Foundation 1 - Equivalence and difference.

The modern Cartesian Meditator

To consider the nature of the world requires first that the Cartesian meditator, in pursuit of rigor and foundation, consider the preconditions for thinking about anything. This accords with Quine (1969) that one should base one’s knowledge on psychology, presumably cognitive psychology, to assure us that what we think we know is at least something for which there is some mental state capable of drawing well reasoned beliefs.

Before you go further, you need to begin at the beginning.

A challenge to Descartes’ epistemology (meaning theory of knowledge) and hence Descartes’ goal of a sturdy ontology (meaning sturdy description of the world) is Hume’s assertion that all ideas come from impressions. If true, then one might ask Descartes where he gains a basis for thinking about anything from an internalist perspective, for all that one thinks of originated in impressions alone yet is simultaneously subject to doubt. Hence even the contemplation of particulars that might be doubted would leave one in a position that no distinction could be made between anything and anything else with any certainty. By this gauge, Descartes’ meditator could mistake one’s spouse for one’s hat (see Sacks 1985).

Hume said that to refute him one need only provide an idea that does not come from impressions. I will argue that the ideas of equivalence and, by extension, difference (inequivalence) are independent of impressions. These ideas must be inherent within humankind and any sentient being for any impression to become an idea. This will provide the initial step toward a Cartesian ontology, although expanding this into a rich ontology is outside the scope of this section of the blog.

Refuting Hume is of little importance. However, in showing that the Copy Principle itself is dependent on recognising equivalence and difference, and that this is innate, provides a foundation for epistemology, for these associated ideas are at the base of well-founded knowledge. Without them we are bereft of the capacity to think, and indeed bereft of any mental objects that might found a world model of any kind at all. That is, ideas of equivalence and difference are the foundations upon which all that one can know, and how one might go about knowing it, are built.

Foundations of thought

In conformity with Quine’s wishes, I begin this area of inquiry with the view of cognitive psychology, that (Kuhn 1991, p. 6):

Thinking entails the internal manipulation of symbols, and as a means of understanding thinking it is therefore essential to understand how symbolic stimuli are attended to, encoded, and operated on.

In this context, it is when this internal manipulation suggests an inconsistency or hole (incompleteness) in an argument that one’s doubts are raised. If there is no hole, such doubt is not justified, and has no place in well reasoned discourse. One hole, identified by Descartes, is that one might be dreaming, in which case the impression itself is suspect. Consequently one cannot from the internalist perspective vest judgment in impressions received from an assumed external source over the top of rationally derived ideas; for it is the thinking that informs the meditator what is or might be true. But this would not be true if Hume (1995, p. 839) is right when he says ‘all our ideas or more feeble perceptions are copies of our impressions or more lively ones,’ for if true, there is no foundation for thought outside received impressions and all our thinking is founded on it.

This contrasts with Aristotle’s belief that another form of knowledge is accessible, a knowledge that establishes the first premise, that anchors demonstrations of other pieces of knowledge as justified truth worthy of belief (Smith 2004).
This would be the view of the Cartesian meditator also, who follows foundationalism.

From a foundationalist perspective there is initially only two ideas that do not come from impressions: the ideas of equivalence and difference. The Humean might argue that it is receiving impressions from the external world that gives one the idea of there being things that are the same and things that are different. I don’t think this is true, and will present two arguments in support of my case: an argument from priority and an argument from dependence.

The priority argument

Without the prior idea of equivalence and difference it is not possible for one to make sense of impressions. Beyond hardwired responses, such as the pain response (which is not understanding in this context) a human’s first impression is essentially meaningless because it has no point of reference, no point of difference, and, insofar as thinking entails the internal manipulation of symbols (Kuhn 1991), no meaningful mental symbol that can be used as a basis for recognition. This equates at an epistemological level to an absence of understanding about the external world. Even the notion of manipulation requires difference, for the idea of manipulating a mental symbol in isolation is as sensible as the sound of one hand clapping. The mere naming of this concept of equivalence and difference comes after its existence in the human mind (this process of naming, of course, applies to received impressions as well).

The dependence argument

Without the internal idea of equivalence, no number of presentations can convey meaning. The Copy Principle (Hume 1995) itself requires that one assign an equivalence status between externally sensed objects and internal mental symbols. Otherwise the world is an internal fiction brought from who knows where and there is no copying at all, but this reverts to the earlier argument of priority.

These priority and dependence arguments gain strength in the recognition that definitions of equivalence and inequivalence first require an internal understanding of the terms.

