Chapter 7: A Mathematics of the Mind

 

Part (1): The statement that ‘the immortality of Descartes was assured when he enunciated his Cogito ergo sum (I think, therefore I am)’, (p 511, The Material World, Rodney Cotterill) raises questions because ‘I think, therefore I am’ reads more like advertising copy than a major theory. Further, it may have some alignment with quantum mechanics, but doesn’t seem to predict anything or do anything!

Conversely, the reverse ‘I am, therefore I think’ sounds nonsensical, but is surprisingly rich under investigation.

 

postscript: the statement is actually a Half-truth, and as shown later, when we think, we create a reality because we have to measure something and we must measure something because we evolved to place a prey or predator in a space-time situation gauging safety for the organism and ‘I think, therefore I am’ reflects this, but ‘I am, therefore I think’ is a consequence of the structure of the brain, which has special quantum mechanical effects built into its structure to produce creativity, thinking etc. and uses the logic upon which the universe is built.

 

‘Afferent information originates at the various sense organs, which are divided into 5 types: touch, hearing, taste, smell and sight. To this traditional classification we could add balance.’ (P 474) The link between the 5 senses and ‘balance’ is organization, which is the purpose of the brain.

 

In other words, the five senses and an ‘organizer’ controls everything we are or do, in a ‘system’ sense. Even further, within the human brain, the motor cortex and the somatic sensory cortex lie side by side and ‘there is actually a spatial mapping of the body’s components’. (p 481)

 

This same organization of the brain is reflected in the ‘lower orders’ of mammals and chordates, even as the cerebrum becomes smaller relative to the size of the animal. A diagram (p 512) shows the similarity of the brains and the gradation of the cerebrum from human to animals to birds to snakes to frogs and to fish.

 

‘Even humans follow ALL the ‘animal, vegetable or mineral’ of the world in a ‘computer-like’ program called ‘life’ in ‘iterations’ of continuous ‘survival of the fittest’ in the Darwinian sense. We even contain within our cells other, ‘complete organisms’ to help us compete better, called Mitochondria ‘which do show a striking resemblance to the bacterium Paracoccus denitrificans’ (p372).

 

Even the simplest elements and chemicals ‘think’, in a ‘system’ sense, driven through quantum mechanical patterns to find the lowest energy states, such as in the molecule H2 or O2 etc, similarly there is a probability of finding an electron in a ‘classically impossible’ position called the tunnel effect etc. Thus the phrase, ‘I am, therefore I think’ seems to be consistent with the world around us, so, as to formulating a prediction, one must ‘construct a bottom-up picture of events. Only then can one appreciate the remarkable fact that complexity can be the result of basic processes that are relatively simple’ (p 405). The prediction is, that everything that, to us, ‘is’ is bound by a set of constraints imposed by our senses, and to ‘appreciate’ ‘new things’ we need to ‘expand our senses’ by using new methods, such as microscopes, telescopes, echo location in bats etc.

 

postscript: we expand our senses and create reality by measuring or cataloguing that that we see. Whatever we see was indeterminate until we saw it and followed its own particular rules, which are, but are not, our rules of physics. We (all organisms) see the same reality because we all have the same architecture in the brain because we need to sense other organisms. An organism that cannot sense another organism is quickly eliminated by predation and this is heritable.

 

However, this is precisely what we have been doing for 3 billion years, and is the source of life’s success! Further, this is precisely what has been happening since the ‘Big Bang’.

 

Conclusion: The above derivation shows that some parts of Philosophy must be time or knowledge dependent, because Descartes’s statement has been accepted for hundreds of years! But, thinking about it, that is obvious, because time and knowledge change the way we interact within our society, and Philosophy is supposed to bring ‘awareness’ to society’s problems, based on the current state!

 

Part (2): Given that all things ‘think’ within their limitations, the question then becomes, what do they think about? I suggest that the vast majority of thoughts pertain to survival of the organism and those few thoughts that cannot be traced to having an end with survival as its ultimate aim, are irrelevant. Notice that I am including all organisms, molecules, elements and sub-atonic particles etc within ‘all things’. ‘Some physicists today are like animists in that they believe everything that exists is alive. It is a sense of our unity with a living universe, the feeling that we are all just parts of that greater life, which is basic to animism’ (Sharmanism, Shirley Nicholson, The Ancient Wisdom in Sharmanic Cultures, interview with Michael Harner) However, it seems to stretch the point that humans ‘think’ in the same way as a molecule, but it is true.

