But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potential infinite.
— Metaphysician Undercover

Care to edit this? I do not understand the last part. — tim wood

Sorry I left off the suffix, 'ly'. Try this:

But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potentially infinite.

That better?

It was synthesis who raised the idea that God is an absolute. I was just saying that this idea, of God as an absolute was in synthesis' mind.

Perhaps you can explain yourself here. — synthesis

The maple trees look like they're dormant, but if you tap them they'll give you sap, so they're really not dormant. Looks are deceiving.

The Relative and The Absolute stand opposed to each other as that which we use intellectually (the Relative) and that which exist outside of our intellect (The Absolute). All things knowable (intellectual) are relative. These things that exist intellectually are constantly changing, exist in time, therefore their relative nature. — synthesis

I think you've got this backward. The absolutes are ideals, they are within your mind. Things outside your mind are relative. "All things knowable (intellectual) are relative." is an absolute which your mind has for some reason produced. "God" is an absolute which human minds have for some reason produced.

Accessing The Absolute is the goal of all spirituality and religion, as this is where the The Truth lies. And although you can never know this Truth, you can be with and part of it, a need that has apparently driven man's behavior for thousands of years. — synthesis

Once you see that the absolutes are within you, you'll have no problem to access them, just direct your attention that way, and learn how to ignore the external distractions.

Instead of bothering you guys, I think I'll go outside and consult with the Oak and maybe the Maple, as well. — synthesis

It's a very good time to consult with the maples, they've got much to offer. I'm about to go make some syrup myself. The oaks appear to be dormant right now so maybe they've got nothing to offer. Come to think of it, the maples appear to be dormant too. Looks are deceiving because you judge your sensations relative to your intentions. And your intentions may be misguided.

But I would say that while you can't take THE OBSERVATION any further, you can improve your intuitions. — Acyutananda

What do you think improving one's intuitions would consist of? Aristotle placed intuition as the highest form of knowledge in his Nichomachean Ethics. He looked briefly at the question of whether intuition is innate or whether it is learned, and decided it was a combination of both.

In western society we generally consider intuition to be instinctual. It is the inherited aspect of knowledge. When you say intuition grasps the truth of "2+2=4", this would mean that we instinctually accept this as true. However, we still need to learn the meaning of the equation. We are taught it in school, so the instinctual aspect is the attitude that we have toward learning. We accept the teacher (authoritative figure) as the authority, we have a desire to learn, we see the usefulness in what is being taught, so it appeals to our intuitions.

How do you propose that it is possible to improve one's intuitions? Would this be a matter of moral training, to improve one's attitude toward authority? Or what do you think?

I will tell him about your claim that infinities play no role in programs and see what HE has to say about that. — Gregory

You seem to have missed the gist of the conversation Gregory. I said that when they are rounded off, or "terminated" in Ryan's words, such as the example in the op, or using pi as 3.14, then the so-called infinities are very useful. But a process which is said to be potentially infinite, which will necessarily be terminated at some point, cannot truthfully be said to be potential infinite.

2' denotes a number. '1' denotes a number. '+' denotes an operation. '2+1' denotes the result of the operation + applied to the numbers 2 and 1. That result is a number. Therefore, '2+1' denotes a number. — GrandMinnow

Ok, we we have the numbers 2 and 1 denoted, and the operation + is denoted. Where is the result of the operation denoted? It seems to me like you're jumping the gun. Jumping to the conclusion, assuming that the some result of the operation, 3, is already denoted when clearly it is not denoted

That's the reason why we need to denote = 3, if we want to denote some result, because "2+1" on its own does not say 3. Otherwise there would be absolutely no purpose to the "=" because everything which 2+1 equals would already be said simply by saying "2+1". Therefore "2+1" would denote an infinite number of things, and that would make interpretation impossible. Furthermore, equations would be absolutely useless because the right side would just be saying the exact same thing as the left side, along with all the infinite other things that are equal. What would be the point to an equation in which the right side represented the exact same thing as the left? You'd never solve any problems that way, because the problem would be solved prior to making the equation. If you didn't know that the two sides signified the exact same thing already (meaning the problem is solved) you could not employ the equals sign.

-The program spits out numbers as it is being executed, so it doesn't need to be terminated to get something useful from it. — Ryan O'Connor

But spitting out numbers is not something useful. Useful is the application of the numbers towards counting or measuring, or something like that. If the computer is tasked with counting something and does not complete the task it hasn't been useful.

-We can discuss the execution of the program without ever running it (e.g. we can say 'if I executed the program, it would be potentially infinite) — Ryan O'Connor

But what good is that?

n the end, I think you're splitting hairs here. What's your point? — Ryan O'Connor

I can't even remember now, but I believe I said it would be good to rid the system of infinities and you said there is no problem with working with infinities so long as we recognize that they are merely potential.

But what's the point to working with infinities? If an infinity represents an uncompleted tasked, then isn't it better to complete the task before proceeding. After a while the unfinished tasks start to pile up and become a little overwhelming. And if it is a task which is impossible to complete, then to give oneself an infinite task is to set oneself up for failure, so we ought to address the conditions by which this happens so that we can avoid it.

π is often written as the solution to a problem - for one it's what they say is the volume of Gabriel's Horn. — Ryan O'Connor

Obviously, that's not a real solution.

Also, who said math had to be practical? — Ryan O'Connor

This is probably the crux. "Math does not have to be practical". There's a fundamental element of free choice which lies at the base of all of our understanding of everything. "Has to be" is thereby excluded. And so, we do not have to do anything, nor do we have to figure anything out, or anything like that. One can refuse to move and die if one wants. However, we choose to try and figure things out, we choose to try and understand the nature of reality, and mathematics plays a very big role here. So we need to choose the appropriate mathematics.

Of the mathematicians, the people who dream up axioms, and produce elaborate systems, some might have the attitude that math does not have to be practical, and others might have the attitude that math ought to be practical. But the idea that mathematics does not have to be practical is just an illusion. Each such mathematician will choose a problem, or problems to work on, as that's what mathematics is, working on problems. And problems only exist in relation to practice, as that's what a problem is, a doubtful aspect of practice which needs to be resolved. Without the influence of practice, the need to resolve the issue does not arise, therefore there is no problem. So the reality of the situation is that since mathematicians work on resolving problems, and problems only exist in relation to practice, math is always fundamentally practical, and this fact cannot be avoided. That's why math is classified as an art rather than a science. Despite the huge amount of theory which goes into it, it is theory which is always directed toward solving problems. Therefore, despite the fact that math doesn't have to be practical, it always is practical. If the people who dreamed up axioms and other systems weren't doing something practical (resolving problems), they would have come up with something other than mathematics.

