Frege: Fun With Classes

The Equivalence Class

There's a logical idea used in the Fregean definition of number: the equivalence class.

Say that we want to define the geometrical term "the same direction". Take also the notion of a line in Euclidean geometry. Does

         ab have the same direction as cd?

We can answer the question

         if and only if ab and cd are parallel.

ab and cd, therefore, must be an example of the well-known geometrical parallel lines. What has this to do with the equivalence class or, indeed, with Fregean number theory? It does so because the direction of ab can be seen in terms of a class or of classes. That is:

            ab is the class of all lines which have the same direction as ab.

In other words, the class of ab (or the direction of the line of ab) has as its members all that have the same direction as ab. So, in that case, perhaps cd is a member of the class ab. It is so, as said, because it is parallel to ab. If any other relation, say yz, is parallel to cd, then by definition it must be parallel to ab as well. So both cd and yz (amongst many other lines) belongs to the class ab – they are its members; though they aren't classes themselves.

It would be better to give our ‘ab’ a better symbolic expression so as to distinguish it from ‘cd’, ‘xy’ and every other member of it.

The equivalence class, then, will "fully identify the extension of the concept: direction of ab". That is, its extension includes all examples of direction which are equivalent to ab or parallel, in this case, to ab. In terms of the concept or predicate expression "direction of ab", the class ab (or the class of ab’s) has as members the extension of that concept or predicate expression. Or both cd and yz fall under the concept [direction of ab] just as horses fall under the concept [horse], etc.

All the above is an example of all the directions instantiated by parallel lines. What of other directions or of "direction in general"? These too can be defined in terms of classes. Instead of the class direction of ab (or the class of all parallel lines), we can now have the class of classes which are equi-directional. In other words, this higher-order class has as its members the members that instantiate direction in general (or are all examples of direction). However, didn’t we say earlier that classes within classes were disallowed on pain of paradox and infinite regress? Now we have a class of classes, and these member-classes themselves must also have their own members and so on. We would have yet another case of the infinite regress or indefinite inflation of classes in Fregean number theory (or his class theory). In addition, the equi-directional class must be an infinite class for another reason. That reason is that it's surely the case that there's an infinite (or innumerable) amount of actual or possible directions in general; especially bearing in mind that one line (or direction) may start off being perfectly straight but then, for example, take a diagonal turn and so on. The permutations of a given line or direction must surely be infinite or innumerable. So this strange meta-class – the class of equidirectionality – must give rise to many infinite regresses and paradoxes, not just the one brought about by having other classes as examples of some of its members.

Numbers

Let’s get back to defining the concept NUMBER.

As we've said, we can define number as classes of equinumerous classes. In the case of the number 6, this number is defined in terms of all six-membered classes (whether the class of six horses or the class of six black persons and so on). Though, again, this equinumerous relation between classes mustn't rely on numbers or counting. Instead, that is, in terms of the one-to-one correspondences between all the members of a six-membered class and a different six-membered class. And this relation of correspondence is brought about by using the logical ideas equinumerosity and the equivalence class.

What of the problem of classes within classes?

To sum up. All the above includes examples of the logicist or Fregean attempt to

"complete the definition [of number] using only logical concepts: concepts whose meaning and extension are determined by the elementary laws of thought" (388).

In other words, it's an example of the many attempts to reduce mathematics (or simply arithmetic) to logic. As we know, this project failed and it was later generally accepted that logic is in fact a branch of mathematics, not vice versa.

0

Let’s go into detail and define zero (or 0) in purely logical terms.

We can do so, to begin with, by considering the predicate "not identical with itself". Clearly, then, there's not a single thing in our universe (or even at any possible world) that isn't identical to itself. It follows, therefore, that nothing falls under the concept or predicate "not identical with itself". In addition, Frege also believed that every predicate has as its extension a particular class and its members. What if nothing falls under "not identical with itself"? We can now say that its extension has precisely no members. Thus we can have an extensional class for our own number zero or 0. We can even say, a la Frege, that the number 0 is the class of all no-membered classes. Thus 0 too has its own extension!

