On Philosophy

October 7, 2007

Pure Formalism

Filed under: Epistemology,Metaphilosophy — Peter @ 12:00 am

Yesterday I “proved” something about logical deduction, namely that it could not extend our knowledge and was thus not the proper tool with which to make claims in any discipline where we strive for knowledge (obviously I have in mind here philosophy specifically, and am refuting the idea that philosophy is to be done by some form of logical deduction in a roundabout way). This may strike you as contradictory. How can I “prove” something about logic deduction, via a process that itself looks a great deal like logical deduction, without contradicting myself? That is a good question, and by answering it I can elaborate on and clarify some of the claims made previously.

Although yesterday’s post doesn’t make it apparent, the arguments made in it are grounded in a very simple theory about knowledge (or at least true and reliable beliefs, which I will call knowledge for the remainder of this post), specifically that it consists of statements/theories of some kind that accurately reflect the subject matter. The key claim is that there is some distance between the statements and the subject matter; the statements do not incorporate the subject matter in some way, the mind does not reach out directly to it, and the subject matter that we are trying to capture that has an objective existence (or, in other words, that there is a standard by which claims could, from some god-like perspective, be definitely determined to be right or wrong). This is not a claim that stems from the meaning of knowledge, simpler axioms, or an intuitive grasp of what knowledge is. Rather it is an empirical generalization from an extremely large number of cases in which we have knowledge about something. All of these cases follow the above pattern and so I reason that knowledge in general follows that pattern. While I could be wrong I have discovered no counter-examples so far (at least no cases in which it is clear that we have knowledge about something, I will discuss mathematics in a bit).

This then is my knowledge about knowledge itself. And obviously that knowledge comes along with a formalism, which is my theory describing it (that it involves statements that accurately reflect a subject matter … and so on). What I am doing then is not making a positive claim about logical deduction, but a negative claim. I am claiming that the formalism of logical deduction conceived of as yielding new knowledge is inconsistent with the formalism that is my theory about knowledge. If we accept the conclusions of that argument then what I am showing by this reasoning is not a claim about some subject matter, but rather a claim about a formalism, specifically the formalism of logic. And that is the kind of conclusion that logical deduction can actually justify. In addition, it is clear that the argument has not extended our knowledge about knowledge itself, the theories that people actually endorse themselves and their relationship to their subject matter. So the conclusions of the argument do not negate the argument itself or its results.

Thinking about knowledge and its subject matter raises an interesting question: what is the subject matter of logic? I would claim that if logic can be said to have a the subject matter it is formalisms themselves, but I lean towards the idea that, properly speaking, logic has no subject matter and is thus a pure formalism. Simple reflection should demonstrate that if logic has a subject matter it is not a physical part of the world, because otherwise logic would be conducted with the aid of some kind of instruments, and logicians would conduct experiments to test their axioms. If logic has a subject matter then it seems likely that the subject matter is formalisms themselves, because logic can produce “knowledge” claims about formalisms, just not knowledge in the sense used here, as about some objective and separate domain. I lean towards the claim that logic should be treated as having no subject matter because there isn’t the appropriate distance between logic and formalisms. The distance is lacking because the rules of logic essentially define proper formalisms from those that are simply mistakes (or, in other terms, that each formalism brings along with it some kind of logic). To abstract away general principles from the subject matter we must frame our claims in some kind of symbolic (representational) fashion, defined by rules that define how this symbolic model “works” to describe the subject matter. To use a mathematical example a function is a symbolic representation of some relationship, and the rules define how to convert inputs into outputs in accordance with that function. So we can’t even have a formalism (or at least not a useful one) without these rules. Thus logic and formalisms come into existence together, and so it is hard to separate them and say that logic is about formalisms, which is why I lean towards the view that logic is a pure formalism. Similar reasoning can be extended to the whole of mathematics. Mathematics does not have a subject matter; numbers and the other objects of mathematics do not have an independent existence, but rather are defined completely by the rules of mathematics. Which is why I would argue that mathematics is not “knowledge” properly speaking, because it has no subject matter.

But just because logic and mathematics are pure formalism doesn’t mean that they are useless. Indeed they are essential components of knowledge, because it is with the formalisms of logic and mathematics that we capture our subject matter. A function by itself doesn’t indicate anything, but the same function can be turned into a general claim about the relationship between rainfall and corn growth. And in this service it is knowledge, not about numbers and their relationships, but about rain and corn. Science is, naturally, the easy case since the subject matter and the relationship of the discipline to it is relatively obvious. I would contend though that the same goes for philosophy, that philosophy has its subject matter and that it must describe it through formalisms. It’s just that the subject matter of philosophy is not so easily poked at with an empirical stick. Consider this very investigation. Our subject matter is ideas (theories) and their relationship to the things in the world that they are supposed to describe. Both the ideas and the things they describe are quite clearly part of the world. And although we may make prescriptive statements as well as descriptive ones, claiming that the ideas would be more accurate if investigations were carried out in one way instead of another, things in the world have never stopped being our subject matter (prescriptive statements are simply involve extending our formalisms to possible cases and comparing the facts about the possible cases to the actual cases). And so philosophy has essentially the same relationship to formalisms as science, philosophy studies the world and the formalisms themselves are best left to the mathematicians (or philosophers of a mathematic bent).

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