When I say potential infinite, I don't mean 'something that might be infinite'. I mean a process that certainly goes on to no end. We are certain that if you begin to write out the digits of pi that that process would never end. — Ryan O'Connor

Why call this "potential infinite" then? If you are certain that the process goes without end, then you are certain that it is actually infinite.

We might however use terms like potential and actual to distinguish between things which have real existence in the world, and things which are completely imaginary. In this case, there is no such infinite process actually going on, so we say that if someone endeavoured to carry out that process they would find it unending, therefore infinite,. But since they would never arrive at "the infinite", we'd say that such an infinity only exists potentially. One could never prove it to be infinite by reaching infinity.

This is why I didn't like your use of "infinity". You used it as if it signified something with actual existence, which one could be approaching. It seemed as if you thought that if you carried out such an activity, then after a designated point you could be said to be "approaching infinity", when in reality you know that you would never be approaching infinity, because you clearly recognize that the process would never end without ever reaching infinity. Therefore that is a deceptive us of words, when you claim to be approaching something which you clearly acknowledge you can never reach.

Morality involves adapting your personal taste, desires, inclinations, and attractions, to socially accepted standards.

Also Tim apparently disagrees with you and or seems to be contradicting himself, so one (or both) of you has misunderstood Collingwood or else he also contradicts himself: — Janus

I've discussed this with timmy before, and I've come to the conclusion that the idea of "absolute presuppositions" as proposed by Collingwood, is itself contradictory. This is what happens when someone pushes the boundaries in proposing a concept, trying to assign to the concept, a function which is impossible.

The north pole of the Riemann sphere. But carry on. — jgill

Amazing, the things which mathematicians will come up with, in an attempt to solve their problems, instead of simply recognizing that the dimensional representation of space is wrong.

You were dismissive of my question without answering it and apparently not even getting the substance of it. — tim wood

I said your hand is not at absolute rest because the earth is moving. And you asked me how do I know this. So I answered that. How am I supposed to know that you were asking me something other than what you were asking me?

If I am at rest in any sense whatsoever, then on your account any acceleration I'm subject to must be in the instant infinite. And that is absurd. — tim wood

It isn't absurd, it just shows that the idea of an "instant", a zero point in time is absurd. The conclusion of an infinite acceleration is only produced from the idea that there is a point in time when a thing goes from resting to moving. Obviously then, what is absurd is the idea of a point in time, not the idea of rest. And the falsity of this idea (of a point in time) is borne out by special relativity which describes simultaneity (being how we determine a point in time) as frame dependent. There are no real points in time, they are arbitrarily assigned according to a frame of reference.

I don't have a problem with saying 'x approaches infinity' in the context of a potentially infinite process. I interpret it as 'the value of x is continuously growing'. 'x approaches infinity provides some information about the journey, even if we never can arrive at some final destination. — Ryan O'Connor

I don't like this phrase "approaches infinity" because it reifies infinity as a thing which is approached. Either the process is designated as infinite, or it is not. Suppose we don't know whether it is or it is not infinite, then we might say that it is potentially infinite, because we don't know. Perhaps, for some reason we are not ready to commit to infinity, like if someone worked out pi to hundreds of decimal points, and was still not convinced that it would go forever. That person would say that it appears to approach infinity, and it is potentially infinite, but I think it might still reach an end at some point, so I won't admit that it's actually infinite.

Don't you think that we have enough evidence to make the judgement, it is infinite? Suppose we say that it appears to be infinite, but it's still possible that it is not. How is this compatible with "approaches infinity"? Then you'd be claiming that there is some actual thing called "infinity", and the values appear to be heading in that direction. But how does this make any sense? What sort of thing could that be, which is called "infinity"?

This is not a typical graph in that it spans all possible values of x and y. Think of it topologically in that it is a single system which maintains its topological properties when undergoing continuous deformations. In this plot, there is a point at (1,1) and a pseudo-point at (∞,0). In this context, it makes sense to say that we're starting at (1,1), travelling along y=1/x and heading towards the pseudo-point at (∞,0). By plotting Gabriel's Horn like this, (∞,0) is no longer out of sight, it's right there in front of us. And because of that we have the ability to use it in some contexts without requiring infinite measuring capacity. — Ryan O'Connor

But what is the meaning of that point which you label as (∞,0)? How can ∞ represent a point? You say it's a "pseudo-point". I assume that this means that it's not a valid point. What's the point in having a non-valid point? I can see how it's useful in practice, but this is an exercise in theory.

You know this how, exactly? — tim wood

Because the wind blows.

So in the end you agree with the notion that existence is contingent on opinion, and you simply differ on which opinions count. You just lost the argument methinks.

And what if I find a metaphysician who, based on two years of dialog with me, clearly hasn't bothered to learn the most elementary facts of mathematics? Why should I trust that individual's judgment about anything? — fishfry

What argument have I lost? "Existence" is a word which is being used here as a predicate. So we need criteria to decide which referents have existence in order justify any proposed predication. Naturally we ought to turn to the field of study which considers the nature of existence, to derive this criteria, and this is metaphysics. Mathematics does not study the nature of existence, so mathematicians have no authority in this decision as to whether something exists or not, regardless of whether it is a common opinion in the society of mathematicians.

