I think that without having descriptive account on "orbits" like those of Ellipses, Parabolas, and hyperbolas that mathematics beforehand supplied us with, it could have been very difficult to observe how the planets moves, and it would be very difficult to predict their movements. Possibly similar things might apply with the uncertainty principle. I don't know really. — Zuhair

This is a very interesting subject which you bring up here, but my opinion is somewhat opposite to what you say. I think that mathematics allows us to make many very accurate predictions based on statistics and probabilities, without having any accurate description of the mechanisms involved. So for example, Thales apparently predicted a solar eclipse in 585 BC. I think it's common that we observe things, take note of the patterns of specific occurrences, thereby becoming capable of predicting those occurrences, without understanding at all, the motions which lead to those occurrences.

So the ancient people observed the motions of the sun, moon, planets, and stars, and described these motions relative to their point of observation, and could make predictions based on those descriptions. But the motions they described were completely different from the motions we describe of the very same bodies, today. And we say that they were wrong. However, we still insist that motion is relative so we don't even really have the right to say that they were wrong.

It all breaks down as limits are approached: — John Gill

That's ironic, the numbers approach infinity (limitless), as the condition approach the limit. What this indicates is that the limit is created by the principles which govern how the numbers are applied. The limit is created by the numbers approaching infinity, and the principles of application dictate when the numbers will approach infinity.

That's ironic, the numbers approach infinity (limitless), as the condition approach the limit — Metaphysician Undercover

Not so for Planck Time. You'll need a real, live physicist to discuss this properly. It used to be that this limit was variable according to some physical features.

This is a very interesting subject which you bring up here, but my opinion is somewhat opposite to what you say. I think that mathematics allows us to make many very accurate predictions based on statistics and probabilities, without having any accurate description of the mechanisms involved. So for example, Thales apparently predicted a solar eclipse in 585 BC. I think it's common that we observe things, take note of the patterns of specific occurrences, thereby becoming capable of predicting those occurrences, without understanding at all, the motions which lead to those occurrences. — Metaphysician Undercover

I don't see where you differ with me. Mathematics can also speak of patterns that had not been yet observed! Because it tackle all possible structures in an unlimited manner. That's what I meant when I said *before-hand*, if we had good mathematics about ellipses, parabola, hyperbola, etc.., even before we observed the movements of planets, that knowledge would make it easier for the astronomer to discover the pattern of movement of those planets, because as I said: many times humans don't see what they don't look for. If you have the descriptive arsenal before-hand, you'll recognize and thereby predict easily the behavior of matters with fewer observations because it would look familiar to what you have experienced in say the platonic world about approximately similar behaviors. So of course we don't need any understanding of the mechanisms which lead the planets to move the way they do, we already had the descriptions of their orbits hundreds of years before we discovered that they are moving approximately along them. Notice that we didn't coin the mathematical structures describing orbits (ellipses, hyperbola,etc..) after the observation had been made, we actually imagined it starting form more trivial observations made in our planet, like of approximately circular objects, then we freely contemplated more alternative variety of similar structures in the platonic imaginary world, this free contemplation is what made us arrive at those orbit mathematical structures way before any application or observation was discovered. And I think that's one of the most important jobs of mathematics, to supply such descriptive arsenal that objects in our world may *possibly* follow. I'd say perhaps, the particle physics objects move along some paths that we don't have the mathematical descriptive arsenal necessary to match them with, that's why we remain in ignorance about them.

Not so for Planck Time. You'll need a real, live physicist to discuss this properly. It used to be that this limit was variable according to some physical features. — John Gill

Actually, what I described is exactly the case with Planck time. The limit (Planck time) is the product of the theories being used. This is from Wikipedia: "The Planck time is the unique combination of the gravitational constant G, the special-relativistic constant c, and the quantum constant ħ, to produce a constant with dimension of time.".

