We are discussing a point as being dimensionless. The stake, blob, pixel, is used to give the point some visibility. The point has to be associated with a physical object in order to be useful. — sandman

But the rationals fail to be Cauchy-complete. For example the sequence 1, 1.4, 1.41, ... etc. that converges to sqrt(2), fails to converge in the rationals because sqrt(2) is not rational. There's a hole in the rational number line. — fishfry

That's exactly why the reals are regarded as a mathematical continuum, and the rationals aren't. — fishfry

surveyor places a stake as a marker/point for a property line. Being dimensionless, you can't see it, but an object is provided in the form of a marker, blob of medium on a surface, pixel on a screen. — sandman

I think we can know what an orange is, and that it doesnt have less volume when cut in half. It can be so divided infintely, so it's infinite AND finite. This is so obvious — Gregory

Likewise anyone can see that objects in the world are both finite and infinite. — Gregory

I don’t think evil will ever “ die off”, because it’s inherent in all of us. — Brett

Of course definitions can be contradictory. — tim wood

Anti-trumpists have cheapened impeachment... — NOS4A2

You're just being ridiculous. — tim wood

omeone explain this "field" idea to me. I think of forces as the interactive properties of mass, and particularity as the classical interpretation of massive objects that is reductive to a certain selection of their perceived features for predictive purposes. — Enrique

So mock me as you like, but you're actually mocking the rest of the world.. — tim wood

Are you prepared to argue that an idea has a size? — tim wood

Now, it seems you cannot keep track of the distinction between theoretical math and the math of the world. In the world, most locations have size, for lots of reasons that have almost zero to do with size. — tim wood

Maybe I should pay more attention to what you've written here. — tim wood

There is no issue between us. — tim wood

Maybe you could cut and paste your definition here. And resolve the difficulty of uncountable infinities of infinite size created when points have any size at all. — tim wood

I’m your huckleberry. — NOS4A2

Three-dimensionality is essentially a product of aggregation effects, concentrations of matter into particularized masses such as solids, liquids and gases, which however never completely lose the quantum dynamism of their fluctuation and motion. — Enrique

ou will notice I generally avoid interacting with Nos as I think he or she might be an agent tasked with disseminating Trump-friendly disinformation on this and various other minor social media sites. — Wayfarer

You've been a driver of this thread through an astonishing 251 pages, but you've done it by exhibiting the agility - and the morals - of a weasel. — tim wood

A point in geometry is a location. It has no size i.e. no width, no length and no depth. — tim wood

Well here's an example of confusion in your thinking. We agree that points and lines are ideas, therefore the proper objects of definitions. The which having nothing to do with true. But you mention consistency of definitions. What definitions? Your definitions? But you want to "add" points. How, exactly, do you add points? — tim wood

am not sure why you think so, Points do exist (both in time and space.) Consider that at any point in time, you occupy certain point in space. So there exist at least some of the points that we can imagine. Points that do not exist cannot be occupied by anything under any set of circumstances. — Magnus Anderson

It was only a matter of time before they impeached Trump for word crimes. It was too difficult for them to find actual crimes... — NOS4A2

(Imo) egg-zackly! That is, everything in and of math is an idea. Being ideas, they're subject to definitions, not reality in any sense at all. You wanna talk my ideas, then you have to know my definitions, and vice versa. As such, there is no "how it is." But there is, however, "how I say it is." — tim wood

But maths are useful in the world. As a consequence, it is necessary - and certainly useful - to not confuse the theoretical/idea side with the practical/reality side. There are "points" and there are "lines" and what either means, or any of a lot of other terms found in both theoretical and practical applications mean, is simply a matter of the application and the relevant understandings, without which nonsense reigns. — tim wood

...then they'd be making excellent arguments - excepting that with the qualification they might be trivial and not worth making. — tim wood

You seem a like a pantywaist. Neither are high or low crimes or misdemeanors. Neither are mentioned in the Constitution. — NOS4A2

What “actual crime” did Trump commit? — NOS4A2

I only used 4 properties to simplify the issue and the number 4, in and of itself, has no bearing on the critical aspects of identicalness which is, quite obviously, based on shared properties. The only method by which we may distinguish objects is on the basis of differences in properties and the only method by which we may say two or more objects are identical is by checking if they share all properties or not. — TheMadFool

