04/05/2018 - I'm surprised by how long the nature of intelligence has interested me! When I originally wrote this, my only relevant background was a semester long high school psychology course, taken a couple years prior and which I got a C in. Reading through this again, I see that I'd worked out the essence of or got quite close to: the symbol binding problem, advantages of being embodied, the utility of a generative model for counter-factual reasoning, the role of compression, reward learning and most surprising to me of all, as I do not think it obvious: that consciousness might not be important! I remember going back and forth on that point. Although my goal had been to work out the meaning of math, I more worked out what kind of agent can appreciate math? Which compacts to: what does it take to be a true intelligence? The nature of math is related to the world being predictable.
I see also that I'd vastly underestimated the enormity of the task I had set myself (invent new math? hah!). Inventing new math is working backwards from some structure you want and coming up with a consistent set of rules which connects smoothly with an existing background of rules. I very much doubt I could invent new math—it requires extensive knowledge and skill—but I've since tried to connect certain ideas. In particular, information geometry from one direction and adjoint functors from another (what are we doing when we are learning?) but nothing coherent even as yet.
Friday, 09 December 2005 - It can be shown that the reality which we experience is one of many possible realities. However, since it is also the case that each reality is in fact but a facet of one true reality then it becomes evident that the term reality is grossly unsuitable for the terming of our perceptual experiences.
We say that it is an amazing fact that the mathematics models reality at all and in fact in not only doing so, but with such remarkable facility as well. However, I will show that this is not at all a surprising result since it is obvious that such a phenomenon should exist. Do not our eyes present to us an image or representation of the world?
Sunday, 11 December 2005 - It should be evident that the reality which we experience, that which we see, is but an abstraction, a slimming down of what actually is. A simple example is one from vision, the image of the world, the physical image that is actually projected of the world from the external reality is actually a highly distorted, inverted, curved and 2 dimensional projection of what actually is. The images below attempt to capture the general idea of the retinal image although they showcase depth and colour constancy and are likely not curved in the same manner as the image found on the retina. Note, the shape of the retina is required to know this for sure.
It is only because of the visual processing abilities of the visual cortex that we can perceive such things as depth, sharp corners and non-inverted images in life. In fact, there was an experiment carried out in which subjects were told to wear vision inverting glasses, a period of a few weeks after which they adjusted and were actually able to reinvert the upside down images and actually see right side up again! A testament to the subjectivity of the sensory experience, if ever there was one. Even sounds, touches and smells are due to stimulations of sensory organs which are then reinterpreted by the special processing units of the brain into the totality of our perceptual experience. It is impossible to show that what we experience is what actually is. For some information of the minds susceptibility to illusions see the site http://psylux.psych.tu-dresden.de/i1/kaw/diverses%20Material/www.illusionworks.com/html/ames_room.html and the references contained within.
Consider touch then, in the scale of the atomic the distance between the object and the hand is great, separated [proportionally] by many miles. The exchange of forces of electromagnetic origin and the interaction of fields (whose true nature we can only speculate on and perceive mathematically) is felt as tension in the muscles and skin. The tactile receptors of the fingers translate various motions guided by the nature (energies, momentums, field strengths etc.) of the interactions into feelings of texture and temperature. These abstract concepts that are very divided from our everyday experiences are translated by the brain into something that is obviously a cup. The essence here is that the reality which we perceive is necessarily apart from that which is and thus is an abstraction. Reality as seen by our senses is an abstraction.
So also in a similar way can mathematics be called an abstracting of reality. Just as is the case of our perceptual experience abstracting out, stripping down, reducing and converting the external stimuli and experiences into graspable, manageable and manipulable precepts; so also can the same be said to be done in the world of mathematics. Save that here, the manipulated entities are in fact the manipulations of the precepts themselves. In fact, it can be shown that mathematics is simply another way of seeing the world that is passed to us by our senses but because the processing unit is more complex – the cerebral cortex that is – it is necessary that the perceptual processing and experience as controlled by the cerebral cortex be richer and more complex and also less assuming (although arguably still fairly presumptive in its operation) than those brought to by the simpler units. The sensory input can be considered to be further processed beyond that done by the simpler units and even arguably, in parallel. But I shall leave that to those more able to discuss the finer details of.
