Thursday, June 28, 2007
The General Principle of Equivalence
1. On First Cause
2. First steps: The General Principle of Equivalence
3. Building a sturdy ontology.
4. The cataclysm that attends the General Principle of Equivalence
5. On the structure available to possible worlds.
6. Boundary conditions of the world - the Cartesian view.
7. On denoting and the laws of thought.
8. Something for nothing, and why this is not absurd.
2. FIRST STEPS - THE GENERAL PRINCIPLE OF EQUIVALENCE
Principles of equivalence are the most important to the disciplines of physics, mathematics and philosophy. In mathematics this is embodied by the Axiom of Identity: A = A. In philosophy it is represented by the first law of thought: what is, is. In physics to have an equation requires that there be an equivalence drawn. For example, one equation is that force equals mass times acceleration, or F = ma. What are we saying here? We are saying firstly that if we know a body's mass and its acceleration, given various units, then we will get a number with a composite unit (in this case Newtons) and the equation remains consistent for all masses and accelerations (let's leave Relativity aside for a while to keep things simple - the concept remains the same).
I am going to show that the most general principle of equivalence is also a model of First Cause. In advance I am going to name this principle the General Principle of Equivalence, and abbreviate it 'GPE'. By 'a model of First Cause' I mean that the proposition I am going to enunciate has a correspondence with a global condition of the world, that is it holds for everything in the world, at every level. By 'the world' I mean whatever there is.
Most importantly, if you want to be certain of your understanding, it is of highest importance that you assume nothing about the world beyond that which is certain in an absolute sense. Absolute knowledge of how the world is ('in itself') is not supposed to be possible according to present philosophy, so this is a rather cheeky supposition, but I'll leave to to decide.
So, for a while, set aside your expectations that there is a computer screen before you, that you have hands with which you type and so forth. Descartes dealt with this and I won't waste your reading time going over his work in detail here. Know only that he showed that one can be completely certain of one's own existence ('existence' being just that you are - which you will soon see follows from the GPE). Further, understand that the reason you can be certain of your existence is because you have no basis to doubt it.
Janet Broughton (2002) discusses this. She points out that the reason one can be certain of one's existence is because, to doubt, one must have a reason to doubt. It is not enough simply to say 'I doubt', for this is not justified, it is just a statement. Nevertheless, virtually everything can be doubted. One might be dreaming, or hooked up to The Matrix and deluded. By this hyperbolic doubt most of the propositions one might make about the world are suspect.
But one cannot doubt a proposition (meaning a set of words that proposes something about the world) if the doubt itself relies on the truth of the proposition for it to be a well-founded doubt. Otherwise the act of doubting undermines the doubt, and supports the initial proposition. So, in Descartes's case, 'I think, therefore I am,' is indubitable because to doubt this, I must exist, otherwise I could not be engaging in the act of doubt. Hence there is no proposition that I can utter that can undermine that proposition. This is good, because it gives me a place to stand, a place from which I cannot be shifted into oblivion.
Now 'I think, therefore I am' is the most particular proposition. It is the center of one's world, but says nothing of the world beyond oneself.
The other proposition is at the absolute other end of the spectrum. To write it with proper meaning requires that I take my perspectives out of the world. Kant talks about these perspectives, and took a long time to say it. In extreme precis what he said was that one cannot know the world-as-it-is-in-itself (as opposed to the world as it seems to be to us) because we see the world through the lens of the mind. For example, the way I see the world, interpret the world, is vastly different to the way an electron might see it. An electron has a more natural interpretation in reciprocal space or phase space and when I use equations to study its motion they are easier to work out in these other spaces. Which space is 'real'? Under the GPE it will become clear that all such spaces are just different interpretations and equally valid given certain conditions that apply under the GPE.
To get around this problem of the lens of the mind, one must take one's perceptions out of the system entirely. Simply this. If I talk about 'things' most people immediately tend to think of objects. You may have a broader view than just objects, and might be thinking mathematically, using 'elements' of sets. But there is no guarantee that this is a proper view. There may be things in this world that humans can never know of, and have no physical or mental condition to be able to be aware of them. Some (many) naturalist philosophers (for example Colyvan and Armstrong) argue that we should believe in only causally active entities, that is, if something has no action in the universe, it may be discarded. I understand why they might think this way, but philosophy has reduced to opinion these days and hasn't moved a half pace forward since Descartes.
To circumvent the problem of the lens of the mind, I introduce a new term:
Omnet - An omnet is a term that refers to anything at all, whether we have prescience of it or not. A soccer ball is an omnet, even if it is just my idea of a soccer ball. So does a flurgal. Now I have never heard of a flurgal, and have no knowledge of what it might refer to other than the term itself. Yet it is an omnet. The whole point is that I make no assumptions about what is. I may refer to 'omnets' (plural) in the future, but let omnet be singular or plural without judgment as to which applies. We don't have a basis for considering singular or plural as yet. Mathematics is as yet unsupported.
I want to make a cognitive distinction within omnets. That is that at present all omnets are just possible omnets, meaning omnets that possibly exist in some reality. Some of these may be found later to be well-founded omnets, meaning that there is well-founded reason for us to believe in certain omnets. Unfortunately, at present we can't know which are well-founded, because the evidence of the senses is not sufficient, for the reasons described by Descartes.
