The universe in one dimension - the one dimensional Harmony Set

The birth of structure in our world.

Trusting to the path of the modern Cartesian Meditator, a world evolves. It has interpretations in all dimensions, but the simplest to understand is the 1 dimensional Harmony Set, shown above. The graph shows successive iterations of the world. Because this world develops iteratively, the evolution provides a natural background metronome for the world (this may not be time as we think of it).

The universe begins as a pair of opposing vectors that lie in superposition, the strength of which is represented by the single bar (purple) at the front of the graph. Then this absorbed into the structure shown behind it (brown). This is then absorbed into the structure behind it, and so forth. Consequently the passed leads to the present. The future is not known until the next iteration arrives. Nevertheless, the form of growth can be considered with a level of certainty. But this is exactly what we find in our own world.

At first view it seems almost fractaline in appearance. Exploring it reveals many evolving numeric series that crowd ever closer to our most revered mathematical constants. The left edge exhibits exponential relations, and the whole structure is pervaded with variations of exponentiality. That such exponential relations are known to be at the heart of the equations that fuel quantum physics is relevant. That the peak of the evolving series grows with a rate that crowds in on the natural log of 2, one of the more important constants in physics, is again of interest. What is missing in this set is our treasured number Pi, but it is evident in higher dimensionalities. Is this not what one would expect?

That the model is a bar graph, rather than a continuum of change will bring many to dismiss it. But this difference is fundamental to allowing the world to work. Without this (which is inescapable if one begins with the implications of the General Principle of Equivalence) Zeno's paradox would bring the world to a sudden halt. Instead, here is a foundation for entropy, a proper foundation that ensures the world will evolve, a direction of time, and everything else.

How this world comes to be at all and comes to be as it is, will be the focus of future posts.

The General Principle of Equivalence

This post identifies the first premise upon which all other premises rely. I call it the General Principle of Equivalence. My argument is, and I will gradually show, that from it all other certain knowledge comes. At the outset it may seem just too simple to say anything worth considering, but this is only because, like so many things, it is so close to us that we do not see them. As you come to understand it, you will see that it is no more than an expression of our innate understanding of equivalence and difference.

Accepting, for reasons already given (see my post Foundation 1: Equivalence and Difference), that our notion of equivalence founds understanding, one can begin to study, from first philosophy, the nature of the world. This will provide insight into the validity and nature of the domain of discourse, namely the terms that refer to possible structures in the possible ontology (see my post Foundation 3: A Constructional Ontology).

Using the terms ‘omnet’ and ‘asset’ permits a fundamental shift in the nature of Leibniz’s Law from an effectively synthetic, contingent proposition to a global tautology, here named the General Principle of Equivalence:

Every omnet that has all the assets of that omnet, is that omnet.

Now this is the simplest of concepts, no more than the global tautology. In some ways, if one assumes the innate idea of equivalence, the proposition may be reduced to a word, ‘omnet’. However, and most importantly, while for any particular tautology, the statement is essentially vacuous to the Meditator, the globalised concept, I believe, is wholly different. It can be shown to be true transcendent of doubt and this changes it from an apparent truth to a truthmaker, a constructor, and a sieve of the possible ontology (explained in Foundation 3). That this is true will become evident in fur


Why the General Principle of Equivalence is immune to doubt.
Descartes introduced radical grounds for doubt to suspend judgment about the whole domain of discourse, with the intention to ‘establish something lasting in the sciences’ (Broughton 2002, p. 97). Emulating an architect’s method, he excavated back to the foundations. He did so in the understanding that, by utilizing the strong maxim that one should doubt all that can be doubted (rationally) Descartes would have ‘the power to go up against the authority of common sense’ (Broughton 2002, p. 97). And it is in common sense that philosophy has foundered for a very long time.

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. Success in this then, ought to be an end to criticism. It then rests with the Meditator to understand, accept and commit to the General Principle of Equivalence (see my post Skeptical Commitment and the Modern Meditator).

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.

The indefeasibility of the General Principle of Equivalence follows directly from this. While the GPE is self-evidently true, such self-evidence is insufficient for our purpose. Rather, the GPE is indefeasible exactly because every proposition (true or false) to be that proposition, must have all the assets it purports to have, and not other than these assets, otherwise such is not the proposition it purports to be. Then each proposition assumes the GPE to be true for it. Then any proposition that expresses doubt against the GPE, is reliant on it being true for it. Hence all propositions (true or false) are epistemologically dependent on the truth of the General Principle of Equivalence.

Immediately we see why tautologies are necessarily true, beyond self-evidence, because the GPE makes it so due to necessity (this does not imply that the supposed content of the tautology has a referent of course, but that is a story for another time).

So, all propositions, at minimum, rely on the truth of the GPE, and all propositions purport to model a condition of the world. Then, firstly, the GPE models an actual condition of the world, and it is ontologically prior to all other conditions of the world. Consequently, the GPE models the origin of all that is.

Future posts will focus on how the GPE implies a unique origin to structure, then explore the nature and evolution of that initial structure.

Foundations 4: Omnets and assets

The epistemological dependence (the contingency) of propositions upon the World-of-Seeming (the world as it seems to be to us) that grounds empiricism, disappears if one reformulates Leibniz’s Law in terms of global concepts, because the proposition then refers to all possible ontologies. These global concepts I capture with the terms ‘omnet’ and ‘asset’.

