That said, it can be very difficult to figure out the latest wrinkles in political usage. Why for example, has the phrase "pregnant women" been replaced by 'pregnant people"? The last time I looked it up, men do not get pregnant — Bitter Crank

It's only a matter of time.

I consider the Paradox an aberration that results from collapsing one dimension to a lower dimension in certain circumstances and insignificant although bizarre. But Wolfram claims that this crops up in Feynman diagrams. It goes to the very nature of lines and points. — jgill

No, I think it is significant and general. — apokrisis

One way of perceiving the sum of all paths in Feynman's path integral for paths that consistently move towards a target is to have the particle pass through a series of finite parallel plates, each with "portholes" where the particle pops through, from one plate to the next in a system where both the number of holes and the number of plates increases without bound, with plates being squeezed closer and closer together.

A path then is zig-zag, roughly like the original diagonal paradox except with random configuration. If all possible paths of this type arise, then there will be some with lengths growing towards infinity that are nevertheless contained in a closed and finite environment and go from left to right, let's say. Like the sequence of contours in my note. One meter or infinite meters? The "same" path. :chin:

I may be wrong, but I have come to the conclusion that the only way to be politically correct towards liberal morals and aesthetics is to not be rigid at all, to not offend and to be 'open-minded' towards everything that does not violate state laws.

I repeat that I may be wrong, but this is what I take modern liberalism for: lack of models & aesthetics that first of all are politically correct (aesthetics which try to include as more qualities as possible, so none might feel excluded and no model may dominate). — Eros1982

That's an interesting assessment of American liberalism. But you haven't stated what comes next under that rubric: condemn those who do not follow this prescription.

"Most notably Hilbert's paradox of the Grand Hotel, but also the following:

Gabriel's horn
Galileo's paradox
Ross–Littlewood paradox
Thomson's lamp
Zeno's paradoxes
Cantor's paradox
Dartboard paradox"

The Diagonal Paradox can be extended in principle to any curve in 2D. For example, a circle of radius 1 has a circumference of 2pi, but if I apply my system of sine curves to the circumference I find that as they converge uniformly to the existing circumference, their lengths tend to infinity. Hence I am staring at what appears to be the simple circle I began with, but I now have one with infinite circumference, and hence infinite area.

Thus infinity is everywhere in plane geometry where it shouldn't be.

I consider the Paradox an aberration that results from collapsing one dimension to a lower dimension in certain circumstances and insignificant although bizarre. But Wolfram claims that this crops up in Feynman diagrams. It goes to the very nature of lines and points.

It's not science. But it's speculating or philosophizing about scientific results.

I hypothesized in my post to jgill [a calculator that can't calculate beyond 5 would display 5 (the arbitrarily large number) for both the queries 2 + 3 = ? and 3 + 29 = ?]; he asked, paraphrasing, how can a finite brain grasp infinity. — Agent Smith

I did? I'm old. Maybe I was closing up shop for the night. :chin:

I just proved an interesting result demonstrating a sequence of sine contours on the interval [0,1] that not only converge uniformly to the line from 0 to 1 while growing towards infinite length, but converge in this way along the interval [0,1] all the way down to the point (0,0) while becoming infinite in length. I mention this only because there is interest in the relationship between 0 and infinity and somehow combining continuity and discreteness.

I'll give details if requested. :cool:

I always thought from my experience that maths types were invariably into classical music. It was the physicists who scaled peaks. — apokrisis

I knew Lester Germer, a multi-dimensional person. He was a fighter pilot in WWI.

My oldest and best climbing friend ,Dave Rearick, was a math prof at the U of Colorado.

I believe that the majority of the harms that death visits on a person are post-mortem. Why? Because the ante-mortem harms seem relatively insignificant compared to the harmfulness of death — Bartricks

↪jgill
You clearly don't understand the case I have made at all. — Bartricks

And thankful of that I am. :roll:

This thread has opened the door to a BIG NUMBER theory, a BIG BANG theory companion!

Oops, looks like there already exists such a Theory :sad:

At what time are the harms of death visited upon us? They are visited precisely when death itself is visited upon us and not a moment earlier or later. . . . Not later - because after death we are beyond the harm of losing life and its benefits, having already lost them. — Cuthbert

:up: Pretty much says it all, unless one wants to speculate about surviving death in some form. That would be a thread to which Houdini might contribute - or not.

Perhaps, it is why philosophy seems to be so much of an area for heated debate . . . — Jack Cummins

Amongst the philosophical minded. :smile:

:clap: Eloquently said!

