“God made the integers, all else is the work of man”
I’m not sure if even that is true. Integer counts and succession seem like human constructs to me. An apple isn’t fundamentally a single unit. If I take a bite out of it, is it still one apple?
Well at least according to the friends I have beer with over the weekends, real world is a subset of math we invented. It is a tool we invented to describe the universe, but I am not sure if everything we derive from the axioms is true. (eg. discontinuous functions make me a little uncomfortable)
Disclaimer: I am very poor at math but eager to learn :)
This quote is neither true nor false. The author called it a "sentiment", but I think when Kronecker made this statement, he took it as some kind of principle to work out the foundation of modern mathematics.
Basically the mathematical notion of integer is trying to make sense of our idea of a discrete unit. Taking a bite of apple makes it non-unit, but I think you would agree that the notion of "one apple" or "two dogs" makes common sense.
Also when reading someone's quote, I would always consider its context. I think Kronecket had the axiomatic construction of real number in mind. The idea of rational number and real number can be investigated by just assuming the idea of increment by one is solid.
These mathematical concepts are usually covered in a undergraduate analysis class. For more information, see Cauchy's construction and Dadekind's construction.
Add: so the idea is that once we have integers, we can construct rationals and even real numbers instead of talking about reals as granted. Now when taking a bite out of an apple, we can say it takes, say 1/3, of an apple, while we only assume the idea of integers.
I think this is a really good comment. Are you implying that the concept of integer is a consequence of the abstraction of an apple, an abstraction which is entirely human? I really like this point of view.
I have been entertaining this idea for years. First time I hear someone else voice it.
Do you have an idea what the philosophical term for this idea is? It seems related to connectionism.
One might assume that abstraction and pattern matching in the brain are required to determine that two "things" are the same. It is perhaps a small step to go from comparing things to comparing integers.
Unfortunately, this line of reasoning quickly goes awry, because one cannot assume even the existence of things to describe brains or pattern matching.
The abstraction from counting to the natural numbers is called "decategorification", in category theory. The abstraction of sameness is called "equivalence" or "isomorphism".
> Unfortunately, this line of reasoning quickly goes awry, because one cannot assume even the existence of things to describe brains or pattern matching.
You might be interested in the univalent foundations approach, where the idea is that mathematics is fundamentally the study of equivalences between constructions, and equivalences themselves are first-class objects that are just as easy to talk about as any other mathematical object.
I don’t. I’ll do some research. The origin of this concept comes from discussions I had on identity, specifically the ability to separate the “self” from the rest of the universe, or to isolate some quantity of stuff and claim it’s an apple. Ultimately the universe has no concept of such boundaries.
[update] the study of this concept seems to be semiotics, which examines the creation and use of symbols. I don’t know what branch of semiotics examines the creation of the concepts of unity and succession.
An apple is an abstraction. It's a combination of chemicals bonded together. Those chemicals are composed of atoms which are bonded together, each of which are composed of electrons, protons, and neutrons. Those particles are composed of quarks, and so on.
An apple is a singular thing because it is useful for us to think of an apple as a singular thing in most contexts. It is not a singular thing when it is more useful to look deeper.
Yes. Not to say nothing exists though or that there is no meaning.
To get really out of the realm of HN, no one thing can be proven to exist outside of our collective hallucination. That hallucination has reliable rules, thankfully, so things can be proven to exist inside it.
> Integer counts and succession seem like human constructs to me.
There's definitely some "elementary" mathematical concepts that seem native to our universe. For example, in outer space stuff tends to come together in ways that approximates spheres. We see lattices in crystalline solids all the time. There's naturally-occurring fractals all over the place. The existence of fractals -- snowflakes, trees, broccoli -- seems to "embed" a successor (to iterate over generations), so I'm not sure if it's a human construct.
Your conclusion doesn't follow your premise. Whether it's still an apple is a question of categorisation and a human construct (1). But once it's established it is indeed an Apple you can count the number of such Apples.
Counting is real and the number you count is "true" because it leads to real predictions. This tale of magic bucket is enlightening. (2)
There are different ways to understand reality. For spacial understanding we have cartesian and euclidean models, the former suggests reality is digital, the latter suggests reality is analog, both give real predictions, both are true? For temporal understanding we have declarative and imperative models, the former suggests infinity exists, the latter doesn't, both give real predictions, both are true?
