
The existence of the square root of two (2001) - sundarurfriend
https://www.dpmms.cam.ac.uk/~wtg10/roottwo.html
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mjburgess
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.

~~~
lmm
> 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.

~~~
mjburgess
> It's a lot more productive and clearer

"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, ..).

~~~
lmm
> "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.

~~~
mjburgess
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.

~~~
lmm
> 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...

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pfortuny
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.

~~~
bjourne
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.

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sebapi747
these questions are not obvious and were only cleared by 19th century number
theory.

See Bolzano weirstrass, or define reals as equivalence classes on converging
series of rationals.

~~~
adrianN
The irrationality of sqrt(2) was known to Pythagoras.

~~~
infinity0
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.

~~~
adrianN
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.

~~~
sebapi747
You don’t need Euclidean space to exist to define reals but you need reals
numbers for distance to exist.

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herodotus
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).

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sundarurfriend
(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](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.

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exabrial
Question from stupid person: If we regularly use sqrt(-1), why would sqrt(2)
be any more or less controversial?

~~~
Sylos
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.

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tzahola
Nice try Wildberger!

