You finally helped me understand why a high school “proof” I wrote almost 20 years ago was marked incorrect for taking two sides of the same equation and manipulating both. The whole “if you start assuming the answer and work backwards you don’t necessarily know that you’ve stayed with the same problem.” Essentially I could take any two expressions and multiple them by zero and add one and get 1=1 so anything could be determined equal if I trivialize properly. Thank you. Loved this!
For anyone else struggling with the "smallest positive number" argument, it's helpful to see how the proof-by-contradiction is supposed to work using ½x instead of x²: Let x be the smallest positive number. Then x ½x for all positive x, therefore contradiction In the x² version we skip questioning whether x
Fun presentation, thank you. Here's one for the books: If our number system defines quantity?; then ZERO divided by ZERO equals ONE and INFINITY simultaneously!....or, in other words; at ONE with the WHOLE. e.g. - (No Apples) divided by (No Apples) equals ONE and an INFINITY of (No Apples)... or; in other words; a WHOLENESS of Apples. ....where the WHOLENESS doesn't lie in the quantitative realm, but rather; it is an expression of the CONCEPT of Apples and the lack of them, ....a calculator lacks consciousness and therefore errors...it is dead and cannot 'CONCEIVE'....and.neither can AGI. The moral of this bit of abstraction is; don't take what a dead thing says as correct or even smart!...how do you like them No Apples, hee!
0/0 is more like zero people sharing zero apples. It is not well defined how many apples each person gets, because there are no people. Division, e.g. a/b, is really asking to find x in the equation a = x * b. If a is non-zero and b is zero, there is no x that can make the equation true. Likewise, if a and b are both zero, any x can make the equation true, and thus there is no unique solution. This is why we are not permitted to divide by zero. Taking x to be infinity or zero is just as arbitrary as taking x to be 7 or pi. It does not offer any profound insights.
@@michaeldamolsen hey, hi there!...thanks for your response, it was just a bit of fun though, i must remind you that i didn't introduce 'zero people' into the equation?......just 'zero apples'.....and of course, like Mathematics in general; it's not to provide profound insights.... we have Meditation, the Arts and Philosophy for that.
There are no numbers that you can pick at random of the street except maybe the natural numbers 1, 2, 3 and 4. Numbers are generally constructed. If you make a measurement you will find only natural numbers depending on what is the smallest margin you can measure. If you measure the diameter and the circumference of a circle you find on the street you will find out the ratio between them is always a rational number containing just a finite number of digits (if your margin is small enough it will always be smaller than some approximation of pi) And if the circle is a great circle like the equator, you will find that the circumference is almost equal to twice the diameter.🤣 Even a simple thing like the square root of 2 has to be constructed. If you measure it, you will simply find a rational number. that is close to it. That is the way it is on the street. Irrational numbers only exist in our minds.
Wait. If there's no boring positive integer, then there isn't anything boring. The first boring moment would be special. So it does not count and the next moment is special too. I get why they call it the optimist theorem.
You finally helped me understand why a high school “proof” I wrote almost 20 years ago was marked incorrect for taking two sides of the same equation and manipulating both.
The whole “if you start assuming the answer and work backwards you don’t necessarily know that you’ve stayed with the same problem.”
Essentially I could take any two expressions and multiple them by zero and add one and get 1=1 so anything could be determined equal if I trivialize properly.
Thank you. Loved this!
For anyone else struggling with the "smallest positive number" argument, it's helpful to see how the proof-by-contradiction is supposed to work using ½x instead of x²:
Let x be the smallest positive number. Then x ½x for all positive x, therefore contradiction
In the x² version we skip questioning whether x
I've always thought the dual integrals of Gabriel's Horn gave an absurd result and are pointing to some fundamental flaw in mathematical theory.
Fun presentation, thank you. Here's one for the books:
If our number system defines quantity?; then ZERO divided by ZERO equals ONE and INFINITY simultaneously!....or, in other words; at ONE with the WHOLE.
e.g. - (No Apples) divided by (No Apples) equals ONE and an INFINITY of (No Apples)... or; in other words; a WHOLENESS of Apples.
....where the WHOLENESS doesn't lie in the quantitative realm, but rather; it is an expression of the CONCEPT of Apples and the lack of them, ....a calculator lacks consciousness and therefore errors...it is dead and cannot 'CONCEIVE'....and.neither can AGI. The moral of this bit of abstraction is; don't take what a dead thing says as correct or even smart!...how do you like them No Apples, hee!
0/0 is more like zero people sharing zero apples. It is not well defined how many apples each person gets, because there are no people.
Division, e.g. a/b, is really asking to find x in the equation a = x * b. If a is non-zero and b is zero, there is no x that can make the equation true. Likewise, if a and b are both zero, any x can make the equation true, and thus there is no unique solution. This is why we are not permitted to divide by zero.
Taking x to be infinity or zero is just as arbitrary as taking x to be 7 or pi. It does not offer any profound insights.
@@michaeldamolsen hey, hi there!...thanks for your response, it was just a bit of fun though, i must remind you that i didn't introduce 'zero people' into the equation?......just 'zero apples'.....and of course, like Mathematics in general; it's not to provide profound insights.... we have Meditation, the Arts and Philosophy for that.
Gonna use this to prove to banks I own them no money.
There are no numbers that you can pick at random of the street except maybe the natural numbers 1, 2, 3 and 4. Numbers are generally constructed. If you make a measurement you will find only natural numbers depending on what is the smallest margin you can measure. If you measure the diameter and the circumference of a circle you find on the street you will find out the ratio between them is always a rational number containing just a finite number of digits (if your margin is small enough it will always be smaller than some approximation of pi) And if the circle is a great circle like the equator, you will find that the circumference is almost equal to twice the diameter.🤣 Even a simple thing like the square root of 2 has to be constructed. If you measure it, you will simply find a rational number. that is close to it. That is the way it is on the street. Irrational numbers only exist in our minds.
Surely £1 = £0.01! Square it, Multiply it but All Calculations are Incorrect.........
Attn: Terrence Howard
0.01 Squared is 0.0001! don't follow your logic......
Terrence Howard ?
1x1=2 discuss lol
Wait. If there's no boring positive integer, then there isn't anything boring. The first boring moment would be special. So it does not count and the next moment is special too. I get why they call it the optimist theorem.
No, as the set of all integers doesn't contain everything, only every integer