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If space is not continuous, then is calculus wrong? 
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#55
Sep1911, 01:29 PM

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#56
Sep1911, 01:55 PM

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I DO understand that there is worth to pursuing the kind of things behind Zeno's paradox, what I object to is the phrasing that says Zeno's paradox shows that motion is not real. NO, Zeno's paradox clearly CANNOT show that motion is not real because motion IS real, so the phrasing should be more like "hey, we have this really nifty, clever way of looking at motion that seems to make it not possible and since it so clearly IS possible, we need to figure out what it is about our way of looking at it that leads to such an absurd conclusion". An it seems to me that exactly that has been DONE a couple of times already in this thread. Zeno had the math wrong. It's Zeno's mistake. 


#57
Sep1911, 02:10 PM

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#58
Sep1911, 02:13 PM

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Another way to look at it: At this site we no longer accept threads that try to argue that [itex]0.999\cdots\ne1[/itex]. Zeno's paradox is exactly the same thing, just in base 2: [itex]0.111_2\cdots\equiv 1[/itex]. Yet another way to look at it is a failing to understand how science works. In a perhaps too condensed a nutshell, mathematicians try to prove mathematical theorems while scientists try to disprove scientific theories. There are (at least) two ways to disprove a scientific theory. One way is to attack the logic that underlies the theory. Scientific theories must be logically sound, mathematically correct. A hypothesis that doesn't add up is invalid. Another way is to attack a scientific theory is from an angle that does not necessarily apply to mathematics. Just because the underlying math of some scientific theory is absolutely beautiful and perfectly sound does not mean the theory is correct. Science has to describe the real world. A failure here (observing just one black swan, for example) means the theory is false or is of limited applicability. This connection with reality can never be proven to be true. Science depends on observation. While one observation can prove that a theory is incorrect, mountains of observation do not prove that a theory is correct. It is merely confirming evidence. That one black swan rule does allow us to rule out a lot, including Zeno's paradoxes of motion. The seemingly naive answer, I just walked from A to B, does it in. 


#59
Sep1911, 02:14 PM

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#60
Sep1911, 03:02 PM

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#61
Sep1911, 04:05 PM

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doesnt infinitely divisible mean continuous? 


#62
Sep1911, 04:27 PM

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Good evening jesse
Is is infinite? Yes Is it continuous? No Is it infinitely divisible? Yes This function is, of course, all the tops or bottoms of a perfect square wave. 


#63
Sep1911, 05:08 PM

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and based on current mathematics, this function is infinitely divisible on the interval (0,1), (2,3), etc, but it is not divisible at all on (1,2), etc... my question is, what if space is not infinitely divisible on any interval?...and in this case we would need to use mathematics which takes this discreteness of space/time into account. I notice many people here are saying how good calculus is as a model of reality. but that is not my point. i do not want to know what MODELS reality, i want to know what IS reality...basically i want to know what math mirrors and PERFECTLY describes reality. 


#64
Sep1911, 05:09 PM

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What seems to be missing from Zeno's paradox is the fact that the successively smaller and smaller distances require successively shorter and shorter times to travel over. If the speed is constant then the time taken is the same whether you divide the total distance by the speed or do it the hard way by summing thsee smaller and smaller times. So it isn't the maths that disagrees with reality. What's wrong is the way that people interpret what the maths is telling them.



#65
Sep1911, 05:14 PM

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I could be wrong. One day I could wake up, having recently died, and find some geezer in a long white beard telling me the exact answer to everything  but I won't hold my breath. There are other views about the purpose and meaning of Science, of course but they haven't yet been proven, any more than my view. 


#66
Sep1911, 05:23 PM

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The question of whether there even is anything that counts as "ultimate reality" is an unknowable mystery. I often think that if we discovered an equation that would fit on a tshirt that explains everything there is to know about the working of the universe ... that would tell us more about ourselves than it does about the universe. The universe is not an equation. Ok enough philosophy for one day. I'm off to San Francisco. But first I have to go halfway there ... 


#67
Sep1911, 05:27 PM

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Yes. Ultimate Reality is a naive goal because it needs, yet, to be defined.



#68
Sep1911, 05:28 PM

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You have to study something as it is not as you want it to be. 


#69
Sep1911, 07:04 PM

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Without getting too philosophical, I don't accept the premise that mathematics has the ability to be "wrong." It can be used/applied incorrectly, but to suggest that it can be wrong is analogous to assigning blame to a tool for being used improperly, rather than the person who used the tool.



#70
Sep1911, 11:45 PM

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The axioms that define the real numbers were inspired by human intuition about positions along a straight line. However, in modern mathematics, an "axiom" isn't "something that's so obvious that it doesn't need to be proved". (This is how my high school teacher defined the word "axiom", but it's completely incorrect). It's just a statement that's a part of a definition. A definition simply associates an English word or a phrase with a set that does satisfy the axioms. So once we have defined the real numbers, it's impossible for theorems about real numbers to be objectively wrong. The theorems will hold for what the definition calls "real numbers". (This would be the members of a set that satisfies the axioms that define "the set of real numbers").



#71
Sep2011, 01:59 AM

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#72
Sep2011, 04:50 AM

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