Knowledge, its use and teaching

Bodies of knowledge are, with a few exceptions, not designed to be taught, but to be used. To teach a body of knowledge is thus a highly artificial enterprise. thus a highly artificial enterprise. The transition from knowledge regarded as a tool to be put to use, to knowledge as something to be taught and learnt, is precisely what I have termed the didactic transposition of knowledge.

Chevallard, Y. (1988, August). On didactic transposition theory: Some introductory notes. In International Symposium on Research and Development in Mathematics, Bratislava, Czechoslavakia.

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What is a mathematical proof?

A dialogue in The Mathematical Experience by Davis and Hersh on what is mathematical proof and who decides what a proof is?

Let’s see how our ideal mathematician (IM) made out with a student who came to him with a strange question.

Student: Sir, what is a mathematical proof?

I.M.: You don’t know that? What year are you in?

Student: Third-year graduate.

I.M.: Incredible! A proof is what you’ve been watching me do at the board three times a week for three years! That’s what a proof is.

Student: Sorry, sir, I should have explained. I’m in philosophy, not math. I’ve never taken your course.

I.M.: Oh! Well, in that case – you have taken some math, haven’t you? You know the proof of the fundamental theorem of calculus – or the fundamental theorem of algebra?

Student: I’ve seen arguments in geometry and algebra and calculus that were called proofs. What I’m asking you for isn’t examples of proof, it’s a definition of proof. Otherwise, how can I tell what examples are correct?

I.M.: Well, this whole thing was cleared up by the logician Tarski, I guess, and some others, maybe Russell or Peano. Anyhow, what you do is, you write down the axioms of your theory in a formal language with a given list of symbols or alphabet. Then you write down the hypothesis of your theorem in the same symbolism. Then you show that you can transform the hypothesis step by step, using the rules of logic, till you get the conclusion. That’s a proof.

Student: Really? That’s amazing! I’ve taken elementary and advanced calculus, basic algebra, and topology, and I’ve never seen that done.

I.M.: Oh, of course, no one ever really does it. It would take forever! You just show that you could do
it, that’s sufficient.

Student: But even that doesn’t sound like what was done in my courses and textbooks. So mathematicians don’t really do proofs, after all.

I.M.: Of course we do! If a theorem isn’t proved, it’s nothing.

Student: Then what is a proof? If it’s this thing with a formal language and transforming formulas, nobody ever proves anything. Do you have to know all about formal languages and formal logic before you can do a mathematical proof?

I.M.: Of course not! The less you know, the better. That stuff is all abstract nonsense anyway.

Student: Then really what is a proof?

I.M.: Well, it’s an argument that convinces someone who knows the subject.

Student: Someone who knows the subject? Then the definition of proof is subjective; it depends on particular persons.Before I can decide if something is a proof, I have to decide who the experts are. What does that have to do with proving things?

I.M.: No, no. There’s nothing subjective about it! Everybody knows what a proof is. Just read some books, take courses from a competent mathematician, and you’ll catch on.

Student: Are you sure?

I.M.: Well – it is possible that you won’t, if you don’t have any aptitude for it. That can happen, too.

Student: Then you decide what a proof is, and if I don’t learn to decide in the same way, you decide I don’t have any aptitude.

I.M.: If not me, then who?

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Uberization of Education

An Uberized education is when…

An Uberized education is when – as in antiquity – one goes to a specific teacher to get lectures, bypassing the university. The students and the teachers are thus matched. If a piece of paper is necessary, it would be given by that teacher, or a group of teachers. It is not too different from the decentralized apprentice model. This already works well for executive “education”. I give short workshops in my specialty of applied probability (I have given a few with PW, YBY and RD, though only lasting 1-2 days), limited to professionals. An Uberization would consist in making longer workshops, say of 2-3 week duration, after which the attendees would be getting a piece of paper of sorts. From my experience, both students and lecturers are more sincere when they bypass institutions. And, as with other Uberizations, it would be much, much efficient economically. A full education would be a collection of such micro-diplomas, which can be done on top of a conventional one. Finally I would personally like to attend such workshops in disciplines outside my specialty. After my experience with Aramaic/Syriac last summer, I have a list of subjects I would be hungry to learn outside university systems…