I've always been interested by the Platonist-vs-anti-Platonist debate. I generally disagree with finitist mathematicians, but I really do find their position interesting, and as a computer scientist I'm not unsympathetic to their perspective.
At the end of the day, though, I always go back to an old joke my undergrad philsophy professor told me:
Analytic Philosopher A: "Hey, do you believe in baptism?"
Analytic Philsopopher B: "Believe in it? Hell, I've been to one."
Constructivism is a less extreme position than finitism. It's hard to describe it, but it's got something to do with not believing that a mathematical object exists until it's been "constructed". I'm not sure if it can adequately be described without referring to formal logic. One motivation for considering this philosophy is that mathematics becomes more "computable" when you do, but there are other reasons as well.
The statement that there are infinitely many integers is true in constructive mathematics: Given any integer, you can construct some integer larger than it. Likewise, the statement that the real numbers are uncountable is also true: Given any sequence of real number, it's possible to construct a real number that's not in the sequence (using Cantor's method).
The claim that all real numbers are computable doesn't have any constructive content, so its truth depends on which version of constructivism you believe in. Personally, having looked into this, I prefer versions of constructivism in which the claim is false, because then you can claim things like every continuous function on R is locally uniformly continuous (which weirdly enough, implies that not all real numbers are computable, without asserting the stronger claim that an uncomputable one exists).
Only to the extent “real” is defined by something being embodied by entities of physics.
Though now we are simply describing a characteristic of what something is not.
But the fact that we are describing what some thing is not, assumes then we are describing something that is, and therefore math is real.
Though like many real things it’s embodiment varies.
By the platonic idea maybe: The shadow from the flower on the cave wall, does not smell like a flower, but the flower’s smell is a real thing.
what are some good resources/books to learn the thoughts/debates/ideas that forced us to create/discover say for example concepts like complex numbers or logarithms or calculus etc.,
try "a tour of the calculus" which dives into the genesis of many mathematical premises in order to explore why calculus had to come into being, before explaining what calculus does.
Wikipedia often has a "History" section on these large/broad math topics. Though sometimes it requires you to be relatively familiar with the lingo to follow along (See the History of Complex Numbers [1]).
Specifically, for complex numbers, the problem of finding roots of polynomials is what led us there. Cubic polynomials (ones of the form ax^3 + bx^2 + cx + d) gave us the first hints of something like an imaginary unit. Even if the root is just a real number, expressing that number in terms of radicals (n-th roots) usually contains some square root of -1 that isn't eliminable. After lots of debate about the "existence" of such things, we eventually discovered the Fundamental Theorem of Algebra.
For Calculus, you just need to look at Newton, Leibniz, etc. This subject was strongly motivated by physicsists trying to solve hard problems of the day.
Logarithms essentially encde a relationship between addition and multiplication, something that has been recognized for quite some time! [0]
The abstract objects which aren't self-defeating [0] are real in this sense. Of course, note that we can't simply quantify over such objects since there is no universal way to decide whether objects are self-defeating, but it is nonetheless a simple criterion.
This kind of reminds me of a question a theistic friend of mine with a masters in math asked me once. Was the fundamental theorem of calculus something discovered by man (i.e created by God) or something created by man?
He thought it was both something created by man, and yet also something discovered.
Our mathematical constructs seem to me to describe things as they are, like discoveries, but it’s like we could only understand the discovery by creating a new way to think about things.
What’s most fascinating is they’re not just constructs that are true in that they’re predictable and can be tested, but they have a. absoluteness to them that is this is true or nothing is true.
"Real" for humans, is whatever we experience directly.
Therefore, colour is as real to a blind man as snow to a person who's born in the equator.
Is anger "real"? is love "real"? is depression "real"? - all those are very real for those who directly experienced them.
Mathematics is only real for those who experienced it directly.
> "Real" for humans, is whatever we experience directly. Therefore, colour is as real to a blind man as snow to a person who's born in the equator.
This sounds like crazy talk to me. Of course those two things are entirely real.
The fact that we can deduce reality and communicate it to others in a truth-preserving manner is a fundamental underpinning of society. I think it's dangerous to suggest that reality is relative.
