## Thursday, December 20, 2007

### Faith in √2

If you are most people, you find out what the square root of two is by punching two keys on your calculator. Mine says it's 1.4142135623730950488016887242097, but even just 1.4 gets you a long way. (1.42 = 1.96, which is within 2% of 2)

Neither one, however, is actually the square root of 2. You can never write down all the digits for it; they are infinite. What our calculator gives us is a coarse approximation. Thirty-two digits might not seem like a coarse approximation, since five digits would probably get you safely to the moon and back, but our 32 digits only represent 0% (zero percent) of all the digits required to represent it exactly. If the digits of √2 are infinite, we can never write it as a ratio of two integers. We can approximate it as 7/5 (accurate to 2 digits) or as the ratio 25712973861329/18181818181818 (accurate to 14 digits), but it's still an approximation. The square root of two is incommensurable. It cannot be represented as the ratio of two integers, which is the definition of an irrational number (i.e. "irrational" literally means you can't represent it with a ratio).

See my earlier Proof that √2 is irrational

The Pythagoreans believed that the universe was made of integers. Their religious dogma said god only created natural numbers (i.e. countable positive integers). When Hippasus, 2500 years ago, was able to prove that a number (√2) existed that couldn't be constructed from integers, the story is he was either exiled or drowned as a heretic.

There are still modern mathematicians who believe something similar to the old Pythagorean views. Amongst other things, some believe that in order for a mathematical concept to "exist", it must be physically constructable in our universe. They argue that Hippasus and Chaitin's arguments suffer the same flaw; proving that a number cannot be made in a particular way does not prove that it can be made at all (i.e. that it can exist). If I can prove that John Travolta isn't made of green jello, that doesn't mean that John Travolta must exist and be made of something else, it only proves that there is not a green jello John Travolta.

How can √2 be physically "constructable"? Let's say you have a right-triangle with two sides of length 3 and 4. The Hypotenuse of that triangle (from the Pythagorean theroem) is √(32 +42) = √25 = 5. The Pythagoreans were OK with this because 3, 4 and 5 are all nice integers and this triangle can be "constructed" and shown to actually exist. However, if you have a similar triangle with two sides both of length 1, the hypotenuse is √(12 +12) = √2 = 1.414....

Try to make this triangle. To construct it from equal integer sides, you keep running either too long or too short on the hypotenuse. If you had 10 units on each side, the hypotenuse would be 14 and a bit; an integer value would not complete the triangle. If you tried to get a little finer grained and make it with 100 units on a side, the hypotenuse would be 141 units and a bit. You could continue this Sisyphean task forever, but no matter how big you made the integer sides, a hypotenuse side made with an integer would never quite finish the triangle.

This is an important question because so far, to the best of our knowledge, our universe is integer. At one time the smallest particles were atoms, then protons, neutrons and electrons, then quarks, now maybe strings. Unless there is an infinite parade of smaller and smaller particles, the universe is necessarily made up of "smallest particles". There is no evidence of a infinitesimally small particles (particles that approach zero dimension), and as far as I know, nobody is seriously considering it.

That means that regardless of how many "smallest particles" with which we attempt to construct the triangle, the number is always finite. The hypotenuse never works out to be a single integer, and so the triangle can't be constructed in our universe.

The only way √2 could be physically constructed would be if the universe were infinite in size and you could line up the smallest particles across an infinite universe (assuming you had infinite particles in that universe) to make an infinitely sized triangle. That's mighty big.

So, in order for √2 to physically exist in this universe, we need one of two conditions: either the universe is infinite in size, or the universe is infinitesimally divisible. If one of those conditions aren't met, √2 is an asymptote, an unreal mathematical abstraction. You should no more believe that √2 actually exists in any real sense than you believe Thor or Santa Claus exists, no matter what your calculator says. I'm surprised that people don't come door to door with pamphlets trying to get me to believe in √2. Maybe I've found something to do on weekend mornings.

