Yes. And for me the issue of whether there are 'really' various infinities leads inexorably what we could mean if we say so. All roads seem to lead to the 'problem' of the meaning of 'meaning. — jas0n

I lost the scent there buddy.

It all depends on how you define "circle-like". — Metaphysician Undercover

:chin:

:broken:

I lost the scent there buddy. — Agent Smith

If I tell you that a tower of infinities actually exists in something like a Platonic realm, what does that mean for you and me? If you tell me that you do believe in $\aleph_0$ but not $2^{\aleph_0}$, what am I to make of that? Does it mean you therefore aren't interested in it? But perhaps a skeptic studies the system to debunk it. On the level of math, it's dry logic, something like a symbol game. This tower exists within that 'fiction,' just as the bishop exists in the rules of Chess. It's not clear what is being denied or asserted when we are talking about the outside of this game. Does the denier mean to indicate that his intuition has peeked into Platonic heaven and only found infinity classic? Or is it a matter of taste? Utility? Maybe a mix of things. In any case, ambiguity.

If I tell you that a tower of infinities actually exists in something like a Platonic realm, what does that mean for you and me? If you tell me that you do believe in but not , what am I to make of that? Does it mean you therefore aren't interested in it? But perhaps a skeptic studies the system to debunk it. On the level of math, it's dry logic, something like a symbol game. This tower exists within that 'fiction,' just as the bishop exists in the rules of Chess. It's not clear what is being denied or asserted when we are talking about the outside of this game. Does the denier mean to indicate that his intuition has peeked into Platonic heaven and only found infinity classic? Or is it a matter of taste? Utility? Maybe a mix of things. In any case, ambiguity — jas0n

I'd say you're taking Wittgenstein a bit too far. There are clear-cut definitions in mathematics which don't allow either ambiguity or vagueness. Mind you I'm familiar with high school math only; perhaps it's different at PhD level.

Remember math is a constructed world and in being that it has an advantage viz. precise definitions which, for me, makes no intuitive sense at all. That's just how the game is played I guess.

There are clear-cut definitions in mathematics which don't allow either ambiguity or vagueness. — Agent Smith

I know, and that lack of ambiguity pretty much continues as one climbs the mathematical ladder. The issue is not in the system of symbols but in the relationship of that system to the rest of the world.
Think of one version (the formal version) of $\aleph_0$ as a dead mark that's moved around according to rules with no other meaning but its relationship to those rules. Then think of what, if anything, this 'infinity' means to you beyond being a dead mark in a dead game. Is there 'infinity' in the real world? That is the zone of ambiguity. What exactly is this real world? Does 'infinity' fit in it somehow?

Remember math is a constructed world and in being that it has an advantage viz. precise definitions which, for me, makes no intuitive sense at all. That's just how the game is played I guess. — Agent Smith

The weird thing is that we value the dead game because it does help us in the real world. On the basic level, a person can tell the government that they have 3 kids on their tax return.

All I can say is you're not incorrect, but as I pointed out, infinity allows approximations that turn out to be useful when dealing with feminine geometric objects (curves). — Agent Smith

The issue is that this type of approximation produces the illusion that we understand what a curve is, when we really do not. There's a fundamental incommensurability between two dimensions of space, which makes things like pi and the square root of two irrational ratios. What it indicates is that we lack a proper understanding of space.

The fact that we are in the habit of reducing straight lines at angles to each other to curved lines through the application of infinity, Is evidence that we simply ignore this deep misunderstanding, and proceed as if we think that we understand. I would argue that the "damage control" which you claim, is basically non-existent, because those employing the principles actually believe themselves to have an adequate understand, when infinity proves useful, therefore wouldn't even seek damage control. The problem is prevalent all through modern physics, with vectors and spins, etc..

Yep! Thanks for letting me know. Metaphysician Undercover will find this tid bit right up his alley. — Agent Smith

Actually "infinite-sided polygon", to me, can only be interpreted as an incoherent object.

