Chemistry: 3mm^3 of H20 was produced in ten seconds.
Biology: The number/mass of bacteria doubled in a day.
Physics: Ten complete oscillations of the pendulum took 14s.
Psychology: In 2 weeks, 75% of the patients given nothing but placebo show improvement.
Economics: The increase in prices by 10% has created an increase profit of 20%.
Math: When I was small, in a time I can now barely remember, I saw that 3 people giving me 4 apples each meant I had 12 apples in total.
They all use empirical data indeed.
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You now what I said, those experiment are totally different because for one, I can test the validity of n^2 - n + 41, without the use of any of my senses! No emprical data gathered. Tell me how you can claim to have a new theory in science without showing data?
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Approximations are made in math, if there's PROOF that it is actually an approximation. Like using taylor series to approximate sin and cosine function.
If you have proof that the approximation is always within a certain range from the real value, and can easily show that some other factor is way beyond that approximation, then it's obviously also way beyond that actual value.
But not once will it be accepted in a mathematical proof that since plotting a graph of the number of primes less than x against x produces a curve close to ln(x)/x, that this is now the official approximation to be used for even greater values of x that haven't been graphed yet!
If someone starts chanting numbers and from the very beginning goes on saying "1, 1, 1, 1, 1, 1, 1,...". Science will accept the next number to be one until something happens, and many theories will be based on this. In math, no theory would ever be based on this and accepted because there's no guarantee that the next number
x is with the range
1-a<1<1+a, for a fixed real positive number
a.
Justifiable approximation (in math) is very different from approximation used in science.
Amadeo wrote:Approximations are made in mathematics all the time. All the time.
Nobody knows how to integrate x^x exactly. Nobody knows the exact values of the normal distribution. Nobody knows the exact solution to x * e^x = 1
It can be proven that x is within a given range from the proposed solution/approximation with use of taylor expansion. And x can be found with arbitrary precision. You want it to 5 decimal places? No problem, 100 decimal places possible. It's an irrational number so trying to find the exact form in decimal form is pointless as it has infinite numbers, just like pi or root of 2.
But here's another place where math differs from science again, the exact value of this solutions isn't that important. It's only important when scientist want to use math to understand their observations! Mathematician are simply creating tools for solving equations through theorems, and it's upto scientists to come up with the usefulness in real world.
Just because I'm creating a car, it's doesn't automatically make me a driver!