Was mathematics discovered or invented?

Zeuzzz

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Zeuzzz
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I was asked this by someone the other day and it rather befuzzled me.

Thoughts?
 
Zeuzzz said:
I was asked this by someone the other day and it rather befuzzled me.

Thoughts?
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The notation and such is invented, of course. But the underlying math is discovered. An advanced alien race would discover pi, for instance, and ignoring irrelevancies like base, would find that it contained all the same digits for however long they desired to compute it. Math is certainly "deeper" than any human invention, and seems to be deeper than the Universe itself. It almost makes one want to be a Platonist.

- Dr. Trintignant
 
Mathematics is completely invented. The reason is pretty simple: mathematics is completely free of any (non-self-referential) semantic content. One can use mathematics to describe many structures in the world, but by itself it doesn't actually refer to anything. It's a cliche that mathematics is a universal language, and it's completely wrong--to carry the same analogy, mathematics is perhaps a universal grammar, but that's a bit less than a language.

Sometimes, mathematicians like to talk about discovering this or that, and indeed they do--but it would be equivocating to say that about mathematics in toto, or even mathematical systems. Surely it is possible for one to discover, say, a new strategy or effective opening in chess, but in no way does it mean that chess has been discovered rather than invented.

So in some sense, mathematical results are discovered, but mathematics per se is still invented.
 
Discovered, but numerology was invented which proves some inventions can be profitable without actually working.
 
Jorghnassen said:
Zah? Would physics stop working if we didn't have the words to describe it?
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Mathematics is nothing more or less than symbolic logic. It works for physics because the world is logical.

Where do mathematical concepts lie for them to be discovered?
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In the world, and the rules by which it operates.
 
The question assumes that an invention is different from a discovery.

Is this true?
 
yy2bggggs said:
The question assumes that an invention is different from a discovery.

Is this true?
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Good question. Could it not be said that an invention is nothing more than a discovery of some entity, concept or system.
In any case, regarding mathematics, the universe would behave mathematically with or without our discovery, invention or our existence, for that matter.
 
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Neither. Or both. In the end, it demonstrates the limitations of our language, not of any complexity in the nature of mathematics.

Athon
 
sol invictus said:
Mathematics is nothing more or less than symbolic logic. It works for physics because the world is logical.
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Mathematics broadly construed is the study of abstract structure. There's no reason to tie it down to any particular logic--it can be the structure of imaginary worlds quite removed from this one (and the myriad of mutually contradictory systems of symbolic logic are good evidence for this). Perhaps that's not as interesting or useful, sure, but that's an issue of what motivates or interests those who do mathematics, not a necessary condition on mathematics in general.

sol invictus said:
In the world, and the rules by which it operates.
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Saying the world is mathematical is quite different from saying mathematics is tied down to describing the world.
 
yy2bggggs said:
The question assumes that an invention is different from a discovery.

Is this true?
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It's a matter of degree. I can encode everything about a Picasso painting with a single number. This number already "exists" in some sense; even existed long before Picasso was alive. But it's a long enough number that it could not come about by accident, or by some universal logical process (in the way that pi could be discovered). Instead, it's the result of a complicated mental process that is unique to an individual--hence, we call it an invention or creation.

- Dr. Trintignant
 
Vorpal said:
Perhaps that's not as interesting or useful, sure, but that's an issue of what motivates or interests those who do mathematics, not a necessary condition on mathematics in general.
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Maybe not, but why is it that some types of mathematics are more interesting than others? Partly because some are more applicable to describing our universe than others, but I don't think that's all of it.

What makes pi more interesting than a number that differs from pi only at the 35,000th decimal place? Either one is sufficient for describing circles in our universe, but somehow the "real" pi is more interesting, and I think most people accept that a hypothetical alien species would also discover pi. How could it be that both parties come up with the same number if it's only an invention?

- Dr. Trintignant
 
Both.

Specific mathematical systems (sets of axioms and operators) are invented. Geometry, arithmetic, symbolic logic, lambda calculus, algebra, Turing machines, set theory, and so forth are all invented.

However, the equivalences between them were discovered. Not invented or otherwise designed, and not expected or planned for by their inventors. And the pervasiveness of these equivalences seems downright bizarre. It appears that you can make up just about any set of rules for transforming strings of symbols or collections of abstract objects into other strings of symbols or other collections of abstract objects, and as long it's not trivially simple, you can eventually prove it's equivalent, in terms of what it can compute or prove, to just about any other axiom system in mathematics.

