Before a quantity is measured, It does not have a well-defined value to be affected. That is why the measure number of length, for example, depends the relativistic frame of reference. — Dfpolis

Relativistic length yes, proper length no. Two observers in the same reference frame as the object always get the same measurement results.

At no determinant point is this true. — Dfpolis

The past is finite. There are simply too many arguments in support of this to deny it (Big Bang Bang, entropy, equilibrium, the measure problem, non-existence of actual infinity, causal infinite regresses are impossible, matter needs a temporal start).

Aristotle did not know about the Big Bang. The heavens appeared fixed to him; part of the source of his confusion, he seems to have associated the unchanging with the infinite.

Possible subdivisions are not actual subdivisions. — Dfpolis

With continuous motion, they are all actual subdivisions.

Accidental causal relationships are undefined in such cases because times less than the Planck time are undefined. If you can't measure the interval between events, space and time are ill-defined. Thhis is a major problem for a quantum theory of gravity. — Dfpolis

I would have thought events would be simply concurrent if there is less than Planck time between them. So it would not effect the normal understanding of causality.

This problem has no effect on essential causality because essential causality does not link separate events, but analyzes single acts. — Dfpolis

I find Aristotle's terminology a little confusing. I am happier with cause always preceding event. I think what Aristotle calls an 'essential cause' is actually a non-temporal conditional and it has nothing to do with the modern view of causality.

Really? So the builder building is not the cause of the house being built? — Dfpolis

Building a house is a number of sub-events. For each sub-event, the cause always temporally precedes the effect.

If God does not exist throughout space-time, He cannot act in time, and all the proofs based on His action in nature are ill conceived. That does not mean that God is bound by or confined to space-time. — Dfpolis

God cannot exist throughout all spacetime; parts of spacetime are receding from each other at FTL speeds; that would mean God is causally disconnected from himself.

God created spacetime; he does not act in spacetime, all the proofs based on his action in nature are indeed ill conceived IMO.

So I think what is coming out is that one has to have a clear definition of God to qualify the term 'atheist'.

Deism is a subset of theism but you could argue that deists are atheistic with regard to a traditional definition of God.

Likewise, because of the more moderate definition of God (no 3Os) employed by deists, you could say atheists are agnostic with respect to a deistic god - in that disproving the existence of such a god is beyond the power of science and reason - so to deny the possibility completely would seem unreasonable.

I'm not sure everyone will agree with this; Dawkins head would explode I'd imagine. But his sort of caustic atheism does the debate no favours IMO.

This is because he sees time as a number, and numbers as existing only in enumerating minds. — Dfpolis

Personally I have numbers existing in the mind only but corresponding/representing to real life quantities. I do not believe that whether some macroscopic real world quantity is measured or not effects its value in any way. The measure of the object is intrinsic to the object and measurement just makes that known to an observer. So if numbers cannot be actually infinite (which they cannot), nor can real world quantities (measured or unmeasured).

There is no finite "every number." The traveler keeps counting finite numbers endlessly -- going forward or backward. — Dfpolis

But with past eternity and a counting, backwards travelling, time traveller, we have a measure of eternity - any number we can think of, the traveller must have counted it. So when we ask what is the length of that past eternity, it must be greater than any finite number. Only actual infinity has that property and actual infinity does not exist IMO.

What is the operation that makes them "happen," if it is not mental or physical? Note that moving is not dividing, even though motion can be divided mentally. — Dfpolis

A particle moving along a real number line continuum must pass through every possible sub-division (sub-segment) of the line over time. So for any sub-division I care to choose, say one 10^-10 in length, it is guaranteed that there are always smaller sub-divisions that compose it that the particle also travelled through. The act of movement - positional change from one moment to the next - creates the sub-divisions. If spacetime is continuous, then there must be sub-divisions in length smaller than any finite number we care to name. For that, one can only appeal to 1/actual infinity... which does not exist IMO.

Only accidental causality is time based, and as Hume showed, it lacks intrinsic necessity. — Dfpolis

I trust my senses and experience more than Hume on this point.

As a side note, there seems to be a point (the Planck time) at which time can no longer be defined. Beyond it, accidental causality is meaningless. Essential causality remains meaningful. — Dfpolis

Two events would not be able to share a cause and effect relationship if they are separated in time by less than Planck time? Concurrent events cannot share a cause and effect relationship anyway. So I still believe that time-based causality cosmological arguments have merit.

