Thoughts on Health and Medicine

Humans are a complex set of equations. If you think about the circulatory system, respiratory system, digestive system, skeletal system, the organs and their functions we are a tidy little packet of interlinked equations - the kind you see in linear algebra and those systems of equations that give rise to matrices. How many equations constitute a human I wonder? I have listed the more obvious systems that every (or pretty much) lay person is familiar.

In systems of medicine that use pulse diagnosis, they are working with the rhthyms of the being in front of them. You don't bother to do a mathematical analysis of music. You just process is it automatically and enjoy the finished product. Similarly these doctors plug into your particular rhythms or equations that define you as a being in this moment and place - space-time location if you will. Their job is to balance out the various equations which will ultimately sustain the life force in the body in front of them. To narrow down to symptoms like constipation or dry skin or tooth ache is to focus on one single thing, maybe a variable in an equation.

In order to begin understanding ourselves we need to see our selves as heat equations and all the other myriad equations we combine in ourselves. We are a delicate balance of these systems. Maintaining harmony and doing complex equational manipulation even in situations of seemingly catastrophic or extreme physical stress is how life energy pulses through the body.

Each culture has its own explanation of the human body. Each culture has its own explanation of the properties of the plants, minerals, animals, and the world itself around them. It would be grand if all this knowledge could be gathered and cross-referenced without bias or prejudice so that we can have a multilayered explanation for our selves and the world around us. For example off the top of my head I have heard of Chinese Traditional Medicine, Tibetan Traditional Medicine, Ayurveda, Unani, Homeopathy, Allopathy, Herbalism, Aromatherapy, Chiropractic systems of medicine. Each has a view of the human body. It would be great to be able to look at the body through each of these lenses and better yet look at the body through all of these lenses at once. I wonder what we would see. I wonder which views would cancel another view out. In physics we learn that light passing through colored lenses at different angles can completely block or prevent the passage of certain spectra in the light wave. Maybe these systems could enhance each other or hinder or obfuscate the information contained in an other system. But certainly it looks like it would be interesting to study the body by juxtaposing these paradigms of understanding the human body. Similarly, analysis of plants can also benefit from this treatment. We can subject plants to chemical analysis, form analysis, functional anyalysis and so on. I think by bringing all these aspects together we get a more complex explanation of what we are experiencing or encountering or viewing. These explanations may be macro-level holistic explanations or more micro-level explanations. It is important to be able to correctly identify the different nature of the various knowledge systems and be able to use them appropriately.

Balcony Garden Slide Show

Today I finally got around to creating a couple of web based photo albums. Initially I wanted to share all the images I had of my former balcony garden. But while I was at it I created another album to save images of one of my sewing projects as well. I wound up using the Picasa technology because it is the first listed option for having a slide show at Blogger. For the moment, I have placed the balcony garden slide show at the bottom of the page for lack of a better idea on where and how to place it.

Some thoughts on Numbers

After thinking about my previous post and also my original intention it seems fair to say that "mathematical narrative" is not exactly accurate in describing what I want to do. At least not in the way I sketched out in my previous post where I would go through each area of mathematics and write a narrative about it. Instead I would like to free myself of any such expectations and simply write about mathematics as I see it in any way that is most convenient at the moment.

Today I would like to share my thoughts on the topic of the organization of numbers. There are a number of issues which immediately spring to my mind. The first has to do with how to count. Which system should I useto count? Should I use a base of 1, 2, 3, 4, 10, 16, 32, etc. Base 2 counting system is the familiar old binary system. Base 10 is our decimal system. Base 16 is the hexadecimal system of counting. If we step back a minute we realize that this base business at least in one respect has to do with the idea of the modulus operator and congruences. All numbers in each system is thought of in terms of that base. Each digit used to express a number in a particular system are the ones allowed by the base number of that system. In binary we have just the two options for digits to express all the numbers: 0 and 1. In decimal we have 0 through 9 to express all the numbers of infinity. This brings up the idea of cycles. We literally cycle through our digit options until we exhaust them. Then we start a new digit location and cycle through our digit options again until we exhaust them. In this way we get the idea of permutations.