The denotation of '2+1' is 3. The denotation is not 2 nor 1 nor the process of adding 1 to 2. — GrandMinnow

In general, "2" denotes a number, and "1" denotes a number, but in this particular circumstance, "2" does not denote a number, and "1" does not denote a number. Therefore you equivocate.

A program written to spit out the natural numbers one at a time is potentially infinite, regardless of whether it's been executed or interrupted. — Ryan O'Connor

We already discussed the difference between the rule ("program" in this case) which sets out, or dictates the process, and the process itself. If the process is interrupted, it ends, and is therefore not infinite. The rule ("program") is never infinite, nor is it potentially infinite, it's a finite, written statement of instruction, like "pi", and "sqrt (2)" are finite statements, even though they may be apprehended as implying a potentially infinite process.

If you have ever seen π as the solution to a problem (instead of, say, 3.1415) then the process hasn't been terminated, it hasn't even been initiated. It's incorrect to say that potentially infinite processes are only useful when prematurely terminated. — Ryan O'Connor

I don't see how you can say this. Pi says that there is a relationship between a circle's circumference and diameter. This information is totally useless if you do not proceed with a truncated version of the seemingly infinite process, such as 3.14. "The solution to the problem is pi" doesn't do anything practical, for anyone, if you cannot put a number to pi.

But you don't know anything about the formulation of classical mathematics.

...

But your account of the meaning of mathematics is not compatible with the ordinary formulation of mathematics, so if your account were to have any consequence, then it would need to refer to some other formulation. — GrandMinnow

As I said, if this point is of relevance then the discussion is pointless.

A contradiction is a statement and its negation. You have not shown any contradiction in what I said. The fact that '1', '2' and '2+1' each denote distinct numbers is not a contradiction. — GrandMinnow

I can't believe that you do not understand the contradiction. Let' take the expression "2+1". Do the symbols "2" and "1" refer to distinct objects. If so, then there are two objects referred to by "2+1", and it is impossible, by way of contradiction, that "2+1" refers to only one object. Do you understand this?

A process is a sequence of steps. — GrandMinnow

This is false. A process may be described as a sequence of steps. The sequence of steps is not the process, it is the description of the process. That this is an important distinction is evident from the fact that the very same process may be described in different ways, different steps, depending on how the process is broken down into steps. That's why different people can use different methods to resolve the same mathematical equation.

Also, you have not answered how other abstractions could be acceptable, such as blueness or evenness or the state of happiness, etc. — GrandMinnow

I don't see any need to consider an abstraction as an object. Abstraction is simply how we interpret things, and there is no need to assume objects of meaning as a fundamental part of the interpretive process.

No they are not different things. '4+2' and '10-4' and '6' are different names for the same thing. — GrandMinnow

You agreed that they are different things which have the same result, or the same value. If they are different things, then having the same result, or the same value does not justify calling them the same thing.

Here's what you said:

We've gone over this multiple times already. 2+1 is the result of adding 2 and 1. 6-3 is the result of subtracting 3 from 6. The value (result) of adding 2 and 1 is the same exact value (result) as subtracting 3 from 6.

One more try to get through to you. What you get when add 2 and 1 is the same exact thing as what you get when you subtract 3 from 6. — GrandMinnow

Are you taking that back now? Why do you want to say that adding 2 to 1 is the exact same thing as taking 3 from 6, instead of what you already agreed, that they are distinct things with the same end result? You know the truth in this matter, why try to deny it?

Properties are not things that are physical objects. — GrandMinnow

Then why treat properties as if they are any sort of object? You treat numbers as if they are some sort of objects, when really they are a property of the thing which is numbered.

I suspect that another big obstacle for you is that you don't understand that usually mathematics is extensional, not intensional. — GrandMinnow

I've argued elsewhere that the axiom of extensionality is a falsity. It is the means by which you say that two equal things are the same thing, which is obviously false. So it's not necessarily that I do not understand extensionality, but I apprehend it as based in false premises.

That is, the principle of "substitute equals for equals" holds. — GrandMinnow

In other words, equal things may be considered as the same thing. And that's clearly false.

If you place iron filings over a magnetic field the filings will take a form in line with the field. While it's true that we only see the filings, it is untrue to say that the field is just a model. It's real. The same goes for quantum fields. — Ryan O'Connor

There is an issue of truth here. There is something there causing the form, and the concept of "field" attempts to account for whatever it is. If the concepts employed are inadequate, then it's not true to say that this is what is there. Here's an example. The ancient Greeks used circles to model the movement of the planets, and Aristotle proposed that the orbits were eternal circular motions. It turned out that these models were wrong, therefore it was not true for them to have been saying that the orbits were circles even though this concept was employed and enabled prediction.

No. If we terminate the potentially infinite process we still get something useful (e.g. the rational approximation of pi on your calculator is a useful button). — Ryan O'Connor

Again, there is an issue of truth here. If the process is terminated then it is untrue to say that it is potentially infinite. And if we know that in every instance when such a process is useful, it is actually terminated, then we also know that it is false to say that a potentially infinite process is useful, because it is only by terminating that process, thereby making it other than potentially infinite, that it is made useful. Therefore t is false to say that the potentially infinite process is useful.

But nonetheless banishing infinity from mathematics is a move of an ostrich — Gregory

No, the opposite is the case. Ignoring the fact that infinities in mathematics is a very real problem, is the type of ignorance which is analogous with the ostrich move.

But he/it doesn't, so the issue of passing particular points is no different from passing any point, and yet all those other points are never mentioned. Why is that, do you suppose? Achilleus - or the Arrow - seems to have no problem whatever passing those. Zeno's then, just an entanglement with words. — tim wood

What do you think "passing a point" means? Do you mean to say that there are physical points out there, which the arrow can be seen flying by? If so, then you ought to be able to show empirically, the physical existence of such points, and I don't think there will be an infinity of them. If these points are just imaginary, then the arrow doesn't really fly by them and you have created a false scenario, by describing the arrow as flying by points.

I propose that the truth is that the points are imaginary. If this is the case, then any method of measuring motion, velocity and such; which employs points, is really giving us a false measurement. We might be able to find real physical points, which if they exist, would validate such a method, but then these points would not be infinite, so that scenario with infinite would become irrelevant, because we'd have to make a new method of measuring velocity based on empirically verified points. As I explained to you earlier, this is pretty much what relativity theory does, but each empirically verified point turns out to be a different frame of reference, and that the points are at rest relative to each other is very doubtful due to the observed phenomenon of spatial expansion.

