Posts Tagged ‘mathematics’

On 0.999… being equal to 1

June 18, 2020 1 comment

We can revise our understanding of numbers such that each number is not a noun but a verb. The numerical value comes not from the digitization, which is only a snapshot representation, but from the act of counting. In other words, quantitative value may ultimately require a direction of flow. In this case, the “…” refers to a positive direction of flow that is unbounded so that it must be considered identical to the value that it reaches by its own definition.

All numbers are parts of a continuum and therefore always ‘almost’ the previous and successor numbers just as orange is defined as the color between almost red and almost yellow.
One interesting possibility from making this fundamental shift is that zero could be understood as an imaginary number between two real numbers “-0.000…1” and “0.000…1”.

Once we have that, we no longer need to see numbers as digital entities added on top of nothingness, but as diffraction bands breaking the continuity of a fluid spectrum of ‘everythingness’. Numbers do not connect to form wholes, they break wholes into parts. This has enormous consequences for our understanding of existence, consciousness, and AI.…


April 11, 2014 2 comments

The question that I pose is whether 1+1=2 because it makes sense, or whether our minds make sense because truths like 1+1=2 exist independently of all experience.

If it is the latter, then 1+1=2 stands in for a fundamental set of rules and relations for which consciousness serves to glorify, either accidentally or inevitably.

If it is the former, then that which ‘makes sense’ stands in for a perceptual acquaintance with qualities of undeniable coherence.

It is significant to notice that when we get down to elementary statements such as 1+1=2, we have slipped beneath the realm of logic and numbers without even realizing it. To say that one can be ‘added’ to one and that they are now equal to a group of two is entirely a matter of naming perceptions. There is no real arithmetic going on, we are saying only that when something is to be considered individually we call its individuality “one”, and when we want to consider the presence of one as being adjacent to one other, we call that adjacency “two”. The underlying properties which are being named are conceptually abstracted perceptions. There is no actual “information” named one or two, rather there is a language through which we generalize stereotypical features of our perception – particularly visual and tangible perception. Trying to apply mathematical models to perceptions like flavors and odors is less ‘informative’. They don’t really add up to be enumerable flavors as much as they involve us in a sensory experience in which flavors are both merged and independent.

Lemon + Lime does not necessarily equal two flavors, but can be just as easily thought of as Lemon-lime. Either lemon or lime could be broken down each into multiple flavors including sweet, sour, and citrus, but there remains an idiosyncratic note as well which identifies lemon as one flavor and lime as a different single flavor. Even if we isolate the compounds associated with these flavors, or synthesize artificial compounds with entirely different molecular profiles, there is a huge variation in our perception of any ‘one’ flavor. Lime jelly bean flavor is not the same as key lime pie flavor, yet in another sense, the similarity is self-evident, especially once we give it the name of ‘lime’. It is not a name that is arrived at through a computation or reasoning. Like ‘one’ and ‘equals’, lime is a subjective experience which we can point to but cannot define through a mathematical function.

Does it make more sense, given that the axioms of mathematics as well as physics are defined by subjective expectations (about objective conditions), that we should rule out the idea that all axioms are intrinsically perceptual? We might also ask, if mathematics and information were truly axiomatic, would it be possible to make errors? If our entire conscious experience were made of trillions of precise mathematical reflexes, why is the subject of mathematics even necessary to teach? Wouldn’t it make more sense that we would be able to perform comparatively simple algebras more easily than we can identify whether the flavor of a lime is natural or artificial?

Syzygy Integrals and Other Neoquantisms

February 26, 2014 Leave a comment

Syzygy Integral

Syzygy Integral with labels

When applying the syzygy integral to a sense modality such as vision, the Δæ would refer to the difference in the microphenomenal qualities, such as pixel hue, saturation, value, or contrast/edge detection, etc.. the entire palette of what I would call entopic or generic visual encounters. As shown in optical illusions, these elemental graphic features depend on their surrounding context, and two pixels or shaded regions which are measured to be optically identical can be perceived quite differently.

