Posts Tagged ‘math’

Esoteric Number Sets

May 30, 2014 2 comments

I made this sound really opaque, but all it consists of is reorganizing the sets of numbers so that it begins with the simplest number (1) and progresses through variations on the theme of one-ness. These variations would be ratios, i.e. fractions, only I’m conceiving of them as more like the feeling of a specific fraction rather than a definitely named number. The feeling of ‘half’ would precede the concept of 1/2, so that the number 2 would be derived from feeling of “one half and the other”. Ok, it’s becoming opaque again, but what I’m going for here is flipping our view of number sets around so that the continuum of numbers is not taken as a space that is filled up with Platonic object generation, but one of a sense-making awareness subtracting and ratios of itself within itself. In this way, multiplication is really a division of (1), and division is a multiplication of those divisions.

The above diagram is borrowed from Math is Fun.

Natural Numbers


Rational Numbers

Real Numbers

Imaginary Numbers

Complex Numbers

If we begin from a primordial pansensitivity model, the entire sense of enumeration would be included as an element. That element is shown below as the number 1.

I see the ability to hold multiple numbers against each other in conceptual space as rationality. Rational numbering (Q) is more of a verb, situated between the transcendental sense of unity and the enumerated sense of static multiplicity, represented by the Natural numbers (N).

This effectively turns the number set relations inside out, so that all numbers are seen to diverge from an intuitive simplicity, and progress into nested complexity and abstraction. Negative numbers extend the natural numbers to Integers (Z) through a numberline concept in which 0 is treated as a kind of mirror. Imaginary numbers, Complex, and Reals take advantage of the original rationality (Q) and its nesting, reflecting, elaborations. In this diagram, the number 0 is a Natural number apart from all others, indicating its status as the representation of the complete absence of (1) within (1).

Dark Math and the Vanishing of Zero

March 29, 2014 2 comments

Some bits of Facebook conversations that I have been waiting to have about math. I don’t claim to be even a little bit competent as a mathematician, but I do feel like there is a chance that these ideas might just happen to be so absurd that they are profound. Among these ideas has been the consequences of making zero disappear, and redefining the number one so that it is the container of mathematics itself.

There was another idea that I proposed today of “Dark Math”. I asked the question “Does math have a language/theory to represent its own opposite (independent of consciousness even, just like imaginary numbers, but imaginary anti-math instead)?”

If we turn Incompleteness around, for example, we get something like intuition. Any informal-non-system contains unanticipated reflections of formality..surprising quasi-truthful insights from out of thin air, like an oracle. If we turn Church-Turing around, we get non-universal, non-machines = unique individuals. My suspicion is that such a language would help define or model previously undefinable phenomenological conditions. Anti-numbers, (names which are intrinsically semi-proprietary?), Anti-operators (metaphorical and synchronistic?)

Then the zero idea came up again…

OH: A complex number z is said to be purely imaginary. If it has no real part, i.e., R[z] = 0. The term is often used in preference to the simpler “imaginary” in situations where z can in general assume complex values with nonzero real parts, but in a particular case of interest, the real part is identically zero. 0 is a pure imaginary number .

S33: If the part is identically zero, but zero is entirely imaginary, does that mean that its identicality is also imaginary? If we carry through the idea that 0 is imaginary, then any time we qualify something as being ‘not’ we are being figurative, and the reality would always be some infinitessimal fragment. Not would literally be ‘almost not’.


S33: But what I’m proposing is that vanishing itself is identically zero, then vanishing may be infinitesimally figurative. Nothing can vanish completely in reality, even the difference between A and A.

Can zero be said then to be ‘that which is not anything, *not even itself*. When we apply this to ontology (and I think we should) it means we must accept that nothing has vanished. What happens instead is that things nearly vanish from some set of perspectives. The gap between nearly vanishing and vanishing is entropy. Entropy is how perception compensates, fudges, fills in, etc so that what is for all practical purposes absent (i.e. the past) becomes elided or removed. Even the removal is not total, not real, its just a delay. Eventually all that has been denied must be revealed as unvanished from some perspective or encounter.