You are free to present a formulation (or at least an outline) of mathematics and then say philosophically what you mean by it. But lacking a formulation, I would take the context of a discussion of mathematics to be ordinary mathematics and not your unannounced alternative formulation. — GrandMinnow

I have no formulation, and no desire to present one. The op asks if something has been proved, therefore we are invited to be critical of formulations which claim to prove that. And there is no need to offer an alternative formulation to point out problems with an existing one.

Please do not misrepresent what I said. I said explicitly that '1' and '2' do each refer to a distinct object. My remarks should not be victim to misrepresentation by you. — GrandMinnow

As I said, you equivocate:

I said explicitly that '1' and '2' do each refer to a distinct object. — GrandMinnow

2+1 is a number. — GrandMinnow

Which is the case, do "1" and "2' each signify distinct numbers, or does "2+1" signify a number? You can't have it both ways because that's contradiction. But I've been trying to go easy on you and settle for the lesser charge of equivocation. If "1" and "2" each signify distinct numbers, then there are two distinct numbers represented by "2+1", so it is contradictory to say that "2+1" represents one number, because there are two numbers represented here.

It could not be more clear. 6 is the number of chairs in your dining room, and 6 is the number of musicians on the album 'Buhaina's Delight', and 6 is the number that is the value of the addition function for the arguments 4 and 2. — GrandMinnow

That the same quantitative value is predicated of the chairs in my dining room, and the musicians on that album, doesn't make that predicate into an object.

The value (result) of adding 2 and 1 is the same exact value (result) as subtracting 3 from 6. — GrandMinnow

Sure, the resulting value of each is 3, but that's not the issue. Your claim is that "=" signifies identical to. 6-3 equals 2+1, but what is signified by "6-3" is not the same as what is signified by "2+1". You agree about this. Therefore it should be very clear to you that "=" does not signify identical to.

If you say that they have the exact same value, then we are using "equal" in the way I suggested. You and I have the exact same value in the legal system, therefore, as human beings we are equal, just like 6-3 has the same value as 2+1 in the mathematical system, but in neither case are the two equal things identical.

One more try to get through to you. What you get when add 2 and 1 is the same exact thing as what you get when you subtract 3 from 6. — GrandMinnow

OK, let's go with this then. If you do something, and derive a result, this is necessarily a process. So you are very clearly talking about two distinct processes represented by "2+1", and "6-3". Two distinct and different processes can have the same end result, and so those processes can be said to be equal. Does this imply, that in mathematics you judge a process according to the end result? If so, then how do you propose to judge an infinite process, which is incapable of producing an end result, like those referred to in the op?

Mathematical objects and mathematical properties are abstractions. They are not theological claims like the saying that there exists a God. Also, properties like 'blueness' and 'evenness' are abstractions. You are free to reject that there are abstractions, but I use abstractions as basic in human reasoning. — GrandMinnow

I don't see the difference. You are invoking an imaginary object represented by "2", just like a theologian might invoke an imaginary object represented by "God". Each of you will try to justify the claimed existence of your imaginary object. You are not showing the necessity required, which the theologians show, so you are not doing a very good job of it.

We prove from axioms that there is a unique object having a certain property, and we name it '6'. — GrandMinnow

This is so contradictory to what you've been arguing. You've been arguing that 4+2 is 6, and 10-4 is 6, and that there is potentially an infinite number of different things which are 6. And it isn't just a matter of different names for the same thing, because "4" and "2" must each name a unique thing, so it's impossible that "4+2" is just a different name for "6". How can you now claim to be able to prove that there is a unique object named "6", when you've been arguing that all these different things are the same as 6, by virtue of equality. You are getting yourself so tangled up in a web of deceit, that's it's actually becoming ridiculous.

How does "sqrt" change from signifying an operation, to signifying an object, simply by qualifying (or quantifying, if you prefer) it with a (2), without equivocation?

It's the same sort of problem which GM has with 2+1. Each of the symbols "2", and "1" refer to a distinct object. But GM claims that in the context of "2+1" there is only one object referred, and "2" and "1" do not each refer to a distinct object. How is this not equivocation?

We measure the car at 60mph and maybe that's accurate to within a small margin of error. — tim wood

I said "faults", and I used "contradiction" as an example of a fault. That there is a "margin of error" is another indication of fault. When a small margin of error is ignored or neglected, as if it doesn't exist, one can fall for a paradox like Zeno's, where that small margin of error is infinitely magnified to produce the appearance of contradiction.

My impression is that you're a finitist, so I presume that you believe our universe had a beginning of time. If particles are fundamental, they must have existed at that initial moment, right? Were they concentrated at a point? I take it that you think a measurement involves the interaction of particles, so at the initial instant wouldn't they all be measuring each other? If so, how would they ever move, given the quantum Zeno effect? — Ryan O'Connor

I really don't get your question. I was talking about points, not particles, so your question has some underlying presumptions which I don't follow.

Consider this: "QFT treats particles as excited states (also called quanta) of their underlying quantum fields, which are more fundamental than the particles." source — Ryan O'Connor

I addressed this already. The so-called "underlying quantum fields" are models produced from observations of particles, and are meant to model the interactions of particles. It is implied that there is an underlying substratum which validates this modeling, but the modeling itself, the quantum field theory, does not represent the underlying substratum, it represents the interaction of particles. Until we get accurate and precise modeling of the particles any speculation concerning the substratum is not well informed.

I think you're splitting hairs here. By rule I assume you mean the 'computer program' and by process I assume you mean 'the execution of the computer program'. If so, then we are in agreement, we can study the rule (i.e. the computer program). — Ryan O'Connor

OK, so we say that the rule calls for the computer to carry out an endless, or infinite process. We know that the computer cannot succeed in carrying out this request, because it will wear out first, so all the time spent will be wasted, for the computer to be trying to carry out a process it can't. So if we turn to study that rule, should we not put our efforts into avoiding this rule, making it so that the rule never comes up, because it's like a trap which the computer will fall into? Therefore instead of pretending to be having success at carrying out infinite processes, which is self-deception, we should be looking at ways to make sure that such rules are banished.