That suggests that certain (as yet undiscovered, or at least unproven) underlying rules of mathematics go, as Dr. T put it, deeper than the universe itself.

Respectfully,
Myriad
 
Dr. Trintignant said:
It's a matter of degree. I can encode everything about a Picasso painting with a single number. ...
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Possibly. It's plainly obvious to me that not all discoveries can be called inventions. What's less clear, however, is whether or not all inventions can be called discoveries, or whether or not the question even makes sense to the degree that it has a definite answer. To me, it seems to be the case that all inventions should be able to be called discoveries in some abstract potential concept space, but then, I could also see that it is, indeed, just a matter of degree here, and that I simply don't quite "grasp" some significance about the distinction.

But I think there's definitely an overlap--things you can call inventions and discoveries. So perhaps it's more appropriate to simply question whether or not mathematics falls into the overlap category.

Another possibility that I think demands consideration is this. "Mathematics" isn't one thing anyway... it's a classification for a large number of things. So, supposing there's some clear enough distinction that I simply can't come up with, such that some things are inventions and not discoveries, some things are discoveries and not inventions, and some things are both. Even then, it's not readily apparent that all math falls into one of these three categories.
 
Vorpal said:
Mathematics broadly construed is the study of abstract structure. There's no reason to tie it down to any particular logic--it can be the structure of imaginary worlds quite removed from this one (and the myriad of mutually contradictory systems of symbolic logic are good evidence for this).
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I suspect that all these logics fit into a unifying, more general structure, which itself describes some aspects of the world.

Saying the world is mathematical is quite different from saying mathematics is tied down to describing the world.
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So you're arguing that only certain kinds of mathematics are discovered, and the rest invented? Maybe. But I think that's a delicate distinction to try to draw.

It's clear that we were led to those other logics by thinking about how to generalize something (I'm lacking a term here, sorry). But if (as I maintain) it was the world that taught us logic in the first place, and these other logics are natural extensions of it, they're not really pure inventions, are they?

For example if you learn how to count by observing how many eggs you put in a basket, the fact that you've never observed 6,123,860 eggs (or that many of anything else) doesn't mean you invented that number, does it?
 
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Not that I care all that much for the semantic disctinction since it won't make any real-world difference, I say that it is a human invention used to describe what already exists in the physical world.

We didn't invent color or taste. If someone wants to know what an apple looks like, show them an apple. If they want to know what it tastes like, let them bite it. We merely invented words to describe it in the abstract.
 
Dr. Trintignant said:
Either one is sufficient for describing circles in our universe, but somehow the "real" pi is more interesting, and I think most people accept that a hypothetical alien species would also discover pi.
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Or maybe 2pi instead or something else instead, but yes, I'd expect their mathematics to be translatable to our mathematics fairly easily.

Dr. Trintignant said:
How could it be that both parties come up with the same number if it's only an invention?
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So hypothetically how would they arrive there? By observing circular objects and measuring them? That's not mathematics; if anything, that's physics. By idealizing the structure of those objects, giving them names, and studying them in the abstract? Then they would not be able actually 'derive' pi (or whatever) without also inventing a plethora of further mental contraptions.

Conundrums like these is why I previously distinguished a sense in which mathematical things can be discovered and the more general sense in which 'mathematics itself' is an invention.

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sol invictus said:
I suspect that all these logics fit into a unifying, more general structure, which itself describes some aspects of the world.
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That may be true...

sol invictus said:
So you're arguing that only certain kinds of mathematics are discovered, and the rest invented? Maybe. But I think that's a delicate distinction to try to draw.
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Not quite what I meant--I would rather than that some math is inspired directly observations of the world, rather than discovered.

I would try to explain the success of physics by first agreeing that the universe most likely has some constant, underlying structure. As mathematics is the study of structure, it's then natural that math is successful in physics, and since we're always immersed in it, it's also natural that much of mathematics is heavily inspired by the physical world.

So my counterargument is basically thus: the underlying universal structure "could have been" different, and mathematics would still deal with it--it's a study (the abstraction) of any structure, not just the ones that actually obtain. Your above response seems to be that we aren't really stepping outside the boundaries of the structure when we imagine the alternatives. That's consistent, but then it also carries a bit more metaphysical baggage than necessary to explain the success of physics.

