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"God created Integers, the rest is work of men." Leopold Kronecker
One friend of mine pointed out this number to me: 0.14285714285714285714285714285714 … Apart from the fact that 142857 is repeated in a sequence there does not seem to be any pattern in that number. Nothing intuitive there. Or is there? In fact, it is a number that we ought to be familiar with day in day out. Ring a bell?
The number is more accurately represented as 1/7 and it could represent a day, which is one seventh of a week. But then what is so intuitive about a week? Or a finer grained element as a day? A day is now accepted as such in the current cultural framework that we all live in. So is it intuitive? What constitutes intuition? Maybe it is just a discipline that we've succummed to…
Ask any parent of little children and most would readily agree that there is no fixed idea to what is known as a day as far as infants are concerned.
I have three kids and their age differences are not too much, so I have experienced this first hand. At various points in their infancy, me and my spouse have had difficult times reconciling the incompatible diurnal cylces of our children with that of society. Getting all the three to eventually agree on the same diurnal cycle was quite an exercise. Even now, my second daughter often holds on to the excitement of the day late into the night and sometimes even succeeds in changing the definition of a day with her big, bright eyes glinting from beneath the dark of the blanket.
Mathematics is only an abstract tool to model the reality around us. The word abstract ought to be emphasized but that normally that does not happen. Often the word "natural" (or its various cousins such as "intuitive", etc.) takes over, and causes much confusion. The preponderance on the "naturalness" of these numbers started with the Greeks. They instilled in us that the concept of natural numbers were natural enough that they should be first used whenever we are at unease in our modeling.
Here is an explanation from Numbers and Geometry (John Stillwell):
Begin Quote:
We all realize that the sequence 1,2,3,4, … continues 5,6,7,8,…, and that we can continue indefinitely adding 1.The objects produced by the act of counting are what mathematicians call the natural numbers. Thus if we want to say what it is that 1,2,3,17.643,100097801, and 4514517888888856 have in common, in short, what a natural number is, we can only say that each is produced by the counting process. This is slightly troubling when you think about it: the simplest, and most finite, mathematical objects are defined by an infinite process. However, the concept of natural number is inseparable from the concept of infinity, so we must learn to live with it and, if possible, use it to our advantage.
:End Quote
So the concept of natural numbers do have a strong connection with the abstract concept of infinity. It may have started somewhere in the mists of time when people felt the need for counting and realized that they had ten fingers to count, and they get to the integers by adding one successively. I wonder what whould have happened if they had counted only to two with the numeric system containing only two symbols representing the left hand and the right hand. Maybe, the computer may have got invented earlier!. In fact there are certain tribes in Africa who do count like this: "one", "two" and "many". I am sure they must be getting along just as fine as the rest of us.
We have to understand that as humans we had a whole morass of squiggly worms of natural analogues that we could have formed our mathematical concepts from. It so happened that the analogue of the act of counting gave us the concept of natural numbers. The reasons for such a development may not be important. Today, we are privvy to several other natural phenomena that could have given us our basic building blocks in mathematics. In natural numbers, the distance between one and the very next one is always a constant of …ummm… one. It is interesting to note that almost none of the other natural phenomena that humans could have used involve such a uniform distance, yet they are still perceived to be uniform. Take for example, our sense of the volume of sound. To percieve one sound to be twice as loud as the next one, the energy required would have to be ten times the first one. The perception of the magnitude of sound follows a logarithmic scale. Similarly, the perception of musical tones involve a geometric scale (roughly). So when the lead guitarist in a heavy-metal rock band tells his rhythm guitarist "give me a fourth, buddy" he is referring not to the number fourth in the natural number scale, but the fourth musical interval which works out to approximately 3/4th of an octave.
If we move out of the realm of human perceptions, there are also other phenomena that could have been used as building blocks in mathematics. For example; the concept of bifurcation that is seen in complexity theory (which involves a constant, an irrational number called the Feigenbaum constant) could have also been used. Anyway, the point is not to say that we missed something deep because we could have used any of those "sophisticated" concepts but to point out that there are several alternatives on the platter today.
