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yanqui
climber
Balcarce, Argentina
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Jul 17, 2017 - 06:38am PT
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So if a step-by-step formally expressible procedure to calculate a function does not terminate, it is not an algorithm?
A failed algorithm, you might say. I doubt many people would buy a computer that just whirred around forever without producing an output.
Examples of algorithms abound throughout the history of mathematics, but the formalization of the concept by Turing (and Alonzo Church) was born in Hilbert's (perhaps unwarranted) optimism about our ability to solve problems. In his 1900 address at International Math Congress in Paris, Hilbert said:
Is the axiom of the solvability of every problem a peculiar characteristic of mathematical thought alone, or is it possibly a general law inherent in the nature of the mind, that all questions which it asks must be answerable?...This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no ignoramibus.
One of the the problems Hilbert proposed at this congress was the construction of a precise finite decision process to decide if a Diophantine equation (a polynomial equation with integer coefficients in n-variables) had an integer solution. Hilbert proposed:
"Given a Diophantine equation with any number of unknown quantities and with rational integral numerical coefficients: To devise a process according to which it can be determined in a finite number of operations whether the equation is solvable in rational integers."
The Fermat equation x^n + y^n = z^n is an example. When n=2 there are lots of solutions and they can easily be generated algorithmically. Here is a list of all positive solutions up to z=10,000:
http://www.tsm-resources.com/alists/PythagTriples.txt
When n>2 we now know the only solutions are the trivial ones that come from putting in 0s,1s or -1s.
Going against Hilbert's expectations, it was shown in 1970 that there doesn't exist any algorithm that can do what he proposed. In other words: no matter what "computer" you build, there will always be a Diophantine equation it can't handle. This seems to go against the vague optimism expressed in Hilbert's "solvability axiom". We also know other specific mathematical problems (like the "word problem" in group theory) that can't be solved by any finite, formal decision process. These problems are "algorithmically undecidable". Of course to prove something like this, one needs a precise formalization of the concept of algorithm. That's what Turing (and Alonzo Church) gave us.
In fact, Turing's formalization of "algorithm" as a kind of machine that computes (first appearing in his 1937 publication), and his subsequent treatment therein, was specifically done to show Hilbert's optimism about the decision problem (Entscheidungsproblem) was unfounded. After using his machine to prove "relatively few" real numbers are computable, Turing turned his attention to show:
"... there can be no machine, which supplied with any ... formulae, will eventually say whether (the formulae) is provable."
The original paper by Turing is quite clear and free online:
http://www.turingarchive.org/browse.php/B/12
Edit for clarity (there may be some confusion about this): the failure of the computer to solve the Diophantine equation is not that the computer gives a "partial" answer or an "approximation" after it runs a while. The question you ask the computer is: "Does this Diophantine equation have an integer solution?". The computer is built to answer "yes" or "no". Although the computer you build may be able to answer yes or no for some Diophantine equations, no matter what computer you build there will always be a Diophantine equation where no answer comes out. That's what "undecidable" means.
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feralfae
Boulder climber
in the midst of a metaphysical mystery
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Jul 17, 2017 - 07:04am PT
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So perhaps God is the 9th-century mathematician Abū Ja῾far Muhammad ibn Mūsa?
Sure, if that makes you happy. But remember that his work relies on earlier advances by Indian mathematicians. I don't see any human as G*d, but that is just my own system of recognizance which puts me in this mindset. As usual with humans, YMMV.
However, my point was less about identifying a god or gods, and more about the wonder of perhaps being in a position to intentionally, consciously, and actively participate in the evolution of our own species. To be able to simply contemplate such a concept is delightful to me.
feralfae
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Largo
Sport climber
The Big Wide Open Face
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Topic Author's Reply - Jul 17, 2017 - 08:33am PT
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My attention, awareness, conscious activity, all the discursive was focused on that problem.
Somehow, I executed a highly complex activity of driving my car, in traffic, to work, and I made it. Apparently this didn't require any of what Largo is talking about distinguishes me from Curiosity, my sentience was not an issue, my awareness was not an issue, my attention was not an issue.
