The resolution of singularities is in part due to the precedence of them turning out to be the result of mistakes in our models. — MindForged

Singularities are nasty beasts, and there's a better reason for eschewing them than past experience: singularities blow up your model in the same way that division by zero does (division by zero is one instance of singularity); they produce logical contradictions.

Of course, singularities are not the only sort of infinities that we deal with. As you said, if we use modern mathematical apparatus, then it is exceptionally hard to get rid of all infinities. A few have tried and keep trying, but it's a quixotic battle.

As for the objection "it's just math, it's not real," then my next question is: what is real? Where and why do you draw the boundary between your conceptual mapping of the world and what you think the world really is? Is there even any sense in drawing such a distinction? Are three apples really three, or just mathematically three? If they are not really three, then what are they really?

Are you aware that denying the actual infinite involves committing to one or the other of the following two propositions?

1. If we travelled far enough through the universe in a straight line we'd end up back where we started

2. The universe has a boundary. In that case, as Aristotle asked, what happens if we go to the boundary and poke a spear through it? — andrewk

It’s a problem I agree but I can think of a way past 2 above: imagine as you get closer to the edge of the universe time slows down and right at the edge time stops. So it’s impossible to poke a spear through the edge of the universe because there is no space time in which to poke the spear.

But whatever mathematical formalism is used, they also make recourse to infinity. — MindForged

That’s not correct, they make recourse to the limit concept which is not the same as actual infinity.

You either think it's true or not — MindForged

I believe and so I thought did everyone that relativity is a close approximation only of the large scale universe. The plank length is very small so reality is approximately continuous hence the theory works so well.

The natural numbers can be put into a one-to-one correspondence with a proper subset of itself. That makes it infinite. — MindForged

But numbers just exist in our mind and our minds have finite capacity so numbers are finite in that sense.

Imagine a spaceship that has an accurate odometer. We set the spaceship in motion and it travels across the universe. The reading on the odometer will always be finite no matter how far the ship goes. — frank

I like that story. Here is something similar for time:

- Imagine an eternal being in eternal time
- You notice he’s counting. You ask how long and he says ‘I’ve been counting always’
- What number is he on?

The being could be on any integer if they have counted all the negative integers up to the one they are now on.

The usual objection to that is to ask - 'but what number did they start on?' to which the answer is 'they didn't start'.

The fact that such things are hard to imagine is no reason to suppose that they could not be the case. If there is such a thing as 'the way the world really is' I very much doubt it is something that could make any sense to we cognitively-limited beings.

The usual objection to that is to ask - 'but what number did they start on?' to which the answer is 'they didn't start'. — andrewk

Yes; so an object with no start is a non-existent object; IE infinite time is impossible. Same argument for infinite space.

The only thing I can think of without a start is the counting numbers from negative infinity to zero. But they just exist in our heads they don’t correspond to anything real.

(I'm not well versed in the mathematical language, so pardon me if my confusion is born out of a misinterpretation) - So far I see a lot of mixed metaphors when it comes to showing the distinction between a mathematical model and the actual world, and the best I can understand is that there is a different connotation for infinity when it applies to the mathematical model (theoretical) compared to infinity in the actual world.
I'm just wondering, if the theoretical and actual worlds do not have common points of analogy, then nothing in one would relate to the other. Therefore, what if you picked one scale of magnitude with respect to relateable points in both representations (theoretical and actual worlds) and then compare how infinity is perceived in both within that identical scale? Does this make sense?

2. The universe has a boundary. In that case, as Aristotle asked, what happens if we go to the boundary and poke a spear through it? — andrewk

What's wrong with the idea that the universe has a boundary? That seems to be a natural and intuitive idea, the universe being a thing, and things have boundaries. You wouldn't be able to poke a spear through it though because that would be to violate the boundary, put a part of the universe beyond its own boundary. That's impossible.

The problem though is the nature of the boundary. Boundaries are not what they seem to be. We sense boundaries, see them for example, as the edge of objects. But in reality objects overlap through things described by fields, like gravity. So the boundary which is seen as the boundary of spatial extension, is not the true boundary because things really extend beyond this apparent boundary.

We find a true boundary in the nature of time, as the boundary between past and future. This is the boundary of physical existence. What you do at the present has everlasting existence in the past, as what you have done. So you may stir the pot of the past, your actions having influence on what has occurred, but you cannot poke your spear into the future as that is impossible.

