Given X, then Y.

"My own suspicion is that the universe is not only queerer than we suppose, but queerer than we can suppose."
~ J.B.S. Haldane

(1) If it is the case, as some suppose, that mathematics is somehow fundamental to the universe, then the above quote is necessarily true. Simply think of all the weirdness of the Reals. For example:

"You can hit a random Real number with a dart and we don't even know how to look at the result, how to differentiate it from all the other ones around it" (see 6:41 in the video below). And note that almost surely (probability = 1) we will hit a number of this sort.

Also consider Gödel's Incompleteness Theorem: "Informally, Gödel's incompleteness theorem states that all consistent axiomatic formulations of number theory include undecidable propositions (Hofstadter 1989)." In other words, there are statements that can be made within mathematics that can neither be proven nor disproven: things queerer than we can suppose.

(2) Further, if it is the case that mathematics represents only some part of the universe and there are things within the universe not expressible in mathematics, then Haldane's suspicion is also necessarily true simply because it would entail the conjunction of all those things within the domain of mathematics, which we know contains things queerer than we can suppose, with all those things not expressible in mathematics; thus, it doesn't even matter if the things not expressible in mathematics are queerer than we suppose because we already have that.

(3) Finally, we can suppose that mathematics has nothing at all to do with universe, but if this is the case, then throw out all contemporary science (and, as a result, all the wonders of modern technology) that is intrinsically dependent upon mathematics for its expression. And this seems a result that is queerer than we can suppose.

__________

(4) Therefore, not only is the universe queerer than we suppose, but it is necessarily queerer than we can suppose.

Related:

Unspeakable Things (thanks +Bill Reed for pointing to this article).

Remember to forget!

Dredging Up the Dead

The following two pieces were written in October and September of 2002 (respectively). They were written in response to someone on a forum I used to partake in and s/he was inquiring into Carl Jung's Septem Sermones ad Mortuos. I dredged these up because someone else recently quoted the second paragraph of Sermo I on G+ and I thought: ah, Jung, I like that--where's it from (i.e., I did not recognize it as the second paragraph of Sermo I)? So I Googled it and found I had already encountered it before. Y'know how it is with memory, or, at least, I know how it is with my memory, heh. Anyway, simply thought I'd reacquaint myself with both Jung's text and my own and figured I might as well collect it here on Drifting Labyrinths.

The first piece is an attempt at answering the forum member's question:

What are your thoughts on the text?

And the second piece is an attempt at answering a different question this person had asked:

What do people think of Jung's concept that alchemy and gnosticism formed a kind of unconscious to the institutional relgions that labelled such beliefs and preactises as heretical?

Here is the text for Septem Sermones ad Mortuos in case you'd like to read it for yourself.

*******

 First Piece

Attempting answers (complement: questions; combination: analysis) at ancient analogies.

First thoughts roll with sevens. Seven days in a week, seven classical planets (recall the celestial bodies which circled round a stationary earth), lucky seven, The Chariot, seven major stages in the alchemical process: calcination, putrefaction, solution, distillation, conjunction, sublimation, and philosophic congelation. And of course, our seven sermons.

The Chariot is driven by the light and the dark engines—it is the two forces which power the single vehicle. If we can extract anything from Sermo I, it is a discussion of opposites, but on several layers. We are told that “[t]he pairs of opposites are qualities of the plemora which are not, because each balance each.” Above this statement we are shown eleven pairs of opposites as examples of the binary pairings which negate one another in the plemora. We will now list these as eleven unordered pairs: {effective, ineffective}, {fullness, emptiness}, {living, dead}, {difference, sameness}, {light, dark}, {hot, cold}, {force, matter}, {time, space}, {good, evil}, {beautiful, ugly}, {one, many}. It is these and other pairings that exist as one and the same thing in the plemora; that is to say, they don’t exist in the plemora because it of itself has no qualities. So we get a further pairing of {existence, non-existence} (which may seem similar, but perhaps not readily identical to {living, dead}), as a non/quality (and here we note the ‘/’ is used to indicate, in this case, both the absence and presence of a quality) of the plemora. We might want to recognize that some of Jung’s work focuses on the pairing {internal, external} with respect to individual; in different words, Jung looks at connections between a person’s psyche and the manifest world.

E.E. Rehmus informs us that the plemora itself is the Gnostic divine being, it is the Universal Soul. He goes on to note that all aeons emanate from it. In a Jungian sense, we might consider this to be the archetype of the Self, and from the Self, we derive selves—personal aeons or the aeon of an individual—in the small. In Sermo I, we are told that the plemora is nothingness and fullness, and I recall from a lecture on Jung that he felt the Self was also empty, yet contained every other archetype and, in this sense, is full. We are also told, in the Sermo, many other contrary things about the plemora. In particular, we are informed that we are not part of the plemora, that “we are from the plemora infinitely removed,” because to be a distinct creature is to be distinct form the other, and in specific, distinct form this Universal Soul, and in this Soul there is no distinction; however, we are also told that “because we are parts of the plemora, the plemora is also in us.” So we see here a further pairing of {plemora, individual} or, perhaps more in a Jungian flavour {other, self}. What we can begin to see is that the plemora and the individual, or the self and the other, share an identity in so far as each pairing, in their non/existence in the plemora, become one thing—we’ve denoted this one object by borrowing from set theoretic notation: the unordered binary pairing {x, y}—which is nothing (but of course, everything = nothing). We can note a numeric example of this pair as framed in computational language: {1, 0}.

More important to notice is that “the plemora is both the beginning and end of the created beings,” and that “the plemora prevadeth altogether, yet hath created being no share thereof.” So here we have further pairings of contraries giving rise to our existence, and yet, canceling out that existence. We are told that the plemora has not created being, but yet, it is the end and beginning of existence, and that it is ubiquitous throughout existence. In the pattern of thought being followed, the plemora has {nothing, everything} to do with non/existence. Or in different, but unsurprising words, the plemora is all and it is nothing, that we are both the plemora itself and entirely distinct from it. It is in our being—our existence—that we continue and create distinction: “[w]hen we distinguish qualities of the plemora, we are speaking from the ground of our own distinctiveness and concerning our own distinctiveness.” Thus, since the plemora is empty and possesses no qualities, which is to say that it is full and posses all qualities, it is we who ‘pull out’ (‘push in’?) the pairings, and we who further divide the elements of the pairings—one here, and one over there. Put differently, it is we who create things, it is we who are the blind demiurges that shape the world, it is we who are the plemora. But if we weren’t blind and could see this in its entirety (its eternity), then this “would mean self-dissolution,” and “[t]herein both thinking and being cease.” Perhaps this would be the end (beginning?) of the Jungian goal: full integration of the shadow into the personality.

