Since when is "mathematical induction" a physical law? (vide Hume, Popper, et al) — 180 Proof

Causality is a physical law (at least at macroscopic level and we are dealing with a macroscopic question here). I referenced mathematical induction merely to demonstrate how that physical law requires a first cause.

Lost me. :roll: I can't decide - post hoc fallacy? compositional fallacy? hasty generalization fallacy? (re: problem of induction, etc) — 180 Proof

Not sure what you mean. There are about 20 physical constants that must be at or near current values for the spacetime to support life. That seems to imply something external to spacetime created spacetime with specific characteristics so that it would support life.

You simply refuse to acknowledge the definitions of terms that others are employing, and thus consistently (and persistently) attack straw men. Actual impossibility does not entail logical impossibility. Mathematical existence is not metaphysical actuality. The infinity of the natural numbers is potential, not actual. Continuity of space does not require an actual infinity of distinct positions. — aletheist

But logical impossibility, which actual infinity is (as demonstrated by the paradoxes/contradictions) does imply actual impossibility.

The set of natural is defined in maths as an actual infinity:

'The axiom of Zermelo-Fraenkel set theory which asserts the existence of a set containing all the natural numbers' - http://mathworld.wolfram.com/AxiomofInfinity.html

A set containing all natural numbers exists both logically and actually... but that leads to logical contradictions... so such a set is not logically possible.

Outside of discursive argumentation, what's wrong with "an infinite regress"? What physical law, or condition, precludes it? — 180 Proof

Mathematical induction precludes it: Assume there is no first cause. If there is no nth cause then there is no nth+1 cause. Then there is nothing.

Why multiply entities unnecessarily (vide Ockham)? Suppose time is "outside of time"? Suppose causality is "beyond causality"? On what grounds should we - do you, D99 - assume otherwise? — 180 Proof

The universe appears to be fine tuned. So there seems to be a need for a fine tuner. Imagining the whole of (the fine tuned) spacetime to exist eternally provides no answer to how it was fine tuned. The assumption of a timeless first cause that caused spacetime works better.

"Something" is either formal or factual. Formal, or abstract, denotes absence of causal relations (i.e. cannot create). Factual, or physical, presupposes (space)time; claiming (it) "created time" merely begs the question, and invites the sort of "infinite regress" the OP seeks "to avoid". — 180 Proof

It is possible that something of substance could exist yet it be not of / beyond spacetime. Such a being would be able to interact with matter to create the universe. I admit I am not sure how such a being could work but it seems a logical requirement.

An atemporal, "eternal" cause of a universe that has a definite age (like 14 billion years) is incompatible with the principle of sufficient reason, since such a cause lead us to expect an infinite age of the universe — there's no sufficient reason that the universe is 14 billion years old and not some other age, any other age in fact. — jorndoe

Interesting point. But the universe must have some age and it cannot be infinite, so why not 14 billion years? If you believe in 4d spacetime (as I do) then spacetime in its whole entirety has some form of eternal existence, we just happen to experience the part of spacetime that is 14 billion years since the BB.

Something strangely "atemporal" would be inert and lifeless. — jorndoe

It seems a logical requirement that such a thing exists and is causally efficacious. I acknowledge I am not sure how it could work though (as are others who have considered the problem down the ages).

I believe it is possible that a mapping between each point in spacetime and the timeless thing could exist so that the timeless thing could express itself within spacetime. As to how it could tie its own shoelaces, I have no answer.

All of these are logically possible, just not metaphysically possible. — aletheist

They are not logically possible as you can see from the argument in the OP - assuming that they are logically possible leads to a contradiction. Or if you don't like that argument, see:

https://en.wikipedia.org/wiki/Ross–Littlewood_paradox

So the logical assumption of the existence actual infinity leads to paradoxes/contradictions. Paradoxes/ Contradictions indicate a logical error has been made, in this case the assumption that actual infinity is a logical concept.