You might say that these theories represent something real in the universe, but they only do so to the extent of our understanding. Any misunderstanding creates a limit to the mathematics which is not representative of a real limit in the physical universe. When this is the case, application of the mathematics to observations will produce an abnormal occurrence of infinities, (as we see in quantum physics) as the things being observed go beyond the limits created by the lack of understanding expressed by the theor.

I don't see where you differ with me. Mathematics can also speak of patterns that had not been yet observed! Because it tackle all possible structures in an unlimited manner. That's what I meant when I said *before-hand*, if we had good mathematics about ellipses, parabola, hyperbola, etc.., even before we observed the movements of planets, that knowledge would make it easier for the astronomer to discover the pattern of movement of those planets, because as I said many times humans don't see what they don't look for. — Zuhair

I don't see how this notion of "before-hand" can be realistic. Before-hand, there are infinite possibilities for spatial shapes. So it could not be practical to produce all these possible models prior to observations, then after observing, attempt to fit a model to the observation. What is really the case, in practise is that we see something, observe it and take notes, then we create a model to represent it. So we work from the purest form of mathematics, simple numbers to represent observed occurrences of events, with the most primitive spatial representation of those events, toward creating a more complex spatial form, or pattern, which fits to those occurrences.

I agree that it is necessary to keep our minds open to "all possible structures" but to approach a problem with all possible structures already apprehended, and developed on paper, is not practical because unrestricted possibility approaches infinity. Therefore we take the information presented to us by the particular problem, and create structures as possible solutions, according to what is required, striving to keep our minds open to many possibilities because once we accept one we tend to close our minds to others. And this is not good, because we never actually obtain "the ideal."

f you have the descriptive arsenal before-hand, you'll predict easily the behavior of matters with fewer observations because it would look familiar to what you have experienced in say the platonic world about those orbits. — Zuhair

This is not true, because the "descriptive arsenal" would have to contain all of the countless possibilities. Then, you would have to compare the observations with each of those countless possibilities to determine which description is the best. This is highly impractical, and not representative of the way that we actually proceed. in reality we create the "platonic world" to represent what we have observed.

Notice that we didn't coin the mathematical structures describing orbits (ellipses, hyperbola,etc..) after the observation had been made, we actually imagined it form more trivial observations on our planet, then we freely contemplated more variety of structures in the platonic imaginary world, this free contemplation is what made us arrive at those orbit mathematical structures way before any application was discovered. — Zuhair

I don't think that this is true either. Kepler noted that planetary orbits were not eternal perfect circles as postulated by Aristotle. This knowledge was produced by inconsistent positioning. Kepler approached this problem with numerous possible curves, and found the elliptical orbit to be most suitable. But I don't see any indication that there are any elliptical orbits available for observation on our planet, from which Kepler could have copied the design, and no indication that the design was created for anything other than the purpose of modeling planetary orbits..

And I think that's one of the most important jobs of mathematics, to supply such descriptive arsenal that objects in our world can possibly follow. I'd say perhaps, the particle physics objects move along some paths that we don't have the descriptive arsenal necessary to match them with, that's why we remain in ignorance about them. — Zuhair

I agree that we are very close to complete agreement on these issues, that's why I have pointed out the specific places of disagreement with "not true", hopefully to help you see that my perspective is better suited. Though you might bring me around to your perspective instead.

So we're back to this question of art (beauty, aesthetic), or utility. Do mathematicians create all sorts of shapes, forms, and structures simply because they are beautiful, and have them lying around for possible use, or do they create them to serve as solutions to particular problems. You seem to choose the former, that mathematicians create a whole arsenal of beautiful shapes, simply because they are beautiful, then physicists and cosmologists might choose from this collection of designs, those which are suitable to them. I think that mathematicians create their forms with purpose, as potential solutions to particular problems.