That differs from how I find 'classical' is used. I find that 'classical' mathematics means all and only those results that can be formalized as theorems of ZFC with classical logic. And classical logic means the first order predicate calculus including the law of excluded middle. — TonesInDeepFreeze

Well, that's interesting. I learned something. Thanks. Classical analysis of course means more or less what I said, going back to Weierstrass and Cauchy - and I forgot, the study of special functions - but in foundations classical has another meaning.

Wiki:

In the foundations of mathematics, classical mathematics refers generally to the mainstream approach to mathematics, which is based on classical logic and ZFC set theory.

:cool:

But maybe you didn't mean that you don't use those sets. But that you do use them, but you don't use the extended real line with its points of infinity? As instead you simply deploy the fact that the reals are unbounded? — TonesInDeepFreeze

The basic set theoretic structure of the reals underlies almost everything I have done, but I haven't used infinity as a "point" (nor the axiom of choice). Infinity is a limit in the language of calculus.

(3) What is the difference you have in mind between classical and modern? Ordinary contemporary analysis is classical analysis. — TonesInDeepFreeze

Classical means the tools of analysis like limits, differentiation and integration and all those entail. Nitty gritty. Actual specific results vs broad generalities. The more modern you get the more abstract the subject becomes with broad generalities and topological arguments. It's vague to an extent.

Hard & Soft Analysis

I don't think it takes an MFA to be a good writer. But an apprenticeship could help. I'm not even sure a BA is required. Lots and lots or reading various sources and comparing styles, use of language, etc. would avoid a college education. Off the top of my head. It's not quantum theory.

I have never used infinity as anything more than unboundedness. — jgill

In Calculus 1 classes, there is not a concern that the subject be axiomatized. But if we are concerned with having the subject axiomatized, then the ordinary mathematical context is one in which there are infinite sets. — TonesInDeepFreeze

I'm not sure how your post relates to my quote. To clarify, I've taught both undergraduate and graduate courses in (mathematical) analysis and published papers in the subject, and I have only very rarely had to resort to a transfinite argument or even read such an argument. In fact, the only time I can recall is the Hahn-Banach theorem in the functional analysis grad course I took many years ago. The proof involves the Hausdorff Maximal Principle (i.e. Axiom of Choice or Zorn's Lemma) and even there if one strengthens the hypotheses just a tad HMP or AOC or ZL can be avoided.

All math majors learn a little naive set theory and some ZFC these days, so mathematicians and students work in an environment underlaid by the fundamentals of the real line. It's just that conversations involving cardinalities beyond ${{\aleph }_{1}}$ don't usually occur in classical or even much of modern analysis.

Wiki says this:

The aleph numbers differ from the infinity ( ∞ {\displaystyle \,\infty \,} {\displaystyle \,\infty \,}) commonly found in algebra and calculus, in that the alephs measure the sizes of sets, while infinity is commonly defined either as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or as an extreme point of the extended real number line.

to organize the unformed Potential of the Singularity — Gnomon

Interesting to learn of the various ways the word Singularity is used. In the simplest math it's just where a denominator equals zero. But in the complex plane the function $f(z)={{e}^{\frac{1}{z}}}$ is a world of trouble for
$z=0$

Sin(x)/x as x approaches zero is an entity itself, a ratio that converges to one. Look at the simpler ratio (x^2)/x as x approaches zero. It's an indeterminate form that reduces to x, so goes to zero.

↪jgill
To make a case for thinking that the harms are post mortem. That would then constitute some evidence that we survive our deaths — Bartricks

So an argument that begins, "Assuming part of us suffers after death . . ." is evidence we survive our deaths? :roll:

We do not know what death does to us. — Bartricks

So why begin this thread?

It is speculated (theorized?) that time began with this event called the BIg Bang, as did space, in the context of spacetime. So there is no "before" the BB. So originally there was complete monism that fragmented as time progressed. Well, just more babble (rubble) I suppose.