If you are arguing otherwise, then show me where mathematics provides criteria for "existence" rather than starting with an axiom which stipulates existence.

There's at rest in a given inertial frame. Which is to say, really, that any acceleration, on your theory, should involve instantaneous infinite acceleration. My hand is at rest on the table. I raise it to type. Space-time not locally crushed in the process. — tim wood

It's "absolute rest" which I said is a problem, because this makes a point in time into a real situation rather than a perspective (reference frame) dependent designation. That's a point when no time passes relative to the thing at absolute rest.

There's at rest in a given inertial frame. Which is to say, really, that any acceleration, on your theory, should involve instantaneous infinite acceleration. My hand is at rest on the table. I raise it to type. Space-time not locally crushed in the process. — tim wood

This "at rest" which you refer to isn't real, because the earth is moving. Your hand is never at rest. So the moving of your hand is just a change in the existing motion of your hand, it is not an act of acceleration from rest. Physicists might represent it as an acceleration from rest, but the point I am arguing is that this is really an incorrect representation, which serves the purpose, just like representing Gabriel's horn as approaching 0 is an incorrect representation, which serves a purpose.

You are equating 'approaching' with 'arriving at'. — Ryan O'Connor

No I'm not equating these two. If there is no such thing as the lowest point, then it is impossible to be "approaching" the lowest point. In the case of the natural numbers, do you see that there is no such thing as "approaching" the highest number? We recognize that there is no such thing as "the highest number", so it doesn't make sense to say that if a person is counting higher and higher, they are "approaching" the highest number. You can never approach the highest number. If you can apprehend this, then why can't you turn it around, and see that when infinity is at the low end, there is no such thing as "the lowest number", and it doesn't make any sense to say that someone counting lower and lower is "approaching" the lowest number?

But if my trip never ends there are some situations where I could still give you some useful information since in some situations I could still tell you which direction I'm pointing (e.g. what I'm approaching). — Ryan O'Connor

OK, this is a good point. The question here is what grounds or substantiates "direction". You imply that direction must be grounded by going toward something, but you forget that it might equally be substantiated by going away from something. In Gabriel's horn we have both, moving away from one axis, and moving toward the other. The axes are artificial confines, imposed as standards of measurement, and through the descriptive term of "infinite", the line of the form is stipulated as going beyond the capacity of the measuring scale. Therefore to understand that line we can no longer employ those measurement axes.

This is the problem we have here. Generally we assign infinite capacity to the measuring tool, and this allows us the capability to measure anything with that tool. The natural numbers are infinite for example, and this allows that we might count absolutely any multitude of objects. In Gabriel's horn, we have turned the table. We propose an infinite shape to be measured. Of course we cannot measure it, because it is defined as infinite, meaning that we cannot measure it, the thing is stipulated as going beyond the capacity of the measuring tool. So there is a trick hidden in the proposal, it's asking us to measure what cannot be measured by the tool. Then when we look at the shape, we see it getting further and further from the one axis, and we conclude, 'that's impossible to measure'. But we also see it getting closer and closer to the other axis, and intuition tells us, 'that's a finite distance which can be measured'. However, we must adhere to the principles of the construction, which state that the shape will appear to approach the axis, to a point beyond our capacity to measure the distance between them. Therefore we must resist our intuition and inclination to say that this distance is measurable.

So we must remove the axes as incapable of giving us the scale required for the measurement. The axes are what produced the form, which is by that construction, infinite and therefore immeasurable. Therefore the axes cannot be used to measure that form, because it has been constructed by them, as immeasurable. Now we have no basis for the terms of "farther from" or "closer to", because these values have been stipulated as going beyond our capacity to measure. What we are left with now, is just theoretical values, to be assumes as spatial distances, values which we acknowledge cannot actually be measured as spatial distances. Now we are really not talking about "farther from" or "closer to" any more, even though the numbering system employed was originally derived from that. We have explicitly gone beyond our capacity to determine farther from or closer to, and all we are talking about now is a higher value and a lower value. If we do not divorce the value from the spatial distance, we are just left with a spatial distance which is impossible to measure.

The point now, is that since we have done what the example requires, and taken the values beyond our capacity for making spatial measurements, we cannot use spatial references to ground or substantiate "direction". All we have now is a higher value and a lower value, and the stipulation that each of these may continue infinitely. The two directions (values) are actually defined in relation to each other. As the one gets a lot larger, the other gets a tiny bit tinier. And so long as we allow that the one can continue to get a lot larger, we must allow that the other can get a tiny bit tinier. But these "directions" must be thought of solely as numerical values, because we have gone beyond the relevance of spatial distances as dictated by the proposed example. So we cannot look at them as spatial directions of "farther" or "closer" or else we just fall back into the stipulated impossible to measure..

