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InfinityMaybe there's no joy there. Still, forcing the unwieldy mass of rational numbers to line up single file to be counted was a master stroke. — Srap Tasmaner
When the measuring stick needs to be measured, it's time to throw away the measuring system completely, and devise anew. Otherwise paradoxes are produced, like Russell's.
Some people reject talking about infinite collections, I think, or reject talking about performing operations on them. — Srap Tasmaner
Of course, an infinite collection by any standard definition of "collection" is nonsense. A collection consists of things which have been collected, not things designated as collectible. And that's problems arise in set theory, "collection" becomes a designated collectible type, rather than the collection itself.
This is how the concept of "the empty set" creates paradoxes like Russel's. A "collection" with no items is not a collection at all. It is only a criterion for collection, therefore an abstract 'type" distinct from an item. Allowing for an empty set means that "the set" itself is not the collection of things (or else an empty set would not be a set), but "the set" is the abstract type, which describes the things to be collected. The things themselves, therefore, the elements of the set, must be categorically distinct from the sets, or else the empty set is the contradictory notion of a collection of nothing. Failing to follow this categorical distinction, which is necessitated by "the empty set", and allowing that a set might itself be an element of a set, produces problems.
But if the collection consist of things designated as collectible, and there is none of them, then it makes sense to talk about an empty set. However, this leaves cardinality as completely unjustified because the elements are just possibly collected, and therefore not counted.
Who would say no to that? How could you get from A to B without arriving at a point that's halfway between? — frank
I think that what Srap is saying is that we cannot reduce motion to a succession of truths. That's what Aristotle demonstrated as the incompatibility between being and becoming. If change is represented as a succession of different states of being, one after the other, then there will always be the need to posit a further distinct state, in between any two. Then we have an infinite regress, without ever accounting for what happens between two states, as the change, or "becoming" which occurs as the transition from one to the next.
So if motion is described as getting from A to B, A and B are the two points of being, you are at A, then you are at B. Since they are not the same, there is distance between, and we can posit a middle point. You are at C. Then we posit a point of being in between A and C. You are at D. Notice, we've reduced motion to "being at a point which is different from the previous point". But this produces an infinite regress without ever addressing the real issue of how you get from one point to the other, what happens in between. This is the real nature of motion, what happens in between, and it cannot be represented as being at a designated point. -
Donald Trump (All Trump Conversations Here)DHS notes a more than 1000% increase in assaults on ICE agents. — AmadeusD
I wonder why that happened. -
InfinitySo you think that "to be is to be the value of a variable" is a platonist principle? — Ludwig V
if being is reduced to value, that's idealism, not necessarily platonist though, but most cases yes. That's classical Pythagorean idealism, the cosmos is made up of mathematical objects.
Except that ordinal numbers don't assign a value; that assigns a place in an order. — Ludwig V
A place in an order, or hierarchy is a value.
No, it isn't. It is about whatever I am assigning a value to. — Ludwig V
What we were discussing was the act of assigning value, counting. That was the subject. Now you are changing the subject to claim that we were not talking about this act, but that we were talking about the thing which you assign the value to. Clearly we were not, as whatever it is assumed to be was not even mentioned.
Not all words refer to anything. That's why there's such a fuss about dragons and the present king of France. — Ludwig V
Why do you allow that sometimes when words refer to ideas (two, three, for example), they refer to things, but sometimes when words refer to ideas (dragons, present king of France), they do not refer to things? Why not just maintain consistency and recognize that these are all cases of words referring to ideas? -
InfinityI think many people believe that if something is referred to, it counts as an object. — Ludwig V
I think that would be an odd use of language, if every word referred to an object. Definitely not suitable for a rigorous logic. For example, we distinguish noun and verb, object from subject, subject from predicate. To disregard these distinctions would incapacitate logical procedures.
So you are right to foreground what we do with numbers - or numerals if you prefer. But I think you slip up when you say that the numeral refers to an idea. That just resuscitates that argument you gave about numbers as ideas. The assignation of value in this context is public and shared, so it cannot be about ideas in our individual minds. — Ludwig V
I actually don't mind when people refer to numbers as objects, that's the way I learned in school. But when people do this they need to respect the ontological consequences.
When you count something publicly, you share your assignment of value. Other people can observe and correct you if they think you make a mistake. This clearly is about ideas in our minds.
I'm getting the impression that your objection is simply to the concept of an abstract object, which you call platonism. Would that be fair? — Ludwig V
Banno was denying that the principles he asserted were platonist, and so I was trying to get him to acknowledge that they are. My objection was to the hypocrisy of publicly rejecting platonism then employing platonist principles.
In the Roman number system "V" counts as five. The Chinese system has 五 (wǔ) for the same number. The ancient greeks used the letters of their alphabet as numerals, so five was the letter epsilon. If you just talk about numerals, you lose the equivalences across different systems. — Ludwig V
That's exactly the reality of translation. In most cases there is no true equivalence "across different systems". The different language games come into being and evolve under different social contexts. The assumption of platonism, produces the idea of eternal unchanging objects which words refer to, and disables us from being able to understand the reality of the nonequivalent aspects. -
InfinityAn abstract object is something that isn't physical, but it's not simply mental either. — frank
This is platonism. The abstract object is independent from minds, but accessed by them.
What of quantification? — Banno
Quantification doesn't require platonism. The proposition that a numeral represents a thing which is a number is platonism. But we can quantify without that premise. For example, we can do a bijection between the numerals and the things to be quantified. The presumption of "numbers" is superfluous in this case. -
InfinitySo math is just language games, right? — frank
You could say that. The point though is that if a numeral refers to a number which is an object, and that object is said to be an idea in someone's mind, then it would be a different object in each mind. We are all distinct individuals with different bodies, different minds, and different ideas. It could not be the case that the idea in my mind (if we call it an object) is the same object as the idea in your mind even if we each refer to our ideas with the same word. We might use the same name "1", and even be trained to describe it with the same words, but it's still not the same idea.
You might consider the beetle in the box analogy. We use the same word, "beetle", and we might even describe it in the same way, but we still have distinct objects. The only way to assume that the numeral refers to the same object for distinct individuals, is to assume that the object is independent. That's Platonism. For whatever reasons, I do not know, @Banno insistently denies the obvious, to say that a numeral refers to a number, which is an object, is classic platonism. No one ought to be surprised by this. Western ideology is firmly based in idealism. -
InfinityIf each individual 1 is a token of the type <1>, you have to say what sort of thing the type is. That's not going to work out. — Srap Tasmaner
I don't understand you. In each instance where 1 is taken to be a token, the type is a symbol. And the type of symbol is mathematical. And the type of mathematical symbol is a numeral. How is there a problem with this?
if they are only in the mind, he owes us a story about how we manage to do things with them in the world. — Banno
I have no problem with that story. we are human beings with minds and free will, and we figure things out and do things. Don't you think that's the case?
Notice that this odd position is blandly asserted, not supported by any argument. — Banno
It appears like I didn't make the argument clear enough for you, when I stated it earlier. So, here it is.