The reverse of this entropic clipping of the infinitesimally unvanished would be what I call significance. An augmentation of sensitivity or motive so that a near-vanished experience is encountered first as fiction. In other words, entropy makes things seem to disappear (like the past, coherence, certainty, etc) which really haven’t, and significance makes things seem to appear, but also significance increases the quality of ‘thingness’ beyond the thing. You could say that entropy masks presence to the point of near absence, and significance stretches near-absence to the point of re-presence.

Non-Well-Founded Identity Principle

September 16, 2013 6 comments

Non-Well-Founded Identity Principle

Here’s a crazy little number that I like to call the Non-Well-Founded Identity Principle. It woke my boiling brain up a few times last night, so I present it now in its raw state of lunacy.

The idea here is “For All A, A equals the integral between A and (the integral between A and not A)”.

This represents a refinement of trivial identity, A=A, to reflect the grounding of all propositions in the Absolute inertial frame of pansensitivity. The nested integral specifies that all integrations are themselves defined as that which is not disintegrated. Any object, subject, or sensory presentation or representation (A) is itself, and it is also the range of all possible relations, literal, figurative, and otherwise, between itself and all that is not itself (≠A).

This comes out of the idea that sense is the Explanatory Gap, i.e. the gap between private experience and public bodies is a non-well-founded set (non-well-founded sets contain themselves as members) in which primordial pansensitivity*defines its nested child sense experiences in a terms which are both unique, generic, and everything in between, depending on how the local perceptual inertia frames it.

*pansensitivity is plain old feeling, sensing, being and doing, but extended and universalized beyond Homo sapiens, as well as physics and arithmetic truth. Ontology itself – being; the is-ness and it-ness of all phenomena can be reduced further through the Non-Well-Founded Identity Principle, under which ontology becomes the nested gap between phenomenology and the sense of its own absence. This is a very tricky shell game, but it is not intended as a trick or a game. Said another way, ‘privacy is the difference between privacy and the difference between private and public experience.’

Applied to philosophy of mind, we would get: Naive realism equals the difference between naive realism and (the difference between naive realism and reductionism). Another one would be Sense equals the sense of the difference between the sense and (the difference between sense and logic). It could be said that X=/(=/≠) X, so that any number is a straight isomorphism with itself, but it is also a superposition of any potential combinations with or relativity upon any and all X that it is not.

The reductio ad absurdum can be seen in this second expression:


in which integration itself is the integral between integration and disintegration. Every set or process is defined by its own self-same initiation and termination.

Is this all insipid tautology? Is it another way of catching a glimpse of Heisenberg uncertainty or Gödel incompleteness through a fun house mirror? I don’t know much about calculus, so there may be a more conventional way of expressing these kinds of relations, but in the mean time, to me, it’s an absolutely interesting way of modeling the absolute: A universal capacity to simultaneously universalize and de-univeralize (proprietize) the universal experience.

Charting It All

September 14, 2013 2 comments


In an effort to provide a more straightforward view of pansensitivity and eigenmorphism, the chart above organizes all phenomena in the cosmos by scale of publicly extended body length and frequency range of privately experienced times. Going left to right (Metaphorically, Occident to Orient), the first and second column denote the public, physical scopes (perceptual inertial frames) according to cardinality and size.  The bottom left frames (Ω) correspond to the outermost types of physical phenomena, i.e. absolutely gigantic or absolutely infinitessimal. This reflects the aesthetic intuition by which the atom comes out having more in common morphologically and dynamically with a solar system than a tree or coral reef, despite being at opposite ends of the scale of our awareness. The Ω range of frames is the envelope of physicality, where physical and mathematical ‘laws’ meet the most universally public perceptions.