If I tell you that a tower of infinities actually exists in something like a Platonic realm, what does that mean for you and me? — jas0n

Only coherent intelligible objects could ever exist in the Platonic realm. Incoherencies are banned by the Ruler of the realm.

The issue is not in the system of symbols but in the relationship of that system to the rest of the world. — jas0n

Actually the issue is not as simple you say. In reality, incoherency is allowed to exist within the system, as the example of "infinite-sided polygon", demonstrates. What happens is that there are problems in the relationship between the system and the world, as you say. But since the problems are associated with the very fundamental aspects of the system, no amount of tweaking the system can overcome the problems. So the only way that the system becomes applicable to the world, is to allow inconsistency into the system, to overrule the problems with the foundational axioms.

The simple reality is that the entire system is flawed, right from the very foundations, so any attempts to make it more applicable will require inconsistency within it. And that's what we see, inconsistency is rampant within the system. The only true fix is to replace the entire system from bottom up, with principles derived from a better understanding of space and time. And that's how this discussion is related to mysticism. We need to turn to mysticism to find that better understanding.

The issue is that this type of approximation produces the illusion that we understand what a curve is, when we really do not. There's a fundamental incommensurability between two dimensions of space, which makes things like pi and the square root of two irrational ratios. What it indicates is that we lack a proper understanding of space.

The fact that we are in the habit of reducing straight lines at angles to each other to curved lines through the application of infinity, Is evidence that we simply ignore this deep misunderstanding, and proceed as if we think that we understand. I would argue that the "damage control" which you claim, is basically non-existent, because those employing the principles actually believe themselves to have an adequate understand, when infinity proves useful, therefore wouldn't even seek damage control. The problem is prevalent all through modern physics, with vectors and spins, etc. — Metaphysician Undercover

All I can say is

Le meglio è l'inimico del bene — Voltaire

Please note mathematicians are under no illusion that a curve is in fact made up of infinite straight lines. They are, as I tried to impress upon you, estimations (not exactly a curve, but close). I'm reading this book on mathematics and there's a chapter on the dome of the Hagia Sophia which required the construction of a square. The architects had to find the length of the diagonal (actually the $\sqrt 2$) and they did that using an ingenious method which involved the use of a rational approxomation much like how Archimedes stopped at $\frac{22}{7}$ as the value of $\pi$.

It would be wise not to underestimate the intelligence of people, mathematicians included.

Actually "infinite-sided polygon", to me, can only be interpreted as an incoherent object — Metaphysician Undercover

That's because your conception of $\infty$ doesn't allow you to to make sense of it. Different strokes for different folks. to each his own, eh?

I'm sure you're aware of this but how different is a curve from a straight line between two points that are infinitesimally close to each other? Try drawing a chord between two points on a circle. As the two points come closer, the chord and the arc subtended by these two points approach each other. Extrapolate that unto infinity and you'll get an idea of what mathematicians are trying to convey here.

The idea, it seems, is to reduce the error to an arbitrarily small value and what better way to do that than using infinitesimally small straight lines which results in a corresponding infinity of straight lines (smaller the straight lines, more of them you'll need to measure the circumference).

Please note, I have only a rudimentary grasp of mathematics; although I love the subject, it's not exactly my strong suit. Just sharing my intuitions on the matter.

Also, you're correct about how no matter how small you make straight lines, they can never be curves. It depends then, doesn't it, how stringent one's criteria are. If you want to split hairs then all mathematics that depend on infinity and infinitesimals need to be scrapped. We would be much handicapped if we were to do that.

The way out of this would be to accept that infinities can be used for increasing the accuracy of our estimates but then we need to make it clear that no matter how good our estimate it still is never going to be the real McCoy.

What say you?

Only coherent intelligible objects could ever exist in the Platonic realm. Incoherencies are banned by the Ruler of the realm. — Metaphysician Undercover

Cantor's system seems to work just fine. Gödel would allow it in, I think, and actually believe there was such a place.