But still, there is nothing that “standeth [as] something fixed, or in some way established from the beginning.” So we speak of “qualities of the plemora which are not.” In other words, even our groupings of these pairs into units, and then into non/existence stemming from, into, through, an empty and full plemora is still creating distinction, it is still our thoughts about the plemora, and thus, “we have said nothing concerning the plemora.” “What is changeable” is the shifting thoughts and experiences by which, even in recognizing an x and its contrary y as an {x, y}, and then negating this in the plemora—which is ourselves, we generate the individual as “fixed and certain…or even as a quality itself.” “The plemora is rent in us”: there appears to be a necessity of something as it exists in the tension between everything and nothing, where {nothing, everything} is {infinite, eternal}, and yet, not that at all. In other words, we are thinking of a way, or a being told ways in which we can think of the unthinkable, by which we can ascertain something outside of our experience. Put differently still, the Sermo is both conditioning, reconditioning, and unconditioning our ways of thinking with respect to ultimate reality: it offers us tools that work on our neurosemantic interpretations of reality, and in effect, tells us at one and the same time that ultimate reality—here framed as the plemora—is both thinkable and unthinkable, knowable and unknowable, that from which our experience derives, but also that which has nothing to do with our experience. In short, it seems a jarring interplay between contraries which may serve to free the reader (or hearer) from his or her own private (or not so private) dogmas and fundamentalisms: nothing is true, everything is possible (?).

I would like to note that there is further binary pairings which can be built from basic units of pairs (this again borrows from set theory). We can see that what is asserted about the plemora can be modeled in the following way: {{plemora, {x, y}}, {not plemora, {x, y}}}. This pairing might shed some light on the “deep structure” of the first Sermo: the plemora is full of all possible binary pairings where these pairings are self-negating (balance each other out—cancel or annul one and other), and the plemora has nothing to do with any possible binary pairings. In a yet larger scope we might construct the following binary pairing: {{{plemora, individual}, {x, y}}, {{not plemora, individual}, {x, y}}}, or perhaps some mixture thereof.

To turn briefly to the closing remarks of the opening Sermo, which concern “the striving after your own being.” Which appears to concern the resolution or dissolution of apparent opposites into complementary unities with respect to the identification of self with other, or with the plemora and the individual. It is suggested that the only real striving is that after our self, which I think Jung would agree with: we are, through our being in the world, seeking after that which we are, which, I think, might be seen as a manifestation of Self. What appears to be asserted is that it is our very thoughts and thought processes which move us a step away from being in what might be seen as purely experiential; that is, it is thought which serves to divide the pairings of complements into pairings of opposites, and this division is what separates self from other (leading, in heightened or more “crystalized” form to acute alienation), which in turn places us a step removed from the “isness” of unanalyzed experience. In different words, our interpretations are simply partial representations which further increase distinction, and the uninterpreted is the undifferentiated reality. In coming to “the right goal by virtue of [our] own being,” we might come to the identification of self with other or come to know the unknowable not through analysis or thought, but by the absence of these. In closing, I’d suggest that Sermo I is linked to the first step of alchemy, calcination, where we burn (or transform) the black into the white.

Variables Vibrate Vicariously. 

*******

Second Piece

One of the things I have appreciated about Jung’s thought regarding institutionalized religion is how it can often become a barrier to promoting a personal religious experience. If I recall correctly, Jung felt that what often occurred in the case of an organized religion was that someone at some point in time, in some context, had some sort of revelatory, mystical, or otherwise transcendent contact with the ‘divine’. If the person who had this experience was moved to attempt to describe it to others, and if others were significantly affected by this person’s relation of the experience, then the budding of a new religious movement could now begin.

What occurs at this point is the codification of the details of the experience. Moreover, the personal religious experience of a single individual now becomes the paradigmatic focal point and touch stone for people who might want to recreate such an experience for themselves. However, as the codification becomes dogma and doctrine, and as the experience of one person becomes structured into a organization, we immediately get issues of power and authority. Who is it that understands what that original experience was really about? We get the rise of those who will now become the interpreters and mediators between the source of some other individual’s experience, and the everyday lives of ordinary people. That is, we get, in an organized religion, those who are, by their position in the power structure, allowed to read scripture (or such) in such a way that they are the ones who will interpret this record of the events of a single individual for other people. They become the ones who will lead the flock (so to speak).

Now, it seems to me that alchemy is the personal pursuit of an encounter with the divine. It is the seeking after Hermes’ Bird, the Philosopher’s Stone (and many other names), but this is nothing other than the understanding that the individual is right there with God in creating the world. As Joderowsky frames it in The Holy Mountain, it is turning your excrement into gold. In existentialist terms, it is deciding the meaning and direction of your life without fear or angst in the face of the idea that there is no meaning or direction. In a sense, the alchemical pursuit is the unification of self and other, especially in terms of the mundane with the divine, or the particular with the universal. This entire pursuit is one that must be carried out by the individual alone, and not something that can be given or learned from someone else. I think that in traditional alchemy it was the alchemist alone who would mix the ingredients in the alembic, and it would be the alchemist alone that would supervise, observe, and participate in the growing of Hermes’ Tree. It seems to me that the idea here, in more Jungian terms, is that it is the individual who must turn his or her shadows into light, or integrate the shadows of his or her being into a whole individual. While people might be able to give guidance and what not, the life of any person is entirely in that person’s hands (whether he or she can admit this or not).