That extension-class is often called the null class. More technically, the number 0 can be defined with variables instead of the numeral ‘0’. We can say that the

"number zero can be defined as the number of things x, such that x is not identical with itself (alternatively, as the class of all classes of things which are not identical with themselves)". [388]

This is another way of saying that there's no x which isn't self-identical; which is itself a logical truth. In terms of classes rather than the variable x, there are no classes of classes that aren't self-identical. However, according to Frege there is a class of classes of things (or members) that aren't self-identical. The point is, all these classes are empty, for the reasons just given. Or surely we must add that the class of all classes of non-self-identical things does indeed have members; though these members must only be classes - if only a single such class. It doesn't, then, have particulars or individuals as members, only an empty class – the null class.
 
This is an even more stark and blatant case of a class that has other classes as members. Not only that: its own member-classes are all empty! So, as with the infinite-regress argument, an empty class that's a member of the class of all empty classes will itself have at least one member-class as its own member. And so on. However, can we really say ‘and so on’ when this ostensible infinite regress is a result of empty member-classes and an empty meta-class – though that too, as said, may or must itself be a member of a higher or ‘larger’ class in a case of infinite inflation rather than infinite regress.

Frege required these empty classes (these null classes) in order to find an extension for the number 0, and therefore complete his reduction of arithmetic to logic, etc. Can we even make sense of an empty class?

S provides us with a logical schema to define the number 0, thus:

                           0 = df. ¬(x = x)

Put in a stark logical language, the proposition of non-self-identity seems even more ridiculous and absurd, if not contradictory or paradoxical. In natural language, the schema above can be translated thus:

For a definition of the number 0, we can say that it's not the case that all things equal themselves. Indeed there's at least one thing that doesn't equal itself.

It follows that according to Frege we can define the remaining numbers in the way we defined the number 0 – we can do so recursively. It appears strange that we can define 1 in terms of 0; or, more precisely, in term of the empty or null class. As S puts it:

"1 is the class of all classes equal in number to the null class (for it is a logical truth that there is at least one and at most one null class)." (388)

1

It seems strange, prima facie, that the number 1 is compared with 0 (or is ‘corresponded’ to the null class). We think this because empty classes seem strange or even impossible. And there we may think that there isn’t a single null class. As we saw earlier, Frege himself believed that there is a null class, otherwise we wouldn't have a logical extension for 0.

So 1 is the class of all classes with no members. And there's precisely one null class, according to Frege. Thus,the member-class of the null class is made to correspond to the class of all one-membered classes. It can do so because although the null class has no members (except, perhaps, another class), the null class is still a class of sorts even without members. So it's still a one or a unity. Despite the fact that there's one or a single null class, the members of all one-membered classes can still be taken to correspond, literally one-to-one, with the null class. It follows, therefore, that the number 1 also has its own extension – the null class or the class with no members. However, again, it's still a one (or a unity) and it can therefore still be used to correspond with all the one-membered classes which themselves belong to the class 1 (or the number 1).

In that case, we must also say that the meta-class (the class of all one-membered classes) must itself contain the null class (alongside all the one-membered classes as its other members). It is, again, a member of the class 1 not because it has one member; but because the empty class itself is a one or a unity. The class 1 contains two different classes: one-membered classes as well as the singular null class. This situation is replicated recursively with the other higher numbers.

2

Now take the number 2.

Instead of saying that 2 is the class of all two-membered classes, we should instead say that 2 is the class of all classes equal in number to the class whose only members are the null class and the class whose only member is the null class. Perhaps we must put it this way because if we talk of ‘one-membered’ or a ‘two-membered’ class we're using numbers in our definitions of numbers. And that's not allowed; primarily because numbers aren't logical but mathematical objects.

The class of all two-membered classes (or the class with these particular concepts that aren't counted, etc.) must include not only the null class but also the class whose only member is the null class. That last class, as we've already seen, is the class we used to define the number 1; just as we used the null set to define it, and, in turn, the null set alone was used to define the number 0. We now have:

              i) the class of two- membered classes

              ii) the class of one-membered classes the null class

As we've already seen, the classes enclosed within the meta-class must also contain classes ad infinitum.

What we correspond isn't the type of member-classes, but only classes qua classes qua members of classes. It doesn’t matter that the class 2 is a class that itself contains disparate classes - including the class whose only member is the null class and the class whose only member (not members) is the null class. What matters are the classes, not the types of classes (as the later Russell might have put it). In terms of counting, we only count classes as members qua classes, not types of classes. In terms of corresponding classes or members one-to-one, we also only take member-classes qua classes, not qua types. And so on. This ‘so on’ is a recursive ‘so on’, as it were.

