Saltar para: Posts [1], Pesquisa e Arquivos [2]

"OH WATERS, TEEM WITH MEDICINE TO KEEP MY BODY SAFE FROM HARM, SO THAT I MAY LONG SEE THE SUN." - Rig Veda

Tags:

"There is another, more interesting, explanation for the structure of the laws of nature. Rather than saying that the universe is very structured, say that the universe is mostly chaotic and for the most part lacks structure. The reason why we see the structure we do is that scientists act like a sieve and focus only on those phenomena that have structure and are predictable. They do not take into account all phenomena; rather, they select those phenomena they can deal with.

Some people say that science studies all physical phenomena. This is simply not true. Who will win the next presidential election and move into the White House is a physical question that no hard scientists would venture to give an absolute prediction. Whether or not a computer will halt for a given input can be seen as a physical question and yet we learned from Alan Turing that this question cannot be answered. Scientists have classified the general textures and heights of different types of clouds, but, in general, are not at all interested in the exact shape of a cloud. Although the shape is a physical phenomenon, scientists don’t even attempt to study it. Science does not study all physical phenomena. Rather, science studies predictable physical phenomena. It is almost a tautology: science predicts predictable phenomena.

(...)

In order to describe more phenomena, we will need larger and larger classes of mathematical structures and hence fewer and fewer axioms. What is the logical conclusion to this trend? How far can this go? Physics wants to describe more and more phenomena in our universe. Let us say we were interested in describing all phenomena in our universe. What type of mathematics would we need? How many axioms would be needed for mathematical structure to describe all the phenomena? Of course, it is hard to predict, but it is even harder not to speculate. One possible conclusion would be that if we look at the universe in totality and not bracket any subset of phenomena, the mathematics we would need would have no axioms at all. That is, the universe in totality is devoid of structure and needs no axioms to describe it. Total lawlessness! The mathematics are just plain sets without structure. This would finally eliminate all metaphysics when dealing with the laws of nature and mathematical structure. It is only the way we look at the universe that gives us the illusion of structure.

With this view of physics we come to even more profound questions. These are the future projects of science. If the structure that we see is illusory and comes about from the way we look at certain phenomena, then why do we see this illusion? Instead of looking at the laws of nature that are formulated by scientists, we have to look at scientists and the way they pick out (subsets of phenomena and their concomitant) laws of nature. What is it about human beings that renders us so good at being sieves? Rather than looking at the universe, we should look at the way we look at the universe."

HERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."

Naturally, we are inclined to smile about the simplicity of the classmate's approach. Nevertheless, when I heard this story, I had to admit to an eerie feeling because, surely, the reaction of the classmate betrayed only plain common sense. I was even more confused when, not many days later, someone came to me and expressed his bewilderment [1 The remark to be quoted was made by F. Werner when he was a student in Princeton.] with the fact that we make a rather narrow selection when choosing the data on which we test our theories. "How do we know that, if we made a theory which focuses its attention on phenomena we disregard and disregards some of the phenomena now commanding our attention, that we could not build another theory which has little in common with the present one but which, nevertheless, explains just as many phenomena as the present theory?" It has to be admitted that we have no definite evidence that there is no such theory.

The preceding two stories illustrate the two main points which are the subjects of the present discourse. The first point is that mathematical concepts turn up in entirely unexpected connections. Moreover, they often permit an unexpectedly close and accurate description of the phenomena in these connections. Secondly, just because of this circumstance, and because we do not understand the reasons of their usefulness, we cannot know whether a theory formulated in terms of mathematical concepts is uniquely appropriate. We are in a position similar to that of a man who was provided with a bunch of keys and who, having to open several doors in succession, always hit on the right key on the first or second trial. He became skeptical concerning the uniqueness of the coordination between keys and doors.

Most of what will be said on these questions will not be new; it has probably occurred to most scientists in one form or another. My principal aim is to illuminate it from several sides. The first point is that the enormous usefulness of mathematics in the natural sciences is something bordering on the mysterious and that there is no rational explanation for it. Second, it is just this uncanny usefulness of mathematical concepts that raises the question of the uniqueness of our physical theories. In order to establish the first point, that mathematics plays an unreasonably important role in physics, it will be useful to say a few words on the question, "What is mathematics?", then, "What is physics?", then, how mathematics enters physical theories, and last, why the success of mathematics in its role in physics appears so baffling. Much less will be said on the second point: the uniqueness of the theories of physics. A proper answer to this question would require elaborate experimental and theoretical work which has not been undertaken to date.