So we're back to this question of art (beauty, aesthetic), or utility. Do mathematicians create all sorts of shapes, forms, and structures simply because they are beautiful, and have them lying around for possible use, or do they create them to serve as solutions to particular problems. You seem to choose the former, that mathematicians create a whole arsenal of beautiful shapes, simply because they are beautiful, then physicists and cosmologists might choose from this collection of designs, those which are suitable to them. I think that mathematicians create their forms with purpose, as potential solutions to particular problems. — Metaphysician Undercover

Honestly I think its both cases. Some structures were actually contemplated due to their own beauty in a platonic world, while others raised secondary to observations and need for application as you depicted. I in some sense do agree with you that we'll have infinite possibilities if we were to contemplate just purely, but there are definitely some scenarios that are more attractive platonically speaking than others.

Example of "mathematics prior to observation" is that the orbit of planets which suits more of an ellipse. Ellipses where there on board since ancient Greek, and their study didn't arise from contemplating planet orbits as you think. No they actually were studies on our earthly structures which are simply about inclined sections of cones. Then Kepler picked what is already available and matched it with observations about planets movements.

Other examples include Riemannian n-dimensional geometry, this was contemplated before relativity theory and other recent theories of physics which use many dimensions. Also non-Euclidean geometry was long contemplated by Al-Tusi and also by various mathematicians long before relativity theory called for their use, and they did arise from the pure study of geometry in the platonic realm, mainly becuase of the non-proof of parallel postulate. Pure Platonic contemplation is not random, and so it pursue interesting alternative structures, and also can pose general mathematics investigating wide array of those structures.

On the other hand I agree with you that there are other situations where the mathematics had been created AFTER the observations had been made, i.e. "observation prior to mathematics" direction: like mathematics about the DNA double-helix structure, and also like quantum mechanics, Dirac delta function, etc...

So in real practice both lines are occurring, the pure investigation of those entities in the platonic laboratory and on the other hand the on-demand construction of mathematical entities to match needed application. We can say that mathematics can work to enrich our knowledge about the world by detecting behaviors in the later that we already knew of in the platonic world (in approximate manner), and also the other direction is also true, that observation in our real world as the source and the motive to contemplate certain platonic structures, so our world enriches mathematics also. It is a bilateral movement. And I think this bilaterality is important. And it (the bilaterality) should be observed if we are to have mathematics help enrich our knowledge about our world.

That said. I think we need to unleash both directions!

Honestly I think its both cases. Some structures were actually contemplated due to their own beauty in a platonic world, while others raised secondary to observations and need for application as you depicted. I in some sense do agree you that we'll have infinite possibilities if we were to contemplate just purely, but there are definitely some scenarios that are more attractive platonically speaking than others. — Zuhair

I wonder if this is even true. Can you bring an example of an imaginary structure, created neither for the purpose of copying something in the world, nor for the purpose of resolving a specific type of problem. I suppose that it would be very difficult to distinguish whether the structure was created purely for beauty, or for utility. And, if you were to go and create one right now, saying you created it purely for beauty, I would argue that you did it for the purpose of your argument. So we might leave this point as unresolved, or even unresolvable. However, we might still argue our opinions, in an attempt to get the other into our own metaphysical camp.

Example of "mathematics prior to observation" is that the orbit of planets which suits more of an ellipse. Ellipses where there on board since ancient Greek, and their study didn't arise from contemplating planet orbits as you think. No they actually were studies on our earthly structures which are simply about inclined sections of cones. Then Kepler picked what is already available and matched it with observations about planets movements. — Zuhair

OK, I think you're right here, but this does not exclude the possibility that the ellipse was created for another purpose. So it doesn't really force the conclusion that the model was produced prior to having an application. We just might not be a ware of the application it was first designed for.