Omnet: Let an ‘omnet’ denote whatever there is, in its most liberal interpretation, in every world, without judgment as to what omnets there are or are not. Omnets might include houses, cats, people, and regular things, but also might include thoughts, nothingness, even things that the human mind has no capacity to have cognizance of, in the particular, such as high dimensionality. While these are all omnets, they must begin in the possible ontology. Only asset founded on indefeasibility can promote these to the actual ontology of the Meditator.

Asset: Let an ‘asset’ denote whatever an omnet has, in its most liberal interpretation. Assets might be parts, properties, attributes, thisness, universals, identity, the name or names we call what is, other omnets, or any specific that an omnet has.

In this work, the number of omnets or assets that are in the world is not assumed, either one or many. For the present, the number of omnets or assets should be left flexible, and the plural be given equal priority of meaning such that the use of either term is context sensitive. The concept of number will become evident as omnets are accepted into the actual ontology.

These terms begin wholly vague and assuming that assets is synonymous with properties is not valid for the Meditator until some correspondence of meaning derives naturally from the givens of the actual ontology. That is, any particular that might be assumed of them begins in the possible ontology and is initially dubious. Nothing may be validly presupposed of them other than that they stand in for items of the possible ontology.

On the choice of terms: Omnet and Asset.

Given the very rich language available to the writer, the reader is justified in asking why a new pair of terms is necessary. Why not ‘object’ or ‘entity’ in place of omnet, and ‘property and relation’ in place of ‘asset’? I am discomfited by having to introduce new terms, which asks so much of the reader. But I am more discomfited in the knowledge that a failure of the reader, due to my own lack of diligence with respect to terms, would lead to false mental constructs. For example, one must not try to think of assets as necessarily being just properties and (or) relations.

For the first part, objects or entities might suffice for some readers but not for others. Some readers may read objects to exclude, for example, properties and relations (Laycock 2005). Strictly speaking, if I accept Russell’s (1905) very general notions of objects and terms, I find that objects are not sufficient for the purpose to which I intend. In particular, Russell, in seeking to be general, includes whatever may be an object of thought, or may occur in any true or false proposition. This might suffice, but what if there are objects that can never be an object of thought, and as such never occur in any true or false proposition? This may be unnecessary, perhaps going too far, but I want to be sure that the term used is absolutely and resolutely global.

For the second part, I use the term ‘assets’ in place of properties and relations, because there may be properties, relations, tropes, parts, and any number of other ways of describing omnets. To quote Locke, ‘the boundaries of the species, whereby men sort them, are made by men...’ (in Robinson 2004, n.p.). Any choice of species may be a proper representation of the world, and I have no way of being sure which applies. It is not beyond the realm of possibility that assets may be perspective based. I am not saying they are, but the history of investigation is littered with the corpses of assumptions.

So I introduce these new terms (omnet and asset) only to, and with the specific intent of, removing preconceptions as to what there is, or is not, or the nature of what attends what there is or how we might have cognisance of such. Existing terms (for example things, objects, substance, being) are overloaded with meaning and presupposition of meaning (see Laycock's 2005 notes on Mill). Additionally these are subject to semantic vagueness (Quine 1960).

These new terms (omnet and asset) permit one to assign new mental symbols whose meaning is ontologically grounded by the global whatever, for omnets, and by what such omnets have, without World-of-Seeming specificity, for assets. In the context of this thesis, a presupposition of meaning, such as assigning similarity of terms to substances, attributes, things, entities, objects, properties, relations, beings, or anything else, beyond the global non-specific, is in error, and diminishes the path to a useable result. This is the world of the Cartesian meditator.

Foundation 3: A Constructional Ontology

Ontology - eh?
Ontology, is the study of reality. For historical (and somewhat odd reasons) an ontology is a chosen view of reality. So, for example, the physicist's ontology is a world of matter and energy. A psychologist's ontology is that there are people, and thoughts and feelings. For the modern Cartesian meditator none of these other ontologies are valid, in that people could be figments of our imagination (they aren't, but the strict Cartesian skeptic cannot accept them into the ontology because there may be no one there).

So, an ontology can be chosen based on how it is make up. A constructional ontology (the point of interest in this post) is one which serves to construct complexes from simples (Fine 1991). I want to show how the modern Cartesian stance (see here for a description of this stance) leads the modern meditator to adopt a constructional ontology. I also want to show how this ontology is constituted and to consider the primary items that populate the Meditator's world.

So this post develops framework from which the Meditator may begin to construct a world from an a priori principle (a principle that needs no reference to the world as it seems to be) rather than relying on Descartes’ non-deceiving God. Identification of this principle is outside the scope of this post.

The roots of a Cartesian ontology.
An ontology consists of all those items which are, in an appropriate sense, accepted (Fine 1991). The criterion of acceptance is that the item should be there (for good reason) not necessarily because someone puts it there. This presents a challenge to the modern meditator. While the Meditator can have—is obliged to hold—belief in those propositions that are immune to doubt (see my post on skeptical commitment) apart from one’s self, all other items that might populate the Meditator's world are initially suspect, so should not be there, at least at the outset of an enquiry.

As such, the Meditator begins with a very sparse world indeed. By comparison, where the physicist's world, or at least ontology, is immediately and densely populated by the evidence of the senses, and comes with a spacetime already built in, and within which measurements can be made as a basis for deriving laws and other theoretical constructs in any number of dimensions, the Meditator begins with only those propositions that are indefeasible, and they are the most abstract of propositions. For example, "I think, therefore I am" establishes that the Meditator exists, but says nothing of that existence. There is no body here, and the mind has no more than an innate idea of equivalence and difference (as argued in this post). All these other ideas come only later, after the other founding proposition is introduced (not in this post - later). But it is a start.