Maths is constructed. One can do with it as one pleases with the symbols involved. We make the rules up as we go, and we can and do go back and change them as we like. They are not tied to instruments or forms or anything other than themselves. — Banno

What “actual crime” did Trump commit? First it was bribery, extortion, “a classic Mafia-like shakedown” as Schiff called It, and of course the media’s obsequious base followed right along. Now it’s...obstruction of congress? Sorry, but there never was any actual crime to begin with, just a snivelling, well funded political corporation known as the DNC living out its fantasies. — NOS4A2

What would it be made of? — tim wood

Is it reasonable (however defined) for philosophers who have not studied mathematics to argue basic principles of the subject? — John Gill

Though you cannot measure how long a point is (since it has no length, as per definition) you can identify a point. And we do so through a complex process that involves the movement of our bodies (if we're talking about identifying points in physical space, that is.) — Magnus Anderson

There are many things that have no size but that nonetheless exist (and are not logically contradictory, illogical or otherwise irrational.) The word "existence" does not imply size. For example, colors and feelings exist, and yet, they have no size. A typical counter-argument is that colors are light waves and that light waves have size (their wavelength.) But light waves are not colors. Rather, light waves are things that cause colors. (This is evident in the fact that light waves can exist without conscious beings whereas colors can't.) — Magnus Anderson

As you guys have gone on and on describing the great need for this impeachment and congratulated one another on each other's rhetoric, you'd think the House would actually impeach this President instead of playing politics. — Hanover

...when he unlawfully takes.. — Hanover

The fact that points have no size (which, by the way, does NOT mean that points have zero-size), whereas line segments (whether continuous or discrete) do, does not mean that line segments are not made out of points. It merely reflects the fact that distance is something that exists between points. — Magnus Anderson

You don't measure the length of a line segment by counting how many points it has, you measure it by counting how many pairs of points-at-certain-distance it has. "This line is 10cm long" means "This line is made out of 10 pairs of points-at-1cm-from-each-other". — Magnus Anderson

But it is made out of points. — Magnus Anderson

For example, if you're charged with theft, there will be a clear law setting forth the several criteria that must be met for a theft to occur. The jury willl then determine if the facts establish a theft.

In an impeachment, there are no such criteria to be met. There's just a vague Constitutional standard that the House then sets forth into a more specific allegation after the offense is committed (ex post facto). Whether the House's articles actually describe a "high crime and misdemeanor" will remain an open question for each Senator to answer. — Hanover

If you agree then imagine four objects A, B, C, and D and that there are only 4 possible properties: round, square, red and black.
A is a red and round
B is red and square
C is red and round
D is black and square.

Using "=" to mean "identical to" we conclude that A = C but A not = B and B not = C and so on. Identicalness, in this sense, means all properties must match exactly. — TheMadFool

t was only a matter of time before they impeached Trump for word crimes. It was too difficult for them to find actual crimes, so they reduced themselves to scouring his statements for transgressions of speech, and then lying about them to make them seem worse than they are. — NOS4A2

Apparently the House, having no respect for the most fundamental of Constitutional principles, namely the separation if powers, refuses to cede the power to the Senate to hold its trial as it sees fit. — Hanover

Republicans do know where they have to stand. During Nixon's time, they were far more confident where they stood. They could throw away Nixon and be confident that they would have enough popular support in election (even if the Dems got Carter later). Now they aren't so confident about themselves anymore, hence they will defend to the last man Trump, even if they hate the guy privately. — ssu

The President has been accused vaguely of "high crimes and misdemeanors" — Hanover

Impeachment is democracy turning in on itself, where our representatives vote out our representatives. — Hanover

So, it seems that God’s guiding principle when judging the fate of humankind is not the amount of evil but the existence of a minimal quantity of righteousness which one might call it the quanta of righteousness. — Jacob-B

I directed the OP towards a (highly online) reference that explains how mathematicians disarmed his or her objections over a century ago. The date is relevant only because the OP ignored this reference and continued to insist that the methods of analytic geometry are unfounded. The relevant foundations were provided by mathematicians operating around the turn of the 20th century. — quickly

(note that when I say that change is more fundamental I’m not saying that there is nothing that temporarily stays the same within the change.) — leo

1. Change is immediately evident to us, you agreed with that, whereas being is not immediately evident. — leo

2. If being (absence of change) was most fundamental then there wouldn’t be change by definition, yet there is change. If being changes it is no more being. — leo

