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Some parts of mathematics are developed by quantifying completely non mathematical relationships, specifying what to count, why to count, way of counting, and sequence of specific calculations. Perhaps, without that "outside mathematics" relationships certain parts of math would not be developed. Example is calculus. It is fascinating, to me, how completely arbitrary, even subjective, sometimes very vague descriptions can lead into specific mathematical operations and calculation sequences. Example can be valuation of financial instruments, Black-Scholes equation and calculus. However, all mathematics can be developed without any outside input. We are used to think that, perhaps, physics, for instance, developed mathematics. But, it is not so. It may only motivate to select certain sets, numbers and sequence of calculations on them. These sequences can be developed within math too. Outside world may motivate us what to count, when, and in what order. Fair to say, it can lead to selection of certain mathematical mechanisms. But, again, those mechanisms can be constructed within math too, independently from any outside interference.
Now, here is one interesting thing. Many of us are trying to find the relation between the real world and mathematics. How we can apply mathematics to real world? We may have a tendency to think that math is linked, that it is dependent on the real world. But it is not always so. Math can not distinguish, can not tell between real and fictional world or concepts. For example, if you have two dragons, and each dragon ate 53 kg of coal, how much coal in total they ate? The answer is 2 x 53 = 106 kg of coal. Here we used mathematics in a fairy tale situation. Moreover, we have used fictional, even untruthful concepts to do calculation. The dragons have nothing to do with physics or with real world! Especially coal eating. Yet, mathematics can not see that. It can not tell if the relationship between numbers you provide to it is fictional or real. Math can not see why you do or ask for calculation. It is you that specify what and why has to be counted, calculated and those requests you plug into math.
You may ask now, how math can be developed independently. Here is one illustration. Let's say a person writes number 7. Then he writes number 25. You don't know whether he counted anything or he just plays with numbers. He may add these two numbers, subtract. Whatever he wants. Let's say he multiplies them. So, 7 x 25 = 175. Looking at this mathematics operations you can not tell what is counted, if it is counted at all. It can be just play with numbers. Yet, it can be 7 train cars with 25 tones of grain on each, or 7 dragons with 25 kilograms of coal they ate, or 7 boxes with 25 CDs in each. The common calculation for both is 7 x 25 = 175. You can reuse this general mathematical result in all those cases. But, again, you can still independently play with numbers, as we have just seen, without keeping track what is counted. You may be interested to work with numbers only. Meaning, you can, just say now, 8 x 30 = 240. Then say that 240 > 175. These investigations of numbers properties constitutes mathematics as independent discipline.
You can read more "How math can be applied to so many different fields and how we can use math in real life".
Some parts of mathematics are developed by quantifying completely non mathematical relationships, specifying what to count, why to count, way of counting, and sequence of specific calculations. Perhaps, without that "outside mathematics" relationships certain parts of math would not be developed. Example is calculus. It is fascinating, to me, how completely arbitrary, even subjective, sometimes very vague descriptions can lead into specific mathematical operations and calculation sequences. Example can be valuation of financial instruments, Black-Scholes equation and calculus. However, all mathematics can be developed without any outside input. We are used to think that, perhaps, physics, for instance, developed mathematics. But, it is not so. It may only motivate to select certain sets, numbers and sequence of calculations on them. These sequences can be developed within math too. Outside world may motivate us what to count, when, and in what order. Fair to say, it can lead to selection of certain mathematical mechanisms. But, again, those mechanisms can be constructed within math too, independently from any outside interference.
Now, here is one interesting thing. Many of us are trying to find the relation between the real world and mathematics. How we can apply mathematics to real world? We may have a tendency to think that math is linked, that it is dependent on the real world. But it is not always so. Math can not distinguish, can not tell between real and fictional world or concepts. For example, if you have two dragons, and each dragon ate 53 kg of coal, how much coal in total they ate? The answer is 2 x 53 = 106 kg of coal. Here we used mathematics in a fairy tale situation. Moreover, we have used fictional, even untruthful concepts to do calculation. The dragons have nothing to do with physics or with real world! Especially coal eating. Yet, mathematics can not see that. It can not tell if the relationship between numbers you provide to it is fictional or real. Math can not see why you do or ask for calculation. It is you that specify what and why has to be counted, calculated and those requests you plug into math.
You may ask now, how math can be developed independently. Here is one illustration. Let's say a person writes number 7. Then he writes number 25. You don't know whether he counted anything or he just plays with numbers. He may add these two numbers, subtract. Whatever he wants. Let's say he multiplies them. So, 7 x 25 = 175. Looking at this mathematics operations you can not tell what is counted, if it is counted at all. It can be just play with numbers. Yet, it can be 7 train cars with 25 tones of grain on each, or 7 dragons with 25 kilograms of coal they ate, or 7 boxes with 25 CDs in each. The common calculation for both is 7 x 25 = 175. You can reuse this general mathematical result in all those cases. But, again, you can still independently play with numbers, as we have just seen, without keeping track what is counted. You may be interested to work with numbers only. Meaning, you can, just say now, 8 x 30 = 240. Then say that 240 > 175. These investigations of numbers properties constitutes mathematics as independent discipline.
You can read more "How math can be applied to so many different fields and how we can use math in real life".
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