That would mean that God could not exist. — Dfpolis

... could not exist in time, but his presence seems necessary, so he must exist outside of time. God cannot have a temporal start or end to his existence. He would just 'be' with no tense. God would be both finite and eternal - which is only possible outside of time.

You maybe right about science, so I will drop the usage of spacetime and just argue on the basis of time. I've given this argument before but it is the basis of my belief that there must be something permanent in existence outside of time (in order for there to be anything at all in existence):

1. Can't get something from nothing
2. So something must have permanent existence (because if there ever was a state of nothingness, nothingness would persist to today)
3. Something cannot exist permanently in time ('always' existing in time implies no temporal start which implies it does not exist)
4. So there must exist a permanent timeless something. This is identical to the necessary being that philosophers have argued for down the ages.
5. The permanent timeless something is the cause of the Big Bang.
6. Timeless things are permanent (they just 'are' - no tense). They are beyond causality so do not in themselves need creating.

I believe the above represents the only credible metaphysical explanation for the origin of everything.

I am not aware of any sound metaphysical arguments that support strong atheism. It is the general incredulity over the possibility of a deity that is used as an argument for strong atheism and there is no logic to back that argument.

I am using the term universe to indicate what is called spacetime, which had a start 14 billion years ago, rather than the totality of everything. So it is perfectly possible that spacetime was created by something external to spacetime. This cannot be ruled out with 100% certainty. So there is a non-trivial probability that spacetime was created. That requires a creator. The creator of spacetime is traditionally called God. Hence strong atheism cannot be justified on the basis of probability.

So would you say that you're not 100% sure when it comes to denying any arbitrary, absurd/fantastical claim that anyone makes? — Terrapin Station

I guess to be precise, we have to give God a definition.

I have a deist view of God in mind as a non-interventional entity that create the universe through non-magical means. I do not believe it is possible to rule out such a God with 100% certainty, so by this definition, there are no atheists.

Your definition of God as something outlandish and unbelievable seems would seem to lead to virtually 100% certainty that such a God does not exist. So with that definition, there are atheists.

I guess your usage of the word athiest is closer to common usage of the word. But I feel my usage is the technically correct usage.

There is a distinction between weak/negative and strong/positive atheism:

https://en.wikipedia.org/wiki/Negative_and_positive_atheism

Weak atheism seems to me to be a kind of agnosticism - a lack of believe in any deity rather than the positive belief that no deities exist.

Strong atheism is a positive belief that no deities exist. How anyone can be 100% sure of that, I do not see.

So it could be reasonably argued there are no strong atheists and the weak atheists are actually agnostic as per your OP.

The series is not convergent according to the definition of convergent:

https://en.wikipedia.org/wiki/Convergent_series

So the series has no sum. This article explains it quite well:

https://plus.maths.org/content/infinity-or-just-112

As far as the physics is concerned it is something called the Casimir effect that is associated with a finite sum of a divergent series. I don't know the maths or physics well enough, but from a layman's perspective, I think some of the maths in QM looks suspect. There is an alternative explanation for the Casimir effect that does not involve questionable maths:

https://en.wikipedia.org/wiki/Casimir_effect#Relativistic_van_der_Waals_force

Or it could be that the physicists have the precise maths wrong but it is close enough to being right that with some questionable manipulations, it gives results that agree with experiment. QM maybe is not quite the finished article.

Aristotle defined time as the measure of motion according to before and after. So, for him, time is a measure. "Unmeasured change" is how I would think Aristotle would describe the unlimited prior history of the cosmos -- as to have any kind of time would require a measurement. — Dfpolis

So is Aristotle saying when we measure it time exists (measured change); when we don't, it does not (unmeasured change), so a past eternity is possible without accepting actual infinity in reality? Putting QM aside for the moment, measuring does not change what is measured. So if something cannot be actually infinite because we can measure it, the same something cannot be actually infinite when we cannot measure it. IMO he should have concluded the past cannot be eternal.