As S puts it: We ‘“build” the numbers from the null class, while making no ontological assumptions whatsoever’ (388). This is another way of saying that Frege’s theory is concerned with classes qua classes, not classes qua types of classes. As with an axiomatic deductive logical system, we derive or ‘infer’ the rest of the numbers from the ‘axiomatic’, as it were, 0, rather than from logical axioms or premises. This is no surprise considering that it was part of Frege’s attempt to reduce mathematics to logic. This deductive system of number is itself logical in nature, not just the definitions of the individual numbers themselves.

The Successor Relation

We can also mention the "successor relation". We define this by means of the existential quantifier -∃. More correctly, for the

"number of the Fs is one more than (i.e. successor to) the number of the Gs if there exists an F such that the rest of the Fs are the same number as all the Gs". [388]

In other words, the members of class F are one more in number than the members of class G. In addition, if class F contains a member or individual F such that when that is excluded, all the other members of class F in fact are the same in number as the previous class G. So, in terms of one-to-one matching of every member of class G with every member of class F bar one. That single anomaly makes class F a different class.

However, in the above we appear to be using number-terms such as ‘number’ itself, ‘one more than’, the ‘same number’, and so on. I thought that number-terms couldn't be used in these Fregean definitions of numbers. However, S writes: "Remember 'same number as' is defined without reference to number.” So S must have used number-terms to translate and simplify the purely logical definitions that, instead, rely only on logical ideas or terms like ‘corresponds with’, ‘equivalence’, and ‘equinumerous’. Without the above translation using numbers, it may be difficult to understand the purely logical definitions of number offered by Frege and others.

In terms of logical recursion in number-definition, we can use a variable and say that

"x is a natural number if it falls under every concept which zero falls under and which is such that any successor of whatever falls under it also falls under it". [388]

In other words, x is a natural number if it falls under a concept which nothing falls under, or has as its extension the null class. This, clearly now, is a reference to the definition of the number 0. In terms of recursion, we can now talk about other concepts or classes under which at least one thing falls or has as its extension a single class. That is, the number 1. It's a recursive definition because the null class is a member of every ‘successor’ concept or successor class. That is, the class 1 contains both the class with a single member, the null class, and the null class with no members. In that case, successor numbers have been defined in terms of prior numbers or prior classes that are now contained in new classes and thus recursively generate newly defined higher numbers. As S puts it:

"… every concept which zero falls under and which is such that any successor of whatever falls under it also falls under it". [388]

To conclude, we logically define the number 2/3 by including a class which itself contains classes as members, such as class 2, class 1, and the null class. Or classes with two members, classes with one member, and a single class with no members – the null class.

Peano's Postulates

This Fregean definition was actually used by Peano to derive his own postulates. In turn, from Peano’s he also derived the rest of arithmetic. Both done recursively. Further, Dedekind and Cantor showed how to derive the whole of number theory from arithmetic. Therefore we have derived mathematics from logic. 

We can represent this recursive edifice in terms of a foundational schema thus:

         i) the whole of number theory

        ii) arithmetic

       iii) Peano’s ‘postulates’

       iv) Frege’s number theory (logical terms)

Or instead:

       i) the whole of mathematics

       ii) the whole of number theory (Cantor, Dedekind)
    
       iii) arithmetic (also Peano)

       iv) Peano’s ‘postulates’

       v) Frege’s number theory

       vi) logical terms and the ‘constants’

We can therefore condense these schemas into the simpler:

      i) mathematics

      ii) logic

Hegel & the Being/Nothingness/Becoming Trinity

According to Houlgate, philosophers had traditionally thought of thought in terms of "logical possibility", "necessity", "contradiction" and "non-contradiction". For Hegel, however, "thinking must at least be the thought of is" (98). This, to put it simply, must come before the more logical ways of seeing thought or cognition. That is

"before we can arrive at a determinate understanding of any possibility, actuality, or necessity – that is, of anything at all – we must at least think of such an undetermined possibility, actuality, or necessity as being whatever it is." (98)

So this Continental interest in being isn't "anti-intellectual" (as it is often seen in Anglo-American analytic philosophy). It's more a position of the prior nature of being before it's conceptualised or embodied within various logical contexts.

To begin with, Houlgate stresses what Being is not. For example, we

"cannot simply assume in advance that it means 'existing' or 'having a certain identity', or 'subsisting over time'.” (98).