For this reason and others, I suggest that the fundamental nature of all phenomena is only definable in terms of specific properties, but of a pseudo-specific quorum of detectable differences. It looks like a lighter grey on the bottom because of the adjacent contrasts, and it is my conjecture that this kind of pseudo-specificity is at the heart of all measurements, particularly those which we have used to define subatomic particles.

On the top of the integral, the Æ would refer to an entirely different, top-down mode of visual perception. Instead of a delta (Δ) to stand for a the difference of generic micro-phenomenal qualia, the nabla symbol () is used to stand for a divergence from a larger perceptual context. This relates to the binding problem, i.e., when we see two dogs walk behind the same fence, we do not perceive them as becoming the same dog – the narrative continuity does for our overall understanding what the ‘illusory’ plasticity does on a microphenomenal level. To see the ) as a smile in the emoticon  : – )  requires both a low level fudging of pixels into a curve, as well as the ability for our expectation of a face to be projected from the top down. The emoticon is a minimalist example, but a better example would be something like this:

Terms like pareidolia, apophenia, simulacra, and eidetic hallucination all have in common this potential to misread a more proprietary, macrophenomenal text on top of a relatively generic, microphenomenal context.

What the syzygy integral is supposed to model is that any given sense modality is a special kind of integration between top-down or holotrophic orientation and bottom up, entropic orientation. In the case of visual sense, the top-down images are encountered like those in an Rorschach inkblot, as endless wells of imaginative psychosexual association. The personal range of the psyche is here encountering influences from the super-personal range of the overall presence of this moment in relation to their lives, and their lives in relation to eternity.

The bottom-up ‘entopic’ confabulation (entopic hallucinations are those which are geometric designs, etc as opposed to eidetic hallucinations which are images such as specific faces) is where the personal psyche encounters the sub-personal influence of neurological, biological, and chemical events as it impinges on them visually. An entopic hallucination presumably maps much more directly to neurochemical patterns in the visual cortex, whereas the eidetic, storytelling hallucinations would be much more obscure and proprietary. A hallucination of Darth Vader or Dick Cheney might be hard to tell apart from looking at an fMRI, but it should not be so difficult to get a fix on zig zag patterns vs concentric circles, etc.

The syzygy integral of vision then would be this continuum between the sub-phenomenal adhesive that holds the graphic canvas together and the cohesive that renders the meta-phenomenal meanings and figures phenomenally visible. It’s not an ordinary integral, since it has an encircled triple bar in the center, which denotes a participatory intent (motive effect), and an aesthetic contour (sense affect). The term syzygy, an old favorite of mine (its a real word), refers to a union of opposites, either figuratively as in yin-yang, or literally as in an solar eclipse where the Moon is opposite to the Sun behind the Earth.

In the syzygy integral for vision, the vast sweep of possible interpretations from the meta to the micro level is interrupted by the inflection point of the moment as it is localized from eternity (the absolute). That which is seen had been both filtered from above and built up from below, but the visual encounter is defined even in opposition to that. The seeing is not the seen. All visual forms are opposed to an equally rich continuum of possible ways to appreciate those forms and images. The syzygy integral is not just a map of what there is ‘there and then’ but the entire domain of what each and every there and then still means ‘here and now’.

As the syzygy integral can be used to describe vision (vision  = the participatory integration of graphic differences and imaginative likeness) or sound (sound = the participatory integration of phonic differences and psychoacoustic likeness), so too should it be able to describe the character of all phenomena. The underlying formula (Grand syzygy ingegral) uses the * asterisk and # pound to denote the limit of infinite figurative unity and the limit of literal, finite granularity respectively. In this case, the encircled triple bar refers to the Primoridal Identity Pansensitivity, from which all other syzygies are diffracted.

Grand Syzygy Integral

The syzygy integral without the contour circle I am calling the information integral.

Information Integral

Unlike the syzygy integral, which defines every piece of information as an aesthetic encounter or re-acquaintance, the information integral refers only to the skeletal functionality of sense. Locally we may experience novel encounters or acquaintances, but some would argue that all experiences can only be re-acquaintances from the absolute perspective. I think that it may make the most sense to think of even that either-or condition as just another superimposed quality of the absolute. Awareness is infinitely novel, infinitely repeating, and paradoxically non-paradoxical. It is only the disorientation of locality which provides orientation.

