The underlying issue here seems to be a conflict between a thin notion of existence and a fat one.
A fat notion includes properties. To say "x exists" is to say that, eg., it has a spatial and temporal location. This is often just baggage brought in from common ways of speaking and thinking.
It is more helpful to reduce "exists" down to its barest sense and discard any properties that often come along with it.
In this "thin" sense, exists only means that you can -- roughly -- successfully talk about it.
"Santa" exists, only that the object "Santa" does not possess the properties of being a man. He is a fictional object.
So "Santa, the man who visits you on christmas," does not exist. I cannot successfully talk about it (ie., i cannot refer to that man, I cannot point him out, there is no way of "locating" him).
In this sense I cannot see how one could object to the existence of the square root of 2. It can be successfully pointed out in the same way any number can be (esp. see the article). Only, say, that it failed to possess some properties of interest.
I guess the hypothetical objector in this article wants numbers to possess the property of "being easily exemplified by stones" (or whatever), in which case sqrt(2) doesn't have that property.
Why it should be that "2" exists because it can be exemplified by fingers, and sqrt(2) doesnt because it cant be -- is a bit strange.
But history is full of people wishing to arbitrarily define a set of properties that count as existence.
The point of the article is to point out the non-obviousness of some true statement that an outsider to mathematics would take to be obviously true, and thus to demonstrate the necessity of formal (or at least careful) reasoning in mathematics. It's posed as the question of whether the square root of 2 exists, but really, it should be posed as the question of whether there exists a unique real number whose square is 2. Of course, if the square root of 2 didn't exist, we could just define it as an object whose square is 2 and be satisfied like we might for the square root of -1. However, if we did that, our new sqrt(2) wouldn't be a real number.
To answer the better-specified question we need first to define the real numbers and what it means to multiply two real numbers together. This would be done in a real analysis class, after which one could properly prove that sqrt(2) exists, and understand what the proof actually accomplishes.
I believe that the article poses the problem as whether sqrt(2) exists and not as whether sqrt(2) is a unique real number because the article is meant to appeal to the outsider who wouldn't see the point of the quantification, at least, not before reading the article.
> In this sense I cannot see how one could object to the existence of the square root of 2. It can be successfully pointed out in the same way any number can be (esp. see the article). Only, say, that it failed to possess some properties of interest.
Well the property of interest is having x^2 == 2. If x^2 isn't 2 then x isn't in any meaningful sense "the square root of 2", and it's much clearer to say "the square root of 2 doesn't exist" than "the square root of 2 squared doesn't equal 2".
> I guess the hypothetical objector in this article wants numbers to possess the property of "being easily exemplified by stones" (or whatever), in which case sqrt(2) doesn't have that property.
> Why it should be that "2" exists because it can be exemplified by fingers, and sqrt(2) doesnt because it cant be -- is a bit strange.
By that logic you wouldn't be able to make any statements about general properties of numbers. E.g. we like to say that all integers are either even or odd. But what if I define a new integer, let's call it Banana, that is both even and odd. It's a lot more productive and clearer to say "Banana doesn't exist" than to talk about Banana not having particular properties.
"There can be no integers which are both even and odd", sure. Such a number cannot exist because the properties can not be co-instantiated.
> Well the property of interest is having x^2 == 2.
Sure, but that property is possessed by some x. The interlocutor in the article seems to resist not that this equation could be satisfied, but whatever satisfied it wouldnt "count" because it didnt possess some touchy-feely "existential" property.
cf., at the end where concession is only given to defining existence within some system as opposed to some prior intuition of "Existence" which was being denied to sqrt(2).
My point is that whatever this intuitive notion of existence is it's basically unhelpful. It is embedding properties into the very criterion of there-being-a-quantity of something.
As in, it is not merely good enough that an x can be defined such that x^2==2 but that it must also possess the Property of Reality (whichever that might be, eg. spatial location, abbacusy, ..).