All these things, therefore, must come after the beinghood of Being is established. They're ways Being can be: being can exist, have a certain identity, or can subsist over time. Being, therefore, is a broader term than, say, existence, identity, etc. Or, to put it another way, Being (or a being) needn't exist and certainly needn't be alive. We can say here that Being is anything than can be the subject of thought before any philosophical or logical determinations.

Not unlike Edmund Husserl later, Hegel was interested in the nature of "presuppositional" thought. As Houlgate puts it, the

"first category we come across when we presuppose nothing whatsoever about thinking – except that it is thinking – is thus the simple category of being". (98)

Or as Hegel puts it, being "without any further determination" (1812-16; 1832).

Clearly, then, this Being (or "pure being"), whatever it is, seems a pretty hard thing to think about. Hegel himself acknowledges this very indeterminateness of Being. In a sense, the point of Being is its very indeterminateness; or, as Tractatus Wittgenstein might have put it, its "un-analysability". Houlgate too accepts this paradoxical nature of pure Being. He says that because 

"of its sheer indeterminateness, the thought of pure being is in fact completely and utterly vacuous". 

So, again, Hegel fully acknowledges the emptiness of pure Being; or, alternatively, of the concept (or notion) of pure Being.

However, that indeterminateness of being had many interesting consequences for Hegel. Again, Houlgate admits that pure Being is "indistinguishable from the thought of nothing whatsoever" (99). But to Hegel this acceptance of pure Being’s essential vacuity has many interesting consequences; primarily its relation to genuine nothingness and the nature of becoming/individuation.

So the first category of Being "thus immediately gives rise to a second category" (99) – that of Nothingness. And from this stark duality of Being and Nothingness, another essential and fundamental category immediately arises: Becoming. So the category of nothingness is very like the category of pure Being. As Houlgate puts it: 

"Nothingness, like being, itself is sheer emptiness and lack of determinacy, and so is itself nothing but indeterminate be-ing." 

Though, again, the Being/Nothingness duality itself brings with it a new trinity: Nothingness/Becoming/Being. In terms of our cognitive position on the Being/Nothingness duality, "thought of being slides immediately into the thought of nothing" (99). So, as a consequence of this, the Being/Nothingness relation is not a genuine duality at all for the Hegelian philosopher. As Hegel puts it:

".... their truth is, therefore, this movement of the immediate vanishing of the one in the other: becoming, a movement in which both are distinguished, but by a difference which has equally immediately dissolved itself." (99)

So just as Hegel rejected all the primary dualisms of Western philosophy; so, in a sense, he also rejects his own Being/Nothingness duality by emphasising the category Becoming. We now have a Nothingness/Being/Becoming trinity; rather than a duality between Being and Nothingness.

To reiterate. In Hegel’s presuppositionless philosophy of thought we begin with the basic category of ‘is’ or Being. Then "turns out that to think is also minimally to think “becomes”’ (99). From this inter-fluidity of basic categories of thought various things applicable to other areas of ontology and philosophy follow. This acknowledgement of the fluidity of categories or concepts (or of what they express) can be seen in the way that Hegel "follows the footsteps of Heraclitus and anticipates the thinking of Nietzsche" (99). Most importantly, this Being/Nothingness/Becoming trinity exemplifies Hegel’s well-known account of dialectical processes. Hegel himself defines his notion of dialectical processes thus: "the dialectical movement as the self-sublation of finite determinations" (1830, 99). Despite this ostensibly pretentious flux of technical terms, Hegel himself writes that he's essentially describing "their [categories] passing into their opposites" (Hegel, 1830). In other words, "finite determinations" (or categories) are always parasitical on other determinations (or categories/concepts). This preempts Jacques Derrida’s stress on the fluidity of what he calls "binary oppositions".

A Basic Overview of Reliabilism

Alvin Goldman is more or less the inventor of reliabilism (if such a thing can be invented). So what does he say about it? This:

• "What kinds of causal processes or mechanisms must be responsible for a belief if that belief is to count as knowledge? They must be mechanisms that are, in an appropriate senses, ‘reliable’. Roughly, a cognitive mechanism or process is reliable if it not only produces true beliefs in actual situations, but would produce true beliefs, or at least inhibit false beliefs, in counterfactual situations.." (87)

This theory could just as easily be called the causal theory of knowledge (or of true belief). This has indeed been stressed by many of its adherents and commentators. It's precisely its stress on causation which perhaps makes this account of knowledge scientific in its credentials. That is, there's no reference to "intuition", the "a priori" or even to "justification" in this account. 