Beginning with some aspect or abstraction from the presenting world, as is done by the physicist, is a very useful way to develop an ontology. However, the challenge for the physicist is not in populating the ontology, but that all theories developed around such are limited to description rather than true explanation. For example, to attribute the attraction of one massive body to another, to gravity, does no more than name the apparent attraction. To then explain gravity as the effect of gravitons or the curvature of spacetime, or some other aspect simply moves the description one step to the left. As one approaches questions as to the actual nature or origin of things such as space or time, the physicist draws up short. Consider Richard Feynman (in Walding, Rapkins and Rossiter 2004, p. 6):

We physicists work with time every day but don't ask me what it is. It's just too difficult to think about.

In fact all the fundamental concepts of physics are similarly affected. Until a well-founded answer is available, the efforts of the physicist, and seemingly everyone else, is incomplete and may be just plain wrong. Certainly, as Popper identified, regardless of protestations by the physicist, no amount of data can verify a theory, but data can falsify the theory (Oddie 2008).

Is the mathematician any better off? While mathematicians mostly agree that mathematics is a discipline built on the a priori, its development is very closely linked with the natural sciences (von Neumann 1966, p. 181) in an empirical sense. Geometry, for example, previously argued by Kant to be a priori, von Neumann finds to have originated as a natural, empirical science. The problem of mathematics as it applies to the physical world is easy to spot. For example, π is found deeply embedded in our physical theories, from circular motion to quantum mechanics. But Heisenberg's Uncertainty Principle imposes strict limits to measurement. Then, under contemporary mathematics, there is an infinity of numbers sufficiently close to π that can be substituted for π in the equations of physics, from an empirical standpoint that would provide solutions experimentally indistinguishable from using that actual π. Consequently an infinity of theories that substitute other numbers for π in our trusted theories would work equally well. One need only consider the number of decimal points required. Around eighty decimal places might do it.

Then, if one ought to accept the simplest theory (Occam's razor) our present physical theories ought to be regarded as false: a rational number of around eighty digits is easier to compute than the irrational π. It may be that using π provides an over-description of the world that is inherently misleading. Indeed the same argument can be applied to the number e, which, through its association with the trigonometric identities would undermine the greater part of quantum theory.

While truth is the aim of inquiry, fallibilism affirms that, typically, our theories are false or very likely to be false, and when shown to be false they are replaced by other false theories (Oddie 2008). This cannot be the case for an ontology built on an indefeasible Cartesian ontology. This provides sufficient reason for the philosopher who seeks truth, to set aside, at least for a little while, the empiricist who demands that the world is as it appears to be, and to walk with the modern Cartesian rationalist.

Seeking to beat the fundamental ontological tradeoff.

Regardless of the near empty ontology that first presents, Descartes’ epistemology is attractive in that, being indefeasible, the items of the ontology cannot be falsified by presentations from a dubious world, so a Cartesian ontology ought to be pursued. Comparing the physicist’s ontology to that of the Meditator, we see what Swoyer (2009) calls the fundamental ontological trade-off: that a rich ontology with its promise of powerful explanatory power is fettered by increased epistemic risk, whereas a more rigorous ontology seems to have a lesser ability to explain our world.

Is this trade-off an illusion? For the first part, the promise of causal explanatory power (as contrasted with descriptive explanatory power) through an empiricist’s ontology is just that; a promise that cannot be realised. For the second part, insofar as the Meditator is obliged to accept indefeasible propositions as a model of an actuality, then, if indefeasible principles imply consequences, these too ought to be accepted into the ontology. This growth of accepted propositions characterises a constructional ontology, to which I shall return shortly.

Initial conditions, Fine’s possible ontology and actual ontology.
Fine (1991) says:

An ontology is actual; it includes everything that it is correct to accept. By contrast, we may talk of a possible ontology, which consists of everything that might be accepted (as the total ontology).

With respect to the idea of a possible ontology, Fine is not here talking about metaphysical possibility as it is usually used, but what might be true of an ontology, rather than what is true. This is exactly in tune with the Meditator's stance of holding no initial belief in any proposition or its negation. Then the Meditator is authorised to survey what might be true.

Fine notes that one might hold that there is no such thing as an actual, as opposed to a possible ontology for there is nothing which counts as the correct ontological stand; there are merely different, equally legitimate, stands. He investigates how absolutist notions might be avoided in the description of an ontology. But such avoidance is not a thing to which this Meditator would subscribe. The Meditator does not begin with absolutist notions, but that does not mean he or she might not end up with one.

