Not faith so much as care and attention. — Banno

No, it's faith.

Have you some alternative? — Banno

Everybody grows the psychological structures they need to deal with the life they have. I can't tell you how you need to think in order to successfully be you. If deep suspicion about mental stuff, coupled with strong faith in the world is the outlook your psyche thrives with, then God bless it.

How rude! :razz:

:nerd:

It's a belief if one thinks that it's so. It's faith if one believes it is so despite the evidence.

But this is about method rather than belief. What is suggested is that if there is an inconsistency we reconsider what it is we are saying about how things are, rather than deciding that the world must be inconsistent.

There presumably is a point at which the world is so confusing that our reconsidering of what we say is insufficient to explain what is going on. But I hope we're not there yet.

The risks are that we are hiding behind grammar, using it as a shield against metaphysical inquiry rather than engaging it, or that we miss phenomena that actually resist conceptual capture. I acknowledge that.

We should remember that we unfortunately have lost Plato's original book, where likely the Eleatic school would have made their own viewpoint. Now we have just the texts of those who were against the Eleatic school, the "mainstream" Socratic-Platonic school. — ssu

That would indeed be of great interest. I wonder if we could construct a reply that they might have made?

I do admire your devotion to the practical. Detaching yourself from it and purely following the contours of the mind will set you out in front of contradictions. — frank

It may well do so. It may also set you in front of outright fantasies that have no connection with any kind of truth. The theoretical stance needs a grounding in ordinary life, if only because there is no escaping ordinary life. Not even philosophers can really escape from it.

Paradoxes occur when we say things incorrectly. The world cannot be wrong, but what we say about it can be. — Banno

There used to be a story that aerodynamics showed that bumble bees cannot fly. Did anyone doubt that bumble bees can fly? I don't think so. I understand that aerodynamics is now clear that bumble bees can fly. But in that case, it was clear how the world is, as opposed to how we thought about it, or described it. Why is it that we don't just point out that the arrow will leave the bow, and that Achilles will catch up with the tortoise? It seems that we cannot simply correct infinity, but have to learn to live with it. Calculus fits in to that project.

I'm not convinced that all paradoxes can be resolved. Some of them, like Zeno's, may be inherent in the project of saying things about the world. Self-reference is another part of our language that we struggle to escape from. Are we sure that we cannot just live with at least some of them?

Everybody grows the psychological structures they need to deal with the life they have. I can't tell you how you need to think in order to successfully be you. If deep suspicion about mental stuff, coupled with strong faith in the world is the outlook your psyche thrives with, then God bless it. — frank

Each to their own, I suppose. But is that how you think about your own views, as well? If that's what's going on, why do we bother arguing with each other?
It is not uncommon for people to believe that the ordinary world is not really real, but is some kind of dream or fantasy or shadow. I doubt they would welcome a pat on the head and permission to believe whatever they need to believe.

insisting that Zeno's infinities are about how the world is and not how we talk about it is question begging. That's exactly what is in question.

But you are right that 's response is just his way of excusing his own views from critique... :wink:

But you are right that ↪frank's response is just his way of excusing his own views from critique.. — Banno

Actually I just assumed my views would bore you.

What we have are ways of talking, language games, a grammar, or a paradigm - whatever you want to call it. Infinity is a mathematical notion that we can use to calculate physical results. It is not an ontology. — Banno

And we do use it. It is, well, essential.

What we don't have is a proof. Or how it fits everything else.

What we have is threads like this constantly coming up. That itself tells something.

Now, am I crazy to argue that there might be something more to be said here? Perhaps I'm annoying in repeating myself, but I think this is a topic worth wile to talk about.

And ontology? Well, what is the relationship of infinity with numbers?

If we define numbers being arithmetic values that represent a certain quantity over all other quantities, what if we skip away the "arithmetic" part? What if we say that infinity represents a certain unique quantity over all quantities also?

nsisting that Zeno's infinities are about how the world is and not how we talk about it is question begging. That's exactly what is in question. — Banno

Let me try to be a bit clearer. I cited the bumble bee just because it was a case where there isn't much, if any, doubt about how the world is as opposed to how we think about it. I wanted to contrast that with the issues about infinity. There are two ways of approaching Achilles & co. One is Zeno's way, the other is simple arithmetic, which one might think is how the world is. But that's not how we respond. I'm not sure I understand why, exactly, except that both are methods of calculation, so both come from the same stable. (Contrast the bumble bee). Possibly, we could choose to stick with simple arithmetic in the Zeno case. So perhaps the reason is that we need it for other calculations, such as the orbits of planets and other issues in geometry. In which case we need both. In other words, this choice cannot really be posed as between how the world is and how we talk about it.

