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Are some aspects of Calculus not real math?

Ahmed.M
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6/13/2012 8:31:26 PM
Posted: 4 years ago
I think some aspects of calculus breaks math rules. When taking limits in calculus such as first principles, as h approaches zero, one has a choice whether making h actually be zero or 0.000001. This breaks rules because of the dual option. The 'zero' portion allows you to get rid of terms such as 2h, 10xh etc in a polynomial equation and the 0.0000001 choice allows you to divide by h when h is the denominator since you cannot divide by zero. Are these dual options justified?

However, I obviously acknowledge how great calculus is and the contributions it has made to solve the tangent problem, provide equations for getting the slope of the tangent, the rules to finding all sorts of derivatives etc etc. The list goes on.
royalpaladin
Posts: 22,357
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6/13/2012 8:34:18 PM
Posted: 4 years ago
"Real math" does not exist. Math is based on a lot of assumptions called "axioms" that we accept as true for no valid reason.

Case in point: you've probably been taught Euclidean geometry. Shift the assumptions and learn circle geometry. It's different :)
Nur-Ab-Sal
Posts: 1,637
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6/13/2012 8:43:00 PM
Posted: 4 years ago
At 6/13/2012 8:31:26 PM, Ahmed.M wrote:
I think some aspects of calculus breaks math rules. When taking limits in calculus such as first principles, as h approaches zero, one has a choice whether making h actually be zero or 0.000001. This breaks rules because of the dual option. The 'zero' portion allows you to get rid of terms such as 2h, 10xh etc in a polynomial equation and the 0.0000001 choice allows you to divide by h when h is the denominator since you cannot divide by zero. Are these dual options justified?

However, I obviously acknowledge how great calculus is and the contributions it has made to solve the tangent problem, provide equations for getting the slope of the tangent, the rules to finding all sorts of derivatives etc etc. The list goes on.

To answer your question: I'm not exactly sure where in calculus you are, but estimates can be made at times in Calculus. In fact, there are entire "methods" in Calculus devoted to estimation.
Genesis I. And God created man to his own image: to the image of God he created him: male and female he created them.
Ahmed.M
Posts: 616
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6/13/2012 8:44:01 PM
Posted: 4 years ago
At 6/13/2012 8:34:18 PM, royalpaladin wrote:
"Real math" does not exist. Math is based on a lot of assumptions called "axioms" that we accept as true for no valid reason.

I'm not trying to get into philosophy and epistemic problems. I'm talking about the the laws of math and how modern calculus bends them. One example is when one is finding the derivative from first principles.

Case in point: you've probably been taught Euclidean geometry. Shift the assumptions and learn circle geometry. It's different :)

What? I'm not talking about geometry, but calculus.

Anyways since you brought up circles and I randomly thought about this. Let me ask you a question: Is a circle really a circle?
Ahmed.M
Posts: 616
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6/13/2012 8:45:36 PM
Posted: 4 years ago
At 6/13/2012 8:43:00 PM, Nur-Ab-Sal wrote:

To answer your question: I'm not exactly sure where in calculus you are, but estimates can be made at times in Calculus. In fact, there are entire "methods" in Calculus devoted to estimation.

I agree with you and that is what I'm questioning.
royalpaladin
Posts: 22,357
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6/13/2012 8:46:38 PM
Posted: 4 years ago
At 6/13/2012 8:44:01 PM, Ahmed.M wrote:
At 6/13/2012 8:34:18 PM, royalpaladin wrote:
"Real math" does not exist. Math is based on a lot of assumptions called "axioms" that we accept as true for no valid reason.

I'm not trying to get into philosophy and epistemic problems. I'm talking about the the laws of math and how modern calculus bends them. One example is when one is finding the derivative from first principles.

The "laws of math" are based on axioms.
Case in point: you've probably been taught Euclidean geometry. Shift the assumptions and learn circle geometry. It's different :)

What? I'm not talking about geometry, but calculus.

Calculus is based on both geometry and algebra, but it really doesn't matter. I was just demonstrating that there is no such thing as "real math" by providing an example from another branch of math in which the rules change entirely if the axioms change.
Anyways since you brought up circles and I randomly thought about this. Let me ask you a question: Is a circle really a circle?