The stance of the Meditator is to hold no belief either in the affirmative or the negative about the existence of conditions of the world unless and until propositions about supposed conditions are shown to be indefeasible (see [Author] 2009b). Then the Meditator is authorised and obligated to accept indefeasible propositions into the actual ontology as a matter of sceptical commitment (see [Author] 2009b) as models of actual conditions of the world. Rather than contrasting a chosen ontology with other possible ontologies, the Meditator begins with all possible ontologies bundled into one, and, through application of the Method of Doubt, sieves through all those available in the hope of finding propositions that must be accepted, due to their indefeasibility. So, rather than beginning with givens, the items of the ontology gradually come into view. Because something is possible if it is a possible view (Fine 1991, p. 268), the possible ontology encompasses the totality of what-is, and what-might-be (but isn’t). Then, the dialectical nature of the possible ontology is no more than an expression of the Meditator's idea of equivalence and difference, for in the possible ontology is every possible proposition and its negation, at least at the outset. The possible ontology is populated immediately through reference to what might be. However, for the Meditator, such reference is metaphysically vague: [T]he world of the modern Meditator begins as a sparse world of barely defined, essentially windowless objects, shrouded in doubt, about which we seek to determine laws that have ontological force, that bring these objects into some level of focus, in so far as each principle defines a characteristic of the system, to which the objects must comply, or they have no place in the system. ([Author] 2009b): From the possible ontology the actual ontology (see Fine 1991) may be populated by sieving through the possible ontology using the Method of Doubt, seeking trustworthy propositions as models of actual conditions of the real world. How one can consider dialectical possibilities, when the objects of the dialectic are initially windowless, is a challenge, but consideration of this must be left to another time. One must first consider the ontological framework, and it does not require objects. The Meditator’s Constructional Ontology. Fine studies how a discipline gets shaped by the positions which are adopted and the strategies that are pursued. The apparent absence of givens in a Cartesian ontology, brought by the Meditator’s position, restricts the Meditator’s strategies, and suggests that any world discoverable from a Cartesian stance might be a constructional ontology. A constructional ontology is an ontology where some, perhaps most, of the objects of the ontology are accepted on the grounds that they are constructed from other objects in the ontology (see Fine 1991). It is an ontology in which complexes are constructed from simples through the action of a constructor. Fine describes the nature of the constructional ontology in terms of givens, constructs, constructees, constructors, items and elements. Givens are those items that are just accepted into the actual ontology; a starting condition or origin of the ontology. Constructs are those objects constructed from constructees and constructors. Essentially, the constructor acts on constructees to create constructs, though, as Fine later notes, there is no rule to say that a constructor requires a constructee to operate on – the constructional ontology might arise ex nihilo. As an example of a constructional ontology, Fine provides the cumulative hierarchy of sets, in which the constructees of any set are its members and the constructor is the set-builder. Constructs and constructees are elements; elements and constructors are the items of the ontology. Fine describes three non-modal principles for a constructional ontology. These are: Inclusion: Each given is to be accepted (within the ontology).
Closure: If the elements el, e2, . . . and the constructor o are accepted and if o is applicable to el, e2, . . . , then the result of applying o to the elements el, e2, . . . is also to be accepted.
Foundation: Any element of the ontology can be constructed from the basic elements of the ontology by means of constructors in the ontology. For the Meditator's ontology, these principles fall away. The principle of inclusion is unnecessary. Strictly speaking, there are no givens in the actual ontology of the Meditator. For example, the Meditator is accepted from the possible ontology due to the indefeasibility of, 'I think therefore I am,' so is not a given. In the same way, the Meditator seeks an indefeasible proposition as constructor. Equally, the principle of closure is unnecessary also because constructors are not applied to elements, as the mathematician does to the null set to produce a hierarchy of sets. The Meditator has no authority to do so. Rather, if there is an indefeasible proposition that refers to some aspect of the ontology, for example to an element or the totality, and the proposition says something of that aspect, then the aspect shall comply. The new aspect exists. That is, the Meditator is obliged to accept that the condition of the world that an indefeasible proposition models has ontological force upon the ontology. Closure therefore is no more than recognition of Sceptical Commitment (see [Author] 2009b). The Principle of Foundation can also be dropped because, for the Meditator, conditions will be what they are, and the Meditator is in no position to create principles unless and until they arise naturally through Cartesian enquiry. In any case, Foundationalism is already part of the stance of the Meditator. If this idea is well-founded, the world will eventually come into view as an evolution brought by constructors. As an example, say there is an indefeasible proposition that says all that exists is bounded. Then each boundary is likewise bounded, and the ontology evolves automatically. Indefeasibility of a proposition that has implications for the actual ontology implies causal power in the real world of the item the proposition models. In the case of the fictitious example, such a proposition would mean that there is a condition of the world that in some way brings an iteration of boundaries into existence. The constructional ontology and foundationalism. Decartes' foundationalism features a foundation of unshakable first principles, and a superstructure of further propositions anchored to the foundation via unshakable inference (Newman 2008). One wonders at the number of first principles that might be needed. Descartes’ foundationalism, in the light of Aristotle’s expectation that a form of knowledge is accessible that establishes the first premises, that anchors demonstrations of other pieces of knowledge as justified truth worthy of belief (Smith 2004) supports the idea that there is some foundational statement upon which all other pieces of knowledge are built. If so, a constructional ontological framework, with indefeasible propositions organised in the manner of a well structured, architectural edifice, would permit the growth of knowledge about what is, independent of empiricism. The task of the modern Meditator is to discover sufficient items to act as constructees and constructors for a model of the world. If Aristotle is right, there might be just one first principle, and as such this would be a constructor acting on what is and what might be but isn’t; hence the possible ontology. This is not very different from the present state of physics. Gibbs (1995) and Wigner (1960) for example, observe that the history of theoretical physics has been a succession of reductions to lower levels: more fundamental, more unified, more symmetrical and ideally simpler. So the empiricist’s world and the rationalist’s world might coexist. First step to identifying Aristotle's foundational statement. Given that the actual ontology has accepted the Meditator, it also includes the Meditator's innate ideas of equivalence and difference (see [Author] 2009a). Consequently, because these ideas are epistemologically prior to all other ideas, while at the same time all impressions are dubious, it is from equivalence and difference alone that the Meditator must develop his or her understanding of the world. A natural step would be for the Meditator to express this idea of equivalence and difference as a singular proposition. Then this proposition ought to be the foundation of the Meditator's world. To do so, one needs to develop a lexicon free of Quinean circularity (see Quine 1960) by which proper, non-judgmental, non-anthropocentric reference might be made to the contents of the possible and the actual ontology, thereby enabling the Meditator to create propositions. Conclusion. Combined with past work (see [Author] 2009a, 2009b) the Meditator has made progress. Knowing of his/her own existence, the Meditator has found a position within the actual ontology from which to consider the possible ontology. Having identified that he/she has an innate idea of equivalence and difference, the Meditator has a basis for thinking about the world. Added to this, the Meditator now has an ontological framework within which a rich Cartesian ontology might be developed. Within this, the idea of equivalence and, by extension, difference, may provide a key to wider knowledge of the world. Reference: [Author] 2009a, Toward a Cartesian Ontology – Equivalence and Difference, Submitted to Nous, Forthcoming. [Author] 2009b, Toward a Cartesian Ontology – Sceptical Commitment, Ontological Force, and the Modern Meditator, Submitted to Nous, Forthcoming. Fine, K 1991, The Study of Ontology, Noûs, Vol. 25, No. 3 (Jun., 1991), pp. 263-294 Oddie, G 2008, 'Truthlikeness', in EN Zalta (ed.), The Stanford Encyclopedia of Philosophy, Fall 2008 edn, http://plato.stanford.edu/archives/fall2001/entries/truthlikeness/. Quine, W.V.O. (1960), Word and Object, Cambridge MA: MIT Press.