Thus mathematics is another way of seeing the world so to speak. Therefore, it should come as no surprise that it represents the world so well. It does an even better job of translating our sensory experiences than the visual cortex and other similar. It is no surprise that it should describe the world “as well as it does”, there is no magic or difficult concept involved here.
Friday, 23 December 2005 – It has been some time since I have written an update, I have been a bit lazy in putting to magnetized oriented ions my thoughts. The next logical order in the progression of the cortex-“ial” line of development is the cerebral. In order that actual “meaning” and sense be attributable to the world, the external entities must further be processed in a manner such that the natural outgrowth be another layer or level of processing that is able to freely manipulate and make constructs and or symbolisms of all the perceptual data that is received by the sensory organs. We see the world and represent it within (and make it as without) with a special structure that is our take or twist on reality. Our limited senses and brain reconstruct internalised representations of reality based on what minutely small subset of reality that is experienced. As advanced creatures we can further give meaning to these visual and other constructs to make them cerebral by building further upon the sensory data by giving them symbolic attributes.
Monday, 26 December 2005 - Just as the visual system can add depth to a flat 2 dimensional picture, so also is a similar thing done in giving meaning to that which is not apparently meaningful. Thus we see that the plethora of stimuli and perceptions experienced must be given a symbolic attribution in order that they obtain meaning to us. The brain must maintain a semi flexible rigid structure in its attribution process. The significance of this is at the moment unexplorable by me. I shall return to this fact at some later date. I also must learn on schemas and grouping mechanisms of Gestalt theory to properly elucidate certain ideas I have to do with the brains ability and manner of making sense of reality by grouping and ordering.
Nonetheless, I can speak on some general features required to make sense of reality. We know that the visual cortex uses different grouping methods and adds attributes such as depth and colour constancy to our perceptions, things which were not a part of the original reality, to make sense of images. It is thus not so far fetched to say that the cerebral cortex adds attributes of a symbolic nature to attach meaning that is not necessarily so. Although we must assume that the meanings attached are not completely arbitrary and make some sort of sense. In order that it be possible that meaning be applicable to perception, such properties as separation, enumerability, grouping/ordering and countability must exist. It is worth noting that under grouping we find that we can more or less relegate most properties as colour, size and shape etc. Here, a conjecture is made that if a manner of symbolism exists (which is a natural outgrowth for a sense to be made of reality) then it is necessary that a method for their manipulation exists. This is called thought and the externalised expression of which is language. Subset of language and due also to the existence of symbolisms and their manipulations is mathematics.
As recap, we state that just as the visual experience is one whose construction is learned as one gains in experience so also is one’s meaningful experience; a set of interlinked symbolisms are attached to a set of diverse sensory inputs and signals to generate our subjective reality. Grouping mechanisms based on the similarities and apparent uniformity of the inputs are used to order the world into colours, furniture, people etc. I argue that large quantities of energetic motion (with the point of comparison being of the molecular scale) is required for anything to be experienced by humans in this world. To identify a set of energies and interactions one sees that they are in motion relative to some background. We see those energies, which move and seem to have constraints in their separation and call them masses. We notice tables, which might seem very unenergetic on the surface but are in fact reacting at a very large level with photons and phonons and other such. Strong fields (that is anything which creates any form of resistance for the human) and not the exclusion principle, is principal to our tactile experience of reality.
Since symbolic meaning is due to a set of interlinked constructs whose generation is due to experience and is relative, it is necessary that not only is our experience of reality subjective and limited, it is limited also to things which can be compared and stated relative to one another. The last is a limit that it is hoped can be consciously overcame once things are cast in a different perspective. Indeed the general idea is that the manipulation of perception, jokingly called takes (as in my Take on that) is the next level of abstraction, which allows one to experience things at a higher level than before. Allowing perhaps, that more and deeper questions be answered by shedding a new and brighter light unto things that are already there but putting to light things that were once in shade. Or at least, that is the hope.
