On Philosophy

November 3, 2007

Philosophy And Formalisms

Filed under: Metaphilosophy — Peter @ 12:00 am

The hardest part about doing metaphilosophy is finding a way to proceed to make metaphilosophical claims that doesn’t presuppose certain claims about how philosophy works, since metaphilosophy seems like a part of philosophy itself, and thus to proceed in a certain way to make claims about it would presuppose that we can make valid philosophical claims in general in that way. One way to overcome this problem is to ground metaphilosophy in some kind of proto-philosophy, that doesn’t itself presuppose anything interesting, and from there arrive at a more interesting way of pursuing philosophy. Or we may attempt to approach the problem negatively, illuminating what philosophy is by showing what it can’t possibly be. Either approach is fine, if we can make it actually work, in some fashion, but it would seem hard to evaluate the entire philosophical method in this way. How could we possibly show that rejecting any step in the method leads to contradiction, or is necessitated by a genuinely vague proto-philosophy? The solution, I claim, is to delegate this work to a discipline outside of philosophy, specifically mathematics.

But before I turn to that problem I must first return briefly to my previous description of mathematics as a pure formalism. A pure formalism, in the simplest possible terms, is simply a system in which the terms are not pinned down such that their referents can be determined. For example, consider the axioms of arithmetic. The axioms of arithmetic apply to anything that those axioms can be said to be true of, under some interpretation. This might be the numbers themselves, if such abstract objects could exist. But the axioms could equally well describe an infinite sequence of pineapples, so long as they were in sequence of succession. The actual theory of arithmetic itself (the axioms, and the rules for deriving consequences from them) thus cannot be said to be about one of these possible interpretations of it and not the others, at least not legitimately. This contrasts to a theory about trees, for example. The theory about trees contains claims about trees, and all the terms in those claims are fixed with respect to what they refer to. “Tree”, to pick a term arbitrarily, is not an empty concept, we have an intentional relation to trees, ultimately reducing to some kind of sensory intentionality, and thus “tree” refers, if it refers at all, to that thing, whatever it is, that we are intentionally directed at. Naturally this doesn’t prevent us from being wrong about trees, or even so deluded that there is nothing we are really intentionally direct at (if, for example, we had a false memory of trees when none existed, and intentionally directed ourselves at the kind of things that caused that impression, when really there are no such things). But it does fix our theory about trees to be about, at most, one kind of thing. Even if some other interpretation could be given to the theory that makes it a true theory about pineapples this does not make it a theory about pineapples, nor can it legitimately be taken as such. And because they have a subject matter that we have some form of access to such theories develop in different ways than pure formalisms such as mathematics, which are bound only by their own rules, and not by some external standard.

This might seem to imply that pure formalisms are deficient in some way, because they seem much less useful than theories that are about some subject matter. In fact it might make them appear to be completely worthless, and mathematicians fools. But pure formalisms are not without their own kind of value, they just don’t tell us anything about the world by themselves. Their worth stems from their completely general nature, a pure formalism can be applied to anything that its axioms can be interpreted to be about. And thus mathematics can be applied to anything that we can count. Now, obviously, whether the axioms are actually true of the domain that we are interpreting them to be about is something that we can never be completely sure of, but that is true of every general claim about a subject matter, since we never have complete and perfect access to anything.

This is relevant because one of the things pure formalisms might be employed to describe is the method of philosophy itself. The idea being that we might overcome our earlier difficulty, with extending our limited resources to yield an entire philosophical method, by managing to describe the philosophical method with a formalism of some kind, and so reducing our task to simply showing that the axioms defining that formalism suitably describe the philosophical method. There are, naturally, many ways of going about this, some easier than others. I suggest we might proceed by developing a formalism that captures whether a proposed method for philosophy conforms to certain standards or not. As such we need to only capture the method abstractly, we don’t have to deal directly with any claims, for example, about what the domain of philosophy is. If the domain of philosophy is mentioned at all it will only be indirectly, with claims made about the structural properties of that domain, but not what falls under it. The goal then would be to show that there is only one method that is suitable given a certain set of standards, or that all the methods that are consistent with those standards yield the same results (at least under ideal conditions and in the long run).

With such a formalism in place we would be left only with two problems left to solve, the assumptions and the standards. I haven’t mentioned the assumptions so far, so allow me to begin by discussing them. The assumptions I have in mind are assumptions about our capabilities relating to the sources of information we have access to, and certain other facts about our intelligence. Assumptions of this nature are represented within our formalism in the sense that they partly determine whether a method considered in abstraction can meet a set of standards. For example, someone might think that we have intellectual access to a mysterious realm of forms, containing all philosophical truths, and evaluate methods under that assumption. Obviously with such an assumption some methods will pass standards that they wouldn’t under the assumption that we don’t have this kind of special access. Or someone might proceed on the assumption that everyone shares certain beliefs, or that under certain kinds of processes everyone’s beliefs converge to a single set. Again, with such an assumption certain methods will satisfy standards that they otherwise wouldn’t. And obviously we can’t determine the validity of these assumptions within the formalism itself. But, fortunately for us, whether these assumptions are actually true is usually a relatively easy matter to test empirically. In both of the examples mentioned we could just observe people using the proposed methods and see whether their conclusions really will converge. This is a simple enough test and, most importantly, settles the issue in a completely empirical way, with no need to rely on philosophy in any guise.

This leaves us with just the standards themselves to consider. An example of a standard we might ask our philosophical theories to adhere to is universality, we expect them to produce the same theory as a result no matter who uses them, under certain ideal conditions (this could be formalized either by requiring that the information the method proceeds on to be sharable or universal in some way, or by insisting on the existence of a way of resolving disagreements as they are found, which essentially amounts to the same thing). Unfortunately there is not an obvious way to decide on the standards we should adopt. It is here then that we must leverage the approaches discussed earlier for overcoming the problem posed by metaphilosophy in general, we must proceed by ruling out standards (or their negations), or we must proceed with some kind of proto-philosophy to decide what they are that doesn’t entangle us in questionable commitments. Previously I have argued for universality as a philosophical standard in essentially just that way, I argued that we couldn’t accept a philosophical method that failed to be universal on the basis of certain simple assumptions about what was required for a philosophical claim to matter. This is essentially a combination of both ways, it employs a negative approach to ague for universality by ruling out the possibility of doing without that standard. And it does so using a kind of proto-philosophy about how philosophical claims might matter to us.

The real difficulty then is extending such techniques to arrive at a sufficient number of standards from which, along with some empirically justified assumptions about the kinds of information we have reliable access to, we can apply our formalism to and arrive at conclusions about various possible philosophical methods. Specifically we would hope to be able to show that some method uniquely satisfied those standards, or that the results it produces must be produced, as a matter of necessity, by all methods that satisfy those standards. With such a method we could leave our metaphilosophical framework behind, and proceed to establish whatever results we are interested in with it instead.


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