> "There can be no integers which are both even and odd", sure. Such a number cannot exist because the properties can not be co-instantiated.
What makes you so sure that having x^2==2 can be instantiated? What about, say, "an even number greater than 4 that cannot be expressed as the sum of 2 primes"? It's ridiculous to say that that exists and we're arguing over whether it has some property or not; far more natural and practical to say that it's an open question whether an integer like that exists.
> Sure, but that property is possessed by some x.
No, you can't say that until you've proven it. If you just assume that such an x exists, that might turn out to be just as incoherent as assuming that an integer that's both even and odd exists.
There's a mathematical urban legend about the Journal of Foo Manifolds; these were manifolds defined in some particular way, and proved the object of much interesting study, so eventually got their own journal. In particular, mathematicians wondered about the Bar property, and published various incrementally improving papers about what kind of Foo Manifolds did and didn't have the Bar Property.
Eventually, a paper was received proving that all Foo Manifolds had the Bar Property, and another proving that no Foo Manifolds had the Bar Property. Both were carefully checked and found to be valid. The final issue of Ruhr Journal of Foo Manifolds consisted of both papers and an announcement that it was closing down.
I agree with you, but I don't think the objector in the article is merely making that point.
His scepticism seems deeper. It isnt merely that he thinks that nothing satisfies that equation, he seems to in addition think, that even if something does it wouldnt count as "existing".
If the objection were merely that "x^2 == 2" isnt satisfied, then I agree, that is equivalent to saying "x does not exist" and it would be appropriate to say that.
> His scepticism seems deeper. It isnt merely that he thinks that nothing satisfies that equation, he seems to in addition think, that even if something does it wouldnt count as "existing".
I think that's just the only way to express scepticism about the existence of a solution to x^2 == 2 in a world where any schoolboy can tell you x = 1.414...
> If you remove all the properties you are left with the bare quantification claim: "there is one of..."
On that view, the square root of two does not exist because there are in fact two different numbers with the property that their square equals two.
Defining square root as "the positive root" doesn't help because then the square root of -1 would not exist. There are two numbers whose square is -1 as well (more if you're dealing with quaternions or octernions), but neither (none) of them is positive.
It's a semantics for the term "exists", not a claim about the properties anything -- including the sqrt of 2 -- must have. Including being unique.
"x exists" just means "there is an x" which just means that we can identify an object not that we can only identify one alone.
There is a number which satisfies x^2==2 is just the claim that "there is an x such that x satisifes x^2==2". The claim is not "there is an x such that x uniquely satisfies x^2==2"
Uniqueness is an additional property to the bare quantification claim.
That my point. All this discussion is about additional properties existing things have.
My point is that the bare term "exists" includes no properties its just, if you like, the beginning of the recipe with the ingredients to follow. Its a placeholder for what you will later say.
The objector in the article seems to be saying the sqrt of 2 doesnt exist not because "there was no recipe" but because it didn't have his favorite ingredient.
To say something doesn't exist is to say "there is no recipe for it". Santa Clause doesn't exist because there is nothing which includes the ingredients "a magical man", etc.
But Santa DOES exist. He doesn't exist as a physical entity made of real atoms, but neither does the square root of two (or any other number). Numbers exist as mathematical constructs, and Santa exists as a fictional character.
> The underlying issue here seems to be a conflict between a thin notion of existence and a fat one.
The fundamental issue is described at the end of the article:
> "I didn't just define the square root of two. Rather, I defined an entire number system and showed, by which I mean actually proved, that the square root of 2 exists in that system.
> "Oh, well if that's what you mean by the existence of the square root of two, then I suppose I accept it."
This seems to be the key point of the article, and it's quite correct. Any concept is meaningless without a framework in which it is defined. We can, by the mechanism described in the above quote, define all sorts of mathematical concepts that demonstrably "exist" in such frameworks.
