Daniel Niv mathematical pondering 1

Comments: Hah! Now this I can read fast.

"If this is a simple question and I am missing something obvious please forgive me, however, I believe that I have recently entered into a very interesting train of thought. Recently in my AP Calculus BC class, I was studying functions, which is a large part of Calc BC. During one of the exercises that we worked on during class and on a quiz it was necessary to enter a stat list and then graph it using a linear, quadratic, exponential, logarithmic... essentially whatever type of regresion that the teacher specified. Since I realize that the graphs are all approximations, I was wondering... If there is a relation that meets the requirements of being a function, so suppose a part of a relation... "

-Given n (finitely many) cartesian points such that NO POINTS have the same x value, we can interpolate the data using an n degree polynomials. http://mathworld.wolfram.com/LagrangeInterpolatingPolynomial.html

Is there, in all cases, a formula for each of these functions. The reason I ask this is because they fit the definition of a function and for this reason must be functions. Sometimes however these graphs seem really complex. For example, when people look at a set of data comparing the year the olymipcs was held with the winning height of the olympic pole vault, (This is just an example), shouldn't there, by the reasoning used earlier, always be a graph that satisfies every single data point. This makes me wonder.

-By lagrange interpolation, as long as the aforementioned condition is satisfied, will guarentee to go through all points. But this shows no trends. The point of statistical regression is to say that it follows an "easy" graph like linear or exponential. Linear graphs such simple correlations (more food causes more weight) and exponentiation shows extreme explosions (such as populations).

"The next step of my logic is that since our formulas for the relationship between the winning heights and the year of the olympics keep changing each year as new results come out, this would imply one of two things. Case A: If our formula fits every point perfectly in the set of data that we already had, then we already have an equation that will exactly predict the winning future heights. "

Not mathematics and not a proof. Just because all perfect numbers less than 1000000000000000000000000000000000000000000000000000000000000000000000 are even, does not mean that no odd perfect numbers exist.

Case B: Case A is empirically proven to be false and so we need to look at the facts again.

Not mathematics...as Doctor House on House normally says, exception to every rule.

"The first true statement which we know is that we have an exact formula for the set of data until the next winning height at the pole vault occurs. What can we do with this information? Well first when we check in the results for the next olympics we see that the earlier formula fails... However, we know the formula was exact for the earlier data and we know that the olympics function does have a formula though. Even if the function is a relation where the winning height is the same for a few years there technically should be a formula. However, first of all the chances of jumping the exact same number of femto meters up in the air is very very minute. "

Ok.. 10^-15.

"Still, there should still be a formula for the relation. So from our current information, our earlier formula didn't work..."

World doesn't always have obviously formulas for everything. I wish I had an explicit formula that when given n would spit out the nth prime number. Or given a year, will tell me what I accomplish that year.

However, we now know that there is a formula for the previous data set plus the year and winning height at the new olympics. From our earlier theorem that every relation has a formula, there exists a different formula which satisfies the previous data and the new point added to the function. This notion is very strange for me for many different reasons.

"Let me explain... Look at the square function for four values x=1,2,3,4. We get the points (1,1), (2,4), (3,9), (4,16)."

Sure.

" I have tried many different times to plug in a different formula for x and still get the same y value in each of the four cases. Each time I tried I did not get a conclusive answer. Now, it is easy to see that, for example, there are all these little patterns that can be used to find a fitting formula that is completely different than the original function. For example, 1x3+1=4; 4x3-3=9; 9x3-11=16. "

That is because it is impossible..function is strictly increasing at an increasing rate as we move to the right.

"Well, we can probably suppose that there exists a formula for the numbers -1 (because we subtracted negative 1), 3, 11... However, if we are just given three numbers there are so many ways to get these three values it seems countless how many different formulas any relation has. "

Correct. There are infinitely many real numbers that 4 numbers. We do not need our functions to be continuous.

