in logical discourse. He took this idea from Frege. Tarski’s "unspoken starting point was the account of reference proposed by Frege, in which truth features both as the aim of discourse, and as the semantic value of successful utterances" (109). This is also the position, it would seem, of Brandom’s inferential holism.


What are the three fundamentals of Tarski’s ‘semantic theory of truth’? –


i) That it should assign truth-conditions to each sentence of our language.


ii) That it should derive those truth-conditions from the semantic values of the parts of a sentence.


iii) That it should meet what he called a ‘condition of adequacy’, namely, that every instance of the following ‘convention’:


(T) s is true if and only if p.


should come out true.


What can we say about the schema above? We can replace the letter s above by a name. Or, more correctly, by ‘the name of a sentence’ (Frege said truth-valued sentences are names – names of truth-values?). Since it is a name of a sentence, and not a sentence itself, it will have inverted commas around it. In terms of the letter p, that will be replaced by the sentence itself – that is, without inverted commas. Now we can have:


(S) ‘Snow is white’ is true if and only if snow is white.


Because of his belief in object-languages and meta-languages, Tarski believed that


"truth could only be defined for each language taken on its own, and moreover that it must be defined not in that language but in another, which is called the “meta-language”’ (110).


Of course we need to ask why Tarski thought that this should be the case. A sentence cannot predicate truth of itself. Therefore a language cannot predicate truth of itself?


We mentioned correspondence earlier. Now we can clarify why the convention explicates correspondence. Such schema "relate a sentence to the fact that it is used to express, by first naming the sentence, and then using it" (110). A sentence is used to express a fact. Why isn’t S using the term ‘truth-condition’ here? Does that mean that a fact is simply a truth-condition? Is there no difference according to Convention T? Anyway, in the jargon, when we write ‘snow is white’ we are naming that sentence (hence the quotes). When we write snow is white we are using that sentence, not naming it.


Because of the intuitive simplicity of convention T, or even its vacuity (according to some), we can know ‘a priori that the sentence “snow is white”… identified the very state of affairs… that makes the sentence “snow is white” true’ (110). We can know this a priori simply because the sentence used is simply the sentenced named with quotation marks. We can't go wrong! Thus this theory can entail every instance of (T) in a language (say, English). And this is "all that can be captured of the idea of correspondence: all that can be captured in language" (110). If someone asks what the correspondence theory of truth amounts to, we can say this:


(T) S is true if and only if p.


I said earlier that some philosophers have called this convention ‘vacuous’. S says that "Tarski simply returns us to the indisputable platitudes about truth" (110). What’s the point of platitudes when it comes to something as deep as truth? This is an alternative to ‘profound metaphysical theories’. Indeed ‘perhaps we should not ask more of a theory of truth’ (110). Perhaps this is all there is to say, even if it's basic. Anything more, one thinks, would be metaphysics, and perhaps that was Tarski’s point. He may have still had logical positivist sympathies, despite not being a member of that school.


Quine took this idea further by considering the predicate ‘true’. This doesn't ‘describe the metaphysical status of a sentence, but simply as what he calls a “predicate of disquotation”’ (111). Does that mean that the predicate ‘snow is white’ is simply disquoted to become snow is white? I mentioned naming a sentence and then using that sentence. In this case, ‘we pass from words quoted to words used: and that, indeed, is its function’ (111). That is the ‘function’ of what? The truth-predicate?


Again, the purpose of Convention T is in its "making the minimum metaphysical assumptions". That was the whole point. That is why it is so simple! Having said all that, Tarski came to believe


"that it was impossible, and that theories of truth could only be devised for artificial languages, and then always at the expense of constructing another language in which to discuss them".(111)


Why, then, did he think that a theory of truth is ‘by no means easy’. Indeed why did he fail in his task (in the case of natural, not artificial, languages)? Does that mean that there is something wrong with (T) above? In that case, what is wrong with it? Is it that, in the end, one can't leave out the metaphysics after all? Perhaps, then, rather than providing the requisite metaphysics, or failing without it, he should have given up on truth altogether and become a elimitivist or naturalist about truth.


Another problem with leaving out the metaphysics of (T) was that "minimalist theories could be embraced by defenders of correspondence and by defenders of coherence" (111). Alternatively, "maybe these are just rival descriptions of the same idea – the idea contained in convention T’"(111). We must ask, then, how the coherentist interprets Convention T. However, it seems pretty obvious how the correspondence theorist will interpret it. (Perhaps on a Tractarian model in which the picture theory tells us that parts of the world, the atomic fact, are pictured by the parts of the sentence.)

