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Post by laughter on Jul 6, 2012 3:31:50 GMT -5
I find it a wonderful ironic expression of the paradox at the root of our existence that statements of absolute truth are only to be had by resorting to the pure conceptual.
Mathematics is the ultimate language of abstraction and but for what comes out of it there are no absolute truths that can’t be debunked. Not only is a mathematician not concerned with the physical world, she goes out of her way to ignore it.
At the root of mathematics is the concept of equality. I’d welcome a correction if this statement is wrong, but as far as I know it’s not possible to define a system of mathematics without the concept of equality … a concept that finds no physical embodiment.
A banker will equate two different piles of $100 bills, but this is just another abstraction. Each bill has a different set of wrinkles and stains, has traveled a different road of wallets, purses and cash register drawers to reach the pile and has been spent and counted a different number of times than the bill above or below it in each pile. Even if the piles are of new bills printed on the same day and with the error of identical serial numbers, each bill came off the press at a different time, and the two piles occupy different positions in space.
By the very fact that there are two piles there is a difference between them, and the banker’s equality is seen to be an abstraction. When applied to the physical, the equals sign doesn’t really mean “the same”, it means “two different entities with similar qualities”.
The piles are the same but are different. Whenever there is more than one, this is the case, and in seeing one there are two.
Using the abstraction of equality the mathematician can speak the absolute truth that the ratio of the circumference of the curve comprising all points at a given radius from a given center point to that radius is the same number regardless of the center point or the radius. Funny thing though, she can’t ever tell you exactly what that number is, because it would take more than all of eternity to calculate it precisely.
So to say there is no absolute truth isn’t quite the whole story, because there is always our mathematician, but we can see that any absolute truth can be expressed only conceptually and not embodied physically, and we here all know about the nature and ultimate value of conceptual expression now, don’t we?
... but then again, in digesting the absolutely true concept with our minds (with embrace, indifference or rejection), don't we, in a sense, give it some physical embodiment? So in saying that "The Tao that can be spoken of is not the Tao" ... even in this, we perhaps go too far ... say too much? Well, sometimes expressing compassion requires words.
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Post by andrew on Jul 6, 2012 4:39:38 GMT -5
Reminds me of this theorem, which I don't understand and yet can still sense there is something very special and beautiful about it. Euler's Identity... www.youtube.com/watch?v=zApx1UlkpNs
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Post by topology on Jul 6, 2012 6:15:26 GMT -5
Laughter, what do you mean by equality having no physical embodiment? Let's suppose John and Bob have crisp brand new $100 bills from the bank. John sets his bill on a table and then goes to another room. Bob, being the prankster that he is, swaps his bill for John's with the same postion and face showing to see if John notices. John comes back and picks up the bill on the table and puts it in his wallet. Bob asks "was that bill on the table yours?". John answers "yes it is." Our perception is pattern based not object based. We assume sameness until a difference is perceived. John didn't pay attention to the serial number or the exact position on the table to know the difference and a continuity of identity was assumed based on the sameness of the pattern experienced. This assumption is automatic and goes on all the time. In our lack of sufficiently detailed perception and not noticing a difference we use the wrong key, pickup the wrong coat from the coat rack, take the wrong lunch bag from the fridge, etc. If this is not a physical enough demonstration of the principle of equality, then let us simply go straight to the physicists subatomic model of the world. There is no difference between one electron and another with thensame spin apart from time and place of observation. en.wikipedia.org/wiki/One-electron_universe. Feynman says that as far as we know there may be only one electron in the universe. Objects vs. Patterns Patterns exist in our perception, but Objects only exist in our theories about a physical reality independent of the phenomena in our perception. Two patterns are the same unless there is a difference perceived. That difference can come in the form of simultaneously perceiving two instances of the same pattern and they are distinguished in location within the perceptive field. Or they can come in the form of being separated over time, a discontinuity in the experience of the pattern, and by inserting a story about why the second pattern is not the first pattern as I did with the story of the $100 bill substitution. When you say equality and sameness are only used by mathematicians in purely conceptual thought, that confuses me. It happens all the time in our perception. If we have a lack of perception of the difference between identical twins, we will continually mistake them for eachother. And this is assuming that there is a perceptible difference between them, with some things there is no perceptible difference.