The information integral strips away all of the mystical trappings – the supertext and subtext contours, and refers instead to the conventional concepts of information theory. Here, the triple bar is still a participant and intentional arbiter of interpretation between signal and noise, but without the aesthetic complication. This is the standard view of information processing as a functional exercise, only with the additional acknowledgement of a core superposition of telic intention and ontic unintention, absolute improbability and immaculate reliability.

Attack and Redemption of Computational Theory of Mind

February 23, 2014 Leave a comment
It is my claim that CTM has overlooked the necessity to describe the method, mechanism, or arithmetic principle by which computations are encountered.My hypothesis, drawn from both direct human experience as well as experience with technological devices, is that “everything which is counted must first be encountered”. Extending this dictum, I propose that

  •     1. There is nothing at all which cannot be reduced to an encounter*, and that encounters can be presented directly (phenomenally) or re-presented indirectly (physically or semiotically).
  •     2. That the nature of encounters can be described as aesthetic re-acquaintance, nested sensory-motive participation, or simply sense.**
  •     3. In consideration of 1, sense is understood in all cases to be pre-mechanical, pre-arithmetic, and inescapably fundamental.

My challenge then, is for CTM to provide a functional account of how numbers encounter each other, and how they came to be separated from the whole of arithmetic truth in the first place. We know that an actual machine must encounter data through physical input to a hardware substrate, but how does an ideal machine (e.g. Turing machine, Universal Machine or program) encounter data? How does it insulate itself from data which is not relevant to the machine?

Failing a satisfactory explanation of the fundamental mechanism behind computation, I conclude that:

  • 4. The logic which compels us to seek a computational or mechanical theory of mind is rooted in an expectation of functional necessity.
  • 5. This logic is directly contradicted by the absence of critical inquiry to the mechanisms which provide arithmetic function.
  • 6. CTM should be understood to be compromised by petito principii fallacy, as it begs its own question by feigning to explain macro level mental phenomena through brute inflation of its own micro level mental phenomena which is overlooked entirely within CTM.
  • 7. In consideration of 1-6, it must be seen that CTM is invalid, and should possibly be replaced by an approach which addresses the fallacy directly.

The next three points have to do with my own hypothesis (Multisense Realism), submitted here only for those who might ne interested.

  • 8. PIP (Primordial Identity Pansensitivity) offers a trans-theoretical explanation in which the capacity for sense encounters is the sole axiom.

PIP is the conjecture that sensitivity is the sole capability that is required for all phenomena. If we wanted to conceptualize a ‘unit’ of this pansensitivity, I suggest the aforementioned terms ‘aesthetic encounter’, ‘sensory-motive participation’, ‘re-acquaintance’, etc, or any other neologism which suggests a pre-monadic generator of pre-self and self-like perspectives.

My conjecture is that self is a type of symmetry within sense. There is no self except for self vs not-self. What is distributed by pansensitivity is not solipsism, but opportunities to modulate self-like symmetries. Selfhood is a particular form of sense distribution in which the symmetry between the absolute and the conditional is recapitulated twice. From the absolute perspective, the self becomes a branch toward isolated locality, while from the conditioned perspective, the self (the condition) stands in for the absolute.

  • 9. CTM can be rehabilitated, and all of its mathematical science can be redeemed by translating into PIP terms, which amounts to reversing the foundations of number theory so that they are sense-subordinate.

Here I am talking about turning the assumptions of mathematical theory on its head. Platonic forms, for instance, would be conceived of as superficial emblems of sense rather than profound and perfect absolutes. Sacred geometry, while imbued with numinous meaning locally to a self (as it reflects the symmetry to which it owes its elaboration), but from the absolute perspective, sacred geometry is akin to a test pattern – crystallized reflections of aesthetic depth, but containing no depths themselves.

  • 10. This effectively renders CTM a theory of mind-like simulation, rather than simulating macro level minds, however, mind-simulation proceeds from PIP as a perfectly viable cosmological inquiry, albeit from an impersonal, theoretical platform of sense.