"Either all the ways of expressing 1, 4, 9, 16 end up in the formula x^2 which would mean that there can not always be a function for any set of data or there is a different function that fits the exact set of points. "

The mapping is only a function if each x determines a unique (one and only one) value f(x). If you are explicitly given all points on the reals there exists only one function. But if you are given any finite or countable set of data, there are an infinite number of possible functions.

"For this reason I wonder, couldn't people, within a math test that asks for a formula of a small set of data, have an infinite number of answers making answer keys useless because people could come up with the most obscure results and they would be correct. "

That is why it is is usually multiple choice and people often petition an alternative solution.

"And there could be no actual answer for the graph in the data. For this reason. I am going to use the axioms of set theory. "

Dude, I am so into logic. but now you need to give me a strict definition of relation because we have our own definition.

"If I know that any relation of n points has an infinite number of exact formulas because there theoretically exists, an infinite number of relations that are exactly the same as the original relation except for the fact that they have one more data point and these all have an exact formula which means all of those exact formulas fit the earlier relation which is one point shorter, and I can then state that as n goes to infinity I can repeat this process on a relation with an infinite number of points. So, for example, in theory the equation x^2 should not be the only solution to the parabola which represents the points (0,0) (-1,1) (1,1) etc... "

Absolutely, because you only gave me a countable number of points. But if you gave all the points on R then it is unique on R (the reals). But you did not give me all the points for C (the complex numbers). So I have infinitely many functions here.

There should be an infinite number of other equations which should be able to satisfy this graph. But this notion is absurd... If this were true... then the graph x^2 is not a unique graph.

It is a unique graph. The confusion lies in the fact that humans are restricted to only listing a finite number of points. But it is defined for the huge continuum of real numbers which humans cannot count even a fraction of.

" or a completely different equation can represent this graph. this thought makes me extremely interested. Because that means there are an infinite number of graphs which may not even rely on exponents which use other operations on x to produce the exact same result as the square function would produce if applied to x. This result is perplexing and astounding. x^2 is not the only function which fits the points that lie on x^2. But if this were true... couldn't there be an infinite number of formulas for any function or relation... This does not seem to make any sense at all... "

Calculators cannot define x^2 on all possible points because they do not have enough memory. The best they can do is fill in a fill points and interpolate using a polynomial to make it look somewhat nice.

"Please help me understand either why this argument is wrong or if it is actually correct. Because if my idea is correct then I have either created a parathox or/and I am completely and utterly confused. "

This is a paradox that has been accepted so I guess it is not a paradox. There are something in the world that we cannot possibly count (uncountable) even if we had all eternity. We are humans living in a continuous world, cursed only to see, hear and talk in the discrete

"To reach this result however one must prove that any relation has a formula which generates every term. Except this seems like an elementary notion. However, if this were true then every function would not be unique... This would deeply affect graph theory (maybe) I think. I am not sure but the notion of two different formulas for the same graph is somewhat perplexing. "

We can have two formulas represented by two different graphs. We can also have two graphs represent the same formula (we call it bisimulation)

"Please give me your opinion on this strange phenomenon because it makes no sense and because I'm wondering either why noboy's noticed this or why it isn't a big buzz right now because I haven't found anything about this topic anywhere. Thank you very much for reading and I hope for a reply as soon as you read this and have time to digest the information. Thank you once again. "

Sorry, but you are more than 100 years too late: http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument. Cantor's argument is FAMOUS.

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Submitted:

