What is the relationship between probability and irrationality in mathematics?

[pinwheel111]

I recently tried to engage one of my professors with the idea that every application of mathematics to the real world can only involve irrational numbers if perfect measurement is allowed (with the notable exception of counting loosely defined or symbolically perfect ideas, like, how many apples are in this basket, or, a football field is exactly 100 yards long, etc.). For example, if you had a cartesian coordinate system set up on a dart board, and threw a dart at it, the coordinates of the impact must necessarily be irrational and nothing else. Or that any two arbitrarily chosen intervals of a continuum, i.e. space or time, must necessarily be incommesurate; say, the ratio of the precise length of an episode of The Golden Girls to the length of the very unit used to measure it--a minute. To refute this, he cited probabilty. I contended that the chance of the result of a coin flip coming up heads is not 1/2, but slightly less because of the unlikely event that the coin lands on its side, and the rediculously unlikely event that the coin lands on the corner joining the side and face. Even certain events that intuitively yield a 100% probability, like the chance that the coin will hit the ground after being flipped are rendered irrational due to quantum weirdness.
Anyway, to avoid a purely philosophical post, this got me thinking, even in purely conceptual terms, where a nickel is a perfect cylinder, is there a way to calculate the true odds of a coin toss coming up heads given all the possible outcomes (faces, side, edge), neverminding quantum weirdness? Also I would like to get some more in depth insight into the aforementioned idea of irrationals.

Thank you.

 

[CRGreathouse]

Actually, the information I've seen on coin flips shows that they're actually very bad random number generators, with high (> 60%) chance of landing the way they were thrown. I wonder if Google can drag something up on that.

 

[Rogerio]

The rational numbers are dense, too. So, I don't see why every application should only involve irrational numbers.

 

[EnumaElish]

is there a way to calculate the true odds of a coin toss coming up heads given all the possible outcomes (faces, side, edge),

You can estimate it by throwing a coin "infinitely" many times.

 

[daveb]

I'm by no means an expert in probability measure theory, but since the measure of the rationals is zero, and the measure of the irrationals isn't, then isn't the liklihood of hitting an irrational on the dart board much greater than of hitting a rational?

 

[CRGreathouse]

I'm by no means an expert in probability measure theory, but since the measure of the rationals is zero, and the measure of the irrationals isn't, then isn't the liklihood of hitting an irrational on the dart board much greater than of hitting a rational?

Yes, it's probability 1 that you'd hit an irrational. That was pinwheel111's point, I think.

 

[Rogerio]

... but since the measure of the rationals is zero,...

Zero? Why?!

(Again: the rationals are dense!)

 

[mathman]

Since the rationals are countable, you can cover each rational by an interval of measure < e/2ⁿ, so all rationals are contained in a set of measure <e, which can be arbitrarily small. Therefore the measure of the rationals is 0.

 

[EnumaElish]

The Lebesgue measure of rationals is zero. But it is possible to define a probability measure that is non-zero on the rationals. Example: an intelligent dart works in two steps. First it picks two natural numbers x and y at random. Then it homes in on point (x,y). With this intelligent dart, there is zero probability of hitting an irrational.

 

[CRGreathouse]

The Lebesgue measure of rationals is zero. But it is possible to define a probability measure that is non-zero on the rationals. Example: an intelligent dart works in two steps. First it picks two natural numbers x and y at random. Then it homes in on point (x,y). With this intelligent dart, there is zero probability of hitting an irrational.

Or a dart that follows a thread to (0, 0). No chance there either.

 

[EnumaElish]

Or a dart that follows a thread to (0, 0). No chance there either.

I read your post as implying that any distribution which ascribes a zero measure to the irrationals is a degenerate distribution. Which I think I agree with. [Edit: Except, is there a chicken and egg problem here?]