Other examples include Riemannian n-dimensional geometry, this was contemplated before relativity theory and other recent theories of physics which use many dimensions. Also non-Euclidean geometry was long contemplated by Al-Tusi and also by various mathematicians long before relativity theory called for their use, and they did arise from the pure study of geometry in the platonic realm, mainly becuase of the non-proof of parallelity postulate. Pure Platonic contemplation is not random, and so it pursue interesting alternative structures, and also can pose general mathematics investigating wide array of those structures. — Zuhair

Again, we cannot really resolve the question this way, because there would always be a reason for speculating about non-Euclidian geometry. You say that it is because the non-proof of the parallel postulate, but as I think I indicated earlier, Pythagoras was dissatisfied with the irrational nature of the square, and we also have the irrationality of pi. These are all good reasons to speculate about non-Euclidian geometry, and it would be difficult to prove that utility is not at the base of this dissatiisfaction.

Pure Platonic contemplation is not random, and so it pursue interesting alternative structures, and also can pose general mathematics investigating wide array of those structures. — Zuhair

So let's assume as we would agree, that pure Platonic contemplation is not random. How could we assign anything other than utility (what Plato calls "the good") as the thing which delivers us from randomness? If we were to contemplate pure beauty, completely devoid of utility, wouldn't this be randomness itself?

So in real practice both lines are occurring, the pure investigation of those entities in the platonic laboratory and on the other hand the on-demand construction of mathematical entities to match needed application. We can say that mathematics can work to enrich our knowledge about the world by detecting behaviors in the later that we known in the platonic world (in approximate manner), and also the other direction is also true, that observation in our real world as the source and the motive to contemplate certain platonic structures, so our world enriches mathematics also. It is a bilateral movement. And I think this bilaterality is important. And it should be observed if we are to have mathematics help enrich our knowledge about our world. — Zuhair

I'm not convinced that there is such a thing as pure investigation in the platonic laboratory. I think this would require that we totally remove ourselves from the necessities of life, and the constraints of the physical world, and this is impossible. That is why Plato himself settled on "the good", as that which makes the intelligible objects intelligible. The good inheres within the essence of the intelligible object therefore, as what gives it the characteristic of being intelligible. If we remove this good, we are overwhelmed by randomness. And randomness might itself be the most beautiful thing there is, such that we would be overcome by the beauty of pure randomness, but such beauty would be inherently unintelligible because of the nature of randonmness, and therefore impossible to be the source of any type of structures.

I wonder if this is even true. Can you bring an example of an imaginary structure, created neither for the purpose of copying something in the world, nor for the purpose of resolving a specific type of problem. I suppose that it would be very difficult to distinguish whether the structure was created purely for beauty, or for utility. — Metaphysician Undercover

I didn't claim an absolute platonic approach to mathematics, definitely not. But it appears to me that once we get to think about some mathematical problem, even if that raised within the context of solving a problem about our world, or certain application, etc.., we can easily figure out many offshoots that can be developed on purely theoretic basis, those offshoots can later on have applications, but I would guess that many were coined even before seeking some application that was raised to instigate them. Lets take a very simple example, lets take the negative integers, those were contemplated by the Chinese as well as the Greek mathematician Diophantus way before the Arabs made full use of it in commerce and related business. Actually Diophantus objected to their existence as being "false", this shows that he knew them but saw no application for them and rejected them along philosophical lines, much as many people rejected imaginary numbers, irrational numbers, and "transfinite" numbers, non-standard naturals etc.. Yes you can start with a shape in application and then try to figure out its rules in the platonic world then problems will raise and their solution can be approached in the mathematical realm and many offshoots of that approach may later turn to have applications. As I said Non-Euclidean geometry was an offshoot of the challenge caused the the "parallel postulate" of Euclid, now this is a pure mental problem, it was not related to utility, many geometricians in their endeavor to prove the fifth postulate actually discovered the roots for non-Euclidean Geometry. The equi-interpretability of non-Euclidean Geometry with the Euclidean was established therefore establishing the independence of the fifth postulate (the parallel postulate), this is a pure platonic problem, however it turned to have a utility later one in Einstein relativity theory. Same to be spoken about Riemannian n-dimensional Geometry, which raised from within solving problems that seem to me to be purely related to theoretic mathematics, rather than being instigated by some particular application in the real world, later on Relativity theory used this multi-dimensional space of Riemann. Regarding shapes ellipses, and others might have been raised from some problem about the real world as contemplating the idea of a shape nearer to that of an egg, or the inclined cut section of a cone, etc.. but that also can be seen as an offshoot of contemplating closed figures beginning from one with ideal symmetry, i.e. the circle, to ones less symmetrical and so on, one can Platonically think of a whole spectrum of these.