Source: The Black Swan Report › An Uberized education is when…

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Cover of Carl Sagan’s The Dragons of Eden: A good example of bad science

carl-sagan-dargons-of-eden-cover

Carl Sagan was a wonderful writer. He wrote many amazing books for popularising science and also championed against pseudo-science prevalent in the society. Like many countless others Sagan’s works have inspired and fired imagination in me. Particularly he decimated the arguments made by Velikovsky in Worlds in Collision. Two of his books which deal with the topics of pseudo-science and anti-science are Broca’s Brain, Demon Haunted World: Science as a Candle in Dark. Sagan is most meticulous when explaining things, and adds disclaimers whereever they are necessary and needed.

When I was reading his book Dragon’s of Eden (a Pulitzer prize winner !) the cover of the book stuck me as unusual. The Wikipedia page says that the cover artist was Don Davis. The cover illustration shows a humanoid animal sitting below a tree (of knowledge?) in a serene landscape with a lake and few herbivored near it. Interestingly, and also problematically the cover also shows a variety of dinosaurs in the area as the hominid. This is rather unsettling. And it is definitely wrong. The dinosaurs for all we know, and Sagan knew this too well (for example, see Demon Haunted World), became extinct long before any humanoid forms came into existence. So showing them existing contemporarily is wrong, and factually incorrect science. This illustration goes against all that is known via fossil records that we have.

I wonder what made Sagan, who otherwise was skpetical and very particular, choose this wrong and factually incorrect illustration for the cover of his book, or that he did not have any say in choosing the cover of the book?

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Implicit cognition in the visual mode

Images become iconified, with the image representing an object or
phenomena, but this happens by enculturation rather by training. An
example to elaborate this notion is the painting Treachery of
Images by Belgian surrealist artist René Magritte. The painting is
also sometimes called This is not a pipe. The picture shows a
pipe, and below it, Magritte painted, “Ceci n’est pas une pipe.”,
French for “This is not a pipe.”

176

When one looks at the painting, one
exclaims “Of course, it is a pipe! What is the painter trying to say
here? We can all see that it is indeed a pipe, only a fool will claim
otherwise!” But then this is what Magritte has to say:

The famous pipe. How people reproached me for it! And yet, could you
stuff my pipe? No, it’s just a representation, is it not? So if I had
written on my picture `This is a pipe’, I’d have been lying!

Aha! Yess! Of course!! you say. “Of course it is not a pipe! Of
course it is a representation of the pipe. We all know that! Is this
all the painter was trying to say? Its a sort of let down, we were
expecting more abstract thing from the surrealist.” We see that the
idea or concept that the painting is a \emph{representation} is so
deeply embedded in our mental conceptual construct that we take it for
granted all the time. It has become so basic to our everyday social
discourse and intercourse that by default we assume it to be so. Hence
the confusion about the image of the pipe. Magritte exposes this
simple assumption, that we so often ignore. This is true for all the
graphics that we see around us. The assumption is implicit in all the
things we experience in the society. The representation becomes the
thing itself, for it is implicit in the way we talk and communicate.

Big B and D

When you look at a photo of something or someone, you recognize
it. “This is Big B!” you say looking at the painting! But then you
have already implicitly assumed that the representation of Big B is Big B. This implicit assumption comes from years of implicit training from being submerged in  the sea of the visual artefacts that surround and drown us. This association between the visual representation and the reality it represents had become the central theme of the visual culture that we live in. The training that we need for such an association comes from the peers and mentors that surround us from the childhood. The meaning and the association of the images is taught/caught over the years, so much so that we assume the abstract association is the normal way things are. In this way it becomes the implicit truth, though when one is pressed, the explicit connections are brought out.

Yet when it comes to understanding images in science and mathematics, the same thing doesn’t happen. There is no enculturation of children into understand the implicit meaning in these images. Hardly there are no peers or mentors whose actions and practices can be imitated by the young impressible learners. The practice which comes so naturally in other domains (identifying actor with a picture of the actor, or identifying a physical space with a photo) doesn’t happen in science and mathematics classrooms. The notion of practice is dissociated from the what is done to imbibe this understanding in the children. A practice based approach where the images become synonymous with their implied meaning is used in vocabulary might one very positive way out, this is after all practitioners of science and mathematics learn their trade.