To then talk about existence in the physical world requires mapping these concepts from their abstract frameworks into the physical world. In the case of the square root of 2, we might point out that the length of the diagonal of a physical version of the unit square "is" sqrt(2), where "is" means something like, "as close to the mathematical definition as it is possible to get given physical limitations."
Separating our formalized abstractions of the world from the physical objects they sometimes are intended to describes helps achieve clarity and avoids confusing platonic abstractions with physical existence.
It seems he confuses a quantinty with its numerical expression. The square root of 2 is just the length of a segment: the diagonal of a square of side 1. As long as “square, side and 1” are allowed to “exist”, sqrt(1) does exist.
No, you cannot write its decimal expansion. But neither can you that of 20/7. However, its continued fraction is trivial.
Your argument rests on the notion of "length" which you don't get for free. :) You won't be able to define "length" without assuming the existence of irrational numbers.
But in this sense, the length of the diagonal exists but you cannot measure it with your chosen unit which is the side of the square. If you take the diagonal as your unit you cannot measure the side either.
You missed the point of the post. Pythagoras could prove sqrt(2) is irrational but the proof already implicitly assumes it exists.
"exists" in mathematics, basically means that it does not contradict the axioms of whatever system you're working under, or result in a logical inconsistency. For example, the set that is the subject of Russell's paradox, cannot exist in any reasonable system of set theory, and its definition shows that naive set theories are logically unsound.
No, Pythagoras started from something that definitely exists: the hypotenuse of a triangle with two sides of length 1. He wanted to find out its length and to his great dismay he discovered that there is no rational number to express its length.
sqrt(2) assumes continuity on a single axis. sqrt(-1) assumes an entirely new axis. (Not arguing there is any problem, just putting into words the thoughts I have heard from many students.)
I would definitely not say that sqrt(2) is more controversial than sqrt(-1).
With sqrt(2), we define it into existence, because with all the surrounding bits, it makes sense for it to exist.
With sqrt(-1), we define it into existence, because under the assumption that it exists, you can do some calculations that you can't do otherwise, or you can do them more easily, and the results of those calculations happen to be correct.
sqrt(2) isn't controversial these days. We might even say it's "obvious", because we're all used to using it from childhood. So you might wonder why maths professors would spend effort proving it in detail. This post is an attempt to remind people that there was a time when sqrt(2) wasn't at all obvious, was controversial, and that it makes sense to see it that way. (And indeed if you go back far enough, -1 was controversial).
The ancient greeks where geometericists not algebraists, so for them the existence of a line was equivalent to the existence of what algebraists think of as a number. A square with unit length had a diagonal, and hence sqrt(2) existed. However, despite the name "Pythagoras' Theorem", sqrt(2) was traumatic for Pythagoras because he wanted to be believe in a rational universe, that is, one in which everything (distance of planets, musical harmonies and many other things) could be expressed as a ratio of whole numbers (rational). And then, damit, the simplest possible example, a unit square, threw a monkey wrench into the works! I have read that Pythagoreans were forbidden from talking about numbers like sqrt(2) (or from eating green beans).
(Original poster here.) If you are interested in this sort of thinking, I recommend checking out the page I found it on: https://www.dpmms.cam.ac.uk/~wtg10/vsipage.html , including the last link on the page that delves a bit into the philosophy behind mathematics and its implications.
That page in turn was found at the end of the author's book 'Mathematics: a Very Short Introduction', which is pretty good exploration of what goes into mathematical definitions and how modern mathematical thinking works, addressed towards the layman but without excessive watering down of the subject.
31 comments
[ 5.1 ms ] story [ 74.5 ms ] threadA fat notion includes properties. To say "x exists" is to say that, eg., it has a spatial and temporal location. This is often just baggage brought in from common ways of speaking and thinking.
It is more helpful to reduce "exists" down to its barest sense and discard any properties that often come along with it.
In this "thin" sense, exists only means that you can -- roughly -- successfully talk about it.
