## Infinite Balls and Jugs [solution]

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gmalivuk
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### Re: Infinite Balls and Jugs [solution]

Wildcard wrote:It does mean that claiming that supertasks, infinite cardinalities and, yes, even complex numbers have objective validity (or objective truth) is simply not true.
What is "objective validity"? Where in objective reality can you find pi? I don't mean some approximation to a few dozen digits, I mean the actual exact value of pi.

PeteP wrote:That is a weird argument imo. I wouldn't call defining the natural numbers an example of an supertask at no point does it require actually going through it step by step and obviously no number requires going through a supertask to reach it. But more importantly you can describe the "ends with infinite balls" argument for the infinite ball thing like a proof by induction, for any step x the step x+1 has more balls and at the end of step 1/at step 2 you have a positive number of balls so at no step greater than 2 will it ever become less than 1. In other words proof by induction is another thing that breaks down when supertasks get introduced so I find it a bit weird call it an example of one.

Achilles is always behind the tortoise:
At step zero, he starts out behind the tortoise.
Suppose at step x, he is behind the tortoise. Then at step x+1, he has covered the distance to where the tortoise was, but the tortoise has moved 1/10 of that distance forward, so he's still behind the tortoise.
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### Re: Infinite Balls and Jugs [solution]

gmalivuk wrote:What is "objective validity"? Where in objective reality can you find pi? I don't mean some approximation to a few dozen digits, I mean the actual exact value of pi.

A circular object can be viewed. It is circular. It has a shape. It has a center. It has a rim. There is space (distance) between the center and the rim. There is distance (extent) across the object through the center (diameter). All of these are real, objective, and observable.

The idea of circumference, radius, diameter, each have a solid objective basis. The idea of comparing the circumference to the diameter is perfectly objective. Decimal representations (symbolizations) of that idea are symbols, and the method of representation (symbol) of the idea of pi is distinct from the actual value (concept) which that symbol represents.

In other words, to see the objective reality of pi, look at a circular object and imagine (think, conceive of) comparing the distance around the rim to the distance across the object through the center.

Thoughts are real; they are themselves a part of reality. A door is composed of matter, it exists in space, it persists through time, and you see it using energy (light)—and it is a manifestation of the thought of a door.

Aside: This discussion reminds me of the philosophic question of whether mathematics is a created (invented) thing or a discovered thing. From a purely philosophical point of view this is an interesting question, but my own answer to it is that one must be much more precise about what is meant by "discovered" and quite clear on the nature of reality before it can even be a meaningful question. In particular, the question contains an implicit assumption that some things exist independently of our creation of those things, and thus can be "discovered" rather than being created. But this is really a separate and deeper philosophic question than the subject actually under discussion, which is the applicability or relevance of mathematics to the observable physical universe. (End aside.)

Mathematical ideas have their own beauty. For instance, addition, multiplication, exponentiation are all based on observable reality. Then extending those ideas to fractional exponents, negative exponents, etc. is a perfectly logical extension and behaves in a beautifully consistent way. In fact I would even agree (and I'm sure you would as well) that if one accepts that fractional and negative exponents are possible, then it would be illogical (dare I say impossible?) for them to behave in any way other than how we agree that they do.

However, the fact that these abstractions can be created and are so beautifully consistent does not conclude the philosophic question of whether these mathematical ideas have objective correspondence to the real universe. You may recall that the very reality of negative numbers (let alone imaginary numbers!) was debated for years by the Greeks.

To bring it back to the particular puzzle under discussion: The point here is that the answer to the puzzle does not have objective reality, despite being phrased in terms of real physical objects such as balls and jugs. To criticize someone's intuition for supplying an "incorrect" answer is disingenuous, because the puzzle is not real. I do in fact agree with the canonical answer mathematically and agree that it is the most valid mathematical answer, but this is not based on any objective "correctness" of the answer; it is based on the greater workability and aesthetic quality (yes, aesthetics can play a part in acceptance of mathematical ideas) of the agreed-upon methods of dealing with abstract infinities.