I addressed this in my post. The position and velocity functions are not differentiable at time zero. So there's no well-defined acceleration. Nor as others pointed out does relativity bail us out. Relative to your own frame of reference, you are at zero velocity at time zero and nonzero velocity a short time afterward. You have to come to terms with that. — fishfry

So the point I'm making, is that zero is completely arbitrary, and represents nothing real, just like in the case of Gabriel's horn. That's why we must decline this idea of "approaching zero". It is extremely useful in practice, yes, for sure it serves the purpose. But this is an exercise in theory, and we need to be able to go beyond what works in practice to be able to see that the principles which we employ in practice mislead us in our metaphysical efforts to understand the true nature of reality. The existence of paradoxes such as Zeno's demonstrate an incompatibility between theory and practice, and these incompatibilities expose where we misunderstand the true nature of reality.

I know one mathematician who thinks the world is discrete and that continuity is a fiction, and then I know another who believes the reverse. — norm

This is a different, but related issue, the difference between discrete and continuous. The issue is not whether the world is discrete or continuous, it is to find compatibility between the two. In practice the world is continuous (time passes continuously), but in theory the world is discrete (represented by distinct units, numbers). Simply modeling the world as discrete, or modeling the world as continuous, is fine either way, until someone approaches you with an example of the other, and makes a paradox jump out at you.

This function is not differentiable at zero. There is no instantaneous velocity at zero and no definite acceleration either. I agree that this is counterintuitive, and your intuition is not uncommon. But it's wrong. Clearly it's wrong. If you experienced infinite acceleration even for a moment, every atom in your body would be flattened like so many pancakes.

I actually agree with you about the intuition. If we're not moving, how do we start moving? It's a bit of a mystery actually, I'm not sure what physicists say about this. Well I guess I do know. If we're a steel ball in Newton's cradle, or we're a ball on a pool table, we start moving when we get smacked by another ball that transfers its momentum to us. But how does our velocity go instantaneously from zero to nonzero? The Newtonian physics works out, but not the intuition. — fishfry

I addressed the issue in my post. There is only a need to conclude infinite acceleration if we assume absolute rest, zero velocity, but relativity denies absolute rest. If something had an absolute zero velocity, and changed from that zero velocity to having a positive velocity, this would imply a point in time (not a short duration) when the thing goes from zero velocity to a positive velocity. At that point in time, since there is no duration, but there is acceleration, the acceleration would be infinite. We avoid this problem with relativity theory which denies the reality of rest, and makes any supposed zero point in time into an extended duration.

I haven't got a good answer. A lot of smarter people than I don't have a good answer either. Have you? — fishfry

There is a special field of study which delves into the nature of being, existence itself, and this is called metaphysics. Metaphysicians, being trained in this field, are best able to say whether something exists or not.

To put it simply, non-dimensional existence, which is represented by the point, 0, is incompatible with our representations of dimensional existence. So 0 cannot enter into our scales for measuring dimensional existence, as a valid measurement point until we establish commensurability between non-dimensional and dimensional existence. This problem with zero becomes very relevant when we start to consider motions, and acceleration from rest. An infinite acceleration is required to go from rest to moving. The problem is somewhat avoided with relativity theory which denies the reality of rest, making acceleration simply a change in direction. But what that does is make the mathematics extremely complex, still working with points and vectors, rather than resolving the problem of how the non-dimensional truly relates to the dimensional.

We both agree that y gets lower and lower (and perhaps you would even agree that the greatest value which y never reaches is 0) but I call that approach and you call that not approaching. Let us agree to disagree on definitions! — Ryan O'Connor

As I explained, by way of example, to assume such a "greatest value", or "lowest value" is contradiction. When we say that the natural numbers are infinite, and therefore have no highest value, it's contradiction to say that 20 is closer to the highest value than 10. Likewise, when there is no lowest value, it's contradiction to say that .01 is closer to the lowest value than .02.

What is misleading in the example of Gabriel's horn, is that the x and y axes are set to converge at 0, at a right angle in relation to each other. This proposed point of convergence creates the illusion that 0 is a valid value, where x and y are 'the same". However, as I described earlier, the spatial representation of two dimensions at right angles to each other is actually a false representation, making the two dimensions incommensurable, as demonstrated by the irrationality of the square root of two. This incommensurability indicates that the two proposed lines, x and y, cannot actually be modeled as intersecting, and sharing a common point at 0.

So this false idea that x and y actually meet each other at that point, 0, is what misleads you into thinking that 0 is a valid measurement.

Yes. We have here the "hope" of wanting our live to improve. Everything needs an effort but previously we do need to have beliefs and then believe in... As you perfectly said previously.
More than a reason I guess is important how to perceive our feith. Sometimes hope and belief are upon the reason itself.
Probably the reason could say to you "do not do it because it is impossible" but the beielfs and feith say to you "let's do it we have another chance" — javi2541997

I've never seen the word "feith" before, and I'll assume that you mean "faith".