One might argue that ideas of sameness and difference have no value without acquaintance, but even if the meditator has no sensory input, there is self awareness. Such awareness is a sense of identity, and identity is the notion of being the same as oneself. If it is any more than this, then one must admit notions of inequivalence, so the counterargument is thwarted. Descartes himself argued that one can be certain of one’s existence, so there is value in this directly. It remains only for the meditator to develop these ideas of self, equivalence and difference into a rich ontology populated by recognisable objects.

If these ideas are so important, why did Hume not notice them himself? As Heidegger (1969) notes, we encounter same and different so unquestioningly, the encounter itself goes unnoticed, and nothing compels us to notice it. Having noticed it, that these ideas echo the Laws of Thought goads one to explore this link further. But more importantly, it would seem that for the Cartesian meditator, the pursuit of all knowledge must be grounded in equivalence and difference.

Conclusion

For the Cartesian meditator there is a basis for thinking about an ontology or domain or discourse. Firstly, the meditator can know he/she exists within the ontology so has a place from which to consider it. Secondly, the meditator has an inherent idea of equivalence and difference by which the objects of the domain might be distinguished. Moreover, because ideas of equivalence and difference are epistemologically prior to all other ideas, it is on these ideas that all well-founded ontologies must be built.

Reference:

Heidegger, M 1969, Identity and difference, Harper and Row, London.

Hume, D 1995, 'An enquiry concerning human understanding', in SM Cahn (ed.), Classics of western philosophy, 4 edn, Hackett Publishing Co., Indianapolis.

Kuhn, D 1991, The skills of argument, Cambridge University Press, Cambridge.

Quine, WV 1969, Ontological relativity and other essays, Columbia University Press, New York.

Sacks, O 1985, The man who mistook his wife for a hat, Summit.

Smith, R 2004, 'Aristotle's Logic', in EN Zalta (ed.), The Stanford Encyclopedia of Philosophy Fall 2004 edn, http://plato.stanford.edu/archives/fall2004/entries/aristotle-logic/.

An every-person's description of the origin.

The seventh step
This is a very rough description of why the universe began, and begins with the seventh step of the order of development that I set out in my introduction. Reiterating - this is just a rough sketch. The full and terse development will be added soon. First read the introduction. Comments are welcome.

In the beginning
Let an omnet stand in for anything at all. Cats, tables, thoughts, nothingness, things we are not and cannot be aware of, or anything else, is an omnet.

Let 'asset' refer to anything and omnet has. Properties, tropes, relations, names we call things, may all be assets. Equally, none of them might be. We have no way of knowing that we have captured a reality by referring to it from experience. For example 'red' is just my way of referring to what I see as a particular color; but this is more likely just my recognition of a particular wavelength of light, and even this is not sufficient, because the wavelength changes if I move relative to the source of emission. So lets ditch properties, relations and so forth, and stick to assets.

For good reason, in the beginning was the simplest omnet possible. This was not a spacetime singularity. This was not infinitely dense, and all matter and energy was not contained within it. To think it might be so is ridiculous. How would the world know what attributes it had? Why mass and energy, space and time, as opposed to something else? Rather, this omnet is essentially windowless. You cannot know much about that simple omnet because its interior is opaque to inquiry. To investigate its 'inner contents' is impossible for then it would no longer be simple.
Nevertheless, this initial simple omnet has the assets that all omnets necessarily have. For example we can be sure it has identity - it has what it has that makes it what it is. That is, this omnet has the assets it has and no other, and you don't need to know which for this to be true.
Equally one can also say that it has 'place', as in Aristotles idea of place, where all things have their place, and place too has its place. It is easy to see that its identity is the foundation of its place, for if it had any different asset, it would have a different abstract place in the scheme of things.

Importantly, this omnet is finite, in that it cannot be infinite, for this would require that it be other than simple, for it would require extension into something that does not yet exist. Such extension requires variation, and variation would require that the simple not be simple, for there would have to be variations within it.

Being finite, this simple omnet has a boundary; a place at which its domain ends. This does not mean that there is 'outside' the boundary. Exactly what a boundary is is unimportant. What is important is that the boundary is different from the simple omnet, and we can know that it has the assets of a boundary. Moreover, that boundary is itself finite, so it too must have a boundary. And so must its boundary have a boundary.

This does not imply an infinity of boundaries all at once, for the existence of the second boundary first requires the first boundary. That is, the second boundary is ontologically dependent on the first.

After the beginning
Consequently there is order to this growth of boundary omnets, and this order is independent of time. Rather, this implies a timelike metronome - one state of the world per boundary iteration. Hence time has a foundation.