I think (though I don't have a proof ) that many mathematics even if initiated in application, would have a pure platonic intermediates bringing many possible structures, then many of those would fade away because they don't have applications, while those that have, will continue also raising problems about them in the platonic world leading to many offshoots, some of which would have applications, and so on..

I don't think the whole of mathematics, i.e. every step of it, was instigated by some utility in a direct manner to solving a problem in the real world, neither do I maintain that the whole of mathematics did or even could have proceeded in an absolutely purely platonic manner. Its a mixture of both that we have.

Can you bring an example of an imaginary structure, created neither for the purpose of copying something in the world, nor for the purpose of resolving a specific type of problem — Metaphysician Undercover

You ask, and it shall be so . . .








I work with this sort of thing all the time.

Later: Interesting, my image came up perfectly, then when I checked part of the code had been deleted (by whom?). It's coming up now. Let's see how long it will last this time.

Well, it didn't last. There must be a rule against linking to images on this site.

What is it? I can't open it.

I can't get the image to stay. Suggestions?

OK. Got the message. $5/month for permission to hot-link images. Maybe, maybe not. We'll see if it's worth it. :sad:

@John Gill, @Zuhair, @Metaphysician Undercover Some 340 posts in. As the current main contributors would you all be willing to summarize in two or three sentences what this thread is about up until this point and what the issues are?

As a professional math guy, here is my opinion regarding infinity and the complex plane:

For me and my colleagues, |z| getting larger and larger without bound means z -> infinity. An actual point at infinity is irrelevant in practice. If I think of time going to infinity, I mean it in this sense. If you look at the projective plane sitting below the Riemann sphere, you can see z moving further and further out, without bound, and as it does so its projection on the sphere moves closer and closer to the north pole, but never reaches it.

www128.pair.com/r3d4k7/Mathematicae7.html

https://en.wikipedia.org/wiki/Riemann_sphere#Extended_complex_numbers


You can't even do links here it seems. This is a page on my website, and it comes up first time, then won't connect. It keeps returning to this forum. I don't know what is going on. The Wikipedia site keeps coming up. Not mine.

I think we've covered much ground in this thread so it would be difficult to summarize. The pivotal issue seems to be the reality of Platonic objects. So we had an extensive discussion concerning what various mathematical symbols are representative of, whether they represent objects, if so, what kind of objects, and particularly the identity of the objects. The law of identity was prominent. . It appears like axioms which treat an infinite collection as an actual object, require the reality of Platonic objects. However, the point I argued is that mathematical objects do not have an identity which is consistent with the law of identity. .

Thank you! Sounds like some interesting points.

the reality of Platonic objects. — Metaphysician Undercover

Quick question: did you-all settle on a definition (at least for your present purposes) of "reality?" Yes or no is fine. If you did I'll go find it. If you did't....

No, we didn't really discuss the nature of reality. We discussed the difference between trying to make true descriptions of objects, and creating imaginary figures. Both of these, I would say, are part of reality. The issue I think is whether the imaginary figures qualify to be called objects. So the question would be to define "object", and this is why I turn to the law of identity. An object has a unique identity.

Is kind of why I asked about a working definition. It seems to me all those matters depend on context and what is required in and for the context. Kind of a shame you-all didn't. A good topic deserves good grounding.

I'm confused by the distinction actual vs potential infinity?