I believe it is very important, in any understanding of belief, which is not to be a misunderstanding, to apprehend the role of faith. Faith relates to the effort required to produce or create a belief. If we take belief for granted, as something which just naturally occurs without requiring effort, then we overlook the necessity of faith. From this perspective we'd have no approach to the cause of belief, thinking that beliefs just pop into existence spontaneously. But when we (correctly) see that belief requires effort, just like memorizing requires effort, then we can apprehend this effort as the cause of belief.

I think that faith relates to the effort required to produce belief. It is the confidence which we have in our efforts, that the efforts will produce results, be successful. But there is a real issue with losing faith, disillusionment, which happens if the goals start to appear as impossible, and the faith starts to look like a false faith. The significance of faith and effort, in the role of producing belief, is the reason why Plato associated belief, and intelligible objects in general, with "the good", rather than with "the truth". Assuming "x is good" leads to effort, while assuming "x is true" often leads to disillusionment.

his I would disagree with. One can take the viewpoint that symbolic math is a human endeavor; and that a thing has mathematical existence whenever a preponderance of mathematicians agree that it does. No Platonism needed. We've seen this standard applied to irrational numbers, negative numbers, transcendental numbers, complex numbers, quaternions, transfinite numbers, and many other now-familiar mathematical objects.

In order to demonstrate that sqrt(2) has mathematical existence, I do not need to posit a mystical Platonic realm in which sqrt(2) lives. If I did, I might be challenged: What else lives there? The baby Jesus? The Flying Spaghetti Monster? Pegasus the flying horse? No, I don't need to sort all this out just to know that sqrt(2) exists. — fishfry

Why give special status to " a preponderance of mathematicians", granting them the capacity to determine the existence of things? Isn't there a preponderance of Catholic theologians who believe in Jesus, and a preponderance of Pastafarians who believe in the existence of the spaghetti monster? Why do you think that mathematicians, simply by believing in something have the capacity to grant that something "existence", while for other groups of people, simply believing in something is insufficient for the existence of what is believed in?

So I guess you want to explain that we can only have the belief hypothesis when some methods and objectives are actually true. — javi2541997

Not necessarily true, but belief requires a reason to believe. Like any form of memorizing, it requires effort, and effort is only made when there is a reason to make it.

What about people who hold irrational beliefs - say paranoid psychotic delusions - that couldn't possibly derive from some type of memory process (because such belief content lies outside of previous experience)? — emancipate

False memories are common. That's what I described as self-deception. When what you remember happened, contradicts what another person remembers to have happened, then one or both of you are wrong. But they are still memories, even if they are mistaken.

Belief is always a living, current, fundamental commitment. — Pantagruel

What you refer to here is the act of believing, which is distinct from, and ought not be called "belief".

Having faith in something we can do comes when we are ready to pursue it. So I think firstly comes the act of "believe" in general terms and then "believe in..." specific terms. — javi2541997

I wouldn't be so quick to judge priority in this manner. Since we are essentially active beings, continually engaged in activities, I would think that it is quite likely that we "believe in" a particular capacity prior to formalizing a belief. So for example, if I have a walking trail which I regularly walk, and there's a water course, a stream or ditch which I must jump across, I gain confidence in my capacity to jump across, prior to gaining the belief that I can jump across. We see this clearly in the scientific method, where certain theories or hypotheses provide us with the capacity to predict, then after obtaining faith in this capacity (believing in it), we proceed toward the belief the hypothesis provides some sort of truth.

Right, the belief is the "attitude of confidence" that is what we are discussing. It is not the memory, and it doesn't have to be "about" memory. Belief is always a living, current, fundamental commitment. — Pantagruel

I don't think that's quite right. A "belief" is a thing, the word used in this way is a noun. That thing is a memory which has been subjected to the process of believing. Believing is an activity and it is produced by the attitude of confidence. The belief is the result of this activity. So the belief is the memory which has been subjected to that process, of believing. It is not the attitude of confidence, nor is it the process (believing) which is produced by that attitude, it is the result of that process.

This would be a vicious circularity. You can't believe something unless you already believed something. Clearly we do begin to believe, which is not an 'historical fact'. — Pantagruel

There is no circle, because I do not equate belief with memory as if they are the same thing. Belief is derived from memory which is prior to belief, as required for it, such that a belief is a particular type of memory. To believe is to have an attitude of confidence toward your memory. Then a belief is the memory subjected to that attitude of confidence.

If I believe I am writing this now, how is that a memory? — Pantagruel

All those words which you are applying in reference to your belief, "I am writing this now", require a memory of meaning. To believe "I am writing this now" is to have confidence in your use of those words, and that requires your memory of how those words ought to be used.

refer to all of those who have the power of believe in something: the next vaccine or reduce the Carbon emissions (for example). — javi2541997

This is a more specialized use of "believe", to say "believe in". It is better represented as having faith in a particular power, or capacity, to overcome obstacles. To simply "believe" is to have faith in one's power of memory, but to "believe in" is to have faith in some capacity to act.

