Now the nature of infinity is an interesting topic to explore!

I have much empathy for your position. Since our lives are finite, it seems impossible to experience an infinite set. But it may surprise you to know that many mathematicians today believe that actual infinite sets exist in math!

Now the nature of infinity is an interesting topic to explore! — Real Gone Cat

I stated earlier in the thread, what I believe to be the reason for the concept of infinite:

What I think, is that we allow "infinite" so that we will always be able to measure anything. If our numbers were limited to the biggest thing we've come across as of yet, or largest quantity we've come across, then if we came a cross a bigger one we would not be able to measure it. So we always allow that our numbers can go higher, to ensure that we will always be able to measure anything that we ever come across. In that way, "infinite" is a very practical principle. — Metaphysician Undercover

But it may surprise you to know that many mathematicians today believe that actual infinite sets exist in math! — Real Gone Cat

That doesn't surprise me at all. I've had numerous discussions in this forum with mathematicians, and I've already been well exposed to the absurd ontology which seems to be exclusive to that cult.

But I accept that you don't consider 'utilitarian' as a correct description of your productivity and outcomes argument. Indeed, my point doesn't rely on the particular rubric 'utilitarian' but rather that I reject your productivity and outcomes argument, whatever rubric it correctly falls under. — TonesInDeepFreeze

It correctly falls under 'contemporary analytic virtue ethics', and is not consequentialist in the substantive sense of the word but the trivial one (the trivial one where all ethical theories are 'consequential' in that Kantians would care about that consequence of violating the imperative, or Aristotelians about virtue, and so on). The substantial sense of 'consequentialism' and 'utilitarianism' (which falls under the former) is not as broad, and the argument translates to the fact that your actions fall short of moral excellence (in that indignancy fail prudence and justice): indeed, no virtue ethicist would ever grant you that your personal frustrations offset this.

That doesn't surprise me at all. I've had numerous discussions in this forum with mathematicians, and I've already been well exposed to the absurd ontology which seems to be exclusive to that cult. — Metaphysician Undercover

Yes, we're a wicked bunch intent on the corruption of the intellects of youth in order to bring them to the alter of our Satan, Paul Erdos RIP. All bow.

What about modulus arithmetic? A simple example is the clock; contrary to the joke about 13 o'clock being the time to buy a new clock, 13 o'clock is 1 o'clock. Time is infinite is the received wisdom and we tamed that beast using modulus arithmetic. Ouroboros/Sisyphusean arithmetic could be another name for it. In Hindy mythology, if movies are to be believed, the wheel (finite) is the symbol of time (infinite).

$0 + 0 = 0$

$\infty + \infty = \infty$

$\infty$, rather $0$ish!

Supertask (James F. Thomson of Thomson's lamp fame); What o'clock is it after $\infty$ time has elapsed?

Yes, we're a wicked bunch intent on the corruption of the intellects of youth in order to bring them to the alter of our Satan, Paul Erdos RIP. All bow. — jgill

Not all mathematicians are the same, just like not all theists are the same. It's just that some are fanatics with a cultlike attitude, who are inclined toward professing absurd ontologies to support their beliefs.

I had to look up Paul Erdos, to see that he is famous for his work on Ramsey theory. Seems like Erdos was very socially active. Is he responsible for the famous notion "six degrees of separation"? Or was he just paranoid about aliens? I see you can still earn money by solving Erdos' problems. Have you ever managed to get any reward?

I gotcha!

$\frac{5}{5}$=1

and

$\frac{8}{8}$=1.

What about $\frac{0}{0}$ or $\frac{\infty}{\infty}$ ? Gotta be 1, right?

0 and $\infty$, rather 1ish!

A little advice, friend : get ya some math learnin'

What surprises me about our math-phobic friends on TPF, is that philosophy majors usually love the esoteric. You would think they would revel in knowing more about mathematics than the Great Unwashed. Instead, they make up their own notions and denigrate 5,000 years of developments by some of the greatest minds that have ever lived. Weird.

Danke Real Gone Cat. I do need to take some good quality math lessons. I've got some books on my reading list, but alas, time is not on my side mon ami, time is not on my side.

I like the patterns you found. They're natural extensions of numerical operations on numbers and they are kinda sorta right. Kudos to you for that.

P. S. Does that mean 0 and $\infty$ aren't numbers like 1, 3, 2938, 10¹⁰⁰?

Does that mean 0 and ∞ aren't numbers — Agent Smith

0 is a number.

I don't understand why you keep skipping my point that using the leminscate as if it stands for an object makes no sense the way you do.