So it is my contention that arguing with those who disagree with the stated solution using the basis that their answer is "wrong" is both fundamentally incorrect, and likely to be fruitless. The answer is not objectively correct. Disagreeing with the stated answer is not wrong. The only real defense for the stated answer is that it is predicated on a consistent abstract approach for dealing with ideas of numerical infinities, and frankly the aesthetic qualities of modern mathematical approaches to infinities have a great deal more bearing on the discussion than any notion of "correctness" or "incorrectness."

---------

Also, I'm greatly enjoying this discussion; much more so than the argumentation which preceded it.
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gmalivuk
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### Re: Infinite Balls and Jugs [solution]

Any representation of the exact value of pi involves an infinity somewhere, no matter how much imagining you do with some approximations of circles.

And what exactly is the distance around a curve? Distances between points I'll grant you, but what's a distance around a curve?

What do you look at in the real world to get negative 1?
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### Re: Infinite Balls and Jugs [solution]

The primary reason I can't fully get behind your argument is that a nonempty jug leads to (many) logical contradictions. I'm not sure on what basis we can say anything is objectively incorrect if that does not qualify.

Wildcard wrote:
gmalivuk wrote:What is "objective validity"? Where in objective reality can you find pi? I don't mean some approximation to a few dozen digits, I mean the actual exact value of pi.

A circular object can be viewed. It is circular. It has a shape. It has a center. It has a rim. There is space (distance) between the center and the rim. There is distance (extent) across the object through the center (diameter). All of these are real, objective, and observable.

Are they though? I mean, the circular plate in my cabinet certainly has those things, and they are objectively real and observable. But the ratio of the circumference of my plate and its diameter is certainly not pi.

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### Re: Infinite Balls and Jugs [solution]

Xias wrote:
Wildcard wrote:
gmalivuk wrote:What is "objective validity"? Where in objective reality can you find pi? I don't mean some approximation to a few dozen digits, I mean the actual exact value of pi.

A circular object can be viewed. It is circular. It has a shape. It has a center. It has a rim. There is space (distance) between the center and the rim. There is distance (extent) across the object through the center (diameter). All of these are real, objective, and observable.

Are they though? I mean, the circular plate in my cabinet certainly has those things, and they are objectively real and observable. But the ratio of the circumference of my plate and its diameter is certainly not pi.

Yes, you're getting it. The ratio is not pi. But pi is a useful abstraction which allows you to predict solutions to quantitative problems in the real physical universe with sufficient accuracy. In other words, as I stated earlier, mathematics is neither true nor false; it is useful or not useful.

Considering real-world concerns such as margin of error (margin of safety), the precise decimal computation of exact answers is seldom actually necessary or useful for most people; it's certainly not so relevant to most real universe quantitative problems as might be inferred from the stress placed on exact calculations in early grade school (e.g. long division).

The best arguments in favor of teaching everyone precise mathematics (rather than only teaching engineers and physicists) are those that appeal to aesthetic sense and the sense of fun, rather than trying to claim that complex numbers are actually useful and applicable for most of the students. (Incidentally, you may think I'm conflating the idea of real-world applications with the frequency of use of particular applications, but there is a fundamental relation between personal viewpoint and information evaluation; or, to quote L. Ron Hubbard's phrasing: "The value of a datum or a field of data is modified by the viewpoint of the observer.")

The idea of infinite digits of pi is not really a very important one (not widely applicable or useful), though it is certainly interesting for its own sake. A much more useful, important concept for actual application is the idea of "significant digits." (Unfortunate that this is only seldom taught to high school students in the U.S.)
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### Re: Infinite Balls and Jugs [solution]

Okay, but complex numbers and limits are immensely useful for (the technology underlying) everyday life, irrespective of how uncomfortable you are with the implications thereof in certain contrived cases.
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### Re: Infinite Balls and Jugs [solution]

gmalivuk wrote:Okay, but complex numbers and limits are immensely useful for (the technology underlying) everyday life, irrespective of how uncomfortable you are with the implications thereof in certain contrived cases.