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/. Swoyer, C 2000, 'Properties', in EN Zalta (ed.), The Stanford Encyclopedia of Philosophy Winter 2009 edn, http://plato.stanford.edu/archives/win2009/entries/properties/. von Neumann, J 1966, 'The mathematician', in RB Heywood (ed.), The works of the mind, University of Chicago Press, Chicago, pp. 180-96. Walding R, Rapkins G, Rossiter G 2004, New Century Senior Physics, Oxford University Press, South Melbourne.

Why science (and empiricism) is not well placed to answer questions about foundations.

In a response to a comment by me to his blog, Alan Forrester said:

Your blog seems to indicate that you think we should start from secure foundations and build everything up from them. The particular foundation you pick is an innate idea that some things are the same and some things are different. This doesn’t seem likely to get us very far. Also it doesn’t seem to me that it matters whether an idea is innate. Innate ideas are just ideas created by biological evolution, which doesn’t guarantee the truth of its products. Also the idea of foundations is itself irrational since you are saying there is something you cannot prove and do not leave open to argument.

The move Popper made is far better. No idea is ever proven, or made more probable and we should stop trying to do this. Instead we should seek to notice problems with our current ideas, propose solutions and then criticise them according to non-justificational criteria, like whether they solve the problems they are meant to solve, whether they conflict with other theories, whether they conflict with experimental results and so on.

This post responds to his issues, as the answering makes my argument both clearer, and stronger. I would say that my dog's nickname is Popper, as a mark of affection. Equally I am not seeking to shoot Forrester, for one should begin with skepticism, as Descartes has urged. Recall that Cartesian skepticism is generally regarded as the higher form of skepticism - indeed the endpoint of the skeptical view - at first one must doubt all that can be doubted.

The bother is, the view expressed by Forrester suggests that he is nowhere near skeptical enough, for the first part, and also he, like most of us, carries a great number of presuppositions about the way the world is. He holds a scientific view, but, at present all science and mathematics is just description. As Popper himself argues, there is no verification in science, only falsification. The same goes for contemporary mathematics. Where we differ, Forrester and I (at present anyway - I hope to call him over to the dark side) is that he assumes that in some way our present theories are right, or at least closer and closer to being right (see Oddie's article on Truthlikeness in the Stanford Encyclopedia of Philosophy). But I say this is not knowledge at all. It is belief, and belief does not make you right (see my introduction). I do not follow Plato's Justified True Belief idea of knowledge (see Analysis of Knowledge (by Steup I think) on the Stanford Encyclopedia of Philosophy for a discussion of JTB). Rather, I advocate a justified truth worthy of belief form of knowledge, and in particular the view of Skeptical Commitment as explained in my post on the subject.

NOW TO ANSWER FORRESTER:
He says:

Your blog seems to indicate that you think we should start from secure foundations and build everything up from them.

Yes, my blog indicates that I think we should start from secure foundations. Are you suggesting that we ought to begin with insecure foundations? No, of course you are not, and let us remember that Critical Rationalism is the endpoint of a long philosophical analysis that arose only because a secure foundation had not been identified, from which a rich ontology - meaning a rich world view - could be developed. It is this which my efforts aim to fix, after which, empiricism, to which critical rationalism is bonded, becomes second philosophy, and metaphysics returns to its state of first philosophy. This must surely be what philosophers want, though they argue against it. This is what Shapiro calls the 'philosophy last if at all' principle. Here I am using Armstrong's sneering quotes, because sarcastic font is not available just yet.

Indeed, since Parmenides and Zeno (may they live forever in our minds), thorough rationalist analysis has always shown that the world at it appears to be to us does not accord with how it ought to be if it accords with reason. Parmenides argued so, Zeno backed this up, Aristotle admitted that Zeno had a valid argument that must be answered (and never has - until my blog of course, but it is very incomplete, so keep watching). Berkeley said so to, and so did Descartes, among others. Fortunately, so long as you are prepared to work through the reasoning, the world finally is returned to us in good order. But my blog is not that far ahead yet.