 

postscript: the above paragraph uses the word ‘think’ which is a Half-truth. Thinking by an organism creates a reality around the organism through its senses and (if they can be) are recognised by the organism. An inanimate thing ‘thinks’ by feeling the universe, such as its continually changing potential energy as it moves around the universe. The colour of a photon is indeterminate but changes until it is absorbed and its colour measured. The universe is probability space with a logical ‘dimension’ called the Half-truth and it needs space-time (which is our brain) to determine it.

 

Offspring can be generated and turned out to battle for a place in the world, such as turtles, bacteria etc, or they can be nurtured by the parent until they can fend for themselves. Then they are dependent on the environment and it’s food supply. Thus, every organism currently alive, due to survival of the fittest, is the best possible at that particular time given the food supply, local climatic conditions etc.

 

Humans are no exception, except that the Neolithic saw us change our environment much more quickly by developing farming. This change would have been relatively gradual over time, as we took other species with us in a progression from hunter gathering to ‘plant and gather’ to ‘living in one place and farming and gathering’ to true farming. The co-opted animals started with dogs, horses, cats etc and the easily propagated fruit plants such as grapes, walnuts, apricots etc. ‘Agriculture has always been an unnatural business’ (50 Genetics Ideas, Mark Henderson, p 128)

 

At this point our genes reflected the local conditions and were set by the local conditions through survival of the fittest. Our genes would change as our food supply changed, gradually. However, with the rise in farming and technology, which occurred at a ‘fast’ rate our genes were left behind. Obviously farming has been successful, but, we now face over-population, global warming and a host of medical and lifestyle problems. We owe our success to the rise in technology, but how do we control the technology that threatens our existence so that we maximize our gains? This is a problem that needs a consensus that a Philosophical Mathematics can give us, because it is based on a logic the everyone can accept. The following will attempt to answer this problem.

 

Part (3): ‘A number system is a method for handling the concept of ‘how many’‘. (50 Mathematical Ideas, Tony Crilly, p 8) This presupposes that each ‘thing’ counted is similar enough to be lumped together. This is very restrictive! What we need to do is ‘sum’ a set of conditions or patterns. This could be done by ‘summing’ a set of matrices with a huge number or elements, or by using a ‘wave function’. Both of these could effectively ‘access’ infinity.

 

It is easier to consider quantum mechanics. ‘Schrodinger showed that the variation of electron probability density is characterized by nodes. …… In three dimensions there are nodal surfaces at which the electron probability density is zero. One thing that can be stated immediately: because an electron must be located somewhere or other in space, there will be a nodal surface at infinity’. (The Material World, Rodney Cotterill, p 47)

 

Thus all matter (and energy, as E=mc2) have probability densities extending to infinity, and the ‘patterns’ are fundamentally similar. For example, in graphical form we need to ‘sum’ patterns A, B, C, D, E and F.

A

E

F                                                              =   ?

C

B                 D

 

These patterns are only ‘determined’ when someone recognizes them as patterns, otherwise they have no relevance, as is similar in quantum mechanics. Also, the Heisenberg ‘uncertainty principle’ ‘is the general problem that confronts any attempt at simultaneous measurement of two properties, such as energy and position, of particles having atomic dimensions’. (p 38) Putting the two factors of position and time into the context of patterns, this suggests that each pattern should contain the element of time and cause the equation, above, to add another dimension.

 

This aspect of time difference is crucial because our social problems started to escalate around 10,000 years ago and continue to the present, whilst our bodies (but not our minds, as will be seen later) are genetically mired at the former time. Thus, technology is ‘advancing’ quickly, whilst our bodies are still expecting the Palaeolithic lifestyle.

 

Mathematics treats patterns in a simplistic way by extracting an ‘essence’ that can be generalized and simplified into a ‘tool’. For example, the set of circles yields ‘pi’ and the set of right-angled triangles yields ‘square roots’. Further, ‘the mathematical message is that this limit, which mathematicians call e, is the amount $1 grows to if compounding takes place continuously’. (50 Mathematical Ideas, Tony Crilly, p25) Also, ’the prize for the most remarkable of all mathematics …

 

i x pi

e                       +     1   =   0

 

is a result attributed to Euler’. (p 27)

 