The issue I am looking at, is not how things are viewed by "ordinary mathematics", it is what is meant by the mathematical concepts. If we adhere to how things are viewed by mathematics, as if this is necessarily the correct view of things, as you seem inclined toward, then the discussion is pointless. You'll keeping insisting that I am not seeing things the way that mathematics sees things therefore I am necessarily wrong. I've been exposed to enough of this already, and see no point to it.

(1) You are still making your use-mention mistake. Yes, '+' represents an operation and '2+1' is a representation of a value, but '2' and '1' are not values, they are representations of values. — GrandMinnow

OK, I'll try to adhere to this formality. It is not the convention I am used to, but I'll try it.

2) As I explained, and as you ignored, + is the operation; 2 and 1 are the arguments; and 2+1 is the value of the function for those arguments. — GrandMinnow

No, "2" and "1" signify values. Or do they sometime signify values and other times signify arguments? If so how do we avoid equivocation? Anyway, I see no way that a function, which is a process, could have a value. That's like saying that + has a value.

You are conflating the meaning of the world 'equal' in various other topics, such equality of rights in the law, with the more exact and specific meaning in mathematics. — GrandMinnow

Look, "2+1" means to put two together with one, and 2+1 equals "6-3", which means to take three away from six. You cannot try to tell me that to take three away from six is the exact same thing as putting two together with one, or I'll tell you to go back to elementary school and learn fundamental arithmetic properly.. Your claim is clearly false, equals does not mean identical to, or the same as, in mathematics.

Ordinary axiomatic mathematics is extensional. Each n-place operation symbol refers to a function on the domain of the interpretation, and functions are objects. The function might or might not be an object that is a member of the domain, but it is an object in the power set of the Cartesian product on the domain. — GrandMinnow

I know that a function is a process. And I also know that the concept of process is incompatible with the concept of object. The two are distinct categories. Therefore it is fundamentally incorrect, by way of category mistake, to say that a function is an object.

It is the mathematical object that is the number of chairs, and is the number musicians on the album 'Buhaina's Delight', and is the value of the addition function for the arguments 4 and 2 ... — GrandMinnow

I really don't know what you could possibly mean by this. The number of chairs is referred to by "6". There is a specific quantity and that quantity is what is referred to with "6". I don't see where you get the idea of an object from here. There are six objects which form a group. The group is not itself the object being referred to, because the six are the objects. Therefore the quantity must be something other than an object or else we'd have seven, the six chairs plus the number as an object, which would make seven.

You are saying that the number of them is 6. — GrandMinnow

More correctly, the quantity is six. You assume that this quantity is an object, a number, which is something other than the quantity. But that's not the case, the quantity is six, the number is six, and both "quantity" and "number" mean the same thing here. Why assume that there is something other than a quantity, an object called 6? That makes no sense, where and how are we going to find this object?.

When we say that 2 is even, we mean that 2 has the property of being even. 2 is the object, and evenness is the property. — GrandMinnow

You're just making an imaginary thing, like God, and handing a property, "even " to that thing, like someone might say God is omniscient. When we use the symbol "2", we use it to refer to a group of two things. like chairs or something. When we say that there is an even number of chairs, this means that the group of chairs can be divided into two groups. But the group clearly cannot be divided by three. If you say that "2" refers to some imaginary object, then you can assign to it whatever properties you like. You could make it infinitely divisible if you want. But if you're not adhering to any principles of reality, this is just useless nonsense. What good is assuming an imaginary object which you can attribute any properties to with total disregard for reality?

With '2+1 = 3', we have the nouns '2+1' and '3', and '=' stands for the 2-place predicate of equality, and indicates in the equation that the predicate of equality holds for the pair <2+1 3>. — GrandMinnow

Now you're doing the same thing again, you're claiming two nouns, 2 and 1, are one noun signified as "2+1". Clearly this cannot be the case without equivocation. Either 2 is a noun and it refers to an object, or it is not. But you can't have it sometimes being a noun, and sometimes not without equivocation.



.

But I am going to ask you to write something - anything - that is true. — tim wood

Sorry tim, but if I already knew the truth, then I wouldn't be looking for it, would I?

Are you gong to argue that the car is not moving at any speed during its traverse of the distance A to B? — tim wood

Yes, that's about it. Speed is what we assign to the car, it is what we say about it, it has speed. In philosophy we must maintain the distinction between what we say about the thing, and what is really the case, to allow for the real possibility that what we say about the thing might actually be a falsity. If the property which we assign to the thing, "speed", in this example, has faults within its conception (contradictions for example), then despite the fact that it has become acceptable to say this, the concept is defective, and it is really not true to be attributing that property.

sqrt is an operation. sqrt(2) is the object that is the result of the operation applied to the object 2. sqrt is the operation, and 2 is the argument to which the operation is applied. — GrandMinnow

No, if "sqrt" represents an operation, then "sqrt(2)" represents that operation with a qualifier "(2)".

Your us-mention is inconsistent there. Yes, '3' represents a value. But so also does '2+1'. — GrandMinnow

This is incorrect, because "+" represents an operation. So there are two distinct values, "2", and "1" represented, in "2+1", along with the operation represented by "+".

'equals' is another word for 'identical with'. — GrandMinnow

No it isn't, that's a false assumption which I've discussed on many threads. You and I are equal, as human beings, but we are in no way identical with each other. "Equals" is clearly not another word for identical with.

A number is an object. If it's not an object, then what is it? If it is something that, according to you, might or might not be an object, then what is that something to begin with if not an object? How can we refer to something that is not an object? — GrandMinnow

Let's start with numerals, which are symbols. Do you agree that a symbol has a meaning, which is not necessarily an object? So there is no need to assume that "2" or "3" represent objects. We'd have to look at how the symbols were being used, the context, to determine whether they represent objects or not. When I say that there are 6 chairs in my dining room, "6" refers to a number, but this is the number of chairs; the chairs are the objects and the number 6 is a predication. The number is not an object, it is something I am saying about the chairs in my dining room, just like when I say "the sky is blue", blue is not an object.

This is why I think the number 3 can exist but not the 'number' sqrt(2). We never actually work with irrational 'numbers', we only work with their algorithms or rational number approximations. So why do we even need to assume that irrational 'numbers' exist? Why not assume that irrationals are the algorithms that we actually work with? — Ryan O'Connor

Isn't this the difference between an object and a process? We'd say "3" represents a static object, a number, but "sqrt(2)" represents an operation. What would you say about "2+1"? Doesn't that represent an operation rather than an object? The difference between "2+1", and "sqrt(2)", is that the one process adequately resolves to an object. The question I see is what does it mean for a process to resolve to an object, and why does this make the process somehow more valid as a process? So, we say "2+1=3", and we are stipulating an equivalence between the process and the object. But we cannot produce the precise object which "sqrt(2)" is equivalent to.

