This whole thread, and others hereabouts, are based on the apparent inability of a number of folk to comprehend the way mathematics treats infinity.

That is, on a failure to teach mathematics well.

The axiom of choice in theory allows one to decompose into pieces a solid sphere of diameter one foot, then reassemble those pieces into a solid sphere of diameter two feet. :gasp:

Could you perhaps enlighten us as to 'the way mathematics treats infinity' and how we should interpret these teachings?

"..." doesn't constitute doing it. And saying "done" doesn't necessarily mean that it is done. Clearly, "..." symbolizes what is not done, not what has been done. The meaning of the ellipsis symbol is "unfinished". So your claim of"'done" is false. — Metaphysician Undercover

Hello, Meta. Sigh.

The harmonic series diverges (very slowly) to infinity. What is it that you think is not done?

Well, that's kind of what my interest is. It would be wonderful if there were philosophical issues with infinity; so i like to keep an eye on these threads jst in case.

But each time, it seems instead that a misunderstanding lies at the heart of each supposed problem.

And what is that misunderstanding in this case?

These are interesting variations on Hilbert's hotel, perhaps.

It seems there are folk who read these and think "but you can't do that!", concluding that there must be an error somewhere. So they go on to philosophy forums and reinforce each other's errors.

Which case? Being specific is important. See, for example, Meta's comments.

Could you perhaps enlighten us as to 'the way mathematics treats infinity' and how we should interpret these teachings? — Devans99

Yes, Banno - write a book for us with the whole of the maths of infinity in it.

Well for example I contend that because a mathematical point has length 0 therefore there are, on a line segment length 1, 1/0=UNDEFINED points, rather than an infinite number of points.

So my thinking is different from the traditional maths explanation. Please explain where I am going wrong?

A neat division, which I think leads astray.

See http://mathworld.wolfram.com/DivisionbyZero.html, especially the flippant answer from Derbyshire.

So mathematicians count points, instead of doing naughty divisions. Which has led to all sorts of interesting developments in dealing with infinities - especially uncountable ones.

The answer to your question, how many points are there on a line segment, is not that there are infinite, not that there are indefinitely many, but that there are uncountably many.

That was an excellent example. Thank you.

That is rather my point - maths with its definition of a point having zero extents - flaunts its own rules - there is an insistence in maths that both of the following are true:

1. 1/0 = UNDEFINED
2. There is an infinite number of points on a line segment length one

That equates to a belief in both of:

1. 1/0 = UNDEFINED
2. 1/0 = ∞

You see that obviously both [1] and [2] cannot hold at the same time... unless UNDEFINED = ∞ ... which is my belief.

Well for example I contend that because a mathematical point has length 0 therefore there are, on a line segment length 1, 1/0=UNDEFINED points, rather than an infinite number of points. — Devans99

"points" are an intriguing notion aren't they? Like "lines" with no thickness. There are theories of time that posit the non-existence of a "present point" But they are not useful in physics.

...and then one gets a reply like this. Your objection was anticipated, and answered:

To the persistent but misguided reader who insists on asking "What happens if I do divide by zero," Derbyshire (2004, p. 36) provides the slightly flippant but firm and concise response, "You can't. It is against the rules." Even in fields other than the real numbers, division by zero is never allowed (Derbyshire 2004, p. 266).

Right, so the mathematical definition of a point is 'against the rules' ... of maths.

Yeah. No. I'm taking this example to my thread on critical thinking, because you are providing me with an excellent instantiation of the issue I was addressing there. .

https://thephilosophyforum.com/discussion/comment/362733

Are you a 100% positive or 99%? I like an opponent that is over confident. Look up the definition of a circle in a geometry book.

Again, the answer is that the number of points on a line segment is uncountable. — Banno

You are implying that:

- 1/0 is illegitimate
- 1/0 is uncountable / infinite

So which is it? It cannot be both.

You are implying that:

- 1/0 is illegitimate
- 1/0 is uncountable / infinite — Devans99

No, I'm not.

Which is it then?

You've set up a false dilemma.

You need to stop kicking people's asses. This is fun to watch though.

You've set up a false dilemma. — Banno

How so?

You state that 1/0 is illegitimate.

You also agree with the mathematical definition of a point as having zero extent.

Then the question of how many points there are on a line segment length one is perfectly valid:

- If its uncountable/infinite, then that suggests that 1/0 is legitimate
- If its undefined, then you are agreeing with me

Which do you choose?

- If its uncountable/infinite, then that suggests that 1/0 is legitimate — Devans99

No, it doesn't.

Thanks Christian! It is getting late and I'm getting nowhere with @Banno so I guess I will say good night to everyone!

hat is rather my point - maths with its definition of a point having zero extents - flaunts its own rules - there is an insistence in maths that both of the following are true:

1. 1/0 = UNDEFINED
2. There is an infinite number of points on a line segment length one

That equates to a belief in both of:

1. 1/0 = UNDEFINED
2. 1/0 = ∞

You see that obviously both [1] and [2] cannot hold at the same time... unless UNDEFINED = ∞ ... which is my belief. — Devans99

0 has no multiplicative inverse, hence 1/0 is an undefined or meaningless symbol. That's true.

That there are an infinite number of points in a unit line segment is true but ambiguous.

The cardinality of a set depends on the notion of bijection. But this is quite different from the measure of a set, which is a generalization of intuitive length. A measure (like standard Lebesgue measure) is a countably additive set function (among a few other requirements). Hence the measure of an uncountable union of points is not the (uncountable?) sum of the measures of the individual points. Indeed, this would lead to a contradiction, as you note, and assign a line segment of positive length (in the intuitive sense and in terms of the measure) a measure of 0.

Banno is frustrated because, well, mathematicians know their business. Any of us can give one of them hell about their philosophical interpretation of their professional discourse (if they bother to have one), but it's highly unlikely that a non-expert will catch them in a genuine contradiction. Mathematicians are experts when it comes to finding and using contradictions.

Math isn't philosophy. It's isn't metaphysics. 'It' can always retreat to formalism and utility.

The harmonic series diverges (very slowly) to infinity. What is it that you think is not done? — Banno

It hasn't reached infinity, so it is not "done".

It hasn't reached infinity, so it is not "done". — Metaphysician Undercover

Yeah, it does - that's what the ellipsis is for.

As I said, ellipsis means unfinished. So using the ellipsis and claiming "it's done" is a false claim.

...ellipsis means unfinished... — Metaphysician Undercover

No. Here it means "keep going like this..."

https://en.wikipedia.org/wiki/Ellipsis#In_mathematical_notation