Further examples of patterns are the Fibonacci series from the ‘problem of rabbit generation: mature rabbit pairs generate young rabbit pairs each month’(p 44) which leads to the ‘golden ratio’, the cattle population problem …….. intermediate stage in the maturation process as cattle pairs progress from young pairs to immature pairs and then to mature pairs’. (p 47) This leads to the ‘supergolden ratio’. Both of these ratios can be found via the ‘golden rectangles’.(p 48-51)

 

Lastly, ‘Pascal’s triangle is famous in mathematics for its symmetry and hidden relationships’. (p 52) In particular, ‘if we substitute 1 for the odd numbers and 0 for the even numbers we get a representation which is the same pattern as the remarkable fractal known as the Sierpinski gasket’. (p 54) ‘The formula which generated the Mandelbrot set was simply x2+c’. (p 101)

 

The first formal attempt to solve general problems was called Philosophy, which through a plethora of subjects has blossomed in the world that we know today. ‘By the Renaissance mathematics had achieved a level of development far beyond the practical problem solving needs of finance, architecture, or engineering. As it freed itself of functional applications, the discipline grew in complexity and nuance and became a branch of knowledge with one foot in philosophy and one in science. (P98, Numbers, the Universal Language, Denis Guedj)

 

To restate the above, I would like to use Philosophy at the time of its ‘golden age’, 3000 years ago, when it was much ‘simpler’ than today, to solve ‘social’ problems that have developed over the last 10,000 years. To do this, we need to solve some sort of pattern equation that will indicate a set of ‘solutions’.

 

Part (4): Firstly, I will indicate the general solution of the equation and then give an example of its use in ‘solving’ a problem. Unlike traditional mathematics, which provides an answer that is the same each time the problem is looked at, there will be changes in the solution because of time dependent and possibly ‘chaotic’ influences, if for no other reason than that time has passed between solutions. In other words, traditional mathematics is time independent.

 

Not only is traditional mathematics time independent, it aims for a ‘unique’ solution. In certain dynamic systems there can be a set of ‘attractors’ to which the particular system tends depending on certain initial conditions. Each ‘attractor’ may be thought of as a ‘solution’ of some sort, that we wish to ‘balance’ with other ‘solutions’ for an ‘acceptable’ solution at some time and place. The mathematics that I am describing is navigating the space between the ‘solutions’ or ‘attractors’ according to an external set of rules that changes with time.

 

As in the case of dancing and poker machines in chapter 6, one attractor may be a dance hall and the other a casino, but there are a set of restraints such as noise, opening hours, licences etc. determines the ‘mix’ that is acceptable in a club situation and this changes with time and location.

 

The general solution:

 

(1) take any ‘situation’ to bring the problem into our ‘perspective’, because everything is linked. However, the closer to the problem, the easier the derivation, but the greater the chance of over-looking relationships.

 

(2) ‘disturb’ or ‘stimulate’ the patterns by using a simple ‘change’ formula composed of (a) the change ‘scenario’, (b) the change ‘agent’, (c) the change ‘director’, and (d) the ‘carrot/stick’.

 

Any change will be resisted, so the change scenario will (most probably) be resisted. Thus a change agent is needed with the authority to implement the change. But, similarly, the authority will resist the change. Hence the change director is the implementor ‘of last resort’ and if he fails to apply enough pressure, nothing will probably happen.

 

The carrot/stick is a set of situations that will force the change agent to effect the change, either by the change director providing a ‘carrot’ to promote the change, or a ‘stick’ to force the agent to act.

 

(3) The change director has to decide whether the possible changes (under consideration) satisfy ‘theory’ theory. This is simply that a ‘new’ theory has to explain all current manifestations and provide a prediction.

 

(4) These predictions (or set of predictions) are our solutions.

This ‘general solution’ is mathematical, but uses ‘Change theory’ from Business Studies and ‘Theory theory’ from science. It is very simple, but then consider Occam’s Razor and 3,000 million years of evolution honing the system to be the most efficient possible. “When we talk about mathematics,” [John von Neumann] wrote towards the end of his life, we may be discussing a secondary language, built on the primary language truly used by our central nervous system.” (The Shallows, Nicholas Carr p 176)

 

Part (5) The example:

 

I have to admit that the above doesn’t make much sense, but an example will show how simple it is! Firstly, the basic problem is to keep the patterns ‘separate’ and to this end, I have used several type-faces to make it easier, and this was done in chapter 6.