The only true fix is to replace the entire system from bottom up, with principles derived from a better understanding of space and time. And that's how this discussion is related to mysticism. We need to turn to mysticism to find that better understanding. — Metaphysician Undercover

Something similar to that has been tried: https://en.wikipedia.org/wiki/L._E._J._Brouwer

Brouwer’s little book Life, Art and Mysticism of 1905, while not developing his foundations of mathematics as such, is a key to those foundations as developed in his dissertation on which he was working at the same time and which was finished two years later. Among a variety of other things, such as his views on society and women in particular, the book contains his basic ideas on mind, language, ontology and epistemology.

These ideas are applied to mathematics in his dissertation On the Foundations of Mathematics, defended in 1907; it is the general philosophy and not the paradoxes that initiates the development of intuitionism (once this had begun, solutions to the paradoxes emerged). As did Kant, Brouwer founds mathematics on a pure intuition of time (but Brouwer rejects pure intuition of space).

Brouwer holds that mathematics is an essentially languageless activity, and that language can only give descriptions of mathematical activity after the fact. This leads him to deny axiomatic approaches any foundational role in mathematics. Also, he construes logic as the study of patterns in linguistic renditions of mathematical activity, and therefore logic is dependent on mathematics (as the study of patterns) and not vice versa.

I think anyone can make up their own version of mathematics, but it'd be hard to get anyone to care. Even intuitionism (Brouwer's & Heyting's) and constructivism (like Bishop's) are mostly ignored in universities.

math is a constructed world and in being that it has an advantage viz. precise definitions — Agent Smith

Well, there are lots of ambiguities in mathematical symbolism. The equal symbol for example, then the idea of transforms and transformations, etc. In advanced math one has to consider context to interpret accurately.

Even intuitionism (Brouwer's & Heyting's) and constructivism (like Bishop's) are mostly ignored in universities — jas0n

Mathematicians in analysis or topology mostly know Brouwer for his famous Fixed Point theorem .

Well, there are lots of ambiguities in mathematical symbolism. The equal symbol for example, then the idea of transforms and transformations, etc. In advanced math one has to consider context to interpret accurately. — jgill

I believe you mean polysemy (a feature) and not ambiguity (a bug).

They are, as I tried to impress upon you, estimations (not exactly a curve, but close). — Agent Smith

And as I tried to impress on you, a curve is not even close to a multitude of straight lines.

I'm sure you're aware of this but how different is a curve from a straight line between two points that are infinitesimally close to each other? Try drawing a chord between two points on a circle. As the two points come closer, the chord and the arc subtended by these two points approach each other. Extrapolate that unto infinity and you'll get an idea of what mathematicians are trying to convey here. — Agent Smith

I think I know what mathematicians are trying to convey, I've been told that numerous times. I simply believe that it's fundamentally incorrect. "Infinitesimally close to each other" is not a standard of measurement which has any rigorous meaning. I mean it's not a distance which is measured.

If you want to split hairs then all mathematics that depend on infinity and infinitesimals need to be scrapped. We would be much handicapped if we were to do that. — Agent Smith

Now you're getting the idea. Yes, I agree, that anyone who scrapped that stuff would be greatly handicapped at this time of scrapping the stuff. But necessity is the mother of invention, and what would develop out of the scrapping, making a fresh start, knowing what we know now, would be a great improvement.

Something similar to that has been tried: https://en.wikipedia.org/wiki/L._E._J._Brouwer — jas0n

That looks interesting.

Mathematicians in analysis or topology mostly know Brouwer for his famous Fixed Point theorem . — jgill

Of the following, I can see intuitively why #2 would be true, but I haven't a clue as to why #1 and #3 are.