If a numeral refers to an object, which is within a human mind, it is a different object for me as it is for you, due to the nature of subjectivity. My thoughts are not the same as your thoughts, so we'd have distinct objects being referred to because we have distinct minds. Therefore, since a numeral is supposed to refer "an object", not a bunch of different objects, and also to the same object for you, as it refers to for Srap, that object must be independent from both of you. The referenced "number", as "an object", must be an independent object This is known as a platonic object. Hence, assuming that a numeral refers to an object called a number, is platonism.
To state it simply, without assuming that the object referred to is an independent, platonic object, it is impossible that the numeral refers to the same object for distinct people, because we each have distinct minds with distinct thoughts. Then the numeral would refer to a bunch of different objects in different minds, instead of "an object", the specified "number". Therefore the assumption that a numeral refers to an object called a number is platonism.
He relies on presuming that all reference must be object-reference, — Banno
That's another one of your very absurd misrepresentations. I explicitly stated, in the passage you quoted, that the symbol might refer to an idea in a mind. Never did I imply that I believe all reference must be object-reference.
What I said, is that if a numeral is taken to refer to an object, a thing called a number, that object must be a platonic object. This is supported by the argument above. However, I do not believe that a numeral refers to an object called a number. I believe that it refers to an idea called a value. I believe that values are not objects, yet they are referred to. Therefore, in no way do I believe that all reference is "object-reference". -
InfinityNumbers are not just ideas in the mind, but are rooted as objects in our shared practices. — Ludwig V
Let's be clear, numerals are objects in our shared practices. Numbers if they are assumed to be objects are nothing other than platonic objects.
The question is, what do you think a numeral like "1" refers to. If you think it refers to an object, in the type of "number", or the "mathematical" type, that is a platonic object. If you think it refers to an idea of quantitative value, or order, in your mind, that is meaning, not an object. If you think it refers to an object of shared practise in your mind, there is no such thing. Numerals are objects of shared practice in your minds, not numbers. -
InfinityBut we need another step - "1 counts as a number" - to get the procedure moving.
...
It's not platonic. — Banno
The only way that "1" can refer to an object called "a number", instead of referring to distinct ideas in the minds of individual subjects is platonism. Platonism is the only way that "1" can refer to the same thing (a number, an object) for multiple people. Otherwise "1" refers, for you, to the idea you have in your head, for me, to the idea I have in my head, and so on. This is the way that values such as mathematical values are presumed to be objective rather than being subjective like many other values. It's known as platonism. -
InfinityDo you mean the premiss that space can be infinitely divided, not merely conceptually, but also physically? — Ludwig V
No, I've repeated this numerous times now, "space" is purely conceptual. it doesn't make sense to talk about dividing space physically. Physically there is substance, and that's what is divided. And representing that substance as "space" which is infinitely divisible is what I called the false premise which produces Zeno's paradoxes.
But a physical limit to the process of division doesn't undermine the conceptual description. — Ludwig V
It means that the conceptual description is false. And, this falsity, because it is a falsity, produces the absurd conclusions which Zeno demonstrates.
We've already left Meta behind, since he has claimed numbers are not ordered... — Banno
As usual, a completely false and utterly ridiculous representation. I said it doesn't make sense to use "next" in a way which is not either spatial or temporal. If we switch the term to "order" rather than "next", this allows all types of hierarchy such as good/bad, big/small, etc.. But the principle of the hierarchy, and the order of things within the category still needs to be defined. There is no such thing as simply "order" in the general sense. And to have a next implies a direction, which implies either a temporal or spatial ordering.
Therefore we cannot avoid expressing the order itself in spatial or temporal terms. If the scale is big and small for example, then for there to be an order one of the two extremes must be prior to the other, and this turns out to be a temporal order. If there was a supposed order which was infinite in all ways it could not be an order, because infinite possibility is disorder.
I was thinking some days ago that, though I'm not sure what the favored way to do this is, if pressed to define the natural numbers I would just construct them: 1 is a natural number, and if n is a natural number then so is n+1. I would define them in exactly the same way we set up mathematical induction. (Which is why I commented to Metaphysician Undercover that the natural numbers "being infinite" is not part of their definition, as I see it, but a dead easy theorem.) — Srap Tasmaner
You just show that it is limitless which is how "infinite" is defined, so there is no difference and you are not getting away from it being so, by definition.
But we need another step - "1 counts as a number" - to get the procedure moving. — Banno
The prerequisite platonist premise.
It's not platonic. — Banno
The usual denial. That "1 counts for a number" rather than signifying a quantitative value, is platonic. That's what platonism does, it makes values which are inherently subjective mental features, into countable independent objects. This is a faulty attempt to portray what is fundamentally subjective (of the subject) as something objective (of the object)
So we get "One counts as a number" and "every number has a subsequent number" and discover that the pattern does not end, and then learn to talk of the whole as being unbounded and that infinite counts as being unbounded... iterating the "...counts as..." to invoke more language games. — Banno
Your statement "every number has a subsequent number" is a stipulation. Therefore it is something produced by design, definition, it is not something that we "discover". So you continue in your misguided attempt to justify mathematical platonism. -
InfinityBecause next can mean two different things.
1) Next in the definition (logical next).
In mathematics, next often just means “the item with the next label in the sequence.” It’s part of how the rule is set up, so if you tell me where you are, the rule tells you what counts as the next one. That doesn’t require anything to be happening in time. — Sam26
The "logical next" is next in time in this context. The only other option is "beside" in space, and this is clearly not the case. "The item with the next label in the sequence" is the one which comes after the other. Therefore the sequence is temporal. Without the separation of before and after, there is no sequence. The rule tells you "what counts as the next one", but unless you follow the rule, and produce "the next one", then the next one never comes. And following that rule is a temporal process. Therefore the sequence is a temporal process.
One might argue, that the order of such mathematical things simply exists, as eternal platonic objects, and that "the rule" is a description of that platonicly existing order. Then we'd have the nontemporal order, without having to fulfil the process of following the rule. But platonism is clearly wrong here. the rule is clearly not descriptive, because the proposed platonic objects cannot be observed to be described. They have no spatial/temporal existence. Therefore the rule is a prescriptive rule, and the sequence only comes into existence by following the process temporally.
"Next" here implies a relation, and mathematics is the study of the relations between its "objects," which it is happy to treat as effectively undefined. — Srap Tasmaner
Yes, "next" implies a relation, as you say. It implies a temporal relation. "Next" has two distinct meanings, a spatial relation, or a temporal relation. In this case it is not a spatial relation, therefore it must be a temporal relation.
You may insist that mathematics keeps "objects" as undefined, But mathematics would be useless if it cannot define its relations. And this is a serious consequence of having "object" as undefined. If we cannot identify an object, how can we formulate relations? In other words, we cannot unequivocally understand the proposed relations between objects if we do not know what an object is.
So, you are asserting that "next" implies a relation. Do you think you could explain what "next" means in the context of a mathematical sequence, without describing it as either temporal or spatial? Otherwise you are simply making an unjustifiable claim.
Empirically, that may be true - especially if you regard a field (gravity, magnetism) as a medium. But setting up a set of co-ordinates does not require a medium in addition, so far as I can see. — Ludwig V
I know we can do that, and that's the point I was making. We can, and do set up sets of co-ordinates without reference to the medium. That is a universal conception of "space", which allows in principle, for infinite positioning. But it is conceptual only. And if one sets up such a universal set of co-ordinates, with infinite possibility, and applies it to a real medium, it is a false representation as the primary premise in the representation which will follow. That false premise is what creates Zeno's paradoxes.
