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Consequences of Mathematical Notation   Rating: (3.4 / 10)    Views: 1,480

Submitted By: Bensen on 10/29/2006. (  |  Share  |  Clikk It! )   

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Clear concise mathematical notation allows our minds to grapple with problems and thought patterns that it would normally be unable to grasp. For instance, before there was a clear number system there was no real way in which to count something. If one wanted to count, for instance how many sheep they had, they literally would have no way of knowing how many sheep they actually possessed.

They could of course develop other means of solving problems, such as figuring out if they had the same amount of sheep, say in the evening, as they did in the morning. This could be accomplished by doing something like having stones in a bag, which represented each individual animal in a one to one correspondence. With this method they could take one of the stones from the first bag and place it in a second bag and in the end if the first bag is exactly emptied when the last sheep walked by, they would know they had the same amount as they had before. In a similar fashion they could deduce greater than and less than properties, but never necessarily know how many more or less they had within a certain limit, as the human brain seems to be able to only naturally count to around five to eight without some other form of representing “numbers” higher than that.

In order to compensate for that inherent weakness in our psyche, we have come up with a numeric system where we can count up to an infinite value, if we have the time. Once we had this system in place, the question arises what does it mean to add one number to another number? From that we figured out that if we have ten sheep and want to add another group of ten sheep we know that the total should add up to twenty sheep and we came up with an operator to indicate the addition operation, namely the “+” symbol. From there examples can get significantly more complicated, such as multiplication, reciprocals, squares etc. All these have a very specific meaning and are a way to quantify certain things in the real world.

Some of these more complicated mathematical representations provoke problems, in that, for instance, what does it mean to say something is the square root of a negative number? Symbolically this might make complete sense in a given set of equations that follow the rules of the language of mathematics, but intuitively it is ambiguous. Such as in the case where you have the equation: x^3 = 15x + 4, which, by substitution, x can be shown to equal 4. However, if you try to solve this problem you will end up coming up with this equation: x = cuberoot( 2 + sqrt(-121)) + cuberoot(2 - sqrt(-121)).

These problems arises from the fact that when you start having equations and symbolic notations that allow for orthogonal manipulations you can end up with expressions that are problematic in that what the expression says and what it actually represents can be ambiguous or problematic in meaning. In order to get around some of these problems that arise in orthogonal manipulations, mathematicians had to invent new mathematical domains of discourse, such as in case of the imaginary and complex numbers.

Similarly in orthogonal programming languages we end up at times being able to represent in code ideas that we previously were unable to think about clearly. It allows us to think in new ways that at times can arouse interesting new problems similar to situations in mathematics. In the end, this also allows us to solve certain problems using simpler methods, or at times even to actually be able to think about and solve a previously “unsolvable” problem, which our normal faculties of thought would not have been able to grasp some given concept. This is characterized by Dijkstra when he said “Lisp has jokingly been called ‘the most intelligent way to misuse a computer’. I think that description is a great compliment because it transmits the full flavor of liberation: it has assisted a number of our most gifted fellow humans in thinking previously impossible thoughts.” (Edsger Dijkstra, CACM, 15:10)


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