Forrester then says:

The particular foundation you pick is an innate idea that some things are the same and some things are different. This doesn’t seem likely to get us very far.

This is not exactly what I say. I say that the foundations for thinking about anything is our innate idea of equivalence and difference. By then expressing this as a general principle of equivalence that applies to all that is, and is also immune to doubt (this post is not yet posted) then one is obliged to accept that the principle is a proper model of a necessary condition of all possible worlds. I have written a very rough exposition of this in my post 'An every-person's guide to the origin of the universe'. It says, let the term 'omnet' stand in for anything (concrete, abstract, nothingness, even that which we have no ability to have cognizance of); let an asset be what an omnet has; then

Every omnet, that has all the assets of that omnet, is that omnet.

This is the General Principle of Equivalence. Be very sure you understand what it means before commenting. Some philosophers have said it is just Leibniz Principle of Identity of Indiscernibles, which is regarded with some suspicion. Other philosophers have said it is a just a global tautology, and so 'does not seem likely to get us very far.' But the PII is a meaty proposition, and a tautology as a particular is thought to be no more than an observation of the obvious. Indeed it is neither, or at least, in being a global tautology takes on a life of its own, if it is indefeasible (it is indefeasible exactly because any doubt that might be raised against it, relies on it being true, if that doubt is to be that doubt. But I will tackle that more rigorously elsewhere).

Forrester said:

Also it doesn’t seem to me that it matters whether an idea is innate. Innate ideas are just ideas created by biological evolution, which doesn’t guarantee the truth of its products.

Firstly, your comment 'Innate ideas are just ideas created by biological evolution' is sophistry, of no greater force than Thales pronouncement that the world comes from water. Comments like this are the hallmark of Hume, who makes imperious comments like this from beginning to end, and we were stupid enough to believe it. If this seems harsh, that is not my intention. Rather, remember your own quotation from Whitehead:

The discussions of every age are filled with the issues on which its leading schools of thought differ. But the general intellectual atmosphere of the time is always determined by the views on which the opposing schools agree. They become the unspoken presuppositions of all thought, (the) common and unquestioningly accepted foundations on which all discussion proceeds

There are many presuppositions implied by your comment, and I'm sure you can see them immediately. Nevertheless:

Firstly, Broughton (2002) identifies correctly that if philosophical discourse is to be rational, then any doubt expressed needs to be based on some rational reason, even if these are hyperbolic doubt, such as Descartes dream argument or his evil genius argument. Your reason for doubt is actually circular. You found your doubt on evolution, but evolution (while likely correct) reduces to belief. Any thoroughgoing philosopher is aware of the tenuous nature of relying on empiricism as a foundation for one's ideas. Like him or hate him, Descartes arguments against empiricism as a foundation for argument are generally regarded as valid. Indeed this is one of the motivating factors for Popper's critical rationalism, though this is almost lost in the mists of time (as Whitehead's comment implies). To put this criticism to bed, let me point out that your criticism, if it is to be that criticism, and not some other (hence a straw man) requires that its import refers to an actual condition of the world, else it might be properly considered false. Whether it is true or not is not important here. Rather, there is no way that you can justify it to be true, unless you first identify a proper foundation. That proper foundation, in the end, will end up being the General Principle of Equivalence, and it, later, implies that there is a unique origin of the universe.

Forrester said:

Also the idea of foundations is itself irrational since you are saying there is something you cannot prove and do not leave open to argument.

I don't actually understand what you are saying here. I didn't say this. Perhaps you could point it out to me. Maybe you mean that all propositions ought to be always open to argument. Not if there they are immune to Cartesian doubt. For example 'I think therefore I am' is immune to doubt, for to doubt it, requires the existence of the doubter. This is not true or false because of any principle (i.e. Principle of Non-Contradiction) but because the existence of the doubter is ontologically prior to the doubt, the doubt is ontologically dependent on the existence of the doubter. This is no longer open to argument, at least within the bounds of rational discourse. The same thing applies to the General Principle of Equivalence.

Forrester says:

The move Popper made is far better. No idea is ever proven, or made more probable and we should stop trying to do this. Instead we should seek to notice problems with our current ideas, propose solutions and then criticise them according to non-justificational criteria, like whether they solve the problems they are meant to solve, whether they conflict with other theories, whether they conflict with experimental results and so on.

I certainly am very respectful of Popper, who, at least, recognized that his program had challenges. As you will already be aware, Popper's efforts are an endpoint of an essentially negative program that began with Hume, more or less - Hume to Kant to Popper. Popper never proved this point, but, rather, put forward some convincing argument. But convincing argument must step aside when indefeasibility arrives. Regardless of the cognitive dissonance that floods the mind when the evidence of the senses demands that the world derived from unshakable principles seem contradictory to the way the world seems to be, yet the true skeptic is bound by skeptical commitment to indefeasible propositions, exactly because they are indefeasible, yet the world as it presents to us, is not. That said, if the skeptic can hold true, the world as it must be, eventually coincides with the world of our experience, contrary to the argument of Parmenides.

The real challenge of philosophical inquiry

I recently read on another blog:

“When you are criticizing the philosophy of an epoch, do not chiefly direct your attention to those intellectual positions which its exponents feel it necessary explicitly to defend. There will be some fundamental assumptions which adherents of all the variant systems within the epoch subconsciously presuppose. Such assumptions appear so obvious that people do not know what they are assuming because no other way of putting things has ever occurred to them. With these assumptions a certain limited number of types of philosophical systems are possible, and this group of systems constitutes the philosophy of an epoch”
Alfred North Whitehead in “Science and the Modern World” (1925).