The point being that we can, and do set up such co-ordinate systems, I'm not arguing against that. What I am saying is that when we apply them they are applied as false premises, As such, they produce unsound conclusions as demonstrated by Zeno. Zeno concluded that motion cannot be real. -
InfinityIn math, process doesn’t have to mean a thing happening in time. It may just mean a rule, a precise recipe that tells you how to get the next step, or how to compute the nth term. Infinity shows up because the rule has no final step. — Sam26
How could "the next step" not imply "a thing happening in time"? -
InfinityFor me, empty space is not a mediium. — Ludwig V
Of course it's not empty space, or else it wouldn't qualify as a medium. That's the point I was making. There is no such thing as empty space between objects. So to make a co-ordinate system which shows the positions which an object could have requires knowing the type of object and the type of medium.
Space is a co-ordinate system, which defines the possibilities where certain kinds of object may be. Objects are distinct from mediums because the latter are found everywhere, but objects have a locating within space. — Ludwig V
So "space" here is completely conceptual. And the point I was making is that it needs to be conceptualized according to the objects which are to be mapped and the medium between the objects. If we make a co-ordinate system which allows any objects to be anywhere (infinite possibility) that produces Zeno paradoxes. It's the faulty conception of space which allows for infinite possibility that creates Zeno type paradoxes. -
Donald Trump (All Trump Conversations Here)so the advice is to remain calm, don't open the door unless they show you a warrant,. — frank
Careful, they may be disguised as Jehovah's Withesses. -
InfinityIt seems to me that the question of a medium in space is secondary. The first move is to set up a co-ordinates and rules for plotting the position of objects on those. (In other words, the concept is defined by the practice.) Once we have co-ordinate and objects, the question of a medium makes some sense. How non-mathematicians develop the concept is another question. But we can be pretty sure it is by interacting with the ordinary world. Mathematics, in my book, is a development of that. — Ludwig V
Well, I can't say I understand exactly what you are proposing, but it seems like you are saying the question of the medium is secondary, but then you explain why it must be primary.
The nature of the medium, in relation to the nature of the substance which is moving, determines the possible positions. So without determining the medium and the substance first, one could set up a co-ordinate system with infinite possible positions, but it would be false if the medium doesn't allow for it. That is also the case with divisibility. The mathematical system could allow infinite divisibility, but in reality divisibility must be determined according to the substance to be divided, and the means of division. So we might start with the co-ordinates and rules for plotting, as you say, but then it would just be trial and error, in application.
So you start out by saying that mathematics ought to be prior, "The first move is to set up a co-ordinates and rules", but then you end with the statement that mathematics is a development from our interacting with the world, which would place it as posterior.
The paradox of Zeno's paradox, for me, is that Achilles is precluded from reaching a point that defines the system - the limit. The first step is to divided the distance from the start to the goal, limit, by 2, and so on. The limit is not an optional add-on, (as it seems to be in the case the natural numbers). — Ludwig V
The problem in this paradox of Zeno's, is the issue which is explained above, as starting with the designation of rules and limits, instead of determining the true limits of the medium and substance first. The rules allow for infinite divisibility, but this does not correspond with the true medium.
Here's a way of looking at it. Suppose the measurement is on the ground, a long tape measure on the ground. Each time Achilles takes a step, the foot is at a new position on the tape measure. And, the section of the tape measure between there and the last step, is never traversed by Achilles. he steps from one position to the next, with a gap in between. So the false premise which Zeno makes is that all the area has to be covered. It doesn't Achilles steps from one spot to the next. Achilles could give the tortoise a short head start, then take one step and be past the tortoise, without ever properly catching up. This is why the nature of the movement and the medium is so important. -
Mechanism of hidden authoritarianism in Western countriesIn particular, these laws are always aimed at suppressing small businesses, because small businessmen are less dependent on the power and can overthrow it. — Linkey
In general, large business is advantageous to the government. There are many reasons, but it's easier and more efficient for the government to have the company rule over its various activities itself, and it's employees, collect taxes etc., and report to the government, then for it to govern over a whole bunch of small businesses.
So in agriculture and food production for example, the government can stipulate that the company must hire inspectors, and maintain a safe food supply, rather than having to send out a whole crew of inspectors around to all the different small businesses. The company does the inspections, but a small business couldn't afford this. It's a matter of efficiency. -
InfinityThe question is, at what level of explanation should this incompatibility be situated? at the physical level, as physics usually assumes, or at the level of the rules of mathematics? — sime
I think it must be both. The theoretical mathematicians who practise what they like to call pure mathematics want to be free from the constraints of the physical world. So, they may produce axioms independently of the requirements of physics, and other sciences. However, the axioms which get accepted and become conventionalized are the ones which are applicable. Then by the time the use of any axioms become standard practise, they have been selected for, by the needs of the scientists.
Therefore we can separate the two in principle only. We put the hypotheses of science in one category and the hypotheses of mathematics in another category. But if we maintain the supposed separation into a description of actual practise, science must have logical priority. Ultimately then, conventional and standard mathematics has been shaped to meet the descriptions of scientists. So the incompatibility is within the descriptions provided by physicists. We must maintain a different, real separation though, and that is between descriptions and the real world. It is not necessarily a feature of the real world, which causes incompatible mathematics to be accepted, but perhaps a mistaken description.
So the conventional mathematics is shaped by demand, and the demand is the sciences. The incompatibility manifests in the mathematics which has been conformed to the descriptions of the sciences. Therefore the descriptions provided by the sciences must be faulty, they require incompatible mathematical principles. Like @Banno says, we cannot conclude that the real world is faulty. So what Zeno demonstrated is that our descriptions of motion are faulty, and the mathematics as applied to these descriptions, reveals this by leading to paradox.
I think we should consider the fact that Newton and Leibniz didn't invent calculus for the purpose of solving Zeno's paradox, but for describing trajectories under gravity. Hence the mathematical definition of differentiation that we inherited from them and use today, isn't defined as a resource-transforming operation that takes a mutable function and mutates it into its derivative; rather our classical differentiation is merely defined as a mapping between two stateless and immutable functions. — sime
What you say here about Newton and Leibniz demonstrates how modern mathematics was fundamentally subservient to physics. Since then, the study of pure mathematics, and number theory have become more distinct and separate from that foundation. This actually provides an advantage toward solving these issues, because it allows us to look directly at the incompatibilities within the mathematics, without being influenced by empirical prejudice. Plato's principle "the senses deceive us" is very important.
I believe this is how the heliocentric model of the solar system was figured out. When we remove the mathematics from the influence of our observations of the empirical world, which the mathematics is formed around, then we can extend it in all directions to see where the infinites appear. Each appearance of infinity represents a problem within the empirical description. (In the case of the solar system eternal (infinite) circular motion was the fundamental problem demonstrated by Aristotle.) Then we can make a map of just the problems themselves, and attempt to correlate them and determine a unified underlying cause. A huge number of problems can actually have one simple cause.