What validates, or grounds numbers definitionally, is quantitative value "2+1" is equivalent to a definite quantitative value represented by "3". Having a definite quantitative value is what makes the number an object. If we do as you propose, and allow processes which do not have a definite quantitative value, to be "worked with", then we allow indefiniteness into our solutions. The solutions will contain indefinite quantitative values. This is counterproductive because the goal when using mathematics is to measure things, which is to assign to them definite quantitative values.

After all we don't know the ultimate nature of reality so who's to say if the notion of instantaneous velocity really makes sense. — fishfry

This is the point. When we use math to figure out things like instantaneous velocity, the volume of a supposed infinitely small tube, etc., it is implied that we know things about reality which we do not. This is a falsely supported certitude.

A car going at constant speed passes point A at stopwatch time=0, then passes point B, one mile further at stopwatch time=one minute. You ask, "What was the speed of the car back there at point A?" Your answer, "It was moving at 60 mph at point A". — jgill

There is a flaw with your example jgil. That the car was "going at a constant speed" is just an assumption, so it may not be the truth of the matter. And your answer as to how fast the car was going at point A requires that the assumption be true. So it needs to be proven.

You might lay out a series of such points, at equal distance, and do numerous similar measurements. If your measuring capacity is precise, you'll find that all the measurements will not be exactly the same. The assumption of "constant speed" cannot be validated. That's what we've found out about the nature of reality, motion consists of spurts and starts. So you'd have to establish trends, and the more measurements you took the better your graphing of the trends would be. But you'd be graphing averages which does not tell you the precise amount at any given point.

I don't agree with this claim so I'd like to see your evidence that supports it. What is fundamental in quantum physics is the wave function, a continuum. Definite states (like points) only emerge when a measurement is made. — Ryan O'Connor

What is real and fundamental in quantum physics is the points where particles appear. The wave function is the mathematical apparatus which predicts where particles might appear. Yes, the wave function is fundamental to the model, but what is being modeled is the appearance of particles at specific points. This is why physicists understand light as photons, because the energy appears at, and causes an effect at a point.

You say that points only emerge from a measurement, but a measurement is an interaction between the energy, and the object which is the measuring device. So, such points exist wherever energy is interacting with objects. What this indicates is that energy, though it is modeled as existing in a continuum, (wave function) only interacts with the physical objects which we know, at discrete points. Therefore our only access to observe whatever substratum there is, which is modeled as wave functions, is through an understanding of these points where we can observe interactions.

Sure, you might say that the continuum, or substratum as I call it, is more fundamental, but from the point of view of the model, and this means the mathematics, the points must be fundamental. This is because we only find a route inward, toward understanding the substratum through a mapping of the points where it interacts with the spatial existence we know, observe, and understand. What must be fundamental, and basic to the model, is what we know the best, and this is the points. The substratum is modeled based on the existence of those points where we can observe it The better we know the points, the more reliable our speculation about the substratum will be.

Assume that there exists a wave function of the universe that spans all of time. This is the fundamental object of our universe and it is a continuum. And until the wave function is measured it is meaningless to talk about who lived when and where because a wave function does not describe what is, it describes what could be. It is only when you make a measurement that all of the potential states collapse into a single actual state. When I say that points are emergent, I mean that they only emerge when we make a measurement. We cannot say things like 'there are infinite points on this line' because we have not actually placed infinite points on the paper...what we placed on the paper was a line. — Ryan O'Connor

The substratum, which is represented by wave functions, may or may not be a continuum. That a wave function represents it as a continuum doesn't mean it is. Furthermore, a measurement is simply the substratum interacting with a physical object. So if this causes a "collapse", there are collapses occurring all the time, all over the place, as the substratum is interacting with physical objects. And, if measurements are only possible at particular points, then we ought to assume that other interactions between the substratum and physical objects are only possible at particular points, and this is most likely a feature of the substratum itself.

Put it this way: a computer program that calculates 2+2 is what I mean by 'process' and such a program can be studied (even if the program is never executed). — Ryan O'Connor

I don't agree. A process which is never executed cannot be studied. It has no existence so it cannot be studied. Let's say that you write out a rule, an algorithm, but the algorithm is never implemented. You can study that rule, but you cannot study the process dictated by that rule, because it does not exist. The rule was never put to work, actualized, it exists only as the potential for the designated process Do you see the difference between the written rule, and the activity which is prescribed or described by that rule? To study one is not the same as studying the other.

A moving body has an instantaneous velocity,../quote]

Yes, because that is the convention, use some math, and figure out the "instantaneous velocity", just like the convention is to place a zero limit on the example of the op. But what these conventions really represent may not be what one would expect from the terms of usage. — fishfry

But why the square root of 2? How about the number 3? That has no more claim on existence than sqrt(2). — fishfry

This is doubtful, and seems to contradict the rest of your statement. If we're talking mathematical existence, I do not think that natural numbers have more claim to existence than irrational numbers. In fact, I think the opposite is more likely the case. "The natural numbers" were in use prior to the Pythagoreans who are supposed to have demonstrated the "existence" of irrationals. So it was only by the work of the Pythagoreans that "existence" was assigned to numbers, and existence was stipulated in order to provide reality for the irrationals. Prior to this the natural numbers were in use without any assumptions that numbers exist, so the naturals are lacking in the claim of existence. There is no need for them to be stipulated as existing.

If we're talking "existence" in the philosophical sense, we'd have to first agree as to what existence means before we might judge whether one type of number has more of a claim to existence than another. If we do not find a definition of "existence" which allows that numbers exist in the first place, then the suggestion is meaningless.

Clearly numbers don't have the same claim to existence as rocks or fish. — fishfry

Why not? I don't see the reason for approaching the question with such a bias. It will only make a true answer more difficult to find. Plato demonstrated the pitfalls of this bias thousands of years ago. It is a mistake to assign a higher degree of being to something apprehended through the senses over something apprehended directly with the mind.

This thread hasn't even begun to touch on the subtleties of that subject. I've seen no decent arguments one way or the other. And if that's what the OP really cared about, they'd have asked if 3 exists. Once you bring in sqrt(2) you are talking about mathematical existence. — fishfry

This again shows some sort of bias toward natural numbers over irrational numbers. If "3" represents a number, and "sqrt(2)" represents a number, then why assume that the question is better asked of "3" than of "sqrt(2)"? That's just admitting that "3" is in some sense a better representation of a number than "sqrt(2)", and in doing this you undermine the concept of "mathematical existence". If some numbers have a better, or more valid "mathematical existence" than others, then there must be ambiguity within the concept which could allow for equivocation.

