...and the difference between "real" and "actual" is...? — Pattern-chaser

The real is that which is as it is regardless of what any individual mind or finite group of minds thinks about it. The actual (or existent) is that which reacts with other like things in the environment. Hence reality and actuality are not coextensive--besides real actualities (e.g., individual events), there are also real possibilities (e.g., qualities) and real conditional necessities (e.g., laws of nature) that cannot be reduced to collections of their actual instantiations.

Assumption 1:

Time is discrete.

From this follows that space must be discrete, where one unit of space equals one unit of time multiplied with the speed of light.

Assumption 2: Every physical process can be expressed mathematically.

Then it follows:

The logical framework that underpins a theory of everything must be based on natural numbers. This means, by the incompleteness theorem, that this system cannot be complete and consistent at the same time. Meaning, there are two options:

There exists phenomena in this universe that cannot be described by a theory of everything,

or

the theory of everything must produce false predictions.

Or, in other words, a theory of everything for such a universe cannot exist.

...what? I don't think time or space are discrete, but even if it were this:

Assumption 2: Every physical process can be expressed mathematically.

Then it follows:

The logical framework that underpins a theory of everything must be based on natural numbers. This means, by the incompleteness theorem, that this system cannot be complete and consistent at the same time — Karl

Is incorrect. Logical theories are not based on numbers, number systems are reasoned about by logical systems. Further, scientific theories use real numbers (decimal numbers) just as much as of not enormously more than natural numbers. Calculus is all about the real numbers, for example. And even further than that, your mention of Gödel's Incompleteness Theorems is probably false. Even in physics, theories aren't fully mathematical and are more like quasi-empirical. Gödel's Incompleteness Theorems apply purely to formal systems capable of expressing arithmetic, a theory of everything is usually thought of as a theory sufficient to explain all of fundamental physics (say by a theory of quantum gravity to unify GR and QM).

Length of 'now'

If we consider the length of ‘now’, it cannot be zero seconds because that gives a divide by zero error when we work out the number of ‘nows’ in any finite interval (eg 1 second gives 1/0=undefined). Also if now had zero length, it would not exist.

Could the length of now be 1/∞ ? Problem with that is no matter how many ‘nows’ elapse, the total time period elapsed is still infinitesimal (1/∞ + 1/∞ = 2/∞ etc…). So with now length 1/∞, we’d be stuck forever at a single point in time no matter how many ‘nows’ elapse.

So the length of now must be greater than 1/∞, IE a finite number (else time would not ‘flow’).

That makes time discrete.

Or "now"--like any other durationless instant--is simply an arbitrary human construct that marks continuous space-time, rather than a real constituent of time itself.

What changes to make ‘now’ ‘then’? There is some measurable quantity we call time that changes. So it is reasonable to discuss the duration of 'now'.

I don't see how you can have a 'durationless instant' surely a contradiction in terms? To say someone eats for zero seconds is to say they don't eat. A durationless instant indicates non-existence.

There is some measurable quantity we call time that changes. — Devans99

No, time is not an independent "thing" that changes, it is the (fourth) dimension of space-time that corresponds to spatial change. As I keep pointing out, motion through continuous space-time is a more fundamental reality than discrete positions in space or moments in time, which we arbitrarily mark for the sake of measurement and analysis.

I don't see how you can have a 'durationless instant' surely a contradiction in terms? — Devans99

Exactly--time is not composed of durationless instants, and space is not composed of dimensionless points. Those are human constructs, which are very useful for certain purposes, but not real.

What changes to make ‘now’ ‘then’? There is some measurable quantity we call time that changes. So it is reasonable to discuss the duration of 'now'. — Devans99

Right, change occurs at the present, now. And since change requires that time passes, it is very reasonable to discuss the duration of now.

Your issue is not with infinity per se but its corollaries.

Who can deny that the natural numbers are infinite. If you do then you'll have to furnish us the greatest possible natural number and I just add 1 to that and add 1 to that and so on.

The corollaries of Cantorian infinity are hard to digest. How can even numbers equal the natural numbers? It's ''obvious'' that the even numbers are half of the natural numbers, the other half being odd.