But if Zeno's paradox is to be exorcised from calculus, such that calculus has a dynamical model, then I can't see an alternative than to treat abstract functions like pieces of plasticine, that can be sliced into bits or rolled into a smooth curve, but not at the same time. — sime
I think the key issue is that we improperly represent time. Modeling time as the fourth dimension of space implies that time emerges from space. So at the Big Bang, there is something spatial, and time emerges. But that's fundamental incoherent because all activity, including "emergence" requires time. So this representation implies that time existed before time. To rectify this we need to model time as the zeroth dimension, and allow that space emerges from time. This requirement is also indicated through an analysis of the pure mathematics, removed from empirical prejudice. The non-spatial, non-dimensional, "point", is very real and necessary to mathematics. Therefore it must be accommodated for in our modeling of activity in the empirical world. Currently, empirical descriptions do not allow for the reality of non spatial activity (time passage without spatial change). This is a very big problem, which makes the modeling of activity at a non-spatial point completely speculative, and somewhat incoherent.
That is part of what @sime was talking about, the incompatibility between representing objects as having position relative to each other, and as being in motion relative to each other. So when we attempt to unite the concepts of space and time, we must alter them. "Space" gets altered by being "curved", so that standard Euclidian geometry doesn't serve. And "time" gets altered by the relativity of simultaneity, so that there is no objective present. -
InfinityWe can be pretty confident that space is not infinitely divisible and yet still use calculus to plot satellite orbits. — Banno
This is an interesting remark. Many would say that "space" is conceptual only. And if it is, how could it be anything other than the way we represent it, as infinitely divisible?
"Space" might be the distance between two objects, but space is not what is measured, distance is. Furthermore, we commonly assume a medium between two objects, air or something. And space is not the air. Clearly, if we were talking about air, we wouldn't represent it as infinitely divisible.
So this is why there is a problem, when we get down to the basics, the medium between the nucleus of the atom and the electrons of the atom for an example, we really don't know what the medium is. The proposal of aether has been dismissed, so we just produce an artificial (imaginary) medium, some fields or something like that. Since the concept of "space", and its accompanying mathematics provide for infinite divisibility, and the proposed medium is simply conceptual, how could the medium be modeled in any way other than a way which is consistent with the concept "space", and the related mathematics, i.e. as infinitely divisible. Without proposing a real medium with real restrictions to divisibility, to propose that the fundamental medium between things is not infinitely divisible, according to how it is conceptualized as "space", is somewhat incoherent. -
Trump's war in Venezuela? Or something?As the Trump administration oversees the sale of Venezuela's petroleum worldwide, Senate Democrats are questioning who is benefitting from the contracts.
In one of the first transactions, the U.S. granted Vitol, the world's largest independent oil broker, a license worth roughly $250 million. A senior partner at Vitol, John Addison, gave roughly $6 million to Trump-aligned political action committees during the presidential election, according to donation records compiled by OpenSecrets. — Stephen Groves, The Associated Press -
Trump's war in Venezuela? Or something?But the bullshit pretext would be easy to see, given it’s Iran. But if Trump says Good was a domestic terrorist hellbent on running officers over— then it gets taken as fact, despite the video evidence. — Mikie
I see the point. If thousands of protesters are being shot, then the "bullshit pretext" is easily seen. If it's just the odd one here and there, it's easily hidden, though the pretext is the same bullshit. And the intended effect, intimidation, is very similar. -
Trump's war in Venezuela? Or something?Immediately after the killing, the President labelled the victim a "domestic terrorist", and blocked a complete investigation. — Relativist
Protesters are domestic terrorists. That looks similar to what the regime in Iran would say.
What most concerns me is the fact that a large segment of the US population thinks it was perfectly fine to execute her because she violated the law. There is an absence of commitment to due process. — Relativist
Look at those poor drug runners, blown up in their boats. Since when is 'shoot on sight' the strategy for dealing with cocaine traffickers? Oh I forgot, they're not cocaine traffickers, they are "unlawful combatants" in "a non-international armed conflict" with the USA. I suppose, "non-international" makes them domestic terrorists so 'shoot on sight' is warranted. -
InfinityNo one ever says either of those things. You're arguing with someone in your head who knows no more about mathematics than you do. — Srap Tasmaner
Huh? Someone in my head knows more about mathematics than I do? Isn't that contradictory?
...there are an infinite number of steps in this description of the distance between 0 and 1, but that simply does not stop it being traversed in a finite time. — Banno
Obviously then, the description is wrong.
Unless Zeno can find a fault in that calculation, it proves that the issue is in the approach to the question, not in the situation as described. — Ludwig V
The approach to the situation, is logically prior to the description. What appears to be the case, is that the description, and consequently the approach, are both wrong.
To represent motion in a way that avoids the paradox, requires a smooth and differentiable continuous topology that doesn't contain points that are in need of traversal, but only open sets that can finitely intersect to create spots, but not infinitely often so as to create points. Yet on the other hand, to represent positions requires a discrete point-based topology of infinitely thin spikes that doesn't blur position information. Hence motion and position require incompatible topologies. — sime
Or, we can represent motion as discontinuous, which is the way that quantum physics seems to demonstrate is the real way. The particle has a position, then it has another position, without traversing the intermediary. I believe, that what happens in between cannot completely be represented as "a smooth and differentiable continuous topology". Issues with the wavefunction demonstrate that this is not quite right. So what happens in between ought to be represented as truly unknown, though it is actually represented in a not very accurate way, as a continuous topology of superpositions.
No need to overcomplicate things. — Banno
Reality is complex, this is philosophy, and the common mistake is to oversimplify. Sometimes Ockham's blade just doesn't cut it. -
InfinityI realize that you see the contradiction as implicit and unavoidable. But you are not recognizing the meaning given to the terms within the system.
"countable" within the system means only that some of them can be counted and we cannot find any numbers in the sequence that cannot be counted. Actually, since we had that discussion, I've come across the term "countably infinite" which I think is much less misleading. — Ludwig V
I don't agree with that interpretation, because along with "countably infinite" there is "uncountably infinite". Furthermore, Banno produced the proof that the set of natural numbers can be in a bijection with itself. And there can be a bijection between the naturals and the even numbers, and so on, so that the cardinality of these types of sets is the same. To assign cardinality is not a case of "some of them can be counted".
And I think that you are not aware of how the term "limit" is used within the system. A limit, in this context, is a value that the series gets closer to, but never reaches. It is not a value derived from the function. It is not the last term in the series.
It does not constrain the series at all. So, in Zeno's paradox, Achilles gets closer and closer to the tortoise but never reaches it. (Forgive my inexpert account.)
From my perspective, the adjusted meaning of terms within the system is one of the biggest differences between us. — Ludwig V
That is exactly the problem. It is treated as, and called "the limit" but it is not the limit. Since it is not the true limit, you don't actually ever get closer to it. There is an infinite number of places between here and the designated "limit", and no matter how many places you proceed through, there is still always an infinite number of places. Therefore you are never actually getting any closer to the supposed limit through that means. It is a faulty means. It's supposed to be the limit, and it's supposed that the series gets closer to the limit, but it does not.
That is why this representation of "limit" is faulty. It is a false representation, because there is always a gap of infinity between where you are and the supposed limit. Therefore it's not the true limit, you're never closer to it, or further from it, by that method. So you don't actually keep getting closer, and it's nothing but an arbitrary, and false designation of "limit".