Our awareness is extended technologically, which broadens our view of the public universe, however, since the awareness being extended is primarily visual and somatic (‘tangible’-kinesthetic rather than tactile), the telescoping of our sensory awareness is also narrowing our depth of field within the private (phenomenological) side of physics. My conjecture is that because of the nature of perceptual relativity, the more we focus on the the outer contexts, not only do we not see the private experience in the universe on these distant scales, but also our entire worldview will, by default, adjust to recontextualize the local experiences of the self. The fallacy of the instrument (if you have only a hammer, everything looks like a nail) might arise through this kind of empathetic feedback loop. This is likely to extend into so-called ‘supernatural’ phenomenon, which explains the increases in magnitude, frequency, and connectedness of coincidence experienced by subjects through altered states of consciousness.  The higher up on the right hand column, the more large patterns of synchronicity, with deeper resonance (A1) are available for direct personal experience (A).

By contrast, the lower down on the Oriental, right hand scale we go, the more the needle of synchronicity tips toward mere statistical coincidence and the more top-down intuition, imagination, and eidetic narrative are collapsed into the stepped logic of bottom-up causality. The numbering schema is confusing, but intentional. The use of A in the middle row on the right side denotes that sense is always anchored centripetally. Perceptual capacity radiates as a range, often a literal circular, conic, or spherical perimeter, of awareness, but also sensitivity radiates figuratively as nested channels or layered modalities of sense.

The use of A1 and Ax above and below A respectively, implies the hierarchical pull from the superlative top, down to the personal, and the plummet from the personal middle down to the bottom. Ax would be the opposite of A1; insignificant, low status, shame and indignity. Looking at the Left side of the chart, the numbering scheme is even more confusing, but it is to emphasize the multiple levels of opposition that characterize the public and private aspects of physics-sense. The A1 range is the most universal private experience, (the ultimate being experience itself), which is meta-phorical. Experiences are associated to each other through metaphor, with the most tightly isomorphic metaphor being imitation or repetition. The higher up on the A column we go, the more latitude there is in recognizing common associations. Pareidolia and Apophenia are examples of having the aperture disproportionately dilated to the super-private, which becomes unsupportable within human society (delusions of grandeur, ideas of reference, mania, etc.)

Back to the Omega column on the left, the Ω1 is a different kind of magnificence than the private rapture of the Absolute. The public side is not centripetally oriented, but linear and circular. There is no radiant center, only jumps and slides. The Alpha/Aleph side in the East fills the gaps, it infers and elides, it puts two and two together. The Ω1 is mutation and fluke. Unintentional singularity. Its uniqueness is simply an inevitable accumulation of imperfectly repeating behaviors, so that the wonder of biology through evolution can be examined correctly in Darwinian terms. These are terms of the exteriors, however. Regardless of how complex and convoluted the patterns, they are patterns of insensitivity rather than awareness, automaticity rather than authenticity. This is individuality from the outside in – stochastic, social, generational rather than individual.

The cells of the bottom row have as much in common as the cells of the top row are polar opposites, but they are also skewed (this is what the arrows in the center are supposed to mean). This is what I call eigenmorphism. A to Ω1 has a half-black, half-white arrow, showing the relation between the human mind and its homid body. This is a maximal polarization, or so it seems to us. The black arrow from the Microscopic Ωx to the Sub-private Ax have in common the x, connoting the strong relation between mechanical appearances on the molecular level and the recursiveness of awareness in its least signifying, quantitative form. This is contrary to the idea that vibrations or energy are what we are, rather vibrate is what the various parts of our sub-private experience do – jiggling or wagging from position to disposition, from incident to co-incident.

The grey arrow from A1 to Ω points to what I call the profound edge of the continuum. This would be the level at which the Totality is an unbroken, Ouroboran monad. This is what happens ‘behind our backs’, hypnotically through evanescence. It is significance reclaimed and re-membered after having been diffracted into the entropy of spacetime. By contrast, the black and white split arrow corresponds to the ‘pedestrian fold’ – the level of the monad which appears most polarized and least evanescent – the terrestrial aesthetic of ‘ordinary’ experience.

In the top chart I have limited cardinality to the public side and ordinality to the private side to show the relation between morphic scale and phoric frequency. Privacy runs first to last (ordinal), publicity places astronomically numerous to few (cardinal).

Compare with Frame Set View:


Why Computers Can’t Lie and Don’t Know Your Name

September 12, 2013 12 comments

What do the Hangman Paradox, Epimenides Paradox, and the Chinese Room Argument have in common?