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You're isolating out one part of the process that involved a long sequence of working up to a phase where you went into full absorbtion, and you were able to operate on auto-pilot as your brain went to work. You're using that phase of processing to suggest that the whole process comes down like that. To understand the process you need to look deeper.
You couldn't be more mistaken when you said: "my sentience was not an issue, my awareness was not an issue, my attention was not an issue.
Next time you go into absorbtion mode, and you're driving a car, try closing your eyes (attention), or unfusing your attention from unconscious processing and rather fuse your attention to the song on the radio.
You just went into machine mode, Ed, which we all do following a long process of consciously sorting out a bunch of variables that put you in a position for your brain to do some data processing. If you would go back to when you first started working on the problem, or ones like it, you will find long periods where you were in anything BUT autopilot. All of those periods set you up to "go off" on the solution. But likely this intense processing was not the end of the line. You probably had to revise and refine what your brain churned out - try and do that on autopilot. It's always a back and forth process.
What you described is essentially what the professional baseball player goes through when he steps to the plate during a game. While he is entirely present in the batter's box, oblivious to the crowd noise, he is also fully on autopilot because the action happens faster than he can think. But what you are seeing is the final product, preceded by YEARS of conscious, painstaking work.
Like I said, consciousness is an ongoing, dynamic process during which determined, time-bound mechanistic functioning is at play with timeless, conscious elements. We constantly ping pong between the two. For example, I consciously work on a story for several hours and reach a sticking point. Often I will go over to the couch, put on a soundproof headset (like they use in construction), cover my eyes with a shirt and let my mind go blank. After ten or fifteen minutes I'll return to the story with a brand new take on the sticking point. In this way I can modulate the swing between conscious and unconscious processing, both of which are essential to the creative process. It starts to get tricky during the revision process - and most all good writing is in the revisions.
So yes, you can isolate out the unconscious processing phase when the brain catches up, but you can't understand the PROCESS only looking at absorbtion.
One of the interesting aspects of absorbtion is that our brain waves increase in coherence all across the spectrum. And during absorbtion we are totally or mostly unaware of time.
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Ed Hartouni
Trad climber
Livermore, CA
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Jul 17, 2017 - 09:30am PT
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it is interesting that you would resort to such a tortured description of what goes on in that "zombie mode"
a more parsimonious explanation would be that there is a lot of what goes on in our nervous system that is not accessible to our "discursive mind," and looks like a more elaborate version of machine behavior, though it be a biological one, that is to say it is "mechanistic." If a part of our "awareness" is machine like, why wouldn't all of it be? and if you posit it is not, what is it then?
And I always like how Largo re-interprets my experiences for me, I'm sure he learned that in his practice from his masters, and has built a particularly robust point of view (as we have seen).
Here is another interesting experience that all of you have had which stretches across many topics.
I can't remember how old I was when I received this present, I think it is a variant of something you can find online called "math dice." The game was pretty simple. A number of 6 sided dice have numbers and math operations printed on them, I can't remember all the operations, but likely the addition, subtraction, multiplication and division were represented. I also can't remember how many dice there were, nor how many operators in total.
The game is relatively simple, though I will probably get the details wrong. A single dice is drawn blind and thrown until a number comes up.
The rest of the dice are then put into a cup and thrown.
The game is to use the operators and numbers that come up on those dies in a formula to form the number on that first dice.
I was pretty good at that game, and it improved my arithmetic skills, if for no other purpose than lots of practice. I could beat everyone in my family, father included, and got really quick at it strategizing how to factor that first number and take advantage of the operators and numbers I had to work with, and with time pressure, since you won by being the first with a correct answer.
I was training my mind to manipulate numbers and execute those sets of operations mentally, writing things down was not allowed by the rules.
A machine could very easily be programmed to play this game, and as we all know machines are very precise at this sort of arithmetic.