Unless you mean the point of infinity in the extended reals, though I don't think you are even that specific. But even as a member of the extended reals, the point of infinity is not necessarily itself an infinite set. It can be any object that is not a real number. Then we define an ordering using the standard ordering on the reals but setting the point of infinity as greater than all reals.

:up:

Means "Whatever, dude", eh?

Agreed that many of our TPF worthies misunderstand what is meant by $\infty$. But as a minor defense, we often do get sloppy with our use of this symbol. In calculus classes, for example, when a limit evaluates as $\frac{\infty}{\infty}$, we tell students to try applying L'Hopital's rule. I doubt AS has this usage in mind.

It's a conceptual issue. He seems to think that, because mathematics is infinitistic, it has a thing that is called 'infinity'. As if the leminscate stands for that thing like the golden arches stand for a hamburger place.

I had to look up Paul Erdos, to see that he is famous for his work on Ramsey theory. Seems like Erdos was very socially active. Is he responsible for the famous notion "six degrees of separation"? Or was he just paranoid about aliens? I see you can still earn money by solving Erdos' problems. Have you ever managed to get any reward? — Metaphysician Undercover

I don't know about the six degrees or aliens. I met him once and talked with him briefly at a meeting in Hungary years ago, but I am not in that select group of mathematicians who have "Erdos numbers". He stayed with my advisor for a week long ago as he traveled around the world, giving talks and working with colleagues. Universities supported him on his visits to their campuses. He lived out of a suitcase and shopping bag and, while his mother was alive, stayed at her apartment in Budapest on and off.

Their is a curious parallel in the world of climbing. Fred Beckey, who died in 2017, was the most prolific climber in American history. Like Erdos he lived a vagabond life, sleeping on the sofa of whoever he was visiting. I knew him slightly and we bouldered together occasionally. I suppose there could be something like a "Beckey number" earned by doing notable climbs with him. :cool:

As if the leminscate stands for that thing like the golden arches stand for a hamburger stand. — TonesInDeepFreeze

:lol:

What surprises me about our math-phobic friends on TPF, is that philosophy majors usually love the esoteric. You would think they would revel in knowing more about mathematics than the Great Unwashed. — Real Gone Cat

Yes, they certainly like to talk about what Kant meant or how Aristotle would know, but math is more precise, with less room to wiggle. :nerd:

I think what is missing from this discussion is context. Given a context, there are of course numbers which you can consider effectively infinite: c for velocities, the age of the earth for history, and so on. But there is no such thing as an effective infinity free of context. This is especially clear when you consider that when you are talking about the real big boys, Graham's Number for instance, you need their context to even mention them. Without the context of their formulation, they are inconceivable.

Another interesting discussion topic (and perhaps what the OP was alluding to) would be the distinction between pure math and applied math. And the usefulness of the former.

And now in support of the much-maligned OP - a quote from Marc Rayman, a chief engineer at NASA (with the coolest name any employee of NASA ever had) :

Let's go to the largest size there is: the visible universe. The radius of the universe is about 46 billion light years. Now let me ask a different question: How many digits of pi would we need to calculate the circumference of a circle with a radius of 46 billion light years to an accuracy equal to the diameter of a hydrogen atom (the simplest atom)? The answer is that you would need 39 or 40 decimal places. If you think about how fantastically vast the universe is — truly far beyond what we can conceive, and certainly far, far, far beyond what you can see with your eyes even on the darkest, most beautiful, star-filled night — and think about how incredibly tiny a single atom is, you can see that we would not need to use many digits of pi to cover the entire range.

OP — Real Gone Cat

He's "The Crank With The Friendly Face".

If we agree that there's a largest number we'd ever need to use, then still, what's the harm in having all the rest of the larger numbers in the attic, just in case we ever feel like looking at some larger ones, even if only to play around with them?

I wonder whether the debate lines up with certain kinds of personalities.

One personality type just cannot stand that there are infinite sets. It terribly rankles them that we would allow such a sweeping abstraction. It goes against their way of looking at mathematics as expression of our experiences as a stream of immediate and concrete based perceptions.

The other personality type just cannot stand limiting mathematics to particulars. It bugs them that we would cut off the numbers at some arbitrary point (or a point merely estimated by current cosmological theories) in the succession of numbers. It goes against their way of looking at mathematics as being of greatest abstract generality.