Cantor's treatment may be to blame. He used an old trick in the book viz. 1 to 1 correspondence. This works well for finite numbers (pls note this). If I pick up a stone each for every sheep I have then 3 stones = 3 sheep. 1 to 1 correspondence is a basic mathematical tool in our bag.

Cantor used the above principle of matching two infinities.

Even numbers = {0, 2, 4,...}
Natural numbers {1, 2, 3,...}

As you can see we match with the formula 2n-2

Natural number ( n )...Even number (2n-2)
1...2(1)-2...0
2...2(2)-2...2
3...2(3)-2...4
.
.
.
n...2n-2

As you can see, or as Cantor would like you to see, every even number has a matching natural number without any unmatched numbers. So, if 1 to 1 correspondence is applicable to infinities then we must conclude that the infinity of even numbers is equivalent to the infinity of natural numbers.

I understand your argument that 4 cm is twice as long as 2cm and it seems quite mad to think they both contain the ''same'' infinite number of points. However, 1 to 1 correspondence says this is true.

Can you explain your point in the context I've provided above? Thanks.

Exactly--time is not composed of durationless instants, and space is not composed of dimensionless points. Those are human constructs, which are very useful for certain purposes, but not real. — aletheist

How do we describe time then? The only models of a continua I've seen have used points or line segments to model it. In both cases its valid to discuss the length of the point or line segment representing 'now'.

Who can deny that the natural numbers are infinite. — TheMadFool

The natural numbers are in our minds only so sure they can be infinite there. Trees can dance in our minds too. But try writing out an infinite list of natural numbers... its impossible.

I doubt infinity can exist in the real world. It's not a quantity so why should real world quantities ever take the value of infinity or 1/infinity? That leads to the suspicion that time must be discrete.

A couple of other points:

- It sounded perfectly reasonable that matter was infinitely divisible but then we found out different. Maybe the same will happen for space and time?

- A continuum has the property that its parts are each equal to its whole. That is a somewhat mad conception. Should not be part of math IMO - too illogical.

Exactly--time is not composed of durationless instants, and space is not composed of dimensionless points. Those are human constructs, which are very useful for certain purposes, but not real — aletheist

You can say time is a human construct but it represents something that does/did exist in reality. There was ‘then’ and there is ‘now’ and there is a non-zero distance between them measured in units of what we call time. They both have a length. It can’t be zero because:

- We would not be able to perceive something of zero seconds long
- Time does not flow if ‘now’ is zero length: now + now = now.

If the length of now was 1/∞, time would still not flow.

Here is a different argument that time exists:

If time does not exist it has no properties.
- Time has the property that it flows from one moment to the next enabling cause and effect
- Time has the property that it slows close to the speed of light
- Time has the property that it slows in intense gravity
So time exists

How do we describe time then? The only models of a continua I've seen have used points or line segments to model it. In both cases its valid to discuss the length of the point or line segment representing 'now'. — Devans99

We describe time as continuous--it is not composed of discrete instants or very short durations. Likewise, we describe a line as continuous--it is not composed of discrete dimensionless points or very short line segments. "Now" is an arbitrary human construct that separates what we call "the past" from what we call "the future," but time itself does not really include any such discontinuity.

You can say time is a human construct ... — Devans99

But that is not what I am saying. Time is real and continuous; a durationless instant is an arbitrary human construct.

There was ‘then’ and there is ‘now’ and there is a non-zero distance between them measured in units of what we call time. — Devans99

Measurement is a human construct. We indeed mark two instants as "then" and "now," and measure the non-zero interval of time between them by comparing it to an arbitrary unit--e.g., one second as "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium-133 atom."

We describe time as continuous--it is not composed of discrete instants or very short durations — aletheist

And what is your definition of a continuum? All the mathematical definitions I've seen use instances or short durations of some form. What do you mean by continuous?

How on earth would you ever construct a continuum? By what magic processes do you construct something with the property 'each part is equal to its whole'? If it is a challenge for us to even conceive of a workable continuum then surely that suggests that nature would incorporate such an illogical concept.

Measurement is a human construct — aletheist

But when we measure time intervals, we are measuring something real. The measurement is not real but the measured is.