The obvious question is this. What 

"kinds of casual processes or mechanism must be responsible for a belief if that belief is to count as knowledge" (87)? 

This is like the theory of direct reference as applied to knowledge (i.e., rather than as applied to names and referential acts). 

The obvious answer to the question above is perception. Perception is, after all, a causal phenomenon in that objects and events cause us to have the perceptions that we have. (Whether or not they correctly represent the objects and events is, of course, another issue.)

Goldman clearly sees the problem with the vague and unreliable word "reliable". What does "reliable" actually mean? That it can be trusted? That it has worked in the past? That it does what we want it to do? 

It can be seen that all these accounts of reliable processes (or mechanisms) involve an appeal to inductive justification. Thus, in that limited sense, reliabilism not only relies on induction: it also relies on induction as a form of justification. The inductive reliability of this or that process (or mechanism) justifies our use it from an epistemic point of view.

In addition, how reliable must a process (or mechanism) be? 100% reliable? 50% reliable? 1% reliable? Clearly, the phrase "1% reliable" is almost a contradiction in terms in that something that is that reliable is not, well, reliable. What about 50% reliable? That's harder to decide. In fact it's reliable 1 out of every 2 times; which is not that bad. Or is it? 70% reliable sounds a lot better. And clearly 100% reliable is actually more than reliable – it is, as it were, foolproof. 

Is anything 100% reliable? Not even my light switch is 100% reliable in that, say, six months ago it failed. Thus our measurements of reliable should bring aspects of time-length and probability ratios. That is, reliable over which time period? A month? A day? One minute?

Anyway, whatever the answers to these questions are, what a reliable process or mechanism must do is "produce true beliefs" (87). Yes; though every time? Most of the time? Half of the time? Or just some of the time? The earlier problems are coming up again.

How would we know that they are "true beliefs"? Can we rely on reliabilist processes or mechanisms to decide when something is a true belief? Indeed can we rely on reliable processes or mechanisms to justify reliable processes or mechanisms in a meta-epistemological sense? What justified our use of what we take to be reliable processes or mechanisms? 

Thus we don’t seem to have escaped from justification even if reliable processes or mechanisms justify reliable processes or mechanisms. Justification is still embedded in the process or reliabilism. 

What about the problem of the meta-epistemological issue of vicious circularity just mentioned? Unless it's a virtuous circularity (as with inductive justifications of induction).

Chappell gives a simple and mundane perceptual example of a reliable process or mechanism:

"I believe that I’m looking at zebras because I’ve gone to the zoo to see zebras, and here I am in front of the pen marked ‘Zebras’; why, I can see the beasts, and they certainly look like zebras to me. And that, says the reliabilist, is all I need for knowledge." (87)

Chappell seems to be saying that he has no reason not to believe that he's looking at zebras. Every bit of evidence you would expect to rely upon in these situations, he does rely on. Therefore he has no reason to suspect his own judgement. 

The above sounds like basic common sense. Perhaps that’s the point of reliabilism. 

For example, Chappell doesn’t even raise the issue as to whether or not he’s a brain in a vat or that an evil demon might have caused him to hallucinate zebras, the word "Zebras" and all the rest. Again, there's no reason for him to think about the possibility that he's a brain in a vat of that the zoo was created five minutes ago replete with fake zebras, fake zoo keepers and the rest.

Chappell then goes on to analyse his judgements about the zebras in specific terms of reliable processes or mechanisms:

"My belief that ‘These are zebras’ is (let’s suppose) true; the mechanisms or methods by which I got that belief – commonsense inference and the use of perception – are reliable ones. That’s it; I need nothing more to be able to claim to know ‘These are zebras’. In particular, says the reliabilist, I don’t need to eliminate every conceivable alternative hypothesis, no matter how crazy, before I can make this claim." (87)

"Commonsense inference" and "the use of perception" may not be glamorous epistemic principles; though they're nevertheless good ones. They work. (Or do they?) Of course, according to the hard-core sceptic, everything Chappell says is up for grabs. 

For example: 

How do I know that commonsense inference works? 

How do I know that this is commonsense inference? 