I agree completely with the main point here (and now that we're talking about usefulness instead of truth! ) but I'm not sure what implications, contrived cases, and discomfort you're referring to.

Incidentally, while I can very comfortably handle complex numbers (and limits, and calculus), I've never been entirely happy with my understanding of the applications of complex numbers. I have heard it vaguely mentioned that they are useful in electrical engineering, but with no specifics. I consider them quite beautiful for their own sake, though.
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### Re: Infinite Balls and Jugs [solution]

Wildcard wrote:
gmalivuk wrote:Okay, but complex numbers and limits are immensely useful for (the technology underlying) everyday life, irrespective of how uncomfortable you are with the implications thereof in certain contrived cases.

I agree completely with the main point here (and now that we're talking about usefulness instead of truth! ) but I'm not sure what implications, contrived cases, and discomfort you're referring to.
The contrived case is the balls and jugs, and the discomfort is your own unwillingness to acknowledge something as "objectively valid" just because it's not intuitive.

Incidentally, while I can very comfortably handle complex numbers (and limits, and calculus), I've never been entirely happy with my understanding of the applications of complex numbers. I have heard it vaguely mentioned that they are useful in electrical engineering, but with no specifics. I consider them quite beautiful for their own sake, though.
I don't remember how they show up in EE, but they're kind of important in quantum mechanics.
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### Re: Infinite Balls and Jugs [solution]

From my incomplete remembrance of my EE classes back in college (though it was never my strongest subject), they tend to show up a lot in phase calculations and frequency-domain analysis, such as Fourier/Laplace transforms. Usually somewhere in the exponent of e, next to a pi.
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### Re: Infinite Balls and Jugs [solution]

Offtopic electronics stuff:
Spoiler:
It comes up in electronics for looking at AC circuits...

If every power source in the circuit is AC with the same frequency, then every voltage/current measurement will be a sine wave of that frequency (since the sum of any two sine waves of the same frequency is another sine wave of that frequency). So any voltage/current looks like "a cos ωt + b sin ωt" for some a and b.

But that's awkward to work with, so we do a basis remapping to complex numbers, and represent this as a + bi. When we map across our operations, addition maps to addition and scalar multiplication maps to multiplication by reals, so that's all pretty simple. But more usefully, derivatives map to multiplying by ωi, and integration to dividing by ωi. Which is useful, because there are a lot of derivatives in electronics (resistors have V∝I, capacitors have dV/dt∝I, inductors have V∝dI/dt). It means that you can turn the linear ODEs that fall out of a circuit analysis into far-simpler linear equations, and all three of those basic components become simple linear impedances.

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### Re: Infinite Balls and Jugs [solution]

Wildcard, I think you are confusing "objective" (vs. subjective) with "concrete" (vs. abstract). You can hold a pie in your hand, but you can't hold pi in your hand. For that matter, you can't hold two in your hand, even if you have two hands to hold it with. However, this is not unique to mathematics - you can't hold 'to' or 'too' in your hands either. Like language, mathematics is abstract, not concrete. But unlike language, math is objective, in the sense that it exists independently of anybody's thoughts about it (or about anything else). And mathematical concepts are true (or false) with equal independence, even if they are not concrete items that can sit on a shelf.

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### Re: Infinite Balls and Jugs [solution]

Wildcard wrote:"Suppose I have infinitely many balls." Even after this one sentence, this hypothetical is already no longer susceptible to observation or experimentation. It thus becomes a question in pure imagination, and as such we can quite reasonably imagine any outcome whatsoever, including the outcome, "All the ducks come." (Which ducks? All of them. Where do they come? They just do.)

"Suppose I have 10100 balls in a container and add another 10100 balls. How many do I have now?" But even after the first phrase ("Suppose I have 10100 balls"), this hypothetical is already no longer susceptible to observation or experimentation. (There are estimated to be only about 1080 subatomic particles in the observable universe.) So, according to your reasoning, there is no single, valid, answer to the question: "All the ducks come." is an equally valid answer!