To say that you actually keep getting closer would be like saying as you count the natural numbers you keep getting closer to the end. Or, as you work out pi to more and more decimal places you keep getting closer to the end. That's false representation. You do not. Likewise, to say that the series gets closer to, but never reaches the limit is equally a false representation, because there is always an infinite gap between the series and the limit. You are not getting any closer. This shows it to be a faulty representation, and a faulty method. Zeno's paradox is unresolved.
. -
InfinityI do not see a formal contradiction. — Esse Quam Videri
I wouldn't call the contradiction "formal", because it's not explicit as a statement of X and not-X. However, i see it as implicit, and unavoidable from the meaning given to those terms within the system. "Countable" implies capable of being put into a bijection, and "infinite" implies not capable of producing a bijection. The closest we can get to resolving this is with platonism, by saying that the bijection simply exists. But then "countable" would need to be replaced with "counted", and that implies that the limitless (infinite) is not limitless. So in my mind, the contradiction is unavoidable. -
InfinityI think we are on the same page now. I personally don't think that the axioms of ZFC are "true" in any metaphysical, transcendental or empirical sense. However, I accept that existence claims derived from those axioms are nonetheless valid within the formal system. This is formal/heuristic truth, rather than metaphysical or empirical truth. — Esse Quam Videri
We're still not on the same page, because I claim contradiction within the system, therefore I cannot conclude that the existence claims are valid. That would require that existing things could violate the law of noncontradiction. I say that there is no coherent way to make whatever is said to be infinite also countable, regardless of how you describe "existence" (except by violation of noncontradiction).
The problem is that "infinite" puts severe restrictions on what type of "existence" the supposed infinite thing can have. Some definitions of "existence" might not even allow for anything infinite. Therefore, if "exist" is to be a relevant term it needs to be defined in a way which would allow for an existing infinity. The specified "infinite" parameter must be limitless, boundless. This makes that parameter impossible to measure. Therefore if the infinite parameter is quantity, it is impossible to count.
So it's not a question of whether or not the said bijection "exists", it's a matter of whether it is possible for it to exist. I argue that the meaning of the terms, "infinite" and "bijection" render it impossible for that specified bijection to exist, no matter how you define "exist", unless you allow contradiction.
There is a connection to theology, which might explain why those approaches survive, though I confess it would not recommend them to me. — Ludwig V
The connection to theology which I get a glimpse of, is the ontological argument, which became unacceptable even in theology. The principal problem which I see with that argument is that it attempted to but a boundary, a limit to the limitless. The "infinite" God was described as "That than which nothing greater can be conceived". Notice how it is a deceptive way of describing what is supposed to be limitless, infinite, (God), as an actual limit, the limit to thought. (The greatest possible conception is a limit to thought.) So it's a sort of self-contradicting premise, because we can conceive of the limitless, "infinity", but then the premise presents this as a limit to thought, 'the greatest possible' conception.
So set theory clearly demonstrates how the ontological argument is faulty. If we represent the limitless, i.e. "infinite" as a limit to our thinking (as the ontological argument does), then our thinking will just think up a way to surpass that boundary. That is because the thinking mind will not allow itself to be limited by such self-deception. So when "infinite" is proposed as a limit, as is the case in calculus, then the mind which refuses such a constraint, will just say 'that's a countable infinity', and surpass it with a new, 'uncountable infinity'. Therefore, when the limitless is proposed, and accepted, as a limit to the thinking mind, the true reality of the thinking mind is that it will not allow itself to be limited in that way, so it simply surpasses that limit, and creates a new limitlessness for itself.
How I see this as a problem is that this method is complex and convoluted, so it needs to be simplified. And, the complexity obscures inherent contradiction, which I've argued for. This is because the way of set theory doesn't actually address the problem, it builds on top, working to cover up. The true problem is with the principles of calculus, where the limitless (infinite) is proposed as a limit. That's where the base of the contradiction lies. Then, instead of addressing this as fundamentally flawed, the mathematicians simply forge ahead, thinking that if the limitless has become a limit to us, we're just going to create a new limitlessness. That's the way the human mind works, if you constrain its freedom it will find itself a way around that constraint. -
Infinity
It's not even an issue of does math, or does math not exist. We are talking about whether we have an accurate description of it. To assign a name "exists" without any principles for understand what that word means, does nothing toward the purpose of describing.
If set theory starts with the premise that numbers are countable objects, and this is false, then it is a theory based on a false premise. We can ignore the question, and say that a number is "an object" in a different sense of the word, from how we commonly use "object", but then we have to ask, in what sense is it "countable" then. Counting is how we measure ordinary objects, and now we are not dealing with ordinary objects. It's incoherent to say that a number is countable in the sense of an ordinary object, but the number is not an object in the sense of an ordinary object. That's mixing apples and oranges. So this type of object, the mathematical object, must be "countable" in a different way from the way that ordinary objects are.
And the matter of infinity confirms this. If we assume the real possibility of an infinity of ordinary objects, they would not be countable. But then we say that an infinity of the type of object, which numbers are supposed to be, is countable. This confirms the problem. We count ordinary things with a bijection. Numbers are not counted in the same way as ordinary things. How are they counted? Or is the act of counting just a pretense? It appears to me to be the latter. Numbers are not counted at all, there is a stipulated formula for determining cardinality.
The matter of how does math work if it does not exist, is a completely different issue. That math works, implies that it is the means to an end. As such it is a technique, a "way of acting". Even though a "way of acting" cannot be said to exist, it can reliably bring about the desired end. A way of acting is perhaps best described as a sort of conditional. If the conditions are X, then the response is Y. So it might be described as a prescriptive rule. It's a way of applying a general principle to the particular situation.
The issue of whether a general principle, a Form "exists" is difficult. When you ask a bunch of philosophers what type of existence a rule has, you get as many different answers as people. But now we go to a further extent, and question the relationship between the general rule and the particular, in application, as a way of acting. It's like asking about the existence of a habit. It's best to avoid "exists" altogether in this context. Plato exposed this difficulty with "the good", as something which does not fit into the realm of existence. So we just judge habits as good or bad, without regard to whether they have "existence".
This seems to be the crux of the issue for you, and I can appreciate the tension that you are raising, but personally I don't see this as an issue. — Esse Quam Videri
That's fair, it appears like most people do not see it as an issue.
I'll try to explain my reasoning as clearly as I can. For me, to say "the bijection exists", is literally to say nothing more than:
(1) the bijection is formally derivable from the axioms of ZFC in combination with the inference rules of classical first-order logic. — Esse Quam Videri
How I interpret this, is that you believe it exists by stipulation. if something is stipulated to exist, then it does. I have a problem with this, because it circumvents the judgement of truth, allowing you to employ premises (axioms) without the requirement of truth. Ultimately the conclusions are unsound.
But I've discussed this with others, and it appears like "pure mathematics" likes to provide itself with the ultimate freedom of being unfettered by empirical judgements. This is advantageous to the mathematician for a number of reasons, but significantly it allows judgement based on results, rather than prior constraints. Therefore "unsound" is not necessarily bad. Forging ahead with unsound principles often produces what is good.