What are you trying to get at?
Nur-Ab-Sal
Posts: 1,637
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6/13/2012 8:46:44 PM
Posted: 4 years ago
At 6/13/2012 8:31:26 PM, Ahmed.M wrote:
I think some aspects of calculus breaks math rules. When taking limits in calculus such as first principles, as h approaches zero, one has a choice whether making h actually be zero or 0.000001. This breaks rules because of the dual option. The 'zero' portion allows you to get rid of terms such as 2h, 10xh etc in a polynomial equation and the 0.0000001 choice allows you to divide by h when h is the denominator since you cannot divide by zero. Are these dual options justified?

However, I obviously acknowledge how great calculus is and the contributions it has made to solve the tangent problem, provide equations for getting the slope of the tangent, the rules to finding all sorts of derivatives etc etc. The list goes on.

Ah I see where you are in Calculus. If you're learning about limits, you will probably be taught to make charts like these:

f(x) = x^2

f(.9)
f(.99)
f(.999)
f(.1)
f(.01)
f(.001)

It's just an example to show how the function is getting closer and closer to a certain number from either side of that number.
Genesis I. And God created man to his own image: to the image of God he created him: male and female he created them.
royalpaladin
Posts: 22,357
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6/13/2012 8:48:05 PM
Posted: 4 years ago
At 6/13/2012 8:45:36 PM, Ahmed.M wrote:
At 6/13/2012 8:43:00 PM, Nur-Ab-Sal wrote:

To answer your question: I'm not exactly sure where in calculus you are, but estimates can be made at times in Calculus. In fact, there are entire "methods" in Calculus devoted to estimation.

I agree with you and that is what I'm questioning.

I'm not sure why you are uneasy with it. It's obvious that according to accepted math, division by 0 lead to infinity/is undefined. Calculus just eliminates that in order to estimate the limit. It's not an exact calculation.
Ahmed.M
Posts: 616
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6/13/2012 8:51:07 PM
Posted: 4 years ago
At 6/13/2012 8:46:38 PM, royalpaladin wrote:
What are you trying to get at?

I'm just asking you a question haha. Stop being so paranoid. Is a circle a circle??

BTW I agree with all your other points. Your getting into epistemology and philosophy. I know that if 2+2=5 was taken as an axiom then everything would change but I was talking about the standard.
darkkermit
Posts: 11,204
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6/13/2012 8:51:28 PM
Posted: 4 years ago
I'm pretty sure that our ability to get to the moon and back using principles of calculus kind of proves its validity.
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Ahmed.M
Posts: 616
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6/13/2012 8:53:58 PM
Posted: 4 years ago
At 6/13/2012 8:46:44 PM, Nur-Ab-Sal wrote:

Ah I see where you are in Calculus. If you're learning about limits, you will probably be taught to make charts like these:

f(x) = x^2

f(.9)
f(.99)
f(.999)
f(.1)
f(.01)
f(.001)

It's just an example to show how the function is getting closer and closer to a certain number from either side of that number.

I know, I'm actually way passed that section. What I'm saying is that you can choose whether to make the number 0.0000000001 or 0. The reason why limits were invented was to bypass the division of zero and to still use zero!!

0.00000001 is very different than 0 in terms of calculations.
Nur-Ab-Sal
Posts: 1,637
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6/13/2012 8:57:09 PM
Posted: 4 years ago
At 6/13/2012 8:53:58 PM, Ahmed.M wrote:
At 6/13/2012 8:46:44 PM, Nur-Ab-Sal wrote:

Ah I see where you are in Calculus. If you're learning about limits, you will probably be taught to make charts like these:

f(x) = x^2

f(.9)
f(.99)
f(.999)
f(.1)
f(.01)
f(.001)

It's just an example to show how the function is getting closer and closer to a certain number from either side of that number.

I know, I'm actually way passed that section. What I'm saying is that you can choose whether to make the number 0.0000000001 or 0. The reason why limits were invented was to bypass the division of zero and to still use zero!!

0.00000001 is very different than 0 in terms of calculations.