Because each boundary is, apart from order, the same as every other, there is an interaction between boundary omnets. In one sense, all boundaries would exist at the same place, there being no difference in assets between them at one level. In another sense, no boundary can exist at the same place due to ontological dependence - there is another boundary already in that place, and two omnets which have this one difference cannot take the same place.

These interactions express themselves in all equivalent ways. One way is in one dimension, and this can be regarded as the simplest expression of the universe. When you set out this system of interactions, with each interaction having an interaction value of 1, in a graphical form, the shape of the universe is rather amazing. The sum of the interaction 'vectors' immediately demonstrates several important constants of physics - ln2, exponentials, and so forth. In equivalent higher dimensional interpretations (and this requires some further consideration before you just rush off and do it) fundamental constants such as i and Pi pop out.

Now there is no Big Bang just yet.
This growth is certainly not a Big Bang. It is a regular growth of structure. But for special reasons this growth is expressed in all kinds of ways, in different kinds of universe that all live on top of one another. We can access these kinds through a newly founded mathematics that is a lot like the present mathematics (and simpler in some ways).

There are several options. One is that, soon enough, physical structures of the kind with which we are familiar coalesce in the expanding space as the world gets bigger. Another possibility is that eventually structures collapse in on themselves and pass through the center of the world. Here 'the center' is defined by the point through which these pass, because in some ways (for reasons a bit involved to go into in this rough description) everywhere is the center of the universe for that point, exactly as we find it to be today. (Yes. You are the center of your world, but don't let it go to your head.) For those who have been reading widely, this is indeed the solution to the paradox brought by Bell's inequality - everywhere is connected to everywhere else in one description of the universe (the graph theoretical description - look it up) so any change is immediately conferred (in one sense) to the world at large (no, this doesn't conflict with Einstein's theories, but, again, we aren't really up to that yet).

That said, its a bit more complicated than that.
If you get the gist of all this, then you can go away happy, but this is a very rough outline. Probably there are things I have said here that are not as clear as I would like. Feel free to ask questions and I will use this as a basis to fill in any blanks. Otherwise you can wait for my more in-depth posts.

Introduction

This is a story about the origin of the world. It is also a story about the origin of the physical universe, why and how it comes to be at all and, in so far as one blog can contain it, why and how it comes to be as it is. It is not a story about the first three seconds. It is not a story about the first seven days. To talk about these latter things we must first find a basis for time, space, physics and mathematics. So this story aims to illuminate First Cause. It aims to answer in an absolute sense (yes, I know this isn't supposed to be possible) the questions:

Is there any knowledge in the world that is so certain that no reasonable person can doubt it? (Russell 1978)

What is the source of all things? (Allen 1992)

and

Why is there something and not just nothing? (Sorensen 2009)

I think there are many people who wonder about these questions, worded in one form or another. For some, the source of all things is God. Asked whether God is the cause or source of himself, they will answer that it doesn’t work like that, and in any case, this is dangerous talk, sacrosanct, and a matter of belief. Perhaps they are right, and the source of all things is God. Perhaps not. But belief does not in and of itself make you right.

Perhaps surprisingly, the same can be said of science. Founded on a mountain of evidence, science provides the illusion of certainty, and foreshadows the most amazing discoveries. Atom bombs! Positrons! It isn’t just space and time, its spacetime! Let me tell you about the first three seconds! If truth is anywhere it is to be found in science!

There are many people who believe that science holds the ultimate truth. Look at all the evidence, they say. Our theories predict! But this depends on what you mean by prediction. For a long time, Newton’s theory of gravity predicted the motion of planets and seemingly all gravitational objects, until one day it didn’t. Mercury, so near yet far, stuck its finger in Newton’s eye. I want to go faster than you say I should, it said. So along came Einstein with the fabulous General Theory of Relativity to sort it out. Ought we hold any greater faith in the predictions of Einstein’s theory than that of Newton? While these theories provide the illusion of truth, they are merely of sufficient complexity that the phenomena of which we are aware happen to coincide with the theories offered, and whenever some confounding phenomena occurs, the scientists pick and choose from the available options to create a new theory.

Until one at least understands the basis of why things are as they are, and how this happens to be, sprung off a well-founded theory of the origin, the truth of such theories remains in doubt. Though physics is my trade, there are other reasons to be suspicious of science, at least philosophically. Indeed, when taken to its limits, as mathematics would dictate, we ought not be here at all, a point to which I will return in another blog, or when someone asks about it. Of course, the best physicists, mathematicians and philosophers already know this, but are content to hold to their faith in science, set theory, or naturalism.