All that I mean here is that because CTM trades in symbols of reflected sense-making rather than sense encounters, it doesn’t ever have a chance of capturing the important features of consciousness. It can, however, capture important features of how consciousness is distributed.

EDIT: Details added for clarity –

*Encounter can be thought of as ‘stimuli’, but need not include any information. It can be thought of as ‘qualia’ but it need not include any subject or object. The intention here is to reduce all phenomena to its absolute minimum – an opportunity for modes and motives for discernment to arise.

**This sounds jargony, for sure, but I’m inventing a precise vocabulary here, so as to avoid being misconstrued as a standard argument for what might be called ‘pseudo-subtance idealism’ (everything is made of energy, love, vibration, fields, etc) or information panpsychism (conscious experiences are produced by complex systems, functional states, etc).

MSR Legend

July 19, 2013 1 comment


A first second draft of a Rosetta Stone for MSR terms, including the debut of new quasi-alchemical code.

Second draft 7/22/13, added Solitrophy and updated the formulas of time and space to clarify.

Our Mathematical Universe

June 24, 2013 2 comments

Our Mathematical Universe

I have not read Max Tegmark’s new book, but the argument that mathematics is the ultimate reality of the universe is a strong one that has been around for a long time. I would agree that mathematics is an irreplaceable tool for understanding the universe, and for understanding knowledge, but mathematics alone is not sufficient to derive the actual universe which we experience.

In my view, mathematics can only be an emergent property of representation and therefore ephiphenomenal. The underlying (and overarching) phenomena of presence or presentation is fundamentally aesthetic and consists of sensory-motor experiences. This is not a biocentric view as inorganic matter is also, by my understanding, a tokenization of aesthetic experiences. The universe is a significance-building machine, where significance is the temporal super-saturation and transcendence of sensory qualities.

As far as comparing mathematics to computation, mathematics seems to be a broader category which would embrace ideas which computation cannot, such as irrational numbers and geometric forms. While computation can be used to drive a sensory experience in which geometric forms are inscribed visually or sculpted tangibly, those outputs are irrelevant to the computation itself and are desirable to us purely for aesthetic reasons.

Computation is, however, closer to empirical realism than other kinds of mathematics, since it is rooted in digital interactions which can be reproduced and re-presented in any solid-body/persistent-position form-function. If there is no discrete fundamental unit which is subject to reliable inspection (which is an experiential and aesthetic property that is generally overlooked ) then computation cannot be initiated or preserved.

I get into this a bit here:

Mathematics requires a mind and a brain while computation requires only a brain substitute. By this I mean that the sense of computation is a low level sensory-motor interaction through which higher level interactions can be transported from one location to another. This transportation offers the opportunity for reconstruction only if the receiver has the appropriate frame of reference to imitate the sender’s intents. We use a computer to listen to music or watch a video, but in the absence of human receivers, there would only product would be disconnected instants of acoustic or optical activity.

Mathematics similarly owes its universality to its exploitation of a low level ‘common sense’ which depends on similarly overlooked assumptions about the validity of conceptual realism. Mathematics depends on sanity in the intellectual and logical sense. It presumes an aesthetic minimalism. Where computation can be more clearly seen to depend on concrete mechanisms of read/write/erase, storage, pattern recognition, loops, etc., mathematics seeks a more anesthetic representation – as Baudrillard might have said, a simulacra: A representation without any presentation. In my understanding, mathematics can be thought of as an ultimate reality only in the sense that all of our intellectual models of reality can be rendered in mathematical ways.

Unfortunately, most people conflate the idea of reality with the experience of it, and have developed a misplaced authority for “information” as the progenitor of physics and awareness. This is, in my view, almost correct, but actually upside down as information can only ride on top of an aesthetic exchange of experiences, which involves public to private extractions of significance and private to public export of entropy. Information, by itself, has never done anything. No byte of data will ever feel anything, be anything, want anything, go anywhere, etc. Mathematics deals in figures, which have no form or function but represent forms and functions. What figures cannot represent is presence itself. There is no substitute for experience, and that is why it is experience which is the ultimate reality – the absolute and authentic substrate of the universe is a unique agenda of aesthetics, not a generic consequence of configured figures.

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