I have nothing against natural numbers. I do not want to arbitrarily replace one abstract system with another one, and then stand by the second one vociforeously with another set of dogmas. We already have a whole system of mathematics built up on the foundations (axioms) of natural numbers, and we can still go ahead with it. My only point is that it is incorrect to think that there is something sacrosanct (or natural or instinctive or intuitive, etc … choose your replacement word ) about using integral intervals. When we model the real world, our concentration has to be focused on getting to the objective (E.g. are the crucial aspects of the model being represented?) If we get waylaid by concepts outside to the modeling process then we may end up artificially defining the concept of precision and pretend that the concept of fuzziness (or the gray areas that are so often seen in the real world around us) can never be represented. Not so. Today, there are many modeling methods available. For example; we can use concepts from Fuzzy logic to handle "imprecise" representation issues.
Lotfi Zadeh's book on Fuzzy Logic establishes the difference between Aristotelian and fuzzy logic: Draw a line segment. Call one point of the segment as "A" and the other as "not-A". Aristotle believed in "A" and "not-A" i.e. he did not believe there could be anything useful in between "A" and "not-A". In fact, Aristotle is famous for his "law of the excluded middle" in logic which translated into common language states: "Maybe there are things that are in between the extremes, but even if there are; they are to be specifically excluded if we have to carry on this conversation further". Much of Western philosophy and approaches can still be traced to this antiquated Grecian dogma. (For example; The law of excluded middle can be seen in George Bush's words when he said "Either you are with us or you are against us" or words to that effect.) Lotfi Zadeh comes across as the proverbial disgruntled anti-establishment fighting off old perceptions in one corner of Berkeley, standing on his head in one corner of his room doing Yoga. When I read the book I got the feeling that he himself was in the excluded middle for quite some time, because of his views! Lotfi explains how Buddha (as a philosopher — not as a "God") , on the other hand, can be seen as someone who is seen sitting smiling benignly plonk in the middle between "A" and "not-A" bringing perfect logical sense into that which was otherwise being excluded.
The belief in integers can be seen in architectural designing quite clearly. I know several architects who are aghast when they find that I do not have a position on how to put column positions in my designs. I keep in mind the fundementals of how structures behave while designing but invariably I leave the details on things like column positions to the structural engineer. On the other hand, I am surprised that some architects still find it necessary to dictate the actual positions of columns. In fact, some of them get shivers if they cannot get their design to fit into a grid. Once again, such an attitude can be traced to the law of the excluded middle handed down to us by Aristotle. Here is an example: In the figure below, it is often assumed that the design for covering a square area, on the left is cleaner and simpler.
But on closer examination, the one on the right may very well do a much better job of covering that space. Putting in columns in-between the grids can lead to counter-balancing forces which will lead to a more efficient structure. The external periphery of the slab is cantilevered, and they would counter-balance the forces developed within the space between the four interior columns.
The approach shown on the left, has only given the architect her peace of mind but may not be actually true to the demands of the design. Note the use of the word "may". I am not advocating the use of this approach over that one. There is no one single right approach. Design is always contextual. In certain situations, it is obvious that use of integers can be the only way: For example; we all know what would happen if a staircase does not have an integral amount of steps. What is critical is that we should not be swayed by mathematical processes. Mathematics is supposed to only guide us and not dictate our hands.
I believe this incongrous application of integers in design is a mapping problem: Humans are pattern seekers. In fact, that is what predominantly distinguishes us from other animals. We tend to map an earlier piece of knowledge to another one so that we are comfortable in handling the problem. Sometimes the act of being comfortable with the pattern put on a problem becomes unconsciously more important than the correct description (i.e. modeling) of the problem itself. There are various ways by which mappings can be effected. E.g. analogy is a mapping device. There are other mappings that designers are caught up in. Some of them are seemingly non-mathematical. For example; architecture is often mapped to language and even there; I can see much inconsistencies. But I'll explore those maps separately.
There are other mathematical concepts (i.e. other than integers) that are used by architects without investigating how they map onto design representations. For example; the mathematics behind music is often quoted in architecture also (But I have yet to see a clear cut example. Most of them are only poetic references, and not rigorous ones). Then there is the age old system involving the use of the Golden Mean Ratio (and its ilk, the Fibonacci numbers) which has guided many architects. I take all of those with a pinch of salt, and I may explore those maps separately in this book.
When Leopold Kronecker made his famous statement, he was referring to the abstract concept called "God". As I am an atheist, I do not have any theological position on that concept, but I can at least agree that the concept called God is abstract, and therefore the basis of integers is abstract. Let us, as humans, understand that thoroughly as we go about creating the "work of men" based on such an abstract concept.
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