More than this, because science is founded on empirical evidence (the evidence of the senses) yet the origin if there was one (as scientists mostly believe) existed before there was physicality, then science cannot be regarded as complete. Again, I will consider this in more detail later. Because these theories are both incomplete and always necessarily falsifiable, our scientific theories require belief to attract adulation. And, as with religion, belief does not make you right.

To explain why and how the world comes to be at all, and why and how it comes to be as it is, is a challenge (some would say impossible, but they are wrong). But this challenge is not as difficult as laying it out in a way that others can follow. My intention is to have all the pages complete before New Year 2011.

I pause as I put this out to the world to judge. It is easy to know one is right in private, but quite another to lay it down before others with sharp philosophical knives, and a lifetime of reasons to stick to their old ideas. I hope that my Cartesian shield (I explain this later) will be strong enough.

Now this is a fairly large task, that will require a little patience on your part. I hope you will rest easy in the knowledge that the volume of my words is tiny compared to the flotilla of books that have been written on the subject, none of which answers the above three fundamental questions.

I can and will answer all these and many others besides. To do so, I must first develop a framework in which a study of reality can be properly carried out (an 'ontological framework'). And to do that, I must first provide a secure foundation for thinking about the world at all. I have several formal papers on this, but these are probably a bit heavy going for a blog. Should someone want to see these, I can forward them if asked. With this in mind it may be that my blogwords will sometimes be open to misinterpretation. Ultimately I stand by the formal papers.

To do so requires a rigid order of going about developing the theory, and Descartes' Foundationalism is the only proper way to do so, as will become evident later. The central insight of foundationalism is to organize one's beliefs in the manner of a well structured, architectural edifice. Nothing may be introduced unless it is either necessarily true (and even self-evidence is insufficient at this level) or follows by unshakable inference from such truths.

As Descartes says:

All the mistakes made in the sciences happen, in my view, simply because at the beginning we make judgements too hastily, and accept as our first principles matters which are obscure and of which we do not have a clear and distinct notion. (Search, AT 10:526, in Newman 2010)

He said once that he could put his arguments in a formal order (in response to a criticism by Gassendi) but when he did, he relied upon definitions, axioms and so forth, any of which, in these days of modern philosophy, would not stand up. I will consider this in more detail at another time.

The natural order of development prevents me from explaining straight up how the world comes to be, except in the loosest fashion. Rather, one must first identify a basis for thinking about the world. So the order becomes:

1. Provide a foundation for thinking about the world. This is not as plain as you might think.

2. Find a place for the thinker to view the world (for a person cannot assume they have a rightful place to stand and look out at the objects that may or may not exist).

3. Develop an understanding of what skepticism is, and how the skeptic can come to develop a well-founded understanding of how the world is, contrary to two and a half millennia of argument to the contrary.

4. Identify a principle of equivalence and prove it to be a truth beyond doubt. Such a truth is a model of a condition of the universe. In this case, the principle applies to all that is, so all that is, necessarily beats to the sound of its drum.

5. Show that there is a unique origin of the world (meaning all there is, including physical things and thoughts, as well as anything else there might be).

6. It will also be clear that all physics, mathematics and philosophy also must be founded off this origin, as does space, time and everything else. We begin with a very sparse universe indeed!

7. Further investigation will show that the unique and simple origin evolves into complex structure and with it comes the ability to reinvent the numbers, geometry and others of our mathematics, but with some important differences. The mathematics is constrained to rationals, and we will introduce a new concept I call 'Block numbers' which fill between the rationals. If you don't understand 'rationals' and aren't too good at mathematics don't be concerned because it starts simple, and grows in difficulty only slowly. Is this not what you would expect of a new universe? How could the universe 'know' how to start out complex? This is similar to Penrose tiles, where simple rules lead to complex results.

8. Finally we will investigate the growth of the new structure, and find that our favorite physical constants appear as if by magic, with no need to introduce this or that parameter, or to jiggle this or moderate that, as the physicists do.

9. Is there a Big Bang in there somewhere? I don't know. Maybe you can help. What I do know is that there is a unique origin, a growth of structure, and that structure homes in very rapidly on the equations that we use in our best physical theories, but now without the problems that these equations have, such as infinities.

How this comes to be written.
In 1996, while studying physics, I bumped into a problem with mathematics. The book I was working with (I think 'Calculus' by Swokowski) said, 'A hammer, falling from 500 feet, passes through all points on the way to the ground.' All very good for the hammer, but how does a point in space trade the properties it had for the properties of the hammer? To do so would require an infinity of infinities of changes, for presumably between one property and any other, or variation of that property there must be an infinity of variations, so the act of change enters an infinite regress. And that is just for one point!