The underlying Symbol Grounding Problem common to all three is that from a purely quantitative perspective, a logical truth can only satisfy some explicitly defined condition. The expectation of truth itself being implicitly true, (i.e. that it is possible to doubt what is given) is not a condition of truth, it is a boundary condition beyond truth*. All computer malfunctions, we presume, are due to problems with the physical substrate, or the programmer’s code, and not incompetence or malice.  The computer, its program, or binary logic in general cannot be blamed for trying to mislead anyone. Computation, therefore, has no truth quality, no expectation of validity or discernment between technical accuracy and the accuracy of its technique. The whole of logic is contained within the assumption that logic is valid automatically. It is an inverted mirror image of naive realism. Where a person can be childish in their truth evaluation, overextending their private world into the public domain, a computer is robotic in its truth evaluation, undersignifying privacy until it is altogether absent.

Because computers can only report a local fact (the position of a switch or token), they cannot lie intentionally. Lying involves extending a local fiction to be taken as a remote fact. When we lie, we know what a computer cannot guess – that information may not be ‘real’.

When we say that a computer makes an error, it is only because of a malfunction on the physical or programmatic level, therefore it is not false, but a true representation of the problem in the system which we receive as an error. It is only incorrect in some sense that is not local to the machine, but rather local to the user, who makes the mistake of believing that the output of the program is supposed to be grounded in their expectations for its function. It is the user who is mistaken.

It is for this same reason that computers cannot intend to tell the truth either. Telling the truth depends on an understanding of the possibility of fiction and the power to intentionally choose the extent to which the truth is revealed. The symbolic communication expressed is grounded strongly in the privacy of the subject as well as the public context, and only weakly grounded in the logic represented by the symbolic abstraction. With a computer, the hierarchy is inverted. A Turing Machine is independent of private intention and public physics, so it is grounded absolutely in its own simulacra. In Searle’s (much despised) Chinese Room Argument – the conceit of the decomposed translator exposes how the output of a program is only known to the program in its own narrow sensibility. The result of the mechanism is simply a true report of a local process of the machine which has no implicit connection to any presented truths beyond the machine…except for one: Arithmetic truth.

Arithmetic truth is not local to the machine, but it is local to all machines and all experiences of correct logical thought. This is an interesting symmetry, as the logic of mechanism is both absolutely local and instantaneous and absolutely universal and eternal, but nothing in between. Every computed result is unique to the particular instantiation of the machine or program, and universal as a Turing emulable template. What digital analogs are not is true or real any sense which relates expressly to real, experienced events in space time. This is the insight expressed in Korzybski’s famous maxim ‘The map is not the territory.’ and in the Use-Mention distinction, where using a word intentionally is understood to be distinct from merely mentioning the word as an object to be discussed. For a computer, there is no map-territory distinction. It’s all one invisible, intangible mapitory of disconnected digital events.

By contrast, a person has many ways to voluntarily discern territories and maps. They can be grouped together, such as when the acoustic territory of sound is mapped to the emotional-lyric territory of music, or the optical territory of light is mapped as the visual territory of color and image. They can be flipped so that the physics is mapped to the phenomenal as well, which is how we control the voluntary muscles of our body. For us, authenticity is important. We would rather win the lottery than just have a dream that we won the lottery. A computer does not know the difference. The dream and the reality are identical information.

Realism, then, is characterized by its opposition to the quantitative. Instead of being pegged to the polar austerity which is autonomous local + explicitly universal, consciousness ripens into the tropical fecundity of middle range. Physically real experience is in direct contrast to digital abstraction. It is semi-unique, semi-private, semi-spatiotemporal, semi-local, semi-specific, semi-universal. Arithmetic truth lacks any non-functional qualities, so that using arithmetic to falsify functionalism is inherently tautological. It is like asking an armless man to raise his hand if he thinks he has no arms.

Here’s some background stuff that relates:

The Hangman Paradox has been described as follows:

A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on Friday, as if he hasn’t been hanged by Thursday, there is only one day left – and so it won’t be a surprise if he’s hanged on Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday night, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.The next week, the executioner knocks on the prisoner’s door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.