It's "above their pay grade" as Obama would have said. So make an argument. Do you think 3 exists? Of course the positive integers have a pretty good claim on existence because we can so easily instantiate the smaller ones. So how about 2googolplex2googolplex? That's a finite positive integer that could in theory be instantiated with rocks or atoms, but there aren't that many distinct physical objects in the multiverse. So make an argument, say something interesting about this. Forget sqrt(2). Do you think that extremely large finite positive integers exist? — fishfry

As I said, there's really no point in making an argument as to the existence or nonexistence of something unless we have a workable definition of "existence". That's why the thread really won't get anywhere because all the members in this forum have wide ranging biases about what constitutes "existence", and a relatively small number of them have any formal training in this subject, so it will end up with people arguing to support their own prejudices.

I would be inclined to define "existing" as having either temporal or spatial extension.

Eternal circular motion is fine. — Gregory

OK then, show me this perpetual motion which you know about.

In fact your earlier point is correct, any measurement is taken over time. — fishfry

That's why velocity is always an average, requiring at least two temporal points. Duration is derived, just like distance is. To infer an instantaneous velocity requires a second derivation.

You must be aware that Aristotle rejected points (infinitesimals) and instants — Gregory

Aristotle also posited eternal circular motion, which is nonsense.

Time for you to develop a new axiomatic system, then, that leads to "Truth". — jgill

It's not time for me to do that, I'm not a mathematician. There's something called the division of labour. The person who puts one's efforts into pointing at the problems in existing systems need not be the one who produces the repair. Of course the people using the system would probably not like the person pointing and would have the attitude of 'if you think you can produce a better system than ours, then do it'.

Pi is a finite number because it's inbetween 3 and 4. But if the length of a circumference is multiplied by pi than you have a length with space corresponding to each number, so the circle has infinite space within a definite finite limit (like being inbetween 3 and 4). Aristotle never understood this stuff — Gregory

At least I'm not alone then, because I haven't got a clue what you're saying.

The fact that your philosophy would result in a weaker mathematics is a red flag that you're on the wrong track. — Ryan O'Connor

You demonstrated that you do not grasp the need for the point to be prior to the line, therefore your claim that it would result in a weaker mathematics is based in misunderstanding. What quantum physics demonstrates to us is that points have real existence, and continuities are constructed.

I made this video on my proposed resolution to Zeno's Paradox. What do you think? — Ryan O'Connor

I don't see how you get from points to continua. You show measurement points, then you assume that there is some sort of continuum connecting those points. The problem I see, is if certain measurement points are actually possible, then these must be represented as real points which the moving object actually traverses and can be measured at. That's why I give priority to these points, as the real features. The supposed continuum might not have any sort of linear existence at all, in fact we might not have the vaguest idea of how the points are related to each other in the underlying substratum of reality, which produces the appearance of a continuum. For all we know, the object might appear at point A, then completely disappear, and then reappear at point B a moment later, and this is what appears to us as motion.

The reason why I say that priority must be given to the points, is that whatever it is about the underlying substratum which produces the appearance of continuity, this 'power' must be constrained by possible points of appearance. If there wasn't such constraints then we'd have the problem of infinite points where the object could be measured. Furthermore, the nature of spatial expansion demonstrates that there must be points where expansion is centered.

So I find the video mostly acceptable, but what you are really showing is a points based motion, points where the object might be measured to be at, and you are assuming that there is some sort of continuum which underlies the points and connects them. Therefore all you need to do to be consistent with my perspective, is put the points as primary, being the real constraints of real space, and allow that whatever continuum emerges from existence at the points it is a creation produced from the relationships between the points, and this set of relationships comprises the substratum.

When I say that processes are valid objects of mathematics, I simply mean that they can be studied in themselves, just as one might write a book entitled 'The Art of Dog Walking'. — Ryan O'Connor

I have doubt in the truth of this. Are processes valid objects of mathematics, or ought they be relegated to physics? Let's start with something simple, assume that a number is an object of quantitative value. So '4' represents such an object, it must be a static and unchanging value to maintain its validity, therefore it cannot be a process. Now let's say that in '2+2', the '+' represents a process. So the inquiry is whether the process represented by '+' is a valid mathematical object to be studied by mathematics. We need to determine what the '+' means. What does it mean to add one unchanging quantitative value signified by '2', to another? Mathematics does not answer this inquiry, it just makes an assumption about how processes like these affect quantitative values. And we can see the same with the other processes, multiplication, division, etc., these process affect quantitative values, but if quantitative values are what are properly referred to as objects, then these processes are something different.

"There exists an object that has the property that its square is equal to 2" is perfectly fine English. — GrandMinnow

I didn't say it isn't perfectly fine English. I said you haven't properly identified the subject signified with "there", to which "exists an object" is predicated.

Existence is the same. If someone's been existing for a few decades they know as much about existence than a philosopher. The philosopher knows the history of what great thinkers have written about the subject. But philosophy does not confer actual knowledge of its subject; only knowledge of what others have said. — fishfry

All I can say is, wow! This is an unbelievable opinion coming from an educated person like yourself. Would you also say that a person who has been breathing for a few decades knows as much about breathing as a biologist?

So going to university and studying a subject of study only provides one with what other great thinkers have said about that subject, but it doesn't provide one with any knowledge of the subject? It only provides one with what those who've studied that subject, say about the subject? What about studying mathematics, wouldn't this be the same thing? Studying mathematics doesn't provide any real knowledge of mathematics, only what others who have studied the subject say about the subject. What do you think knowledge of a subject consists of, if not what those who study the subject say about the subject?

In particular, a philosopher who knows hardly anything about mathematics is in no position whatsoever to comment on mathematical existence. Many philosophers of mathematics are in this position. They simply don't know enough math to comment intelligently on the subject of mathematical existence. — fishfry

The problem with this perspective is that "mathematical existence" means something completely different than "existence" in the philosophical sense. The op does not ask about "mathematical existence", it asks about "existence". If it asked about the mathematical existence of irrational numbers there would be nothing to discuss. Clearly irrational numbers are used by mathematicians therefore they have mathematical existence.

