I understand that already, Dave, but thanks. Did what I said make any sense?
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Smiling Dave: Autolykos, The right Math course answers all your q's. They have been chewed over by great minds, and they came up with answers.
Autolykos,
The right Math course answers all your q's. They have been chewed over by great minds, and they came up with answers.
But it doesn't matter who says what when reason is the key to understanding and knowledge. Albert Einstein could have said 6+1=10 and guess what, he's wrong no matter how you slice it.
Aristotle was brilliant in determining the elements of rhetoric in argumentation: ethos, pathos and logos. When seeking truth, though, as I believe our mission as philosophers ought to be at all times, logos is all that matters. One can always question credibility and emotion. Take my above equation. No matter who says 6+1=10, no matter how many guns are pointed at you forcing you to agree that 6+1=10, 6+1 will never be 10, and we know this because of logos.
I believe a sufficient refutation of Zeno's aforementioned paradox is that mathematics is incorporeal and completely subjective. While brilliant things can be accomplished by using it, there will never be anything amounting to a number in nature. In essence, Plato was wrong.
triknighted:Aristotle was brilliant in determining the elements of rhetoric in argumentation: ethos, pathos and logos. When seeking truth, though, as I believe our mission as philosophers ought to be at all times, logos is all that matters. One can always question credibility and emotion. Take my above equation. No matter who says 6+1=10, no matter how many guns are pointed at you forcing you to agree that 6+1=10, 6+1 will never be 10, and we know this because of logos.
You're talking about base-10 notation, right? Because 6+1=10 in base-7 notation.
triknighted,
You seem to think I was appealing to authority in what I wrote to Autolykos. No. I was recommending he peruse their works, as taught in sophomore or junior year of university math. \\
When your refutation of Zeno's paradox is that math is incorporeal and completely subjective, you throw out the baby with the bath. It's like solving a chess problem by overturning the board.
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It's easy to refute an argument if you first misrepresent it. William Keizer
Autolykos: triknighted:Aristotle was brilliant in determining the elements of rhetoric in argumentation: ethos, pathos and logos. When seeking truth, though, as I believe our mission as philosophers ought to be at all times, logos is all that matters. One can always question credibility and emotion. Take my above equation. No matter who says 6+1=10, no matter how many guns are pointed at you forcing you to agree that 6+1=10, 6+1 will never be 10, and we know this because of logos. You're talking about base-10 notation, right? Because 6+1=10 in base-7 notation.
Hah hah ok I have no idea, not a math wiz. How about the classic 1984 example of 2+2=5? I think you know what I meant though mate, good one though.
Smiling Dave: When your refutation of Zeno's paradox is that math is incorporeal and completely subjective, you throw out the baby with the bath. It's like solving a chess problem by overturning the board.
How exactly?
Well, you could simply say that in terms of steps taken, Achilles makes far more in the same amount of time passed. If you slow down time enough (like, to milliseconds or nanoseconds), you can simply state that Achilles is making up the distance in the brief period where the tortoise isn`t moving or is about to move. Hence, he overtakes it while it isn`t moving.
Because it throws out all of the accomplishments of civilization that are based upon Mathematics.
It might make sense, but it is not the direction humanity has taken ever since Euclid. I think it behooves us to find an answer to Zeno's Paradox consistent with a few millenia of the universal thinking of the mathematical and scientific community.
Smiling Dave: Slicing into infinity has been clearly defined. It does not imply that there is nothing in reality itself, only that a single point in space has no dimension. Existing and having a dimension are not the same thing.
Slicing into infinity has been clearly defined. It does not imply that there is nothing in reality itself, only that a single point in space has no dimension. Existing and having a dimension are not the same thing.
So a point can have no length, height or depth yet exist? If I told you I had a burger which had no beef, no bun and no cheese you would say I have no burger. We commonly call such things as non-existent, the same goes for the point. Unless you argue that the point in question exists as an abstract object then fine but then it is not immediately clear how abstract objects relate to the physical universe in question.
I really don't care whether something has been "universally accepted" for thousands of years. If it is false, it is false. It may be the case that assuming infinite divisibility has been useful for practical problems.
The atoms tell the atoms so, for I never was or will but atoms forevermore be.
Yours sincerely,
Physiocrat
Mr Buddy: Well, you could simply say that in terms of steps taken, Achilles makes far more in the same amount of time passed. If you slow down time enough (like, to milliseconds or nanoseconds), you can simply state that Achilles is making up the distance in the brief period where the tortoise isn`t moving or is about to move. Hence, he overtakes it while it isn`t moving.
QFT.
At last some one else who can have a fresh look at a problem.
If I told you I had a burger which had no beef...
Space and time are different, being intangibles. Even a cubic foot of space and an hour of time have no beef, no bun, and no cheese. So the poor point is not to be picked on and bullied out of existence because of that.
I really don't care whether something has been "universally accepted" for thousands of years. If it is false, it is false.
The question is, does Zeno's Paradox [or anything else] prove it is false, or is there a way of resolving the paradox.
It may be the case that assuming infinite divisibility has been useful for practical problems.
Geometry and the underlying mathematical assumptions of physics [aka calculus] are also studied as logical systems independent of reality. Like everyone assumes 2+2=4 and the other theorems of arithmetic are true independent of the existence of apples and pears. Mathemeticians would have to abandon Geometry and Calculus, or at least spend many sleepless nights, if they proved to contain a paradox.
Well, you could simply say that in terms of steps taken, Achilles makes far more in the same amount of time passed.
Nobody denies this. Zeno's Paradox is not resolved by this assumption.
Achilles is making up the distance in the brief period where the tortoise isn`t moving or is about to move.
The tortoise is assumed to be travelling at a constant velocity [as is Achilles, though Achilles is faster]. Implicit in that assumption is that in any interval of time, the tortoise has moven some distance, however small. He never "isn't moving" or "about to move". We are not talking about some lazy tortoise or donkey who slacks off every once in a while.
Smiling Dave: Space and time are different, being intangibles. Even a cubic foot of space and an hour of time have no beef, no bun, and no cheese. So the poor point is not to be picked on and bullied out of existence because of that.
I agree that space and time are different in that they are not matter. In fact all three are needed. Matter to move, space to move through and time to allow it to move.
Is the point matter, space or time?
Smiling Dave: The tortoise is assumed to be travelling at a constant velocity [as is Achilles, though Achilles is faster]. Implicit in that assumption is that in any interval of time, the tortoise has moven some distance, however small. He never "isn't moving" or "about to move". We are not talking about some lazy tortoise or donkey who slacks off every once in a while.
That misunderstands the point: if space is discrete then when something is moving it is always moving little bits at a time and if between two atomic seconds (defined here as the basic units of time) any object that appears to move is in fact stationary unless it is at literal maximum speed i.e. moving between two atomic blocks of space in two atomic seconds.