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
It all depends on how you define "circle-like". — Metaphysician Undercover
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 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
There are clear-cut definitions in mathematics which don't allow either ambiguity or vagueness. — Agent Smith
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
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
Yep! Thanks for letting me know. Metaphysician Undercover will find this tid bit right up his alley. — 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? — jas0n
The issue is not in the system of symbols but in the relationship of that system to the rest of the world. — jas0n
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
Le meglio è l'inimico del bene — Voltaire
Actually "infinite-sided polygon", to me, can only be interpreted as an incoherent object — Metaphysician Undercover
Only coherent intelligible objects could ever exist in the Platonic realm. Incoherencies are banned by the Ruler of the realm. — Metaphysician Undercover
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
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.
math is a constructed world and in being that it has an advantage viz. precise definitions — Agent Smith
Even intuitionism (Brouwer's & Heyting's) and constructivism (like Bishop's) are mostly ignored in universities — jas0n
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
They are, as I tried to impress upon you, estimations (not exactly a curve, but close). — Agent Smith
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
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
Something similar to that has been tried: https://en.wikipedia.org/wiki/L._E._J._Brouwer — jas0n
Mathematicians in analysis or topology mostly know Brouwer for his famous Fixed Point theorem . — jgill
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 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
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
I believe you mean polysemy (a feature) and not ambiguity (a bug). — Agent Smith
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.
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
Now, you're joking, right? :smile: — Agent Smith
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
Yes, curious isn't it? A problem is that this is an existence theorem. — jgill
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. — 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. — Metaphysician Undercover
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
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
So, something that is not perfect is deeply flawed? — jgill
Mathematics consists of ideals — Metaphysician Undercover
What I am saying is that the reason why perfection is impossible is that the tool (mathematics) is fundamentally flawed — Metaphysician Undercover
Need to get a better tool, properly designed for the job — Metaphysician Undercover
there is no true measurement — 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. — Agent Smith
What is a true measurement to you? — 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 — Metaphysician Undercover
No one says math is perfect. But it is interesting and useful. — jgill
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? — Metaphysician Undercover
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