I suppose my terms are that no decision is random ergo, no decision is truly free because it is the direct consequence of something that happened before.
This is fundamentally what I can't disprove. I hope that makes sense. — Barondan

Here's what you can do to prove the reality of free will to yourself. This is something which you must prove to yourself, because the idea that someone could prove it to you is counterproductive because that presupposes cause and effect. You can take an object, any object but preferably unbreakable, and hold it above the floor. Decide for yourself, that you will drop it to the floor, yet refrain from dropping it, knowing that you will drop it, but not at any specific time. Wait for a while, then drop it at a time determined only by your mind without any other influence. If you are capable of doing this, then you know that you have free will.

What is a belief, other than a memory? Nothing. And to believe is to have confidence in what is believed, i.e. the memory. Confidence in one's memory is the assumption of sameness, that the thing recalled is the same as the thing which was remembered. So to believe is to be confident in one's capacity for maintaining sameness.

You can see how self-deception is very relevant when we allow ourselves to conform our memories for various different purposes.

Did you read the rest of my post? What I'm saying is that 0 is not even relevant. That's what I've been trying to explain, that to describe the value of y as approaching 0 is a false representation. Y is always infinitely far away from zero, because zero is impossible on that line. The value for y never "approaches 0". It is correct to say that the value gets lower and lower, but it is incorrect to say that it approaches 0, because no matter how low it gets it never approaches 0. 0 is not at all relevant to this line.

To say that y approaches zero is an inaccurate simplification, nothing but a rounding off in your description. The real description is that the value of y gets lower and lower without ever approaching zero. Of course the true description is "not workable", that's the nature of any infinity. The appearance of infinity is the result of something being not workable. To make an infinity into something workable is to provide a false representation.

In short, I believe our disagreement is simply the result of us having a different definition of limit. — Ryan O'Connor

I think where we disagree is in the role that zero can play in this measurement. You think 0 can play a role, as the value which y approaches. I think that since the line has no start nor end, 0 is not an applicable number.

Perhaps I don't grasp it, or perhaps I just don't agree with it. Let's assume it's the former. Please tell me whether the following points aligns with your view:

1) One can travel along y=1/x in the positive-x direction, without bound.
2) The limit of the journey corresponds to the final destination, which if anything would be (∞,0).
3) The point (∞,0) does not exist (since ∞ is not a number) therefore there is no limit. — Ryan O'Connor

I surely disagree. There is no "final destination." That's MU's error, why are you amplifying it? — fishfry

Yes that's what I'm saying, there is no final destination, so to even produce any representation (such as ∞,0), as if it is a final destination, is a misrepresentation amounting to contradiction.

There is a particular line we are talking about, and #1 ought to state that this line is extended "without bound", which means "there is no limit". So #3 ought to read "the point (∞,0) does not exist because there is no limit". Now we could add #4: "∞" is a description of the entirety of the line (not a point on the line), and 0 is completely unrelated to the line, therefore irrelevant.

Here's another way to look at the position of zero. It is an ideal, like infinite is an ideal. The two are opposing ideals, like hot and cold are opposing ideals. The line takes the characteristic of the one ideal, the infinite, therefore the opposing ideal, zero, is excluded. It's just like if we were talking about the absolute, ideal hot, cold would be completely excluded.

A question to you: what exactly is the difference between the living and the non-living? — TheMadFool

If we take Aristotle's perspective, described in "De Anima", On the Soul, the difference is found in the capacities, or powers of the soul, potentia. A living thing is defined by a soul, which is a principle of actuality, or activity. This is similar to vitalism. The living being, by virtue of having a soul, may have a range of different capacities, ranging through self-nutrition, self subsistence, self-movement, sensation, and intellection. The different capacities are understood as residing in the material aspect of the living body. The non-living, lacking a soul as a principle of activity, cannot be said to have these various capacities, which though they reside in the material body are properties of the soul.

So, the suggestion that living organisms can't be wholly understood through the objective sciences implies 'the supernatural'! — Wayfarer

The need to assume a "supernatural" is produced by the materialist tendency to dissolve the division between the natural and the artificial. When we maintain this division, we see that the artificial is created by intention, and the natural is not, and there is no need to invoke a supernatural. But when the intentional, the artificial, is conceived of as being a feature of the natural (such as emergence), rather than the inverse, which is to see "the natural" as a category created and produced by the intentional human mind, then "the natural" becomes fundamentally unsupported. This produces the need to assume a supernatural to provide substance for the reality of the natural.

Of course the ancients had no conception of fields which appear to enable sub-atomic particles to exert force over a distance. — Wayfarer

The Pythagorean "aether" is comparable to the modern understanding of "fields". In the Pythagorean cosmology, the heavenly bodies are a manifestation of the aether, just like in the modern system fundamental particles are a manifestation of the field. The principles which apply toward understanding the fields, are principles derived from the understanding of soundwaves, just like the principles applied toward understanding the aether were ratios derived from the understanding of soundwaves.

