Thursday, June 28, 2007

The General Principle of Equivalence

Page Navigator>>
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.


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,

Swoyer, C 2000, 'Properties', in EN Zalta (ed.), The Stanford Encyclopedia of Philosophy Winter 2000 edn,

Sunday, June 24, 2007

On first cause

Page Navigator>>

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.

1. Introduction: why is there something rather than nothing?

The first question that should rightly be asked, according to Leibniz, is 'Why is there something rather than nothing?' This is the biggest question, and has been asked since the beginning of scientific and mathematical thought. Even the Ancients asked this question. And, it would seem you too are asking this question and probably getting a heap of unhelpful answers. You will have read various other works on the matter and found one or more of the following proposed answers:
  • The non-answer:
    This is not a question that can be answered by humans for we cannot know the world as it is in itself. This is the philosopher's answer, based on Kant's work, and follows from Hume's work; or
  • The boggle-with-science non-answer:
    The universe began with a vast explosion that generated space and time and created all matter and energy in the universe. Of course, while this may be true, this is not an explanation (don't be fooled) it is only a description. Most of us freeze at the maths involved and so take it on faith. I know this maths - it is cool, but it is still just a tool of description, as is physics. Describing an electron in terms of quantum mechanics for example, is just a description. It says no more than 'an electron has certain properties'. It does not explain why these properties are the way they are. While physics is extremely useful, and likely 'mostly right' this does not make the above an explanation, so it is not an answer.
  • The shifting the answer one more step away non-answer:
    This is also included in the physics answers: that observed phenomena are 'explained' by forces of nature, and these are explained by something (say quantum mechanics or relativity), that are explained by something (say string theory) ... a potentially infinite regress. One might include recourse to God under this bullet point because God is something, so one might ask why is there a God and not just nothing. I am not making any claims for or against God here. You will have to make that call after I have explained first cause to you and you properly understand it. There are many other shifting methods available, from spiritual concepts to claims that the world is implied by the Laws of Thought (ie logic). But one must first show why the Laws of Thought exist, even as conditions of the world (I'll touch on these elsewhere, or you can look it up in Wikipedia or some other source).
In contrast, I can give you the answers, proven beyond Cartesian doubt (the most stringent of skeptical positions that one can have). But before you proceed, I want you to understand that gaining this knowledge comes with a price:
  • You will have to think deeply about the world, and come to a higher, more skeptical level of understanding about the way the world actually is, and your perspective of it, and your position in it.
  • You must promise yourself not to shy away from unexpected or unwanted conclusions. You are now bound by 'skeptical commitment', which is that if you have found a proposition that can only be true, you cannot simply bat it away because some less well-founded, but appealing concept presents (such as the world as it appears to be to us).
  • In compensation, I can say that the end view is a more beautiful world than the world you presently inhabit, because you will see it with a better perspective, as an expression of a simple, necessarily existent condition: a principle of equivalence. This is the most mature world view, and manifests as a sudden recognition of the beauty of nature and occasional epiphany such as why the numbers we find exemplified in the physical universe happen to be as they are (say 'e') and not some other number.
The second biggest question is 'Given that there is something, why is it the way it is and not some other way?' The answer to the second question is in some ways more amazing than the first. The answer to both can be traced to the same root. That is, if there is a truly first cause then all things must find their origin in it. That is, the world and all its well-founded descriptions - physics, mathematics, philosophy - all must show how they justify a place in our thinking by showing that they follow as an implication or expression of first cause.

There may be many questions that immediately come to mind. Here are some:
  1. How can there be a first cause? Doesn't it require that it have a cause? Doesn't the idea come as an infinite regress? Didn't Kant and Hume, and bunch of other people say this can't be done?
  2. If there is a first cause, how can I know that it, and not some other, is the first cause?
  3. Isn't this promising something for nothing?
  4. What about God? Isn't he first cause?
There are probably many other questions. I cannot answer all at once. Rather, I want to show you first cause and why it is first cause and why the proposition that I use to model it is a justified truth worthy of belief (I provide the justified truth, the belief is not my prerogative). Then the first two questions just go away. The third question is an odd one to ask. Given that both are just terms that refer to some possible state of affairs, why should one find a possible transition from nothing to something (or the reverse) any more odious than a transition from one state of something (the universe, say) to another state of something. Both are states of being. But this is a deep matter to which I will return when the ground has been plowed.

As for God, well. If there is a God, he instructs me to live by the truth, for otherwise I am living a falsehood. If I find an absolute truth (ie can't be false) and that leads me to find that there is a God, well and good. If the same reasoning shows there is no God, then God must step aside for fear of breaking his own rules about truth.

The world in a nutshell.

Eugene Wigner (1960), one of the twentieth centuries smartest mathematical physicists asked
How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?
How would one know which phenomena should receive preference? Does any phenomenon deserve priority of importance over any other? Simply, no. In the phenomena of everyday experience there is no hint of the subtle results that forced a radical rethink of our scientific theories that led to Special and General Relativity, and Quantum Mechanics. So if we are not justified to elevate any particular phenomenon, how can we conceive of any phenomenon as a preferred place to start. In particular, all physical phenomena are subject to doubt, for we cannot exactly measure them, neither can we be sure that we are not mistaken. Consider the work of Descartes if you doubt me here, for I will be building my case from his method of doubt. If you are prepared to trust my word on what he said, then we can proceed.

Loosely speaking, I will show that there are two things about which one can be absolutely certain: that one exists, and that what is, has what it has. These may seem fairly obvious, but I aim to show that these are concepts (properly formalized) that are transcendent of doubt. Removing the possibility to rationally doubt these concepts has several very serious consequences to the implications of these propositions. The first provides us with a place to stand and consider the world. The second, though seemingly no more than a global tautology (a proposition like 'A bachelor is an unmarried man') but this global tautology implies directly that there is a condition of the world that applies for all worlds, and also implies that all possible worlds have a unique origin. This origin is a 'thing' that has unknowable and irrelevant properties beyond having abstract identity and an abstract boundary (explained later).

I will then show that this first state is not stable because the boundary itself has a boundary (just as every meaning has a limit of meaning), but iteratively. First there is one boundary, then two, then three, and so forth. As the boundaries evolve, so too the interactions between them, all based on the principle that brought them to be in the first place (the General Principle of Equivalence). Taken together this implies the generation of structure. The structure exhibits many of the important mathematical characteristics we find in the universe, such as 0, 1, i, e, and Pi. We will also find that the structure is pervaded with an inherent uncertainty that cuts off irrational numbers, but retains a complete system with no gaps between points. This concept I have not seen anywhere else, I had difficulty understanding it myself when first it arose. I have studied this kind of structure through my post-graduate years. If you are a mathematician, physicist or philosopher, you might find this quite entertaining.

I will then show that there are many different perspectives that may be brought upon this system, all of which are equivalent under the General Principle of Equivalence (which is the final arbiter). This means that the system has interpretations in 1, 2, and higher spacetime dimensions (and possibly other kinds of dimensions that are not included for the present). In the 3space-1time interpretation early indications are that this is the right kind of conditions for life, for the structure contains structures that are closely associated with our beloved quantum physics.

The fundamental purpose of this blog is to elicit support for this world-view. I believe with just a little more work, all the bothers associated with quantum mechanics and our empiricist limitations simply fall away.

Now you are ready to consider the next step. Click on 'The General Principle of Equivalence'.