Hmmm I can’t do maths sorry, but maybe a continuous example might be energy turning into something else continuously in space. Retract, collapse and expand than repeat?
It’s hard to imagine or explain my point sorry.
Over trillions of years of density in space, tiny particles called atoms start forming.
Atoms start grouping and knocking into each other, causing friction (heat)
Random nebula clouds from clustered atoms.
Atoms compact further into solid forms and gas forms (asteroids and stars)
One part in space is unusually warmer and bigger than the rest, but is dying out from its mass.
Heat cools down, star collapses on its self under its own gravity so quickly to the point atoms bind.
Causing a sudden shock outwards! To this day that explosion is still expanding, and it’s gravity is still strong, any moment our universe can retract so quickly that we wouldn’t even know what happened.
This one point is a singularity. Literally meaning 1. There will always be 1 something. Except in maths

2. Similarly, the point in time ’now’ cannot have length=0 (if it exists for 0 seconds, it does not exist) — Devans99

From the perspective of the foton, no time has passed, does that mean fotons (and thus light) doesn't exist? time is relative, so sure it can exist for 0 seconds. to light itself there is no time, hence light is timeless, There is no time, there is only spacetime. separating the two is creating a false dichotomy that is what is causing your troubles understanding reality.

What do you mean by continuous? — Devans99

Exactly what I said before--not composed of discrete parts. If we were to "zoom in" on a continuous line, we would never "see" anything other than a continuous line.

How on earth would you ever construct a continuum? — Devans99

Who said anything about "constructing" a continuum? It is the more fundamental reality.

By what magic processes do you construct something with the property 'each part is equal to its whole'? — Devans99

Who said anything about such an alleged property?

Who said anything about "constructing" a continuum? It is the more fundamental reality. — aletheist

Geometry reflects reality. If we can't construct it geometrically, its probably does not exist.

Who said anything about such an alleged property? — aletheist

If you sub-divide a continuum you get two continua identical to the one you started with - the parts are equal to the whole. Thats a unique and illogical property of continua.

Geometry reflects reality. If we can't construct it geometrically, its probably does not exist. — Devans99

"Construct" implies building something up from discrete constituents, which cannot be done in the case of a true continuum. I have never claimed that true continua exist, only that they are real. I have already given a specific example of a true continuum in geometry--a line.

If you sub-divide a continuum you get two continua identical to the one you started with - the parts are equal to the whole. Thats a unique and illogical property of continua. — Devans99

Who (besides you) has attributed any such property to a continuum? What I said was that if we were to "zoom in" on a continuous line, we would never "see" anything other than a continuous line.

You have a point. I agree there's a difference between a tree in the real world and an imagined dancing tree.

Is this analogy apt to the issue?

Mathematical infinity is an abstract concept. It doesn't claim any physical representation, does it?

What is time anyway?

Is it real (then your argument works) or is it too an abstract concept (your argument is faulty)?

Do you know of planck time? It is, supposedly, the smallest unit (ergo discrete) of time in physics.

If we think along the lines of planck time then, since we are limited by our biology to a smallest discrete unit of time that is sensible to us, time can be thought of as discrete.

Can any human ever experience a millisecond? If you get shot in the head you die without experiencing the pain of the bullet because our nerves don't conduct information fast enough. We're dead before we know it.

That's as far as I'm willing to go on discrete time.

Mathematical infinity is an abstract concept. It doesn't claim any physical representation, does it? — TheMadFool

But nature is logical so maths can explain it because it is logical also. Actual infinity is not a logical concept so does not fits in maths or nature.

Is it real (then your argument works) or is it too an abstract concept (your argument is faulty)? — TheMadFool

Thats the question. We can measure time, does that make it real? I can't think of anything we can measure that is not real.

Who (besides you) has attributed any such property to a continuum? What I said was that if we were to "zoom in" on a continuous line, we would never "see" anything other than a continuous line — aletheist

If we were to try that with a real line, we'd see discrete atoms.

If we start with the common sense notion that there must be more points/intervals in a large line compared to a small line then a continua immediately violates this with ∞ = ∞. Continua are illogical, reality is logical, hence continua don't exist in reality.