The idea that something can exert force instantaneously over a great distance is extremely old, as the sun appears to do this on the surface of the earth. Heating and cooling of the earth's surface appears to be instantaneous as the sun goes in and out of the clouds. But that idea is deeply counter-intuitive, and when we see an event at a distance, and hear it at a delayed time, we question the medium, and the means of transmission, of these force. The fact that the timing of the force of sound is distinct from the timing of the force of light makes us conclude that light has a different medium from sound. But the media are believed to be similar in the sense of both being understood by wave principles, so that the knowledge of the movement of the slower force, which is more easily understood, is applicable to the faster. In modern times, physicists have proposed a number of distinct fields each can be apprehended as a distinct medium by which force is exerted. But these proposals are extremely primitive, and it is quite likely that the distinct forces, and their appropriate media, have not been accurately individuated and identified, because the theories employed do not allow for distinct speeds at high levels. This is evident from the confusion which remains in that field of study.

Your notion of 'close' that is based on the number of points between A and B can only have value in a number system which is not dense in the reals, such as the integers. For example, since there are 3 integers between 0 and 4 but 6 integers between 0 and 7, we can conclude that so 4 is closer to 0 than 7. If you want to restrict your mathematics to the integers then your notion of 'close' is suitable. However, such a mathematics is far less powerful than orthodox math so I suspect that you'd have a very tough time convincing anyone to adopt your view. — Ryan O'Connor

You don't seem to quite grasp why I reject "closer". The line is known to never reach the limit, that's the point with "infinite". So it makes no sense to say that it is getting closer to the limit. It is impossible for the line to reach the limit. so it is impossible that one point is closer to the limit than another. The proposed limit is outside the parameters by which the line can be measured.

The issue is how to best measure the shape which is Gabriel's horn. Orthodox mathematics, while it is very good at other things, fails here, as is evident from the appearance of the paradox. The reason it fails, I believe, is because it describes a feature which very clearly cannot be described in terms of limits, as getting closer to a limit, and that's nonsensical.

So I don't agree with your resolution, because you still want to apply limits, where the shape denies the application of limits. If measurement necessarily involves the application of limits, then we regard this as having created a shape which cannot be precisely measured, just like a circle.

But feelings are not exactly like noses.

Asking if someone else has the very same feeling as I do is treating feelings as if they were noses or mobile phones. It's taking that language and misapplying it; feelings are not a something, and not a nothing, either.

"Are your feelings exactly the same as mine?" is less like "Do you have the same mobile phone as I do?" and more like "Have you stopped beating your wife yet?". — Banno

I really don't see the problem with talking about feelings as things. This allows us to compare our feelings, and understand each other. In fact, if we did not describe our feelings as things, we would not be able to understand our feelings at all, because that is how they appear to us, as things which we can describe, talk about. Then we'd have no way of knowing whether our feelings were exactly the same as each other, because we cannot see them to judge the differences, as we can see all the different noses. Therefore it is essential that we talk about our feelings as things which can be described, so that we can understand the differences between us.

Those who insist that feelings are not things which we can talk about, like noses or cell phones, are just creating a problem where there ought not be a problem. We should all be encouraged to talk about these things which we call feelings.

Sure. Here you have an interesting link related of how supposedly masons contributed to the build of cathedrals back in the Middle Ages: http://pedro-mundodebabel.blogspot.com/2014/12/pasajes-de-la-historia-xviii-el-origen.html — javi2541997

As builders, (engineers), the masons preserved and maintained all the mathematical principles which are the secrets to the universe. The Church took mathematical principles for granted, as given by God, and allowed the masons providence over them. So there was a fundamental division between the roles of the Church who saw the need to maintain the purity of, and defend against corruption of moral principles, while the masons received the task of maintaining the purity of mathematical principles.

So we have distinctly divergent subjects of importance here, a division which is fundamental to the structure called the division of labour. But the Church needed to be the highest in the hierarchy, to maintain the absolute authority of God, and this meant that the mathematical principles guarded by the masons could not change, or else the "given by God" status is challenged, along with the authority of the Church, which relied on that relationship with God. Of course this questioning of authority did happen with the Copernican revolution. If you read Plato's "Republic" he foresees the downfall of the State as a change in numbering system.

Example 1. United States Dollar. (1 $). If you check the emblem in the pyramid it says: nevus ordo seclorum This literally means "new order established". It is interesting to point out how a clearly meaning from masonry is in the most famous currency of the world... Right? — javi2541997

What does the pyramid with an eye on top of it signify?

