There are no rational numbers in nature

So what do I mean by this obscure phrase? It comes from a thought I had the other day. It started with pondering certain famous irrational numbers like square rout 2, the natural logarithm, or pi.

I think what these numbers stem from is the fact that they are derivable from the geometric properties of space itself as opposed to all rational numbers which are derived from human thought. That is to say, whereas neat and discrete quantities like the integers and the ratios between integers (which is what all rational numbers are) are idealizations that we tend to impose on the natural world to the full extent of precision we can muster, the irrational numbers are already given by nature, and they are packaged in the geometric properties of space itself. In fact, nature only gives us irrational numbers.

To prove my point, I propose the following thought experiment. Suppose you’re taking a measure of something. keep it simple - say the length of your arm. Suppose you measure it to be 1.5 meters. Now, how precise are you being? I mean, is it exactly 1.5m? Probably not. It might be more like 1.53m, or maybe 1.56m, or something thereabouts. But then again, maybe even greater precision is needed to get at exactly how long your arm is. So if we zoom in further, we may find that it actually measures 1.539m or 1.562m. Well, how precise is precise enough? We could take this to extremes. We could say that your arm actually measures 1.5392396293742893721398m. In fact, I’ll bet (and this is the key point) that you could keep zooming in ad infinitum and keep tacking on more digits to the decimal expansion without end.

But isn’t it possible to measure something out to be exactly 1.5m or any rational quantity? Well, supposing you did, how would you know it’s exactly 1.5m? You could say that you’ve measured it out to the greatest of your precision abilities, but this is one thing; it’s quite another to say that a measure is some quantity exactly in an absolute sense - that is, to infinite precision. To say the latter is essentially to say that no matter how much you zoom in, the reading of 1.5m, or whatever it was, will never change - that is, there will never be additional digits tacked onto the decimal portion of the number. Now, to say such a thing technically isn’t paradoxical, but it does sound absurd. I would imagine that if you kept zooming in forever, the appearance of extra digits to your decimal expansion is an inevitability. It may take an extremely long time, but it will happen. And if it happens once, it can happen again. That is, if, when you final zoom in enough to discern an extra digit that needs to be tacked onto your decimal expansion, you keep zooming in further, it will be another inevitability that you discern yet more extra digits. So because there is no end to how far one can zoom in (at least in principle), there is no final digit that marks the righthand end of the decimal expansion.

Now that’s measurement. Measurement is the way we glean quantitative values from nature herself. In other words, nature gives us her measurements; we don’t impose them. The only thing we impose are our idealized rational numbers, and we only take it for granted that certain measures in nature can be captured precisely by ration quantities. Put if you want absolute precision in any of your measures of nature, you’ll have to settle for irrational numbers. Pi, the natural logarithm, and the square root of 2 are irrational because they aren’t derived from our measurements. They are derived by the immutable properties of geometrical shapes and forms - that is, the properties of that part of nature we call space.

Well, the old standard for a meter was literally a stick made of platinum or something like that. It was a meter long because that’s what it was defined as. It was 1.50000000000000000000000 meters forever and always, even if it changed! Same thing with the standards for grams and liters and such. Some standards are more scientific now…I know the standard length is measured by the distance light travels in a vacuum for a set period of time or something like that, but the point is if you define something to be the standard, it is exact.

Also, easily enough like Tortoise said, quantities are easily rational numbers.

You’re absolutely right that it would be highly improbable to make something exactly the same length as something else.

As far as irrational numbers go, they are the infinite series of ratios of rational numbers, so they are dependent on rational numbers. Unless of course you were to define pi = 1, or e = 1, or something crazy like that, in which case our rational numbers would be irrational!

The idiotic dichotomy between irrational numbers vs rational numbers is ridiculous to begin with. Irrational numbers are numbers and functions the reason why nature is filled with irrational numbers is because time and energy flow is always moving, whenever we measure something we are never measuring absolute stasis, we are measuring flow, hence the irrational numbers, if you froze time and motion you could rationalize all numbers in nature by multiplication.

How do you figure?

Think about it.

3.33 times 100 becomes 333, what you are doing is changing the scalar, to see this go get a 3D program like 3DS max or a free version of truespace, and play around with segmenting shapes into smaller and smaller pieces (adding lines of resolution/vectors, etc). An easy way to think of it is to imagine a graph in which you keep adding lines and points of resolution by making ever larger and larger numbers/functions, an easy way to think about it is to use the concepts of coloring - all filled, all empty, etc.

3.33 is not an irrational number. Multiplying any irrational number by a rational number will still result in an irrational number

You’re missing the point completely irrational number is a number AND a function, if you stopped time the "irrational number* would no longer be irrational (i.e. go on forever), because to function you need time (i.e. a vector to put the next number), I forgot to state that. Irrational numbers are functions (are the results of ratios of movement in time).

Numbers must have a location in a vector space for each unique digit (i.e. 3.14, the first 3 would be 3, 1’s, which it really is, , etc, etc).

Could you explain why an irrational number is a function or at least point somewhere for more info? I’ve never heard that before. How can it be a function of time? The value of pi is not rational, and it does not change with time, so how would stopping time suddenly make the value rational? The definition of an irrational number is a negative one, meaning that irrational numbers are defined as any real number that is not a rational number. There are an uncountable number of irrational numbers.

