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.

But metaphysicians don't know any more about existence than the rest of us. — fishfry

It seems like you don't know anything about metaphysics, which is the study of existence. Why would you think that someone who has not studied existence would know as much about existence as someone who has studied existence?

Metaphysics studies questions related to what it is for something to exist and what types of existence there are. Metaphysics seeks to answer, in an abstract and fully general manner, the questions:[3]

What is there?
What is it like?
Topics of metaphysical investigation include existence, objects and their properties, space and time, cause and effect, and possibility. Metaphysics is considered one of the four main branches of philosophy, along with epistemology, logic, and ethics.[4] — Wikipedia: Metaphysics

The entire history of mathematics is filled with examples, starting from the discovery of irrational numbers right through to the present day. — fishfry

So, where are your examples? Where is the criteria for existence found in mathematics?

We can do so much with potentially infinite processes. Not only can we interrupt them to produce rational numbers, but we can work with the underlying algorithms themselves. For example, the following program to outputs the entire list of natural numbers. This program can never be run to completion, but it still is a valid program...I'm talking about it after all and it makes sense even though I've never run it. The same can be said about irrational processes. We need to embrace potential infinity for what it is, not reject it. — Ryan O'Connor

I'm in complete agreement that an infinite process is a logically valid process. That's what I said, but it's counterintuitive to think that any process could 'live' forever. This is the same issue which the ancients had with the immortal soul. It's valid logic, but there's something wrong with the premises, which makes the conclusion unsound, despite the fact that the logic is valid. Furthermore, the "immortal soul" served as a very useful moral principle, just like the "infinite process" serves as a very useful mathematical principle, but usefulness does not necessitate truth, if truth is what we are ultimately after..

To say that a process is infinite is to say that it will run forever. That is what you are claiming with "infinite processes". Notice, that to say "X process is infinite", is to say "it will run forever", and that's a statement of necessity, just like to say "the soul is immortal" is to say it will, necessarily live forever. But of course you respect the possibility that the process may for some reason, at some time, cease. Therefore you say that it is "potentially" infinite. It could run forever, if it's allowed to. There is a clear difference between "it will run forever" and "it could run forever".

What I am requesting is that you recognize the constraints of the physical world, to acknowledge that an infinite process is not possible within our universe. Therefore both "it could run forever", and "it will run forever" are excluded as impossible, therefore false premises. From this perspective we see that any problem which requires an infinite process to resolve, is actually not solvable. If a question is not solvable, it is not properly posed, it is not a valid question.

You say: "I have no doubt that math works, it's 'why' that has puzzled me for many years." The simple answer is that it works because it conforms to the constraints of our universe. We can dream up a seemingly infinite number of logically possible axioms which will be completely useless in our universe. It's correspondence with the physical reality which makes them useful. Some people will make a distinction between coherence and correspondence, claiming that coherence is all that is necessary within a system of logic, but coherence itself is fundamentally a correspondence. Fundamental laws of coherence, like non-contradiction, work because they correspond with our universe. So a coherent logical structure has a basic correspondence simply by being coherent.

You might think that this completely contradicts what I said above, "usefulness does not necessitate truth". However, we must maintain the distinction between sufficient and necessary. Proof requires necessity. So despite the fact that correspondence (truth) works, we need to also be aware that there are other reasons why principles, like mathematical axioms, work. And this is dependent on the end, the goal which we have in mind, by which "works" is judged. If the goal is not itself truth, then the axioms are formulated toward that alternative goal, and "work" for that alternative purpose. I find that in the modern sciences, which are the principal users of mathematics, the goal has shifted from truth to prediction, and these two are not the same.

So I think you need to adjust your enquiry from 'why does math work?', to 'what does math do?' The point being that so long as people are applying it, it will work, otherwise they wouldn't be using it. The reason it works is because it's designed, shaped, conformed to the purposes which it is put to, like a finely honed tool. But if that purpose is something other than giving us truth, then sure it "works", but is what it's doing really good?

In my continuum-based constructions there are still points, it's just that there are only ever finitely many of them and they are not fundamental. Can you expand on a situation where points need to be fundamental? — Ryan O'Connor

The issue here is with the way that I conceive of the relation between space and time. It is not conventional. With evidence derived from modern observations of phenomena like spatial expansion, it becomes clear to me that we need to give time priority over space in our modeling, to allow that space itself changes with the passing of time. This means that we cannot model spatial existence with a static 3d representation, adding time as a fourth dimension, because we need to allow that the principles for geometrical figures which model 3d space must actually differ being time dependent.