While I am not particularly well versed in Gnosticism, if I recall, one of the driving ideas is that the world was created by an evil demiurge, and that the real God was so far removed from this world that he or she had no contact with us, nor we with him or her. So let us put this into the context of what I’ve been spinning out so far. Alchemy is the individual quest for unification with the divine, for participation in the creation and transformation of the world. Gnosticism seems at least partially correct so far as contact with the divine is not an everyday part of most people’s lives, and that contact with the divine appears very difficult or might be impossible. So, if we have organized religion arising out of an individual’s encounter with the divine, then the individualized encounter was something unique, difficult, and not likely to be duplicated. Thus, we see how there might be elements of alchemy and Gnosticism providing the origins of an organized religion in so far as the origins are founded in an individual’s experience. As the experience itself becomes lost in the religion’s codification and increasing rigidness of formalization, the original experience which shares these elements of the alchemical pursuit and certain ideas pertaining to Gnosticism does become like the unconscious underpinnings of the institutional tradition.

At least part of the reason that such beliefs and interests in things like alchemy and Gnostic ideas become viewed as heretical is because these ideas would or could serve to promote in the individual a unique and difficult encounter with the divine for that individual. Such an encounter will likely be counter to some or all of the codified ways of the institution, and so, would work to undermine or at least question the authority of those “higher up” in the power structure. If everyone could promote his or her own encounter with God, then what would be the need for a church (in the traditional sense)?

So, in a sense, I agree with Jung that the institutionalization of an individual’s religious experience often seems to prevent others from having their own encounter with the divine. Such an encounter must be based upon, and stem from, those things which become labeled as heretical under the authority of a power structure, and this structure, paradoxically, owes its existence to these same heretical experiences, which become the “unconscious” (in the sense of unrecognized or forgotten) of that institution. 

This Isn't Quantum Physics. Wait. It Actually Is. OK, Maybe Not Really...

I don't know why I sometimes feel inclined to take decent, working, knowledgeable scientists to task, but there could be something Freudian there. Y'know: we don't really know ourselves, and we don't know what motivates us to do the things we do. Of course, that doesn't really explain much, does it? And perhaps claiming that "...really everything is quantum physics" doesn't explain much either or maybe it's simply a mistaken point of view to take.

I mean, it seems to me here's an example of mistaking the map for the territory. Sure, yeah, it seems that, when observed in a suiting manner, the universe and all the stuff in it is "... all happening by the interactions of tiny particles," which also happen to sometimes be waves (again, seemingly dependent upon the mode of observation)--a result which, in part, led to the development of Quantum Theory. Yet, "...the rules of quantum mechanics" are the (current) map by which we assess, experiment with, and otherwise observe the going-ons of the stuff in the universe in its seemingly "fundamental (by which we seem to actually mean "smallest scale")" expression. In other words, the interactions of the stuff of the universe--currently observed at very small scales as "tiny particles" and waves--is the territory and Quantum Mechanics is the map. And it's likely a "category error" to say that the interactions of the stuff of the universe are identical to Quantum Mechanics: they are not the same thing. One is a bunch of stuff, the other is a bunch of symbols: territory and map.

For a long time--maybe around two-thousand years--many of the people inclined to study such things thought that Aristotelian physics could explain most (or maybe some thought all) of the stuff of existence. Of course, a few people came along and ended up knocking all that down, and what ended up in its place was Newtonian physics. And maybe some people thought that Newtonian physics could or would explain most or all the stuff of existence. And of course they were mistaken since Quantum physics came along and supplanted the Newtonian version of physics. And no, they are not the same thing.

So, maybe once upon a time some people thought that everything was explainable by Aristotelian physics, and we know now that they were wrong. And then once upon a different time it's likely that some other people thought that everything was explainable by Newtonian physics. Well, we know now that they were wrong too. Now it so happens that currently there seems to be some people who feel like everything is explainable by Quantum physics (or some slight modification of it--gravity being the main trouble-maker here), and we know now that they are...?

History repeats itself, (ad infinitum) as some say, and I'm willing to wager everything I own (although myself and any takers on the bet may not be alive to collect the payoff) that some day something else--some other theory--will explain, well, some things anyway. Because, let's be realistic here, when we ask, "[w]hat is not quantum physics," well, there's a whole slew of things that aren't. And many of those things--at least seemingly so--can be collected under the general heading known as Qualia.

For instance, Quantum physics has nothing at all to say about my taste of, for example, chocolate--let alone whether or not my taste of chocolate is identical to yours. And what's Quantum physics got to say about my like or dislike of Tarantino films: it certainly won't predict if I like his next one or not. And y'know those days where you simply feel like pulling the covers back over your head and staying in bed (well, I know those days, anyway)? I don't think Quantum physics has much to say about that either.

Hossenfelder writes, "[i]f it wasn’t for Pauli’s exclusion principle, you’d fall right through the ground."

Y'know, I don't recall learning about anyone ever having a problem with falling through the ground before Pauli formulated the exclusion principle, but maybe that's a result of some sort of historical revisionism covering up some grand conspiracy or other? I mean, what if people used to fall right through a flat earth or something? But I digress...

Now I can't speak for Hossenfelder's point of view in general (not really knowing her personally or anything), but this particular article appears to reflect a kind of scientific reductionism that I find distasteful, and I suppose that neither Quantum physics nor Freudian psychology are going to explain that particular aversion. Perhaps this hostility towards scientific reductionism--at least in some self-reflective and (semi) self-conscious sort of manner--goes some way to informing my apparent compulsion to take decent, working, knowledgeable scientists to task.

Shrug. I dunno'.

Science is cool, science is fascinating, science is a worthwhile human endeavour and has clearly aided the human race (yet also helped our seemingly insatiable appetite for self/other destruction as well...yet that's not science's fault--or is it? Perhaps if we hold that everything is quantum physics, then, yes, maybe science can be held morally accountable for all the atrocities it's helped to create...oh, sorry, again, my tendency to sometimes digress). But, at least currently, science isn't everything--can't explain everything, at any rate. And when it comes down to the fine grained details none of the stuff of existence is science. Science seems to be a method which produces a body of knowledge: together these inform investigations over a specific and limited range of human-based experiences.

Maybe one day science can and will explain everything there is to human experiences, but I guarantee that from our current position in the timeline we would not understand whatever that will be as science: its theories would likely seem to us as gibberish and its results as some form of sorcery. As it currently stands with or without Quantum--or any other formulation of physics--there are still milkshakes, weight loss, basketball, and Noam Chomsky. Sorry, Sabine.