That is not Aristotle's view. I also think it is factually incorrect. We do not do division into parts (which is an intellectual operation) when we run a race, and if we did, it would take forever to do the actual dividing which is why Aristotle is denying actual numerical infinities. The same applies to time. You can only measure from a beginning to an end, and if change has no beginning, you can't actually measure all of it. — Dfpolis

But our bodies do the division of space into parts for us in a race. So Aristotle is saying because we are not conscious of the division of space, it is not happening?

I think actual infinity cannot be regarded as a purely intellectual construct; it represents a fundamental characteristic of the continuum. If the continuum exists (which I doubt), then actual infinity is a fundamental part of reality and every movement we make is a division of space into actually infinitely small components. The fact we do not compute the divisions mentally does not mean they are not happening in reality. The fact the divisions took place in the past I suppose could be argued that actual infinity is not realised in the present, but it is realised in the past which is as bad to my mind - the past happened and was real.

It is like he is saying actual infinity is an artefact of the measuring process, along with number in general I suppose. He seems to be classing actual infinity as a human construct only. But maths mirrors reality and true continuity of spacetime surely requires something physically equivalent to actual infinity?

It is not inconsistent to hold that something can be potentially infinite, but always actually finite. That is how counting is. There is no intrinsic limit to a count, but actual counts are always finite. — Dfpolis

Counting extends forward into a potentially infinite future so I agree that it is always actually finite. But it is an infinite past that I contend requires actual infinity. We can, as a thought experiment, imagine an ever-lasting time traveller travelling backwards in time whilst counting. From our perspective, the past is completed, so the time traveller must have counted every number if the past is infinite. But there is no largest number so we can only conclude the traveller counted to actual infinity somehow. So past eternity seems to require actual infinity.

There are cosmological arguments based on accidental causality, such as the Kalam argument popularized by Craig, and arguments based on essential causality, such as those of Aristotle and Aquinas. The Kalam argument is persuasive, but logically unsound because, as Hume argued, accidental causality has no intrinsic necessity. — Dfpolis

I see all the cosmological arguments as either explicitly or implicitly time-based. Causality and time are inextricably linked; movement and time are likewise linked. From the second way:

"The second way is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible"

Aquinas is talking about efficient causes being time ordered IMO. The 3rd way is temporal in its phraseology, the first way is all about motion hence time.

Ontologically prior (first in order of actualization), yes. Temporally prior, no. There is can be no logically necessary connection between events at separate times and places. This is because there is always the possiblity of intervention. There is no possibility of intervention with essential causality because the agent actualizing the patient is (identically) the patient being actualized by the agent. (The builder building the house is identically the house being built by the builder.) — Dfpolis

Can you explain how the actualisation order could be different from the temporal order?

The possibility of intervention by God? I thought that Aristotle had God as external to the universe, existing in the heavenly spheres - a deist view of a non-interventionist God.

Asserting that something has no beginning does not entail that it does not exist. — Dfpolis

If something never started existing, it does not exist. Time is like space in this regard: if something has no beginning in space, it does not exist. Likewise if something has no temporal start, it has no temporal start + 1, start + 2, etc..., so by induction, it does not exist. A beginning is also required for instantiation of innate attributes, like the mass/charge of a particle. Without a beginning, matter would not have innate attributes and would be null and void.

I think that in virtue of his discussion of quantity in the Metaphysics, Aristotle would say that infinity is a property of numbers, and unmeasured change, however extended, is not a number, but a measurable -- and therefore not an actual, but a potential infinity. — Dfpolis

Time is not IMO 'unmeasured change'... time has a start so it must be physical. But assuming for arguments sake that time is 'unmeasured change', then it still measurable and has a quantity associated with it. Infinity is a property of quantity in general (as well as number). If past time is postulated to be eternal then the associated quantity of past time must be actually infinite... hence Aristotle is holding a contradictory view.

That is how he resolves Zeno's half-the-distance paradox. He argues that while the distance to the goal is infinitely divisible, it is not actually infinitely divided — Dfpolis

The very act of attaining the goal would seem to me to infinitely divide the distance to the goal - it is not possible, after all, to teleport over portions of the distance so to avoid infinitely dividing it. I am not convinced that Aristotle resolved Zeno's paradoxes. The solution is probably discrete spacetime.