There is a need to produce two dimensions of time, one consistent with our present modeling of space as a static continuity of 3d existence, and the other to allow for the changes which occur to space, they require time as well, but this cannot be the same dimension of time. The finite points you refer to are the points where the two dimensions of time relate, or intersect with each other. So there is a continuum of spatial existence, extended in time, that is the classical 3d modelling. The points within that continuum need to be more fundamental because they represent where the other dimension of time intersects, thus constituting the real possibility for spatial existence. If we propose an infinite continuum of space, and we want to map onto this continuum real finite points of possible existence, then the limited location of those points must be derived from something real. That "real thing" must be more fundamental, because it represents real existence whereas the infinite continuum is the artificial map produced by us. The real points are points of spatial expansion (in this proposed model), which dictate the contortions to classical 3d spatial representations required to account for spatial expansion.

As for your continuum ideas, almost twenty years ago Peter Lynds wrote a paper appearing in Foundations of Physics Letters that postulated time having no instants and instead being composed of intervals. — jgill

The problem with this "intervals" of time is that some sort of instants are still require to separate one interval from another. Anything posited to break the apparent continuity of time would require a distinct aspect of time, calling for a second dimension. And if the intervals in any way overlap then there is also a need for multidimensional time.

I'll have the corned beef.

Small point. I have asked you how you now something, and you have just exhibited that you do not know it, but instead accept it as the consequence of an argument, which is neither an answer to my question nor, if the argument is otherwise flawed, defensible in that way either. — tim wood

I give up. I've tried numerous times to answer your question, and it appears I have no idea what you're asking. It seems like you're having the same difficulty with the word "absolute", which you had the last time we discussed the use of that word. You just carry on as if the word isn't there, and insist that it has no meaning. So, onward with your diatribe if it makes you feel good, tim.

Philosophy is a quest for knowledge. The true quest for knowledge starts from a lack of knowledge. That's why Socrates professed to not knowing. The "presupposition" is a bias which interferes with the true quest for knowledge, because it's an assumption of already knowing certain things. So the philosopher must do everything possible to rid oneself of such presuppositions in order to enjoy a true philosophy.

To insist that one must start from a presupposition of some sort is simply counterproductive, an attempt to justify not making the effort to free oneself from the influence of bias, and do everything possible to approach with an open mind. And to argue that the assumption that one has rid oneself of such biases, is itself a presupposition, is a failure, because we all know that we cannot completely rid ourselves of them, so we do not presuppose such perfection.

The point to a good philosophy is to make any such presuppositions (biases) as irrelevant as possible, having as minimal as possible influence on the philosophy. So any philosophy which sees presuppositions as playing a significant role in philosophy is simply a misguided philosophy.

Which is why we're so thankful that you deigned to comment on it. — Pantagruel

Your very welcome, the gratitude is much appreciated.

What calculus does is describe the potential of that process. And I believe that when calculus was made rigorous by going from numbers (infinitesimals) to processes (limits) some 'infinite-like' numbers (irrational numbers) were left behind. I believe that to complete the job, we need to reinterpret irrational numbers as irrational processes. Calculus is the study of potentially infinite processes. In my view, the math is the same, dy/dt=3t2-10t+9. It's just that the philosophy is different. — Ryan O'Connor

This is very good insight. My belief is that we need to go one step further, and apprehend an infinite process, or "irrational process" as actually impossible. But since this process is a potential process, as you describe, this means that it is a possible process which is actually impossible. Therefore the infinite process must be rejected as logically invalid, because it's contradictory.

Rejection of the impossible needs to be recognized as the greatest epistemological tool which human beings possess. It is the basis for certainty in knowledge. In our quest for an understanding of "what is" we narrow the field of possibilities by rejecting any proposed possibility which can be determined as impossible.

In the case of the infinite process, we have a very difficult judgement. It appears to be a reasonable and logically valid possibility. But for some reason it's extremely repugnant to intuition, and the reason why it is repugnant cannot be properly identified, so as to prove logically that it is impossible. It appears as absolutely impossible to show, or demonstrate the impossibility of infinite possibility, because it cannot be done with an empirical demonstration. There is however a logical demonstration commonly known as the cosmological argument, in Aristotle's Metaphysics, and I assume that other forms may be produced.