We can be pretty confident that space is not infinitely divisible and yet still use calculus to plot satellite orbits. — Banno

This is an interesting remark. Many would say that "space" is conceptual only. And if it is, how could it be anything other than the way we represent it, as infinitely divisible?

"Space" might be the distance between two objects, but space is not what is measured, distance is. Furthermore, we commonly assume a medium between two objects, air or something. And space is not the air. Clearly, if we were talking about air, we wouldn't represent it as infinitely divisible.

So this is why there is a problem, when we get down to the basics, the medium between the nucleus of the atom and the electrons of the atom for an example, we really don't know what the medium is. The proposal of aether has been dismissed, so we just produce an artificial (imaginary) medium, some fields or something like that. Since the concept of "space", and its accompanying mathematics provide for infinite divisibility, and the proposed medium is simply conceptual, how could the medium be modeled in any way other than a way which is consistent with the concept "space", and the related mathematics, i.e. as infinitely divisible. Without proposing a real medium with real restrictions to divisibility, to propose that the fundamental medium between things is not infinitely divisible, according to how it is conceptualized as "space", is somewhat incoherent.

Many would say that "space" is conceptual only. — Metaphysician Undercover

Space is an aspect of gravity. Mass tells space how to curve, space tells mass how to move.

If there are an infinite number of natural numbers, and an infinite number of fractions in between any two natural numbers, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and an infinite number of fractions in between any two of those fractions, and... then that must mean that there are not only infinite infinities, but an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities. and an infinite number of those infinities, and... (infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and that infinitely times. and...) continues forever. and that continues forever. and that continues forever. and that continues forever. and that continues forever. and.....(…)… — an-salad

When people say, “there are infinitely many fractions between 1 and 2,” they mean you can always find another fraction in between, no matter how close two numbers are. For instance, between 1 and 2 you can pick 1.5, and between 1 and 1.5 you can pick 1.25, and you can keep doing that forever. It never ends.

But that doesn’t automatically mean there are “different levels” of infinity stacked on top of each other. It’s the same idea repeating you can keep splitting the space and still find more numbers.

If you want a real case where infinities truly differ in size, you have to switch from “fractions” to “all decimal numbers,” like numbers with endless digits. There are way more of those than there are whole numbers or fractions. That’s where mathematicians say one infinity is bigger than another.

And if I put on a “Wittgenstein hat” for a second, he migft say: don’t let the word “infinite” hypnotize you. Most of the time it just means “this process can continue without end,” not “we’ve discovered a weird tower of endless infinities.”

The question is, at what level of explanation should this incompatibility be situated? at the physical level, as physics usually assumes, or at the level of the rules of mathematics? — sime

I think it must be both. The theoretical mathematicians who practise what they like to call pure mathematics want to be free from the constraints of the physical world. So, they may produce axioms independently of the requirements of physics, and other sciences. However, the axioms which get accepted and become conventionalized are the ones which are applicable. Then by the time the use of any axioms become standard practise, they have been selected for, by the needs of the scientists.

Therefore we can separate the two in principle only. We put the hypotheses of science in one category and the hypotheses of mathematics in another category. But if we maintain the supposed separation into a description of actual practise, science must have logical priority. Ultimately then, conventional and standard mathematics has been shaped to meet the descriptions of scientists. So the incompatibility is within the descriptions provided by physicists. We must maintain a different, real separation though, and that is between descriptions and the real world. It is not necessarily a feature of the real world, which causes incompatible mathematics to be accepted, but perhaps a mistaken description.

So the conventional mathematics is shaped by demand, and the demand is the sciences. The incompatibility manifests in the mathematics which has been conformed to the descriptions of the sciences. Therefore the descriptions provided by the sciences must be faulty, they require incompatible mathematical principles. Like @Banno says, we cannot conclude that the real world is faulty. So what Zeno demonstrated is that our descriptions of motion are faulty, and the mathematics as applied to these descriptions, reveals this by leading to paradox.