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Epistemicide!

When I read the word for the first time it invoked a very intense and intentional pun in my mind. The word was coined by a Portuguese sociologist Boaventura de Sousa Santos in his multi-volume project Reinventing Social Emancipation. Toward New Manifestos.
In this post I will be elaborating on this term, for my own future use and reference.

Episteme is a philosophical term derived from the Ancient Greek word ἐπιστήμη, which can refer to knowledge, science or understanding, and which comes from the verb ἐπίσταμαι, meaning “to know, to understand, or to be acquainted with”. Plato contrasts episteme with “doxa”: common belief or opinion.

(from Oxford Dictionary of English)

Further more the suffix cide is combining form

  1. denoting a person or substance that kills: insecticide | regicide.
  2. denoting an act of killing: suicide.

So combining the two we get the word epistemicide.


What epistemicide essentially is then is an act of killing certain knowlege, or understanding or acquaintance. It is argued that the English academic discourse which is dominant world over has killed other ways of understanding, or acquiring or transmiting knowledge. To control or invade another territory physically may still keep the invaders and their culture away from the people who are invaded and their knowledge. But with an epistemicide this invasion is complete. For the invaders have successfully dissociated the people they have invaded from their own knowledge and replaced it with the dominant discourse.

For the way that a particular culture formulates its knowledge is intricately bound up with the very identity of its people, their way of making sense of the world and the value system that holds that worldview in place. Epistemicide, as the systematic destruction of rival forms of knowledge, is at its worst nothing less than symbolic genocide.

Epistemicide works in a number of ways. Knowledges that are grounded on an ideology that is radically different from the dominant one will by and large be silenced completely. They will be starved of funding, if the hegemonic power controls that aspect; they will remain unpublished, since their very form will be unrecognizable to the editors of journals and textbooks; and they are unable to be taught in schools and universities, thus ensuring their rapid decline into oblivion.

In the name of freedom and justice, he set about destroying all opposition…

(Bennett, 2007)

Are we performing an epistemicide in our classrooms by only promoting a certain way to learn and teach and worse a centralised way to evaluate and assess that learning? Teaching things which are dissociated from the immediate real world environment of the children? Perhaps we are. This post was just to keep a reference of this term and its meaning. I will explore this further in later posts.

 

References:

Bennett, Karen (2007) Epistemicide! The Translator 13(2)

Oxford English Dictionary (2010)

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Mascot Mantis

mantis-01.JPG

A mascot is any person, animal, or object thought to bring luck, or anything used to represent a group with a common public identity, such as a school, professional sports team, society, military unit, or brand name. [Source]

A few months back, while I was riding my Motorcycle to work, I noticed that a small green praying mantis was stationed on the speedometer. I thought it would move away once the bike starts. So didn’t give it much thought and started the drive. I have to cover three traffic signals to go to work. Now when I stopped at the first signal I noted that the little mantis was still there, holding on to the rim and the glas of the speedometer. As the signal went green, we started the journey again, this time we had appreciable speed as we  were on the highway. I was keeping a tab on the mantis, thinking that it would be flicked away by the air flow. It crouched and held on, no matter how much I accelerated. Its antennae went back with the wind and at times it really struggled to keep on the position. It looked as if it was determined to come with me (or lead me) to the workplace. A personal mascot for me! Leading me through my journey of life. When I finally came to my office, I placed it over the nearby shrub hoping the best for it. (Unfortunately, I did not have my camera at that time, so could not take any photos. )

Having a mantis as a mascot is not a bad thing at all. In the manga Baki Son of Ogre (Vol 2), the hero shadow fights with a mantis for practice, as by weight they are perhaps the strongest of animals. It is claimed that if the mantis was ~100 kg they could hunt a full grown African elephant singly (Note that manga are read from right-to-left). That is the ratio of their weights and size to that of their preys. Since a mascot is supposed to represent you and your qualities, these are no bad at all.

So I am all for a mantis as a mascot, after all, it choose me!