For meaning to exist there must be a concept of memory, construction, significance and the ability to divorce, separate, become more independent. The nature of memory is well studied and I shall not go into it here beyond stating that having a set of linked, grouped and ordered set of various sets of experienced stimuli for later manipulation is pivotal for an entity’s intellectual ability. Of significance one might suppose that a method for attachment to or ability of having (dis)preferences for a set of stimuli for which an ordering could possibly be a representation of tree or Jack exists. Concurrently, we might assume that the concept of hurt might lead to one of satisfaction to happiness and its opposite and then to full fledged emotional ability developing. Why, one might even make the tentative suggestion that in order to do mathematics most proficiently, one must be an emotionally complex entity. One imagines that the ability to create without external stimuli, to divorce entirely from the external sensory inputs to create a representational what if world would prove most advantageous to a creature’s survival. It could be that a set of these representations, created from a set of interlinked stored stimuli and various past experiences for which there existed a strong positive emotion. Such a creature could use sets with different configurations and choose one which resonates strongest with some hidden significance attribution process. This would allow one to ”do” without doing, a most useful ability. The ability to dream shows that the brain is capable of independent construction or manipulations of stimuli regardless of external influences to construct a sensory experience independent of, but strongly based on the external world. A construction within the construction. One necessarily more flawed than the first, indeed one might wonder if the disparity between imaginings and reality might shed a glimpse into that between reconstructed reality and actual reality. Perhaps they are proportional? A look into the constraints… but I digress and get ahead of myself.
The key is in the ability to divorce into higher and higher levels of independence. For example, one might not require spacial reconstructions but simply un-solidified concepts of where. The gained flexibility allows one to deal with and reapply solutions into strange situations most effectively. Indeed, if the manipulations focused less on stimuli but rather on the totality of the stored experiences – the manipulations of which create the strongest resonances or feelings of satisfaction – one might argue that meaning emerges within a hop, a skip and a step. But then, where does consciousness enter into this and is it required to do mathematics? This I cannot answer but there is a line of reason which may be explored that may shed some light into this. Among the other mentioned processes, memory is key to one’s ability in mental operations. One might also assume that with manipulations of mental processes - especially combinations of those which result in meaning - that awareness might occur. Indeed, one might presume a line of development which encompasses such phenomena as self preservation and avoidance behaviours to conceptions of satisfaction, contentment and preference, to delight and happiness in one’s accomplishments. Properly acquiring a trickily located piece of food being an example perhaps.
With such complicated processes and their manipulation, it is not unlikely that consciousness might occur. Nonetheless, the placing or development of consciousness is not required for an accurate theory of the underlying nature of the mind’s ability with mathematics. Awareness and an ability to appreciate, simply, dictate the ability to do mathematics that is personally meaningful. The ability to divorce or abstract and assign significance to resonant ideas (implying ability for preferences and internal resonance with developed conceptual structures) as well as to construct and manipulate conceptual sets, which are operated on beyond the bounds of the existent concept sets which are most directly related to reality are all that are requisite for creative mathematics to be done. Where or how and if consciousness plays a part or is even required at all is unknown to me. There is however the supposition that if all the afore mentioned requirements for meaningful and creative mathematics are met, then it is highly probable that a conscious mind exists.
Friday, 06 January 2006 – As stated, the ability to disconnect such that a language exists must imply that the entity is able to manipulate its experiences and present them in a generic form (condensable to mediums such as text, music etc.) and mapped unto any capable device or mind. Ability for such abstraction allows that such a mind can do mathematics. There exist for man a preference for resonant sounds and strong beats; one can only conjecture that such is an indication of the underlying organizational structure of the human mind. The creation of music involves the manipulations of our perceptions of sound and our experience of it, specifically the combinations most preferable or resonant to us. This fact is something worth exploring, music points to some internal mental logical structure and hence a look into an ability to do math among others. The same statements can as well be said about art. Art is a direct projection of our internal visual experience, indeed, the so called abstract pieces (despite my dislike for them) are projections of one level of abstraction beyond the visual. They are the presentation of the manipulation of emotional and preferential precepts presented in a form and structure which itself is an indication into the structure of our mental working. In fact I say here that the interpreting of abstract art is silly since it is a personalized process. The work of the artist is mapped unto the map of the individual, where the projection (of the artists internal experience) is the art, since the minds are not isomorphic and the projection is many magnitudes of dimensions less than the artists experience, the reconstruction in the brain is a reflection of the self and differences in one’s mental preferences. I note also that the artist need not have a well developed meaning, the work may simply be a direct projection and one not done necessarily with special meaning beyond expression. One level of abstraction beyond art (although not necessarily abstract art and highly complex works of music) is that of written works. Although seeming simpler, due to our constant use of it, language constructs involve the manipulations of the very symbolisms which have been attached to give meaning and thus the entire sensual experience. With poetry one finds the focus is placed on audio resonance, with creative works, the invocation of complex visuals and finally non-fiction the ordering and reordering of our schemas or ordering process itself.