Some thoughts on this:

1) The conclusion “I won’t be surprised to be hanged Friday if I am not hanged by Thursday” creates another proposition to be surprised about. By leaving the condition of ‘surprise’ open ended, it could include being surprised that the judge lied, or any number of other soft contingencies that could render an ‘unexpected’ outcome. The condition of expectation isn’t an objective phenomenon, it is a subjective inference. Objectively, there is no surprise since objects don’t anticipate anything.

2) If we want to close in tightly on the quantitative logic of whether deducibility can be deduced – given five coin flips and a certainty that one will be heads, each successive tails coin flip increases the odds that one the remaining flips will be heads. The fifth coin will either be 100% likely to be heads, or will prove that the certainty assumed was 100% wrong.

I think the paradox hinges on 1) the false inference of objectivity in the use of the word surprise and 2) the false assertion of omniscience by the judge. It’s like an Escher drawing. In real life, surprise cannot be predicted with certainty and the quality of unexpectedness it is not an objective thing, just as expectation is not an objective thing.

Connecting the dots, expectation, intention, realism, and truth are all rooted in the firmament of sensory-motive participation. To care about what happens cannot be divorced from our causally efficacious role in changing it. It’s not just a matter of being petulant or selfish. The ontological possibility of ‘caring’ requires letters that are not in the alphabet of determinism and computation. It is computation which acts as punctuation, spelling, and grammar, but not language itself. To a computer, every word or name is as generic as a number. They can store the string of characters that belong to what we call a name, but they have no way to really recognize who that name belongs to.

*I maintain that what is beyond truth is sense: direct phenomenological participation

Orthomodular Panprimordialism

August 23, 2013 Leave a comment

Playing around with a more math-friendly look and feel. Multisense Realism’s quant-flavored twin…Orthomodular Panprimordialism

I am really pushing it with the neologisms, but I am liking both of the recent adds, pansensitivity and now panprimordialism.

Pansensitivity is used to emphasize a position beyond panpsychism, idealism, and materialism where sensitivity becomes a palatable common capacity for all phenomena on its native scale.

Panprimordialism is used to emphasize the distribution of ‘one-ness’ across all phenomena, the relocating of all quantities to interrelated diffractions within the a sole, absolute singularity. This constitutes a figure-ground pivot from arithmetic assumptions which place zero or null as an absolute, such that all null values are considered a hypothetical representation of the sense of one’s self nullification.

The orthomodular lattice above gives some idea of that relation, with each node acting as the nexus for orderly juxtapositions within the overall monad.

Philosophical Gender

August 23, 2013 2 comments

Like sexual gender, the psyche tends to favor hovering around one aesthetic preference at a time. So much of philosophy seems to be rooted in just that, the aesthetic preferences of the psyche. How else should we explain why we are so often personally attached to our philosophical views – why we are in fact attracted to them, and to writers and speakers who have espoused similar perspectives.

Many are traditional in their philosophical tastes, and find that even the thought of experimenting with other views makes them very uncomfortable. Others find it natural to consume philosophy of all sorts, the more the better, but at the same time they may favor one particular flavor, or they may get sick of the whole intellectual-masturbatory scene eventually.

Philosophy engenders a feeling of firm orientation within it, despite the many other options available which might directly contradict it. That’s sort of the hook. A particular way of looking at things makes you feel that you are on the right track, maybe for the first time. It can change the way that you feel about other ways of acting and thinking. Like hitting puberty, what was once merely charged with social naughtiness and furtive mystery becomes irrepressibly intense. Childish ways of behaving, especially those which cross gender or leave it undeveloped are often discarded in shame and become repulsive, at least publicly. Gender is suddenly unexpectedly prominent, and exaggerated to the point of caricature.

Philosophy is almost inevitably tied to politics. Views on what the universe is, though seemingly esoteric and remote, have a way of filtering into our attitudes about everything from nature and technology, to society, personal responsibility, money, possessions, art, drugs, literature, etc.  Early math is practically inseparable from philosophy.