I'm just wondering, if the theoretical and actual worlds do not have common points of analogy, then nothing in one would relate to the other. — BrianW

I agree. The best language clarification I know is Aristotle‘s: https://en.m.wikipedia.org/wiki/Actual_infinity

Relating this to maths, a potential infinity is most close to the limit concept. Actual infinity occurs in set theory.

We are indeed lacking analogy between maths and reality. I can think of no actual example of real world Actual Infinity. On the maths side we have very little also. Set theory basically uses an axiom equivalent to ‘actual infinity exists’ (check here if you are brave https://en.m.wikipedia.org/wiki/Axiom_of_infinity)

So mapping two non-defined concepts is hard... sort of the point of this thread... Actual Infinity is nonsense...

Thanks :up:
It actually makes it easier to follow the thread.

- Imagine an eternal being in eternal time
- You notice he’s counting. You ask how long and he says ‘I’ve been counting always’
- What number is he on? — Devans99

That's paradoxical. But I don't see how finitism solves any paradoxes. Do you think it does?

Yes the paradox is due to the use of actual infinity in a logical argument.

Imagine a timeless being existing permanently; he would be finite but permanent in the sense he is outside time. That solves the paradox as far as ‘god’ goes; a timeless, permanent, finite being.

That’s not correct, they make recourse to the limit concept which is not the same as actual infinity. — Devans99

No no no, calculus makes use of multiple legitimate infinities, namely having the reals be larger the naturals. This is absolutely indisputable.

I believe and so I thought did everyone that relativity is a close approximation only of the large scale universe. The plank length is very small so reality is approximately continuous hence the theory works so well. — Devans99

If you accept relativity as pretty close to the truth you necessarily must accept that space is infinitely divisible (basically true in quantum mechanics too). Hell, a large chunk of quantum mechanical interpretations are relativistic as well so I don't even see the objection here.

But numbers just exist in our mind and our minds have finite capacity so numbers are finite in that sense. — Devans99

Numbers do not just exist in the mind, that's silly. I mean, your argument can easily be inverted. If the mind is finite and mathematics requires infinity (it does), then mathematics can't be dependent on the mind.

Singularities are nasty beasts, and there's a better reason for eschewing them than past experience: singularities blow up your model in the same way that division by zero does (division by zero is one instance of singularity); they produce logical contradictions. — SophistiCat

It's a fair point. I would just like to point out the leeway I gave myself. I said the elimination of singularities was part of the reason, not all of it. ;)

Yes; so an object with no start is a non-existent object; IE infinite time is impossible. Same argument for infinite space. — Devans99

I guess we're just supposed to take it as a given that infinite time is impossible, yeah? There's no contradiction in postulating an infinite series of moments, even into the past. It can even be given a simple description sans-contradiction:

For every moment before time "t" there is another moment.

We can even get simpler by just pointing out the infinite divisibility of time:

Between any two moments of time there's another moment.

Those are not contradictory, so how is it impossible?

No no no, calculus makes use of multiple legitimate infinities — MindForged

Yes I was discussing this briefly with a mathematician. The foundations of calculus do you indeed make use of actual infinity as defined in set theory. But set theory merely states that the actually infinite exists as an axiom; it does not prove anything. So the foundations of calculus rest on rather shaky ground.

It's an axiom because no one has found any contradiction that is provable from it. The definition of infinity is pretty clear, it's extremely useful in mathematics and science, and it introduces no contradictions into the theorems. If that makes a "shaky ground" I have no idea what use your standards of a good foundation is to anyone.

That's paradoxical. — frank

Not paradoxical, just undefined. Let's tweak the story:

- Imagine Donald Trump
- You notice he’s counting (you can tell because he is muttering and holding up his fingers). You ask how long and he says ‘I’ve been counting for ten minutes’
- What number is he on?

So put this way, this is a pretty dumb counterexample, but there are actually many puzzles involving infinities where you might think there ought to be a definite answer, but there isn't, such as Thomson's Lamp for example. There are also genuine paradoxes, where an imaginary setup that seems like it ought to be possible, in principle, leads to contradictions. But in each of these cases you have an option to reconsider your starting assumptions: Are you sure that there must be a unique answer? How do you know? Are you sure the setup itself is coherent? How do you know?