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Post by laughter on Jul 6, 2012 11:07:21 GMT -5
In our lack of sufficiently detailed perception and not noticing a difference we use the wrong key, pickup the wrong coat from the coat rack, take the wrong lunch bag from the fridge, etc. Those are situations where the the mis-perception is corrected by subsequent experience. In your example involving John and Bob, John's mis-perception is never corrected. Now you can argue with me that since there is no such correcting event in that case that it follows that you've established the equality of the two bills, but I'll argue in turn that all you've established is that the numerical value (and any other salient quality, such as you pointed out, the initial position on the table) associated with each of the bills is the same. I'm not going to argue with you about 100 = 100 but you elicit gales of me if you hold forth that Mary-Kate and Ashley are the same person. Laughter, what do you mean by equality having no physical embodiment? The fact that there are two dollar bills means that they aren't the same bill. Your point about sub-atomic particles is another matter but the explanation ultimately remains the same. This is a profound result, and maybe there really is only one electron -- perhaps our measurement of the value of currents in the lab that imply otherwise is some trick of the way things are that awaits the explanation of some future model. That's an interesting speculation. But mistaking an electron for something objective is similar to mistaking it for a pattern. It's actually both, and absent said future model every other model I know of treats electrons the same as unserialized newly printed Benjamins -- that is to say, they are interchangable, but we still need to resort to multiplicity to get from Physics to Chemistry (think the periodic table). If there is more than one, then they occupy different positions -- the differentiating characteristic is not intrinsic to the entity, but it is there nonetheless. The two electrons in every molecule of water are indeed similar in every detail, but if they were the same you would have hydrogen peroxide instead. It's really just the distinction between sameness and similarity. Physically, sameness can always be focused into the weaker cousin .... but the statement x=y implies, for instance, that there is only one "2".
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Post by laughter on Jul 6, 2012 11:32:25 GMT -5
Reminds me of this theorem, which I don't understand and yet can still sense there is something very special and beautiful about it. Euler's Identity... www.youtube.com/watch?v=zApx1UlkpNsI've read presentations of √-1 as a model for "the illusion of identity" (aka, "the liars paradox" ... think of the idea of "recursive thought") and have corresponded with people who are into Harding's work that find significance in e i∏ ("turn the direction of attention around" ... or, in other words "look inward") Fun places for the mind to go, no doubt!
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Post by topology on Jul 6, 2012 12:23:58 GMT -5
In our lack of sufficiently detailed perception and not noticing a difference we use the wrong key, pickup the wrong coat from the coat rack, take the wrong lunch bag from the fridge, etc. Those are situations where the the mis-perception is corrected by subsequent experience. In your example involving John and Bob, John's mis-perception is never corrected. Now you can argue with me that since there is no such correcting event in that case that it follows that you've established the equality of the two bills, but I'll argue in turn that all you've established is that the numerical value (and any other salient quality, such as you pointed out, the initial position on the table) associated with each of the bills is the same. I'm not going to argue with you about 100 = 100 but you elicit gales of me if you hold forth that Mary-Kate and Ashley are the same person. Laughter, what do you mean by equality having no physical embodiment? The fact that there are two dollar bills means that they aren't the same bill. Your point about sub-atomic particles is another matter but the explanation ultimately remains the same. This is a profound result, and maybe there really is only one electron -- perhaps our measurement of the value of currents in the lab that imply otherwise is some trick of the way things are that awaits the explanation of some future model. That's an interesting speculation. But mistaking an electron for something objective is similar to mistaking it for a pattern. It's actually both, and absent said future model every other model I know of treats electrons the same as unserialized newly printed Benjamins -- that is to say, they are interchangable, but we still need to resort to multiplicity to get from Physics to Chemistry (think the periodic table). If there is more than one, then they occupy different positions -- the differentiating characteristic is not intrinsic to the entity, but it is there nonetheless. The two electrons in every molecule of water are indeed similar in every detail, but if they were the same you would have hydrogen peroxide instead. It's really just the distinction between sameness and