This may seem like a trivial difference, but I believe that with this continuum-based view (as opposed to the standard points-based view) many paradoxes are no longer paradoxical. In fact, I can't even think of a paradox with this view (especially given our refined intuitions developed through quantum-mechanics). — Ryan O'Connor

I believe that this distinction between the continuum perspective and the points perspective is a very good start, but I don't think it's an either/or question. We need to allow for both. It is the application of both, the two being fundamentally incompatible, which leads to infinity, and the appearance of paradoxes. However, we cannot simply exclude one or the other as unreal, and unnecessary, because there is a very real need for both non-dimensional points, and dimensional lines. You cannot remove the points because this would invalidate all individual units, therefore all number applications would be arbitrary.

What I propose is a fundamental division between numerical arithmetic and geometry, which recognizes the incompatibility between these two. Numbers refer to discrete units, and geometrical constructs refer to a continuum. We ought to recognize that reality consists of both these aspects, and the metaphysical question which we are faced with is to determine in which circumstances each system is applicable. Of course the need to establish a system of correlation between the discrete and continuous will never go away, this is mathematics, but we need to get a firm handle on which aspects of reality are discrete, and which are continuous, before we can axiomatize that correlation in a way which might serve us adequately.

If you look at the definition of a limit, it's actually timeless. For all epsilon > 0 , there exists a delta > 0 such that ETC. So there is a leap from the intuition of the potentially infinite approximation process. The fundamental question is something like: what are we approximating? A limit is a real number, a point, and not the process (in the mainstream view). Different processes can converge to the same point. (Subsequences make this obvious, but it's not only subsequences.) — norm

The problem I see here, is that a process is fundamentally continuous. If it were not continuous we would identify it as a number of different processes. We apply a point, to limit or individualize a process, marking a beginning or ending. But if these points are arbitrarily applied, i.e. not in reference to real points in the apparently continuous world, the numbering system will not provide us with truth (in the sense of correspondence).

When we look at the physical world empirically, it appears like we find true limits in spatial extensions, the boundaries of objects. The reality of these spatial limits have always justified individuals, units, numbers, and quantity, with distinct spatial objects forming discrete entities. However, now that we've come to look closer at these entities, the boundaries have become vague, and when we take the physical object right down to the micro level, those boundaries are not at all valid. So we find that the empirical evidence has really been misleading us, we think that there are spatial boundaries, and distinct entities when there really is not, spatial existence is continuous.

Likewise, if we look at time rationally, (because we cannot look at it empirically), we find the exact opposite situation. Our intuition tells us that time is continuous, that's how it seems to be, time is continually passing. However, we know that there is a big difference between past and future. Since there is such a difference, it is necessary to assume a boundary between past and future. If such a boundary exists, (and it is logically necessary that it does, to maintain a difference between past and future), then this boundary will provide the necessary principles for discrete units of time.

Yes, and now for the third time you have - I have to presume - deliberately evaded the question. Which is unfortunately par for the course for you. Which earns for you a change of tone. How the F do you know, you ******* ******? — tim wood

Do you understand the principle, that if you have two premises which you do not know whether they are true or not, and they lead to a logical conclusion which is obviously false, then one or both of the premises must be false? The two premises are, rest may be absolute, and there is motion. These two premises lead to the conclusion of infinite acceleration. The conclusion is obviously false. The second premise, "there is motion" looks true, so the first must be false.

This presupposes that RGC claims otherwise. He doesn't. Absolute presuppositions are but one part in the field of study.

Read the paper. — creativesoul

This presupposes that logic precedes thought. — creativesoul

This is why I have no inclination toward reading the paper. It appears to inspire all sorts of nonsense like this, which I would simply reject and have no part of. Therefore it would just be a waste of my time.

I am still waiting for your explanation of your claim that you know that nothing is at absolute rest. And this not a claim there is such a thing or place, but instead how it is that you know that there is not. — tim wood

What I said is that I know your hand is not at absolute rest. And, I also said that if there is something at absolute rest, that thing would have to go through infinite acceleration to start moving. You, and some others here maybe, are the ones claiming infinite acceleration is absurd. I suggest to you, that instead of thinking about the absurdity of infinite acceleration, you simply apprehend absolute rest as absurd.
Then there is no need to think about infinite acceleration because it is only the idea of absolute rest which produces that idea. Make sense?

...and what is that problem? — InPitzotl

That problem is, the deficiency in our capacity to measure. Obviously, if the thing appears to be infinite, this means that we do not have adequate capacity to measure it.