The photo at the top has another story to it. One evening I was just strolling on a terrace when I found this impressive specimen. I placed my camera on the ground to get closer and better photos. The mantis became aggressive
(too friendly??) and came on the camera itself, just after this photo was taken.

 

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Multiple citations, biblatex and APA

Issues with multiple citations of an author and biblatex-apa

Recently I had to refer an author multiple times in a document. I had recently shifted to biber and biblatex. The American Psychological Association (APA) style was the recommended style.

And in the final references

The author name was, let us, was Wilbur-Markus Rowland, and he had about 10-11 articles. The author had many co-authors, in some cases this author was the first author in other cases he was the second or even third one.
Now the issue that I was facing was this: The same author name was getting cited differently at different places.
For example, in some cases it appeared like
W.-M. Rowland, in other cases Wilbur-Markus Rowland, and in some cases even as Rowland. This was very confusing. And on top of all that the final references had entries like

Rowland W.-M. (W. M.)
Rowland W.-M. (Wilbur-Markus)
Rowland W. M. (Wilbur-Markus)
Rowland W. M. (W.-M.)

I checked and rechecked the bibtex entries in hope of finding some error but it was not to be found. I must have done a Clean All a dozen times, after editing the bib file.. I tried adding same name entries for Rowland as first author to be the same, but it didn’t work. This was really frustrating.

Similarly a couple of references with two different authors with a common surname were giving intials in the main text. For example, D. R. Cook and M. P. Cook. Now all other refernces (except the Rowland one) were coming as per the APA requirement, which is (Author, Year), so ideally they should have been (Cook & Weston, 1999) and (Cook, 2004). But rather they were being displayed as (D. R. Cook & Weston, 1999) and (M. P. Cook, 2004). Now no change in the bib file was changing any of this. I thought there was some issue with the bib entries. I deleted the entries and entered them again manually, but no avail.

Then I chanced upon this link

 even though the year of publication differs in the two Campbell (Cook in our case) references, the lead author’s initials should be included in all text citations, regardless of how often they appear.

So the Cook mystery was solved. biblatex was compiling correctly as per the APA guidelines, the initials for the two different Cook entries must be there. This is because

 Including the initials helps the reader avoid confusion within the text and locate the entry in the reference list.

Now for the Rowland entry, the next part of the blog gave me a hint towards the possible problem:

Although this rule seems straightforward, one thing that trips up some writers is how to proceed when different lead authors with the same surname are also listed in other references in which they are not the lead author.

After this I checked the entries where Rowland was second or third author. These entries differed from the entries where he was the lead author. And this was causing the problem. For each different entry style of his name, biblatex was considering him as a separate author. Hence the initials and the different references entries with names in brackets. Once all the entries for Rowland were made consistent the problem disappeared. Phew!

This issue bugged me for almost a couple of working days, to find the cause and subsequent addressing was most rewarding.

Note that the bibtex references that I had were taken from google scholar, hence differing styles in author name. Please make them consistent. This is a warning for future me and others who are reading this post.

 

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Hymn of Creation from Rig Veda

This wonderful Hymn of Creation one of the oldest surviving records of philosophic doubt in the history of the world, marks the development of a high stage of abstract thinking, and it is the work of a very great poet, whose vision of the mysterious chaos before creation, and of mighty ineffable forces working in the depths of the primeval void, is portrayed with impressive economy of language.

“Then even nothingness was not, nor existence.
There was no air then, nor the heavens beyond it
What covered it? Where was it? In whose keeping?
Was there then cosmic water, in depths unfathomed?

“Then there were neither death nor immortality,
nor was there then the torch of night and day.

The One breathed windlessly and self-sustaining.
There was that One then, and there was no other.
“At first there was only darkness wrapped in darkness.
All this was only unillumined water.

That One which came to be, enclosed in nothing,
arose at last, bom of the power of heat.
“In the beginning desire descended on it
that was the primal seed, bom of the mind.

The sages who have searched their hearts with wisdom
know that which is is kin to that which is not.
“And they have stretched their cord across the void,
and know what was above, and what below.

Seminal powers made fertile mighty forces.
Below was strength, and over it was impulse,
“But, after all, who knows, and who can say
whence it all came, and how creation happened?