On the Mistakes of AI Supporters and Detractors

This is a response to a post on Google+ by +Singularity 2045. The original post can currently be viewed here. This post (if no longer available at the time of reading my response below) was based on a link to this page, apparently.

When it comes to pro or con arguments with respect to "machine intelligence," or "AI," the logic on both sides of the argument is frequently lacking.

In the OP Singularity 2045 references a statement of Mark Tegmark, which is presented as:

One thing is certain, and that is that the reason we humans have more power on this planet than tigers is not because we have sharper claws than tigers, or stronger muscles. It’s because we’re smarter. So if we create machines that are smarter than us there’s absolutely no guarantee that we’re going to stay in control,

and Singularity 2045 subsequently critisizes it in the following:

Analogies regarding tigers are only valid if tigers had created the human race via intelligent AI engineering of human brains, or AI design of precursor human brains. The point is our intelligent engineering of AI makes humans utterly different to any unintelligent species below us unable to create higher intelligence.

It is easy to see how, regardless of Tegmark's use of analogy or whether Tegmark is pro or con AI, this statement is possibly true:

If we create machines that are smarter than us there’s absolutely no guarantee that we’re going to stay in control.

The thing here is we need to understand what we mean by "artificial intelligence."

If we merely mean really smart machines that are still machines, that is, they are still entirely constrained by their programing and what we command them to do via that programing, then we can not really "lose control over them" in any meaningful sense.

Things could go wrong with the programing, sure, there could be glitches and bugs, sure, but ultimately by definition, and in virtue of, our programming of these machines, we are always going to be "in control" of them even if our control of them goes sideways due to human error (via glitches and bugs in programming). Put differently, when a machine (as we currently understand them) breaks down or malfunctions or whatever else, our temporary "loss of control" is metaphorical. It's not that we suddenly found ourselves facing a machine that chose to go "out of control," rather, the degree of our control over the machine simply reached a low we would rather it had not.

Put differently still, when we "lose control" over a given machine, we do not suddenly think we are living in the world of Maximum Overdrive.

By means of a simple analogy: we can literally "lose control" of our pet dog, for example, but we cannot literally "lose control" of our pet rock.

However, if by "artificial intelligence" we mean the creation of a machine that can also think for itself, which entails, it can choose how to act and respond to its environment, and is unconstrained by its programing--that is, it is free to alter its programming and is not dependent on our commands--then it is entirely possible that we will "lose control" over it. Indeed, by definition we have little to no control over any "free agent," and if we can exercise control over a "free agent," then such an agent is not truly "free."

Now here's an analogy that is actually workable. We have no idea about other human beings. We think we might "know" them, and we think we might be able to predict their behaviours based on our knowledge of them with respect to previous interactions with them under such and such a circumstance or another.

However, people can be unpredictable. They can surprise us. This is because either:

1) They have something known as "free will" and can choose to act or respond differently with respect to how we predict they might respond, or

2) There is no such thing as "free will," but our universe, in its complexity, is a nonlinear deterministic system, which entails that we can not always predict results; put differently, a nonlinear deterministic system is unpredictable, by definition.

(There is, of course, a third option here in that the universe is a deterministic linear system: this is highly unlikely, in my opinion, but if that is the case, then no one is "controlling" anything, and everything in the universe is merely the playing out of predetermined results based on previous causes, which we could, in theory, anyway, predict if we could figure out what those initial conditions, in fact, were, but this is an aside.)

Now, if we are unable to always and accurately predict the behaviour of other human beings--whose general biological and psychological framework are similar and familiar to our own--then how much less so will we be able to predict the behaviour of a machine intelligence that either has "free will" or is itself able to act, as we do, along the lines of a participant in a nonlinear deterministic system?

In other words, we have a difficult enough time trying to control and predict the behaviours of complex systems (human beings) that are reasonably similar and familiar to ourselves, so there is no reason to think that we can have any greater success in controlling or predicting a machine entity that we have never encountered before.

Therefore, the only logical conclusion to the debate as to whether or not a machine intelligence (of the "free" or "nonlinear" variety) is going to be beneficial to humankind or malevolent is that currently such a question is undecidable.

Logically speaking, we have no good reasons to suppose it is either. Taking a side at this point in time (where we have ZERO experience with such a "machine intelligence") is simply speculative navel gazing based on our own particular prejudices and biases. We have no actual empirical evidence on which to base our conclusions.

We can guess and speculate all we want, but until such an entity is actually in the world, there is nothing upon which to ground our guesses and speculations. It is exactly like trying to decide if alien visitors to the planet would be a boon or a misfortune to humankind: until we actually have alien visitors (assuming we have none already and assuming that if we do, their impact on the world has been hidden from us and, therefore, most of us have no way to adequately assess the matter), there is simply no way to predict the outcome.

What Is an Interpretation?

I've been meaning to write on this subject for awhile now, but recent conversations seem to expedite the necessity of getting clear on what I mean when I use the word 'interpretation'. So, without further fan fare, hand waving, or other particularly useless and time wasting linguistic devices, let's go!

We all interpret. Simple right? We all have a way of looking at the world that frames our experiences along certain lines, which then enables us to express things about how our experiences and observations are related to one and other, and we generally express ourselves using some "natural language" or another. Our natural languages tend to obscure the formal language which underlies our statements. When we express ourselves in a natural language, we are actually also using an underlying layer of reasoning and logic which is necessarily about our interpretative strategies. Put differently, if we are making any sort of statements beyond mere logical formalisms, then reason and logic apply only within our interpretations, but we are unable to use reason or logic in the absence of an interpretation with respect to our use of a natural language about the world.

So, what do we mean by an interpretation? Well, we sort of mean the way we use the word 'interpretation' colloquially: like, for example, when we say things such as "That's only your interpretation" or "Let me offer my interpretation of this painting," and, in some sense, we also are using an interpretative structure when we translate languages from one specific language to another specific language. This latter, though, is more of a "meta" interpretive structure: an interpretation as a  function that assigns elements in one interpretation to elements in another interpretation--we are less concerned about this sort of interpretation than we are about specific interpretations, like those we most often refer to when we use the word 'interpretation' in common speech.