Ah, I see your issue. It’s a function, but not of time. Infinte series work off a generic variable, call it whatever you like, but it’s not time. Freezing time, whatever that would mean, would not change the value of an irrational number, or stop it in it’s tracks.

You guys are both right. I think my OP title was a little too grandeose. I think it should have read “there are no rational numbers in space” since it has more to do with geometry than countable objects. In other words, my argument works better with quantities that describe “how much” of something there is (like distance, angle, volume, anything spatial, etc.) as opposed to “how many” there are. And I guess you could put all these things in terms of units so that a question of “how much” volume there is can be rephrased as “how many” cubic feet there are, but I think the point here is that you might still end up with an irrational number even when measuring a certain number of units, but not necessarily. When you stick strictly to the “how much” terminology, it is necessary (not logically, but practically).

Well, by definition, measurement is an estimate. Any real scientific measurement includes not only the estimation itself (length, mass, etc), but also a margin of error, and measure of confidence.

Think of it this way, if knew the value we were looking for, we wouldn’t have to measure it, would we?

I wonder if the quantum length would come into play at some point and stop the irrationality of a measurement. But, for all intents and purposes, I’d say you are right; measurements are irrational numbers. The probability that a measurement just stops at a certain point, like 2.1231231490551340980134810329841240000000000 meters seems unlikely.

And that is where you are incorrect, we are talking about numbers made of stuff, you can’t have a number that is not made of something, when you think of a number it is made of concept-data, therefore, in order to calculate you have to be made of something, you can see this by grabbing some playdough and creating numbers out of it, once you run out of playdough you cannot create more numbers.

You’re still under the mistaken idea that math exists ‘apart’ from nature, when math is nothing more then a description of geometric structures. Again all calculation requires time, and if you represented and irrational number in binary the binary string would be growing, and each bit in the string needs a vector and location in space to be stored, if you stopped time that growth function would stop, period. So no, it has everything to do with time, if we froze time all calculation would stop.

Do you know anything about calculus? An irrational number just exists, it’s only the calculation that requires an effort on our part. Freezing time does nothing to the number. You can say that you only want to take the first ‘n’ terms of an infinite series, but that has nothing to do with time.

Here, this video should sum it up for you: youtube.com/watch?v=WrjwaqZfjIY

This was a thought that actually occured to me a few years back and I’m glad I’m not alone in the thought. My final conclusion then was that even though nothing in nature really seems to have infinite zeros after the decimal point, if we could instead measure the smallest particles (my assumption was electrons/protons/neutrons or maybe even quarks) we would have a rational number since there’s only a finite number of particles. So our measurements would be precise to the point and rational. Does this seem logical?
And to back up anthem here, math is a concept irrelevant to time so stopping time would have nothing to do with it as infinite, or finite for that matter, has no “speed(units/time)”, which seems to be your assumption superculture.

Thanks Rouz

Yes it does, but I think the number of particles falls into the “how many” camp as opposed to the “how much” camp. When it really comes down to it, I think the “how much” camp can be reduced to only continuous quantities such as space and time (in other words, not discrete “things” that exist).

What you’ve said still doesn’t deal with my point: In the real world these numbers would go on forever, and they would have to be made of stuff, if you were to expand an infinite series, it would take up all matter in the universe pretty quickly, and you’d run out of stuff to make the next number in the series. i.e. they are numbers and functions, you have to get used to the idea that numbers are self-recursive functions, and when time stops these functions also stop

If the ratio 1/3 is a function, then it’s opposite 3/1 is a function and since even “irrational” numbers have a base, that means they must by definition be numbers and functions, for math to remain consistent, remember all numbers are made of the first, that is, 1. Therefore:

Pi is actually Pi / 1 (which is a function), you can only see it by representing numbers in total expanded form using shapes, and ‘stuff’ to make numbers out of. Numbers are not made out of non-existence, they are made of stuff that exists.

What you’ve said is pure bullshit, numbers are functions, and functions only function if there is time, numbers are not made of non-existent stuff, I want you to grab apiece of paper and cut it into equal pieces and for each element of a number (say, 2 there is 2 squares, 3 there is 3 squares and so on), soon enough you’ll run out of stuff to make numbers out of, your fallacy is that you think numbers are made out of non-existent stuff, that is why you can’t grasp what I am saying and it is not an assumption it is a fact, you can’t get a non-existent number from non-existence, period.

There needs to be a distinction between counting and measuring. You can count the number of atoms along the length of a piece of wood, and get a rational number, but how long is each atom? You end up with the same problem. And as we deconstruct particles further we find that the smallest parts may not even really exist at all times ( quantum physics is crazy). It’s murky ground still, but the point is we can’t get exact measurements because we still don’t understand what goes on at the smallest levels

Ok, are you saying that you need to time to count digits, so that if time were stopped there would be no time to count digits and therefore the digit after you no longer can count no longer exist? That makes no sense to me. And why insist that numbers are made of physical stuff? Counting, and assigning numbers to things is a human concept. The fact that we use base 10 to count, or bases 2 and 16 in computers is an arbitrary human decision. Numbers, counting, units of measurement - all are human concepts that we use to describe what we see. Numbers don’t have to exist in nature, only in our heads, and if that’s the case, then there is no reason to limit them