No one is claiming otherwise. When mathematicians state that the natural numbers "exist," they are not thereby calling them an actual infinity, only a potential infinity. — aletheist

A potential infinity is like a limit - something approaches but never actually reaches that limit. Actual infinity is equivalent to the claim that the natural numbers exist - the axiom of infinity says they actually exist - not potentially. That leads to the conclusion that there is a set that exists with a greater than any number of elements.

Incorrect--it is logically possible, just not metaphysically possible. — aletheist

It is not logically possible to reach the end of something that has no end.

This indicates a confusion between existence in mathematics and actuality in metaphysics. They are not synonymous or equivalent. Everything that "exists" in mathematics is merely logically possible, not actual. — aletheist

An actual infinity of naturals (IE a set with a greater than any number of elements) is impossible.

This indicates a confusion between logical possibility and metaphysical possibility. Again, they are not synonymous or equivalent. It is logically possible to choose balls from an infinite number of bags, even though it is not metaphysically possible; i.e., it is actually impossible. — aletheist

It is not logically possible to complete a task that has no end.

I know you saw this, from fishfry in the bijection thread: — tim wood

You are missing the point I'm making. I believe that the naturals and reals are purely mental constructs. They exist in our minds only (where the impossible is possible). They have the same status as talking trees and levitation - illogical/impossible things can exist in our minds but they cannot exist in reality.

In the instances of the axiom of infinity and axiom of choice, maths departs radically from reality and that departure leads other folks astray (physicists, cosmologists, philosophers). That is why I raised this thread.

If you disagree, then please give an example of something that has the structure of the naturals from nature.

Why are you on about this? Everyone agrees with you and no one disagrees. As was said in my parochial school to the girls, do you want a medal or a chest to pin it on? — tim wood

I'm glad we are in agreement. I think though that not everyone agrees with us. For example, the bedrock of maths:

1. Axiom of infinity. It claims that the set of natural numbers exist. They exist in our minds where the impossible is possible, but there is nothing like it in reality, so maths should not claim 'they exist'.

2. Axiom of choice. It claims it is possible to choose balls from an infinite number of bags. In reality, one cannot complete an infinite task, so it is impossible to make the infinite selection of balls. Hence maths should not claim it is possible.

I believe that the universe is both finite and discrete (=free of actual infinities). Plenty of people disagree so I think there is still a discussion to be had.

Every video and article I read about one-to-one-correspondence is garbage. They arbitrarily move infinity, place the first units together, send them off into infinity (without proving anything yet about uncountable vs countable) and exclaim "they are the same!". Nevermind ALL infinities are composed of units and you can do this trick with ALL infinities — Gregory

As explained above, you can use mathematical induction to see that all the bananas in each sequence are in one-to-one correspondence. So for all finite numbers n, the bananas are in one-to-one correspondence. At 'actual infinity' are the bananas in one-to-one correspondence? I think not because actual infinity is not a well defined concept. It is an illogical concept that cannot occur in reality IMO.

And while you're at it, please provide an epistemological ground for your "doubt." — tim wood

If space is continuous then my hand moves through an actually infinite number of intermediate positions. But actual infinite leads to contradictions. So I doubt that it can exist.

Also there is an information based argument. True continua would be structurally identical no matter what the size. So a millimetre of space would have the same structure as a light year. Suggesting the same information content. That is hard to swallow.

The materialist option says that time started with the first motion. The first cause was gravity in the first motion. — Gregory

But what was the cause of time starting? It seems it cannot be gravity as gravity requires time to express itself? Perpetual motion is impossible so there must have been a start of motion. What caused a start of motion? Call it X. What caused X. Say Y. What caused Y. Say Z. So we are in an infinite causal regress - which requires something from beyond time/causality to start the whole sequence.

You need to give up the the Newtonian idea of time — Gregory

I believe in spacetime. It is finite and it has a definite shape. So it can be said to have a start as all definitely shaped objects have identifiable start points.