From wikipedia I get:

Potential infinity is a never ending process - adding 1 to successive numbers. Each addition yields a finite quantity but the process never ends.

Actual infinity, if I got it right, consists of considering the set of natural numbers as an entity in itself. In other words 1, 2, 3,.. is a potential infinity but {1,2, 3,...} is an actual infinity.

In symbolic terms it seems the difference between them is just the presence/absence of the curly braces, } and {.

Can someone explain this to me? Thanks. — TheMadFool

The principle of induction is intimately related to the recursion theorem. The principle of induction states that if $\phi$ is a predicate such that $\phi(0)$ and $\phi(n) \rightarrow \phi(n + 1)$ for all $n \geq 0$, then $\phi(n)$ is true for all natural numbers $n$. A proof of the induction principle is a recursive function that transforms a proof of $\phi(0)$ and a proof of $(\forall n \geq 0)(\phi(n) \rightarrow \phi(n + 1))$ into a proof $(\forall n)\phi(n)$ for any predicate $\phi$.

The induction principle is interesting because it allows us to conclude that $\phi(n)$ is true for every natural number $n$ without checking that $\phi(n)$ is true for every natural number $n$. In other words, it allows us to conclude that every member of an infinite set of objects possesses some property by performing a finite amount of work. There are analogous induction principles for other infinite sets. For example, some proof assistants automatically generate induction principles for arbitrary datatypes satisfying certain technical conditions.

In other words, there is a systematic relationship between certain infinite sets and inductive proofs (recursive functions), the latter of which are "potential infinities" in your terminology. In other words, induction principles (recursive functions) systematically translate between statements about "potential infinities" and statements about the elements of certain infinite sets. This has nothing to do with curly braces or other arbitrary syntactic phenomena unless you are committed to some kind of hardline formalism about mathematics.

Here is another example: consider the "potential infinity" defined by the Fibonacci sequence. You can generate every Fibonacci number using a recursive function defining the sequence. In other words, the recursive function defines the first, second, third, and so forth, Fibonacci number. However, you can always consider the collection of elements generated in this way by saying: "suppose that $n$ is a number in the Fibonacci sequence." What you are talking about, in the latter case, is an infinite set of objects - there is no limit to the number of objects that satisfy this condition, although there are restrictions on the kinds of objects that satisfy the condition.

Is kind of why I asked about a working definition. It seems to me all those matters depend on context and what is required in and for the context. Kind of a shame you-all didn't. A good topic deserves good grounding. — tim wood

The nature of reality is not an issue here. The nature of an object is. "Reality" is the more general concept, so there is more to reality than just objects. What we are interested in here, is objects.

Here is another example: consider the "potential infinity" defined by the Fibonacci sequence. You can generate every Fibonacci number using a recursive function defining the sequence. In other words, the recursive function defines the first, second, third, and so forth, Fibonacci number. However, you can always consider the collection of elements generated in this way by saying: "suppose that nn is a number in the Fibonacci sequence." What you are talking about, in the latter case, is an infinite set of objects - there is no limit to the number of objects that satisfy this condition, although there are restrictions on the kinds of objects that satisfy the condition. — quickly

Now the issue, which we discussed already in the thread, is whether or not a written numeral necessarily represents an object. In actual usage, the numeral might be used to represent an object, or it might not. If it doesn't represent an object, then any supposed count is not a valid count.

Your example seems to create ambiguity between the symbol, and the thing represented by the symbol. So you would have to clarify whether there is actually existing numbers, existing as objects to be counted, otherwise the claim of "an infinite set of objects" is false. As proof, it doesn't suffice to say that it is possible that a numeral represents an object And actual usage of symbols demonstrates that it is possible that the symbol represents an object, but also possible that it does not. To present the symbol as if you are using it to represent an object, when you really are not, is deception.