His stance seems to be that to have an actual infinity requires someone to actually count or measure and infinite quantity. We might say that applying "infinite" to something that is not an actual count or measure is a category error. — Dfpolis

His stance seems strange. Does a falling tree make a sound if no-one is present? Does time or space have duration or distance if no-one measures it? Surely yes to both questions. I think actual infinity, if it existed, which it does not, would exist independently of any count or measure, so it is inconsistent to hold a believe in past eternity but to deny actual infinity.

That is the error Kant makes in criticizing the cosmological argument. The argument is based on essential, not accidental, causality. — Dfpolis

The cosmological argument is fundamentally a time-based argument so we are talking about a time based infinite regress - which is impossible - which is what Aquinas says in the 5 ways.

On the other hand, as Hume noted (and as was known to the Scholastics), accidental or time-sequenced causality has no intrinsic necessity. Thus, "proofs" based on accidental causality lack necessity. — Dfpolis

The fact that X exists means that it is intrinsically necessary that a prior cause of X existed. Thus we can trace backwards all along the causal chain establishing that each node is necessary to establish the state of affairs as today.

Aquinas says explicitly that there is no philosophical reason to reject Aristotle's view that the cosmos is indefinitely old. Creation in time is, for Aquinas, an article of faith, not a conclusion of reason. — Dfpolis

I do not see how Aquinas can reject a time ordered infinite regress and maintain a belief in an eternal cosmos - the second implies a time ordered infinite regress. The cosmos cannot be infinitely old, matter cannot exist 'forever' - that would imply matter with no temporal start, which in turn implies the matter does not exist.

The Wikipedia is wrong. Aristotle believed that each circular motion mathematical astronomers were then finding was caused by a distinct "intelligence," later Christianized into angels. He is quite clear that the intelligences do this because constant circular motion is the closest they can come to the nature of the unmoved mover. Further, he calls the causality linking the intelligences to the unmoved mover "desire," thus seeing it as a species of intentionality. — Dfpolis

That makes sense, thanks.

Should not Aristotle have distinguished between:

- Past eternity. The past is complete, it has actually happened. Past eternity implies a greater than any finite number of days has elapsed - an actual infinity - which is impossible.

- Future eternity. For most models of time, the future is not complete - so it is a potential infinity.

Accepting eternalism, the two are clearly quite different?

My understanding of Aquinas is that he rejects a time ordered infinite regress. From the prime mover argument:

"If that by which it is put in motion be itself put in motion, then this also must needs be put in motion by another, and that by another again. But this cannot go on to infinity, because then there would be no first mover, and, consequently, no other mover; seeing that subsequent movers move only inasmuch as they are put in motion by the first mover; as the staff moves only because it is put in motion by the hand"

The 2nd of the 5 ways contains a similar argument against a time ordered infinite regress:

"Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God."

I think Aristotle's message on a time ordered infinite regress in not clear. From Wikipedia:

"Aristotle argued against the idea of a first cause, often confused with the idea of a "prime mover" or "unmoved mover" (πρῶτον κινοῦν ἀκίνητον or primus motor) in his Physics and Metaphysics. Aristotle argued in favor of the idea of several unmoved movers, one powering each celestial sphere, which he believed lived beyond the sphere of the fixed stars, and explained why motion in the universe (which he believed was eternal) had continued for an infinite period of time. Aristotle argued the atomist's assertion of a non-eternal universe would require a first uncaused cause – in his terminology, an efficient first cause – an idea he considered a nonsensical flaw in the reasoning of the atomists."

https://en.wikipedia.org/wiki/Cosmological_argument

So it seems he argues for a time ordered infinite regress on the basis that a first uncaused cause is impossible?

I'm in agreement with Aquinas here - a time ordered infinite regress is clearly impossible. The objection of what caused the first cause goes away if time has a start - then the first cause is timeless and thus itself beyond causality - it in itself does not need a cause.

Its an interesting piece of math, but it is wrong at the same time - its all based on Grandi's series evaluating to 1/2 which is wrong - Grandi's series does not converge and so it limit is undefined. Once you start playing around with something that is undefined, you get all sorts of nonsense results coming out. Actual infinity is undefined IMO and playing around with it leads to nonsense results (Hilbert's Hotel etc).