Referring to my notes from Logic I with Dr. Mark Gardiner taken many, many moons ago, we will set out on a more than less thorough presentation of the formal and rigorous definition of an interpretation. Don't worry too much though, I will do my best to translate along the way from the formal definition into a more natural language understanding of such a thing; so, simply plow through the formalisms as best as you are able, and it is my hope that once we come out the other side of the natural language interpretation of the formalisms, the reader will be able to go back to the formal definition with a deeper sense of understanding. So, fasten the seat belts on your brains, then, we're in for a bit of a bumpy ride.

In logic, an interpretation is defined in the following way:

1. An interpretation requires a set of linguistic symbols.

In logic these are referred to as 'wffs' or, as I pronounce it, and I think others do as well, "woofs." This is simply an abbreviation for "well formed formulas." Now, without trying to teach or instruct the reader from scratch, what we mean by a "woof" is simply a string of linguistic symbols that also satisfies the rules of their construction. In symbolic logic, for example 'A & B' is a "woof" but 'AB &' is not a "woof." Another example, 'A -> B' is a "woof" but '->AB' is not a "woof." What we mean, then, by a "well formed formula," is a string of linguistic symbols that conform to specific rules in their construction in such a way that they are able to represent a clear and precise statement that we are then able to use within our system of logic. And what we mean by "a set of linguistic symbols" is all and only those strings of linguistic symbols that conform to the rules governing their construction.

Now, to be entirely as clear as I am able, let's look at the same idea as applied to English.

The sentence, "Sally had a nice day at the beach," would count as a "woof" in English. The sentence "Beach a the at Sally nice had day" is not a "woof" in English. Another example, "Seven is the number that occurs between eight and nine," is a "woof" in English (even though we might observe it is a formula that, when interpreted in terms of English qua mathematics, we see is not true--more on that in a bit). The sentence "Eight the is seven nine occurs and number between that," is not a "woof." So what we mean by a "well formed formula" is that it follows the rules of a grammar such that the sentence is sensible within our structure of interpretation; that is, the string of linguistic symbols occur with a specific sort of order that allows them to be interpreted.

Getting the picture? Good. Let's continue.

2. An interpretation requires a structure.

What the heck do we mean by that?

Here is the formal definition of a structure, U:

U must consist of precisely two things:

1. a nonempty set of individuals or elements, which together constitute a Universe of Discourse also known as a Domain.

2. one or more relations defined on this Universe of Discourse.

Any clearer? No, not really, right? And in order to present this formally and in such a way that is understandable, the reader really needs to have sat through a formal logic class. And it is not my intention to subject anyone to that against his or her will. So, I will refer again to my notes taken a few days later, where we discuss these formal matters in a somewhat less formal way. These notes, by the way, were taken on April Fools day in 1999 (probably the right year, anyway--I can't be particularly bothered to figure it out exactly): make of that what you will. This section of my notes bears the title "Metaphysical Musings."

And I quote:

A structure consists of only two types of things:

1. objects, and

2. sets of objects.

End quote.

So, in reference to the formal definitions in (1) and (2) above, the "nonempty set of elements" is simply the things that are counted as things within our Universe of Discourse. That is, it is the scope of things that our Universe of Discourse is able to "talk" about. In a natural language such as English, this is all the objects that our words refer to: dogs, cats, stars, protons, people, places, black holes, clouds, the weather, colours, and on, and on, and on, etc..

Now with regards to (2), what we mean by "one or more relations defined on this Universe of Discourse" are the relations that can be expressed in such a way that all and only those things which bear a specific property occur in some specific set of objects; in other words, the definitions of the relations that establish that the set of objects actually have some specific property.

To return to the formal definitions in terms of Predicate Logic, which is a more complex and developed version of Symbolic Logic, we define these sets of objects in two ways:

(I) By Intension, and

(II) By Extension.

"By intention" is simply a list of a set of necessary and sufficient conditions which any object in our Universe of Discourse must satisfy in order to have the property in question.

"By extension" is simply the set of every object that has the property in question.

Now, to return to some simplified versions of these things with respect to the natural language of English, we can consider the following as ways to understand what we mean by intensional and extensional definitions of objects that are in turn collected in some set of things.

Let's use the property of "blueness," as in, things (objects) that have the property of being blue.

Now, a definition by Intension (and this is by no means to be considered as a complete definition of the Intension regarding the property of blueness, but merely to give an idea of what such a definition would consist of) might include things like:

(A) The objects in question must reflect light at wavelengths that fall between 450–495 nm and at a frequency of 606–668 THz that have a photon energy of 2.50–2.75 eV.

(B) The objects in question must conform to what we mean by "being blue."

(C) The objects in question are in fact coloured blue.

We see that by using this list we offer a set of conditions which any object must satisfy in order to be a member of the set of all things that have the property of blueness. The objects that do not satisfy the definition by intension are not to be taken as members of the set of all things which have blueness.

A definition by extension, then, is simply the list of all the objects in our domain that also have the property of blueness. This set includes, for example (and again, not to give a complete list, but only to give the reader an idea of what is meant by "definition by extension"): bluejays, blue balloons, the billiard balls typically numbered with either a '2' or a '10', cars that are blue, the sky (sometimes), lakes (sometimes, but more and more less frequently, perhaps), garments that are blue, and so on and so on, etc..

So, it is my hope that the reader has a bit more sense of what we mean by the formal definition of a structure. We mean, then, the set of all objects that our Universe of Discourse is able to discuss and all the subsets of this set that are defined in terms of the objects having specific properties. We can note, then, that the subsets are a way of dividing up the objects in the Universe of Discourse in many different ways, which will necessarily entail that some objects are members of several subsets, but not all objects that are members of a given subset will necessarily occur in all the same subsets as one and other. For example, and returning to our subset of things with blueness: our blue billiard balls occur in the same subset of blueness that bluejays occur in, but bluejays do not occur in the set of all round things in the same way that billiard balls do not occur in the set of all birds. Some of the elements in the set of balloons that are blue will occur in the set of all round things along with the set of all billiard balls that are blue; but, the set of all balloons that are blue will not occur in the set of all things we use to play billiards, where, obviously, the set of all billiard balls, regardless of their colourings, will occur as a member of that set.

Getting the hang of this? Good.

Finally, but certainly not least:

3. An interpretation requires a set of rules for associating symbols with things.

This is not nearly as mystifying as it might seem, although when we look at it formally below, it will seem quite mystifying indeed.