How do I know that you have validly inferred y from x? 

How can I trust perception? 

How do I know that this was an act/process of perception?

How do I know what the word "perception" means? 

How do I know what "means" means? 

How do I know I know anything? 

And so on.

Isn’t that the very point of epistemology? That is, to counteract (or even refute) these sceptical possibilities? Aren’t these sceptical scenarios at the very heart of epistemology and therefore the meat of our knowledge-claims? 

If we ignore these sceptical possibilities, then we effectively stop doing epistemology and start doing the descriptive science of, say, belief-acquisition (as the naturalisers of epistemology do). 

Of course in one sense the reliabilist is correct. It's literally impossible "to eliminate every conceivable alternative hypothesis, no matter how crazy" (87). Does that mean that we should ignore each conceivable hypothesis - even one that's a genuine threat to knowledge? Especially since the reliabilist is supposed to be an epistemologist of sorts! Isn’t the reliabilist taking the sceptical attitude towards scepticism which the layperson takes? Either that, or he is simply adopting the approach of a descriptive and empirical science (a la Quine and the rest)?

Not Knowing that One Knows P

That last point about reliabilism’s anti-epistemological (not just anti-sceptical) epistemology is brought into clearer view when we consider its approach to knowing, or not knowing, that we know something:

"Reliabilism interestingly implies that I can know things without knowing that I know them. Indeed, according to reliabilism I can know things without knowing very much else at all." (87)

This too goes against epistemology itself. How can you know something without knowing that you know this something? This would mean that one hasn’t consciously attempted to acquire knowledge by using the correct epistemic principles, no matter how unclear these principles may be to you. One must simply know p without knowing that one knows p. How is that even possible? If knowledge literally requires little, or even no effort, then how can knowing that p be distinguished from believing that p? Isn’t that effort, whether a justification or whatever, part of the knowledge process and thus part of the constitution of p as a piece of knowledge? Chappell gives a concrete example of what the reliabilist actually means by what he claims by using his zoo example again:

"In the zoo case, for instance, I might have no idea that commonsense inference and the use of perception are reliable methods of acquiring knowledge, and I might have grave doubts about whether these beasts are in fact zebras at all. Provided I still manage to form the true belief that ‘These are zebras’ on the basis of those reliable methods, I still count as knowing ‘These are zebras’." (87/8)

If one didn’t know that "commonsense inference and the use of perception are reliable methods for acquiring knowledge" (87), why would one use them at all? Of course one may not be able to formulate such methods in a language that would satisfy the professional epistemologist; though that shouldn’t matter much – the metaphysician may have a problem with epistemological language. That is, the layperson will not even use the phrase ‘commonsense inference’ or even ‘perception’ in these or any contexts. We can call it implicit or tacit knowledge of epistemic principles or even of reliable methods. The layperson, nevertheless, is still acquiring knowledge. Indeed he is still using ‘reliable methods’. Not only that: he knows that he is using reliable methods and he would even know what those reliable methods are if asked about them. Again, he may not, or would not, use the language of the epistemologist to explain what it is he does when he acquires knowledge. But he has acquired knowledge. For example, the next day at a pub quiz he may now be able to answer a question correctly about zebras. He wouldn’t answer the question at all if he didn’t trust his implicit or tacit epistemic principles.

However, what is important, to the reliabilist, is that this person uses reliable methods to acquire knowledge. It doesn’t matter what else he thinks about his cognitive processes or anything else for that matter. It doesn’t even matter that he’s now sure that ‘"these beasts are in fact zebras at all" (87). If he has used commonsense inference and perception, then that’s all that matters. Indeed if he has used these methods then, basically, they must be zebras and not, say, horses.

What about his knowledge that they are in fact reliable methods? Does he know that commonsense inference and perception are reliable methods for acquiring knowledge? If they are, perhaps one can use commonsense inference and perception to make judgements about one’s methods of commonsense inference and perception. Would this by circular reasoning? Would it be vicious or virtuous?

So there are indeed certain things which I must know that I know. I must know that I know, in these examples, that commonsense inference and perception are indeed reliable methods for acquiring knowledge. In that case, the reliabilist or layperson is a partly self-conscious epistemic agent after all.