Do you believe in √2?

Burton MacKenZie http://www.burtonmackenzie.com/

Greg said...

Just because you can't line up particles so that they form a line of length √2 does not mean that √2 does not exist. Actually it proves that a right triangle whose legs have length one and whose sides are composed of single rows of particles does not exist. This is rather useless knowledge. You say that the universe is integer, but then demonstrate that matter is integer. These are separate propositions. The idea that space is integer seems axiomatically false. If it were, that would mean that whenever a particle moves it, actually ceases to be in one location and appears in another, without crossing the intervening space. The problems this presents are left as an exercise for the reader.

burton mackenzie said...

Hi Greg, thanks for the comment. I enjoy getting feedback.

The whole article is about whether or not √2 can be constructed in this universe, then ponders the case that if something can never be constructed in reality, can it be said to "exist".

I state at one point "the universe is integer". You interpreted this statement to mean "the physical dimensions of the universe are digital", but it was not intended in that vein. My statement in context means "the contents of this universe are digital". The original statement could easily be taken to mean either, but in this context, I assumed the understanding would be the latter.

I do disagree strongly with one statement you made: "This is rather useless knowledge". I assert that no knowledge is useless knowledge. It may be useless to you, or even most people, but every little scrap of knowledge we can gather absolutely beats back the unknown exactly by the amount of that little scrap.

Please feel welcome to come back and argue some more! That's the subtitle of the blog! :-)

Starhawk said...

Of course I believe in √2 I am a Platonic idealist and naturally think all mathematical conceptions exist!

As to your blog it is so full of misinformation I don't even know where to start, so bear with me as I'll have to spread this over several rather long comments.

For now lets examine the Pythagoreans:

According to the Stanford Encyclopedia of philosophy, "What were the beliefs and practices of the historical Pythagoras? This apparently simple question has become the daunting Pythagorean question for several reasons. First, Pythagoras himself wrote nothing, so our knowledge of Pythagoras' views is entirely derived from the reports of others" and else where we read "even in the fifth century there was debate within the Pythagorean tradition itself as to whether Pythagoras was largely important as the founder of a set of rules to follow in living one's life or whether his teaching also had a mathematical and scientific dimension."

Moreover the same Encyclopedia list 4 different ways of even defining what "Pythagoreanism" is so that your statement "The Pythagoreans believed that the universe was made of integers." is not particularly clear nor particularly meaningful and almost certainly historically FALSE. The followers of Pythagoras certainly felt mathematics was useful in describing the universe and many were probably Platonic in philosophy so may have felt that only mathematical objects can be said to exist and our material world being either a shadow (ie illusion or Maya) or an example of one particular mathematical object. However number played almost no role in their mathematics or physics, it for the most part being synthetic geometry!

And further your claim "Their religious dogma said god only created natural numbers (i.e. countable positive integers)." is likewise terribly confused. Certainly the Greek Pythagoreans were PAGAN and while may have had deeply philosophical ideas about theology most certainly they were polytheist and or pantheistic. So if there was any truth at all to your statement it would be necessary to clarify and identify which god or goddess and to emphasize that this is no Christian concept of God! And even worse is the question of creation, I don't think greek paganism had the same conception of creation as modern christians and as even the Gods themselves were BORN it seems birth is a totally different concept than creation. Nature and reality both conceived of as ORGANIC. And even given the modern myth of Pythagoras his great mythological vision was that All is number (ie mathematics) which clearly implies that even the gods themselves are mathematical objects so in poetic terms one might say that the natural numbers gave birth to the Gods and to all of reality, not the other way around. For the record it was Leopold Kronecker (1823-1891) who said "God created the natural number, and all the rest is the work of man."

hope this helps ;)

Gary said...

If you are saying that there is nothing that exists in the universe that is measurable as being √2 (or √2 of something), then this might well be the case* and an equilateral right angled plane of side √2 cannot exist either.