 

postscript: chapter 6 reads like a normal derivation, which it is, but it was written before this chapter. The postscript is to be read after the whole book has been read, so that these comments are relevant, but will only make sense when the book has been read. The comment is that whilst chapter 6 appears normal, the act of thinking, and all the other language words, thoughts, nouns, verbs etc. is the fifth dimension, which is the Half-truth and is the dimension of the universe that we use without acknowledging it.

 

Secondly, the example starts off with a remark, and ends with a surprising number of ‘solutions’ both to my problem but also shows the need for more government control of clubs. This effect is a ‘form’ of quantum computing.

 

Thirdly, the initial ‘problem’ is linked to a simple means of controlling poker machine use in the community, and a means of increasing the well-being of the public!

 

The ‘psychology’ behind the urge that some people have to put their life savings through the machines must be known to the makers of the machines, but it is definitely not known in the wider world! Andrew Wilkie used his balance of power in the Parliament to try to limit people’s losses and he did not succeed as he had hoped! The solution is obvious when it is pointed out! But, not obvious is the additional requirements that must be satisfied to produce that solution, and they may be none, one or more!

 

Lastly, the example shows how some clubs may be taking advantage of their members, in way that the ACT politicians should do something about.

 

 

 

Part (6): A quick over-view of the brain to show how its working relates to the mathematics above:

 

As mentioned above, natural selection has honed living organisms to be extremely ‘efficient’. Indeed, ‘perhaps the biggest surprise to emerge from this work has been the indication that the human genome comprises only about 30,000 different genes’ (Cotterill p 365).

 

Using ‘the Shallows’ by Nicholas Carr, ‘the chain of linked neurons form our mind’s true “vital paths”. Today, scientists sum up the essential dynamic of neuroplasticity with a saying known as Hebb’s rule:”Cells that fire together wire together”’. (p 27) ‘Evolution has given us a brain that can literally change its mind – over and over again’. (p 31) ‘Their brains had changed in response to actions that took place purely in their imagination – in response, that is, to their thoughts’ (p 33)

 

Consider Artificial Intelligence, “every aspect of learning or any other feature of intelligence can in principle be so precisely described that a machine can be made to simulate it.” (P 175) This is ‘a fallacy born of our desire to explain phenomena we don’t understand in terms we do understand. John von Neumann warned against falling victim to this fallacy’. ‘Whatever the nervous system’s language may be, “it cannot fail to differ considerably from what we consciously and explicitly consider as mathematics”’. (p 176)

 

postscript: the logic of the nervous system is based on the fifth dimension (Half-truth), which is the dimension of the universe and is a logic dimension which our mind uses to create language and anything logical. The Mathematics of the Mind uses the five senses, but mathematics dispenses with the mind’s logic by using a set of theorems and placing the mind outside of mathematics. Formal logic is similar and only uses part of the Logic of the Half-truth.

 

“Every indication is that, rather than a neatly separable hierarchy like a computer, the mind is a tangled hierarchy of organization and causation. Changes in the mind cause changes in the brain, and vice versa”. (p 176) “Short-term memory produces a change in the function of the synapse, strengthening or weakening pre-existing connections; long-term memory requires anatomical changes”. (p 185) ‘An implicit memory “is recalled directly through performance, without any conscious effort or awareness that we are drawing on memory”. ‘Explicit memory encompasses everything that we say we “remember” about the past’. ‘Consolidation of explicit memories involves a long and involved “conversation” between the cerebral cortex and the hippocampus’. (p 188)

 

‘The hippocampus …. an important role in weaving together the various contemporaneous memories – visual, spatial, auditory, tactile, emotional – that are stored separately in the brain but that coalesce to form a single, seamless recollection of an event’. ‘Many of the connections between memories are likely forged when we’re asleep’. (p 190)

 

‘Biological memory is in a perpetual state of renewal’ (p 191) ‘With each expansion of our memory comes an enlargement of our intelligence. (P 192) ‘The Web is a technology of forgetfulness’. (p 193) ‘To have the self-awareness and the courage to refuse to delegate to computers the most human of our mental activities and intellectual pursuits, particularly ”tasks that demand wisdom”’. (p 208)

 

postscript: it will be seen later, that the computer wiring is insulated, whilst the wiringof the brain allows leakage of thoughts (stimulation of action potentials in adjoining dendrites) to produce creativity.

 

 

Clearly, from the above, we are what we have learned and experienced and our senses are continually updating our brain and we remember what we choose to remember allowing for the fact that ‘nature and nurture “actually speak the same language”’ (p 28) In other words, our memories are patterns which change in time, can be implicit or explicit and coexist with our sensory inputs.