One result of this exclusive concern with reliable methods, processes and mechanism and not with our knowledge of our knowledge is that John McDowell points out that

"in the purest form of this approach, it is a matter of superficial idiom that we do not attribute knowledge to thermometers." (88)

After all, thermometers are reliable and they do give us accurate information about the temperature. What more would the reliabilist want? He certainly wouldn’t require that the thermometer knows why it knows that temperature is hot or cold. Then again, the reliabilist doesn’t require this of persons either – not even, perhaps, of epistemologists!

Another often-used example of these cases of knowledge without knowledge of knowledge is reported by Chappell about chicken-sexers:

"Apparently, as Linda Zagzebski reports, there are professional chicken-sexers ‘who can determine the sex of baby chicks without knowing how they do it or even if they do it correctly… Philosophers with strong externalist intuitions about knowledge have no hesitation in saying that such people know the sex of the chick." (88)

Again, this is another case in which knowledge, if it is a case of knowledge, simply doesn't require any cognitive effort. These chicken-sexers simply know what sex the chicken is. That’s it. They do not know how they know, why they know or ‘even if they do it correctly’ (88). It this just a case of some kind of tacit or implicit knowledge? After all, there must be some reason why or how they know what sex the chick is. Surely we are not saying that it is a case of a priori or even supernatural knowledge. This would be a strange thing to have either a priori or supernatural knowledge about! What other alternative is there?

Chappell mentions ‘philosophers with strong externalist intuitions’. These philosophers, to put it simply, simply don't care about what goes on in the minds of chicken-sexers. The only things that matter to them are the things that go on in the external world. And what goes on in the external world appears to be, or is, reliable. These sexers do get the sex of chicks correct it most cases. So who cares what goes on in the minds of sexers if they are getting reliable results. Why should we want more from even an epistemic point of view?

One definition of ‘certain’ or ‘being certain’ would be in the case that we have refuted or eliminated every case of not-p against our p. That, however, is impossible. So we either have to redefine what ‘certainty’ means to make it less strict or simply say that we do not need certainty outside, say, logic and mathematics (perhaps not even in these cases). The other alternative is to accept that we have knowledge even if we don’t have certainty or certainty that we have a case of knowledge.

Of course the sceptic can simply ask the reliabilist the following questions: How do you know that these processes or procedures are reliable? How do you know that they have worked in the past? In other words, all the standard sceptical questions can be asked here. The reliabilist either simply ignores the sceptical scenarios or ‘begs the question against scepticism’ (89).

Not Knowing that One Knows

Again how can we know without knowing that we know? Surely knowing entails knowing that we know. How can we just know without knowing that we know – without cognitive effort? Or, as Chappell puts it, without ‘epistemic feedback’? –

"It seems to be corollary of this that it is important not only to know, but also to know that you know. The reason is that you need epistemic feedback. If you know something but don’t know that you know it, you can’t use the fact that you know it to help you to modify your epistemic behaviour in the direction of greater accuracy than before." (89)

Knowing that you know is not just ‘important’, it is necessary for knowledge. Knowledge requires epistemic effort, surely. However, I know that there have been many arguments in epistemology against this (e.g., David Lewis’s arguments). Perhaps if you don’t know that you know something, what you really have is truth; though not knowledge. That is, you can know that p is true without knowing why it is true. It may still, however, be true. It may be true that it is 9 O’ clock but you don’t go through any process in order to recognise this truth. You just know that ‘It’s 9 O'clock’ is a true statement. In the past you may have acquired the knowledge required to be able to tell the time correctly; though now this past knowledge is no longer needed. However, perhaps, as with Lewis, past knowledge does not need to be re-justified or learnt or whatever. I can simply rely on the fact that I once did acquire knowledge as to how to tell the time correctly. Now I just can tell the time without cognitive or epistemic effort.

Chappell finishes off his critique of reliabilism with a rather predictable conclusion:.

"So it looks like knowledge is not ‘true belief acquired by a reliable method’. If so, reliabilism is not the right analysis of knowledge." (91)

Why assume that an ‘analysis of knowledge’ will determinate at all? Why presuppose a definite and determinate answer to the question, What is knowledge? Indeed perhaps there is nothing that pre-dates our analyses and stipulations. Perhaps there is no such kind or thing that is knowledge until we decide what it is, or what it should be. Why be an epistemological ‘realist’ in the Michael Williams sense of that term? Why assume the mind-independent existence of the true theory or true concept of knowledge? Why assume the same about knowledge itself?.

 

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?