More than this, given that there is an infinity of points between some point on the hammer and the ground and that same point on the hammer and any other point in that line, how can the hammer go to the next point, when there are an infinity of next points to choose from? Indeed it implies a scaling problem for the universe, where every distance is the same as every other, at least this is what contemporary mathematics would argue, and the mathematicians nod their head sagely. It's all just a bit crazy, a fact that the greatest mathematicians have already identified, for example Riemann and Hilbert. In fact the problems associated with dropping a hammer makes me very suspicious of hammers. I did not know it then, but I had crashed into Zeno's paradox, and the riddle of Parmenides. If you do not know what these are, you might click on the hyperlinks, gain a quick overview, then come back.

Now, philosophy is divided into two halves: before Parmenides, and after Parmenides. This is because before Parmenides, philosophy was little more than opinion. For example, Thales, the first recorded philosopher said that the world comes from water. One might take from this that he meant that water is the basic element from which all else is made. Instead, Parmenides used reason. From reason he developed a metaphysical argument that has earned him a reputation as early Greek philosophy's most profound and challenging thinker (Palmer 2008). Under his guidance, his follower Zeno developed a series of paradoxes that show that reality does not accord with reason. That is, the world as it appears to be does not accord with how it ought to be, if the world accords with reason.

Most philosophers believe that Parmenides rejected pluralism and the reality of any kind of change. I don't think so. While he argued that what is, is one indivisible, unchanging reality, and any appearances to the contrary are illusions, and contrary to reason, in actuality he asks future philosophers to show how the paradox of change in a world that ought not be able to change can be resolved.

Unsurprisingly his philosophy found many critics, who ridiculed his argument because it flies in the face of some of our most basic beliefs about the world ('Look! There is change everywhere. Hence you are wrong!') But then along came Zeno. Imagine, said Zeno, the flight of an arrow. At any moment in time, it is frozen in time. Indeed, at that moment, everything else is frozen in time. Indeed, no matter how you model it, all that might change is frozen, and there remains nothing that can bring the arrow to move forward.

Aristotle saw deeply into this paradox:
'Zeno's difficulty demands an explanation; for if everything that exists has a place, place too will have a place, and so on ad infinitum. (Aristotle Physics, in Huggett 2008).'

From Aristotle's time to now, no satisfactory answer has been forthcoming, despite protestations to the contrary (see Huggett 2008 for an array of methods that purport to solve Zeno's paradoxes) and the presentation of mathematics that shifts the problems out of sight.

In any case, Zeno's paradoxes are just a few of an entire class of related problems that imply that the world ought to fall apart; yet it doesn't. How this is, and that it provides a proper foundation for understanding the world is my task to explain.

It is an unfortunate if often useful fact, that physics, philosophy and mathematics have had a very long time to develop. Useful, in so far as these disciplines are the foundation of our present standard of living and do show us a rich landscape that stands in for well-founded knowledge of the universe.

While useful, this is unfortunate in that my task is made more difficult simply because of the enormous number of perspectives provided by a large number of philosophers, mathematicians and physicists, since 2500BC, most of whom will be wanting to challenge the view. They have the advantage of many followers. I have the advantage of being right, at least up to the point of Cartesian skepticism (I'll use USA spelling for this) which all agree is the most rigorous level of skepticism available to those of us who think deeply about such things.

Given the journey that lies ahead, I think it only fair that I provide an every-person's explanation of why and how the world began. This us a loose description only. It will be the focus of my next post. Comments are always welcome.

References:

Allen, RE, ed. 1992, Greek philosophy: Thales to Aristotle, The Free Press, Maxwell Macmillan International, Sydney.

Huggett, Nick, "Zeno's Paradoxes", The Stanford Encyclopedia of Philosophy (Fall 2008 Edition), Edward N. Zalta (ed.), .

Newman, Lex, "Descartes' Epistemology", The Stanford Encyclopedia of Philosophy (Summer 2010 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/archives/sum2010/entries/descartes-epistemology/.

Palmer, John, "Parmenides", The Stanford Encyclopedia of Philosophy (Fall 2008 Edition), Edward N. Zalta (ed.), .

Russell, B 1978, The problems of philosophy, Oxford University Press, Oxford.

Sorensen, Roy, "Nothingness", The Stanford Encyclopedia of Philosophy (Spring 2009 Edition), Edward N. Zalta (ed.), http://plato.stanford.edu/archives/spr2009/entries/nothingness/