There are many polarities and nested polarities within philosophy, especially philosophy of mind. I often focus on reducing those polarities (reductive vs non-reductive…there’s another polarity) to a single hetero-normative gender which I am lately calling Anthropmorphism and Mechanemorphism, but have also referred to it as ACME and OMMM, Oriental Animism and Western Mechanism, Public-entropic and Private-holotrophic, and for those with a symbol fetish, ((-ℵ↔Ω) ºt)  and (ωª (H←d)).

In the course of studying this swirl of gender, it became apparent that the swirl itself could be transcended philosophically. While the battle between mind-firsters and body-firsters rages on forever, the battle itself can be seen as their most powerful overlap. Somehow even in the antiquated writings of long dead thinkers (well, the thinkers who were deemed white enough and male enough to be published anyways), fresh controversy can be sparked. It’s remarkable, really.  The enduring conflict, a perpetually circulating difference of opinion on everything, the difference between differences, and different ways of defining difference, and defining definition.

Is philosophy is a strange attractor?

“The Lorenz attractor is an example of a strange attractor. Strange attractors are unique from other phase-space attractors in that one does not know exactly where on the attractor the system will be. Two points on the attractor that are near each other at one time will be arbitrarily far apart at later times. The only restriction is that the state of system remain on the attractor. Strange attractors are also unique in that they never close on themselves — the motion of the system never repeats (non-periodic). The motion we are describing on these strange attractors is what we mean by chaotic behavior.”

If there were an ‘end’ to all of this (which is what exactly?) I think that it can be found not in being and nothingness or in difference and repetition, but in recognizing the commonality of conflict – the sense which discerns between differences and which also is the motivation which negates indifference. Carrying this principle into physics, and then mathematics, what it looks like is that a revolution of the most primordial propositions should be considered:

  • The number ‘one’ should be reinvented and restored as the root of the number line.
  • Zero should be regarded as neither a real or imaginary number, but rather an imaginary absence of all number.
  • The Big Bang singularity should be reinterpreted to reflect this new understanding of 1 and 0, beginnings and endings.

Here is the main insight: Since the difference between a difference (1) and indifference (no difference = 0) is in fact a difference, the two concepts are not perfectly symmetrical negations of each other, but rather, indifference, like nonsense or disorder, is a qualifier of difference. Zero is just 1 minus itself. In a universe of just the concept of 1 and subtraction, 1 would have to reproduce itself once in order to have another one to subtract, and then reproduce itself once more in order to carry out the subtraction. One cannot disappear, and zero cannot generate any numbers or operations. They don’t cancel each other out, they nest within each other in strange loops.

In this way, as I have posted before, the Big Bang must never be considered an explosion in space at some particular moment in the past, but rather it is the frame of all events, and all spaces. We are within the Big Bang, which was not a 1 emerging from 0, not a Universe from Nothing, but the opposite. What I call the Big Diffraction is “The Universe Within Everything”. The whole of physics can be seen as pieces to the puzzle which is getting more piece-ier and peace-ier as it goes on. The whole of mathematics can be seen as taking place within the number one, transformed, non-Euclidean style into the Absolute set of all sets. One is not an object, it is a primordial language of experience, of sense and sense making – a singularity not only of quantity, but of ontological-psychophysical gender.

But wait. Sense is not just a matter of being and knowing, it is also a matter of sensing and thinking, of comparing. It does not resolve the Material and the Experiential as being ‘the same thing’, it resolves them as both being equal to the same thing (1) in the opposite way. The Big Bang is not just 1, it is more like “=1”. This is a more primordial opposite even than being and nothingness, since nothingness can only be imagined by something. The relation of = to 1 is as opposite of that of 1 and 0 but more subtle. Just look at the characters. Parallel horizontal lines compared with an arrow-like stroke of singular effort. I guess I’m getting too into this, but whatever, consider it a piece of Suprematist art. It’s a before and after, an open canal, and an erect figure. An invitation and an expression. There’s a whole philosophy lurking in there just in the shapes of arithmetic symbols. Hmm.

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