The op is asking a philosophical question about the existence of certain mathematical objects, not whether those mathematical objects have mathematical existence. That would be self-evident. So mathematicians who hardly know anything about existence, yet think they do because they know something about mathematical existence really seem to have very little to say about the philosophical question of whether certain mathematical objects which obviously have mathematical existence, have existence.

If you reject potentially infinite processes as valid mathematical objects then you must reject calculus, and nobody will buy into your philosophy. — Ryan O'Connor

I'm not looking for people to buy in, I'm looking for truth. If others are looking for the same thing, they might like to join me. Otherwise I don't really care if people might deceive themselves into thinking that they are engaged in infinite processes. Many think that the soul is eternal, and this doesn't both me either. I consider those two beliefs to be very similar.

The mathematical object is the process itself. — Ryan O'Connor

There is a fundamental incompatibility between an object and a process, which was demonstrated by Aristotle. If an object changes, it is no longer what it was. We assume a change (process), to account for the object becoming other than it was. So we have object A, then a process, then object B, whereby object A becomes object B. If we represent the intermediary between A and B as another object, C, then object A becomes object C which becomes object B. Now we need to assume a change (process) to account for object A becoming object C, and a process to account for object C becoming object B. We might represent the intermediaries between A and C, and C and B, as objects again, but you can see that we're heading for an infinite regress. So we ought to conclude that "objects" and "processes" are distinct categories.

Yes, I read the rest of your post, as reading the rest of my post should have indicated to you.

The other Wheels turned on, they were parts of an elegant piece of machinery, and they had been made to turn. But there was a hole, the bored Wheel had left a fault in the perfect system by leaving. The Wheels that had been beside it turned and turned, and they stretched to fill the holes, widening their teeth and reducing the space between teeth. — New2K2

You cannot replace a missing wheel in this way. The cogs on one side of a wheel move in the opposite direction as the cogs on the other side. So if you stretch to fill in a hole created by a missing wheel, the new motion will be in the opposite direction of the old motion, before the wheel removed itself.

Finally the other Wheels touched again, here was a productive use of thought they said to each other, and began to grind again, on and on they ground on each other, grinding nothing, but this time the stretched wheels grated on each other, they scraped and ground and grated, at first this was horrible but soon they had smoothened each other out, scraping off the flecks that caused the grating. — New2K2

A bit of lubrication might have fixed this problem, but nothing would have fixed the problem of the end motion being in the wrong direction. Do you think that the one wheel leaving started a process of backward thinking?

The equations of special relativity entail that nothing can accelerate up to or beyond the speed of light, taken as the constant c, since the logical consequence would be a division by zero. — jkg20

What about the effects of spatial expansion? When spatial expansion increases as time passes, and things start speeding away from each other faster than the speed of light, does this not qualify as acceleration?

Can you explain this to Metaphysician Undercover and @Ryan whose handle doesn't show up when you use the @ button? — fishfry

The point being, that you cannot take the arrow at a particular moment in time. This is an impossibility because time is always passing, and this would require stopping time at that moment. So, despite the fact that using mathematics to figure hypothetical conditions at particular moments is a very useful thing to do, what it provides us with is a representation which is actually a falsity. Then if people start talking about this situation, with the underlying implication that this mathematics provides us with some sort of truths about these situations, this talk is really a deception or misinformation.

But actually it's a good question. Suppose there were such a thing as an instant of time, modeled by a real number on the number line. Dimensionless and with zero length. So the arrow is there at a particular instant, frozen in time, motionless. Where does its momentum live? How does it know where to go next? — fishfry

This is the key point to understanding temporal continuity, inertia, Newton's first law, and the overall validity of inductive reasoning in general. We observe that things continue to be as they were, as time passes. Intuition tells us therefore, that they will continue to be as they were, unless something causes them to change, and this intuition is what validates inductive reasoning. However, there is a very real, and very big problem, and this is the reality of change. We see that human beings have the capacity to interfere with, and change the continuity of inert things. Because of the reality of change, we are forced to accept the fact that this continuity is not a necessity. This appears to be the hardest thing for some people, especially those with the determinist mentality, to accept, that the continuity of existence, which we observe, is not necessary. This means that the supposed brute fact, which underlies all inductive principles as supportive to those principles, that things will continue to be as they have been, is itself contingent, not necessary.

When we take a law, like Newton's first law, we view it as something taken for granted. The law tells us the way things are, and it's assumed that it's impossible for things to be otherwise, that's why it's "the law". However, when we apprehend that this is not necessary, then we can grasp the fact that there is a need for a reason why the law upholds. Through principles like the law of sufficient reason, we see that if there is a temporal continuity of existence, described as momentum or inertia, and this feature of existence is not necessary, then there must be a reason for it, a cause of it.

What Aristotle did was posit "matter" as the cause of the temporal continuity of existence. So contrary to the common notion that matter is some sort of physical substance, "matter" by Aristotle's conception is really just a logical principle, adopted to account for the observed temporal continuity of physical existence. It is, in a sense, a placeholder. He didn't know the cause, but logic told him there must be a cause, so he identified it as "matter". In your example of the arrow, we do not know "how does it know where to go next", but we do know that it does. Aristotle attributes this to its "matter", or more precisely he posits "matter" as what causes it to go, where it does go, next. Therefore the theoretical points in time are in reality connected to one another by what is called "matter".

Ok. Maybe. Let me put to you a hypothetical. An object moves with constant velocity. Does it have a velocity at a given instant? — fishfry

No, because "a given instant" is not anything real which can be adequately identified. We can attempt to arbitrarily assign an instant to time, to mark a point for the purpose of measurement, but that assignment becomes much more difficult than it appears to be, at first glance. To mark a temporal point in one process or activity, requires a comparison with another process or activity, thus requiring a judgement of simultaneity. According to special relativity such judgements are dependent on the reference frame. Therefore any "given instant" may not be the same instant from one frame to the next, and the question of what a thing's velocity is at a given instant is rather meaningless because it depends on what frame of reference you measure it in relation to.

I'm kind of done with this topic, the point I'm making isn't worth all this ink. You don't need calculus to do analog measurements. And yes physical measurements depend on time, even if those intervals are tiny. There aren't any actually physical instants as far as we know. Or as far as we don't know. The matter is not answered by current science. — fishfry

I don't think you've adequately considered what is required to produce accuracy in a time related measurement.

Actually, I don't think you have.. You simply used "exists" as a verb, and verbs refer to actions which must be predicated of a subject to say anything truthful. So "there exists..." really doesn't say anything meaningful because you haven't properly identified the subject referred to when you point with "there".