The blind man can hear a color described, but that is not very useful for something that is visual.
The person born in the equator can have the feeling of snow described and by walking into a walk in freezer they can understand cold, they probably already understand water, examples can be made to them. Furthermore they can see videos of major snowstorms from other places.
Thus there are different levels of real between the color for the blind man, and the snow for the person born in the equator.
For Plato mathematics was real, but not only that, the stuff we see around us was not.
In the Republic he therefore complains about mathematicians "... always talking in a narrow and ridiculous manner, of "squaring" and "extending" and "applying" and the like - they confuse the ways of geometry with those of daily life; whereas knowledge is the real object of the whole science."
I find it amazing that Plato wrote this 2000 years before it was taken up again (quite radically) by Brouwer's intuitionism, where "squaring", "extending", "applying" are the only way of doing mathematics (and proof by contradiction, one of Plato's favourite devices, is not admissible)
The article's conclusion "Just about everything [except basic number theory] in mathematics is determined by the society in which you live" is rather disappointing, and not at all contradicts the reality of mathematics, as those societies might just have made different discovery journeys in Plato's realm of eternal mathematical truth.
What is more real - a concrete object, or the abstraction describing it. You could argue that number 5 is more real than for example 5 apples since you see it everywhere.
I'm quite surprised intuitionism is not brought into this conversation.
> Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966). Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds. [^0]
The intuitionistic logic, through Curry-Howard-Lambek correspondence, serves an important role in proof theory and model theory.
This question can be applied to any concept and to me it translates roughly to "is language a good representation of reality?"
The answer I like: any model that is useful for practical applications I consider being real. You only have to remember that it is a model and thus has serious limitations. Take it out of context of the application is has been designed for and turns to nonsense immediately. So you must be careful to strictly limit model usage. But the same limitations also make it possible to attack any concept with solipsism / deconstructionism - which is a mistake in the opposite direction imo. Guess this is enlightenment vs postmodernism.
One-ness is a physical property and that's why we can see it. The magic happens when we call them all one, give it a symbol, and make them all equal. Now "one" has a unique existence of its own on a piece of paper on a mathematicians desk, scribbled alongside other unique symbols, whose relationships come complete with proofs. And the only question then is whether what was scribbled says something true, as in, something that aligns with the reality it came from, because that would make it useful.
Math is a map. Maps are only real insofar as being a map, and only useful insofar as being true.
The drawing of the streets may be drawn with your pencil from memory, but the truth your friend relies on to get them where they need to go is real. The paper and pencil are real. And the physics of the informative truth that transcends from the streets to the paper is also real. Computers are the machines we've built based on the physics of logic, abstraction, and meaning. A computational value is something that means something to something else.
> One-ness is a physical property and that's why we can see it.
We can see it, but does that make it a physical property or something that we impose upon the universe?
For the rest of your statement, I don’t know if we can easily define “true” or “useful”. Epistemology has been working on those concepts for a long, long time, and I don’t know if they’ve made much progress recently.
One-ness is a shape of a thing, much like triangle-ness or circle-ness. Shapes are physical, but they can be copied, unlike the things themselves. And when meaning is associated with a shape, that meaning can then be copied and read. So we can take "one-ness" which is physically measurable and identifiable, associate a new shape such as "1", and now we have a shape with meaning detached from its origin that we can copy, read, and communicate. The source is still reality, and this physical mechanism is what makes communication possible. Gather enough of these symbols and you have a language. Combine these symbols and you have a statement. Have anything that reads and reacts to statements and you have consequences. From DNA to protein, from grocery list to grocery bag, from tweet to insurrection.
Any combination of words can form a statement. So someone comes along and says let's agree on grammar. Then someone comes along and says let's confirm with nature whether statements actually align with nature before we call it a fact. This is modern science and the scientific truth. Scientifically, nature is the single source of truth and is all true. False only exists in the gap between a statement and nature.
I think facts, truth, unity, etc, are fundamental building blocks of our perception of reality, not reality itself. That doesn’t make truths or facts invalid. We all share some common perception of the universe, at a very low, fundamental level. I believe knowledge can be built on that foundation.