I'm not sure how much is known about human arithmetic operations, those of us with the skill believe it could be very "mechanical" though the exact biological algorithms are not available to us, we somehow conjure up an answer. And arithmetic can be a passing fancy to someone like me, to someone like Ramanujan it could be a formidable tool making some very sophisticated mathematics possible. But even Ramanujan felt it was a "gift from the gods," he had no idea where this ability originated. It is a well known skill for savants, who lack any other recognizable mental faculty.
There are well known algorithmic processes for doing these arithmetic operations, we learn many of them in grade school 'rthmetic, lining up the numbers so that the power of tens form columns, notating the number or "caries" to the next power of ten, memorizing multiplication tables, learning how to "long divide."
For more complex mathematical operations we might apply the Taylor series expansion, and if we're lucky we only have to deal with rapidly converging series. There is a famous story of Feynman racing an abacas operator to get the answer of some problem posed... Feynman was doing the expansion in his head and continued to provide ever increasingly precise updates.
But the interesting question, I think, is this: in spite of training to do arithmetic for as long as I could read I still make arithmetic errors. It is (annoyingly) very common. Sometimes it can lead to days and even weeks of work on something I didn't bother to check carefully. So the incentive to get the arithmetic right is tremendous, and more so as I am getting old and don't have those days to "waste."
How is it that I get the answer wrong?
Generalizing it to learning anything I'll ask a related question. When I was assisting the teaching of large "service courses" at UMass I remember the delight of the head instructor when the distribution of grades on some test turned out to be a Gaussian, aka Normal, distribution. Now we learn in our statistics that this would be expected if the measure of precision of the distribution was finite, and the individual "measurements" are independent.
What I wondered then (and still do now) is why should we expect the learning outcome to be normally distributed? and if we can attribute that to some acceptable explanation, then what of the process of learning, and taking a test, is random?
The invocation of the central limit theorem (in addition with the "law of large numbers") does not help answer this question.
My feeling is that this fact, the lack of precision in arithmetic being the most obvious example, reveals something about how we actually "think," how learning happens and how we apply those lessons and gets at the underlying issues of a "mechanistic" model of mind.
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Dingus McGee
Social climber
Where Safety trumps Leaving No Trace
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Jul 17, 2017 - 12:48pm PT
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Largo,
big Assumtption you made,
preceded by YEARS of conscious, painstaking work.
likely most of the work was quite unconscious in all that training you mention for the batter example.
Consciousness does't do sh#t in motor skill learning -- Just go thru the motions.
Largo, remember conscious awareness capacity is 7 bits and this is not near enough info processing to control even a robotic arm. Get up to speed in control theory and you will know more than you lead on to know. You computer wizard!
You just gave Ed a great fiction talk of what you think [know/ hardly] was going on during his absorbtion? drive. Does doing this kind of walk thru chatter without any elements of the brain science help your consciousness prose writing? Simply a fodder explanation without experimental backing of what goes on during these experiences.
Ed
And I always like how Largo re-interprets my experiences for me
Exactly and you can bet the friction story he makes up conforms to his: the one and only theory of conscious that resorts to no mechanisms and achieves nothing. We are talking electrochemical here.
Largo,
see SciAm July 2017 Neuroscience Memory's Intricate Web pgs 30 - 37. A link to this article will cost you $$ but the story would enlighten you a bit about doing the science of neuroscience and the nano tools they have made along with the one molecule chemical two event data recorder molecule used in their brain research.
A microscope fits into a single brain cell and watches the one molecule chemical sensor for another event recording.
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WBraun
climber
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Jul 17, 2017 - 12:51pm PT
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in spite of training to do arithmetic for as long as I could read I still make arithmetic errors.
One is forced to make mistakes.
There is absolutely no escape from this fact when materially conditioned .....
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Marlow
Sport climber
OSLO
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Jul 17, 2017 - 01:06pm PT
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Much thesis and anti-thesis on this thread. Not much synthesis.
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paul roehl
Boulder climber
california
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Jul 17, 2017 - 02:28pm PT
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My point is simply that within the world of science there is a predilection for the worship of nature that is a vestige of romantic notions from the 19th century and even before.