If we change our focus and look at numbers again we can use the following characteristics to get some sort of initial handle on them. For example, when we think of integers (positive and negative numbers) we get the ideas of reflection and mirroring and the notion of using zero as the location for the "line" of reflection. If we think about rational numbers we get the idea of containment, precision, discreteness, compactness, well-defined. If we think about irrational numbers we get what cannot be well-defined. If we think about odd numbers and even numbers again we are visiting the ideas of modulus operator but this time in a more sophisticated guise of an equation. With Complex numbers again we use an equation to describe numbers: we have the real numbers (a and b in the equation to follow) and the imaginary number (i) combined to give us the classic definition of a composite number,z: z = a + bi.

With this slight exploration I feel there is a rhythm , pattern for mathematical concepts, structures, and techniques. There are so many concepts I haven't delved into and woven into my discussion but I think I will leave that for a later date. For example there are the concepts of shape, space, equation in addition to the numerous details for each of the ideas I have introduced which I have left out.

The Need for Narratives in Mathematics

In order to better understand the discoveries of mathematics and aid in the assimilation of their ramifications to my conceptual vocabulary I have a need for narratives that give the big picture of each subfield in a typical mathematics curriculum. What I mean by this is that for each of the courses say in a mathematics undergraduate program, all the axioms, theorems, corollaries, lemmas, and vocabulary are brought together to create a story of what these things mean and how they relate to each other and what they portend for other areas of mathematics. What i envsion is someone (probably myself) taking a look at something like Calculus or Linear Algebra or Abstract Algebra and collecting together the key ideas captured by the vocabulary, theorems, and other matter that have been discovered and established. Reading such a narrative I can then assimilate the ramifications of the material easily. By understanding the larger picture during the actual detailed study of the material for each subfield in mathematics I can look at each of the actual theorems, corollaries, lemmas, and appreciate how they contribute to the development of the subfield. It also clearly answers questions of what is known and what is not known in this field as well as understanding any cross connections if any.


I think then a narrative of the sort that I envision would have to be arranged by the questions which have motivated research in that area of mathematics. There is a book on prime numbers titled "The Little Book of Big Prime Numbers". The author does a wonderful job of organizing the research and discoveries pertinent to Prime Numbers by questions such as "What are Prime Numbers", "How are Prime Numbers Distributed?". What really struck me and remained with me after looking through the book and looking at the last chapter was that the primary questions regarding prime numbers is in some deep sense still a mystery. Work has been done by many great minds at coming to grips with the Prime Numbers but so much still remains unanswered about them.

In my attempts to unearth my understanding of what I consider mathematics, I have discovered tht narrative is of prime importance to me. This narrative is driven by my questions such as "What is the point of Abstract Algebra? What does it aim to do? What are its goals? What does it seek to establish? What does it establish? What are its discoveries?" The answers to these questions can in part be derived from looking at the established knowledge: vocabulary, axioms, theorems, corollaries, lemmas. I suppose by diegesting and assimilating this knowledge I can accomplish the task of creating some sort of narrative.

I envision this narrative as being far more deep than vaguely pointing to the existence of structures that link senses and mathematics or more prosaically algebra and topology as I do in my earlier posts. I hope to be able to show by doing this that there are some recurrent themes, for lack of a better alternative word to "structure", that lends itself to a mathematical organization of reality. The relevance of such work to me resides in the promise of deep practical applications to my life.

Nature of Numbers

If we start off with a consideration of the data that we perceive we recognize that numbers are useful in describing our experience. But this raises the question about what numbers themselves are and how they are even useful.

In our experience of the world, of being and existence, we notice that numbers are useful in describing counts and characteristics. Pleasing sounds are made by harmonizing/marrying together the soundwaves - modulation of sound through amplification, dimunition or cancellation. This process can be modelled by plugging numbers into functions. Functions are higher order abstractions than numbers so we will put aside our questions about the nature of functions for the moment and give our focus entirely over to numbers. But I think it would be fair to say that functions are predictive and characteristic of a feature of our experience. Counts are so familiar that it will suffice for me to say that we are always counting even if it is nothing other than the number of dollars in our bank account.