Whatever QM says, I feel that in the macroscopic world, 'everything in time has a cause' still holds. The origin of things involves huge amounts of matter/energy so is a macroscopic question, so I'm inclined to regard the (possible) acausal nature of matter/energy at a microscopic level as not relevant.

It takes say 1 second for my hand to pass through 1 meter of space and thereby pass through, if space is a continuum, an actual infinity of intermediate points. This is what is referred to as a supertask. I believe all supertasks are impossible - they lead to paradoxes such as:

https://en.wikipedia.org/wiki/Ross–Littlewood_paradox
https://en.wikipedia.org/wiki/Thomson%27s_lamp

Paradoxes indicate we have a wrong assumption somewhere, in this case, the assumption that it is possible to complete an actually infinite number of steps in a finite time is suspect. So I doubt that true continuity is possible.

I don't think we will reach agreement on this point.

So the boundary is other universes. Time is relative change. Change in each universe is relative to the other, so time and space would encompass the multiverse, not just one universe. Then, your infinite collection would be universes. — Harry Hindu

Beyond the boundary is nothingness IMO. Nothing cannot be actually infinite because it is nothing. If it is other universes then they cannot be actually infinite because it would lead to the absurdities referenced in the OP. Or see here for another example of the absurdity of actual infinity:

https://en.wikipedia.org/wiki/Ross–Littlewood_paradox

No, continuous motion is the reality, while discrete positions are creations of thought to facilitate describing the motion. — aletheist

If my hand moves from position 0 to 1, it is guaranteed to pass through positions 0.5, 0.25, 0.125, etc... Or are you saying it somehow skips over intermediate positions? That would be discrete movement.

No, treating space as continuous does not require an actual infinity of positions, only a potential infinity of positions. — aletheist

But movement is something that actually happened in the past - my hand in the past moved through all possible positions - so that must be an actual infinity of positions.

Positions 0 and 1 do not exist unless and until we arbitrarily mark them as such, and the same is true of any and all intermediate positions between them. — aletheist

But the action of movement does mark positions 0 and 1 and all positions in-between. We know that our hand actually passed through all those positions in the past, so if space is a continuum then motion actualises an infinity of positions.

As pointed out in the OP, actual infinity is absurd, so therefore space is not a continuum.

Are you saying that there is something beyond space and time? You're implying a boundary, but a boundary with what? Space and time are infinite. Energy/matter isn't. — Harry Hindu

I think that there maybe pure nothing beyond spacetime - no time at all so nothing can be in any way whatsoever.

Time probably started 14 billion years ago. All expanding universes have a start in time and our universe has always been expanding.

Space has been expanding at a finite rate for 14 billion years so it must be finite in size.

No, it would not. If space is truly continuous, then it is not composed of distinct positions. We arbitrarily impose distinct positions on space for various purposes, including measurement. They are entia rationis, creations of the mind, not constituents of reality itself. — aletheist

I don't understand you. If my hand passes from position 0 to 1, then it passes through positions 1/2, 1/4, 1/8, 1/16, ... 1/∞. So movement in a continuum actualises infinity, which is why I think everything must be discrete.

I bow to your maths knowledge, but I'd still maintain that there is nothing in our universe to which the axiom 'when it is changed, it does not change' applies. Maths may produce alternative realities in which such an axiom may hold but these realities are not our reality.

It is very convenient to regard the naturals, the reals as actually infinite sets, but this is merely a mental convenience to allow us to reason with all the naturals/reals. It does not mean that anything with the structure of the naturals/reals can exist in reality - they are a purely mental construct.

We can imagine all sorts of things in the mind - levitation, talking trees, actual infinity - but only a subset of what we can imagine is possible in reality.

I have a similar reaction to the axiom of choice. How can balls be selected from actually infinite bins? It is not possible to complete such a selection - the selection process goes on forever so it is impossible to complete. So again, we have something that is impossible in reality, but we can imagine it in our minds where the impossible is possible.