Steven Yablo's 'Identity, Essence, and Indiscernibility' (1987)

 

 

 

 

"If the requirements for being β are stricter than the requirements for being ά, then β ought to have a ‘bigger’ essence than ά…Thus, more is essential to the Shroud of Turin than to the piece of cloth [which was used as the Shroud], and the Shroud of Turin ought accordingly to have the bigger essence.” (Yablo, 1987)

We can admit that it's “necessary that the Shroud of Turin is the Shroud of Turin” (according to Steven Yablo’s paper), and that it wasn't necessary that the cloth of Turin (which was used as the Shroud) actually became the Shroud of Turin. (Therefore the Shroud has a property that the cloth didn’t have, according to Yablo, and so they aren't necessarily identical' but only “contingently identical”.) So isn’t it also necessary that the cloth of Turin was the cloth of Turin, in the same manner it's necessary that the Shroud of Turin is now the Shroud of Turin? If, on this count only, we can say that the Shroud hasn’t yet got a ‘bigger’ essence than the prior cloth.

 

How do we decide these essences in the first place? (So as to thereby decide which object has the ‘bigger’ essence.)

 

For example, it might well have been necessary that the cloth could clean things (i.e., have a functional essence); otherwise it wouldn’t have been a cloth. (Let’s take the cloth of Turin to have been a cloth created to be used as a cleaning implement.) It's not necessary, on the other hand, that the Shroud can clean things because, after all, it's now a shroud and not a cleaning cloth. Therefore it must follow that the cloth had an essence or property that the Shroud doesn't have.

 

Similarly, it might well have been necessary (via the sortal cleaning cloth) that the cloth wasn't black; but white instead (i.e., so that it showed up the dirt). Again, surely it's not necessary that the Shroud is white rather than black.

 

Yablo extracts a ‘bigger’ essence from the Shroud by treating its function as part of its essence (i.e., the function sortalised by shroud for a dead body). He disregards the cloth’s own possible functions, one of which might have been cleaning. And even if Yablo’s Turin cloth was never a cleaning cloth (but only a piece of material used for garments), it would still have had an essence/property that the Shroud doesn't have which belongs to the sortal garment material.

 

For example, the cloth might have been used for garments (not shrouds) and therefore it shouldn't (or couldn't) have made its wearers itch. And it might have also kept them warm too. However, a shroud, or the Shroud of Turin, needn't have these properties because the dead don't suffer from itches or cold.

 

So not only is Yablo’s belief that the Shroud’s essence is ‘bigger’ than the cloth’s somewhat arbitrary, it may also be the case that all deemed ontological essences are always somewhat arbitrary and also stipulated via sortals rather than discovered ontologically.

 

The Yablo example somewhat parallels the oft-quoted Quine example of the rational, two-legged mathematician and cyclist.

 

Mathematicians are, in this example, deemed to be necessarily rational: sortalised by necessarily rational being. (Does this automatically make computers capable of difficult mathematical calculations and the discovery of new proofs rational?) Cyclists, on the other hand, are deemed to be necessarily two-legged – sortalised by two-legged beings. (Although a no-legged cyclist could free-ride down hills and push the cycle up hills.) However, what if we have a mathematician who's also a cyclist – a being who falls under the two sortals: rational being and two-legged being? Quine asks:

 

"Is this concrete individual necessarily rational and contingently two-legged or vice versa?" (1960)

Perhaps, according to Yablo, the mathematician cyclist has a ‘bigger essence’ than a mathematician who isn’t a cyclist. (Perhaps because he has no other interests either.) This mathematical cyclist would fall under the sortals: rational being and two-legged being. But Quine thinks all this is silly. He says:

 

"There is no semblance of sense in rating some of his attributes as necessary and others as contingent. Some of his attributes count as important and others as unimportant, yes, some as enduring and others as fleeting; but none as necessary or contingent." (Word and Object)

The essences of the cloth of Turin and the Shroud of Turin depend on sortal specification. The cloth turned out to have a smaller essence than the Shroud simply because Yablo didn't specify it in any way; except by saying that it was the cloth of Turin and that it became the Shroud of Turn. However, Yablo does specify the Shroud (via that very sortal shroud) by saying that it shrouded the dead Christ. Again, the cloth could be specified via its material makeup. A cloth must necessarily be made up of certain materials (e.g., wool, etc.), or that it must necessarily be woven or that it mustn't retain water. Quine, therefore, had this to say on essentialism:

 

"An object, of itself and by whatever name or none, must be seen as having some of its traits necessarily and others contingently, despite the fact that the letter traits follow just as analytically from some ways of specifying the object as the former do from other ways of specifying it…This means adapting an invidious attitude towards certain ways of specifying x…and favouring other ways…as somehow better revealing the ‘essence’ of the object." (From a Logical Point of View, 1953, pp. 155-6)

As Gibbard (1987) might have said: The Shroud is specified via two sortals: cloth and shroud. The cloth, on the other hand, is only specified via one sortal: cloth. So, in this scheme, essences come via the sortals of objects, not the objects themselves. Indeed the Turin Shroud could come to us (or we to it) via a sortal that Yablo didn’t use.

 

For example, the Shroud could have been specified via the sortal objects that bear an imprint. This is a genuine sortal because there are other members of the sortal, objects that bear an imprint, other than the Shroud (e.g., white walls with their hand prints). Again, the Shroud could be specified via a sortal that the cloth certainly didn’t have: historical artefact. However, as has been said, the cloth of Turin could have come to us, or we to it, via sortals not specified by Yablo, say, cleaning cloth or garment material.