For every moment before time "t" there is another moment. — MindForged

The above refers to future which is potentially infinite which is not the subject of this thread.

Past infinite time is however an Actual Infinity so is disallowed. For example this argument:

- Time is a series of moments
- The moments so far must be an actual number not infinity
- So time has a start

The above refers to future which is potentially infinite which is not the subject of this thread. — Devans99

Did you miss the word "before"? That was talking about a past series of infinite moments.

Past infinite time is however an Actual Infinity so is disallowed. For example this argument:

- Time is a series of moments
- The moments so far must be an actual number not infinity
- So time has a start — Devans99

Premise 2 is the obviously question begging premise. Nothing about the concept of "moments of time" precludes an infinite past series of moments. I repeat:

For every moment before this very moment, there is another moment.

That might well be false in the universe we are in (it looks like it has a first moment of time), but the sentence entails no contradictions unless infinity is a contradictory concept; it isn't, ergo there are no inconsistencies.

For every moment before this very moment, there is another moment. — MindForged

I believe that leads to contradictions. For example, how could we ever reach today if the past stretches to negative infinity (no matter many moments you add to negative infinity you still get negative infinity).

There's no force to this objection. We're not at an end point. Ignoring our actual universe, if the past were infinite then we would just be at an arbitrary part of the sequence of time. I don't see the issue of "reaching today" any more than it's a problem that I can reach 100 despite there being an infinite number of real numbers (decimal numbers) between 0 and 1 alone.

No, numbers are a figment of our minds so can be actually infinite. Time however is part of the physical universe so it can’t be actually infinite.

Numbers aren't part of the mind. And besides which, you're contradicting yourself. Previously you said that the mind was finite. But you're saying numbers can be infinite, yet you also said numbers are figments of our minds (which are finite). That's just inconsistent. Anyway, you haven't given an argument for why time can be infinite, you're just presupposing it.

Numbers aren't part of the mind — MindForged

So you are a Pythagorean? All is number? But you believe in actual infinity too? So that means you believe the physical world is actually infinite?

Give me an example of the actually infinite from nature.

I'm not a Pythagorean, that's just a silly response. Even if I were a mathematical platonist, nothing about that in particular makes me think some aspects of reality might be infinite. Besides, I've given you one example already: Space is infinitely divisible. This is born out in Relativity and in relativistic quantum mechanics (probably in non-relativistic QM too, but I've only doen classical mechanics) , and there are even good strictly logical arguments for the infinite divisibility of space. That aside, my constant objection has been that you haven't given a single argument against the possibility of actual infinities that was question begging. Look:

I believe that leads to contradictions. For example, how could we ever reach today if the past stretches to negative infinity — Devans99

Not a contradiction.

Time however is part of the physical universe so it can’t be actually infinite. — Devans99

Assuming the thing being discussed, namely, that something in the world could be infinite.

That's all you've really done so far, these aren't serious arguments IMO.

Space is infinitely divisible — MindForged

If it is, it’s a potential infinity rather than an actual Infinity (you do understand the distinction?).

The division of space takes time, first we must cut one inch, then 1/2 an inch, then 1/4... No matter how many cuts we make we never get to actual infinity, just some small number.

If it is, it’s a potential infinity rather than an actual Infinity (you do understand the distinction?).

The division of space takes time, first we must cut one inch, then 1/2 an inch, then 1/4... No matter how many cuts we make we never get to actual infinity, just some small number. — Devans99

I understand the distinction, you do not understand the point. I'm not talking about the temporal process of looking at ever smaller slices of space. I'm saying that the nature of space itself is such that it is infinitely divisible already; for any two points in space there are in actuality points in between them. You'll never reach a base unit of space because no such thing exists, it's a continuum.

I'm not exactly sure what you mean by an actual infinite, but I can definitely state that the idea of the infinite does not hinder science. After all, if the infinite did not exist, then is there really a largest natural number? Also, many equations cannot be solved directly and setting up an infinite series is a way to do so. In fact, the use of infinite sequences and series are used throughout science and engineering.

I'm saying that the nature of space itself is such that it is infinitely divisible already. — MindForged

But it’s impossible to construct a smallest possible distance (1/infinity) - we can merely construct successfully smaller distances in a process that tends to but never reaches 1/infinity. That’s the definition of potentially infinite. I asked for an example from nature that is actually infinite...