German astronomer Johannes Kepler (1571-1630), after spending many year working on data provided by Tycho Brahe was essentially ridiculed in 1614 by Galileo, who held his writing to be, 'so obscure that apparently the author did not know what he was talking about.' It was only in 1618, when he published the first two of his three laws athat he was hailed as a hero. Let us be clear here that his hero status came, not as a result of his work exactly, but because there were others who wanted to do away with the old geocentric (Earth-centered) model of the universe passed down from Aristotle and Ptolemy.

Someone once noted that it is not our efforts in themselves that lift us up, but rather, we are lifted up by those around us. One of the most gifted mathematicians of the past century was Kurt Godel, yet when he first put out his famous Incompleteness Theorems, only the well respected von Neumann ('von Noyman') understood what his work implied. In many ways it was von Neumann's efforts that brought Godel's work to light and approbation.

The most brilliant scientist, mathematician, or philosopher may have the most brilliant ideas. Yet all comes to nothing unless others are willing to listen without judgement, either in the affirmative or the negative until the ideas being put are properly understood. For the modern Cartesian meditator, the time of judgement only occurs when an argument becomes indefeasible.

In this context, I find that the chief obstacle is not in developing the world of the modern Cartesian Rationalist, but rather, in the challenge of brushing away centuries (indeed millennia) of cloudy thought. For example, most philosophers these days are Naturalists. But this is just a belief system, and is not well-founded, exactly because the objects of the natural world are known to us only vaguely, as Descartes has already argued. Indeed we may be dreaming, and what we think is a real person out there in the street might be a mere fiction. Our minds cannot reach out and touch the presenting objects of the world. So Naturalism reduces to dogma and, these days, to sophistry.

Equally, if I challenge Hume, then, like Kepler my work is shut out. This occurred recently when I sent a paper to the Hume Review. The response was in many ways similar to the comments made by Galileo, and they would not even give it proper consideration. Hume held that all ideas come from impressions. He says that to refute him I need only point out a single counter-example; yet when I point and say 'Look, equivalence and difference are ideas and they are innate. Indeed without them there is no way to copy impressions into your head at all (Hume's Copy Principle) for the act of copying requires that one draw an equivalence relation between the impression and the model of it held in your head. By the response of the editor, it seems the Humean asks for more proof than Hume himself.

It's a funny world, this world of philosophers who have spent a lifetime building castles in the sky, and will protect such by all means, even at the expense of well-founded philosophy itself.

Here is a link to my first page. You need to begin with this, as a matter of foundationalism.

Foundation 2: Skeptical Commitment and the Modern Meditator

This post aims to provide the reader with a philosophical stance suitable for exploring the origins of the world. Without it, you cannot proceed with certainty. With it, all is certain.

There is only one sure stance.
Berkeley picked out the challenges for one who seeks knowledge:

[N]o sooner do we depart from sense and instinct to follow the light of a superior principle, to reason, meditate, and reflect on the nature of things, but a thousand scruples spring up in our minds concerning those things which before we seemed fully to comprehend. Prejudices and errors of sense do from all parts discover themselves to our view; and, endeavouring to correct these by reason, we are insensibly drawn into uncouth paradoxes, difficulties, and inconsistencies, which multiply and grow upon us as we advance in speculation, till at length, having wandered through many intricate mazes, we find ourselves just where we were, or, which is worse, sit down in a forlorn Scepticism.

There is only one stance of sufficient strength to stand against the confusion that rushes in when one tries to make reason match with the presenting world, and that is the stance of the modern Cartesian meditator. This stance is a significant variation on Rene Descartes' stance. Where Descartes relied on the existence of a non-deceiving God, the modern Meditator has no such reliance. I want to develop the stance of the hard-forged skeptic. I will leave it to the academics to decide whether my Meditator is that of Descartes or not.

On conviction.
Should you just accept what you are told? Of course not. Even if the person telling you something has the best intentions (like me) they might be mistaken (to consider this in more detail, consider Newman , Vogt or Groarke). Equally, can you trust the impressions you receive from the world? No; at least not if you are seeking certainty. You could be dreaming; you might be in a Matrix, or be a brain in a vat under the influence of an evil genius. Indeed, that the theories of the universe change every few years, is good reason for suspicion. Indeed there may be many reasons why you ought not trust the world as it appears to be.

The question arises - what method might be employed to ensure us that what we think is correct, is indeed correct? Descartes’ said that knowledge is conviction based on a reason so strong that it can never be shaken by any stronger reason (1640 letter, AT 3:64-65 in Newman 2008). He argued that the only way to identify knowledge is to begin by doubting everything. This is known as the Method of Doubt. Throw everything out, he said; start again. Then, if you find something that is immune to doubt, that cannot be rationally doubted, in this you can be sure. No matter what assails you, you can say that this at least is trustworthy.

These days philosophers think that this certainty is achieved in stages. They say: 'As my certainty increases, my doubt decreases; conversely, as my doubt increases, my certainty decreases.' I fundamentally do not agree. Descartes’ Meditator does not come to certainty - to knowledge - bit by bit. For Descartes’ meditator a proposition is either fully justified or it is not justified. There ought not be any intermediate assignation of belief either for or against some proposition unless and until it is found to be indefeasible. Descartes devised the Method of Doubt exactly to help set aside preconceived opinions.