So in the reply to frank above, I described mathematics as "a way of acting". The way of acting is judged relative to the end, the consequences, rather judging the premises. This is pragmaticism, if it produces good results then the way of acting is itself good. We don't need to address the truth or falsity of the premises (axioms), and if you analyze the numerous different ways of acting you'll find that often the basic rules being followed cannot even be identified. At the base level, we have habits, and when a person acts by habit one cannot say that there is a rule which is being followed. So the pragmatist perspective renders the exact nature of the rules, and principles which are followed in a procedure, as fundamentally unimportant, because success is what is desired and therefore the focus.
This means that my strategy of attacking the premises, axioms, is rather pointless, as you and others will say "personally I don't see this as an issue". Even if there are blatant contradictions, it wouldn't be an issue, because success is what is important, and that's what frank pointed to. Contradiction at the base level is unimportant if the system reliably produces success. This means that the true test, the real judgement requires a focus soley on "success". In ethics this is consequentialism. However, it requires that we clearly define the intended goal, the end, in order to determine whether there really is success or not.
As a philosopher, I look beyond all the worldly goods, GPS mentioned by frank, computers, internet, AI, all these things, to say that the ultimate goal is to understand the true nature of reality. If this is the case, then as sime pointed out, mathematics delivers us a problem known as the uncertainty principle. This is a roadblock which stands in the way of success, under that description. Since this problem has been revealed, and is demonstrably the result of the mathematics, it is incumbent on us philosophers to analyze the axioms and premise of the mathematics, to determine where we are misguided. Since success is lacking, we need to take issue with the rules. -
Infinity've been meaning to return to this for a while now, but just haven't had time. — Esse Quam Videri
I'm glad you're back.
The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". — Esse Quam Videri
The issue of platonism is more about the existence of any bijection in general, and the question of whether a measurement exists without requiring someone to measure it. It's a form of naive realism, which in our conventions and educational habits, we tend to take for granted. We look at an object, a tree, a mountain, etc., and we assume that it has a corresponding measurement, without requiring that someone measures it. Then, when someone goes to measure it, the correct measurement is assumed to be the one which presumably corresponds with the supposed independent measurement. This type of realism requires platonism because there must be independent numbers and measuring standards which exist independently from any mind, in order that the object has a measurement before being measured.
But if we understand that numbering conceptions, and measuring conceptions are products of the human mind, then it's impossible that an independent object could have a measurement prior to the measurement being made by a human being. This rules out the possibility that the natural numbers could have a measurement, or be countable because we know that human beings could not count them all.
The argument about measurement that you provided in your reply is interesting, and I can see how it is relevant to question of whether (or in what sense) a countably infinite set can be said to "exist". But the word "exists" can have different meanings depending on the context. Within the context of ZFC set theory, to say that a countably infinite set "exists" doesn't imply that it exists in some Platonic heaven. That's not to say that you couldn't interpret it in a Platonic way, just that nothing in ZFC itself forces this interpretation. — Esse Quam Videri
So we could say, that numbers "exist" in a way other than platonic realism, but we must consider what would be meant by this. We need to ask, what is the criteria for existence. Consider the difference between the following two statements. 1. "the set of natural numbers between 0 and 5". 2. "{1,2,3,4.}". We might say at first glance that they both say the very same thing, and they both necessitate the existence of those four integers, but this would not be correct. That is because the first is a formula, and the existence of the named integers requires that the formula be carried out correctly. So we need to respect this difference, the existence of the integers in the first example is conditional, or contingent, on "correctness", and in the existence is necessary. And when we say 1, 2, 3, ..., or use the successor function, the existence of those numbers is conditional, contingent on correctness. Now we have the problem mentioned above, the natural numbers cannot have a measurement, because the procedure cannot be carried out to the end, "correctly".
The inclination might be to deny that distinction between necessary existence and conditional existence, which I provided. But we cannot do this because we need to account for the reality of human error. A formula does not necessitate the existence of numbers because error may arise in a number of different ways. The formula might be carried out incorrectly, or it might be a mistaken formula in the first place. So we might add, "the designated numbers exist if the the formula is properly formulated, and if it is carried out correctly. But that makes it conditional.
So to say that "a countably infinite set exists" is just to say "ZFC ⊢ ∃x CountablyInfinite(x)". The actual derivation follows very simply from the axiom of infinity in combination with the definition of "countably infinite". — Esse Quam Videri
So you are talking about a conditional existence. The supposed existence of the natural numbers, is dependent on the correct procedure. The issue with the definition of "countably infinite" is that the procedure cannot be carried out. The formula states something which is impossible to correctly finish, therefore the numbers cannot exist.
Furthermore, platonism doesn't solve the problem because the infinite is defined as being impossible, so the numbers cannot even exist in a platonic realm. That would be like saying that the full extension of pi exists in a platonic realm, when this has been demonstrated to be impossible.
I presently suspect that the structure of the uncertainty principle, that concerns non-commutative measurements, is a logical principle derivable from Zeno's arguments, without needing to appeal to Physics. — sime
I agree. Many people conclude that calculus solved Zeno's paradoxes. I've argued elsewhere, that all calculus provided was a workaround, which was sufficient for a while, until the problem reappeared with the Fourier transform.
I'm sorry. I should have said "separates", not "divides". — Ludwig V
I don't think this makes any difference.
Can you think of a form of measurement that is not counting - apart from guessing or "judging"? — Ludwig V
Sure, I believe measuring is fundamentally a form of ordering. So most comparisons which are intended to produce an order are instances of measuring. Get a bunch of people, compare their heights, and order them accordingly. That's a form of measuring.
As Frank points out,
It really comes down to which view best accommodates what we do with math.
— frank
And Meta's view undermines most of mathematics, despite what we do with it. — Banno
You mention "what we do with math", but are neglecting something very important, "what we can't do with math". This is the limitations, like the uncertainty principle mention above. We do a lot with math, sure, but there is a lot more we would be able to do if we work out some of the bugs. Then there's the even worse problem of the many things that people believe we do with mathematics, which we really don't. Some people think that calculus has solved Zeno's paradoxes. It has not. Some people think that mathematics allows us to determine the velocity of an object at a single instant in time. It doesn't. Some people think that mathematics has provided a way to make infinite numbers countable. It has not. That's what I'm talking about. To have the attitude that math is perfect, ideal, therefore it is wrong to subject it to philosophical skepticism is the real problem.
I guess Meta is a math skeptic. — frank
I like to apply a healthy dose of skepticism to any so-called knowledge. Nothing escapes the skeptic's doubt. -
Infinity(The finitude of an object's exact position in position space, becomes infinite when described in momentum space, and vice versa. Zeno's paradox is dissolved by giving up the assumption that either position space or momentum space is primal) — sime
i don't consider this to be a solution, because the result is the uncertainty principle. What you indicate is two distinct concepts of space which are incompatible, "position space", and "momentum space".
It depends, as I explained earlier, how you define "countable". I don't say that it's just all just a matter of definitions, but it's probably a good idea to get those agreed so that we can be sure we are talking about the real issues. As it is, we don't agree and so we never get to identify and discuss the real issues. — Ludwig V
We went through the common definition of "countable" provided by jgill, and the contradiction remained. So I really don't know what type of definition of "countable" you might be thinking of.
I'm not sure what you mean by "serves as a medium". — Ludwig V
"Medium is commonly defined as "something in a middle position". If something is between two things, it is distinct from each of the two as in the middle.