I'm going to take a look at this to understand it better:

https://www.youtube.com/watch?v=YuIIjLr6vUA

With Grandi's series, at any finite point in the series, its sum is either 1 or 0. So the sum of the infinite series (if it has a sum) must be 1 or 0. But mathematics cannot be sensibly evaluate Grandi's series - standard methods produce 1/2 or contradictory results.

A more physical demonstration of Grandi's series is Thompson's lamp - the lamp is on for the 1st second, off for the next 1/2 a second, on for the next 1/4 of a second, and so on. After 2 seconds, is the lamp on or off? It must be one or the other, it cannot be 1/2 on.

Both problems are examples of 'supertasks'. In both cases, knowledge of whether actual infinity is odd or even is required, which requires knowledge of the size of the associated sequence representing the series. But an actually infinite collection of objects is an impossibility - so the sequence cannot be said to have a size - so we cannot evaluate it.

I think these are illustrations that all supertasks are impossible - anything with an actually infinite number of steps cannot happen in finite time. In the case of the lamp, something physical in reality would limit the number of steps in the supertask to a finite number - probably time is discrete - planck time for example - then the series could not go on forever and the lamp would definitely be on or off at the end of 2 seconds. Discreteness addresses Zeno's paradoxes too (which are also examples of supertasks).

I think the fact that maths cannot evaluate the series is supportive of the view that actual infinity does not exist as a (sound) mathematical concept and does not exist in reality.

Are we the standard by which we measure infinity? Isn't this like saying tortoises that live upto 300 years don't exist because humans only live upto 100 years? — TheMadFool

Numbers exist in the mind only so I guess you are correct - we should not limit our consideration to the human mind. An alien with a larger mind would be able to mentally realise larger numbers than a human. But they would still be limited to conceptualising finite numbers as per the storage capacity of their minds.

Minds exist in reality so all minds are a finite collection of storage elements (I hope we agree that realising an actually infinite collection in reality is impossible). So whatever the actual size of the mind, the largest possible number is still finite.

Assuming this is correct, which given your penchant for error and misstatement is going some. But given this, let's just consider the stronger which I assume will carry the lesser, actual infinite sets. If there are actual infinite sets then it would seem to follow a fortiori that there wold be potentially infinite sets.

Question: how many numbers are there? It's a fair question and a meaningful question. Answer!

Does the answer mean that you could produce a bean for every number and thus have a very large pile of beans that extended out of sight? Of course not! But the concept for the purpose is complete.and actual. QED. — tim wood

Numbers exist in our mind only. Actual infinity can exist in our minds only along with other things that are not logical / do not exist in reality like square circles. But just because you can think of actual infinity does not mean you have realised/completed actual infinity in your mind.

How many numbers are there?

- First thoughts: there are a potential infinity of numbers. We can go on counting forever so its a potential infinity. But of course we would lose track after a while. So this potential infinity has a finite, mind related limit.

- Second thoughts: in no way is the human mind capable of realising an actual infinity of numbers - the mind in finite after all - there is a limit on the largest number than can be conceptualised - the number of bits representing the number must fit into the storage capacity of the human mind. A very large number then, and dependent on the mind that's conceptualising it.

So actual infinity is not realisable in our minds or in reality.

Past eternity is infinite rather than unbounded. I have no issues with the 'no boundary' model... maybe it could be true... the point is it is a finite model so I cannot rule it out.

Actually infinite is completely different from unbounded - the example I gave (CCC) has an actually infinite past - it has nothing to do with unbounded.

It is the concept of an actual completed infinity in the real world that I am arguing against. There are two basic classes of infinite objects:

- Potential Infinities. The limit concept in calculus could be regarded as an example
- Actual Infinities. As represented by infinite sets / transfinite numbers in set theory

Finitists (of which I am one) might object to the first usage, but always object the second usage. Strict finitists do not even allow for potential infinities. In fact some even hold the number system should be restricted to only those numbers that are comprehensible to the human mind. Classical finitists allow potential infinities and do not believe there is a ‘greatest number’. I am a classical finitist.

So for example, I do not object to future time potentially going on forever as that would be an example of a potential infinity.

But I would object to the idea of an actual completed infinite set existing in the real world. So a collection of objects whose number is not finite I would object to.

Hope this is clearer...