What do you mean by this? Where do you choose the product of your calculation?
Genesis I. And God created man to his own image: to the image of God he created him: male and female he created them.
Ahmed.M
Posts: 616
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6/13/2012 8:59:41 PM
Posted: 4 years ago
I explained in the OP that for first principles you can choose whether to make 'h' 0.00000000001 (as many zeros as you want) or 0. When you want to cancel terms such as 4h you make it zero but when you are dividing by 'h' you are forced to make it 0.0000000000001 which plays and bends the rules. Is it 0.0000000000000001 or 0? You choose.
Ahmed.M
Posts: 616
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6/13/2012 9:00:32 PM
Posted: 4 years ago
At 6/13/2012 8:48:05 PM, royalpaladin wrote:

I'm not sure why you are uneasy with it. It's obvious that according to accepted math, division by 0 lead to infinity/is undefined. Calculus just eliminates that in order to estimate the limit. It's not an exact calculation.

Actually calculus determines the exact slope of a curve using the equation of the derivative.
Nur-Ab-Sal
Posts: 1,637
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6/13/2012 9:03:43 PM
Posted: 4 years ago
At 6/13/2012 8:59:41 PM, Ahmed.M wrote:
I explained in the OP that for first principles you can choose whether to make 'h' 0.00000000001 (as many zeros as you want) or 0. When you want to cancel terms such as 4h you make it zero but when you are dividing by 'h' you are forced to make it 0.0000000000001 which plays and bends the rules. Is it 0.0000000000000001 or 0? You choose.

I have been through two Calculus courses (AP AB & AP BC) and have no idea what you are talking about.
Genesis I. And God created man to his own image: to the image of God he created him: male and female he created them.
Nur-Ab-Sal
Posts: 1,637
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6/13/2012 9:04:47 PM
Posted: 4 years ago
At 6/13/2012 9:00:32 PM, Ahmed.M wrote:
At 6/13/2012 8:48:05 PM, royalpaladin wrote:

I'm not sure why you are uneasy with it. It's obvious that according to accepted math, division by 0 lead to infinity/is undefined. Calculus just eliminates that in order to estimate the limit. It's not an exact calculation.

Actually calculus determines the exact slope of a curve using the equation of the derivative.

Calculus isn't just about taking the derivative of a function (which does not necessarily need to be a curve). There are plenty of other 'parts' of Calculus... integration, summation, etc...
Genesis I. And God created man to his own image: to the image of God he created him: male and female he created them.
darkkermit
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6/13/2012 9:05:21 PM
Posted: 4 years ago
At 6/13/2012 9:00:32 PM, Ahmed.M wrote:
At 6/13/2012 8:48:05 PM, royalpaladin wrote:

I'm not sure why you are uneasy with it. It's obvious that according to accepted math, division by 0 lead to infinity/is undefined. Calculus just eliminates that in order to estimate the limit. It's not an exact calculation.

Actually calculus determines the exact slope of a curve using the equation of the derivative.

We know that calculus works and has useful application, so what difference does it make? If calculus can make predictions about the real world, then its enough for me.
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Ahmed.M
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6/13/2012 9:10:33 PM
Posted: 4 years ago
At 6/13/2012 9:03:43 PM, Nur-Ab-Sal wrote:
At 6/13/2012 8:59:41 PM, Ahmed.M wrote:
I explained in the OP that for first principles you can choose whether to make 'h' 0.00000000001 (as many zeros as you want) or 0. When you want to cancel terms such as 4h you make it zero but when you are dividing by 'h' you are forced to make it 0.0000000000001 which plays and bends the rules. Is it 0.0000000000000001 or 0? You choose.

I have been through two Calculus courses (AP AB & AP BC) and have no idea what you are talking about.

First principles is f ' (x)= lim h->0[f(x+h)-f(x)]/h

is it 0.0000000000001 or 0? You are allowed to choose whenever it helps you.
royalpaladin
Posts: 22,357
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6/13/2012 9:12:56 PM
Posted: 4 years ago
At 6/13/2012 9:10:33 PM, Ahmed.M wrote:
At 6/13/2012 9:03:43 PM, Nur-Ab-Sal wrote:
At 6/13/2012 8:59:41 PM, Ahmed.M wrote:
I explained in the OP that for first principles you can choose whether to make 'h' 0.00000000001 (as many zeros as you want) or 0. When you want to cancel terms such as 4h you make it zero but when you are dividing by 'h' you are forced to make it 0.0000000000001 which plays and bends the rules. Is it 0.0000000000000001 or 0? You choose.