All this really means is that there is some function that, for a given structure U, takes the objects and sets of objects and maps them to the strings of symbols which make up our language.

Formally, this is called an "interpretation function," and here is its formal definition:

For a given structure U, an interpretation function assigns:

1. to each constant of Predicate Logic an individual belonging to the Universe of Discourse of U, and

2. to each k-ary predicate symbol of Predicate Logic, a set of k-tuples consisting of individuals belonging to the Universe of Discourse of U.

Yes, because that is so clear now, right? Again, this is the formal definition and without knowing what the heck we mean by things like "k-ary predicates," "constants" with respect to PL, and "k-tuples," well, it's pretty much senseless. Again, without actually having sat through a formal Logic class, it is difficult to really get a grasp on these things formally.

So, let's turn back to the section we called "Metaphysical Musings" and see what the notes have to say on this matter.

I quote:

An interpretation bridges "language" and "the world" by:

1. Associating single objects with constants, and

2. Associating sets of objects with predicates.

End quote.

So, what we are getting at by the formal definition of (1) above regarding the constants of PL and the individual members of the Domain of Discourse of U, is simply a symbolic representation in language that picks out only the given object we are considering.

In the natural language of English we might, from the general set of all bluejays in our Domain of Discourse, give a specific name to a specific member of that set, let's say 'Bob,' so that when we discuss "Bob the bluejay" we know that we are referring to the same and only that particular bluejay we've called Bob.

Back to the informal talk about the formal language, an interpretation function assigns a name, say 'c1' to one and only one object from the Domain of Discourse so that there is no confusion as to what we are referring to when we come to evaluate the statements that are made in our formal language by way of operators acting in conjunction with predicates and the constants associated with these predicates. And this is, essentially, related to our association of sets of objects with specific predicates, which, more formally, is about the "k-ary predicate symbols" and the set of "k-tuples."

To go back to English, then, what we mean is that when we say something like:

"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree."

We have a series of words that correspond to not only objects or sets of objects (Bob, bluejays, birds, branches, trees) in our Domain of Discourse, but also words that correspond to predicates, which are properties, (blueness, prettiness, sitting, thickness, tallness) in our Domain of Discourse. We also have "logical operators" which link up various individual claims or statements regarding the objects and their properties. This gets quite complex when we go from a formal language to a natural language. Indeed, some logicians spend some of their time trying to translate philosophical statements from the natural language in which they occur into a formal Predicate Logic version in order to accurately and mechanically evaluate their truth claims. Most efforts are difficult, slow going, and open for debate regarding the translations--especially with respect to which logical operators might best suit a natural language sentence.

In regards to the sentence above, there is basically only the repetitive occurrence of the logical operator & (or at least we will agree to look at it this way for ease of presentation and understanding), which we in natural language, funnily enough, call "and." Taking apart the above sentence and representing all the occurrences of & in it, is, given the simplicity of the statement, pretty much straight forward, so here it is:

1) Bob is a bird & he is blue.
2) Bob is a bird & he is blue & he is a bluejay.

3) Bob is a bird & he is blue & he is a bluejay & he is pretty.

4) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting.

5) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch.

6) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch & that branch is thick.

7) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch & that branch is thick & that thick branch is part of a tree.

8) Bob is a bird & he is blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch & that branch is thick & that thick branch is part of a tree & that tree is tall.

See how much is contained in a relatively simple statement? Bet you had no idea that when you utter a simple phrase there is a whole bunch of background formalism present in terms of what we mean by logic or logical structure.

We are now, I feel, in a position that we can, in some sense, understand what we mean by an interpretation and how an interpretation functions in terms of things like: making claims, assessing truth, discussing objects, and etc..

An Interpretation that would allow us to make sense of the statement:

"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree,"

needs the following objects to be present in its Domain of Discourse:

{the set of birds, the set of trees}.

The Interpretation must also have the following predicates:

{blueness, prettiness, thickness, tallness}

The Interpretation must have a function which assigns strings of symbols to these objects and the set of objects that have any given specific property. Let's call this function in this example E.

Now, E takes things from our set of objects and maps them to a set of words.

E, in our example, maps from the set of all birds that are also bluejays, to the set of words that make up our English language such that:

'Bob' picks out a specific bird, that is also a specific bluejay, from the set of all birds that are also bluejays.

E would also pick out the predicates, or properties, associated with Bob, namely:

That Bob is a bird, that Bob is a bluejay, that Bob is pretty, and that Bob is sitting.

Notice the interplay between the name "Bob," how it is we identify Bob from the set of all birds, and how the predicates as they relate to Bob help us in deciding which bird is meant by 'Bob'. Taken altogether, we hope we have an Interpretation that is rich enough to point to that specific bird that we are calling, in this instance, "Bob." In other words, our "interpretation function" is such that it allows us to pick out what we have named "Bob" as distinct from all other things named "Bob," in such a way that we understand some things about what Bob actually is and how Bob actually is, namely, Bob is pretty, Bob is sitting, and etc..

OK, let's turn to what we mean by Truth and we shall see how (1) truth only exists inside an interpretation and (2) what is taken to be a "fact" only exists inside an Interpretation relative to the truth-value that the Interpretation establishes in regards to statements about objects that are recognized in the Domain of Discourse of the Interpretation.

Back to our relatively simple statement:

"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree."

We have seen how this statement breaks down into a series of conjuncts, the totality of which is reasonably captured by:

Bob is a bird & he is a blue & he is a bluejay & he is pretty & he is sitting & he is sitting on a branch & that branch is thick & that thick branch is part of a tree & that tree is tall.

A "conjunct" is simply any of the statements which occur on either side of any given occurrence of the logical operator &. So, the conjuncts of our statement are:

1) Bob is a bird.

2) Bob is blue.

3) Bob is a bluejay.
4) Bob is pretty.

5) Bob is sitting.

6) Bob is sitting on a branch.

7) That branch is thick.

8) That branch is part of a tree.

9) That tree is tall.

In order for our statement to be true in our Interpretation, each of the conjuncts also has to be true. If any one of the conjuncts is false, then the whole statement is false by the rules of logic. The interpretation, then, requires that we are able to consider each term with respect to the conditions that make it either true or false. This is to say, that for each of the conjuncts we need to be able to understand what makes any one of them true or what makes any one of them false. Let's simply look at one of the conjuncts, namely, "Bob is a bird."