That does not follow. 1 is closer to 0 than 2 is, despite the infinite number of points between 0 and 1. — InPitzotl

That does not follow. 1 is closer to 0 than 2 is, despite the infinite number of points between 0 and 1. — InPitzotl

I told you, you need to stop thinking about the numbers which make up the coordinate system which produces the shape. They are outside the shape, irrelevant, and insufficient for measuring the shape. There is no zero point to start from in our measurement, nor is there a zero ending point. The value of one dimension is allowed to increase indefinitely such that there is no highest value, and correspondingly, the value of the other dimension is allowed to decrease indefinitely such that there is no lowest value. Zero does not enter the picture. It is excluded, (just like "highest value" is excluded, so is "lowest value", or zero) and therefore cannot be a part of the measurement scheme.

Sure it does; but it's a bit more precise than this. The limit specifies that it's possible to get arbitrarily close to 0. "Arbitrarily" here is used in a strong sense that includes all positive distances at once. — InPitzotl

See, zero is irrelevant. At any point on the shape, the y value is "arbitrarily" close to zero. So it is false to say that it gets any closer, at any point, because it's always the same, "arbitrarily close". This arbitrariness indicates that zero is completely irrelevant to any valid measurement.

It appears to me, like you want to allow the x value to increase indefinitely, without limit, assuming no highest number, but you will not allow the y value to decrease in the corresponding way. You want to limit the y value's decrease with an imposed zero. This removes the symmetry from the shape, and is a false representation of it.

You seem to be imagining a hypothetical number "so big that" 1/x dips below 0. — InPitzotl

I didn't say anything about dipping below zero. Where did you get that idea from? I said it doesn't ever get any closer to zero. There is always an infinitude of values between it and zero, so it's really not ever getting any closer to zero. Zero is off the scale, it's literally not part of the scale, as it is excluded by virtue of being impossible. So there is always an infinity of values between the value of y and zero. Since there is always that infinity of values between any given value of y, and zero, it makes no sense to say that it is getting closer to zero.

Given that y continually approaches 0 as x increases, the limit is 0. — Ryan O'Connor

As I explained y does not in any way approach zero. It is always infinitely far away from zero, no matter what value it has, therefore it never gets any closer to zero, and cannot be said to approach zero. Zero is imposed as an arbitrary limit, on something which, by definition, has no limit. If the line came to an end at a particular value, we could say that value is the limit. But it doesn't, the line continues onward without limit. It doesn't reach zero, yet continues infinitely, so zero is right off the scale, irrelevant as unobtainable.

Consider this example.
First proposition: The natural numbers are infinite, therefore there is no highest number.
Second proposition: 20 is closer to the highest number than 10 is.
Do you see how the second proposition contradicts the first? We have the very same type of contradiction when we say that x can increase infinitely without y ever reaching zero, yet we also say that it is getting closer to zero. Zero has been excluded from the scale as an impossibility, just like the highest natural number. So we can't say that one point is closer to zero than another. How does that make any sense? It's just like saying that one number is closer to the highest number than another when there is no highest number. Here it's the lowest number, we're talking about and that's not zero because there's an infinitude of numbers to go through, making zero impossible, just like the highest number is impossible.

That's not the proper way to speak, to say that one number is closer than another to the highest number. One number is higher than another, but it is not closer to the highest number, because there is no higest number. Likewise, with the value of y, one value is lower, and another higher, but we can't say that one value is closer to the lowest number because there is no lowest number, just like there is no highest natural number. There is just more and more numbers, and zero cannot be posited as the lowest of those numbers, because it is not one of them, as unobtainable, impossible, outside the bounds.

But that is really what's going on at 6:20; — InPitzotl

Actually I referred to 6:26, when he says one over infinity, that's zero.

But there is a limit to how small the value can be; for any real number > 1, 1/x cannot be less than 0, whereas it can be less than any other positive real number. — InPitzotl

Let's remove this necessity of a "real number", maybe that's what's misleading you. Is there any limit to how small the value can be? No, that's what's meant by infinitely small, we can conceive that there is always a further value, smaller than any value which we put a number to. That's the same as what's meant when we say that the natural numbers are infinite, only in the inverse direction. We can conceive that there is a further value, larger than any value which we put any number to.

Therefore, what is meant is that there is no limit to how small the value can be, just like in the natural numbers there is no limit to how big the value can be. This, I think, is what's misleading you. You keep thinking that there must be a limit to how small the value can be, but what is indicated by "infinite" is that there is no such limit. If a limit was intended, we'd employ "infinitesimal", which indicates that there is a smallest possible. But "infinite" indicates that there is no limit to how small we can go.

The limit in its definitive form can be used to show that this is only true for 0; it is not true to say that the farther out you go, the closer you get to 1 billionth. It is only true to say that the farther out you go, the closer you get to 0 (arbitrarily so). — InPitzotl

What is arbitrary is the choice of "0" here, as the representation of some non-existent limit. There is no limit, the value can keep getting smaller and smaller, always beyond any numerical representation which you might give it, that's what's indicated by "infinite". So there is no point in representing this value as getting close to some imaginary limit, "0". The value keeps changing without ever reaching that proposed limit, so in reality it never gets any closer to that limit. There is always infinite more values to cover before it gets there. The value really never gets any closer to the proposed limit, it's always infinitely far away, so the limit is completely irrelevant. Therefore "the farther out you go, the closer you get" is not an accurate representation at all. because the whole point in saying "infinite" is to say that there is no end, so it's impossible that the end is getting any closer.