However, by your reasoning, an equilateral right angled plane of side (√2*√2) cannot apparently exist even though 4 is a perfect square! Your argument that √2 cannot be drawn is true therefore of any other length.

You confuse “meaning” and “existence.”

“Big” has a meaning but, of itself, has no existence. A big molecule is minute; a big star is unbelievably huge.

“Chair” has a meaning and many existences but, with all the time and ink in the universe, you would not be able to accurately define “Chair” to the exclusion of all possible confusion and/or exception. Why should √2 be any different?

All numbers have a meaning within the mathematical system. This does not imply that they exist of themselves.

So, at the end of the day, a definition of √2 exists and thus √2 has meaning, whether it, or any other number, “exists” is a meaningless question.

*If you are still worried about the question, let me tell you that √2 does exist – after working for well over 5 minutes, I have developed a unit of measurement such that the length of this reply is exactly √2 units long top to bottom (but only at 0ยบ K).

Gary said...

My apologies for the multiple post (if they appear) This is the first time I have used the site and the repetitive request for "word" verification caught me out.

Starhawk said...

"The square root of two is incommensurable." WTF a number can not be incommensurable.

According to Euclid, "Those magnitudes are said to be commensurable which are measured by the same measure, and those incommensurable which cannot have any common measure." Admittedly a piss poor definition by modern standards but one can easily see this is equivalent to A and B are commensurable magnitudes iff there exist a magnitude M such that a*M = A and b*M = B for a and b elements of N. And thus A and B are said to be incommensurable iff A and B are not commensurable.

Now what is a magnitude? A modern might associate the magnitude of a line or arc segment with a real number and the same goes for other geometrical magnitudes such as area or degree measurement but this obscures the synthetic geometric meaning of the term. we can without introducing real or rational numbers totally develop the algebraic and topological properties of magnitudes and put the whole subject on the firm axiomatic grounds. ( See R.F. Jolly's classic synthetic geometry) The resulting algebraic field is only isomorphic to the real numbers if we assume as an axiom something like making the Dedekind's cuts however we may assume some weaker form of continuity such as the circular continuity principle and end up with a Field of magnitudes isomorphic to the surds. Likewise nothing stops us from assuming a stronger continuity principle such as introducing infinitesimals and hence end up with what is called non standard analysis. There is some argument as to what Euclid actually meant as by modern standards Euclid is kinda vague. lol

My point being the statement "The square root of two is incommensurable" is assigning to a number a property of magnitudes. Numbers are not magnitudes But with the right choice of scale numbers can be used to represent magnitudes but a different choice of scale results in a different numeric representation of the same magnitude. A magnitude may or may not be incommensurable with respect to a given scale but we usually do not talk about numbers being incommensurable And besides even if we did √2 would be incommensurable with say 1 or 2 but it would be commensurable with √2/2. The concept is a binary relation!

burton mackenzie said...

Sorry I've been offline. Replies coming soon. :-)

a2plusb2equalc2 said...

First: You assert that √2 cannot be represented in a finite number of digits. The trivial counterexample is to use base √2. Then √2 = 10.

Second: You assert to construct a triangle of hypotenuse √2 in our universe the universe must either be infinite or infinitessimally divisible. Correct. However, given Heisenberg Uncertainty, Quantum Mechanics and wave functions there exists some small scale at which the (fuzzy-)particle that you add to the hypotenuse has some probability of being exactly the correct length to complete the triangle.

burton mackenzie said...

a2plusb2equalc2,

First, one could reasonably assume that we're talking about an integer radix given the context of the article. Saying we can represent an irrational in a finite number of digits if we use an irrational base (e.g. √2 = 10 in base √2) is trying to do an end run around its irrationality using countable (sub-infinite) units.

Second, the particles do not vary in size. They are particles that propagate a probability wave representing a sum across histories (Thanks, Feynmann!), but they themselves are not waves. The particles are not fuzzy, the location where you will observe them is.