 

’ “Stationary or unchanging objects become part of the scenery and are mostly unseen. But as soon as “something in the environment changes, we need to take notice because it might mean danger – or opportunity.”’ (p 64) Movement is seen through the eyes and brings a pattern to mind and the patterns within the explicit memories decide on the outcome and the implicit memories within the body, brainstem and spine move the animal in response. Similarly, routine pursuits such as driving a car, playing a musical instrument or dancing etc frees the mind to a limited extent.

 

Thus the working of the mind, which encompasses the brain and the body, does not conflict with the mathematics above. A change to one of the senses initiates the sequence, the mind determines the action, the agent is survival, the director is life or death, and the carrot/stick is food/safety.

 

A couple (for want of space) of predictions emerge, such as the futility of looking for intelligence via computers when the human brain has so much potential and the ‘Mozart’ effect where the young brain is extended by application and the resulting (possible) detriment to the individual, and so on into more philosophical questions.

 

Part (7):

 

(1) The above chapter is another example of the general form of the solution, in that it starts at a point, Cogito ergo sum (I think, therefore I am), and ends with a set of solutions to diverse problems, as shown below.

 

(2) ‘Philosophical mathematics’ is a means of getting everyone (especially between countries) to agree on ‘something’, using the techniques of science and logic.

 

(3) Time changes the patterns and so the ‘answers’ change. There are no axioms or derivations on which to build. Even Cogito ergo sum (I think, therefore I am), which has been accepted for hundreds of years has reached its ‘use-by date’ because of increased knowledge.

 

(4) There is a lot for Philosophers to do in guiding modern society, especially in the light of changing patterns, as in (3). Untold generations under survival of the fittest conditions over 3 billion years have produced a mind that ‘works’ using a ‘mathematics’ that can be ‘extended’ to help solve civilization’s problems, so that everyone can agree, on the basis of logic and philosophy that a ‘set of actions’ should be implemented to maximize the world’s population’s ‘situation’ at the present and for the future.

 

 

postscript: it is important to remember that the Mathematics of the Mind arranges the concepts as attractors (or columns), and the contexts as ‘scissors’ that trim the columns to remove the unwanted parts, leaving the most relevant to be considered by the mind as well as applying the other rules.

 

What are we doing? Mathematics strips away the context and misses out on the understanding behind the measurement or result, and chapter 28 shows how fundamental physics changes when another view is taken. To indicate the power that context has within the brain, I would like to offer the following quotation.

 

‘But the most striking finding of all from these early studies of chess experts was their astounding memories. The experts could memorize entire boards after just a brief glance. And they could reconstruct long-ago games from memory. In fact, later studies confirmed that the ability to memorize board positions is one of the best overall indicators of how good a chess player somebody is. And these chess positions are not simply encoded in transient short-term memory. Chess experts can remember positions from games for hours, weeks, even years afterward. Indeed, at a certain point in every chess master’s development, keeping mental track of the pieces on the board becomes such a trivial skill that they can take on several opponents at once, entirely in their heads.

 

As impressive as the chess masters’ memories were for chess games, their memories for everything else were notably unimpressive. When the chess experts were shown random arrangements of chess pieces – ones that couldn’t possibly have been arrived at through an actual game – their memory for the board was only slightly better than chess novices’. They could rarely remember the positions of more than seven pieces.’ (Moonwalking with Einstein: The art and science of remembering everything, Joshua Foer, p 65)

 

‘The chess experiments reveal a telling fact about memory, and about expertise in general: We don’t remember isolated facts; we remember things in context. A board of randomly arranged chess pieces has no context – there are no similar boards to compare it to, no past games that it resembles, no ways to meaningfully chunk it. Even to the world’s best chess player it is, in essence, noise.’ (p 65)

 

Obviously something happens when we add context, and this brings to mind the Philosophers’ stone, where, if the success of the technological side could be augmented with context to invigorate the social side as mentioned previously, the flowering of technology might be brought to bear on the social side that the world needs. That is the purpose of the Mathematics of the Mind.

 

postscript: it is shown in the postscript of chapter 2 that three space, one time and one logic dimensions define everything, in my opinion. They are the space-time dimensions of our world and the logic (Half-truth) of the universe, which of course contains us, and for simplicity of use, everything may be thought of as forming a trinity, for similar reasons to that described in the Bible, which predated mathematics, and by necessity used a Mathematics of the Mind, as above.