...and how are you fixing that? — InPitzotl

I told you already. We need to revisit our spatial representations, from the beginning. Go back to the fundamental principles, armed with what we now know about space and time, derived from modern science, and rework them all, from the very beginning, starting with the most basic relationship between the non-dimensional point, and the dimensional line. The fundamental (Euclidean) axioms of geometry provide us with inadequate modelling principles which incapacitates our attempt to understand a large portion of spatial existence. Ryan gets it:

The real issue is with our assumption: that time can be broken down into a collection of instants. Or more generally, that a line is composed of infinite points. — Ryan O'Connor

And you even admit that your view is shrouded in mystery. Why not consider the alternative...that a line is not composed of points, but instead points emerge from lines? Why won't you consider my...line...of thought? — Ryan O'Connor

I have a slightly different way of looking at this issue. I give priority to non-dimensional existence, represented as points. Points can be related to each other through lines, but these are still non-dimensional, meaning that these lines do not correspond to real physical, dimensional existence, they are ideals, by which we relate other ideals, points. To establish a correspondence between the non-dimensional, ideal, the point, and the dimensional, real physical existence, we need to bring time in as the first "dimension". Time must be the most fundamental mathematical representation as simply order. Order can be expressed without spatial reference. An understanding of the passing of time will determine the relationships which points can have with each other in real dimensional existence, and geometrical figures must be produced in accordance with the principles derived.

Yes - any proposed example can be rejected. That's the whole point. It's not a matter of fact, it's a matter of choice. — T Clark

That's exactly the reason why "absolute presuppositions" cannot serve the purpose of underlying any field of study, or any knowledge in general. If they can simply be accepted or rejected at will, they have no capacity for creating the coherence which we actually find within knowledge. To adequately account for the existence of knowledge we need to understand the power which logic may have over will. And the idea of "absolute presuppositions" essentially denies the role of logic in producing the fundamental metaphysical principles which serve as the basis for epistemology. In reality the fundamental principles are produced by reason, and we adhere to them because we have faith in the capacity of reason.

Have you made that argument elsewhere in this thread. If so, I've missed it. I'd be interested in taking a look. — T Clark

No, I had a lengthy discussion with tim wood (to spell the name properly) before on this subject, and I'm just not interested any more. I concluded that Collingwood simply misunderstands the grounding of epistemology, trying to assign to it something (absolute presuppositions) which just cannot serve the purpose. This is why there is so much disagreement amongst readers of the work in this thread, as to what exactly the term refers to. It is just a fictional thing made up by Collingwood, which he believes must exist in order for knowledge to exist. I find it's quite similar to Wittgenstein's epistemology.

When we start to look around at existing knowledge, and try to identify these absolute presuppositions, we find that it really can't be done. For one reason or another, any proposed example can be rejected. So we must conclude that people like Collingwood and Wittgenstein really didn't understand what supports our knowledge, and their proposed epistemologies are misguided.

In case you didn't quite get it, what you call "multiple infinities" is what I called "fix after fix". Instead of addressing the problem which is the question of what causes the appearance of an infinity, the mathematicians create a "fix" to deal with the infinity.

Incidentally, GR and SR use dimensional representations of space. This includes with GR the use of Penrose diagrams, which have multiple infinities. — InPitzotl

Exactly, look at the problems which envelope modern cosmology and quantum physics, due to the use of inadequate spatial representations, and a massively complex and fractured numbering system with imaginary numbers etc., required to cope with these problems.

The problem with Platonism (Pythagorean Idealism), is that once we assign to mathematics the status of objects, the objects obtain the status of unchangeable, and therefore eternal truth. Then it appears impossible that these mathematical constructs could be wrong. So when a problem emerges, something which cannot be figured out under the current mathematical system, a fix must be created, axiomatized, and added to the system. These fixes are nothing other than exceptions to the fundamental rules, causing the whole mathematical structure to get extremely top heavy with fix after fix. The fundamental rules which ought to be seen as faulty if they require such exceptions, cannot be apprehended as such, because they are already axiomatized as objects, eternal truths of Platonism.

Some time ago, at this forum, there was a discussion concerning the validity of Euclid's fifth postulate, the parallel postulate. It was argued that the geometry required by GR rendered the parallel postulate invalid. To any rational human being, this would indicate that the fundamental postulates of dimensional space need to be revisited and recreated. But human laziness inclines us to avoid this task, and simply make exceptions to the rules in an attempt to establish compatibility between what science now tells us about spatial existence, and what the rules created thousands of years ago say about spatial existence. Then we rationalize our laziness by saying that mathematical principles are eternal Platonic objects, and couldn't be changed even if we wanted to change them.

But it's not socially unacceptable to be attracted to someone of a different race. Many though, would argue that it is morally unacceptable to be attracted to someone solely by their visual appearance.