"Santa" exists, only that the object "Santa" does not possess the properties of being a man. He is a fictional object.
So "Santa, the man who visits you on christmas," does not exist. I cannot successfully talk about it (ie., i cannot refer to that man, I cannot point him out, there is no way of "locating" him).
In this sense I cannot see how one could object to the existence of the square root of 2. It can be successfully pointed out in the same way any number can be (esp. see the article). Only, say, that it failed to possess some properties of interest.
I guess the hypothetical objector in this article wants numbers to possess the property of "being easily exemplified by stones" (or whatever), in which case sqrt(2) doesn't have that property.
Why it should be that "2" exists because it can be exemplified by fingers, and sqrt(2) doesnt because it cant be -- is a bit strange.
But history is full of people wishing to arbitrarily define a set of properties that count as existence.
To answer the better-specified question we need first to define the real numbers and what it means to multiply two real numbers together. This would be done in a real analysis class, after which one could properly prove that sqrt(2) exists, and understand what the proof actually accomplishes.
I believe that the article poses the problem as whether sqrt(2) exists and not as whether sqrt(2) is a unique real number because the article is meant to appeal to the outsider who wouldn't see the point of the quantification, at least, not before reading the article.
Well the property of interest is having x^2 == 2. If x^2 isn't 2 then x isn't in any meaningful sense "the square root of 2", and it's much clearer to say "the square root of 2 doesn't exist" than "the square root of 2 squared doesn't equal 2".
> I guess the hypothetical objector in this article wants numbers to possess the property of "being easily exemplified by stones" (or whatever), in which case sqrt(2) doesn't have that property.
> Why it should be that "2" exists because it can be exemplified by fingers, and sqrt(2) doesnt because it cant be -- is a bit strange.
By that logic you wouldn't be able to make any statements about general properties of numbers. E.g. we like to say that all integers are either even or odd. But what if I define a new integer, let's call it Banana, that is both even and odd. It's a lot more productive and clearer to say "Banana doesn't exist" than to talk about Banana not having particular properties.
"There can be no integers which are both even and odd", sure. Such a number cannot exist because the properties can not be co-instantiated.
> Well the property of interest is having x^2 == 2.
Sure, but that property is possessed by some x. The interlocutor in the article seems to resist not that this equation could be satisfied, but whatever satisfied it wouldnt "count" because it didnt possess some touchy-feely "existential" property.
cf., at the end where concession is only given to defining existence within some system as opposed to some prior intuition of "Existence" which was being denied to sqrt(2).
My point is that whatever this intuitive notion of existence is it's basically unhelpful. It is embedding properties into the very criterion of there-being-a-quantity of something.
As in, it is not merely good enough that an x can be defined such that x^2==2 but that it must also possess the Property of Reality (whichever that might be, eg. spatial location, abbacusy, ..).
What makes you so sure that having x^2==2 can be instantiated? What about, say, "an even number greater than 4 that cannot be expressed as the sum of 2 primes"? It's ridiculous to say that that exists and we're arguing over whether it has some property or not; far more natural and practical to say that it's an open question whether an integer like that exists.
> Sure, but that property is possessed by some x.
No, you can't say that until you've proven it. If you just assume that such an x exists, that might turn out to be just as incoherent as assuming that an integer that's both even and odd exists.
There's a mathematical urban legend about the Journal of Foo Manifolds; these were manifolds defined in some particular way, and proved the object of much interesting study, so eventually got their own journal. In particular, mathematicians wondered about the Bar property, and published various incrementally improving papers about what kind of Foo Manifolds did and didn't have the Bar Property.
Eventually, a paper was received proving that all Foo Manifolds had the Bar Property, and another proving that no Foo Manifolds had the Bar Property. Both were carefully checked and found to be valid. The final issue of Ruhr Journal of Foo Manifolds consisted of both papers and an announcement that it was closing down.