Philip, I have a really interesting question I was wondering about. If this is a simple question and I am missing something obvious please forgive me, however, I believe that I have recently entered into a very interesting train of thought. Recently in my AP Calculus BC class, I was studying functions, which is a large part of Calc BC. During one of the exercises that we worked on during class and on a quiz it was necessary to enter a stat list and then graph it using a linear, quadratic, exponential, logarithmic... essentially whatever type of regresion that the teacher specified. Since I realize that the graphs are all approximations, I was wondering... If there is a relation that meets the requirements of being a function, so suppose a part of a relation... Is there, in all cases, a formula for each of these functions. The reason I ask this is because they fit the definition of a function and for this reason must be functions. Sometimes however these graphs seem really complex. For example, when people look at a set of data comparing the year the olymipcs was held with the winning height of the olympic pole vault, (This is just an example), shouldn't there, by the reasoning used earlier, always be a graph that satisfies every single data point. This makes me wonder. The next step of my logic is that since our formulas for the relationship between the winning heights and the year of the olympics keep changing each year as new results come out, this would imply one of two things. Case A: If our formula fits every point perfectly in the set of data that we already had, then we already have an equation that will exactly predict the winning future heights. Case B: Case A is empirically proven to be false and so we need to look at the facts again. The first true statement which we know is that we have an exact formula for the set of data until the next winning height at the pole vault occurs. What can we do with this information? Well first when we check in the results for the next olympics we see that the earlier formula fails... However, we know the formula was exact for the earlier data and we know that the olympics function does have a formula though. Even if the function is a relation where the winning height is the same for a few years there technically should be a formula. However, first of all the chances of jumping the exact same number of femto meters up in the air is very very minute. Still, there should still be a formula for the relation. So from our current information, our earlier formula didn't work... However, we now know that there is a formula for the previous data set plus the year and winning height at the new olympics. From our earlier theorem that every relation has a formula, there exists a different formula which satisfies the previous data and the new point added to the function. This notion is very strange for me for many different reasons. Let me explain... Look at the square function for four values x=1,2,3,4. We get the points (1,1), (2,4), (3,9), (4,16). I have tried many different times to plug in a different formula for x and still get the same y value in each of the four cases. Each time I tried I did not get a conclusive answer. Now, it is easy to see that, for example, there are all these little patterns that can be used to find a fitting formula that is completely different than the original function. For example, 1x3+1=4; 4x3-3=9; 9x3-11=16. Well, we can probably suppose that there exists a formula for the numbers -1 (because we subtracted negative 1), 3, 11... However, if we are just given three numbers there are so many ways to get these three values it seems countless how many different formulas any relation has. Either all the ways of expressing 1, 4, 9, 16 end up in the formula x^2 which would mean that there can not always be a function for any set of data or there is a different function that fits the exact set of points. For this reason I wonder, couldn't people, within a math test that asks for a formula of a small set of data, have an infinite number of answers making answer keys useless because people could come up with the most obscure results and they would be correct. And there could be no actual answer for the graph in the data. For this reason. I am going to use the axioms of set theory. If I know that any relation of n points has an infinite number of exact formulas because there theoretically exists, an infinite number of relations that are exactly the same as the original relation except for the fact that they have one more data point and these all have an exact formula which means all of those exact formulas fit the earlier relation which is one point shorter, and I can then state that as n goes to infinity I can repeat this process on a relation with an infinite number of points. So, for example, in theory the equation x^2 should not be the only solution to the parabola which represents the points (0,0) (-1,1) (1,1) etc... There should be an infinite number of other equations which should be able to satisfy this graph. But this notion is absurd... If this were true... then the graph x^2 is not a unique graph; or a completely different equation can represent this graph. this thought makes me extremely interested. Because that means there are an infinite number of graphs which may not even rely on exponents which use other operations on x to produce the exact same result as the square function would produce if applied to x. This result is perplexing and astounding. x^2 is not the only function which fits the points that lie on x^2. But if this were true... couldn't there be an infinite number of formulas for any function or relation... This does not seem to make any sense at all... Please help me understand either why this argument is wrong or if it is actually correct. Because if my idea is correct then I have either created a parathox or/and I am completely and utterly confused. To reach this result however one must prove that any relation has a formula which generates every term. Except this seems like an elementary notion. However, if this were true then every function would not be unique... This would deeply affect graph theory (maybe) I think. I am not sure but the notion of two different formulas for the same graph is somewhat perplexing.

Please give me your opinion on this strange phenomenon because it makes no sense and because I'm wondering either why noboy's noticed this or why it isn't a big buzz right now because I haven't found anything about this topic anywhere. Thank you very much for reading and I hope for a reply as soon as you read this and have time to digest the information. Thank you once again. - Daniel Niv