Except that you have refused to acknowledge that I explicitly made clear I was not asking for what anyone merely believed. What, may I ask, is your problem? — tim wood

I think you are splitting hairs between 'believe' and 'aver'. Cantor states he has 'firm conviction' in the belief of the existence of the actually infinite. Aristotle believed/averred time had no start because:

- Time had no start because for any time, we can imagine an earlier time.
- Time had no start because everything in the world has a prior cause.

Newton held this belief also and many people still hold this belief. And such a belief leads to a belief in actual infinity. Similarly, the commonly held belief in space/time being a continuum again leads to a belief in actual infinity.

OK. But the problem remains: removing half the elements from the second sequence results in a sequence that is identical to the first sequence. We have changed something and it has not changed. I am not a mathematician as you can tell but to me the maths seems to lead to a straight forward absurdity, so there must be something wrong with our assumptions. I feel the problem is that we have assumed that actual infinity can exist. If we remove actual infinity and conduct the same operations on finite sequences, the absurdity does not occur. Hence my assertion that actual infinity is an illogical concept that can only occur in our minds and is not a feature of the real world. Hence the axiom of infinity should not claim that 'there exists' an actually infinite set - because such a set does not exist in reality.

I have already given you two clear demonstrations that some people believe in actually infinite collections:

- Infinite past time (leading to a belief that an actual infinity of moments has occurred in the past)
- Space is a continuum (leading to a belief that motion actualise an infinity of intermediate positions)

Not sure what else I can say here :sad:.

Do you understand the word
"aver"? Do you understand "theory? Do you understand "assumption"? When did Hawking became a "fringe" cosmologist? And Cantor apparently invoked God: are you prepared to demonstrate what exactly Cantor meant by the term in terms of any reification of his own ideas? — tim wood

Lets look at what Cantor said again:

"Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. - Georg Cantor

I've highlighted the relevant piece - he is talking about actual infinities in our universe.

I'll try a different form of the question: are you aware of any respectable demonstration that there is any actual infinity of anything? — tim wood

Yes, if you believe space is a continuum (which many people do), then simply moving your hand from left to right actualises an infinity of intermediate positions.

They don't imply; you infer. They hold, if you're referring to a laymen's Hawking, that time is unbounded, like the surface of a sphere, & etc. — tim wood

You are just choosing one (fringe) theory of cosmology; many theories do assume past time is actually infinite. See CCC by Roger Penrose as an example.

Where did any sane person aver that there were any actual infinities of anything? — tim wood

I already gave you an example (Cantor). You need look no further than the philosophy forum for more examples: many folks on here argue that time has no start or that space is a continua, see https://thephilosophyforum.com/discussion/comment/367423 for example - and my response https://thephilosophyforum.com/discussion/comment/367425.

By contrast, I would argue that true continuity is a top-down construction in which the whole is ontologically prior to the parts. Positions are only actual if and when they are marked for a purpose, such as measurement; otherwise, they are strictly potential. — aletheist

Assuming for the sake of argument space is a continuum (which I do not believe), when I move my hand from left to right, it would pass through an actual infinity of distinct positions. Every intermediate position is therefore actualised by motion and we are left with the conclusion that the particles in my hand passed through an actual infinity of intermediate positions.

I believe instead that each particle in my hand performs something like a quantum jump down at a microscopic level and that there are no true continua in our universe.

So you mean they're identical collections, but their elements aren't equal? :S — fdrake

You are confusing me. Every element in both sets is indistinguishable and they are lined up with each other (in one-to-one correspondence):

{b, b, b, b, ... }
{b, b, b, b, ... }

We know all elements are identical 'b's and the collections both have the same infinite cardinality. The first elements are in one-to-one correspondence. If the nth element is in one-to-one correspondence, then so is the nth+1 element. So the two collections are surely equal?

I use the term 'collection' because I'm not a mathematician I'm afraid - I mean a group of (in this case) identical objects. I'm not familiar with multisets but it sounds like the same concept as I'm thinking of, IE:

{b, b, b, b, ... }

Is a multiset where 'b' has multiplicity=∞ I guess?