 

[pinwheel111]

But with the intelligent dart and especially with the thread, the accuracy ascribed to them will be qualified by the familiar notation, "+/- x units" to allow for error. Beyond this, the coordinates are irrational. Even if you walk up to the dart board and put the dart in (0,0), the actual coordinates will be something like (.000023459820349877291..., -.000031056931957129857...). Data from Star Trek might be able to place the dart at (6.29234567194579...* 10^-20, 9.29529634919005...*10^-19), and no matter how good we get millenia from now, a googleplex digits beyond the decimal point, the number will begin to diverge from zero. Anyway, accuracy is cut off way before that point by Heisenberg and Planck. The chances of the coordinate being an infinite repeating string are zero, no matter how the dart is put there.
By the way, I can hit a neutrino from a billion lightyears away, which is only infinity times larger than a mathematical point. Just kidding. I'm not that good.

 

[matt grime]

If you stop conflating reality with a mathematical model then you might get somewhere.

 

[EnumaElish]

With any continuous, non-degenerate distribution over the real plane (as your original post seemed to imply) (e.g. bivariate normal distribution), Prob(.000023459820349877291..., -.000031056931957129857...) = Prob(6.29234567194579...* 10^-20, 9.29529634919005...*10^-19) = Prob(0,0) = 0.

What I hear you saying is Prob(Set of irrationals) = 1, which I agree with. That is a different statement than saying you or Data would or could hit particular irrational points. You are just as unlikely to hit $(\pi,\pi)$ as (0,0).

You should be careful, because it is easy to turn your "reality" argument around to point to the opposite conclusion. For example, I could say that in reality, perfect measurement is an impossibility. You started with an abstract thought about perfect measurement, but when you were held to it, you immediately turned around and started about how reality is subject to errors. Can you make up your mind? Which is it?

 

[CRGreathouse]

By the way, I can hit a neutrino from a billion lightyears away, which is only infinity times larger than a mathematical point. Just kidding. I'm not that good.

Aw, you're just being modest.

 

[pinwheel111]

"What I hear you saying is Prob(Set of irrationals) = 1, which I agree with. That is a different statement than saying you or Data would or could hit particular irrational points. You are just as unlikely to hit $(\pi,\pi)$ as (0,0)."You're just screwing with me. Aren't you. You are. I can tell. I thought it was pretty clear that what I meant by "the actual coordinates might be something like (.000023459820349877291..., -.000031056931957129857...)" was that the actual coordinates might be something akin to (.000023459820349877291..., -.000031056931957129857...); that is, irrational, and non-zero, but close to zero, just like those numbers. Clearly I didn't mean exactly like them. (Sarcasm)(Why don't you criticize me for not putting down actual irrational numbers instead of their rational approximations?)=1. And certainly the point was clear enough as to be consistent even if I said, "Data might be able to place the dart at (6.29234567194579...* 10^-20, 9.29529634919005...*10^-19)..." (which was meant simply to imply that he would be more accurate) instead of, "In his attempt to hit (0,0), Data might hit (6.29234567194579...* 10^-20, 9.29529634919005...*10^-19)" Which is (clearly, I thought) how it should read, especially in relation to the following context. (The word, "might" in the previous sentences having the meaning, "in a hypothetical situation, the actual outcome of which is impossible to know, may or may not..." rather than "could possibly...")

"You should be careful, because it is easy to turn your "reality" argument around to point to the opposite conclusion. For example, I could say that in reality, perfect measurement is an impossibility. You started with an abstract thought about perfect measurement, but when you were held to it, you immediately turned around and started about how reality is subject to errors. Can you make up your mind? Which is it?"

Because of three factors: 1, the imprecision of measurement produces errors no matter the accuracy; 2, the incredible resistance irrational numbers have to simply being written; and 3, quantum mechanics prevents us from pinning down the point at which the board is struck; we can never accurately convey to one another exactly where the dart is. Nevertheless the dart does have a precise location with irrational coordinates with respect to the origin (of the mathematical model drawn on the dart board--i.e fake math space mapped onto real space). I'm sorry, but I don't see how the previous two sentences are mutually exclusive. I started out by saying that perfect measurement is a requisite to knowing what that irrational number is.But seriously, I can hit $(\pi,\pi)$. And surely we all agree now as we did before.

 

[EnumaElish]

Assume, you have perfect measurement and know the irrational number you hit. You were then given two examples of a probability distribution, each of which ascribes zero measure to irrationals. Your answer could have been "these are degenerate/discontinuous distributions, I was assuming a non-degenerate/continuous distribution," which would be a sensible answer if only because it could have made you realize that you are not positing mathematical statements, you are actually positing that the reality is a continuous distribution (and vice versa). A continuous distribution is a mathematical construct. Ergo you are in the realm of metaphysics.