In looking at numbers what we discover is that they themselves fall into patterns and categories and structures. One such has to do with selecting which method we will use to count. We have heard of binary, decimal, hexadecimal, etc. These numbering systems presuppose the architecture of the modulus operator, a ubiquitious presence in a college math class on Number Theory. So what does this mean? What is the nature of the relationship between this level of structure and numbers.

Another area of cleavage in numbers is type. There are so many types of numbers: odd/even, positive/negative, and then the sets: natural numbers, integers, rational, irrational, real, and complex.

In abstract algebra we notice that the procedures of identity, inverse, addition/subtraction, multiplication/division are used again to define/characterise sets. So we use the language of groups, rings, homomorphism and other useful descriptive terminology.

In topology, we look at three dimensional objects like doughnuts and describe them yet again in terms of functions and other similar characteristics.

In linear algebra/vector algebras we look at functions and manipulation of functions which in turn model some aspect of reality.

In calculus we again model reality using functions and manipulating these functions to predict and add to our understanding of the world around us.

So in surveying this landscape it looks like there are techniques of manipulating functions and techniques of organizing numbers/functions using characteristics of similarity.

It seems that increasing levels of abstraction makes it easier to predict what to do with the lower levels of mathematical knowledge. So the question that comes to my mind are there other levels, higher levels of abstraction and organization which will make it trivial to understand mathematics. In other words by understanding a larger concept will the others (namely techniques of manipulation and ordering) also follow? Or maybe a better way to express my intuition is that we already have the abstractions of abstract algebra and the techniques of the other disciplines but we still don't have something which will make these features obvious and trivial. Is there something that will make all the complex techniques we have and make them irrelevant because there is an easier way to get to certain mathematical truths simple and complex.

Numbers and Structures and Functions

In my previous post I had mentioned that there are structures which repeat but with differences depending on the dimension or sense (sight, sound, smell, touch, taste, and the yet to be established sixth sense). We are all familiar with points, lines, shapes from geometry, groups, rings, homomorphisms from Algebra and even things like knots, chaos and synchronicity. All of these structures are organized by what mathematics knows as functions which in turn are defined by numbers. For example time and space parameters are necessary to observe the phenomenon of synchronicity. Events, actions occur in the world. These actions occur in groups/simultaneously or independently. For example on a highway on would think there are cars more or less evenly distributed along its length. But in fact what is noticed is that as one is driving along, cars travel together in clumps, groups, and in between these groups there will be a few lone cars travelling on the highway. Depending on what the action is it can be seen (or not), felt (or not), etc. I wonder which sorts of numbers and which of these numbers are relevant and which structures organize this phenomenon.

What is Mathematics?

What is Mathematics? I experience Mathematics as life itself in this body that I inhabit. It is the underlying language that can be used to express all of my experiences. In fact as I exist in my body and experience being I notice that being entails interaction (passive/active) with my environment. My body is a sophisticated array of mechanisms for experiencing the various dimensions of reality. My different senses open ways of understanding reality in terms particular to it. For example my sense of sound gives my the rational numbers and my sense of sight gives me the irrational numbers. Music and sounds in general can be captured by fractions. But sight is more subtle. What can be perceived by sight is infinitesimal blinking into emptiness. Relating these two senses to these two particular sets of numbers may be premature and inchoate but for me without doubt mathematics does not stand outside of life or the universe but is in fact the very mechanism which gives shape to our experience, it is how we draw meaning and understand the structures of reality.

What I find of note is that in my being all of these sets of numbers come together or assume discreteness depending on whether I am just being or focusing along a particular dimension. In full experience all the number sets are present as an integrated whole the way math text books define how the Composite Set contains the irrational and rational etc. I am automatically coordinating data of various kinds to experience a seamless event.

I wonder exactly how the six sets of numbers match our senses:

senses: sight, sound, smell, touch, taste, and the mysterious sixth sense
sets of numbers: natural, integer, rational, irrational,real, composite.

It seems to me that there are certain structures in reality which repeat. The peculiar representation of the repeat depends on which set of numbers we are looking at.