Last weekend, we had a village get together. Someone had put up a marquee in one of the large gardens that all the houses in the village have. They had put out plastic tables and chairs, and provided food and drinks. We all sat around talking, enjoying the sunshine and the country idyll. Fortunately, unlike the fictional Midsomer, the enjoyment was not punctuated by any murders.
I sat at a table, under the shade of one of the many large trees in the garden, and during the course of the afternoon different residents of the village, and some of its former residents, I had become acquainted with during what was supposed to have been my temporary sojourn amongst them, but has now continued for more than eight years, came to sit and chat.
My neighbour who is a Professor of Mathematics found his way to the table, and we always end up in a more intellectual discussion than those that characterised most of the rest of the day. The discussion came around to differential calculus. I was certain that I did not know more about it than he did, indeed the discussion had arisen, because I had said that I was thinking of doing an A level Maths course, for my own illumination. Of course, I offered tentatively, therefore, you will know that originally mathematicians were reluctant to accept the calculus, because it breaks the laws of Aristotelian logic in relation to the syllogism. He looked at me a little puzzled. Well, it was put forward in two versions, he responded, one by Newton and the other by Liebnitz. A former villager had come to listen to the discussion at this point. In order to include him, as a native Irishman, my interlocutor added, one of your compatriots, Berkeley, had objections to it, but his objections were not well founded.
Berkeley it was, he, added, who as an idealist philosopher, had raised the question as to whether, if a tree falls in a forest, it makes a sound, if there is no one there to witness it.
I returned to my own point. What I was talking about, I added, was that where a tangent touches the circumference of a circle, the angle of the two is equal. "Yes, that's the definition of a tangent," he replied. "So, at that point, then, a straight line is also a curved line, and vice versa. It is two things at the same time, it is A and not A" I insisted. Obviously, uncomfortable with that conclusion he responded that he was not sure that it meant that it was two things at the same time, rather than just a question of notation, and the way zero is dealt with.
Well, here is the whole point, I responded. The way mathematics has got round the contradiction that the differential calculus implies is to consider the point of tangency to, in fact, be a point that can only exist ideally, rather than in reality. The reality is that no matter how small a straight line you take, or how small a curved line, in reality they have a start and end point; they are not equal to zero. A line which has zero length is only one that does not exist in reality, and is, therefore, just as much a product of an idealist philosophy, as the Bishop Berkeley's formulations.
There was one philosopher, for whom that contradiction posed no problem, I noted, and that was Hegel, for whom such a contradiction was simply an illustration of the existence of the dialectic. "When I was a young man," he said, "I remember being told that anyone who brought Hegel into the discussion was dangerous to know". But, then, at the time he was studying at Cambridge, in the early 1970's, it was still thought to be a hotbed of Russian spies!
In an earlier discussion with two of my other neighbours I had said that I'd been focussing more on my Yoga recently, which it been sparked by suffering with a bad back that had inflicted itself on me for some unknown reason. As well as the physical exercises, I pointed out, I had also been focussing on the breathing and mental exercises. The latter are divided into exercises to train concentration, which today is taught as "mindfulness", meditation, and contemplation. The first requires attention to be exclusively focussed on some subject, be it a thing, or a concept, noting everything associated with it; the second requires exclusive attention on the interrelations between all of these different aspects of the subject; and the last requires seeing all of these elements simultaneously, which should thereby provide a new and deeper understanding of the subject. I've pointed out the similarity of this approach with Marx's own method in analysing capital.
This brings me back to the earlier discussion on the differential calculus, the dialectic and materialism. In orthodox economics, the differential calculus plays a fundamental role, as indeed it does in most sciences. In Economics it is the basis, for example, of the use of Indifference curves. An indifference curve plots all the possible combinations of two goods that a consumer might prefer to have, on the basis of their marginal utility for them. The more of something I already have, according to this theory, the more I will value it less compared to the alternative. This is the basis of the so called Water-Diamond Paradox I have discussed before. The optimum solution is found by inserting a budget constraint line, which basically takes the price of each good, and thereby determines, on each axis of the graph, the maximum of each good that could be bought. The point of tangency of this budget constraint line with the highest possible indifference curve is the point that enables the consumer to obtain the greatest utility.
There are many, many problems with marginal utility theory, as previously discussed, but the point that concerns me here is the same one discussed above. That is, at each point where I am indifferent between good X or Y, but where, in fact I choose to buy one rather than the other, I have shown that I was not indifferent after all, but preferred one rather than the other. At least, that is what the syllogism would insist. Its one reason the Austrian School, explain consumer behaviour instead by simply saying "people act". But, as soon as we dispose of the syllogistic approach, and instead view the situation dialectically, the problem disappears. We can accept that the contradiction is real. I am indifferent to both X and Y, and the question of whether I will in reality choose X or Y, is uncertain. It becomes a question of probability and observation. In fact, we have moved from Economics to Physics, and Heisenberg's Uncertainty Principle. The best known version is Schrodinger's Cat.
And, here we return to Bishop Berkeley and the tree falling in the forest, because here observation itself becomes a determinant of the outcome itself. What Quantum theory suggests is that when we get down to these limits of the very small, the normal laws of what we think constitutes reality do not apply. Things can be two different things - a wave or a particle - at the same time, they can be at two different places at the same time - as with an electron, and so on. What this means is that in one reality, a tangent might be a straight line, whereas in another it is curved; in one reality I choose good X, but simultaneously in another reality, I choose good Y.
The Physicist and Yogi, Ernest Wood, describes matter as that principle in Nature which brings the past into the present, and he describes the mind as the means by which the future is brought into the present. So, when we use our mind to plan some future event, we then act upon it so as to achieve that result. We also picture in our mind's eye the future, and sometimes we do that instinctively and unconsciously. When I was a kid, I used to sit in our backyard, throwing a ball against the wall with one hand, and catching it with the other, like The Cooler King, Steve McQueen in The Great Escape. But, I didn't have to look where the ball was going to catch it. My hand moved instinctively to where my mind had already told it to go, based on where it calculated the ball would be in time and space. It did all that in milliseconds, and numerous studies have shown that we often make better decisions based upon instinct than on the basis of reason, where such fast responses are required.
What we are learning is that reality is far more complex than it appears to be. To go back to Wood's point, matter itself appears to be intangible, the more we investigate it, and itself taking on more and more the characteristic of a mathematical formula, or set of mathematical rules, which govern the universe, or more correctly universes. The latest theories suggest that rather than the four dimensions we are familiar with, their are 11 dimensions of space-time. We can comprehend that via these thought processes, but we do not experience reality in that way, as opposed to the 4 dimensional reality we are familiar with. But, that is nothing new.
We are familiar with various optical illusions where whereby our brain attempts to make sense of the nerve impulses sent to it, by the optic nerve, which are themselves triggered by the signals sent to it from the retina as a result of light hitting it, but is thereby led into error. Similarly, if you take a video camera, and rapidly pan from side to side, you will get a blurred image, but if you do the same with your head, you don't. That is because the brain engages in a load of additional processing.
What we experience as a 4 dimensional, linear reality is, in fact far more complex. It is a reality of more than those 4 dimensions, and where time itself is not linear. It is a reality where the past and the future coexist simultaneously, in the same way that a tangent is both a straight line and a curved line at the same time, but it is our consciousness that provides continuity and linearity, which is merely the phenomenal manifestation of this underlying reality, we cannot grasp.
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