I think you do not know much about science, and you probably do not know many scientists.
“In his book The Revenge of Gaia (2006), the pioneering climate scientist James Lovelock presents his own worldview, which is markedly similar to the Romantic’s “one life” metaphor, as a scientific hypothesis. This living Earth, to which Lovelock refers throughout his book as “Gaia,” is the “whole system of animate and inanimate parts” that enables a deeper enlightenment regarding the role of humans in respecting their environment (15). Lovelock describes Gaia metaphorically as a living organism, struggling to maintain its homeostasis under the growing stress of climate change. Fellow environmental writer and scientist, Rachel Carson, adopts a similar worldly perspective in her revolutionary book, Silent Spring (1962), which effectively launched the modern environmental movement. Although Carson does not specifically reference the “one life” hypothesis in her book, she consistently uses a “web of life” (69) metaphor which also emphasizes Earth’s living qualities through both Romantic descriptions and scientific analysis. This raises the question of how the “one life” metaphor, which is both unmistakably Romantic, as Abrams asserts, and yet also critically scientific, supports discussion of present environmental issues.”
I have no real problem with this idea except insofar as it begins to denigrate the remarkable nature of human thought and achievement. Something found repeatedly on this and the religion thread.
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yanqui
climber
Balcarce, Argentina
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Jul 17, 2017 - 02:41pm PT
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Much thesis and anti-thesis on this thread. Not much synthesis.
"The dogge must lerne it, whan he is a whelpe, or els it will not be: for it is harde to make an olde dogge to stoupe." John Fitzherbert, The boke of husbandry, 1534.
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eeyonkee
Trad climber
Golden, CO
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Jul 17, 2017 - 05:20pm PT
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Hey so I said up-thread that a big issue that I see with my world view vs. Largo's, is our starting points at looking at mind. Here's a little sketch I drew that more or less portrays our differences.
You probably need to click on it to see it well.
Each stage of our evolution introduced both new experiential apparatus and a legacy of emotional biochemical algorithms that underlie both our conscious and unconscious states as humans today. These stages can be represented as a tree because we are living organisms with ancestors.
When your starting point is "I can only know what I experience", you somehow circumvent all of that knowledge in the tree of life that we have been able to deduce based on scientific principles.
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MH2
Boulder climber
Andy Cairns
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Jul 17, 2017 - 05:36pm PT
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Thanks for the explanation, yanqui.
I once did read Turing's original publication. A layperson such as myself can follow most of the logic. What I could not be sure of was whether there were non-trivial questions that were undecidable. You make it clear that there are.
Also, it is easy for a person like me to confuse Turing's decidability result with Gödel's on completeness. There is a connection but also important differences.
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MH2
Boulder climber
Andy Cairns
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Jul 17, 2017 - 06:01pm PT
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I read a book by a guy who devised a test for birds.
He used dried salami tied to a string and hung from a perch in the aviary. The salami was two and a half feet below the perch and six feet above the ground.
The salami was too tough in consistency for the birds to grab it and tear off a piece and they could not perch near enough to it to peck off pieces.
The birds had been hand-reared in the aviary and had not seen such an arrangement before, but they knew and prized salami.
The birds were cautious. At first they hopped up and down on the perch and watched the salami jiggle. After time spent just looking they slowly approached the string. One bird gave a peck at the loop tied around the perch and quickly jumped back. One gave the string a few tugs but could not break it free. Then the birds seemed to lose interest. End of the first trial.
A few days later the guy set up the test again.
One of the birds flew up and perched beside the string. It reached down with its beak, grabbed the string a few inches below the perch, pulled it up, placed one of its feet on the pulled-up loop of string to hold it in place, reached down with its beak again for another length of string, and repeated the process until it had pulled the salami up to where it could grab it with its foot and tear off pieces with its beak.
What does this test tell you about the mind of the bird?