(Or the petulant response: stop trying to make me into some kind of physicist!)
 
Vorpal said:
Or maybe 2pi instead or something else instead, but yes, I'd expect their mathematics to be translatable to our mathematics fairly easily.
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Right. The specifics don't matter too much. The point is just that there is something there that is universally recognizable.

Vorpal said:
Then they would not be able actually 'derive' pi (or whatever) without also inventing a plethora of further mental contraptions.
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True enough. But one can arrive at certain mathematical truths (like pi) by a variety of means. Perhaps the aliens wouldn't invent calculus (although I think they would, since it was invented more than once by humans), but pi pops up in enough places that they would be guaranteed to find it.

Conundrums like these is why I previously distinguished a sense in which mathematical things can be discovered and the more general sense in which 'mathematics itself' is an invention.
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But you also said that it's free of any semantic content. But your statement seemingly ignores the observation that Wigner called "The unreasonable effectiveness of mathematics in the natural sciences". One could, in principle, develop a mathematics in complete isolation, and eventually come up with pi. So why is it that circles in the real world, approximately speaking, have a circumference pi times the diameter?

There is a very mysterious connection between math and the real world. I can't explain it except to posit that math is in some sense more real than the universe itself, and we are just here to discover it.

- Dr. Trintignant
 
Dr. Trintignant said:
But you also said that it's free of any semantic content.
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I'm not seeing a contradiction. Discover an interesting phenomenon -> become inspired -> invent a mathematical system -> discover things about that system. The step of assigning mathematical terms and quantities a physical interpretation (thus providing non-self-referential semantics) is not part of mathematics per se.

Dr. Trintignant said:
But your statement seemingly ignores the observation that Wigner called "The unreasonable effectiveness of mathematics in the natural sciences".
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I just don't it as unreasonable that a tool that's applicable in a very wide range of circumstances also works in a particular case. If the variety of abstract structures that mathematics is in principle able to study is richer than the variety that actually obtains in the world, it would not at all be surprising that it is effective.

On the other hand, a large chunk of the mathematics we actually do have has applications precisely because those who create those mathematical systems are immersed in the world.

Dr. Trintignant said:
One could, in principle, develop a mathematics in complete isolation, and eventually come up with pi. So why is it that circles in the real world, approximately speaking, have a circumference pi times the diameter?
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So, what you're saying is that it is in principle possible to randomly create abstract structures without any empirical input whatsoever, and then after the fact find out that one has stumbled upon something that has a correspondence to the real world? What's so strange about that, exactly?

What would be far stranger is that if it was the case than any mathematical structure one concocts in such isolation has some corresponding part in the real world. I suppose it could be true, but it's such a hefty claim to accept without very good reasons. And if it's not true, then we don't need to bother with the heavy metaphysics imposed on us by declaring that mathematics is discovered.
 
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Dr. Trintignant said:
....snip... and I think most people accept that a hypothetical alien species would also discover pi. How could it be that both parties come up with the same number if it's only an invention?

- Dr. Trintignant
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(Ignorant of Maths beyond A level so I am just musing).

Does pi exist in the "real world", isn't it more of a "platonic ideal" of what a circle should be rather than what a circle is?
 
Vorpal said:
So, what you're saying is that it is in principle possible to randomly create abstract structures without any empirical input whatsoever, and then after the fact find out that one has stumbled upon something that has a correspondence to the real world? What's so strange about that, exactly?
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I wouldn't say "randomly". Real mathematicians are inspired by things like symmetry and elegance, and build systems based on these principles. Now, as it turns out, the laws of the universe are also symmetrical and elegant, but I'm not at all certain that the mathematicians simply copied what they observed (since in many cases, the mathematical symmetry came first).

This isn't like some lottery, where sombody has to "win" despite there being no underlying strategy. It very much seems like if you build consistent mathematical systems, you will find that an "unreasonably" high percentage of them have some applicability to reality.

What would be far stranger is that if it was the case than any mathematical structure one concocts in such isolation has some corresponding part in the real world. I suppose it could be true, but it's such a hefty claim to accept without very good reasons. And if it's not true, then we don't need to bother with the heavy metaphysics imposed on us by declaring that mathematics is discovered.
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Well, I don't think that's true, if for no other reason than that we can't be sure that this universe is the only one. Maybe there is a universe where the truth of the axiom of choice matters. It doesn't seem to be this one, so we're able to invent math beyond what matters to physical law. Anyway, your situation would be very strange indeed, but I don't see how it makes the current one not strange.