But the point of a succession is that every step (apart, perhaps, from 0) has a predecessor and a successor. That is what it means to say that n is between n-1 and n+1. It is not wrong to say that 2 unites 1 and 3 and it is not wrong to say that 2 divides 1 and 3. But it is wrong not to say both. — Ludwig V
Yes, every step has a successor, but the succession is described as a continuous process. No individual step can serve as a division between the prior step and the posterior, as each is continuity, not a division. To say that one step is a division would produce two distinct successions, one prior one posterior. then the one which served as the divisor would have no place in either of the two successions.
So I dont't understand what you are saying here, especially what you mean by "2 divides 1 and 3". One divided by two produces a half, and three divided two produces one and a half. But it doesn't make sense to say that two acts as a division between one and three in the way that you propose.
This just turns on your definition of what it is to count something.
Using a ruler to measure a (limited) distance means counting the units. Obviously, we need enough numbers to count any distance we measure. So having an infinite number of numbers is not a bug, but a feature. It guarantees that we can measure (or count) anything we want to measure or count.
I maintain that if you can start to count some things, they are countable. You maintain that things are countable only if you can finish counting them., It's a rather trivial disagreement about definitions. But I do wonder how it is possible to start counting if I can only start if I can finish. — Ludwig V
So it looks like you disagree with my premise that counting is a form of measurement. Since you claim that starting to count something is sufficient to claim that it is countable, then if we maintain consistency for other forms of measurement, puling out the tape measure would be sufficient to claim that the item is measurable. Since this is obviously not true, it seems you are claiming that counting is not a form of measurement at all. How would you define "countable"? -
InfinitySo your argument is that 2 is not between 1 and three. — Banno
That's exactly right. To say that 2 is between 1 and 3 is to say that it serves as a medium. However, in the true conception and use of numbers, 1, 2, 3, is conceived as a unified, continuous idea. This unity is what allows for the simple succession representation which you like to bring up. No number is between any other number, they are conceived as a continuous succession. To say that 2 comes between 1 and 3 is a statement of division, rather than the true representation of 1, 2, 3, as a unity, in the way that the unified numbering system is conceived and applied.
Well, no. You claimed there is a contradiction, repeatedly, but never showed what it was. So go ahead and quote yourself. — Banno
"Infinite" means limitless, boundless. The natural numbers are defined as infinite, endless. limitless. All measurement is base on boundaries. To say a specific parameter is infinite, means that it cannot be measured. Counting is a form of measurement. Therefore the natural numbers cannot be counted. To propose that they are countable, is contradictory, because to count them requires a boundary which is lacking, by definition.
This is why "open sets" are used to justify unmeasurable spaces, resulting in an incoherent concept of continuity. Incoherent "continuity" is the result of the false opinion that there exists numbers "between" numbers, instead of representing the numbers as a unified concept.
Look, to say that something is infinitely heavy or light means its weight cannot be measured. To say that it is infinitely long or short means that its length cannot be measured. To say that it's infinitely hot or cold means that its temperature cannot be measured. To say that it is an infinitely large or small quantity means that its quantity cannot be measured.
Why do you think the proposition that the natural numbers is countable does not contradict the proposition that the natural numbers are infinite, in the way I explained?
But in addition to the usual thngs nominalism rejects, Meta rejects the notion that numbers as values of variables. while nominalists say numbers aren’t abstract objects, they undersntad that they can still be quantified over. Meta says that numbers aren’t things at all — they’re modifiers like “pink”. That blocks: — Banno
Correction. There is no difference between a number and a numeral, the number is the symbol. What the symbol represents is an abstract value. It's a category mistake to say that a value is an "object" unless we define "object" in the sense of a goal.
Regardless of what you assert, to say that the value represented is an object called "a number" is platonism. Calling it an imaginary, or fictional object, doesn't fulfill the ontological criteria of "object". Therefore we'd have to treat it as an idea because treating it as an object would be a false premise. We cannot truthfully treat a fictitious object as an object, because it is an idea and the existence of ideas is categorically distinct from the existence of objects. -
InfinityI think you're discounting the importance of community. If it's not stretching your spine out of shape, you can go along with the rest of the phil of math and write it as platonism. It's a little nod to the deep bonds that hold us together over the millennia as our brothers and sisters try to take freakin' Greenland and what not. — frank
This issue is more complicated though. The Neo-Platonists took Plato's name and claimed to continue Plato's school, but their ontology is consistent with what you call platonist. Aristotle's school claimed to be the true Platonists but the Neo-Platonists took the name. So you have to take on the Neo-Platonists, and tell them that they should call themselves Neo-platonists, as not true Platonists. But this problem has been around for millennia, and they do not like being accused of misrepresenting Plato, they like to claim the true continuation of Plato's teaching.
We can make it simpler for you: How many whole numbers are there between one and three? — Banno
i say it's a loaded question, like "have you stopped beating your wife?". If we give up on the idea that there are numbers in between numbers, we get rid of an infinity of problems from infinitely trying to put more numbers between numbers. This supposition that you have, that there are numbers between numbers is very problematic.
Set the supposed contradiction out. — Banno
I did it all ready in this thread, numerous times. If you're truly interested go back and reread my posts. But I'm tired of it. And I know you, you'll just deny anyway so what's the point?
“Countable” is defined as “there exists a bijection with ℕ (or a subset of ℕ).” I bolded it for you — Banno
Right, begging the question. "There exists a bijection with N" is explicitly saying "N is countable". Are you kidding me in pretending that you don't see this? -
InfinityI would ask one favor though. Stop capitalizing the P in Platonism. The phil of math view of platonism. Plato pitted opposing ideas against each other, so for instance, in Parmenides, he outlines a lethal argument against the Forms. That's why they use a little p: platonism. — frank
I agree. but my spell check doesn't like little p platonism. And, I count the distinction as unimportant because there really would be no such thing as big P Platonism if we maintained that distinction. Plato pitted ideas against each other so there's no real ontological position which could qualify as big P Platonism. So they end up being the same meaning anyway. -
Donald Trump (All Trump Conversations Here)
I'm kind of surprised that he didn't just make his own.Looks like Trump has a Nobel Prize... — NOS4A2 -
Infinity
I can't really follow anything you are saying.
Now many integers are there between zero and five? — Banno
Again, "integer" is a faulty concept, because it assumes that "a number" is a countable object. That's exactly the problem I explained to you. We ought not treat an idea as an individual object. Providing more examples of the same problem will not prove that the problem does not exist. The problem of Platonism is everywhere in western society, even outside of mathematics, so the examples of it are endless.
Mathematics on the other hand takes a bijection between two sets A and
B to mean there is a rule f such that each element of A is paired with exactly one element of B, and each element of B is paired with exactly one element of A. — Banno
This does not address the point. A rule can contradict another rule within the same system. Saying that there is a rule which allows a specific bijection doesn't necessarily mean that there is not another rule which disallows such.