1. Take two sheets of graph paper of equal size with coordinate systems on them, lay one flat on the table and crumple up (without ripping or tearing) the other one and place it, in any fashion, on top of the first so that the crumpled paper does not reach outside the flat one. There will then be at least one point of the crumpled sheet that lies directly above its corresponding point (i.e. the point with the same coordinates) of the flat sheet. This is a consequence of the n = 2 case of Brouwer's theorem applied to the continuous map that assigns to the coordinates of every point of the crumpled sheet the coordinates of the point of the flat sheet immediately beneath it.
2. Take an ordinary map of a country, and suppose that that map is laid out on a table inside that country. There will always be a "You are Here" point on the map which represents that same point in the country.
3. In three dimensions a consequence of the Brouwer fixed-point theorem is that, no matter how much you stir a cocktail in a glass (or think about milk shake), when the liquid has come to rest, some point in the liquid will end up in exactly the same place in the glass as before you took any action, assuming that the final position of each point is a continuous function of its original position, that the liquid after stirring is contained within the space originally taken up by it, and that the glass (and stirred surface shape) maintain a convex volume. Ordering a cocktail shaken, not stirred defeats the convexity condition ("shaking" being defined as a dynamic series of non-convex inertial containment states in the vacant headspace under a lid). In that case, the theorem would not apply, and thus all points of the liquid disposition are potentially displaced from the original state.[citation needed]'/quote] — Wikipedia

And as I tried to impress on you, a curve is not even close to a multitude of straight lines. — Metaphysician Undercover

Now, you're joking, right? :smile:

Now you're getting the idea. Yes, I agree, that anyone who scrapped that stuff would be greatly handicapped at this time of scrapping the stuff. But necessity is the mother of invention, and what would develop out of the scrapping, making a fresh start, knowing what we know now, would be a great improvement. — Metaphysician Undercover

Show us then a different method of measuring the length of a curve if not using infinitesimally small straight lines. I bet you can't and so infinitesimals and infinity it is. Nevertheless we'll wait, with baited breath, for you to discover a new way of tackling curves.

Of the following, I can see intuitively why #2 would be true, but I haven't a clue as to why #1 and #3 are. — Metaphysician Undercover

Yes, curious isn't it? A problem is that this is an existence theorem. I've never used it for this reason, going instead with Banach's theorem that incorporates a procedure for actually finding such a point (which is unique). In the Brouwer theorem the function f(z)=z means all points are fixed points.

I believe you mean polysemy (a feature) and not ambiguity (a bug). — Agent Smith

Merriam-Webster:

Above the level of molecular biology, the notion of "gene" has become increasingly complex. The chapter in which Ridley addresses the ambiguities of this slippery word is an expository tour de force. He considers seven possible meanings of gene as used in different contexts: a unit of heredity; an interchangeable part of evolution; a recipe for a metabolic product; … a development switch; a unit of selection; and a unit of instinct.

But if you like polysemy, be my guest. Although, to me, it seems unseemly. :roll:

Merriam-Webster:
Above the level of molecular biology, the notion of "gene" has become increasingly complex. The chapter in which Ridley addresses the ambiguities of this slippery word is an expository tour de force. He considers seven possible meanings of gene as used in different contexts: a unit of heredity; an interchangeable part of evolution; a recipe for a metabolic product; … a development switch; a unit of selection; and a unit of instinct.

But if you like polysemy, be my guest. Although, to me, it seems unseemly — jgill

Polysemy is necessary, our memory can't handle so many words as there would've been if it were not a feature of human language.

Ambiguity, a result of polysemy, nevertheless is to be avoided to the extent possible. It causes confusion.

I suppose it's a trade-off: to make language less memory-intensive we need to make a sacrifice, befuddlement.

Mathematicians in analysis or topology mostly know Brouwer for his famous Fixed Point theorem . — jgill

Ah yes! I'm guessing that's part of his work he didn't find metaphysically sound. I'm pretty much OK with mainstream math. The outsider versions are intriguing though.

mysticism in a philosophical forum?
The Philosophical Method is an exercise in frustration, not the pursuit of comforting ideas.