It is true there are different definitions. Maths has a definition relating to a one-to-one mapping between a set and a proper subset that is quite odd and different to the normal usage. Personally I use Aristotle's definitions of actual and potential infinity. A common usage is something bigger than anything else possible.

Whatever the exact definition, actual infinity leads to problems... how can a subset be the same 'size' as its containing set? The whole is greater than the parts. How can something be bigger than anything else possible? How can something go on forever? IMO we are in the realm of make believe if we countenance these... unicorns, pixies, magic and actual infinity are all of the same mould.

Lots of cosmology models use it. CCC for example is eternal in time (past eternity is an example of actual infinity):

https://en.wikipedia.org/wiki/Conformal_cyclic_cosmology

Any model of the universe that is actually infinite in space or eternal in past time is using actual infinity. Wikipedia summarises the situation - an actually infinite universe is an open question in cosmology:

https://en.wikipedia.org/wiki/Infinity#Cosmology

Truth is usually defined in philosophy as justified true belief. I guess this definition of truth must refer to what is true in the real world by default. It could refer to truth within some abstract axiomatical system also, but I think the first usage is the common usage.

Any truth in an axiomatical system is only as true as it's axioms (and the axiom of infinity in set theory seems false) whereas true in the real world means more than that - it means it fundamentally reflects the way the real world is.

So set theory or any axiomatical system can be consistent but not true at the same time. But it is (or was) the habit of mathematics, physics, logic to choose axioms that are very likely true in the real world. IMO set theory has an axiom that is very likely false in the real world. It should come with some sort of health warning attached because it has confused a lot of people into believing that actual infinity is a physical reality.

There are other axiomatical system that may or may not be true in the real world. For example non-euclidean geometries. But set theory stands out as choosing an axiom that is clearly false in the real world - so it has no real world applications because of that (the part of set theory relating to infinite sets I mean, finite set theory is useful).

I feel that axiomising something that is clearly false in the real world (IMO) is the basis of my beef with set theory.

Infinity is used in cosmology which is a science.

You are an idiot to believe everything you are taught - you swallow it like a fool without questioning.

It is a fact that a large proportion of what we were taught at school is plain wrong. You have to learn to be skeptical. Have you never heard of group think?

You are downright rude, aggressive for no reason and ignorant.

You are not a very well informed person - finitism is quite a widely held view and quite a broad church:

https://en.wikipedia.org/wiki/Finitism

Lots of people doubt the reality of actual infinity. It just happens that not so many finitists frequent this site.

Actual infinity means in a strict mathematical sense a number greater than any other. But there is no greatest number. So there is no actual infinity.

Actual infinity in a physical sense means something that goes on forever. But only in our minds can things go on forever / be infinity dense etc... these things are impossible in reality.

Once again, I am saying that the standard contemporary reading of Aristotle is at odds with the approach that I have pointed to. Quoting Wikipedia ignores that distinction. — Fooloso4

I'd be interested to hear what you think is wrong with Wikipedia's summary. I can't claim to be an expert on Aristotle myself.

But I don't think Aristotle is your main concern here. What interests you here is the same thing that interests you in everything you post on every forum I have seen you post on - presenting and defending your own views on time, eternity, etc. — Fooloso4

So you don't present and defend your own views yourself then? Seems to me that is an essential part of philosophy. It is the clash of differing opinions that can lead to a productive discussion and progress.

On my view that's more nonsense that we're making up in order to avoid having to revise theories on a more foundational level. — Terrapin Station

Science seems to show the pattern of two steps forward, one step back - for example they seem to have it badly wrong on infinity IMO - so maybe science could be wrong with the nature of empty space too. A lot of what we know now will surely turn out wrong in the fullness of time.

The concept of virtual particles is a little worrying - they do not exist for long enough to be directly detectable so we infer their existence from theory and side effects. But the side effects, like the casimir effect have other viable explanations. So it seems that there is no proper empirical support for virtual particles... more like a metaphysical theory than a physics theory.

But I think the expansion of the universe is in better shape - we have the redshift evidence and the CMB radiation.