For those familiar with Descartes work, I wish to add a caveat. I do not accept all of his reasoning and especially the major body of his conclusions at all, for they were not well-founded. Rather, I accept only Broughton's interpretation of his method of doubt, and not her conclusions either.
So, the world is omnet(s). Let me add a further term:
Asset - An 'asset' is any omnet that an omnet has. So, if a chair is an omnet, it assets might be its legs, back, seat. Or it assets might be its color, mass, shape etc. The fact is that I have no basis for knowing what its assets are at this stage. Its assets might be its atoms; who knows? It doesn't matter because I again leave the term as wholly unspecified other than that asset are what omnets have.
I can now formulate the General Principle of Equivalence:
Every omnet that has all the assets of that omnet is that omnet.
The condition of having all and only one’s assets, then, holds implicitly that the identity of an omnet (what an omnet is) is founded on its assets (what an omnet has). As developed, ‘identity’ is the condition of having all one’s assets and none other than one’s assets. What it means ‘to have’ one’s assets, is based on the ontology (the study of what it means 'to be') of the system. I place a restriction on the terminology that what an omnet is, is conformal with what an omnet has, such that if there is some place where the two do not conform to the same place in the language, or one’s idea of it, that the meaning of ‘is’ and ‘has’, implies a different meaning not intended here. This paragraph may be confusing for some, and is not important if you can see what the statement means and that it captures all possible worlds.
I named the above proposition ‘The General Principle of Equivalence’ because it describes the basis for notions of equivalence and inequivalence, in so far as the complexity that ‘what is’ has, corresponds to equivalence and inequivalence (similarity or difference) of assets.
Why the GPE is immune to doubt.
As mentioned above, up to the limits of rational discourse, to doubt, one must have rational grounds for doubt, even if these are radical or ‘hyperbolic doubts’ (Broughton 2002; Newman, L. 2005). Such doubts include for example those put by Descartes: the dream argument, the evil genius argument, the lunacy argument and the fate or chance argument (Broughton 2002). However, even if one addresses each argument in turn, there is no guarantee that another basis for doubt will not be raised at some future time, unless all such arguments are struck down universally, so that there is no possibility of raising rational doubt.
The great value in Broughton’s interpretation of the foundation of Descartes’s method of doubt, whether this was what Descartes meant or not, lays in the recognition that one is not justified in holding a doubt about some proposition if that doubt relies on the truth of the proposition for the doubt to be valid. So, if there is a proposition upon which every possible act of doubting relies on the truth of that proposition, than that proposition is immune to doubt, and the Cartesian Meditator is honor bound to accept it as a necessary truth - a model of an actual condition of the world (see Building a Sturdy Ontology for rules of acceptance).
The indefeasibility of the General Principle of Equivalence follows directly from this. For any doubt to be valid, it must at least be internally consistent, which requires as a minimum that as an omnet the doubt must at least have all its assets, and only its assets and nothing different from the assets it has (which by the construction of the founding terms is all that such a doubt can be). In this way such doubts require that the General Principle of Equivalence be true. Then a doubt relies upon the General Principle of Equivalence and is self defeating. As such, the GPE is immune to Cartesian Doubt. It is transcendent of doubt for no doubt can be properly raised against it.
Further, because all that is, is an omnet, the idea holds globally - there is no 'outside the system'. It is a universal truth.
It is upon the GPE that I intend to build my case. The fundamental shift is effected by trading reasonably assigned, but not well-founded meanings, such as things and properties (as based on the evidence of the senses), for the global non-specific omnet and asset, that refer to all that is, but makes no supposition as to what the nature of the world is (for example the suppositions that attends the idea of substances and attributes) beyond that expressed in the statement itself.
Such universality and apparent triviality has been previously held to be blighted by the ‘fundamental ontological trade-off’ (Swoyer 2000, n.p.): that a simple but believable ontology (meaning proposed idea of what there is and how it is) seems to have a poor explanatory value, as compared with a rich ontology having great explanatory value, but in which it becomes difficult to believe in the ontological ‘machinery’. This trade-off falls away if a simple founding ontology has sufficiently broad implications. Then, by establishing the most global proposition – the General Principle of Equivalence – beyond Cartesian doubt, everything that follows from it may be rightly viewed as a reflection of the way the world is.
To continue further, we need to provide a system that is supported by the GPE. This will enable a sturdy ontology. Then we can consider the nature of the world, and will find that the world has a unique beginning.
Broughton, J 2002, Descartes’s method of doubt, Princeton University Press, Princeton.
Newman, L 2005, 'Descartes' Epistemology', in EN Zalta (ed.), The Stanford Encyclopedia of Philosophy Summer 2005 edn, http://plato.stanford.edu/archives/sum2005/entries/descartes-epistemology/.
Swoyer, C 2000, 'Properties', in EN Zalta (ed.), The Stanford Encyclopedia of Philosophy Winter 2000 edn, http://plato.stanford.edu/archives/win2000/entries/properties/.