The thoroughgoing Meditator has no business dealing with conviction unless and until it is unshakable, exactly because if there is room for doubt, our ideas may be simply wrong, no matter how attractive they may seem.

20th Century Physics is littered with the corpses of attractive theories. Descartes (Search, AT 10:526 in Newman 2008) himself 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.

I agree. Empiricism (the view that the well-founded knowledge can be founded on the world as it seems to be to the senses) is too hasty: look at the world, guess at something, form a belief, write a theory, see it wither, throw it out. It is all description of more or less dubious objects anyway, so the laws we trust are hardly trustworthy beyond statistics. This is Popper’s legacy to empiricism—that no finite amount of data can verify or probabilify an interesting scientific theory, but data can falsify the theory (see Oddie 2008).

Skeptical commitment
By contrast, the modern Meditator, as hard forged skeptic, is fettered by a particular and important kind of commitment, an endpoint to Ancient Skepticism. Call this ‘Skeptical Commitment’. According to Sextus the skeptic is someone who has investigated the questions of philosophy but has ‘suspended judgment’ (either in the affirmative or the negative) because he or she is unable to resolve the differences among the contrary attitudes, opinions and arguments he found. Instead of adhering to a definite philosophical position, the skeptic is someone who continues to investigate (Groarke 2009).

Skepticism is a natural response to the apparent infinite regress of justification required to establish a truth, as well as the apparent inconsistency of our rationally derived understanding of how the world ought to be, with the way our impressions demand it must be, such that the cognitive dissonance is too great to be accommodated. Examples include arguments that lead to Zeno’s paradox, Parmenides denial of change in the face of change at every level, and the Problem of Bundling, among a plethora of other problems of philosophy.

However, if this founds Skepticism, then the hard-forged skeptic is obliged by skepticism itself to accept indefeasibility as a proof of the truth of the proposition. By indefeasible I mean indubitable in Broughton’s (2002) sense that indubitability:

[C]oncerns not “the power of the human mind to enter into a state of doubtfulness about a proposition,” but instead the condition whereby “it is impossible both that the proposition be false and that I be doubting whether it is true” (p. 100)

Another way to think of this is (and this is the basis of my proofs): One has no basis to rationally doubt a proposition if that doubt relies on the truth of the proposition for the doubt itself to be properly put.

Descartes' proposition 'I think therefore I am' is necessarily true to each meditator because one must exist to be able to doubt one's existence.

Broughton’s interpretation reduces Descartes’ need to rely on clear and distinct impressions, to simple rational thought, founded on inherent ideas of equivalence and difference, that underpins all understanding. This method then defines the modern Meditator and refines the intent of Descartes’ meditator. To the modern Meditator, a proposition that is indefeasible must be accepted as true, meaning a statement that properly models a condition of the world.

Not only does an indefeasible proposition have the power to stand against any argument brought from empiricism because such arguments are founded on dubious evidence, it has ontological force, meaning that the Meditator can be certain that any implication that follows from such proposition, is also a model of a condition of the world to which the objects of the world must comply.

Should this rational analysis be at odds with the presenting world, for example say that it implies the Bundling Problem or Parmenides concerns about change, how ought this affect the Meditator’s stance? Because the modern Meditator holds no belief in the affirmative or the negative about the reality of impressions that might seem to contradict any propositions that have been shown to be indefeasible, there is no need to judge these at all.

Rather, the Meditator can stand steady in the knowledge that any dichotomy will resolve itself as the internally developed world-view develops, for surely an indefeasible proposition cannot truly be contradictory to reality. This ought to goad the Meditator to continue in developing his rationally derived world-view.

Given hyperbolic doubts (Newman 2008) about the presenting world, it would seem that the only available source of knowledge is achievable through an Internalist, Indefeasibilist, Methodist (hence Foundationalist), Rationalist stance.

Implications
Sitting in the lap of the modern Meditator provides a fundamentally different perspective of the world, which is the exact opposite to that of the scientist. The scientist is revisionist. The scientist’s world is immediately populated by the evidence of the senses—by reliance on impressions. But when the world presents with data that conflicts with the laws of the universe, the laws must yield. By contrast the world of the modern Meditator begins as a sparse world of barely defined, essentially windowless objects, shrouded in doubt, about which we seek to determine laws that have ontological force, that bring these objects into some level of focus, in so far as each principle defines a characteristic of the system, to which the objects must comply, or they have no place in the system. The goal of the modern Meditator then is clear—find sufficient indefeasible propositions to build a rich world.

References:
Broughton, J., 2002, Descartes’s method of doubt, Princeton University Press, Princeton. Groarke, L., 2009, 'Ancient Skepticism', in EN Zalta (ed.), The Stanford Encyclopedia of Philosophy Spring 2009 edn, http://plato.stanford.edu/archives/spr2009/entries/skepticism-ancient/.
Newman, L., 2008, 'Descartes' Epistemology', in EN Zalta (ed.), The Stanford Encyclopedia of Philosophy Summer 2008 edn, http://plato.stanford.edu/archives/sum2008/entries/descartes-epistemology/.
Oddie, G., 2008, ‘Truthlikeness’, in EN Zalta (ed.) The Stanford Encyclopedia of Philosophy Fall 2008 edn, http://plato.stanford.edu/archives/fall2008/entries/truthlikeness/.
Vogt, Katja, "Ancient Skepticism", The Stanford Encyclopedia of Philosophy (Summer 2010 Edition), Edward N. Zalta (ed.), URL = .