With that, we return to the example of kinematics. In addition to an internalised visual representation of the world, there exists a physical one, one whose laws are noted and ordered to work with one’s perceptual experience. The shooting of a basketball involves intense calculations, precision and control of various things as trajectories, acceleration, forces and impulses – all done after multiple transformations of spaces. Thus a brain’s innate ability to perform advanced calculations innately is displayed. If there exists a language and a reasoning process with, implying meaning exists in the world and if this language has been developed to an even higher level of abstraction where the language has been given a symbolism that allows a projection unto external media then we find that the development of a mathematics is almost inevitable. Err, writing exists.
With a culture capable of creating artistic works which are harmonious to one’s conception of beauty and an ability to disconnect and relate to such a level that writing exists then it is likely that a curiosity for the environment might develop. With consciousness or awareness an inquisition into why events occur to the self might be undertaken and with the ability to communicate meaningfully an interest into what else might be capable of such. Nonetheless the point of focus is in an interest in the environment, perhaps motivated by all this and also a need to manipulate to better solve problems. An arithmetic notion is inevitably the first work of a culture capable of writing. The concept of language is extended to the symbolisms which pertain to one’s ordering of the environment.
If music, art etc. can be described as the manipulation and ordering of one’s experience of elements of the set of perceived sounds (or visuals, with periodicity and harmonics being a frequent theme) then mathematics would be the manipulation of one’s representation of the physical behaviour of reality and the very underlying schema that allow for understanding itself. The physical experience of reality is made bare for the mind, this physical understanding underpins our very experience of reality. And here finally it becomes possible to state the thesis of this work. The totality of experience is a synthesis of the sum of our perceptual experiences mapped unto a internalised physical representation of which the relating structure (from physical to visual to audible etc.) is our conceptual [logical] orderings. To do mathematics beyond geometry one must also be able to manipulate the abstract logical ordering. This relational structure is more directly connected to the external reality as it is the manipulation of said precepts and thus speaks more on reality than our visual experience of it. Indeed after an arithmetic, a geometry is almost always developed, I argue, and that although a geometry is often strongly biased towards a culture (with the strong emphasis being the visual aspect of the physical) an algebraic method is often most general. Thus a geometry is biased but an algebra or algebraic geometry very much less so, the same bias again for a topology but not an algebraic topology etc. Indeed the task will be to develop a method to describe these concepts of mind and reducing all mathematical concepts to these. The first task will be done with crisp sets and then again, more accurately with the concept of subdefinite or vague sets. Mathematics is a highly abstract conceptual Take or Hue (similar to a space) that places for manipulation one’s experience of reality. It is the manipulation of the constituents of our perceptual experience and also the underlying structural organizations of a relating structure whose patterns must be some harmonic of an external reality. That which we feel directly is a subset of such a concept. Thus mathematics is an ultimate manner of perception –one capable of making statements on itself. Mathematics is not invented or discovered, it is experienced, it is the human experience revealed in its totality.
And so I have developed these ideas to a state that which I am satisfied with. I can now return to these idea and give a far more coherent and mathematical treatment to lay out a new mathematics I think. Worth exploring is the breaking down of different mathematical concepts to the constructs of this theory. As a first year university math student I am struggling to the droll nature of exercises, I see this as a fun synthesis that should make interesting many things. I will start with limits. :D