Let me start this blog by saying that I am not stupid or crazy as you might have thought after reading the title of this blog. The purpose of this article is not to question the basics of the arithmetic, but to show some new ways of understanding mathematics. As I have learned through many conversations with friends, people often perceive mathematics as sort of a scary beast locked in a castle full of ignorant, boring and incomprehensible mathematicians. But it's really not. Let me illustrate my point by an excerpt from an ancient legend of the Ulysses.
"When Ulysses had left the land of the Cyclops, after blinding Polyphemus, the poor old giant used to sit every morning near the entrance to the cave with a heap of pebbles and pick up one for every ewe that he let pass. In the evening when the ewes returned, he would drop one pebble for every ewe that he admitted to the cave. In this way, by exhausting the stock of pebbles that he had picked up in the morning he ensured that all his flock had returned."
So from the very beginning mathematics was based on association: individuals of the flock could be matched, one by one, with those of a heap of pebbles so that both were exhausted together, then the two groups were equal. The same principle was applied when inventing numbers like 1, 2, 3...etc which eventually became just symbolic representations of quantitative meanings attached to them. In fact, if the Mesopotamian man had decided that 2 has to be "one", then today we would be using 2 instead of 1 to represent the unit quantity. And in fact there is no reason why we shouldn't do that. After all, numbers are just symbols, placeholders we created in order to simplify the world surrounding us. Indeed mathematics is just another language we are operating in order to systemize, rationalize and conceptualize the complex world we are living in. Of course, mathematics can get complicated sometimes, but it's only because of the complexities underlying it. Yet, there is also a direct link of mathematics to a conventional understanding of the language - mathematical logic. This branch of mathematics serves like a translation program converting the language we use every day into its mathematical representation.
Since in logic we deal with statements or propositions which have some meaning, every such proposition is either true or false. We can assign the truth value T=1 when the proposition is true, and T=0 when it is false. Let's consider proposition A, which is true either if statements X or Y are correct. Let's also consider proposition B, which is true if statements W and Z are correct. Following this logic, our arithmetic additions would be as following.
Statement A: X+Y=A, 1+0=1, 1+1=1, 0+1=1
Statement B: W+Z=B, 1+0=0, 1+1=1, 0+1=0
As weird as it looks from the "common sense" it is still true, according to the logical rules we set, that one plus one is one! And in fact, we are not doing anything very complicated here, Polyphemus used exactly the same association technique when he was putting a pebble for each ewe, and we are "putting" "1" for every statement which is true.
So here we are probably at the point where you are asking yourself, if you have just wasted 3minutes of your life reading some impractical, nonsensical mathematical daydreaming. Well, not really. The fact that you are reading this blog means that you are reaping the fruits of the direct application of the mathematical logic described above. How? Just think about computers. In their very early days, they were used just by mathematicians as computation systems spitting out some preprogrammed answers (outcomes of the logical propositions) in numbers, which could later be translated into everyday language. When those systems grew more complex, computers "were taught" to do more complicated task like monitoring patient's heartbeat, connecting us to the internet and many more. But at the end, it's still the same 1's and 0's, pebbles and ewes, making those entire monster Pentiums exist.
"When Ulysses had left the land of the Cyclops, after blinding Polyphemus, the poor old giant used to sit every morning near the entrance to the cave with a heap of pebbles and pick up one for every ewe that he let pass. In the evening when the ewes returned, he would drop one pebble for every ewe that he admitted to the cave. In this way, by exhausting the stock of pebbles that he had picked up in the morning he ensured that all his flock had returned."So from the very beginning mathematics was based on association: individuals of the flock could be matched, one by one, with those of a heap of pebbles so that both were exhausted together, then the two groups were equal. The same principle was applied when inventing numbers like 1, 2, 3...etc which eventually became just symbolic representations of quantitative meanings attached to them. In fact, if the Mesopotamian man had decided that 2 has to be "one", then today we would be using 2 instead of 1 to represent the unit quantity. And in fact there is no reason why we shouldn't do that. After all, numbers are just symbols, placeholders we created in order to simplify the world surrounding us. Indeed mathematics is just another language we are operating in order to systemize, rationalize and conceptualize the complex world we are living in. Of course, mathematics can get complicated sometimes, but it's only because of the complexities underlying it. Yet, there is also a direct link of mathematics to a conventional understanding of the language - mathematical logic. This branch of mathematics serves like a translation program converting the language we use every day into its mathematical representation.
Since in logic we deal with statements or propositions which have some meaning, every such proposition is either true or false. We can assign the truth value T=1 when the proposition is true, and T=0 when it is false. Let's consider proposition A, which is true either if statements X or Y are correct. Let's also consider proposition B, which is true if statements W and Z are correct. Following this logic, our arithmetic additions would be as following.
Statement A: X+Y=A, 1+0=1, 1+1=1, 0+1=1
Statement B: W+Z=B, 1+0=0, 1+1=1, 0+1=0
As weird as it looks from the "common sense" it is still true, according to the logical rules we set, that one plus one is one! And in fact, we are not doing anything very complicated here, Polyphemus used exactly the same association technique when he was putting a pebble for each ewe, and we are "putting" "1" for every statement which is true.
So here we are probably at the point where you are asking yourself, if you have just wasted 3minutes of your life reading some impractical, nonsensical mathematical daydreaming. Well, not really. The fact that you are reading this blog means that you are reaping the fruits of the direct application of the mathematical logic described above. How? Just think about computers. In their very early days, they were used just by mathematicians as computation systems spitting out some preprogrammed answers (outcomes of the logical propositions) in numbers, which could later be translated into everyday language. When those systems grew more complex, computers "were taught" to do more complicated task like monitoring patient's heartbeat, connecting us to the internet and many more. But at the end, it's still the same 1's and 0's, pebbles and ewes, making those entire monster Pentiums exist.
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