But instead, you started about accuracy and errors (which are beside the point if you are ready to discuss probability distributions, qua math objects). All of this makes me think that that was your way of positing "reality is not a degenerate/discontinuous distribution," except you were stating it indirectly.

 

[pinwheel111]

You're right. I totally didn't even realize that you thought I meant an idealized theoretical construct of THE GOLDEN GIRLS and not the actual T.V. show. But I thought it might be a problem, which is why I began the second paragraph of my original post with, "Anyway, to avoid a purely philosophical post..." What I meant by that was, "I realize that these aren't mathematical statements, but rather philosophical statements about the way the world works, so let me get to the math question..." So it wasn't a real dart board your intelligent dart was going to?

And yes, NUMBERS are a mathematical construct. I realize. Nevertheless we count things. The circle is a mathematical construct. It exists independently of human thought despite the fact that there are no examples of circles in the universe. Mathematical constructs have that property of being sown into the fabric of reality, WHERE WE DISCOVER THEM.

 

[pinwheel111]

Maybe I was just kind of confused as to how you'd do that to an actual dart board. You know. The whole degenerate/discontinuous business. Actually the whole example about darts and the Golden Girls had absolutely nothing to do with probability distributions. I got into that when I suggested that REAL probabilities worked in the same way. I.e. they too are incommensurate. What I suggested was FOR EXAMPLE, NOT FOR REAL: the probability of a coin toss coming up heads MIGHT BE SOMETHING LIKE: 0.4999999999999999999999999999999999999999999999999999999999999999992034960276106972300367... That is, irrational, not .5, but close to it. Ergo, huh? Nice edit. Looks smarter. Quod erat demonstrandum. This is my final post.

 

[matt grime]

So it wasn't a real dart board your intelligent dart was going to?

The implication in my mind is that you think you have a REAL dartboard with every point labelled by real number coordinates in a way you can actually measure, and that your REAL dart actually hits precisely one of these points. That ain't no real situation I can think of.

And yes, NUMBERS are a mathematical construct. I realize. Nevertheless we count things.

That's the natural numbers, not the real numbers.

 

[EnumaElish]

I got into that when I suggested that REAL probabilities worked in the same way. I.e. they too are incommensurate. What I suggested was FOR EXAMPLE, NOT FOR REAL: the probability of a coin toss coming up heads MIGHT BE SOMETHING LIKE: 0.499999999999999999999999999999999999999999999999 9999999999999999992034960276106972300367... That is, irrational, not .5, but close to it.

You seem to be thinking along the lines of http://planetmath.org/encyclopedia/AlmostContinuousFunction.html in the context of probability distributions, except the phrase in your case would be "Almost Discontinuous Function." A paper that uses that phrase without defining it can be found at www.astro.lsa.umich.edu/~cmullis/papers/Ebeling_et_al_2000.pdf

[Also see https://www.physicsforums.com/showthread.php?t=179963.]

Mathematical constructs have that property of being sown into the fabric of reality, WHERE WE DISCOVER THEM.

For an opposing view, see Critique of Pure Reason.

 

[pinwheel111]

I thought that would be it. Holy crap. If I go somewhere worse than this when I die, I am going to be real pissed. Mr. Grime, I do have a real dart board, and a real dart. Are you telling me that if I draw a Cartesian coordinate system on the dart board, and throw a dart at it, that I am not going to hit any points on the graph? Don't answer that. I know I will. Otherwise what the hell would we use graph paper for? That point has irrational coordinates. That's all I'm saying. That's all I've ever said. I have NEVER said anything different. The conflation of reality and construct is kind of the point and is exactly where I'm trying to get. That spacetime continuum thing, is just that--a continuum. Just like a number line. Or a Cartesian coordinate system. I seriously don't know how anyone could gather anything else from my posts, unless...I mean I know that math people stereotypically have dry senses of humor...