Not only do we share some common perception, we're also all real. We are part of the universe. So to say people are subjective or that imagination is imaginary or that reality is an illusion for the purpose of dismissing them from reality is all wrong. You need a model that is more inclusive that describes it all with physics as the premise. For example, a dream requires a sleeping biological brain full of memories drawn from physical experiences.
Truth isn't reality itself in the physical sense. But the word "apple" isn't an actual apple either in the physical sense. Yet, for the purposes of language, it is. And that is the exact extent to which truth is reality in language. When we claim truth, we claim to be making a statement of reality. If that statement includes God, then your reality includes God. If it includes only physical evidence, then your reality is based on physical evidence. We can align our realities scientifically by aligning them with science. But again, based on your life, your work, and your beliefs, your reality will deviate. That's just part of what makes us all unique.
But as you say, our unique realities are still valuable and real. Though error prone as they may be, they are real reality drawn from a real life. It's not separate just because they contradict. That's why our ability to communicate our truths is cital, and Free Speech is sacrosanct.
"Thus humans do not invent mathematics, but rather discover it, and any other intelligent beings in the universe would presumably do the same."
I pretty sure it's true also for other areas, e.g. engineering. I believe that first vehicles created by some alien civilization would work on a similar principles as ours.
I believe that first vehicles created by some alien civilization would work on a similar principles as ours.
The way vehicles work on Earth is mostly a function of the environment - we use wheels because they work well in our gravity, we use fossil fuels because the history of Earth has made them abundant, we use fixed-wing flight because it works in our relatively dense atmosphere. On an alien planet with much lower gravity they might never have invented the wheel, or one with a much less dense atmosphere they might never have managed to make a wing big enough to support flight. It assumes a lot about an alien work to suggest they have anything like the same sort of problems to overcome that we have.
I wonder why we ask this of mathematics and not, for example, physics.
We all kinda understand how real Physics is: the universe does weird things, we find rules and equations that fit these things. If planets go in circle (the Real - the Territory), we create orbital equations, Kepler's laws, etc (the Made up - the Map).
Isn't it the same with Math? In this universe, some fundamental rules seem to be obeyed: (at macro scale) objects tend to keep their delimitations, they add up and substract nicely, etc. So we make up words, language, equations to match.
Maybe it's just that the constructs Math tries to Map are more fundamental, harder to imagine a universe where they are different (a universe where 2+2 things makes 5 things)
60 comments
[ 3.2 ms ] story [ 161 ms ] threadI’m not sure if even that is true. Integer counts and succession seem like human constructs to me. An apple isn’t fundamentally a single unit. If I take a bite out of it, is it still one apple?
Disclaimer: I am very poor at math but eager to learn :)
Basically the mathematical notion of integer is trying to make sense of our idea of a discrete unit. Taking a bite of apple makes it non-unit, but I think you would agree that the notion of "one apple" or "two dogs" makes common sense.
Also when reading someone's quote, I would always consider its context. I think Kronecket had the axiomatic construction of real number in mind. The idea of rational number and real number can be investigated by just assuming the idea of increment by one is solid.
These mathematical concepts are usually covered in a undergraduate analysis class. For more information, see Cauchy's construction and Dadekind's construction.
Therein lies the rub! Succession is a human concept. All math is a magnificent city built upon that one stone.
Do you have an idea what the philosophical term for this idea is? It seems related to connectionism.
One might assume that abstraction and pattern matching in the brain are required to determine that two "things" are the same. It is perhaps a small step to go from comparing things to comparing integers.
Unfortunately, this line of reasoning quickly goes awry, because one cannot assume even the existence of things to describe brains or pattern matching.
You might be interested in the univalent foundations approach, where the idea is that mathematics is fundamentally the study of equivalences between constructions, and equivalences themselves are first-class objects that are just as easy to talk about as any other mathematical object.
[update] the study of this concept seems to be semiotics, which examines the creation and use of symbols. I don’t know what branch of semiotics examines the creation of the concepts of unity and succession.
An apple is made up of constituent parts, but it is a singular thing.
If you argue against an Apple being singular then you are essentially professing that nothing is singular.
When you take a bite out of an apple you now have a subset of an apple “an apple with a bite out of it”.
To me saying God made the integers is more akin to saying, there is and there isn’t, and all else is built upon the difference.