Imagine going to an alien world, and finding that one of the creatures created a perfect replica of a Picasso painting, down to the finest detail. You can prove by some means that each painting was created in perfect isolation.

So why should we be surprised at this, while not being surprised at them having discovered pi?

- Dr. Trintignant
 
Darat said:
(Ignorant of Maths beyond A level so I am just musing).

Does pi exist in the "real world", isn't it more of a "platonic ideal" of what a circle should be rather than what a circle is?
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Yes, and that's what I find so strange about it. I can write a little computer program to generate a few million digits of pi. The program doesn't seem to have any real relationship with the universe--it's not building a circle in memory and somehow measuring it, or anything like that. And yet, if I go out in the real world, and build bigger and more precise circles and measuring them, I end up converging on the same number.

Worse, pi pops up in far more places than just circles, such as statistics. Why is it that the same number comes up in such disparate topics?

I bring up pi often because it's familiar to most people. But there are many other mathematical objects that also have strange connections. A famous one in math/physics circles is the "Monstrous Moonshine Conjecture", which relates some abstract mathematical group theory with string theory in physics. And the connection was discovered by observing that some coefficients in an equation (the number 196884, specifically) matched the number of dimensions in the so-called Monster group (cue the math people correcting my description :)). It's as close to numerology as you can get in the real world. I find this connection rather weird, since the two fields were developed quite independently.

- Dr. Trintignant
 
Fire was evidently "discovered", but we can say that many techniques for making fire have been "invented".

The same should be true for mathematics too: the techniques (formulas) that we used in math do not exist as such, they have been invented. But some basic concepts like 1, 2, 3 could be said to exist regardless of human understanding of them, so they have been "discovered" rather than "invented" by mankind.
 
JJM 777 said:
Fire was evidently "discovered", but we can say that many techniques for making fire have been "invented".

The same should be true for mathematics too: the techniques (formulas) that we used in math do not exist as such, they have been invented. But some basic concepts like 1, 2, 3 could be said to exist regardless of human understanding of them, so they have been "discovered" rather than "invented" by mankind.
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Unless the universe distingiuished the order or ascending or descending numbers, then that remains purely creation on our behalf.
 
Dr. Trintignant said:
I bring up pi often because it's familiar to most people. But there are many other mathematical objects that also have strange connections. A famous one in math/physics circles is the "Monstrous Moonshine Conjecture", which relates some abstract mathematical group theory with string theory in physics. And the connection was discovered by observing that some coefficients in an equation (the number 196884, specifically) matched the number of dimensions in the so-called Monster group (cue the math people correcting my description :)). It's as close to numerology as you can get in the real world. I find this connection rather weird, since the two fields were developed quite independently.

- Dr. Trintignant
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This I'd like to hear more about!

(going to bed now, but hoping that some exposition might be forthcoming even in absence of immediate responses :) )
 
In fact, the question of whether any of the math is representationally true of the world at large, reminds me of this qoutation by Einstein:

''"Equations are more important to me, because politics is for the present, but an equation is ... "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality. ...'
 
Dr. Trintignant said:
I wouldn't say "randomly". Real mathematicians are inspired by things like symmetry and elegance, and build systems based on these principles.
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If the distribution is skewed in some ways, that's fine.

Dr. Trintignant said:
Now, as it turns out, the laws of the universe are also symmetrical and elegant, but I'm not at all certain that the mathematicians simply copied what they observed (since in many cases, the mathematical symmetry came first). ... It very much seems like if you build consistent mathematical systems, you will find that an "unreasonably" high percentage of them have some applicability to reality.
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Inspiration can be far from copying. Mathematicians frequently use mental models that are laden with their experience of the world, be it visualization or otherwise. The primary purpose of the brian is to deal with the world, and it does so practically all the time. Is it really so unreasonable that the mathematical systems it produces are skewed in that direction?

Dr. Trintignant said:
Well, I don't think that's true, if for no other reason than that we can't be sure that this universe is the only one. Maybe there is a universe where the truth of the axiom of choice matters.
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It might matter in this one. The Hahn-Banach theorem seems like a natural candidate, for example. Although I don't know enough QM to say with any certainty.