Aristotle seems to have believed past time was infinite:

"The ancient philosopher Aristotle argued that the world must have existed from eternity in his Physics as follows. In Book I, he argues that everything that comes into existence does so from a substratum. Therefore, if the underlying matter of the universe came into existence, it would come into existence from a substratum. But the nature of matter is precisely to be the substratum from which other things arise. Consequently, the underlying matter of the universe could have come into existence only from an already existing matter exactly like itself; to assume that the underlying matter of the universe came into existence would require assuming that an underlying matter already existed. As this assumption is self-contradictory, Aristotle argued, matter must be eternal."

https://en.wikipedia.org/wiki/Eternity_of_the_world

I think that it is possible that matter could have existed timelessly and and thus not requiring a 'substratum' to exist. Or there is the zero energy universe hypothesis - matter was created in exchange for negative gravitational energy - again not requiring a substratum. Also his argument implies that matter must exist forever in time which I would argue is not possible - the matter would have no start / no coming into being - meaning it could not exist at all.

The continuum hypothesis or its negation can be added it set theory without making it inconsistent. So that implies something false can be added to set theory without changing its consistency. Consistency and logical correctness do not seem to be the the same thing.

I think that something can be consistent and illogical at the same time. For example, if I define a simple maths system with only one number: 1 and one operator: + then I can axiomatically define 1+1=1. Its consistent but not logical. That sums up my feelings about actual infinity in set theory; maybe the axioms are such that its consistent (or appears to be consistent)... but its not logically correct.

I mean that nothing illogical exists in nature so if maths is to mirror nature, then nothing illogical should exist in maths either.

There are differing definitions of the actually infinite, one is: 'limitless or endless in space, extent, or size; impossible to measure or calculate'. The definition itself essentially precludes its existence in maths.

There are lots of reasons why actual infinity is illogical. Looking at just one - the division operation on transfinite numbers:

We can imagine the division operation as compressing space: for example division by 2 would compress the numbers on the real number line by a factor of 2: 10 would become 5, 4 would become 2 and so on. The convention is that ∞ / (any number) = ∞, so division by infinity does not decrease the 'overall' size of the real number line - the endpoints at 0 and ∞ are unaffected, but every finite number on the real number line must move closer together - this is clearly illogical from a physical standpoint.

I agree maths works very well for modelling reality in most instances. Actual infinity is not one of those instances though. I think perhaps the problem could be that actual infinity is not even a sound mathematical idealisation? Maths mirrors nature; both should be constrained to the logically sound. Presumably that is why maths models nature well. But actual infinity is illogical (Hilbert's Hotel etc...) and so is not found in nature, maybe it should not be found in maths either?

The maths that physicists use (calculus mainly) should only need depend upon the concept of potential infinity; actual infinity should not be needed anywhere in the physical sciences.

I guess so, but it comes back to the problem of people taking ideas from mathematics and applying them to the real world. Idealisations are just that; they are not real world entities. A mathematical point, a plane, actual infinity cannot and do not exist in the real world.Yet we have cosmologists that are assuming time and/or space is actually infinite - idealising the real world and misleading other people in the process.

Well bearing in mind Aristotle believed in infinite time, it seems to me he would have had no option but to also believe in some sort of immortal-in-time being. Such a being, a necessary being, would be required. Causality absolutely requires a first cause.

Aristotle's conclusion is presumably that such a being must be itself uncreated, IE immortal-in-time.

My personal view is that there was a start of time and that the first cause is timeless so beyond causality - hence not needing to be created itself - it just is.

I was given the impression in school that a line segment contains an actual infinity of points. The problem seems to me the mathematical definition of a point is non-sensical. Anything with all dimensions having length zero cannot exist. So bearing in mind that definition of a point, an undefined value for the number of points on a line makes sense - undefined and actual infinity are one and the same IMO.

If we instead use a more sensible definition of a point, say length 0.1, then we can reason about it - we get 10 points on a line segment length 1.