The gods themselves are later than creation,
so who knows truly whence it has arisen?
“Whence all creation had its origin,
he, whether he fashioned it or whether he did not,

he, who surveys it all from highest heaven,
he knows— or maybe even he does not know.

From – The Wonder That Was India – A. L. Basham

 

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Review of I Am A Strange Loop by Douglas Hofstadter – Part 2

Part 1

The toilet flush is one of the simplest and common feedback mechanisms that we find. There is a float which rises with the water level which controls the inflow of water. After a certain height is reached the water inflow is stopped. Do we attribute intentionality to the water flush? We usually do not. And this is the theme that Hofstadter explores in Chapter 4 Loops, Goals and Loopholes.

But what kinds of systems have feedback, have goals, have desires? Does a soccer ball rolling down a grassy hill “want” to get to the bottom? 52

We anthropomorphize objects and impart them our human attributes. Adding a “purpose” or a “goals” to any system is considering it from a teleological perspective. Teleology is the explanation of phenomena by the purpose they serve rather than by postulated causes. Considering examples of variations on this theme, we can say that answer to the above question is not clear cut. There are no black-and-white answers but are judgment calls. We tend to move towards the idea of teleology and intention for a system when the feedback mechanisms are not directly perceptible.

Among other examples, Hofstadter considers plants which in normal time will appear to be static and without any “goals”. But a time-lapse of the same would show that they have “goals” and “intentions” and use strategies to achieve them.

The question is whether such systems, despite their lack of brains, are nonetheless imbued with goals and desires. Do they have hopes and aspirations? Do they have dreads and dreams? Beliefs and griefs? 53

The claim is made that presence of a feedback loop in a system, triggers in us a response which shifts the description from a goalless level of mechanics to a goal-oriented level of some cognitive mechanism. Things have the desire to move!

So far we have considered basic feedback loops. Now we move onto a more complex idea of a positive feedback loop. In a positive feedback loop, a part of the output of the system goes into increasing the output by a certain factor. With each iteration the output increases, which causes the next output to increase even more. A small change in input can cascade into a very large change (exponential) in output.

Perhaps the most common example of a positive feedback loop is the unpleasant, high pitch sound one hears in an auditorium or a meeting. This happens when a microphone gains some of its output as an input and produces an ever increasing pitch and volume of the input sound. An example is given below:

Now one can imagine that due to the exponential nature of growth, any little disturbance in such a system might lead to a sound which will eventually destroy everything.

In theory, then, the softest whisper would soon grow to a roar, which would continue growing without limit, first rendering everyone in the auditorium deaf, shortly thereafter violently shaking the building’s rafters till it collapsed upon the now-deaf audience, and then, only a few loops later, vibrating the planet apart and finishing up by annihilating the entire universe. What is specious about this apocalyptic scenario?

But this is a fallacious argument. The first fallacy is the physical nature of the setup and the amplifier in our scheme of things. If the roof falls, it will destroy the amplifier too! The second case is the nature of the amplifier, it doesn’t amplify in an unlimited way. After a certain gain, due to the physical design, the amplification becomes equal to unity and the system stabilizes at its natural frequency. It so happens that the natural high frequency of an audio amplifier is close to a high pitch scream. This is achieved by the system tends to go towards that pitch in series of rapid iterations. These are the screeching high pitch oscillations that we hear. It seems the systems “wanted” to go there, the stable point of its existence. Thus we see that

Similarly, we can also “see” visual feedback loops, when the output of a camera is given back to the camera. This can be most easily setup by pointing the camera towards a screen which is showing a live output of the camera. The cover image of the book is one such image, captured during Hofstadter’s “experiments” with the visual feedback system. One of the difference, in this case, is that the camera is not an amplifying device, it just transmits. Yet the pictures it produces are bizarre and beautiful. Seeing images of video feedback gives one a sense of mystery and wonder. There is some inherent beauty in it, yet it seems un-natural to watch.