I called the groups of bananas 'collections' rather than 'sets' to get around the issue of sets having to be composed of distinct objects. Hope that is clear. Sorry for any confusion caused by the use of '{' and '}'.

Cantor has had a huge impact on the way we regard actual infinity. It is due to his work that actual infinity has gained legitimacy in maths and that legitimacy has spread to other fields like physics, cosmology and philosophy - to the detriment of those fields IMO.

A little more rigour, please. First, what "some folks believe" is no standard for anything (than perhaps that some folks may believe anything). — tim wood

For example, many cosmologists hold that time has no start, implying an actual infinity of past time. All those past moments actually happened, so it would count as an instance of actual infinity, regardless whether you hold a presentist or eternalist viewpoint.

Then there is the axiom of infinity. It states that there exists a set with an actually infinite number of members (the natural numbers). As you have pointed out, no such set exists (outside our minds where the impossible is possible).

Back to the line segment. It's just a line. Is there "an actual" infinity of points? Depends on your purposes and definitions - but then you're beyond what it is. — tim wood

I feel a line that has existence outside our minds must be constituted of something - points or sub-segments or such. The most common definition is that a line is a set of actually infinite points.

Help me out here: where did any sane person aver that there were any actual infinities of anything? — tim wood

Do you consider Cantor sane? If so:

"Accordingly I distinguish an eternal uncreated infinity or absolutum which is due to God and his attributes, and a created infinity or transfinitum, which has to be used wherever in the created nature an actual infinity has to be noticed, for example, with respect to, according to my firm conviction, the actually infinite number of created individuals, in the universe as well as on our earth and, most probably, even in every arbitrarily small extended piece of space. - Georg Cantor

I'm not claiming {1} = {2}. I am saying that {1} = {1}. And that {1,1}={1,1}. And that, by induction, {1,1, ...}={1,1, ...}.

Help me out here: where did any sane person aver that there were any actual infinities of anything? — tim wood

For example, some folks believe that past time is actually infinite; implying an actually infinite collection of moments in the past.

Or some hold that space is continuous, implying an actually infinite collection of distinct spacial positions in a unit of space.

Firstly, if all the elements of the sets are identical, then they just have one element. Sets are defined by what distinct elements belong to them; a set is a collection of distinct objects — fdrake

You should have really read the posts above. I did acknowledge my mistake here:

https://thephilosophyforum.com/discussion/comment/367343

And I've also updated the argument in the OP accordingly. Sorry.

Two sets being in a one to one correspondence says nothing about whether they are identical sets. The odds are in a one to one correspondence with the evens, but even numbers are necessarily not odd. — fdrake

Two collections of identical objects in a one-to-one correspondence are, by mathematical induction, identical collections.

The point I am trying to make with the bananas is that there can be no actually infinite collections in reality because it leads to contradictions (that something can be changed and yet not change).

I believe time, space, matter/energy are all finite and discrete.

Numbers exist in our heads only, bananas exist in reality. The point of my argument is to demonstrate that actual infinity can exist in our heads but not reality.

So I think we are in agreement?

What about the collection/set of natural numbers? They start at 1 and go on forever.

How does one add a banana at the beginning of a row of infinite bananas? There is no beginning, and therefore no second banana, in a infinite row of bananas. There is no beginning or end with infinity. You're simply misusing terms. — Harry Hindu

What I mean is:

{ b, b, b, b, ... }
^
new banana is inserted here (at the start).

So the infinite collection has a start but no end.

I have made a mistake with this argument... sorry.

A set is a set is a well-defined collection of distinct objects, so my identical bananas cannot be said to form a set.

I still maintain however that my argument highlights the absurdity of actual infinity.

You can use mathematical induction to see the two sets of bananas are identical:

1. The first pair of bananas is in one-to-one correspondence
2. If the nth pair is in one-to-one correspondence so is the nth+1 pair
3. So all bananas are in one-to-one correspondence
4. So the sets are identical