I have been through two Calculus courses (AP AB & AP BC) and have no idea what you are talking about.

First principles is f ' (x)= lim h->0[f(x+h)-f(x)]/h

is it 0.0000000000001 or 0? You are allowed to choose whenever it helps you.

0.0000000000001 is an approximation.
Ahmed.M
Posts: 616
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6/13/2012 9:13:46 PM
Posted: 4 years ago
At 6/13/2012 9:05:21 PM, darkkermit wrote:

We know that calculus works and has useful application, so what difference does it
make? If calculus can make predictions about the real world, then its enough for me.

I already acknowledged everything you are saying. The difference is pointing out a mistake or an error in reasoning. Wouldn't you want people to point out a mistake even though the difference would be small. Everything matters.
Nur-Ab-Sal
Posts: 1,637
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6/13/2012 9:15:47 PM
Posted: 4 years ago
At 6/13/2012 9:10:33 PM, Ahmed.M wrote:
At 6/13/2012 9:03:43 PM, Nur-Ab-Sal wrote:
At 6/13/2012 8:59:41 PM, Ahmed.M wrote:
I explained in the OP that for first principles you can choose whether to make 'h' 0.00000000001 (as many zeros as you want) or 0. When you want to cancel terms such as 4h you make it zero but when you are dividing by 'h' you are forced to make it 0.0000000000001 which plays and bends the rules. Is it 0.0000000000000001 or 0? You choose.

I have been through two Calculus courses (AP AB & AP BC) and have no idea what you are talking about.

First principles is f ' (x)= lim h->0[f(x+h)-f(x)]/h

is it 0.0000000000001 or 0? You are allowed to choose whenever it helps you.

I'm aware of the formal definition of a derivative. Where are you plugging in '0' or '.000001'?

The only way to get a derivative of '.00001' is an original function of 'f(x)=.0001x'. Otherwise, taking the limit as h approaches zero of the function plus h minus the function, over h, is derived from the slope formula. I seriously haven't a clue where this choosing comes in. Taking a derivative, whether through the formal definition or one of the rules, is a fairly straightforward process.
Genesis I. And God created man to his own image: to the image of God he created him: male and female he created them.
Ahmed.M
Posts: 616
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6/13/2012 9:19:33 PM
Posted: 4 years ago
At 6/13/2012 9:12:56 PM, royalpaladin wrote:
At 6/13/2012 9:10:33 PM, Ahmed.M wrote:
At 6/13/2012 9:03:43 PM, Nur-Ab-Sal wrote:
At 6/13/2012 8:59:41 PM, Ahmed.M wrote:
I explained in the OP that for first principles you can choose whether to make 'h' 0.00000000001 (as many zeros as you want) or 0. When you want to cancel terms such as 4h you make it zero but when you are dividing by 'h' you are forced to make it 0.0000000000001 which plays and bends the rules. Is it 0.0000000000000001 or 0? You choose.

I have been through two Calculus courses (AP AB & AP BC) and have no idea what you are talking about.

First principles is f ' (x)= lim h->0[f(x+h)-f(x)]/h

is it 0.0000000000001 or 0? You are allowed to choose whenever it helps you.

0.0000000000001 is an approximation.

I know but we are dealing with an approximation with zero which is a completely different number. Zero is a very unique and different number from 0.00000000000000001, so I think it presents a small problem when they become interchangable.

What circles really are (click on links in order):
http://www.google.ca...
http://www.google.ca...
http://www.google.ca...

Get it?

Anyways I just posted this thread because this topic piqued my interest while doing my work even though no one probably cares about it.
Ahmed.M
Posts: 616
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6/13/2012 9:21:44 PM
Posted: 4 years ago
At 6/13/2012 9:15:47 PM, Nur-Ab-Sal wrote:

I'm aware of the formal definition of a derivative. Where are you plugging in '0' or '.000001'?

for h. You can choose between 0.0000000000000001 and zero.