So, what do you think? How long until he's in jail?

This is not really what I'm saying. What I mean is that if the value on the x axis, or y for that matter, is proposed as infinite, then it is wrong to assign a limit, such as zero, to the perpendicular axis. By saying that the one is infinite you say that there is no limit to how small the value of the other can be, and zero is an incorrect representation. In other words it is incorrect to speak of a limit here, and to say that infinity is a limit is nonsense.

What I think is that there is a fundamental incommensurability between two distinct dimensions of space, demonstrated by the irrationality of pi and the square root of two, which indicates that representing spatial forms with perpendicular axis is a sort of misrepresentation. Spatial forms, as we know them, cannot be accurately represented this way.

So if we look at the difference between a straight line and a curved line, we see that one requires two dimensions, fundamentally, while the other defines (or establishes the limits of) one dimension. The curved line, even at an infinitesimally small length requires two dimensions to be mapped by a straight-lined coordinate system. This would mean that even the most simple thing in a spatial reality (a point particle for example), would require a multi-dimensional representation. And, the multi-dimensional representation would get it wrong because of the incommensurability between two distinct dimensions.

Therefore the curved (real) line will never be commensurable with the straight (artificial) line. And, when we map the curved line with a straight-lined coordinate system, the incommensurability shows as infinity. Gabriel's horn is a curved line being mapped by a straight-lined coordinate system, and the incommensurability is evident.

The further question, which comes to my mind, is which is the proper way to represent space. Which way represents how space really is? Is the proposed "real" curvature just an illusion created by deficiencies of observation, and there is no such thing as a true arch, or perfect circle. Or does all of space consist only in multi-dimensional, non-straight relations, and our artificial dimensional straight-lined coordinate systems are incapable of giving precise representations of multidimensional existence? Or is the apparent incompatibility due to something else, which we have not yet grasped? Perhaps we ought not be representing multi-dimensional lines with perfect curves, archs, and circles. Maybe we need to banish this type of object from geometry as not properly representing reality.

And the paradox (re)surfaces because we're using limits (which were introduced to avoid the paradoxes of actual infinity) to describe something that is actually infinite. — Ryan O'Connor

Wow, someone who actually understands.

I'm inclined to believe that Gabriel's Horn doesn't exist any more than the "number" 1/∞. — Ryan O'Connor

Of course it's a fictitious object.

Well then how do you know the area under the curve is infinite then? — fishfry

Because it cannot be measured. That's what infinite means.

If the extension is infinite, the volume cannot be figured. You can only figure the volume by assuming that there is an end, a limit, and this is rounding off. But then you are not figuring the volume of an infinite extension.

That does not follow. — InPitzotl

I see that we have a fundamental difference of opinion concerning the logic of spatial areas. I think that it is illogical to believe that a 3d spatial form with an infinite extension in one dimension could have a finite volume. You disagree.

It's not a presumption; it's the result of a calculation. The fundamental issue here is that you're critiquing the methods without understanding what they are or why they are employed. Take your critique of Tom at 6:20 in the video for example. There, Tom is calculating the results of an integral. Tom has an improper integral, and this describes the method for its evaluation:
1. If limtaf(x)dx exists for every t>a then∫∞af(x)dx=limt→∞∫taf(x)dxprovided the limit exists and is finite.1. If limatf(x)dx exists for every t>a then∫a∞f(x)dx=limt→∞∫atf(x)dxprovided the limit exists and is finite.
— Paul's online notes
When Tom says: "Well one over infinity, that's zero", that's okay, so long as we know it's a shortcut for:
limx→∞1xlimx→∞1x
...which is exactly 0, as shown previously by definition of that limit. You have confused this with saying that 1/infinity=0. That's baseless; infinity is not a real number; the domain of the integral is the same domain as the x axis, and infinity isn't even in that domain. The method isn't "plug in infinity", and there's a reason it isn't. — InPitzotl

Well you distinctly said it was a presumption. "Those presumptions are based on the fact that the quantity of the surface are "on the outside" is infinite while the quantity of volume "on the inside" is finite..."

We've been through all this. Your so-called calculation, "limx→∞1xlimx→∞1x...which is exactly zero by definition..." is nothing but a rounding off. See, it's zero by definition, not by calculation.

Freemasonry may have played a role in the separation between church and state. However, that separation ought to be viewed as a revolt against the authority of the church, therefore the organization would necessarily have been secret to avoid the wrath of authority. Documenting such a history would be impossible.

If that was the role of freemasonry, then it is historic, and it would not hold influence in modern society.

But to be clear, there is no one understanding of the type-token distinction. It's a bit muddled. Think I mentioned that. — Banno

Evidence That subjectivity is a very real aspect of language use?