I think we should consider the fact that Newton and Leibniz didn't invent calculus for the purpose of solving Zeno's paradox, but for describing trajectories under gravity. Hence the mathematical definition of differentiation that we inherited from them and use today, isn't defined as a resource-transforming operation that takes a mutable function and mutates it into its derivative; rather our classical differentiation is merely defined as a mapping between two stateless and immutable functions. — sime

What you say here about Newton and Leibniz demonstrates how modern mathematics was fundamentally subservient to physics. Since then, the study of pure mathematics, and number theory have become more distinct and separate from that foundation. This actually provides an advantage toward solving these issues, because it allows us to look directly at the incompatibilities within the mathematics, without being influenced by empirical prejudice. Plato's principle "the senses deceive us" is very important.

I believe this is how the heliocentric model of the solar system was figured out. When we remove the mathematics from the influence of our observations of the empirical world, which the mathematics is formed around, then we can extend it in all directions to see where the infinites appear. Each appearance of infinity represents a problem within the empirical description. (In the case of the solar system eternal (infinite) circular motion was the fundamental problem demonstrated by Aristotle.) Then we can make a map of just the problems themselves, and attempt to correlate them and determine a unified underlying cause. A huge number of problems can actually have one simple cause.

But if Zeno's paradox is to be exorcised from calculus, such that calculus has a dynamical model, then I can't see an alternative than to treat abstract functions like pieces of plasticine, that can be sliced into bits or rolled into a smooth curve, but not at the same time. — sime

I think the key issue is that we improperly represent time. Modeling time as the fourth dimension of space implies that time emerges from space. So at the Big Bang, there is something spatial, and time emerges. But that's fundamental incoherent because all activity, including "emergence" requires time. So this representation implies that time existed before time. To rectify this we need to model time as the zeroth dimension, and allow that space emerges from time. This requirement is also indicated through an analysis of the pure mathematics, removed from empirical prejudice. The non-spatial, non-dimensional, "point", is very real and necessary to mathematics. Therefore it must be accommodated for in our modeling of activity in the empirical world. Currently, empirical descriptions do not allow for the reality of non spatial activity (time passage without spatial change). This is a very big problem, which makes the modeling of activity at a non-spatial point completely speculative, and somewhat incoherent.


That is part of what @sime was talking about, the incompatibility between representing objects as having position relative to each other, and as being in motion relative to each other. So when we attempt to unite the concepts of space and time, we must alter them. "Space" gets altered by being "curved", so that standard Euclidian geometry doesn't serve. And "time" gets altered by the relativity of simultaneity, so that there is no objective present.

Actually I just assumed my views would bore you. — frank

Never!

(If the ever did, I'd just not respond.)

What we don't have is a proof. Or how it fits everything else. — ssu

A proof of what, and to what ends? We know it's consistent and we do have rigorous axiomatisations...

I'm not following this, since

And we do use it. It is, well, essential. — ssu

seems to be saying that it does fit in with everything else...

What would count as success here? hat doubt would be removed, what practice would change if the "proof" were given?

The case of the bumble bee, if true, shows that the theory of flight was incomplete, and now, if the account of how they fly works, has wider application.

One is Zeno's way, the other is simple arithmetic... — Ludwig V

Limits, as against calculating velocities? Let's be clear, these two descriptions are quite consistent with each other. If you are pointing out that Zeno's description is incomplete because he doesn't include the bit where Achilles passes the tortoise, I think we agree.

Good to have you drop past, even in AI form.

Yep; and bringing Wittgenstein in explicitly is interesting. There's a fine line between quantifying over infinities and hypostatising them.

Would it help for us to consider the axiomatisation of calculus?

It just assumes first order logic and extensional equivalence over a domain of the reals. We could go into the difference between a limit and a least upper bound, between ∞ and ω, and how infinity never appears in the axiomatisation - except as shorthand for a process.

Might be novel to consider this stuff in detail. I have only a hazy memory of it from first year pure maths.

Would it help for us to consider the axiomatisation of calculus? — Banno

Sounds interesting. Let's do.

Ok, you first.

:wink:

:gasp:

Even in set talk, nothing magical follows: we can define endlessly many infinite sets (like the rationals between a and b), but that’s just a feature of our notation and rules. The mistake is to treat those definitions as a tower of new “infinities,” instead of keeping “infinite” tied to the procedure.