What if that baby grows up to be the next Hitler or Stalin? — Harry Hindu

Moses?

Yet, it still HAS a velocity, wouldn't you agree? — fishfry

Sure, the object is described as moving, it must have a velocity. But it cannot have a velocity at an instant, if no time passes at an instant, just like a point has no spatial extension. That's why points and lines are incompatible, and a line is not composed of points, but points mark off line segments.

So the solution to the issue with velocity, is not to say that it has no velocity, it is to say that there is no such thing as the instant. Time is not composed of instants. So the arrow, or car always has velocity, all the time that it is moving, but that time has no instants. The instant is just an arbitrary point which we insert for the purpose of making a measurement.

Still, would you at least grant me that velocity over a short but nonzero distance exists? — fishfry

Sure, but the whole point I am arguing in this the thread is that the inclination to reduce the nonzero distance to zero, or even define it as somehow related to zero, produces theoretical absurdities. And this is well demonstrated by these Zeno type paradoxes which speak of time as consisting of instants.

"There exists a unique x such that x^2 = 2." — GrandMinnow

The problem of course, being that it is debatable whether there is such an x.

In short, your objection is valid, but overly general. We can't measure any physical quantity at all by your logic. What if I want to measure the wavelength of a beam of light? Well I use a spectrometer, but all that really measures is the prism or the glass or however spectrometers work. — fishfry

You seem to be missing the point fishfry. Velocity is a measurement of motion, and motion only occurs when time is passing. At an instant zero time passes. Therefore there is no motion at an instant, and no velocity at an instant.

A measurement of velocity requires a determined distance over a determined duration of time. It requires two instants, to determine a duration of time, one to mark the beginning of the period of time, the other to mark the end of the period of time, just like it requires two points to determine a distance. One instant (point in time) is insufficient for a determination of velocity, just like one point is insufficient for a determination of distance.

My newest guess is, that there is a layer for all objects (quants) that can interact
(except interacting with gravity),
and if some interact than the layer for all objects is increased.
This way we get in the layers a kind of time arrow since the big bang,
and properties (even in math like prime decomposition)
can depend from it and change with time. — Trestone

This is similar to what I was telling Ryan in the other thread on Gabriel's horn. The classical way that mathematicians apply numbers to spatial representations (Euclidian geometry) assumes an eternally continuous, and static, space. But modern observations have produced a new concept called spatial expansion. Therefore we need to allow that space itself changes with time, and this means that the assumption of a static space is incorrect. So if we propose a number of points in space, and these points, if connected with lines, make a shape such as a triangle or square, and then we propose some passage of time, then these same points in space will no longer make the same shape.

No, actually. Not even a computer program doing the same. Rather, there is a little induction motor attached to the driveshaft. The faster you travel, the faster the drive shaft spins, the faster the induction motor turns, the more current it outputs. And that current directly drives the needle of your speedometer.

Your speedometer is not a mathematically derived average. It is in fact a direct analog measurement of the instantaneous velocity of your car; subject of course to slight mechanical error common to any physical instrument. The velocity is an actual, physical quantity that can be directly measured -- that IS directly measured -- without recourse to any formal mathematical procedure. — fishfry

Oh come on fishfry, you're smarter than this. The current you refer to is just measuring revolutions of the driveshaft. Then the speedometer of the car is scaled to how many revolutions are required to cover a specific distance. It is not measuring the instantaneous velocity of your car. What happens when you use the wrong size tires?

A potentially infinite process is one which will not end (unless prematurely terminated). Does this work for you? — Ryan O'Connor

I understand this, but my point is that due to the nature of our universe, any such "potentially infinite process" will be prematurely terminated. So it doesn't make any sense to say that such and such a process could potentially continue forever, because we know that it will be prematurely terminated. Therefore, if we come across a mathematical problem which requires an infinite process to resolve, we need to admit that this problem cannot be resolved, because the necessary infinite process will be terminated prematurely, and the problem will remain unresolved.

Well, can't the answer to the question simply be the infinite process? — Ryan O'Connor

I don't think so, because the process is the means by which the answer is produced. If the answer requires an infinite process, and the infinite process will be prematurely terminated, then the answer will not be produced.

For instance, consider the question 'what is the area of a unit circle?' Is this a valid question? In one sense, I think you're right since no rational number will do. But in another sense, I think you're too strict in only accepting rational numbers. I think it's valid to say that the answer is pi, which I believe corresponds to a potentially infinite process. (Well my beliefs are changing a bit as I talk here with norm but I think you get what I'm saying). — Ryan O'Connor

The reason I am "too strict", is that I don't believe in coincidence, when it comes to mathematics. Call me superstitious, but I believe that in mathematics, there is a reason for everything being the way that it is. So when it turns out, that a circle cannot have a definite area, then I believe that there is a reason for this. The most likely reason, is that the circle is not a valid object. By "valid" here, I mean true, sound, corresponding with reality.

Here's a sort of anecdote. Aristotle, in his metaphysics posited eternal circular motions for each of the orbits of the planets. Motion in a perfect circle could continue forever because there could be no beginning or ending point on the circumference of the circle, as each point is the same distance from the centre. Of course we've since found out that the orbits are not perfect circles. What we can learn from this, is that despite the fact that the circle is an extremely useful piece of geometry, there is something fundamentally wrong with it, as a mode for representation. It is not real. And, with the irrational nature of pi, the circle actually indicates directly to us, that it is not real. So if we ignore this fact, insisting that we want the circle to be real, or that it must be real because it's so useful, and then we work around the irrational nature, creating patches, and fancy numbering systems to deal with all these seemingly insignificant problems which crop up from employing perfect circles, we are simply deceiving ourselves. We end up believing that the real figures which we are applying the artificial (perfect circles) to are actually the same as the artificial, because all the discrepancies are covered up by the patches.

And your claim that there's no such thing as instantaneous velocity is falsified by your car's speedometer. — fishfry

Velocity is always an average over a duration of time. So-called "instantaneous velocity" is just a derivative from an average. Since velocity is a measure of change, and change without a duration of time is impossible, then also true "instantaneous velocity" is also impossible. It's just a term of convenience, to be able to say that at x point in time, the velocity was such and such. What is really taken is an average over a duration, and from that we can say that the velocity at any particular point in time within that duration was such and such. But you can see from the applicable formula, that "instantaneous velocity" is really just another average. And it's quite obvious that the idea that something has velocity at a point in time, when there is no duration, is nonsensical.