The only way to get a derivative of '.00001' is an original function of 'f(x)=.0001x'. Otherwise, taking the limit as h approaches zero of the function plus h minus the function, over h, is derived from the slope formula. I seriously haven't a clue where this choosing comes in. Taking a derivative, whether through the formal definition or one of the rules, is a fairly straightforward process.
Nur-Ab-Sal
Posts: 1,637
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6/13/2012 9:26:12 PM
Posted: 4 years ago
At 6/13/2012 9:21:44 PM, Ahmed.M wrote:
At 6/13/2012 9:15:47 PM, Nur-Ab-Sal wrote:

I'm aware of the formal definition of a derivative. Where are you plugging in '0' or '.000001'?

for h. You can choose between 0.0000000000000001 and zero.

The only way to get a derivative of '.00001' is an original function of 'f(x)=.0001x'. Otherwise, taking the limit as h approaches zero of the function plus h minus the function, over h, is derived from the slope formula. I seriously haven't a clue where this choosing comes in. Taking a derivative, whether through the formal definition or one of the rules, is a fairly straightforward process.

No you cannot choose. The only way to get the dervative is for the distance between the two points for the slope to be zero. But since we are dividing by zero, we calculate its value using a limit, specifically one where the distance 'h' approaches zero. The slope of a line tangent to any point on f(x)=x^2 is 2x... No approximation there. Using .000001 will give you an approximation of this slope, because there is still a definite distance between the points from which the slope is calculated.
Genesis I. And God created man to his own image: to the image of God he created him: male and female he created them.
darkkermit
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6/13/2012 9:27:46 PM
Posted: 4 years ago
At 6/13/2012 9:13:46 PM, Ahmed.M wrote:
At 6/13/2012 9:05:21 PM, darkkermit wrote:

We know that calculus works and has useful application, so what difference does it
make? If calculus can make predictions about the real world, then its enough for me.

I already acknowledged everything you are saying. The difference is pointing out a mistake or an error in reasoning. Wouldn't you want people to point out a mistake even though the difference would be small. Everything matters.

How is it an error of reasoning, if it works out extraordinary well and makes good predictions?
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tBoonePickens
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6/14/2012 11:08:14 AM
Posted: 4 years ago
At 6/13/2012 9:27:46 PM, darkkermit wrote:
How is it an error of reasoning, if it works out extraordinary well and makes good predictions?
Precisely!
WOS
: At 10/3/2012 4:28:52 AM, Wallstreetatheist wrote:
: Without nothing existing, you couldn't have something.
tarkovsky
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6/14/2012 11:51:02 AM
Posted: 4 years ago
At 6/13/2012 8:31:26 PM, Ahmed.M wrote:
I think some aspects of calculus breaks math rules. When taking limits in calculus such as first principles, as h approaches zero, one has a choice whether making h actually be zero or 0.000001. This breaks rules because of the dual option. The 'zero' portion allows you to get rid of terms such as 2h, 10xh etc in a polynomial equation and the 0.0000001 choice allows you to divide by h when h is the denominator since you cannot divide by zero. Are these dual options justified?

However, I obviously acknowledge how great calculus is and the contributions it has made to solve the tangent problem, provide equations for getting the slope of the tangent, the rules to finding all sorts of derivatives etc etc. The list goes on.

I think you may need to be more specific here in order for me to understand you, but maybe you have misunderstood the exact meaning of a derivative.

One of the simplest definitions of a derivative is 'the limit of a the secant line'. That is to say, 'as the secant line goes to the tangent line, where does it go'? Also, that the derivative is defined in terms of the limit.

I see you are referring to "h" without mention of the limit. This is known as 'the difference quotient' and, strictly speaking, the difference quotient isn't 'the derivative'. What's the difference between the derivative and the difference quotient? The derivative is the limit of the difference quotient:

lim f(x+h)-f(x)/h
h->0

At 6/13/2012 8:31:26 PM, Ahmed.M wrote:
This breaks rules because of the dual option... Are these dual options justified?

I don't understand here what you mean by 'dual option'. When taking the limit of something, you are not, as it were, plugging that value into the equation, though in some instances doing so makes the problem much easier. You're finding the behavior of that function around that point.

Choosing to evaluate the function at a certain 'specific' value is only done to help determine the behavior. It's no different from evaluating a function with increasing values of x in some interval to determine the behavior of the function on that interval (increasing, decreasing, maxima, minima). Doing so doesn't break any rules of math, the question is asking something that requires more than one evaluation.