So we have first order logic, =, +, <, and the Reals. We have the necessities for doing calculations, and a relation "<" and appropriate rules, and unlike the rational numbers, no gaps.

Stealing from ChatGPT,

• A. Field structure (algebraic axioms)
  There are two operations, + and ·, and two distinguished elements, 0 and 1, satisfying:
  
  • Addition is associative: (x + y) + z = x + (y + z)
  • Addition is commutative: x + y = y + x
  • Additive identity: x + 0 = x
  • Additive inverse: for every x, there exists −x such that x + (−x) = 0
  • Multiplication is associative: (x · y) · z = x · (y · z)
  • Multiplication is commutative: x · y = y · x
  • Multiplicative identity: x · 1 = x
  • Multiplicative inverse: for every x ≠ 0, there exists x⁻¹ such that x · x⁻¹ = 1
  • Distributive law: x · (y + z) = x · y + x · z
• B. Order axioms
  There is a relation < on ℝ satisfying:
  
  • Transitivity: if x < y and y < z, then x < z
  • Trichotomy: exactly one of x < y, x = y, or x > y holds
  • Compatibility with addition: if x < y, then x + z < y + z
  • Compatibility with multiplication: if x < y and 0 < z, then x · z < y · z
• C. Completeness (least upper bound property)
  Every non-empty subset of ℝ that is bounded above has a least upper bound (supremum) in ℝ. Formally:
  
  • ∀ S ⊆ ℝ, (S ≠ ∅ ∧ ∃ M ∈ ℝ: ∀ s ∈ S, s ≤ M) ⇒ ∃ L ∈ ℝ: (∀ s ∈ S, s ≤ L) ∧ (∀ L' < L, ∃ s ∈ S, L' < s)
  This is what allows limits, convergent series, and calculus to exist without introducing actual ∞.

This last says, informally, that every non-empty subset of R that is less than some number has a smallest number that is bigger than it... and is what distinguishes the reals from the rationals. SO it's perhaps where our attention might dwel.

I'm with you so far.

If you are following along, I'd encourage you to drop the text in to ChatGPT and have it explain any complexities.

If you are following along, I'd encourage you to drop the text in to ChatGPT and have it explain any complexities. — Banno

:up:

Some resources:

Axioms of Real Number System: an explanation of field structure, order and completeness.

An Introduction to Real Analysis: a University of California text, as a PDF. We're looking at getting to 6.1 and perhaps 8.1.

I suppose the thought here is to show that the limit is not so much made up or defined, but sitting there waiting to be found within ℝ. We construct ℝ then find these interesting results.

Since the concept of "space", and its accompanying mathematics provide for infinite divisibility, and the proposed medium is simply conceptual, how could the medium be modeled in any way other than a way which is consistent with the concept "space", and the related mathematics, i.e. as infinitely divisible. — Metaphysician Undercover

It seems to me that the question of a medium in space is secondary. The first move is to set up a co-ordinates and rules for plotting the position of objects on those. (In other words, the concept is defined by the practice.) Once we have co-ordinate and objects, the question of a medium makes some sense. How non-mathematicians develop the concept is another question. But we can be pretty sure it is by interacting with the ordinary world. Mathematics, in my book, is a development of that.

Limits, as against calculating velocities? Let's be clear, these two descriptions are quite consistent with each other. If you are pointing out that Zeno's description is incomplete because he doesn't include the bit where Achilles passes the tortoise, I think we agree. — Banno

I never intended to suggest that they were in some way inconsistent. On the contrary, the point is that they are both in order. So the question is, why do we prefer to use one rather than the other. Your suggestion is plausible - narrow focus in an analysis can be very helpful, but also very misldeading. The paradox of Zeno's paradox, for me, is that Achilles is precluded from reaching a point that defines the system - the limit. The first step is to divided the distance from the start to the goal, limit, by 2, and so on. The limit is not an optional add-on, (as it seems to be in the case the natural numbers).