An apple is a singular thing because it is useful for us to think of an apple as a singular thing in most contexts. It is not a singular thing when it is more useful to look deeper.
To get really out of the realm of HN, no one thing can be proven to exist outside of our collective hallucination. That hallucination has reliable rules, thankfully, so things can be proven to exist inside it.
There's definitely some "elementary" mathematical concepts that seem native to our universe. For example, in outer space stuff tends to come together in ways that approximates spheres. We see lattices in crystalline solids all the time. There's naturally-occurring fractals all over the place. The existence of fractals -- snowflakes, trees, broccoli -- seems to "embed" a successor (to iterate over generations), so I'm not sure if it's a human construct.
Counting is real and the number you count is "true" because it leads to real predictions. This tale of magic bucket is enlightening. (2)
(1): https://www.lesswrong.com/posts/aMHq4mA2PHSM2TMoH/the-catego...
(2): https://www.lesswrong.com/posts/X3HpE8tMXz4m4w6Rz/the-simple...
> Whether it’s still an apple is a question of categorisation and a human construct
At the end of the day, though, I always go back to an old joke my undergrad philsophy professor told me:
Analytic Philosopher A: "Hey, do you believe in baptism?"
Analytic Philsopopher B: "Believe in it? Hell, I've been to one."
Professor: I am suddenly concerned that my PhD student does not believe in induction
Me: I am suddenly concerned that my PhD supervisor does not know the difference between converse and contrapositive
The statement that there are infinitely many integers is true in constructive mathematics: Given any integer, you can construct some integer larger than it. Likewise, the statement that the real numbers are uncountable is also true: Given any sequence of real number, it's possible to construct a real number that's not in the sequence (using Cantor's method).
The claim that all real numbers are computable doesn't have any constructive content, so its truth depends on which version of constructivism you believe in. Personally, having looked into this, I prefer versions of constructivism in which the claim is false, because then you can claim things like every continuous function on R is locally uniformly continuous (which weirdly enough, implies that not all real numbers are computable, without asserting the stronger claim that an uncomputable one exists).
By the platonic idea maybe: The shadow from the flower on the cave wall, does not smell like a flower, but the flower’s smell is a real thing.
Specifically, for complex numbers, the problem of finding roots of polynomials is what led us there. Cubic polynomials (ones of the form ax^3 + bx^2 + cx + d) gave us the first hints of something like an imaginary unit. Even if the root is just a real number, expressing that number in terms of radicals (n-th roots) usually contains some square root of -1 that isn't eliminable. After lots of debate about the "existence" of such things, we eventually discovered the Fundamental Theorem of Algebra.
For Calculus, you just need to look at Newton, Leibniz, etc. This subject was strongly motivated by physicsists trying to solve hard problems of the day.
Logarithms essentially encde a relationship between addition and multiplication, something that has been recognized for quite some time! [0]
[0]:https://en.wikipedia.org/wiki/History_of_logarithms
[1]:https://en.wikipedia.org/wiki/Complex_number#History
https://en.wikipedia.org/wiki/Universal_property
[0] https://terrytao.wordpress.com/2009/11/05/the-no-self-defeat...
Contradiction != self-defeat.
Is just side effects. A becoming not-A.
I do not exist.
He thought it was both something created by man, and yet also something discovered.
Our mathematical constructs seem to me to describe things as they are, like discoveries, but it’s like we could only understand the discovery by creating a new way to think about things. What’s most fascinating is they’re not just constructs that are true in that they’re predictable and can be tested, but they have a. absoluteness to them that is this is true or nothing is true.
This sounds like crazy talk to me. Of course those two things are entirely real.
The fact that we can deduce reality and communicate it to others in a truth-preserving manner is a fundamental underpinning of society. I think it's dangerous to suggest that reality is relative.
The person born in the equator can have the feeling of snow described and by walking into a walk in freezer they can understand cold, they probably already understand water, examples can be made to them. Furthermore they can see videos of major snowstorms from other places.
Thus there are different levels of real between the color for the blind man, and the snow for the person born in the equator.
In the Republic he therefore complains about mathematicians "... always talking in a narrow and ridiculous manner, of "squaring" and "extending" and "applying" and the like - they confuse the ways of geometry with those of daily life; whereas knowledge is the real object of the whole science."