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feralfae
Boulder climber
in the midst of a metaphysical mystery
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Jul 17, 2017 - 06:18pm PT
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What does this test tell you about the mind of the bird?
1). That the bird is not bird-brained :) ;
2). That the mind of the bird is a problem-solving system and structure;
3). That the mind of the bird is a malleable synaptic system, capable of leaping wide chasms in a single hop;
4). That salami has the potential to alter the quantum-mechanical structure and problem-solving design of the bird's brain. :)
ff
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WBraun
climber
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Jul 17, 2017 - 06:21pm PT
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Instead of studying their own selves the fool scientists study everything outside of their own selves.
No wonder they're so lost, clueless, dazed and confused, all while constantly masquerading themselves as advanced ......
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feralfae
Boulder climber
in the midst of a metaphysical mystery
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Jul 17, 2017 - 06:46pm PT
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Werner, for my comments, you can just substitute "life form" for "bird" and all the statements will be applicable responses. The brain of life-forms has a fairly standard design based on QM, with photons jumping all over the place, making molecular changes at the atomic level. Our human brains just happens to be the latest, most sophisticated model.
Think of all the things that have wheels: they are of a fairly standard design and structure, which is to roll across a surface aided by power sources of all imaginable applications. Finally we get up to a Tesla, and think of all the sophistication...but still the same design to roll across a surface aided by power sources.
So there is this incredibly sophisticated problem-solving brain—by design and structure. (Well, for some of us, anyway. Others may be operating with older models, but probably no one in this discussion.) And we have trained our minds so that our response to stimuli is perhaps not the same as that of, say, a duck or a chimpanzee, or the same response as any other human, come to that.
And I am not a fan of zombie states: even when Ed is driving along, apparently immersed in maths, he is able to volitionally shift his attention at the leap of a photon. So even if he does not understand the entire process, and does not trigger it on a conscious level, he has trained his "driver brain" synaptic routing to trigger such shifts in subsequent routing commands—which easily includes this shift of attention to focus fully to his driving , if needed.
He has volitional consciousness and his brain are capable of this complexity because it is not a linear machine, but a multi-dimensional, active quantum state of excitation, engaged with itself and with all all else in its fields of awareness.
Enough of my nattering.
Thank you.
feralfae
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feralfae
Boulder climber
in the midst of a metaphysical mystery
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Jul 17, 2017 - 06:47pm PT
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It tells me the mind of a bird would be useful at hauling the pigs on a wall climb. Too funny.
One could hire the Eagles' Union :)
ff
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Ed Hartouni
Trad climber
Livermore, CA
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Jul 17, 2017 - 06:59pm PT
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Lovelock describes Gaia metaphorically as a living organism, struggling to maintain its homeostasis under the growing stress of climate change.
not sure where this quote comes from (you wrote it?) but Lovelock might not have been using a metaphor, rather he considered a model where the entire Earth was described as a living organism and pushed that idea to the limits, scientifically, to see where it would lead him.
But then, I've only read Lovelock's papers on search for extra-terestial life.
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Ed Hartouni
Trad climber
Livermore, CA
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Jul 17, 2017 - 07:05pm PT
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What is the supposed quantum mechanical characteristics of the brain/mind?
As Schrodinger pointed out even before DNA was known, the most important aspect of quantum mechanics with regards to the biochemistry, is the fact that you can consider all the reactions occurring at zero temperature, thus conveying a stability to the important macro-molecules.
As far as an extended, macroscopic quantum mechanical system at room temperature, there is no known example.
There are very good reasons why this is so, and why macroscopic quantum systems occur at very very low temperatures (e.g. superfluid Helium, super-conductors, and the like).
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MH2
Boulder climber
Andy Cairns
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Jul 17, 2017 - 07:15pm PT
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Pretty nice, eeyonkee. I like the free spaces. For the mind to conjure.
Glad to have you on board, feralfae. My comment on God the ultimate algorithm was only a joke on the supposed origin of the word "algorithm." As someone said about humor, "If people laugh, it was funny. If they don't, it wasn't."
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