Now the issue, which we discussed already in the thread, is whether or not a written numeral necessarily represents an object. In actual usage, the numeral might be used to represent an object, or it might not. If it doesn't represent an object, then any supposed count is not a valid count.

Your example seems to create ambiguity between the symbol, and the thing represented by the symbol. So you would have to clarify whether there is actually existing numbers, existing as objects to be counted, otherwise the claim of "an infinite set of objects" is false. As proof, it doesn't suffice to say that it is possible that a numeral represents an object And actual usage of symbols demonstrates that it is possible that the symbol represents an object, but also possible that it does not. To present the symbol as if you are using it to represent an object, when you really are not, is deception. — Metaphysician Undercover

I don't understand why skepticism about the meaning of mathematical language is relevant to the discussion about potential and actual infinities. The notion of potential infinity is the notion of a process that can be repeated indefinitely. For example, one can always consider successively larger values of the Fibonacci sequence. The language of sets (types, classes, etc.) provides one way of talking about the elements generated by such processes. The notation used is entirely irrelevant.

You've described a potential infinity, but not an actual infinity. To understand an actual infinity we need to understand the actual existence of the elements represented by mathematical language.

This has nothing to do with curly braces or other arbitrary syntactic phenomena unless you are committed to some kind of hardline formalism about mathematics. — quickly

I read a little of the Stanford Encyclopedia of Philosophy article on the topic.

It seems that Aristotle thought of an actual infinity to be akin to the infinite divisibility of, for example, a finite length in that it is "complete" and potential infinity to be something like the non-terminating process of adding 1 to any number and getting the next greater number.

You've described a potential infinity, but not an actual infinity. To understand an actual infinity we need to understand the actual existence of the elements represented by mathematical language. — Metaphysician Undercover

When you say "the set of natural numbers," you mean "the set of all objects that can be generated from zero and the successor function and which satisfy the Peano axioms" (or something similar). In other words, I can use the language of sets to talk - all at once - about every object satisfying these conditions. If that doesn't satisfy your condition that someone "understand the actual existence of the elements represented by mathematical language," then perhaps we simply have different conceptions of the purpose of mathematical language (e.g., the language of sets).

The Difference Between Actual and Potential Infinity

Imagine the real numbers in the interval [0,1]. Is there an actual or potential infinity of them?

Well one answer (the wrong answer) is we can go on dividing forever by 2 (say) so there must be an actual infinity of reals in the interval.

But we can only go on dividing forever in our minds - if we tried this in reality, we'd never finish dividing (process goes on forever - we'd never finish) - so the possibility of infinite division is just a figment of our imagination (like its possible to levitate in your imagination - but not in reality).

There are therefore (according to Aristotle and I agree with him), a potential infinity of real numbers in the interval [0,1]. When we perform division by 2, we actualise the number 1/2. When we perform division by 2 again, we actualise 1/4 and 3/4. And so on. At no point in this process is there ever an actual infinity of real numbers.

So in summary, actual infinity is a purely imaginary concept . It is sometime mentally convenient to regard (say) the set of reals as actually infinite - but that is not telling us anything about reality any more than our ability to imagine a square circle or a fairy.

What is the difference between actual infinity and potential infinity?

I'll attempt to answer the actual question.

In my opinion it's 37.54, but my uncle here says 289, and my aunt insists it's only 9.

Well one answer (the wrong answer) is we can go on dividing forever by 2 (say) so there must be an actual infinity of reals in the interval.

But we can only go on dividing forever in our minds - if we tried this in reality, we'd never finish dividing (process goes on forever - we'd never finish) - so the possibility of infinite division is just a figment of our imagination (like its possible to levitate in your imagination - but not in reality). — Devans99

You are rejecting the idea of "forever" which translates to "infinitely long time". So you reject the very concept of infinity because your premise rejects the very concept of infinity.

We've been down this road once before. ("What time does the clock show that has been going on since infinite time.")