I find it amazing that Plato wrote this 2000 years before it was taken up again (quite radically) by Brouwer's intuitionism, where "squaring", "extending", "applying" are the only way of doing mathematics (and proof by contradiction, one of Plato's favourite devices, is not admissible)
The article's conclusion "Just about everything [except basic number theory] in mathematics is determined by the society in which you live" is rather disappointing, and not at all contradicts the reality of mathematics, as those societies might just have made different discovery journeys in Plato's realm of eternal mathematical truth.
> Intuitionism is a philosophy of mathematics that was introduced by the Dutch mathematician L.E.J. Brouwer (1881–1966). Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds. [^0]
The intuitionistic logic, through Curry-Howard-Lambek correspondence, serves an important role in proof theory and model theory.
[0]: https://plato.stanford.edu/entries/intuitionism/
The answer I like: any model that is useful for practical applications I consider being real. You only have to remember that it is a model and thus has serious limitations. Take it out of context of the application is has been designed for and turns to nonsense immediately. So you must be careful to strictly limit model usage. But the same limitations also make it possible to attack any concept with solipsism / deconstructionism - which is a mistake in the opposite direction imo. Guess this is enlightenment vs postmodernism.
Also see: https://en.wikipedia.org/wiki/Map%E2%80%93territory_relation
[1] https://www.youtube.com/watch?v=AmySxYHqQCQ
Math is a map. Maps are only real insofar as being a map, and only useful insofar as being true.
The drawing of the streets may be drawn with your pencil from memory, but the truth your friend relies on to get them where they need to go is real. The paper and pencil are real. And the physics of the informative truth that transcends from the streets to the paper is also real. Computers are the machines we've built based on the physics of logic, abstraction, and meaning. A computational value is something that means something to something else.
We can see it, but does that make it a physical property or something that we impose upon the universe?
For the rest of your statement, I don’t know if we can easily define “true” or “useful”. Epistemology has been working on those concepts for a long, long time, and I don’t know if they’ve made much progress recently.
Any combination of words can form a statement. So someone comes along and says let's agree on grammar. Then someone comes along and says let's confirm with nature whether statements actually align with nature before we call it a fact. This is modern science and the scientific truth. Scientifically, nature is the single source of truth and is all true. False only exists in the gap between a statement and nature.
Truth isn't reality itself in the physical sense. But the word "apple" isn't an actual apple either in the physical sense. Yet, for the purposes of language, it is. And that is the exact extent to which truth is reality in language. When we claim truth, we claim to be making a statement of reality. If that statement includes God, then your reality includes God. If it includes only physical evidence, then your reality is based on physical evidence. We can align our realities scientifically by aligning them with science. But again, based on your life, your work, and your beliefs, your reality will deviate. That's just part of what makes us all unique.
But as you say, our unique realities are still valuable and real. Though error prone as they may be, they are real reality drawn from a real life. It's not separate just because they contradict. That's why our ability to communicate our truths is cital, and Free Speech is sacrosanct.
I pretty sure it's true also for other areas, e.g. engineering. I believe that first vehicles created by some alien civilization would work on a similar principles as ours.
The way vehicles work on Earth is mostly a function of the environment - we use wheels because they work well in our gravity, we use fossil fuels because the history of Earth has made them abundant, we use fixed-wing flight because it works in our relatively dense atmosphere. On an alien planet with much lower gravity they might never have invented the wheel, or one with a much less dense atmosphere they might never have managed to make a wing big enough to support flight. It assumes a lot about an alien work to suggest they have anything like the same sort of problems to overcome that we have.
We all kinda understand how real Physics is: the universe does weird things, we find rules and equations that fit these things. If planets go in circle (the Real - the Territory), we create orbital equations, Kepler's laws, etc (the Made up - the Map).
Isn't it the same with Math? In this universe, some fundamental rules seem to be obeyed: (at macro scale) objects tend to keep their delimitations, they add up and substract nicely, etc. So we make up words, language, equations to match.
Maybe it's just that the constructs Math tries to Map are more fundamental, harder to imagine a universe where they are different (a universe where 2+2 things makes 5 things)