The bijection is not assumed, it is demonstrated. — Banno
That's false, it's not a demonstration, at best, it's begging the question. You have no definition of "countable", so your conclusion, "Therefore N is countable" does not follow. It would follow, if you provide a definition of "countable" which begs the question. The proper conclusion is N is countable if N is countable. Then a definition of "countable" could be provided which contradicts the infinite nature of the natural numbers, making "N" "countable". Voila, begging the question with contradiction. I think Magnus already explained this to you, so you're just continuing to demonstrate your dishonest denial. I don't see any point to further discussion, you'll only continue to refuse to look at what is shown to you, and rehash the same faulty arguments. -
InfinitySo you do know that the series is infinite without completing the count of them all. — Ludwig V
Of course, why would say that? it's defined as infinite. That's the whole point. It is infinite and infinite is defined as boundless, endless, therefore not possible to count. So any axiom which states that it is countable contradicts this.
And yet, Frodo Baggins exists - in the way that fictional characters exist. They can even be counted. Similarly, numbers exist - in their way. — Ludwig V
Yes, a finite number of fictional thigs is countable. But infinite is defined as endless, therefore it is impossible to count an infinite number of fictional things.
I'm not quite sure that I understand you. I think that it is not necessary for the infinite number of numbers to exist in my mind. — Ludwig V
I'll explain again. If numbers are assumed to be independent Platonic objects, we can assume that bijections simply exist, without needing to be produced by a human being. However, the infinite bijection is a matter of contradiction, even if bijections simply exist. Therefore it ought to be rejected as incoherent. If, on the other hand, numbers are assumed to be fictional objects, created by human minds, the same contradiction still remains. It is the idea that numbers are infinite, yet countable as objects, which is incoherent. So it does not matter how you validate the existence of numbers as objects which can be counted, the incoherency cannot be avoided.
All I need to have in my mind is S(n) = n+1. — Ludwig V
I don't understand why, so many people on this thread seem to think that if they can make symbols which represent something incoherent, this somehow makes it coherent. When we speak contradictions, that's what we do, use symbols to represent something incoherent. Why would you think that writing it somehow makes it coherent? I can say RNR stands for the thig which is both red and not red at the same time, but how does symbolizing it make it coherent? How does "S(n) = n+1" make you believe that an uncountable number of objects is countable? I truly cannot understand this.
It turns out that the disagreement turns on a metaphysical disagreement. Tackling that needs a different approach. — Ludwig V
It's not a metaphysical problem directly. As a matter of contradiction between basic axioms, it is an epistemic problem within the mathematical system (set theory). As explained above, it doesn't matter which metaphysics you use to validate the existence of numbers as countable objects, the problem remains.
I do propose that it could be resolved with a metaphysical solution though. The solution is to reject the ontology which supports the idea that numbers are countable things, along with the mathematics which follows (set theory). An idea is not a thing which can be counted, and that is a basic flaw in the ontology which supports set theory.
Notice, it's not a metaphysical problem in itself. We can assume that numbers and all sorts of ideas are objects, and maintain that ontology. The problem is epistemic. We think that since numbers are objects then they ought to be countable. that's what produces this problem. To resolve the problem we might say that numbers are a type of object which is for some reason not countable, but that creates a problem with the concept of "object". Therefore it's better, and actually provides a better foundation for understanding what concepts are, if we deny that numbers are objects.
If you think Meta has convincingly shown that numbers do not exist, then I suppose that's an end to this discussion. And to mathematics.
But I hope you see the incoherence of his position. — Banno
The point is that a number is not a thing which can be counted, it is something in the mind, mental. I think you understand the difference between physical, sensible things which can be counted, and mental thoughts which are not individual things that might be counted. You did read Wittgenstein's Philosophical Investigations didn't you? Did you learn anything from it?
There is a very significant error in the idea that a measuring system could measure itself.
No, I don't think that Meta has shown that numbers don't exist. I'm inclined to think that he doesn't believe that, either. He has been explicit that he rejects what he calls Platonism, but I don't think it follows that he thinks that numbers do not exist. I'm not sure he even rejects the idea that there are an infinite number of them - since he realizes that we can't complete a count of the natural numbers. I do think that we can't get to the bottom of what he thinks without taking on board the metaphysical theory that he has articulated. — Ludwig V
The point is that "numbers" do not exist as individual countable things. This is a misrepresentation of what a number is, and the problem becomes evident when we allow the infinite capacity of numbering, and then try to count those numbers. So it doesn't matter if you represent the number as an independent Platonic object, or an object of human construct, either way is faulty. A supposed individual number is really an idea, which is dependent on other ideas for its meaning, and cannot be accurately represented as an individual object. -
InfinityYour view is called finitism. It's from Aristotle. — frank
No, that's not quite right. I reject the assumption of any "mathematical objects" finite or infinite, as Platonism, and unacceptable. It just happens that the absurdity of assuming such objects becomes very evident with supposed infinite objects. In other words, the reality of "infinite", and "infinity" serves very well to demonstrate the falsity of Platonism.
A value can ‘have being’ within a formal system, a constructive framework, or a model, without existing independently as Plato would claim. — Banno
In case you think I did not address this claim, 'having being within a system' is fiction. It can be said that Frodo Baggins has being within a system, but this type of being is well known as "fictional".
Being within a mathematical system, is fictional being. Therefore it is a false premise rendering any supposed "proof" as unsound. -
Infinity
As I said, denial!
When numbers are assumed to be mathematical objects, these objects simply exist independently of any human mind. The supposed object is not in my mind, nor your mind, because it would be in many different places at the same time. Under the assumption of Platonism, a bijection does not need to be carried out, it can be represented, because it is assumed to already exist independently of any minds, so we just need to reference it. Likewise, the natural numbers can be represented with "{1, 2, 3, ...}" but only if they exist independently of any minds.
Do you understand the difference between the representation of a set of objects, and a formula for the procedure which you called "assigning a value"? Could you read 1, 2, 3, ... as a formula for assigning a succession of values?
1 is a number, and every number has a successor. That's enough to show that the natural numbers exist. — Banno
What you stated her is blatant Platonism.
If however, "1, 2, 3..." signifies to you, a formula for the process of "assigning a value" in a specified sequence, rather than representing an infinity of numbers, this is not Platonism. Can you apprehend the difference? I think you can.
And I think, that's why you switched form "to be is to be the value of a bound variable", to, "assigning a value". right after you stated "Platonism is indeed unacceptable". You know the mathematical principles you argue are thoroughly Platonist, and you feel ashamed of this. So you tried to cover this up. Why the dishonesty? Accept mathematics for what it is, and get on with it. Shameful deception and attempts to disguise your ontology get us nowhere.
Every time that you say 1, 2, 3... represents an infinity of numbers, that is blatant Platonism. It is absolutely necessary that the referenced infinity of numbers must have independent existence because it is absolutely impossible that they could exist within any minds. -
InfinityNo, Meta. Quantification or assigning a value does not require Platonic commitment. A value can ‘have being’ within a formal system, a constructive framework, or a model, without existing independently as Plato would claim. — Banno
You were claiming that numbers "exist", and how to be, is to be a value. Now you've totally changed the subject to "assigning a value".
Formally, set theory is just a system of rules. — Banno
Sure, and those rules are axioms about "mathematical objects". When you were in grade school, were you taught that "1", "2", and "3" are numerals, which represent numbers? Notice, "2" is not a symbol with meaning like the word "notice" is. It's a symbol which represents an object known as a number. In case you haven't been formally educated in metaphysics, that's known as Platonism.
Guess it's back to ignoring your posts. — Banno
And I'll opt to believe that you willfully deny the truth, rather than simply misunderstand.
Metaphysician Undercover
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