Now, you're joking, right? :smile: — Agent Smith

Of course I'm not joking. Let's assume that two straight lines is "close" to being a single curved line, two being "close" to one. The curved line is a single line, the two straight lines is two distinct lines. Now you seem to think that the more straight lines you put together, 3, 4, 5, 6, the closer you get to being a single line, such that as you approach an infinity of straight lines, it becomes one curved line. Can't you see that you're going the wrong way? Instead of getting closer and closer, you're getting further and further. Producing a larger and larger multiplicity does not somehow produce the conclusion that the multiplicity is getting closer and closer to being a single entity.

Show us then a different method of measuring the length of a curve if not using infinitesimally small straight lines. I bet you can't and so infinitesimals and infinity it is. Nevertheless we'll wait, with baited breath, for you to discover a new way of tackling curves. — Agent Smith

I use something flexible like a string, bend it around in the curve to be measured, then I lay it out straight and measure it as one long length. The single curved line is effectively converted to a single straight line, then measured as such, in that way. You have to be careful though because the "inside" measurement is always distinctly shorter than the "outside" measurement, and this is a problem which cannot be avoided. So the inside of the string will complete the circle in a shorter length than the outside of the string.

This thing you refer to, "using infinitesimally small straight lines", is not itself an act of measurement, because as I told you, an infinitesimal length cannot be measured, and therefore cannot be used as a unit of measurement.

Yes, curious isn't it? A problem is that this is an existence theorem. — jgill

The question then, is there a way to determine whether it is true or not? For example, if you stir a coffee, is there a way to determine that there is a molecule or something like that, which remains in the same place after stirring as it was prior? Or is this just a principle which is useful for some purposes, but is not really true? I guess it depends on what is meant by "continuous function" and whether it is true to think of things in terms like this.

Of course I'm not joking. Let's assume that two straight lines is "close" to being a single curved line, two being "close" to one. The curved line is a single line, the two straight lines is two distinct lines. Now you seem to think that the more straight lines you put together, 3, 4, 5, 6, the closer you get to being a single line, such that as you approach an infinity of straight lines, it becomes one curved line. Can't you see that you're going the wrong way? Instead of getting closer and closer, you're getting further and further. Producing a larger and larger multiplicity does not somehow produce the conclusion that the multiplicity is getting closer and closer to being a single entity. — Metaphysician Undercover

You're a perfectionist and so the mathematics of infinity and infinitesimals won't make any sense to you.

You're a perfectionist and so the mathematics of infinity and infinitesimals won't make any sense to you. — Agent Smith

That's exactly the problem. I thought mathematics was supposed to provide us with precision, perfection in our understanding. Then I was disillusioned, realizing that it's all a facade, and deep misunderstanding lies behind.

That's exactly the problem. I thought mathematics was supposed to provide us with precision, perfection in our understanding. Then I was disillusioned, realizing that it's all a facade, and deep misunderstanding lies behind. — Metaphysician Undercover

You're too quick to pass judgment. The precision is there, it's just that your way of looking at math (without infinity and infinitesimals) doesn't allow mathematicians to show you how the margin of error can be reduced to an arbitrarily small value.

Imagine if the true value of a measurement is 4.5879... units. I can get very, very close to that value and that should be more than enough. Note mathematicians are fully aware of this rather embarrassing state of affairs. Irrational numbers were called incommensurables.

That's exactly the problem. I thought mathematics was supposed to provide us with precision, perfection in our understanding. Then I was disillusioned, realizing that it's all a facade, and deep misunderstanding lies behind. — Metaphysician Undercover

So, something that is not perfect is deeply flawed?

What I am saying is that the reason why perfection is impossible is that the tool (mathematics) is fundamentally flawed.

Imagine if the true value of a measurement is 4.5879... units. I can get very, very close to that value and that should be more than enough. Note mathematicians are fully aware of this rather embarrassing state of affairs. Irrational numbers were called incommensurables. — Agent Smith

It's not an issue of there being a "true value of a measurement" and we can get very very close to that, it's a matter of there being no true value, because the way of measuring is fundamentally flawed. So you have no justification to your claim that you "can get very very close" to the true value. If you could show the true value, to show how close you are to it, you wouldn't accept the "very very close" value, you'd accept the true value instead.