But nature is logical so maths can explain it because it is logical also. Actual infinity is not a logical concept so does not fits in maths or nature. — Devans99

Infinity is counterintuitive, is the comment I usually encounter. That it's illogical may not be true. How many natural numbers are there?

Thats the question. We can measure time, does that make it real? I can't think of anything we can measure that is not real. — Devans99

Well, I think time is the exception to the rule that measurement entails something is real and the converse may not be true as well.

Time, as far as I know, is ''measured'' in terms of oscillatory phenomena - pendulum swings, vibrations of the Caesium atom, etc.

What are we actually ''measuring''? Looks to me like we're just counting the rhythmic beats of the pendulum or the Caesium atom. There's nothing like an object to which we put a measuring scale and say it's x cm/inches long.

If we were to try that with a real line, we'd see discrete atoms. — Devans99

You mean a physical line, which is not the same thing. If you were to "zoom in" on space-time itself--not any physical object within space-time--you would never "see" anything other than continuous space-time.

Continua are illogical, reality is logical, hence continua don't exist in reality. — Devans99

Continua are perfectly logical, just not in strict accordance with the logic of discrete quantity. It straightforwardly begs the question to insist that the latter is the only version of logic that corresponds to reality.

Infinity is counterintuitive, is the comment I usually encounter. That it's illogical may not be true. How many natural numbers are there? — TheMadFool

Whats logical about ∞ + 1 = ∞ (implies 1 = 0)? In fact infinity is invariant under all arithmetic operations; what's logical about something that when you change it, it does not change?

In the the mind there are an infinite number of natural numbers, but infinity itself is an illogical concept, so that conception is flawed and cannot exist in reality. The natural numbers do not exist in reality so talking about how many there are in reality makes no sense. How many exist in our mind? A potential rather than actual infinity exist as thats all we can ever visualise.

Looks to me like we're just counting the rhythmic beats of the pendulum or the Caesium atom. There's nothing like an object to which we put a measuring scale and say it's x cm/inches long. — TheMadFool

We use rhythmic beats to measure time but time could exist independently of the beating mechanism and indeed enable the beating mechanism. So time enables motion rather than motion defines time.

So think of a clock and next door empty space, I'm saying I think time flows equally for both even though there is movement only in one.

If the world only had 3 dimensions, everything would be static. Time is a degree of freedom even if you don't class its as a first class dimension. So in that sense, time is as real as space.

But nature is logical so maths can explain it because it is logical also. Actual infinity is not a logical concept so does not fits in maths or nature. — Devans99

Nature isn't 'logical', logic is just useful in understanding nature when used appropriately. Mathematics is used to give us models of nature but no model ever really captures things perfectly because nature is just too complicated. And an actual infinity is perfectly consistent no matter how many times you just claim it not to be, whether or not it can be physically instantiated.

If we were to try that with a real line, we'd see discrete atoms. — Devans99

A real line is not composed of atoms or anything else so this is nonsense. A line is an abstract object, you have to investigate it's properties mathematically. And in basically any geometry you like a line is not finitely divisible.

If we start with the common sense notion that there must be more points/intervals in a large line compared to a small line then a continua immediately violates this with ∞ = ∞. Continua are illogical, reality is logical, hence continua don't exist in reality. — Devans99

Lines are not composed of points. In a real-valued n-dimensional space, points are defined by distance from the original. But crucially, *points don't have a width* so an infinite sum of points would never give you a line. Points can be used to mark the beginning and end of a line but they cannot define them.

Further, all you're really saying is "If we assume my position that everything is discrete, then opposing views are incoherent", which, well, who cares? You're position is less credible because you're not telling anyone why they should accept your assumptions and your criticism of continuum and infinities has yet to start with accurstely representing infinity.

Whats logical about ∞ + 1 = ∞ (implies 1 = 0)? In fact infinity is invariant under all arithmetic operations; what's logical about something that when you change it, it does not change? — Devans99

That does not imply 1=0. Finite additions to an infinite number *by definition* cannot change the sum. It's the definition of infinity that it does not change by finite modifications. And saying infinity is invariant under all arithmetic operations is patently false. Take the smallest infinite number Aleph-null. Now take the power set of Aleph-null. The Cardinality has increased, it's size is now that of the continuum, Aleph-One.