similarity. Physically, sameness can always be focused into the weaker cousin .... but the statement x=y implies, for instance, that there is only one "2". I know nothing about a physical reality independent of the patterns occurring in my experience. If you do, then you know more than I. I was taught physics from High-school and throughout undergraduate years by drawing pictures on a chalk board and saying "this is how to think about the 'physical' world". This method of teaching is no better than teaching mythology as history. The $100 dollar bills in the story are only two because I told you there were two in the story. If I told you the story from John's perspective, there would have been just 1. The same problem can go in reverse. A common technique in movie making is to use a blue screen and multiple recordings to insert the same person multiple times into a scene. Is it the same person or are they different due to appearing in different locations at different times? How are we to ever decide there is mis-perception if there is never a conflicting experience or conflicting testimony? How are we ever to decide there is not mis-perception? In chemistry there is the notion of concentration of solute in water. Shake the glass, and even though you've displaced an re-arranged the molecules, its still the same concentration. Replace a board on the hull of a ship, its still the same ship. Do this kind of replacement to every component of the ship. en.wikipedia.org/wiki/Ship_of_Theseus You can similarly replace every component to a computer with exact replicate parts and copy the memory configurations from one hard-drive to another. The computer will be treated and thought of as the same. en.wikipedia.org/wiki/Identity_(philosophy)sameness and similarity are the same operation, just a matter of where you place the boundary on what you are comparing. I take a week vacation and I come back to a house filled with a wife and two kids. They are the same because they are similar to when I left. Having enough of the same memories, facial features, etc. etc. In order to preserve identity over time we allow sameness to be relaxed and allowed to vary. The same "river" is taken to be the same due to its location despite every component of water having moved down-stream. Identity is pattern based, not object based.
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Post by laughter on Jul 6, 2012 13:35:37 GMT -5
Identity is pattern based, not object based. Spoken like a true mathematician ... try telling that to a programmer who forgets to implement the necessary copy-constructor.
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Post by topology on Jul 6, 2012 13:55:32 GMT -5
Identity is pattern based, not object based. Spoken like a true mathematician ... try telling that to a programmer who forgets to implement the necessary copy-constructor. Yes, well you're trying to duplicate the pattern so that they can have their patterns diverge under different transforms going forward. If you'd program with a functional language like SML/NJ which does copy by value instead of copy by reference, you wouldn't have as many issues. OO isn't always the best abstraction layer. Still too much representation overhead for my preference. After I get my damned PhD, I want to do research into Domain Specific Languages.
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Post by arisha on Jul 7, 2012 2:42:10 GMT -5
What about Lobachevsky's ideas regarding the concept of equality? If parallel lines cross how does it relate with equality? I think this concept turns out to be different from what the idea about equality is. Even divine numbers are not perfectly divine. 2=2? Maybe 1=1, but 2 is not equal to 2.
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Post by arisha on Jul 7, 2012 2:51:54 GMT -5
There is no difference between one electron and another with thensame spin apart from time and place of observation. en.wikipedia.org/wiki/One-electron_universe. Feynman says that as far as we know there may be only one electron in the universe. Maybe this electron is God which is everywhere?..
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Post by living on Jul 7, 2012 6:58:39 GMT -5
There is no difference between one electron and another with thensame spin apart from time and place of observation. en.wikipedia.org/wiki/One-electron_universe. Feynman says that as far as we know there may be only one electron in the universe. Maybe this electron is God which is everywhere?.. George Carlin called It the Big Electron, too.
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Post by arisha on Jul 8, 2012 11:26:11 GMT -5
Numbers are the highest known to humans abstraction besides God.
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Post by laughter on Jul 15, 2012 7:52:30 GMT -5
What about Lobachevsky's ideas regarding the concept of equality? If parallel lines cross how does it relate with equality? I think this concept turns out to be different from what the idea about equality is. Even divine numbers are not perfectly divine. 2=2? Maybe 1=1, but 2 is not equal to 2. Thanks for the reference arisha, Lobachevsky seems worth a read, interesting notion ... sometimes not even 2=2 ... good stuff.
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