If you can't imagine it, fine. But arguing that infinity does not exist because you can't imagine it is on hand a weak argument, on the other hand, a subjective argument.

A past infinity of time is an impossibility. For example: perpetual motion is impossible, we have motion, hence time must have a start. There are a several other good arguments that the past must be finite that I won't repeat again here. Future time is obviously potentially infinite only. So I don't see you can use past/future time to justify the existence of actual infinity?

I would suggest you instead focus on elapsed time. Is there, for example, an actual infinity of moments in a second? That is a more interesting question. A few thoughts:

- There must be a temporal difference between 'now' and 'then' else 'now' would be 'then'
- The temporal difference can’t be zero / infinitesimal else 'now' would be 'then'
- So there is a finite difference between 'now' and 'then'
- Could that finite difference be infinitely divisible?

A similar question is: 'when you move your hand, do the particles of your hand pass through an actually infinite number of positions?'. Or do they do something similar to a quantum jump of an electron, on a tiny Planck level scale?

Obviously, this is all related to Zeno's paradoxes, which I think are indicative that time and space are discrete. I can't prove it though obviously.

We have been considering models of infinite divisibility here:

https://thephilosophyforum.com/discussion/7320/continua-are-impossible-to-define-mathematically/p1

It seems to me that there are no sound mathematical models of infinite divisibility? So that may lend some weight to the idea that space/time/motion are discrete?

If space and time are creations then they must be finite and discrete (impossible to create anything infinitely big or small). But I feel a more direct proof of the discrete nature of space/time/motion is what is required...

@Meta,

I've read through much if not most of your writings in this thread. I think I understand where you're coming from. I found two remarks I can get some traction on.

From my perspective, ZFC has unsound axioms concerning the nature of objects, as we discussed earlier. Therefore any proof using ZFC is unsound. — Metaphysician Undercover

ZFC is unsound. Well yes. Of course ZFC is unsound. Its deductions are valid, meaning that each theorem follows from the axioms and prior theorems according to the purely syntactic rules of deduction.

But it's unsound, in the sense that its premises are not necessarily true. It's not even clear that they are meaningful. Personally I don't think they are. I don't personally believe that sets, as understood in ZFC, exist in the real world. You're arguing that they don't but of course they don't. I'm in total agreement with you on this point.

I'll go you one better. ZF is unsound. No need to invoke the axiom of choice. In ZF we have the axiom of infinity, which states that there is an infinite set. Nothing in the physical world corresponds to that as far as we know. So ZF is unsound.

So ok. ZF[C] is unsound. This is a commonplace observation. Might you be elevating it to a status it doesn't deserve? Are you thinking of it as an endpoint of thinking? What if it's only the beginning of philosophical inquiry?

Here's what Bertrand Russell, who knew a thing or two about both mathematics and philosophy, said in 1910. This quote is usually given as a one-liner but the entire paragraph deserves repeating.

“Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.” — Uncle Bertie

https://www.goodreads.com/quotes/577891-pure-mathematics-consists-entirely-of-assertions-to-the-effect-that

Since this understanding is over a century old, it seems reasonable to take it as a starting point for the study of the philosophy of mathematics; and not the endpoint, as you have done. Mathematics is not literally true. It's a language for expressing physical theories; and it's a discipline unto itself that must be taken on its own terms.

Let's take Russell's quote one step farther. In 1960 the physicist Eugene Wigner published an article called, The Unreasonable Effectiveness of Mathematics in the Natural Sciences.

The title says it all. Mathematics is so fundamentally divorced from reality that it is unreasonable that it should have anything to tell us about the real world.

And yet it does.

That, I submit, is one of the key issues of the philosophy of math. We agree that math isn't true, but it nevertheless seems intimately related to the real or the actual.

This is what I mean when I say that "ZCF is unsound" is the starting point of philosophical inquiry into the nature of mathematics; not the end of it. I get the feeling you think it's the last word. It's only the first.

Next: Part 2.