Aristotle, like Plato and Socrates, is a skeptic when it comes to the divine and questions of the beginning or arche and the whole. He knows that no one knows such things, but if he left it there he leaves it open to the theologians, those who make claims regarding the gods, origins, and the whole — Fooloso4

The greek pantheon and the stories associated with the individual greek gods are quite unbelievable - its a wonder anyone believed in it at all - Aristotle was too sensible for that. What Aristotle propounded is much closer to what deists believe I think:

"Aristotle argues, in Book 8 of the Physics and Book 12 of the Metaphysics, "that there must be an immortal, unchanging being, ultimately responsible for all wholeness and orderliness in the sensible world""

https://en.wikipedia.org/wiki/Unmoved_mover

Because stuff is invented (space as a separable thing) to make it work. It's akin to epicycles re planetary motion. That theory fits what we observe, too. It's just that it's wrong. But it was adopted so that we wouldn't have to change the theory. — Terrapin Station

But it does seem, with galaxies receding from each other faster than the speed of light, that space itself expanding is the only possible explanation. Also, as already discussed, the expansion of the universe does not fit with a normal explosion-type expansion.

Empty space appears to have properties - such as quantum fields and (maybe) dark energy. It seems to be something substantial.

Not familiar with those works. What is the reasoning behind holding a belief in eternity but not actual infinity?

OK, let alone BT. In my opinion there are no interesting results that depend in an explicit way on the fact that a continuum line is defined as an uncountable set of points. Are there? — Mephist

A point has length zero. How many length zero things can you fit on a line segment length one?

1 / 0 = UNDEFINED

Anything with length zero does not exist, so its like asking how many non-existing things fit within an existing thing. An answer of UNDEFINED seems right.

Theory should explain observations, not dictate what can be observed, and then require fanciful inventions to not have to discard the theory.

We can say that the movements are "as if" the objects in question were on the surface of a balloon, but to then posit the balloon as a real, independent thing isn't justified. The "as if" is simply to help us picture/understand what we're observing. — Terrapin Station

The theory predicts what we observe:

- The theory of inflation; the ballon is a 2D analogy used to describe what is happening to 3D space
- The observation is of a homogeneous universe which is expanding the same rate everywhere

The two are in agreement.

Clearly a very foundational figure in the history of philosophy. For any particular question, it seems to be best to first consider what Aristotle had to say about it. It’s a fact that the most straightforward metaphysical arguments were be discovered first in human history and are therefore are to be found in ancient texts. Occam's Razor tells us to prefer straightforward arguments. So it is unwise to ignore the works of the ancient philosophers. Aristotle's departure from Plato on the theory of forms was very wise. Aristotle seems like a switched on, realistic sort of guy. He was the first I believe to come up with the cosmological argument for God.

I would take issue with him on one point: he held seemingly contradictory views. On the one hand he argued against the existence of the Actually Infinite and on the other he argued for eternal time (which is a form of Actual Infinity). Aristotle’s arguments for Eternal/Infinite time:

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

The first argument can be countered by examining the overall structure of eternal time; it has no initial starting moment. For each individual moment, we can imagine a prior moment, but we know the system as a whole has no overall starting moment so it cannot exist as a whole (the first moment defines all the others). We can also appeal to modern physics and the Big Bang singularity - a moment for which it appears there was no preceding moment.

The 2nd argument can be by countered affirming causality: his argument leads to an infinite regress which undesirable and also impossible - there must logically be something which did itself not have a cause - the first cause - which itself must be beyond causality (timeless) - else nothing else can logically exist.

If space itself is not expanding and it is a regular explosion instead then I would have thought it would have to fit in with one of the following patterns:

- The edges of the explosion are moving faster the the centre parts. In which case matter might be evenly distributed (homogeneous) like we observe, but we do not observe differing rates of expansion.
- Everything is expanding out at the same rate from the centre of the explosion. In which case the galaxies would form a spherical shell, which is not what is observed.

If galaxies were moving apart themselves then there should be a centre to the expansion. With a centre to the expansion, the rates of expansion would vary according to distance from the centre - things on the edge of expansion moving faster that things close to the centre.

But astronomers say the rate of recession of all objects is simply proportional to their distance from earth. I would have thought that would be impossible unless by chance earth happens to be the centre of the universe? Thats very unlikely so for a regular explosion we should be not at the centre so the rate of expansion should be asymmetrical as observed from earth. Thats not what is observed.

If it is the objects in space that are moving further apart then that seems to imply a regular explosion type expansion. But such an expansion would have a centre - but astronomers insist that the expansion of the universe does not have a centre.

So the ballon analogy is confusing - it is only the points on the balloon surface that correspond to galaxies. The contents of the balloon are not part of space in this analogy. But it is the balloon itself that is expanding...