And if I put on a “Wittgenstein hat” for a second, he migft say: don’t let the word “infinite” hypnotize you. Most of the time it just means “this process can continue without end,” not “we’ve discovered a weird tower of endless infinities.” — Sam26

I'm sure he would. But it is not so easy to rest content with "this process can continue without end". On one hand, we think that the result of the function for each value is "always already" true. On the other hand, we feel that the result is not available until the function has been applied to each value. What makes this game even more puzzling, is that it seems we can know things about the whole sequence without working out the results of the whole sequence. The first example of this is that we can know that the process can continue without end.

I suppose the thought here is to show that the limit is not so much made up or defined, but sitting there waiting to be found within ℝ. We construct ℝ then find these interesting results. — Banno

We are not comfortable with the fact that rules have consequences when they are surprising or not what we want.

I'm sure he would. But it is not so easy to rest content with "this process can continue without end". On one hand, we think that the result of the function for each value is "always already" true. On the other hand, we feel that the result is not available until the function has been applied to each value. What makes this game even more puzzling, is that it seems we can know things about the whole sequence without working out the results of the whole sequence. The first example of this is that we can know that the process can continue without end. — Ludwig V

Ya, I agree it’s hard to rest content with “the process can continue without end,” i.e. we feel a real pull in two directions. On the one hand, once the rule is fixed, we want to say the value at each input is “already settled.” On the other hand, we want to say the value is not actually there for us until we run the rule at that input. This is exactly where the philosophical itch lives.

Wittgenstein, as I read him, is to separate “already settled” from “already computed.” The rule determines what counts as the correct next step, and in this sense the sequence is fixed, but it doesn't follow that the whole infinite list exists as a finished object waiting to be inspected. The “always already” feeling comes from the grammar of the rule, not from possession of an infinite completed totality.

The point about knowing things about the whole sequence without grinding through each case is just more of the same. We can know global facts because they are proved from the rule, for example, “this can go on without end” isn't discovered by checking every term, it’s a consequence of how the procedure is defined. The puzzle is real, but the solution isn't to posit a hidden, completed infinity in the background. It’s noticing what proofs actually license us to say about a rule-governed practice.

A rule can fix the standards for correctness without implying that the entire infinite list exists as a finished thing. We often feel “it’s already there” because the rule is firm, but what’s “already there” is the method, not a completed infinite inventory.

It seems to me that the question of a medium in space is secondary. The first move is to set up a co-ordinates and rules for plotting the position of objects on those. (In other words, the concept is defined by the practice.) Once we have co-ordinate and objects, the question of a medium makes some sense. How non-mathematicians develop the concept is another question. But we can be pretty sure it is by interacting with the ordinary world. Mathematics, in my book, is a development of that. — Ludwig V

Well, I can't say I understand exactly what you are proposing, but it seems like you are saying the question of the medium is secondary, but then you explain why it must be primary.

The nature of the medium, in relation to the nature of the substance which is moving, determines the possible positions. So without determining the medium and the substance first, one could set up a co-ordinate system with infinite possible positions, but it would be false if the medium doesn't allow for it. That is also the case with divisibility. The mathematical system could allow infinite divisibility, but in reality divisibility must be determined according to the substance to be divided, and the means of division. So we might start with the co-ordinates and rules for plotting, as you say, but then it would just be trial and error, in application.

So you start out by saying that mathematics ought to be prior, "The first move is to set up a co-ordinates and rules", but then you end with the statement that mathematics is a development from our interacting with the world, which would place it as posterior.

The paradox of Zeno's paradox, for me, is that Achilles is precluded from reaching a point that defines the system - the limit. The first step is to divided the distance from the start to the goal, limit, by 2, and so on. The limit is not an optional add-on, (as it seems to be in the case the natural numbers). — Ludwig V

The problem in this paradox of Zeno's, is the issue which is explained above, as starting with the designation of rules and limits, instead of determining the true limits of the medium and substance first. The rules allow for infinite divisibility, but this does not correspond with the true medium.

Here's a way of looking at it. Suppose the measurement is on the ground, a long tape measure on the ground. Each time Achilles takes a step, the foot is at a new position on the tape measure. And, the section of the tape measure between there and the last step, is never traversed by Achilles. he steps from one position to the next, with a gap in between. So the false premise which Zeno makes is that all the area has to be covered. It doesn't Achilles steps from one spot to the next. Achilles could give the tortoise a short head start, then take one step and be past the tortoise, without ever properly catching up. This is why the nature of the movement and the medium is so important.