In our Interpretation there are two possibilities: either "Bob is a bird" is a true statement or "Bob is a bird" is a false statement. So, if our Interpretation correctly assigns the word 'Bob' to the object that is a bird, and this bird is exactly the one we mean by the string of symbols 'Bob', then on our interpretation the statement "Bob is a bird" is true. However, if our Interpretation instead assigns the word 'Bob' to our uncle, then "Bob is a bird" is false in our Interpretation, and so, with regards to our whole statement:

"Bob the blue bluejay is a pretty bird and he is sitting on the thick branch of that tall tree,"

our Interpretation would evaluate it as a false statement, since Bob is a human being and not a bird. Now, every other conjunct about Bob our uncle could be true: he could be blue, he could be pretty, he could be sitting on a branch of a tall tree, but since he is not a bird and specifically not a bluejay, our claim about Bob as a blue bird that is also a bluejay that is sitting, pretty, and etc., is in fact false.

Without an Interpretation which assigns words to things and establishes the relationships about the things in terms of their properties, well, there simply isn't any sense of "truth" to the matter. We'd have only words and only things but no bridge between them; that is, without an interpretation, we have no way to make claims about anything because we can't talk about the things and we can't associate things with their properties in any sensible, as in utterable, manner.

We can easily see how "facts" fall out of this. If there is a correspondence between our language and the objects in our language's Domain of Discourse and the statements we make about these objects are also true, then that is what we call a "fact." If Bob really is a bird, and if by the word 'Bob' and 'bird' our Interpretation establishes a relationship between those things such that is is true that "Bob is a bird" corresponds to Bob actually being a bird, then we say that it is a fact that Bob is a bird.

Again, without an Interpretation, then, there are no facts of the matter, there are only objects that have properties which are not in any way sensibly able to be discussed; put differently, the facts of the matter are uninterpreted without an association between our language and its objects of Discourse. What we mean by "fact," then, is something that only comes to be when considering true statements within an Interpretation.

To go way back to the beginning, consider the statement:

"Seven is the number that occurs between eight and nine,"

We know this to be false simply because we have an Interpretation that establishes that the object referred to by the word 'seven' occurs in a mathematical structure such that the object seven has its place between the objects referred to as "six" and "eight". Thus, it is our Interpretation that establishes that the fact of the matter does not correspond to our statement about seven in terms of the property of it occurring between the numbers we call "eight" and "nine". Without the Interpretation we have no facts of the matter because we have no way to understand the properties corresponding to the object seven, and we have no way to assess the statement about that object in its relation to other objects.

Thus, we see clearly how truth and facts are entirely dependent upon an Interpretation.

So, turning to reason and the use of logic, we can also see clearly that we are only able to reason about things insofar as these things correspond to some Interpretation which allows us to understand things about the objects in the Universe of Discourse in such a way that we are, in fact, able to reason about these things in terms of the referents and the properties that these objects have as picked out by our Interpretation. Logic is simply the rules by which we understand how the relationship between statements and the statements' objects and properties are evaluated in terms of being either true or false. Without some set of sentences and rules by which we can logically manipulate and assess those sentences, well, there's no logic to be had at all.

An important thing to note is that all interpretations are open to the use of reason and logic, but we are also able to use reason and logic on an uninterpreted set of sentences whereby the "woofs" do not correspond to any objects of a domain of discourse other than the "woofs" themselves. Such practices are called "pure logic" in the sense that the analysis and results that come out of such a study are about Logic itself. This, however, is a purely formal exercise and is not associated with any natural language; that is, when we reason about logic and use logic to analyze itself we must necessarily not be referencing anything that has to do with objects beyond the objects of logic itself.

So, when I make the claim that "reason and logic apply only within our interpretations, but we are unable to use reason or logic in the absence of an interpretation," that is not quite the case, it is, in a sense, a colloquial way of speaking, of being a bit loose with the language. What I mean is that any time we are reasoning about things in the world or using logic with regards to our statements about things in the world, then we must necessarily do so with respect to an interpretation. Without an interpretation reason and logic can only ever be about themselves.

To put this more concretely, then, when I make a claim like:

 "I do tend towards having a distaste for certain 'breeds' of atheists who declare logic and reason as their allies and claim these support their lack of belief. They are mistaken. Logic and reason apply only within interpretations and their interpretations assume their lack of belief..."

 What I mean is that whenever a person is using logic and reason to assess their beliefs, he or she can only use it insofar as to reason from the assumptions within their Interpretation and use logic to assess the truth-value claims of how these assumptions are interconnected via logical operators (such as 'and', 'or', 'not', 'if...then...', and 'if and only if') with respect to the assumptions of a given interpretation. Logic and reasoning alone cannot establish anything about any belief or lack thereof as without an interpretation there simply is no such thing as belief or lack thereof.

In order to preempt atheists who might wish to say, due to their misunderstanding of the above presentation, that since lack of belief entails that there is no belief, we can thus use pure reason and pure logic alone to support that lack of belief since you (being me) asserted that belief does not exist in terms of reasoning about pure logic alone. I will simply point out that reasoning about logic with logic has nothing to do with any of the other beliefs or facts that exist only in their Interpretations by which they have derived or otherwise assumed the conclusion regarding their lack of belief.

If you, the reader, actually made it to the end of this article, congratulations: you now probably have a better grasp on logic, reasoning, and interpretation than most of your peers, at least, those of your peers who have never taken a course on Logic and have not been trained in ways of reasoning in a formal sense (i.e, mathematically, for example). I hope you found the ride not too bumpy and that your brain made it through without too many bruises. Now, amidst a short burst of fanfare with maybe an accompanying drum roll and perhaps some hand waving, I bid you adieu.

Caring Less: Could or Couldn't

I can be a bit of a stickler about language sometimes. We use language to convey thoughts and feelings, opinions and meanings, and I tend to feel that if we want to express ourselves through language and also want the language we use to adequately convey what we mean, then it is best for us to use language correctly or at least appropriately in order to accomplish this.

