Mathematics and modern science

It has been almost another year now since I wrote my last post.

I suddenly start to wonder how our modern science has evolved since Galileo time. Also I am curious if the path we have gone so far is the right or the only path we could take.

First of all, I personally believe that mathematics is the mother of all science and without it, there would be no modern science at all. Most of the scientific theories have been written in one or another form of mathematical expressions. The reason of using mathematical expression to describe a theory is that the theory, once verified, could be used to predict the future accurately. However, faced by quantum theory, the last mathematical tool left for us is the statistical expression that could only describes group behaviours rather than individuals. In other words, we are no longer capable of calculating and predicting individual objects such as electrons or any other particles as the way we used to. So what is going on? Is that the mathematical tools we have developed is not adequate? Or is that the individual reality does not exist at all? If so, then a reality only appear when we observe that in turn the reality is created by our observation. One possibility is that an event so called observation does not have any meaning.

Before I discuss if observation means anything, lets examine what mathematics has done for us. I believe the modern mathematics originated from the study of numbers and geometries. For numbers, it took humans a very long time to have the concept of “1” In fact, number “1” is only a abstract concept that does not have any physical representation, or in other words, the concept of number “1” only exists in human’s mind. One may argue that there could be “1” apple that does not need the existence of human mind. However, if we really have a close look of “1” apple, the concept of apple become fuzzy. The reason is that we have to ask ourselves where the boundary of “1” apple is. At any given moment, the apple is exchanging molecules with surroundings so that “1” apple at a given time has different compound then another time. We could only omit the details and call that apple “1” apple, but without our mind to abstract the concept of “1”, the physical “1” apple does not exist at all.

Same principle can be applied to geometry. A concept of a circle, a triangle or a rectangle is also an abstracted concept reside in human’s mind. For example, a circle is made with a line that all the points on it have equal distance to a single point e.g. the centre of the circle. The line has zero width that cannot exist in the physical world. One may find a physical object that has similar shape as a circle, but the concept of a circle is absolutely an abstract one. In other words, without human brain, a circle does not exist.

By having a concept of a number or geometry, we still don’t have mathematics yet. To do that, we need to define certain rules. In the ancient days, people count the numbers by tying knots or carving sticks. By then, the concept of adding did not yet exist. Someone on someday in the distance past invented a simple mathematical rule called “adding” and found this rule very useful. The “adding” rule says that when one number is added to another, the result of that operation gives a new number that equals to the sum of the two numbers. When I say “useful” I mean that the rule makes the number counting much easier. Subsequently another rule was created called “subtract” Once again, people found the new rule also very helpful. Of course, many rules have been invented afterwards and now we have our modern mathematics. However, the point here I am trying to make is that the rules are invented by humans for the reason of usefulness. Everyone could also invent new rules. For example, I could invent a new rule called “nonsense” When a number is nonsensed, the result would swap the highest digit of the number with the lowest one. e.g. Nonsense(13886) = 63881. If we study this rule very hard, we would find many interesting properties that might lead us to a new territory where no one has explored. On other hand, we may find that this new rule does not give us anything useful. Quickly, we abandon this rule and start to invent new ones. Another good example of inventing new rules is Mr. Stephen Wolfram who wrote a book called "A new kind of science" In the book, instead of using numbers, Mr. Wolfram developed a new tool called automata. With a set of new rules, Mr. Wolfram explored many very interesting properties of automata. He was also trying to use this new mathematical tool to describe our nature even including quantum phenomena. How successful this new method will be? Time will tell us in the future. To me, mathematics is invented by human just like the steam engine invented by Thomas Savery, Thomas Newcomen and James WattWatt. Without human, mathematics does not exist in nature. In other words, mathematics is invented rather discovered.

The reason I spend a long paragraph to state if the mathematics is invented or discovered is that the modern science development is based on a tool, a language i.e. mathematics created by human. The language helps us to comprehend our perceptions. Just like any other language including linguistic ones, mathematics could only describe the reality to a certain degree while the true reality is far beyond.

One of the interesting phenomena in quantum mechanics is that an object could appear at two different locations at same time. This is only a mathematical illusion and has nothing to do with the reality. In my opinion, the problem is mainly from the concept of location. The properties such as location and time are only ideas that are comprehensible by our brains. In reality they do not exist at all.

More writings are coming.