If you were to "zoom in" on space-time itself--not any physical object within space-time--you would never "see" anything other than continuous space-time. — aletheist

That's because "space-time" is purely conceptual. If you "zoom in" on a concept defined as continuous you'll never see anything other than continuity, otherwise the concept would be contradictory.

Come on. Spacetime is not conceptual, not under any model in physics. You'd have to be seriously in denial to think models saying space is curved and correctly predict gravitational lensing and predicts that simulateneity is relative to reference frames is also saying that thing is not part of the world

Nature isn't 'logical' — MindForged

Can you give an example of something illogical from nature/reality?

A line is an abstract object, you have to investigate it's properties mathematically. And in basically any geometry you like a line is not finitely divisible. — MindForged

Yes but you cannot actually infinitely divide a line - it would take forever. So thats a potential infinity rather than an actual infinity you can describe at best geometrically. It's impossible to describe actual infinity geometrically, mathematically or otherwise so/as it does not exist.

Can you give an example of something illogical from nature/reality — Devans99

The point was logic is about abstract objects, it has nothing to do with nature. I didn't say nature was illogical, it's non-logical. Some models work better than others at explaining it, but that's the best one can do.

Yes but you cannot actually infinitely divide a line - it would take forever. So thats a potential infinity rather than an actual infinity you can describe at best geometrically. It's impossible to describe actual infinity geometrically, mathematically or otherwise so/as it does not exist. — Devans99

This is your fundamental confusion. No one is saying you can actually do a calculation an infinite number of times in a finite period because that's a process that is defined in terms of an activity done in small periods of time. But we know mathematically that the cardinality of the continuum is such that it can be put into a one-on-one correspondence with a proper subset of itself. That's the definition of infinity, hence a line (which is defined in terms of real numbers, i.e. the continuum) is infinite.

This categorization has absolutely nothing to do with some mechanical process. Your argument would be like saying the natural numbers are finite because I cannot count to a number called infinity. This is just a misunderstanding on your part on what these words mean.

But we know mathematically that the cardinality of the continuum is such that it can be put into a one-on-one correspondence with a proper subset of itself. — MindForged

But the one-on-one correspondence procedure yields nonsense like Galileo's paradox. And the continuum does not have a cardinality... Cantor should never have made such numbers up. It's down to a deficiency in the core of set theory; the polymorphic definition of set supports two different object types: finite sets and descriptions of set. The first have a cardinality, the 2nd do not. They are different kinds of objects with different properties and need to be treated differently. Cantor tried to shoe-horn both objects into a common facade and ended up making up magic numbers for cardinality - definitely not the right approach.

But the one-on-one correspondence procedure yields nonsense like Galileo's paradox — Devans99

Begging the question. You're saying it's nonsense because it results in infinities having the same size when you think it shouldn't, no argument given on your part. That's the very thing under debate, you cannot point to it as if it supported your point at all.

And the continuum does not have a cardinality... Cantor should never have made such numbers up. It's down to a deficiency in the core of set theory; the polymorphic definition of set supports two different object types: finite sets and descriptions of set. The first have a cardinality, the 2nd do not. They are different kinds of objects with different properties and need to be treated differently. Cantor tried to shoe-horn both objects into a common facade and ended up making up magic numbers for cardinality - definitely not the right approach. — Devans99

This is ridiculous and incoherent. The semantic method of defining a set is easily proven to be coherent. I definine a set P as the set of all red objects in my room. As it happens, that set has three members, so its cardinality is 3. This is just as coherent as defining a set P explicitly with the members {My phone case, my pen, an apple}. In fact, defining sets semantically is far more efficient and is used by people every day all the time. You give no argument that semantically defined sets don't have a cardinality and by any reasonable means they do have cardinality.

There is nothing "shoehorned" here, you object to it because it's patently obvious that semantically defining sets let's one define infinite sets as easily as one defines finite sets, e.g. the set N of natural numbers is defined as having every whole number 0 and greater as a member. No one is confused by what members populate that set, and the cardinality is infinite per Dedekind and Cantor.