For me, ultrafinitism is ugly that way. I'd rather study mathematics that is not embroiled with a bunch of messy physical stuff like how small an atom is and what is it's rounded-off length, given in a bunch of base ten digits. It's ugly to me to say, "The greatest number is 492^(3327989025)" or whatever. It's so ... choppy.

Oh, I'm no finitist. As I've pointed out, finitism (or worse, ultrafinitism) leads to some odd results : you have to truncate $\pi$ (which turns circles into polygons), you have to deny irrationals, you destroy the foundations for calculus, lines no longer consist of an uncountably infinite set of points, etc.

You may be onto something about personalities. I think it's also possible that one of the sources for the OP may have been some half-remembered quote such as the one from the NASA engineer, above. But perhaps I'm being too generous.

NASA telling me what's the largest number I can use is like the Department Of Agriculture telling me how many taste buds I can have.

As I've pointed out, finitism (or worse, ultrafinitism) leads to some odd results : you have to truncate ππ (which turns circles into polygons), you have to deny irrationals, you destroy the foundations for calculus, lines no longer consist of an uncountably infinite set of points, etc. — Real Gone Cat

That's a list of features, not bugs. It's long past time to drive the Platonists out of maths. Pragmatism makes for a sounder metaphysics when it comes to how one would model reality. :smile:

I don't believe that infinitistic mathematics requires a platonist commitment.

You might believe it, but can you supply a formal proof of that claim?

Of course not. It's not a formal claim.

Anyway, I think the burden of argument is on the side saying that infinitistic mathematics does require a platonist commitment. There are powerful arguments for that side. I very much respect that. But there are two different questions:

(1) Can one have a cogent philosophy of mathematics with infinite sets but that is not platonist? I don't have a firm opinion on that.

(2) Can one work in ZFC without committing oneself to platonism? That's more an empirical question. Such mathematicians as Abraham Robinson do*. And Robinson's own explanation might be pretty good. (but If I recall, I found it not to be entirely glitch-free). For myself, even though I am not a mathematician, I happily study ZFC without having the platonist commitment that the abstract objects of mathematics exist independent of mind.

* A fair number of set theorists do, as does Robinson, say that the notion itself (not even just the existence) of infinite sets is literally nonsense, yet they work in ZFC and recognize its fruitfulness.

Of course not. It's not even a formal claim. — TonesInDeepFreeze

I was teasing. If you gatecrash a comment, you could at least have the courtesy to set out your reasons for your assertions.

For myself, even though I am not a mathematician, I happily study ZFC without having the platonist commitment that the abstract objects of mathematics exist independent of mind. — TonesInDeepFreeze

So you reveal yourself as a pragmatist. Infinity is a useful idea as far as it goes in the real world of doing things – like deciding whether some circle is in fact a very fine construction of flat lines, or a polygon is in fact a very fine construction of flattish curves.

Truncating pi is practical. One can be a finitist and it looks exactly the same as being an infiinitist. Outside the culture wars of philosophy of maths, who could tell the difference? It becomes a difference that doesn't make a difference.

Of course not. It's not even a formal claim.
— TonesInDeepFreeze

I was teasing. — apokrisis

Okay, you were joking with the 'formal' part. Maybe because you perceive me as asking posters to back up with formal proofs? Or you think I can be charactured that way? I don't know. Anyway, of course I don't ask people to provide formal proofs of informal assertions.

If you gatecrash a comment, you could at least have the courtesy to set out your reasons for your assertions. — apokrisis

Oh come on. I didn't "gatecrash" anything. You posted essentially a one-liner on the subject, itself not an argument. That's fine. And it should be allowed that one may reply in kind. And even if a poster replies tersely to a longer argument, that's not "gatecrashing" or necessarily even rude or whatever.

So you reveal yourself as a pragmatist. — apokrisis

I don't have a philosophy of mathematics; and not one that could be called anything, including 'pragmatism'.

Infinity is a useful idea as far as it goes in the real world of doing things — apokrisis

I am sympathetic to that idea. But I don't personally stake my own understanding of infinitistic mathematics primarily to it.

Truncating pi is practical. — apokrisis

Of course no one expects engineers to write an infinite sequence of digits.

Oh come on. I didn't "gatecrash" anything. You posted essentially a one-liner on the subject, itself not an argument. That's fine. And it should be allowed that one may reply in kind. And even if a poster replies tersely to a longer argument, that's not "gatecrashing" or necessarily even rude or whatever. — TonesInDeepFreeze

Oh come on. You yourself have said you have no philosophy to defend on this forum, just a self-appointed need to police it for its mathematical thoughtcrimes and disinformation campaigns.