See? You started an entertaining discussion that drew in some pretty good thinkers. Probably better than paying a PhD student. :cool:

Once again, calculus is about LIMITS, — jgill

True, but in many a calculus problem and theorem the limit IS infinity. — god must be atheist

Normally $x\to \infty$ arises in the following context:

$\underset{x\to \infty }{\mathop{\lim }}\,f(x)=L\text{ }\Leftrightarrow \text{ }For\text{ }\varepsilon \gt0\text{ }\exists M\gt0\text{ }such\text{ }that\text{ }x\gt M \Rightarrow \left| f(x)-L \right|\lt\varepsilon$

or $\underset{x\to \infty }{\mathop{\lim }}\,\,f(x)=\infty \text{ }\Leftrightarrow \text{ }For\text{ }N\gt0\text{ }\exists M\gt0\text{ }such\text{ }that\text{ }x\gt M\Rightarrow \left| f(x) \right|\gt N$

Infinity as a mathematical object is not used in calculus with one exception that I can think of, in complex analysis (calculus of complex variables) where the "point at $\infty$" corresponds in a projective sense with the north pole of the Riemann sphere.

That can't be true. Calculus is all about infinity — T Clark

I was thinking the same thing. — god must be atheist

Once again, calculus is about LIMITS, as my mathematical genealogical ancestor, Karl Weierstrass would have explained.

That damn lemniscate and the problems it produces . . . :roll:

'Error' is constitutive of disorder & foolery, no? — 180 Proof

I type an "a" instead of an "s". Am I displaying disorder or foolery? Do I compare with Putin's error in Ukraine? Are "error" and "mistake" synonymous? The latter can be a verb as well.

That makes the real numbers a challenging and intriguing subject. — jgill

Maybe not as challenging as you think. — keystone

I was speaking of currently accepted set theory, not challenges of it.

Here is a start. But it's pretty simplistic.

A good topic for TPF I would think.

I have never used infinity as anything more than unboundedness. — jgill

That can't be true. Calculus is all about infinity. — T Clark

$x\to \infty$ means x gets larger without bounds. Limits

Perhaps I should have written that I believe it is impossible to imagine assembling points to form a continuum. A bit of magic is needed to make the leap from a finite collection of points forming nothing to an infinite collection of points forming a continuum. — keystone

That makes the real numbers a challenging and intriguing subject.

God = ∞. — Agent Smith

The tone of the OP does not suggest Cantor's theological nonsense.

Thanks and sorry for posting a topic which is not a typical discussion — keystone

And refreshing it is for these times. Not a mention of God or Jesus or climate change.

I was a professional mathematician for many years, focusing on complex analysis, teaching and writing a few papers, and still do modest research. And although I accept ideas like the set of real numbers and associated cardinalities, I have never used infinity as anything more than unboundedness. To all intents and purposes my mathematics has been infinity free.

The forum has had a number of discussions about this topic, but that's no reason for you to avoid bringing it up in a new thread or resurrecting an old thread. There are some sharp people here.

Is it even a matter of set theory? — Michael

Starting off with a proposed "set of everything" I would say yes. Beyond that the discussion is mostly the typical banter about the definitions of words seen on the site. No big deal. Philosophy of mathematics? Questionable.

Whatever you put into a sewer, what you get out is sewage, Is that really the analogy you were looking for? — Janus

I never heard this one, but from CS there is the equivalent and more or less standard that's been around forever: Garbage

I notice the set theorists on TPF are keeping their distance from this thread. :cool:

I have no regrets studying literature and social sciences. Academia is like a sewer: what you get out of it depends on what you put into it. — Bitter Crank

And there lies a tale to be told :wink:

Suppose that all that exists forms a set. — Kuro

I question whether that is actually a set in ZFC. A neighbor of Russell's paradox perhaps.

Now that you have taken your slice of humble pie and realized that you don't know what death does to person, stop discounting the possibility that you survive it and suffer terribly. — Bartricks

:scream: :rofl:

I believe that the majority of the harms that death visits on a person are post-mortem — Bartricks

To be harmed or hurt requires feeling physical or emotional pain. You seem to be saying that a person is alive to these sensations after they die. "harms the death visits" is poetic, but confusing.

↪jgill
So the hermit's death is a great harm to the hermit. It won't harm anyone else. It will harm him.

And it will harm him, will it not, even if he has no plans that killing him with thwart.

And it will harm him even if he isn't particularly enjoying his life.

So, it will harm him even if it deprives him of nothing.

Thus, the harm of death cannot reside primarily in what it deprives a person of. For it harms those it deprives of nothing worth having. — Bartricks

Sounds like Clint Eastwood in The Unforgiven. Good points. What I am saying is it does no harm to him after he ceases to exist.

The thing about lucid dreams is the spectrum they constitute. On one end there's a normal dream you seem to be able to control slightly. At the other end is an experience in which you wake up and move around as if in a heightened reality, feeling and seeing things as solid objects and fully aware. This is an astounding experience that will stay with you to the end of your days.