And Enuma, tell me who invented the circle. I mean, you put my quote up there, but did you read the sentence before it? How can any opposing view be correct? Ah, I'm willing to bet that your "opposing view" is really a view that supports what I'm saying, but just like my posts, you read it and got the exact opposite meaning out of it than what the author was trying to say.

"[Kant] reasoned that statements such as those found in geometry and Newtonian physics are synthetic a priori knowledge and wanted to establish how this could be possible." -- A quote from your link.

Again, you have demonstrated your incredible knack for missing the point. The author of the wiki article says that Kant is saying that geometry and Newtonian physics are ideas woven into the fabric of reality, the constructs of which we arrive at or "discover" via reason and logic. That is, we have but to think about it, and the working of the universe reveals itself to us. As opposed to a posteriori--requiring measurement or observation. Ok, Ok, yes, Kant says (indirectly) the only place these constructs can "exist" is in the mind (my point wasn't that they didn't), nevertheless they are independent, being inherent to the nature of the reality in which we exist (my point in the previous post).

Does anyone else post in this forum and feel like they're in the Twilight Zone?
Just kidding boys. Thanks for the fun.

Mike.

 

[matt grime]

I thought that would be it. Holy crap. If I go somewhere worse than this when I die, I am going to be real pissed. Mr. Grime,

It's Dr if you want to be formal.

I do have a real dart board, and a real dart. Are you telling me that if I draw a Cartesian coordinate system on the dart board,

You reckon you can draw a cartesian coordinate system on it? Precisely? You reckon your dart will hit precisely one point? Geez, that's an infinitesimally thin dart you've got there. Space is infinitely divisible, right...

and throw a dart at it, that I am not going to hit any points on the graph? Don't answer that. I know I will. Otherwise what the hell would we use graph paper for? That point has irrational coordinates.

With probability 1, in a model, yes. But reality isn't like the model. It doesn't really make sense to claim these things of the real life situation. How did you precisely define 1 unit length such that all other measurements can be made with respect to it?

If you want to, you can mail your dartboard to someone with the points (pi,e), (1,sqrt(2)) and some others clearly marked on for us all to verify...

 

[EnumaElish]

Mike,

tell me who invented the circle

"A line of ancient peoples (Sumerians, Akkadians, and Babylonians) who lived in Mesopotamia (now southern Iraq) invented writing, observed the skies, and invented a 360-degree circle to describe their findings" according to WonderQuest with April Holladay.

I'm willing to bet that your "opposing view" is really a view that supports what I'm saying, but just like my posts, you read it and got the exact opposite meaning out of it than what the author was trying to say.

I never disagreed in principle with what I thought you were trying to say. Stated formally: for any continuous measure P defined over a real domain, P(set of irrationals) = 1.

But, a probability distribution can produce rational outcomes with positive probability even when defined over a real domain. Any such distribution is discontinuous (because it skips over irrationals, as well as some rationals) and degenerate (because it has mass points).

Thanks for clarifying your earlier remark that math objects are embedded in the nature, waiting to be discovered. I understand that it has never been your position to contradict the Kantian synthesis, which holds that one must have an a priori concept of space (i.e., geometrical principles) before "filling" it with experiences.

 

[EnumaElish]

See http://en.wikipedia.org/wiki/Cantor_distribution as a possibly more accurate representation of reality.

 

[Extropy]

Again, you have demonstrated your incredible knack for missing the point. The author of the wiki article says that Kant is saying that geometry and Newtonian physics are ideas woven into the fabric of reality, the constructs of which we arrive at or "discover" via reason and logic. That is, we have but to think about it, and the working of the universe reveals itself to us. As opposed to a posteriori--requiring measurement or observation. Ok, Ok, yes, Kant says (indirectly) the only place these constructs can "exist" is in the mind (my point wasn't that they didn't), nevertheless they are independent, being inherent to the nature of the reality in which we exist (my point in the previous post).

Actually, you have missed the point of what Kant said. From this article, "Immanuel Kant provides a good start with his declaration that the best way to view time and space is not as features of reality that the mind tries to comprehend, but as features of the mind we use to comprehend reality." In other words, Kant is saying that such things like space and time are not things within reality themselves, but only in our minds, which we use to try to comprehend the things which are in reality themselves.