Feedback — making a system turn back or twist back on itself, thus forming some kind of mystically taboo loop — seems to be dangerous, seems to be tempting fate, perhaps even to be intrinsically wrong, whatever that might mean. 57

Shifting gears, we get a Hofstadter’s introduction to Gödel when he was fourteen. What intrigued him was the thought that one could have an entire book about a single book. The book was Nagel and Newman’s Gödel’s Proof, published in 1958. Hofstadter wrote the introduction to the new imprint in 2001. He was fascinated by footnote on formal use of quotation marks.

So here was a book talking about how language can talk about itself talking about itself (etc.), and about how reasoning can reason about itself (etc.). I was hooked! I still didn’t have a clue what Gödel’s theorem was, but I knew I had to read this book. 58

This is something that happens to me too. Some time back (almost a decade now) I had posted about books attracting me. Perhaps it happens to many people.

We next look at famous Russel’s Paradox. One of the examples derived from it is Barber’s Paradox

The barber is the “one who shaves all those, and those only, who do not shave themselves.” The question is, does the barber shave himself? [.]

There is also a loop here and there is contradiction too.

This loophole (the word fits perfectly here) was based on the notion of “the set of all sets that don’t contain themselves”, a notion that was legitimate in set theory, but that turned out to be deeply self-contradictory. 60

Russell tried to overcome this by formally re-defining the concepts of sets to save this, but it didn’t work out well. Rather it became too complex, though built on solid, atomic (in the mathematical sense) ideas.

In Principia Mathematica, there was to be no twisting-back of sets on themselves, no turning-back of language upon itself.  61

But why is self-reference considered problematic? Here Hofstadter quotes from his column Metamagical Themas (an anagram of Martin Gardner’s Mathematical Games) in Scientific American on Self-Referential sentences. But all were not receptive to the idea, some of the readers were sceptical about the utility of self-reference and denied any meaningful output of such activities.

In the next chapter On Video Feedback we explore the theme of video of video feedback and Hofstadter’s experiments with it. He explores and explains many of the images which were made by adding slight things in the image, fox example, truncated corridor, endless corridor, helical corridor etc. The common element in all these video feedback is the repeating of the primary image in scaled down fashion till the resolution of the screen can support (theoretically infinite). During one the experiments, he covers the lens and then removes his hands. During this, the movement of his hand is captured and forms an endless image which is moving, even when the hand is removed. This action has formed a loop and is feeding itself in a cyclic setup.

A faithful image of something changing will itself necessarily keep changing! 67

A similar phenomenon is that of dogs barking in sync. Some dog somewhere, starts to bark for something that is passing near it. Now, other dogs pick up and start barking too. And the chain goes on. Once setup, it doesn’t matter what was the reason for the first dog to bar, it may have gone away. But the chain of barking sustains itself. During one the flights, I have seen this happen with small babies. There were about 5-6 babies on the flight. It so happened that one of them started to cry for some reason. Then the rest joined in one-by-one. Perhaps the others were crying because the heard another one cry. And the event became self-sustaining. This went on for quite some time.

This is one of the core idea of an emergent phenomenon, once

In general, an emergent phenomenons omehow emerges quite naturally and automatically from rigid rules operating at a lower, more basic level, but exactly how that emergence happens is not at all clear to the observer. 68

The video explorations led to some fantastic images, many of which are reproduced in color in the central pages of the book. In the last part of the chapter, Hofstadter drives towards one of the central themes which we will explore in the remaining book. The idea is that strange and robust (self-sustaining) structures can emerge from the process of looping.

Once a pattern is onthe screen, then all that is needed to justify its staying up there is George Mallory’s classic quip about why he felt compelled to scale Mount Everest: “Because it’s there!” When loops are involved, circular justifications are the name of the game. 70

Some of the images I myself have collected are shown below:

The locking-in gives rise to abstract phenomena at higher levels.

In short, there are surprising new structures that looping gives rise to that constitute a new level of reality that could in principle be deduced from the basic loop and its detailed properties, but that in practice have a different kind of “life of their own” and that demand — at least when it comes to extremely finite, simplicity-seeking, new level of description that transcend the basic level out of which they emerge. 71

Whether we will be able to actually do it, or want to do it is another question. This reminds me of the saying: In theory, there is no different in theory and practice, in practice there is.

Here are a few more:

 

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