His scepticism seems deeper. It isnt merely that he thinks that nothing satisfies that equation, he seems to in addition think, that even if something does it wouldnt count as "existing".
If the objection were merely that "x^2 == 2" isnt satisfied, then I agree, that is equivalent to saying "x does not exist" and it would be appropriate to say that.
I think that's just the only way to express scepticism about the existence of a solution to x^2 == 2 in a world where any schoolboy can tell you x = 1.414...
http://blog.rongarret.info/2015/02/31-flavors-of-ontology.ht...
If you remove all the properties you are left with the bare quantification claim: "there is one of...".
Taking "exist" to be the bare quantification claim, and the properties as something in-addition to this, clarifies much of ontological discussion.
"There is one of X and it is like A, B, C"
Thin + A-Fat, B-Fat, C-Fat
On that view, the square root of two does not exist because there are in fact two different numbers with the property that their square equals two.
Defining square root as "the positive root" doesn't help because then the square root of -1 would not exist. There are two numbers whose square is -1 as well (more if you're dealing with quaternions or octernions), but neither (none) of them is positive.
"x exists" just means "there is an x" which just means that we can identify an object not that we can only identify one alone.
There is a number which satisfies x^2==2 is just the claim that "there is an x such that x satisifes x^2==2". The claim is not "there is an x such that x uniquely satisfies x^2==2"
Uniqueness is an additional property to the bare quantification claim.
That my point. All this discussion is about additional properties existing things have.
My point is that the bare term "exists" includes no properties its just, if you like, the beginning of the recipe with the ingredients to follow. Its a placeholder for what you will later say.
The objector in the article seems to be saying the sqrt of 2 doesnt exist not because "there was no recipe" but because it didn't have his favorite ingredient.
To say something doesn't exist is to say "there is no recipe for it". Santa Clause doesn't exist because there is nothing which includes the ingredients "a magical man", etc.
The object "the fictional character santa" exists.
The object "the magical man who comes at christmas" does not exist.
There is no such object that satisfies the claim, "there is an object which is magic and which is a man and which comes at christmas...".
The fundamental issue is described at the end of the article:
> "I didn't just define the square root of two. Rather, I defined an entire number system and showed, by which I mean actually proved, that the square root of 2 exists in that system.
> "Oh, well if that's what you mean by the existence of the square root of two, then I suppose I accept it."
This seems to be the key point of the article, and it's quite correct. Any concept is meaningless without a framework in which it is defined. We can, by the mechanism described in the above quote, define all sorts of mathematical concepts that demonstrably "exist" in such frameworks.
To then talk about existence in the physical world requires mapping these concepts from their abstract frameworks into the physical world. In the case of the square root of 2, we might point out that the length of the diagonal of a physical version of the unit square "is" sqrt(2), where "is" means something like, "as close to the mathematical definition as it is possible to get given physical limitations."
Separating our formalized abstractions of the world from the physical objects they sometimes are intended to describes helps achieve clarity and avoids confusing platonic abstractions with physical existence.
No, you cannot write its decimal expansion. But neither can you that of 20/7. However, its continued fraction is trivial.
See Bolzano weirstrass, or define reals as equivalence classes on converging series of rationals.
"exists" in mathematics, basically means that it does not contradict the axioms of whatever system you're working under, or result in a logical inconsistency. For example, the set that is the subject of Russell's paradox, cannot exist in any reasonable system of set theory, and its definition shows that naive set theories are logically unsound.
With sqrt(2), we define it into existence, because with all the surrounding bits, it makes sense for it to exist.
With sqrt(-1), we define it into existence, because under the assumption that it exists, you can do some calculations that you can't do otherwise, or you can do them more easily, and the results of those calculations happen to be correct.
That page in turn was found at the end of the author's book 'Mathematics: a Very Short Introduction', which is pretty good exploration of what goes into mathematical definitions and how modern mathematical thinking works, addressed towards the layman but without excessive watering down of the subject.