Yin-Yang Symbol by Dr. Stefan Jaeger is licensed under a Creative Commons Attribution 3.0 Unported License.

When referring to this Yin-Yang video, please cite this paper, which contains more information about the mathematical daylight model used for creating the different Yin-Yang symbols:

Stefan Jaeger (2011). A Geomedical Approach to Chinese Medicine: The Origin of the Yin-Yang Symbol, Recent Advances in Theories and Practice of Chinese Medicine, Prof. Kuang Haixue,  Heilongjiang University of Chinese Medicine, China (Ed.), ISBN: 978-953-307-903-5, InTech,  Available from: http://www.intechopen.com/articles/show/title/a-geomedical-approach-to-chinese-medicine-the-origin-of-the-yin-yang-symbol

Stefan Jaeger (2011). A Geomedical Approach to Chinese Medicine: The Origin of the Yin-Yang Symbol, Recent Advances in Theories and Practice of Chinese Medicine, Prof. Kuang Haixue,  Heilongjiang University of Chinese Medicine, China (Ed.), ISBN: 978-953-307-903-5, InTech,  Available from: http://www.intechopen.com/articles/show/title/a-geomedical-approach-to-chinese-medicine-the-origin-of-the-yin-yang-symbol

More information, including nice illustrations, is available on Allen Tsai’s webpage.

Merry Christmas and a Happy New Year!

When we look at the root of mathematics, where it all begins, we find sets and numbers. The entire mathematical edifice, no matter how complex it gets, builds upon these elementary concepts. For instance, natural numbers allow us to do basic things like counting and arithmetic. They also play an important part in theoretical computer science, where they can represent computer programs or theorems. Mathematicians spent a great deal of time investigating natural numbers, and yet, I think natural numbers are still not fully understood.

The Peano axioms, named after the Italian mathematician Giuseppe Peano (1858-1932), are a typical axiomatization  of natural numbers. They allow us to construct the natural numbers and derive theorems. Here are a few examples of these axioms: 

• There exists a natural number 0 (The first Peano axiom).
• Every natural number n has a successor S(n) that is a natural number.
• The natural number 0 is not the successor of any natural number.
• Distinct natural numbers have distinct successors.
• …

The first Peano axiom is special in the sense that it guarantees the existence of at least one natural number. This number serves as the root number from which all other numbers can be derived via a successor function. It is exactly here, where we encounter intrinsic uncertainty. With no way of knowing whether the root number really exists, the assertion made by the first Peano axiom is intrinsically uncertain. In fact, the first Peano axiom subsumes the uncertainty about existence and non-existence into one single axiom. Unfortunately, we cannot get rid of this uncertainty. Removing the first Peano axiom from the definition of natural numbers merely shifts the uncertainty from the root number to other numbers.

I suspect that Peano and his colleagues had hoped to get rid of intrinsic uncertainty once and for all when they introduced the first Peano axiom. What they overlooked is that the first Peano axiom only subsumes the uncertainty; it does not help us avoid uncertainty. In my recent paper “Computational Complexity on Signed Numbers,”, I exploit this subtlety to tackle the P/NP problem. If any of the readers knows about papers dealing with that intrinsic uncertainty aspect of natural numbers, I would appreciate hearing from you.

While these considerations are valid from a philosophical point of view, they raise several questions:

1. Where is the best place to introduce intrinsic uncertainty into mathematics?
2. How can we incorporate intrinsic uncertainty into mathematics?
3. What are the implications for the current state-of-the-art?

I addressed these questions in a recent article entitled “Computational Complexity on Signed Numbers,” which I presented in an earlier post on the P/NP problem. Let me elaborate more on these questions here and in the following posts.

As for the first question, all places where mathematics explicitly postulates the existence of “something” are good places to introduce intrinsic uncertainty. When mathematics resorts to postulating the existence of things, it usually means that there is no way to prove their existence with the axioms of mathematics alone. Introducing axioms that postulate existence is therefore an act of desperation more than anything else. Unfortunately, mathematicians have been too careless with axioms postulating existence. I think that intrinsic uncertainty is a fundamental principle. It should play a dominant role and form the basis of mathematics.

As for the second question,  there are two ways to introduce intrinsic uncertainty into mathematics. One way is to introduce uncertainty explicitly. However, adding explicit uncertainty requires statistical models that are in themselves not uncertain. In fact, statistical models, such as the normal distribution, are well-defined and so is their behavior for large numbers. The other way to introduce uncertainty is to make uncertainty an intrinsic part of mathematics. For instance, removing an axiom that postulates the existence of an entity, a set for example, is a simple way to introduce intrinsic uncertainty into mathematics.

In my following posts, I will discuss this in more detail. I will show exactly where I think is the most appropriate place to introduce intrinsic uncertainty into today’s mathematical edifice.

One night, Chuang Tzu dreamed of being a butterfly — a happy butterfly, showing off and doing things as he pleased, unaware of being Chuang Tzu. Suddenly he awoke, drowsily, Chuang Tzu again. And he could not tell whether it was Chuang Tzu who had dreamt the butterfly or the butterfly dreaming Chuang Tzu. But there must be some difference between them! This is called ‘the transformation of things.’

As probably many before me, I have wondered about its meaning. In my humble opinion, the dream exemplifies the distinction between existence and non-existence. We would typically think of the person having the dream as an existing entity, and the world he is living in as reality, while the dream world would be fiction and thus non-existing. However, Chuang Tzu’s butterfly dream tells us that we do not know which is which; meaning we cannot distinguish between the existing and non-existing worlds. Actually, in some sense, both worlds are existing (or non-existing). In our effort to identify one of both worlds as ultimate reality, which the dream tells us is not possible, we are constantly switching between both worlds, living in either one and taking it for real.

This difference between existence and non-existence is a classical Yin-Yang opposite. And yet, because existence is regarded as so fundamental, many would rather abandon the concept of Yin and Yang than to give up their sense of reality. Too strong is our desire to identify the real world and to classify Chuang Tzu as being part of reality. Only few dare to accept the fact that our dreams are as real as our bodies, although this is exactly what Chunag Tzu’s dream tells us in my opinion. Scientists, and mathematicians in particular, are no exception to this. Mathematicians would shudder with horror at the mere thought of performing calculations on objects (sets) that do not exist. In mathematics, everything needs to exist. On the most fundamental level of mathematics, where formal proofs are a rare guest, mathematicians have introduced a plethora of axioms to guarantee the existence of sets, numbers, etc. This clearly shows their ignorance about non-existing things, which are not worth to be considered simply because they do not exist. However, I think that Chuang Tzu’s butterfly dream tells us to incorporate non-existing objects into our formal considerations; and that mathematics needs to embrace the intrinsic uncertainty between existence and non-existence.

The removal of the first Peano axiom introduces intrinsic uncertainty into the encodings of natural numbers, which obviously has drastic consequences on computability and computational complexity. Every computation, and every proof, is uncertain under this relaxed set of Peano axioms. However, the first Peano axiom cannot remove this uncertainty, it can merely subsume it into one statement. I think this fact and the consequences it has on computational complexity has been underestimated in the literature so far. I’m going to elaborate more on this here in the coming months.

Anyway, looks like we have some work to do until the end of the year.