To the rest of yez, I'm still hurting for time to respond to comments, but they're still on my todo list!

Rafa said...

Hi, nice article. But I mostly agree with your detractors:
-Sqrt(2) is in fact know as a constructible number, that means you can construct it from an unmarked ruler and compass. On the other hand, Sqrt(Pi) is not (this solves the squaring of the circle). Ok, maybe I'm taking advantage of semantics here since that doesn't in fact means that you can construct it. However, it is clear that you can construct anything that measures Sqrt(2), by simply having two things that measure the same, and putting them on a straight angle.
-I don't know that much quantum physics, but as far as I know, the interpretation is that matter is both particle and wave, not that it's particles that behave like waves in some cases.
-This discussion was further expanded by Kronecker (mentioned by Starhawk)when Cantor "invented" Set Theory. Cantor show the existence of infinite "numbers" each greater than the previous ones (see ordinal number). This started a big controversy, as Set theory seemed to provide a basis for all mathematics. This further developed on Godel's Incompleteness Theorems, as well as Turing's Halting Problem, which you might have heard mention by Chaitin.
-Finally, something that has been developing philosophically lately, specially by Wolfram (creator of Mathematica), is the idea that we might explain our world better as a computer. Under this ideas, you might be information as a fundamental part of the world, next to matter and energy. I'm guessing, that in this case, ideas such as irrational numbers (or at least algebraic numbers) have a place, as they will be part of our Universe "program".
Again, great article

burton mackenzie said...

Hi Rafa, thanks for the feedback. In the spirit of my recent posting on listening the most to those who disagree with you, I paid special attention to your arguments.

You wrote:
it is clear that you can construct anything that measures Sqrt(2), by simply having two things that measure the same, and putting them on a straight angle.

Anything we measure in this universe (even by your great example) is still a finite measurement. That is, the distance between the two ends of the 90deg angle is an approximation of √2, valid only to the resolution of the atomic (or smaller) material composing the device with which you are measuring. For instance, let's say you get down to the atomic level where the atom at the tip only has a probability of being in one place versus another. The lower limit on the resolution of your instrument is the limit of uncertainty. However, because √2 is irrational and infinite (in an integer base), any finite measurement will always be rounded. It doesn't matter if it's to two digits (1.4), twenty digits (1.4142135623730950488), or two hundred billion digits - it's still a finite resolution resolving an infinite range. Thus, unless we can measure the infinitesimal or infinitesimal, we can't measure anything but an approximation of √2. That is not truly measuring √2.

-I don't know that much quantum physics, but as far as I know, the interpretation is that matter is both particle and wave, not that it's particles that behave like waves in some cases.

By the standard model, matter is a particle. The matter itself is not a wave, but the probability of where to find it is. When detected (i.e. its existence defined) matter is a particle.

I've lightly examined Wolfram's ideas. He's more qualified than I am to talk about it, but I'll shoot anyway. I think he's probably right that stuff in our universe can be modeled by cellular atomata, but that doesn't mean that it is that way. It might just be a nice way to approximate one type of computation by conformally mapping it into a different type of computation. Beats the hell out of me. Even if he's not right, he's probably not wholly wrong, either.

A friend of mine suggested to me that the entire universe is a computer, and the propagation delay of the information in our universe computer is the speed of light. You can't move information faster than that because that's the limit of the metaphorical "data bus". I haven't been able to come up with anything to refute this idea. :-)

Thanks for the commentary!

hidortro said...

we can still use math even if it is not "doable" in the real world it is still a fun thing to do.

P.s you said that a number with an infinite digits was irantinol what about 1/3.

burton mackenzie said...

@hidorto: I didn't say a number with an infinite number of digits is irrational, I said an irrational number has an infinite number of digits.

Think of it this way, a tiger is a cat, but all cats aren't tigers.