Dr. Trintignant said:
Anyway, your situation would be very strange indeed, but I don't see how it makes the current one not strange.
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It might not make it less strange, but it does make it more palatable in comparison, because those situations oppose each other. If mathematics must be discovered, then my strange situation is exactly what obtains.

Dr. Trintignant said:
So why should we be surprised at this, while not being surprised at them having discovered pi?
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Because if you repeat a pattern enough times, even the dumbest sod will get it, and that's a common part of experience. I'm not objecting that the physical pattern is there, that it can be described by mathematics, or that it inspires mathematical development. I'm just insisting on the difference between physical patterns and mathematical abstractions, and saying that the former is discovered while the latter is invented, because that avoids the metaphysical quagmire of considering both a discovery. (The problem of success of physics I've already addressed, although feel free to questions my given reasons.)
 
I vote discovered.

It was well known for quite some time that there existed mathematical paradoxes that could yet not be explained (Zeno's Paradox) which eventually become solvable through limits and calculus. The solution to the problem did, of course, exist in Zeno's time ... but yet lie undiscovered, or thought of. Because developments in Mathematics require new and sometimes counter-intuitive thinking, one can be led to believe it was invented, but I considerer it discovery in a way of thinking as opposed to the conventional definition of coming upon a physical thing.
 
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Vorpal said:
If the distribution is skewed in some ways, that's fine.
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Vorpal, what's an example of something in math that according to your argument is very far from the "real" world, something very much an invention? One of these non-standard logics?

If so, I have a question for you: when you prove something with one of those logics, I would think (although I've never tried) that the proof proceeds in very much the same way a proof using standard logic would. Presumably there are some axioms you're not allowed to use, or perhaps some new ones, but bearing that in mind you still proceed by manipulating expressions with valid operations, one after the other, aiming to produce some useful output. So the actual procedure is very very similar to a procedure using standard logic, or doing integer arithmetic for that matter.

Similarly someone checking your proof would follow much the same set of procedures they would to check any other proof.

So my question is: what about those procedures? If these non-standard logics are so removed from the world, why can we study them using the same techniques we apply to "real-world" logic? Is it possible that those logics are analogous to, say, base 10 decimal notation with arabic numerals (i.e arbitrary choices of some specific form), while it's the underlying rules we use to actually do mathematics that are the discovered part, the part that we learned from observing the world?

Feel free to answer "no". :)
 
Vorpal said:
Mathematics is completely invented. The reason is pretty simple: mathematics is completely free of any (non-self-referential) semantic content. One can use mathematics to describe many structures in the world, but by itself it doesn't actually refer to anything.
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I disagree, thoroughly. A lack of non-self-referential content has nothing to with the question. Mathematics has an internal consistency that's the same no matter who discovers it. Whether it merely describes the real world, or is inherent in the running of it, is a separate question from whether it's discovered or invented.

The fact that the same mathematical truths are frequently found by people of different cultures working separately, to me speaks volumes that mathematics is a science of discovery. Yes, people may choose to explore different areas, but to say it's invented would imply that they could come up something based on their own personal whims, when that is clearly not the case.

But then, by your sig line, maybe we're both wrong :)
 
Did Euler discovered or invented this relation between Pi and the natural integers ?

euler.gif


I believe the relation was true way before Euler could have "invented" it.

Since it was true before Euler was even born my guess is that he discovered it.

nimzo
 
Look at mathematics in its most primitive form: the act of counting or adding.
2 + 2 = 4 is a basic truth. One cannot invent a mathematics with a different answer. Two added to two yields four, stones or spears or abstractly. Primitive man did not invent that; they discovered it. One can build from there to a more complex chain of abstraction. x + 2 = 4, yields x = 2. From there, another level of abstraction leads to Galois theory, and so on. After many levels of abstraction, it appears to be the result of invention, but that is a deception: it is all a long and complex chain of step by step discovery.
 
Singularitarian said:
Unless the universe distingiuished the order or ascending or descending numbers, then that remains purely creation on our behalf.
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The Universe does not need to know nor care about these things for them to have always existed/been true, no more than it does not know nor care how particles interact for them to have existed long before we existed and became aware of their behaviour.

If anything, we recreate on the facts by talking about them and studying them. But that's just a semantic issue and has nothing to do with actually creating something, regardless of the similarity of the words.
 
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