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."

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Also, I'm greatly enjoying this discussion; much more so than the argumentation which preceded it.

There's no such thing as a funny sig.