3. Change cannot be an illusion because it would be a changing illusion, and thus there would be change. Whereas absence of change can be an illusion, it can be change appearing to be unchanging. — leo

4. An experience is made of parts, for instance there can be simultaneously a feeling and a thought. There may be one part that is seen to be unchanging, but in order to see it as unchanging there is another part that is changing, for instance a thought that is interpreting some part of experience as unchanging. The thought itself is changing, if it wasn’t changing it wouldn’t come to see the other part as unchanging, it would remain stuck on a past thought. So within experience there is always change, the experience as a whole is changing. — leo

6. Even if we can find regularities within the change, these regularities wouldn’t exist without the change. Even if something remains temporarily the same within the change, that doesn’t make it more fundamental than the change. There can be sameness within change but there can’t be change within sameness. — leo

8. As a reason against seeing change as fundamental, you said you think if change was fundamental then everything would change and the world would be complete chaos and randomness. However I disagree, because why would we have to assume that everything would be changing randomly? Smooth change still counts as change. — leo

9. You say that “the idea of a thing which never changes comes from the necessity of ending the infinite regress”. As to how that infinite regress arises, you say: “Change is a difference in relations between things. So if a thing changes, the relations between its parts have changed. But if a part can change, then it must be composed of parts, and so on to infinite regress.”. But notice that in this reasoning you assume in the first place that there are unchanging things that exist, which is what you end up concluding. If you don’t assume that unchanging things exist fundamentally, then there is no infinite regress and so no necessity to conclude that there are things which never change. — leo

I see why it means: democracy is not perfect, but I'm not sure why you're saying it doesn't work... — VagabondSpectre

I know how that sounds, but the rest of the world has ostensibly been watching Trump get away with apparently criminal acts, and since America is supposed to the best and baddest and most free democracy around, it sends the message that democracy doesn't work. — VagabondSpectre

A couple of months ago the forum was infested with bad theology. Now it's bad maths. — Banno

I'm saying that your objections are more than a century out of date. — quickly

I think everything is unique in some way or other. If two things appear to be identical they would still be different if they are in different locations for example. — believenothing

It is conceivable, however, that the wardrobe chest I just got delivered to my house made of compressed wood contains two identical wardrobe chests, both occupying the same space at the same time and in the same respect. — god must be atheist

I agree that in order to experience change there has to be something that stays the same in relation to what is changing, for instance a thought, because if everything was changing randomly including our thoughts we wouldn’t even have a static thought that would tell us we are experiencing randomness, we wouldn’t have any memory and so on. But even though some things temporarily don’t change in relation to some other change, we don’t have to assume that it is necessary that some thing never changes. We have no evidence that something can never change forever, while we have evidence of change. — leo

We are certain of change but not of being. — leo

What we interpret as not changing might be simply change that is not perceived, for instance something might seem unchanging and yet by looking more closely we see change. Also, if one part of experience is not changing while another part is changing, the whole experience is changing as a whole, so again change appears as more fundamental than being. For these reasons I think it will be more fruitful to see change as fundamental rather than being. — leo

Would you agree with the idea that fundamentally what we call a “relation” is a thought, an experience? The idea that “change occurs” follows from experiences that are seen to change. Where does the idea of a “relation” come from? Doesn’t it come from seeing that some part of experience is correlated with some other, that the two parts change not independently from one another, but together in some way? — leo

Where I’m going with this is that you were saying that a thing without parts cannot change on its own, but if you agree that a relation is fundamentally an experience, a thought, a thing, then again why would that experience or thought or thing change on its own? If you say that this relation is made of parts, and that this is why the relation can change, then we’re back to asking why do the parts of that relation change in the first place? — leo

Basically it seems to me that you can’t escape the fact that a thing without parts can change, that it can become something different than it was, which again leads to the idea that change is more fundamental than being. It seems to me that it is a circular reasoning to say that “a thing with parts can change because the relation between the parts can change”, because in saying that you’re essentially saying that the relation can change on its own, or that the parts of the relation which are themselves not made of parts are changing on their own. Do you see what I mean? — leo

f we see change as more fundamental than being then a thing is simply an absence of change in relation to some other change. And then we don’t have to explain how a thing changes, change is what’s fundamental, a partial absence of change is what has to be explained, and we can explain it simply by seeing it as two opposite changes that cancel one another (or as several changes in equilibrium). Would you agree with that? — leo

I'm not sure why you think it'll be any different in the Senate. — Hanover