What you're not grasping, is that when measurement is impossible, which is what "incommensurable" implies, then there is no true measurement, and no such thing as close to the true measurement. What is evident is that the measuring tool is inadequate for the job. Need to get a better tool, properly designed for the job

So, something that is not perfect is deeply flawed? — jgill

Mathematics consists of ideals. An ideal which is not perfect is deeply flawed.

Mathematics consists of ideals — Metaphysician Undercover

It's an immense subject - Wikipedia has over 26,000 mathematics articles - consisting of a lot more than Platonic ideals. There's a tremendous amount of material concerning how various entities relate to one another. How one theorem gives rise to another, etc. You have a peculiar understanding of the subject.

Measurements work pretty well for something "fundamentally flawed". Try to not be fascinated by the word "incommensurable". It's not the death knoll of math.

I am mostly a constructivist in that my theorems are generally not indirect, but include processes for obtaining mathematical objects. Certain fundamental objects might be considered ideals, but arguing them into new results doesn't make those results ideals. Especially if one creates rather than discovers math - a can of worms.

No one says math is perfect. But it is interesting and useful.

What I am saying is that the reason why perfection is impossible is that the tool (mathematics) is fundamentally flawed — Metaphysician Undercover

And I'm saying we don't have an option. Infinity and infinitesimals are the best available tools we have to study curves. Maybe some day we'll discover something better. Until that happens, we're stuck with what we have.

Need to get a better tool, properly designed for the job — Metaphysician Undercover

:up:

there is no true measurement — Metaphysician Undercover

You mean to say the diagonal of a square has no true measurement? What is a true measurement to you?

And I'm saying we don't have an option. Infinity and infinitesimals are the best available tools we have to study curves. Maybe some day we'll discover something better. Until that happens, we're stuck with what we have. — Agent Smith

I have a slightly different opinion. I think we will not discover something better until we reject what we have. As I said, necessity is the mother of invention. Everyday I pick up my hammer and bang some nails. I think this tool's just fine, it serves the purpose well. I will never replace it unless I am dissatisfied with it. And until someone shows me, look it's got this problem and that problem, I'll continue to think it's just fine.

What is a true measurement to you? — Agent Smith

I don't know, that was your words. You were talking about being close to the true measurement. If you talk about "true measurement in that way, then there is no such thing. But I think any measurement is a true measurement, how could it not be a true measurement, yet still be a measurement? The issue is the measuring technique.

I have a slightly different opinion. I think we will not discover something better until we reject what we have. As I said, necessity is the mother of invention. Everyday I pick up my hammer and bang some nails. I think this tool's just fine, it serves the purpose well. I will never replace it unless I am dissatisfied with it. And until someone shows me, look it's got this problem and that problem, I'll continue to think it's just fine — Metaphysician Undercover

This is poor logic. I've used a block of wood on nails, when I couldn't find my hammer. When I did find my hammer, I threw away the wood.

True measurement, to me, simply means the correct value of (say) the length of a line. So, a square has a diagonal whose true measurement is $\sqrt 2$.

No one says math is perfect. But it is interesting and useful. — jgill

:up:

@Metaphysician Undercover $\uparrow$

True measurement, to me, simply means the correct value of (say) the length of a line. So, a square has a diagonal whose true measurement is 2–√2. — Agent Smith

Then how were you distinguishing between the true value of a measurement, and a measurement which is close to the true value?

Suppose there's a line. Do you think the line has a measurement without being measured? How could it?

Then how were you distinguishing between the true value of a measurement, and a measurement which is close to the true value? — Metaphysician Undercover

There are ways...

We could calculate using algebraic techniques. In the case of squares, Pythagoras' theorem shows us that the diagonal is $\sqrt 2$.