One of my pet peeves about language use revolves around 'caring' and centres on whether we 'could' or 'couldn't' care less with a specific focus on what we intend to mean when we use a string of language containing some combination of these words. It seems to me that there are those who want to say one thing, but instead use language to say another. When I see people write, or hear people say, "I could care less," my immediate typical response to the meaning of this string of words is twofold. I wonder how much the person actually cares and want to ask, "well, why don't you care less?"

An example of this might be someone who is entirely uninterested in a particular subject, let's say baseball (because I find baseball extremely boring and uninteresting, heh). The wrong way to express this disinterest is to write or say something like, "I could care less about baseball." If you don't care at all about baseball, then what you want to say is "I couldn't care less about baseball." The latter implies that the language user has absolutely no care whatsoever about baseball--that he or she is at the very bottom of the scale that registers one's degree of caring towards, in this instance, baseball. The former implies that it is entirely possible that the language user could have less care on the scale of caring, which actually means there is some degree of caring about baseball.

So, if we want to convey our particular and absolute disinterest in baseball--or anything else--we don't want to say or write that we "could care less," rather, we want to say or write that we "couldn't care less." In other words, if a particular language user wants to be understood as having no care at all for a particular thing--that he or she is at the very bottom of the scale when it comes to his or her amount of caring--then he or she want's to negate all possibility of caring; thus, the language user needs to include the negation in his or her language use, namely, "could not."

Omitting the negation implies that some caring is present, but not to what degree. So, if we "could care less" about a specific thing, then we could actually care quite a lot about this specific thing. Indeed, of all the things we care about, we can always say that we "could care less" about these things and be uttering something true relative to our particular set of cares: it is entirely possible that we could care less about the things we care about the most.

Given this bit of analysis, it might be understandable why I've become accustomed to having my teeth set on edge when I see people use the phrase "could care less" in their employment of language. It has become for me habitual to experience some small amount of annoyance as soon as my eyes scan any text that include the phrase "could care less." So, I was quite surprised and delighted the other day when someone used "could care less" in a way that actually conveyed what he or she meant--that there was absolutely no caring present at all--and it made me do a double take of the sentence and brought a grin to my face. The language user wrote:

No one could care less.

In my entire reading history I am quite sure this is the only occasion I have ever seen someone use the phrase 'could care less' in conjunction with other parts of language to convey what he or she actually meant: there is no amount of caring at all--there is precisely zero caring.

As for the rest of the language users that have no care at all about something, but write or say that they actually do have some degree of care--regardless of how large or small the amount of caring: smarten up. If you don't care at all about something and also desire to accurately convey this to other language users, then, please, use language in a way that actually means what you intend otherwise you might find that people couldn't care less about what it is you are trying to say.

"Holographic" Universe?

There's a recent article that I've seen circulating around Google+, namely:

Simulations back up the theory that Universe is a hologram, Published on Nature.com, Dec. 10, 2013.

This is a bit of a synchronicity for me in some ways as I've been revisiting some ideas regarding notions of a "holographic" universe and, indeed, have been planning on doing some review/analysis here on Drifting Labyrinths of a few YouTube videos on the subject, which I have watched this past week. But this is an aside--now let's get to the matter at hand.

I've seen some comments regarding this article, and I think that some people misunderstand what the article is actually saying. It does not say that we are living in a hologram and it certainly does not say we are living in something that is similar to The Matrix. What it does say is that certain interpretations of string theory--ones which use nine spatial dimensions with one dimension of time (for a total of ten dimensions)--can be seen as a projection, or "hologram," of a specific sort of one dimensional model of quantum mechanics. From the article:

The mathematically intricate world of strings...would be merely a hologram: the real action would play out in a simpler, flatter cosmos where there is no gravity.

and the article later says of this "simpler flatter universe":

The lower-dimensional, gravity-free [universe] has but a single dimension and its menagerie of quantum particles resembles a group of idealized springs, or harmonic oscillators, attached to one another.

What some people seem to miss in the article is where theoretical physicist Juan Maldacena is paraphrased as saying:

Neither of these model universes...resembles our own.

What the article is actually reporting on is the fact that there are problems/solutions which are translatable from the more complicated and thus, mathematically more difficult, ten dimensional model of superstrings to this simpler one dimensional model. In the article, these problems and solutions concern properties of certain kinds of black holes.

Now, the fact that a specific one dimensional model has some sort of equivalency with a ten dimensional model does not seem particularly surprising in this instance: in string theory the strings themselves are seen to be one dimensional objects and these one dimensional objects are "vibrating" in the ten dimensions. It seems at least intuitively possible that a simplified model of the ten dimensions of certain formulations of string theory would be able to be expressed in a single dimension--perhaps not much differently from how we can think of and express four dimensional "light cones" in two dimensions, how we can explore four dimensional geometry by the projection of four dimensional objects onto a two dimensional surface, or how we create a three dimensional visual experience from the two dimensional representation of images on our retinas. What I am getting at here is that it seems there are several instances of translating phenomena from such and such a dimensional configuration to another, and so, it's not like such translations are something new.

Moreover, it's not like the discoveries reported in this article are conclusive that certain ten dimensional string theories are entirely translatable into this specific one dimensional model. As the article itself says the research being reported upon is not a proof but provides "...compelling evidence that Maldacena's conjecture is true."

What was this conjecture? It's not that we are living in a hologram or The Matrix that's for sure; rather, the conjecture is that there exists a model of the universe such that a "...model of the Universe in which gravity arises from infinitesimally thin vibrating strings [i.e., in the context of the content of the research reported on in the article, specific models of ten dimensional string theory] could be reinterpreted in terms of well-established physics [and this would be, apparently, the physics of the one dimensional model reported on in the article]". In other words, the "compelling" evidence is in regards to the fact that there is a simplified model of a more complex model.

And that's the so-called "rub" here: we are talking about models of the universe and not the universe itself. Pretty much by definition all models are going to be simplified versions of whatever it is the model is modeling. Further, since all these models in current physics are essentially mathematical in nature, I can't help but think that any model is going to run up against some problematic manifestation similar to Gödel's Incompleteness Theorem, by which I mean that, for any given model of the universe, it will be impossible for it to be both complete and consistent. Put differently, it seems likely that there will always be truths of the universe that are unable to be fully represented within a physical model of the universe.

There are more things in heaven and earth, Horatio,
Than are dreamt of in your philosophy.